a presentation of doctoral dissertation

20-May-21 | 1

A PRESENTATION OF DOCTORAL DISSERTATION

Presenter Cong Phat Vo

Department of Mechanical Engineering University of Ulsan Korea

Ph D candidate Cong Phat Vo

Advisor Professor Kyoung Kwan Ahn

Title

A Sensorless Reflecting Control for

Bilateral Haptic Teleoperation System based on

Pneumatic Artificial Muscle Actuators

Fluid Power Control and Machine Intelligence Laboratory

20-May-21 | 2Presenter Cong Phat Vo

CONTENTS

Modeling and platform setup based on a APAM configuration

Adaptive Gain ITSMC with TDE scheme for the position control

Adaptive Finite-Time Force Sensorless Control scheme with TDE

Model-free control for Optimal contact force in Unknown Environment

Optimized Human-Robot Interaction control using IRL

Force Sensorless Reflecting Control for BHTS

Conclusion and Future works

Introduction

2

7

3

4

5

6

C

H

A

P

T

E

R

1

7

8


CONTENTS

Modeling and platform setup based on the APAM configuration

Adaptive Gain ITSMC with TDE scheme



Optimized Human-Robot Interaction force using IRL



2

7

3

4

5

6

C

H

A

P

T

E

R 7

8

Introduction1

1 2 3 4 5 6 7 8


hellip

Performence

Perception

Transparency

1 INTRODUCTION

Overview of the BHTS structure

Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x

Figure 11 The bilateral haptic teleoperation architecture

1 2 3 4 5 6 7 8

RobustnessThe controllers are required to be robustly stable with respect to a

specified set of uncertainties introduced by the different components

(operator environment communication channel as well as sensors

introduce uncertainties)


hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8

RobustnessThe main purpose is to provide technical means to successfully perform a

desired task in the environment (achieved a high task performance)

For evaluation physically accessible quantities (task completion time

measurement error applied forces or dissipated energies)


hellip

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8

Robustness

PerformenceIt refers to the operatorrsquos feeling of being there Ideally the operator can

distinguish a remote environment that presence is correlated with task

performance in a positive causal way It implies that by improving the feeling

of presence task performance is improved as well

Perception


hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8

RobustnessCompared to perception it is a quantitative objective Transparency and

robustness are conflicting objectives (trade-off) A measure for

transparency is fidelity It describes the ability to exactly display the

environment to the operator


1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


Variable Impedance Control [5 6]

Model-mediated Teleoperation [40]

Movement Estimation [78 79]

Passivity [27] Prediction [39] hellip

Robustness

Performence

Perception

Transparency

Environment condition

Operatorrsquos characteristics

Communication channel

1 2 3 4 5 6 7 8

classify

evaluate

achieve


1 INTRODUCTION

PAM actuator

Fig 12 (a) Structure and (b) working principle of the PAM

KNEXO robot [22] Wearable Haptic Device [23]2-joint manipulator [16 17] Forcepsrsquo manipulators [20]

Rubber hose Fitting connectionLoose-weave

fibersUn-Pressured state

Pressured state

Stroke

(a) (b)

Rubber hoseLoose-weave

fibers

Fitting

connection

1 2 3 4 5 6 7 8


1 INTRODUCTION

Research Objectives

Propose Position

Tracking Controller

An integral terminal-style SMC is designed with the combination of an adaptive gain and TDE technique

0102

Propose Model-free Control for

slave-environment interaction03

Propose Force Sensorless refleting

control for bilateral haptic teleoperation

Propose Finite-Time

Force Controller

An adaptive gain Fast ITSMC-TDE scheme is established with a friction-free disturbance technique in finite-time

Using IRL to optimize the prescribed impedance model parameters to assist minimum human effort

Propose optimal impedance for

human-master interaction

05

04Applying RL approximation method to learning the optimal contact force (minimal) online

The new AFOB is proposed to estimate the force signalsAchieving great transparency (position force feedback)

1 2 3 4 5 6 7 8


CONTENTS







Introduction

7

3

4

5

6

C

H

A

P

T

E

R

1

7

8

Modeling and platform setup based on the APAM configuration2

1 2 3 4 5 6 7 8


2 MODELING AND PLATFORM SETUP BASED ON THE APAM CONFIGURATION

Mathematical modeling

n turns

nπD

bL

1 t

urn

s

D

2 2

2

3

4

dV dV d b LF P P P

dL dL d n

minus= minus = minus = minus

The tensile force of each PAM [7]

Fig 21 A cross-sectional view of the simplified geometric model

The motion equation of the system

1 2Mx Bx F F+ = minus +

Parameter Description Value UnitPs Supply pressure 65 bar

B Viscous damping coefficient 50 Nsmm

M Mass of the plate 05 Kg

L0 Original length of the PAM 175 mm

n Number of turns of the thread 104

b Thread length 200 mm

Table 21 Parameters of the studied PAM


0 0( ) ( )x f x x g x u d= + +

where f0(x) g0(x) are the nominal value of f(x)

g(x) d is a total of unknown disturbances

( )2 2 2

00 0

2 2

0 0

33 ( ) ( )

2

( ( ) ( )) ( ( ) ( ))

L x bP L x Bxf x g x

n M M n M

d f x x f x x g x g x u

+ minus = minus minus = = minus + minus +

1 2 3 4 5 6 7 8



Experimental modeling

The approximated PAM model based on its

pressure and contraction strain

1 0

1 0

1 0

( ) ( ) ( )

( )

( ) ( 1 2)

( )

k k k k k k k k k k

k k k k k

k k k k k

k k k k k

M b P k P f P

b P b P b

k P k P k k

f P f P f

+ + =

= +

= + = = +

Parameter Value Units

Spring

elements

kk1 11732 Nmbar

kk0 68 Nm

Damping

elements

bk1 962 Nmsbar

bk0 085 Nms

Force

elements

fk1 9122 Nbar

fk0 473 N

0-100

6

0

100

55

200

Fo

rce

[N

]

Contraction length

[mm]

300

400

4

Pressure [bar]

500

103 2 151

Fig 23 Experimental model for certain separate pressure

Table 22 List of APAM model parameters

Load

PAM 1 PAM 2Plate

Linear encoder

Proportional pressure

regulator valves

Card

PCI-1711

Card

PCI-QUAD04

Pressure signal

Control signal

Position signal

Pulley

Fig 22 Block diagram of the experimental system

1 2 3 4 5 6 7 8



Platform setup

Fig 25 The APAM-based (a) master and (b) slave subsystem configuration

Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle

Actuators Interactive

Loadcell

Pulley

Front viewBack view(a)

Proportional valve

Loadcell DamperSpring

Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping

Contact Force Signal

PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8


2 MODELING OF THE BILATERAL TELEOPERATION SYSTEM

Slave robot

Environment

Master robot

Control stick

Device Description

PAM

Festo MAS-10-200N-AA-MC-O

Nominal length 200 mm

Inside diameter 10 mm

Max pressure 8 bar

PPR ValveFesto VPPM-8L-L-1-G14-0L8H

Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711

AIAO 12 bits (resolution)

MEASUREMENT PCI-QUAD04

Linear

encoder

Type Rational WTB5-0500MM

Resolution 0005 mm

Table 23 Specifications of the exp devices

Fig 26 Photograph of the experimental apparatus of the overview BHTS

Platform setup

1 2 3 4 5 6 7 8


CONTENTS

Adaptive Finite-Time Force Sensorless Control scheme


Optimized Human-Robot Interaction force control using IRL



Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8

Adaptive Gain ITSMC with TDE scheme for the position control3


1 2 3 4 5 6 7 8


3 AN ADAPTIVE GAIN ITSMC-TDE SCHEME FOR THE POSITION CONTROL

Proposed control idea

3The control effectiveness is verified by experimental trials in

different challenging work conditions (compare several controllers)

2

The stability of the controlled system is theoretically analyzed

The first time the AITSMC-TDE scheme

is proposed for a PAM system1

1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d

minus= minus minus +

+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =

A terminal sliding surface is chosen

A proposed integral sliding surface is designed

An adaptive law for the robust gain

A robust control signal can be redesigned

Proof See more detail in Appendix 32 33

1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer

Coordinate Integral Terminal

Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d

d x t t f x t t x t t

g x t t u t t

= minus minus minus minus

minus minus minus

( ) ( )1

1ˆ

n su t u u g x dminus

= + minus

The control law can be rewritten

where

Proof See more detail in Appendix 31 34-35

Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus

minus minus minus minus

The auxiliary variables are defined

The nominal control signal can be designed as

1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters


Experimental validation

for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911

for adaptative gain μ = 10 δ = 005 σ = 01

for case study 3 KP = 15 KI = 18 KD = 005

for other case KP = 35 KI = 28 KD = 01

Proposed control

PID control

Fig 31 Performance response in case study 1


Case study 1 (no load) xd (mm) = 8sin(04πt)

Case study 2 (no load) xd (mm) = 8sin(πt)

Case study 3 (no load) A multistep trajectory

Case study 4 (with load) xd (mm) = 8sin(πt)

in turn 2-kg-load and 7-kg-load

1 2 3 4 5 6 7 8




Controller PID ITSMC Proposed

Sine 02 Hz ndash 8 mmL2[e] 04102 01737 01574

L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247

Sine 1 Hz ndash 8 mm

with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584

Table 31 A quantitative comparison between the tracking performances








1 2 3 4 5 6 7 8



Summary

Robust performance against uncertainties and disturbances1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Finite-time convergence of the tracking errors

Reducing the chattering phenomenon

Eliminating the reaching phase

Fast transient response

Next works will be directed to the control using the force properties

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8


Adaptive Gain ITSMC with TDE scheme3

Adaptive Finite-Time Force Sensorless Control scheme4

1 2 3 4 5 6 7 8


4 ADAPTIVE FINITE-TIME FORCE SENSORLESS CONTROL SCHEME

3 Verified the reliability and superiority of the proposed control

demonstrated in different challenging work conditions

2

The FITSMC guarantees the finite convergence new adaptive gain and TDE

are deployed to handle online (uncertainties and chattering phenomenon)

The proposed FSOB scheme obtains

force information eliminated noise and

static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +

1ˆ ˆ ˆ( ) ( ) ( ) ( )minus minus = minus minus minusd d c e dd t d t t u t t K t t

The FITSM surface is defined


2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e

An adaptive switching gain law An estimated lumped disturbance Force observer based-LeD [40]

APAM system

Force based

Time-delay

Estimator

Integral Terminal

Sliding SurfaceControl law

Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+

The proposed control

approach

2k

Plant + Environment

Force Observer

e

Proof See in Appendix 41 ndash 43

Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control

Fig 42 Performance response in Scenario 2

Scenario 1 τd (N) = 75sin(02πt- π2)+625


Scenario 3 A multistep trajectory with a LPFrsquos bandwidth

of 02(rads)

Observersfor FSOB ω = 500 rads ξ = 0001 α1 = 20 α2 = 12

α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain

Controller PID ISMC FITSMC-TDE Proposed

Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685

Table 41 A RMSE comparison between the tracking error performances


- Improved the control performances (fast response high

accuracy and robustness)

- Guaranteed the finite convergence

- Cancelling the lumped uncertainties via TDE solution and

the switching gain in the FITSMC online

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8



Model-free control for Optimal contact force in Unknown Environment5

1 2 3 4 5 6 7 8


5 MODEL-FREE CONTROL FOR OPTIMAL CONTACT FORCE IN UNKNOWN ENVIRONMENT

3 Verified effective for robot interaction control in

unknown environments

2 The stability of the closed-loop control is theoretically demonstrated

by the Lyapunov approach

The RL method is used to learn on-line

the optimal desired force1


1 2 3 4 5 6 7 8


( ) ( ) ( ( ))= minus + minuse e e eF t C x t K x x t

( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control


The environment is modeled as

a spring-damper system [81]

The optimal impedance model The position control in task space

( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d

Task-specific Outer-loopRobot-specific Inner-loop

Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution

The main objective of this work is to determine the optimal desired force

Fd which is the minimum force that minimizes the position error er

( ) ( )=d e rF t e t

The discounted cost function

( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt

J e t F t e Se F RF d

The Algebraic Riccati equation

0+ + + =T T

eA PA S P A PB

The feedback control gain

( )1

1 minus

minus= minus +T T

e B PB R B PA

IRL approximationRL solution

Implementation procedure

1 2 3 4 5 6 7 8

offline soliton





LQR solution RL solution

impossible for all time

The Hamilton-Jacobi-Bellman equation

( ) ( ) ( )min ( ) ( ) ( ) 0

minus minus + minus =

d

t

s rtF

V e t r t r e dt

The new Algebraic Riccati equation

( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =

The optimal desired force 1

22 21( ) ( )d rF t H H e tminus= minus

( ) ( ) ( ) ( ) ( )= = T

r d s rQ e t F t V e t X HX

IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t

online soliton using the off-policy RL to

find the solution of the given new ARE






minimize temporal difference error

The approximator of the Q-value function

( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=

The parameter update law

11( ) exp

2

T

i r d r i d i i r i d ie F e c F c e c F c minus = minus minus minus minus minus

( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation

Q-function update law based on IRL

( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T

r r d dr t e t Se t F t RF t= +

The reward function of the utility function


1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t


Fig 51 RL in unknown environments during the

learning process

Fig 52 Learning curves


κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation

PositionForce controlKPf = 0001 KIf = 002 λ = 02 ηΔ = 01 ρ1 = 05

ρ2 = 20 v = 911 α = 57

EnvironmentCe = 05 Nsmm-1 Ke = 2 N mm-1 r0 = 165 mm

xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8


Fig 53 Desired forceposition tracking response


Three steps are required to complete

the experiment

1-the identification of the environment

2-obtain the ARE solution using the

environmental estimation

3-the complete experiment for the

positionforce controller

IRL approximation

Achieving near-optimal performance without

knowledge of the environment dynamics

LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8



Optimized Human-Robot Interaction force control using IRL6

1 2 3 4 5 6 7 8


6 OPTIMIZED HUMAN-ROBOT INTERACTION FORCE CONTROL USING IRL

3 The unknown human dynamics as well as the optimal overall

human-robot system efficiency is verified in experimental trials

2

Find the optimal parameters of the impedance model to

assure motion tracking and also assist the human with

minimum effort to perform the task in real time

The stability of the closed-loop control is theoretically

demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K

The control torque in task space

1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=

= minus minus = minus

T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +

An approximation of the continuous functionThe prescribed

impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s

The human transfer function [99] Purpose τh rarr ed

0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX

A new augmented state

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

IRL based on LQR solution

Control design

1 2 3 4 5 6 7 8



The performance index

( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +

10 T TA P PA Q PB R BPminus= + + minus

The algebraic Riccati equation [83]

0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus


IRL solutionRL solution

An optimal feeback control

To minimize

solve

The infinite-horizon quadratic cost

Control design

The IRL Bellman equation for time interval ΔT gt 0

Proof of convergence See in Appendix 62

1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T

tV X t X Q K RK X d V X t T

+

= + minus +

( )7 ( ( )) ( ) ( )T T T T

tV X t X Q K RK X d X PX

= + =

( ) ( ) ( )T TA BK P P A BK Q K RKminus + minus = minus +

where P is the real symmetric positive definite solution

( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T

X t PX t X Q K RK X d

X t t PX t t

+

+

+ + +

The IRL Bellman equation can be written as



Fig 61 Continuous-time linear policy iteration algorithm

1 Start with an admissible control input

0 0

1= minuseu K X

2 Policy evaluation given a control policy ui find the

Pi using off-policy Bellman equation

1 1

1

+ minus= minusi T i

eu R B P X

3 Policy Improvement update the control policy

Online implementation

Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i

X t P X t X Q K RK X d

X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=

Solving for the cost using least squares

1|| ||i ip p minusminus

0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i

eu R B P Xminus minus= minus

Policy update

End

No

Yes

1i irarr +



Fig 62 The tracking error of the trajectory of the robot system and

the prescribed impedance model

Fig 63 Reference trajectory versus the state of the robot systems

during and after learning

κe = 00025 S = 1 R = 10 T = 0001

A = minus4006 B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation

Position controlKs = 20 F = 10 G = 1 μ1 = μ2 = 5 k = 01

= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8



Fig 64 The output trajectory and impedance model in the inner loop

Fig 65 The desired trajectory and output trajectory the outer loop

Fig 66 Interaction force

Perfectly performance of the human-

robot cooperation

A little deviation between the desired

and actual trajectory

The human interaction force is reduced

after learning and the optimal set of the

prescribed impedance model is found by

the outer-loop controller

Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8



Force Sensorless Reflecting Control for BHTS7

1 2 3 4 5 6 7 8


7 FORCE SENSORLESS REFLECTING CONTROL FOR BHTS

3 The effectiveness of the proposed control method is verified in

the different working conditions

2The stability and finite-time convergence characteristic of the closed-

loop control are theoretically analyzed by the Lyapunov approach

The first time the hybrid control based

on a fast finite-time NTSMC and AFOB1


1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus

The external torque estimation

[ ]

1 2ˆ( ) = + + minus minus minus +iv

i i ir i i i i i i i i i i idM C K sign s s s

( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+

An adaptive FSOB gain

where

where

The torque control design by style-FNTSMC

The modified FNTSM surface design

Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8



Fig 71 Torque estimation performances

λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control

PID control KP = 38 KI = 25 KD = 015

Observersfor NDO Lh = Le = 10 for RTOB β = 350 rads

for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10

RMSEs associated with NDO RTOB and proposed algorithm

are calculated of 0211 0192 and 0076 respectively

Parameters

Scenarios- The free space situation θd (t) = 5π(1 - 018t)sin(072πt)

- The contact and recovery situation soft environment Ke = 500 Nmm-1 Be = 4 Nsmm-1

stiff environment Ke = 2800 Nmm-1 Be = 20 Nsmm-1

Validation results

1 2 3 4 5 6 7 8



Fig 72 Position tracking performances

Fig 72 Position tracking error performances

Fig 72 Transparency performance in ldquosoftrdquo

Fig 72 Transparency error in soft contract

Fig 72 Transparency performance in ldquostiffrdquo

Fig 72 Transparency error in stiff contract

Validation results

1 2 3 4 5 6 7 8



Finite-time convergence of the

tracking errors

Eliminating the uncertainties the

noise effect in the obtained force

sensing via new AFOB

Robust performance against uncertainties

and disturbances in the system dynamics

Stability of the overall system and the

transparency performance can be

attained simultaneously


Summary

The great transparency performance and global stability of the BHTS under the proposed

control scheme are achieved despite the uncertainties and in different working conditions

1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7



Conclusion and Future works78

1 2 3 4 5 6 7 8


8 CONCLUSION AND FUTURE WORKS

Verified an effective position tracking controller via the AITSMC with TDE technique under

various working conditions (Its uncertainties and disturbance are handled)

Validated the significant effectiveness of finite-time force tracking problems based on the

FITSMC combined with a new adaptive gain and a friction-free disturbance (FSOB)

Based on the online learning ability of the RL approximation method the model-free controller

for optimal contact force ensures great position and force tracking performance

Optimized a prescribed impedance model parameters by the IRL approach to provide little

operator information in yielding highly effective position control purpose

Achieved good transparency performance with both force feedback and position tracking

simultaneously by the new adaptive force estimation combined the separately FNTSMC schemes

Conclutions

1 2 3 4 5 6 7 8



Future works

To improve the control performance in joint-space the robot system model can be

improved by robust identification approaches

Effectiveness of the proposed algorithm can be further increased by developing an automatic

tune method for both the estimation and control gains

Apply optimized strategies on both master and slave sides Note also that the computational

complexity should be reduced by the data size

Based on the developed prototype in this thesis it is possible to develop a high-quality

commercial teleoperation device in the rehabilitation field with a safer solution

1 2 3 4 5 6 7 8

20-May-21 | 53




Thank you for attending

1 2 3 4 5 6 7 8


CONTENTS

Modeling and platform setup based on a APAM configuration

Adaptive Gain ITSMC with TDE scheme for the position control






Introduction

2

7

3

4

5

6

C

H

A

P

T

E

R

1

7

8


CONTENTS








2

7

3

4

5

6

C

H

A

P

T

E

R 7

8

Introduction1

1 2 3 4 5 6 7 8


hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8






hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8






hellip

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8

Robustness





Perception


hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8






1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x






Robustness

Performence

Perception

Transparency




1 2 3 4 5 6 7 8

classify

evaluate

achieve


1 INTRODUCTION

PAM actuator





Pressured state

Stroke

(a) (b)


fibers

Fitting

connection

1 2 3 4 5 6 7 8


1 INTRODUCTION

Research Objectives

Propose Position

Tracking Controller


0102





Propose Finite-Time

Force Controller





05



1 2 3 4 5 6 7 8


CONTENTS







Introduction

7

3

4

5

6

C

H

A

P

T

E

R

1

7

8


1 2 3 4 5 6 7 8




n turns

nπD

bL

1 t

urn

s

D

2 2

2

3

4

dV dV d b LF P P P

dL dL d n














0 0( ) ( )x f x x g x u d= + +



( )2 2 2

00 0

2 2

0 0

33 ( ) ( )

2

( ( ) ( )) ( ( ) ( ))


n M M n M



1 2 3 4 5 6 7 8






1 0

1 0

1 0

( ) ( ) ( )

( )

( ) ( 1 2)

( )

k k k k k k k k k k

k k k k k

k k k k k

k k k k k

M b P k P f P

b P b P b

k P k P k k

f P f P f

+ + =

= +

= + = = +


Spring

elements

kk1 11732 Nmbar

kk0 68 Nm

Damping

elements

bk1 962 Nmsbar

bk0 085 Nms

Force

elements

fk1 9122 Nbar

fk0 473 N

0-100

6

0

100

55

200

Fo

rce

[N

]

Contraction length

[mm]

300

400

4

Pressure [bar]

500

103 2 151



Load

PAM 1 PAM 2Plate

Linear encoder


regulator valves

Card

PCI-1711

Card

PCI-QUAD04

Pressure signal

Control signal

Position signal

Pulley


1 2 3 4 5 6 7 8



Platform setup


Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle


Loadcell

Pulley


Proportional valve


Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping


PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


CONTENTS








2

7

3

4

5

6

C

H

A

P

T

E

R 7

8

Introduction1

1 2 3 4 5 6 7 8


hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8






hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8






hellip

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8

Robustness





Perception


hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8






1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x






Robustness

Performence

Perception

Transparency




1 2 3 4 5 6 7 8

classify

evaluate

achieve


1 INTRODUCTION

PAM actuator





Pressured state

Stroke

(a) (b)


fibers

Fitting

connection

1 2 3 4 5 6 7 8


1 INTRODUCTION

Research Objectives

Propose Position

Tracking Controller


0102





Propose Finite-Time

Force Controller





05



1 2 3 4 5 6 7 8


CONTENTS







Introduction

7

3

4

5

6

C

H

A

P

T

E

R

1

7

8


1 2 3 4 5 6 7 8




n turns

nπD

bL

1 t

urn

s

D

2 2

2

3

4

dV dV d b LF P P P

dL dL d n














0 0( ) ( )x f x x g x u d= + +



( )2 2 2

00 0

2 2

0 0

33 ( ) ( )

2

( ( ) ( )) ( ( ) ( ))


n M M n M



1 2 3 4 5 6 7 8






1 0

1 0

1 0

( ) ( ) ( )

( )

( ) ( 1 2)

( )

k k k k k k k k k k

k k k k k

k k k k k

k k k k k

M b P k P f P

b P b P b

k P k P k k

f P f P f

+ + =

= +

= + = = +


Spring

elements

kk1 11732 Nmbar

kk0 68 Nm

Damping

elements

bk1 962 Nmsbar

bk0 085 Nms

Force

elements

fk1 9122 Nbar

fk0 473 N

0-100

6

0

100

55

200

Fo

rce

[N

]

Contraction length

[mm]

300

400

4

Pressure [bar]

500

103 2 151



Load

PAM 1 PAM 2Plate

Linear encoder


regulator valves

Card

PCI-1711

Card

PCI-QUAD04

Pressure signal

Control signal

Position signal

Pulley


1 2 3 4 5 6 7 8



Platform setup


Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle


Loadcell

Pulley


Proportional valve


Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping


PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8






hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8






hellip

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8

Robustness





Perception


hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8






1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x






Robustness

Performence

Perception

Transparency




1 2 3 4 5 6 7 8

classify

evaluate

achieve


1 INTRODUCTION

PAM actuator





Pressured state

Stroke

(a) (b)


fibers

Fitting

connection

1 2 3 4 5 6 7 8


1 INTRODUCTION

Research Objectives

Propose Position

Tracking Controller


0102





Propose Finite-Time

Force Controller





05



1 2 3 4 5 6 7 8


CONTENTS







Introduction

7

3

4

5

6

C

H

A

P

T

E

R

1

7

8


1 2 3 4 5 6 7 8




n turns

nπD

bL

1 t

urn

s

D

2 2

2

3

4

dV dV d b LF P P P

dL dL d n














0 0( ) ( )x f x x g x u d= + +



( )2 2 2

00 0

2 2

0 0

33 ( ) ( )

2

( ( ) ( )) ( ( ) ( ))


n M M n M



1 2 3 4 5 6 7 8






1 0

1 0

1 0

( ) ( ) ( )

( )

( ) ( 1 2)

( )

k k k k k k k k k k

k k k k k

k k k k k

k k k k k

M b P k P f P

b P b P b

k P k P k k

f P f P f

+ + =

= +

= + = = +


Spring

elements

kk1 11732 Nmbar

kk0 68 Nm

Damping

elements

bk1 962 Nmsbar

bk0 085 Nms

Force

elements

fk1 9122 Nbar

fk0 473 N

0-100

6

0

100

55

200

Fo

rce

[N

]

Contraction length

[mm]

300

400

4

Pressure [bar]

500

103 2 151



Load

PAM 1 PAM 2Plate

Linear encoder


regulator valves

Card

PCI-1711

Card

PCI-QUAD04

Pressure signal

Control signal

Position signal

Pulley


1 2 3 4 5 6 7 8



Platform setup


Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle


Loadcell

Pulley


Proportional valve


Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping


PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8






hellip

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8

Robustness





Perception


hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8






1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x






Robustness

Performence

Perception

Transparency




1 2 3 4 5 6 7 8

classify

evaluate

achieve


1 INTRODUCTION

PAM actuator





Pressured state

Stroke

(a) (b)


fibers

Fitting

connection

1 2 3 4 5 6 7 8


1 INTRODUCTION

Research Objectives

Propose Position

Tracking Controller


0102





Propose Finite-Time

Force Controller





05



1 2 3 4 5 6 7 8


CONTENTS







Introduction

7

3

4

5

6

C

H

A

P

T

E

R

1

7

8


1 2 3 4 5 6 7 8




n turns

nπD

bL

1 t

urn

s

D

2 2

2

3

4

dV dV d b LF P P P

dL dL d n














0 0( ) ( )x f x x g x u d= + +



( )2 2 2

00 0

2 2

0 0

33 ( ) ( )

2

( ( ) ( )) ( ( ) ( ))


n M M n M



1 2 3 4 5 6 7 8






1 0

1 0

1 0

( ) ( ) ( )

( )

( ) ( 1 2)

( )

k k k k k k k k k k

k k k k k

k k k k k

k k k k k

M b P k P f P

b P b P b

k P k P k k

f P f P f

+ + =

= +

= + = = +


Spring

elements

kk1 11732 Nmbar

kk0 68 Nm

Damping

elements

bk1 962 Nmsbar

bk0 085 Nms

Force

elements

fk1 9122 Nbar

fk0 473 N

0-100

6

0

100

55

200

Fo

rce

[N

]

Contraction length

[mm]

300

400

4

Pressure [bar]

500

103 2 151



Load

PAM 1 PAM 2Plate

Linear encoder


regulator valves

Card

PCI-1711

Card

PCI-QUAD04

Pressure signal

Control signal

Position signal

Pulley


1 2 3 4 5 6 7 8



Platform setup


Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle


Loadcell

Pulley


Proportional valve


Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping


PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


hellip

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8

Robustness





Perception


hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8






1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x






Robustness

Performence

Perception

Transparency




1 2 3 4 5 6 7 8

classify

evaluate

achieve


1 INTRODUCTION

PAM actuator





Pressured state

Stroke

(a) (b)


fibers

Fitting

connection

1 2 3 4 5 6 7 8


1 INTRODUCTION

Research Objectives

Propose Position

Tracking Controller


0102





Propose Finite-Time

Force Controller





05



1 2 3 4 5 6 7 8


CONTENTS







Introduction

7

3

4

5

6

C

H

A

P

T

E

R

1

7

8


1 2 3 4 5 6 7 8




n turns

nπD

bL

1 t

urn

s

D

2 2

2

3

4

dV dV d b LF P P P

dL dL d n














0 0( ) ( )x f x x g x u d= + +



( )2 2 2

00 0

2 2

0 0

33 ( ) ( )

2

( ( ) ( )) ( ( ) ( ))


n M M n M



1 2 3 4 5 6 7 8






1 0

1 0

1 0

( ) ( ) ( )

( )

( ) ( 1 2)

( )

k k k k k k k k k k

k k k k k

k k k k k

k k k k k

M b P k P f P

b P b P b

k P k P k k

f P f P f

+ + =

= +

= + = = +


Spring

elements

kk1 11732 Nmbar

kk0 68 Nm

Damping

elements

bk1 962 Nmsbar

bk0 085 Nms

Force

elements

fk1 9122 Nbar

fk0 473 N

0-100

6

0

100

55

200

Fo

rce

[N

]

Contraction length

[mm]

300

400

4

Pressure [bar]

500

103 2 151



Load

PAM 1 PAM 2Plate

Linear encoder


regulator valves

Card

PCI-1711

Card

PCI-QUAD04

Pressure signal

Control signal

Position signal

Pulley


1 2 3 4 5 6 7 8



Platform setup


Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle


Loadcell

Pulley


Proportional valve


Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping


PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


hellip

Performence

Perception

Transparency

1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x


1 2 3 4 5 6 7 8






1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x






Robustness

Performence

Perception

Transparency




1 2 3 4 5 6 7 8

classify

evaluate

achieve


1 INTRODUCTION

PAM actuator





Pressured state

Stroke

(a) (b)


fibers

Fitting

connection

1 2 3 4 5 6 7 8


1 INTRODUCTION

Research Objectives

Propose Position

Tracking Controller


0102





Propose Finite-Time

Force Controller





05



1 2 3 4 5 6 7 8


CONTENTS







Introduction

7

3

4

5

6

C

H

A

P

T

E

R

1

7

8


1 2 3 4 5 6 7 8




n turns

nπD

bL

1 t

urn

s

D

2 2

2

3

4

dV dV d b LF P P P

dL dL d n














0 0( ) ( )x f x x g x u d= + +



( )2 2 2

00 0

2 2

0 0

33 ( ) ( )

2

( ( ) ( )) ( ( ) ( ))


n M M n M



1 2 3 4 5 6 7 8






1 0

1 0

1 0

( ) ( ) ( )

( )

( ) ( 1 2)

( )

k k k k k k k k k k

k k k k k

k k k k k

k k k k k

M b P k P f P

b P b P b

k P k P k k

f P f P f

+ + =

= +

= + = = +


Spring

elements

kk1 11732 Nmbar

kk0 68 Nm

Damping

elements

bk1 962 Nmsbar

bk0 085 Nms

Force

elements

fk1 9122 Nbar

fk0 473 N

0-100

6

0

100

55

200

Fo

rce

[N

]

Contraction length

[mm]

300

400

4

Pressure [bar]

500

103 2 151



Load

PAM 1 PAM 2Plate

Linear encoder


regulator valves

Card

PCI-1711

Card

PCI-QUAD04

Pressure signal

Control signal

Position signal

Pulley


1 2 3 4 5 6 7 8



Platform setup


Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle


Loadcell

Pulley


Proportional valve


Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping


PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


1 INTRODUCTION


Slave

control

Master

control

Co

mm

un

icatio

n c

ha

nn

el

Monitor

Operator

Master

Haptic interface

Slave

teleoperator

Remote

environment

Vision

sensor

hf

m

m

x

x

sdw

s

s

x

x

efmdwm

m

x

x






Robustness

Performence

Perception

Transparency




1 2 3 4 5 6 7 8

classify

evaluate

achieve


1 INTRODUCTION

PAM actuator





Pressured state

Stroke

(a) (b)


fibers

Fitting

connection

1 2 3 4 5 6 7 8


1 INTRODUCTION

Research Objectives

Propose Position

Tracking Controller


0102





Propose Finite-Time

Force Controller





05



1 2 3 4 5 6 7 8


CONTENTS







Introduction

7

3

4

5

6

C

H

A

P

T

E

R

1

7

8


1 2 3 4 5 6 7 8




n turns

nπD

bL

1 t

urn

s

D

2 2

2

3

4

dV dV d b LF P P P

dL dL d n














0 0( ) ( )x f x x g x u d= + +



( )2 2 2

00 0

2 2

0 0

33 ( ) ( )

2

( ( ) ( )) ( ( ) ( ))


n M M n M



1 2 3 4 5 6 7 8






1 0

1 0

1 0

( ) ( ) ( )

( )

( ) ( 1 2)

( )

k k k k k k k k k k

k k k k k

k k k k k

k k k k k

M b P k P f P

b P b P b

k P k P k k

f P f P f

+ + =

= +

= + = = +


Spring

elements

kk1 11732 Nmbar

kk0 68 Nm

Damping

elements

bk1 962 Nmsbar

bk0 085 Nms

Force

elements

fk1 9122 Nbar

fk0 473 N

0-100

6

0

100

55

200

Fo

rce

[N

]

Contraction length

[mm]

300

400

4

Pressure [bar]

500

103 2 151



Load

PAM 1 PAM 2Plate

Linear encoder


regulator valves

Card

PCI-1711

Card

PCI-QUAD04

Pressure signal

Control signal

Position signal

Pulley


1 2 3 4 5 6 7 8



Platform setup


Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle


Loadcell

Pulley


Proportional valve


Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping


PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


1 INTRODUCTION

PAM actuator





Pressured state

Stroke

(a) (b)


fibers

Fitting

connection

1 2 3 4 5 6 7 8


1 INTRODUCTION

Research Objectives

Propose Position

Tracking Controller


0102





Propose Finite-Time

Force Controller





05



1 2 3 4 5 6 7 8


CONTENTS







Introduction

7

3

4

5

6

C

H

A

P

T

E

R

1

7

8


1 2 3 4 5 6 7 8




n turns

nπD

bL

1 t

urn

s

D

2 2

2

3

4

dV dV d b LF P P P

dL dL d n














0 0( ) ( )x f x x g x u d= + +



( )2 2 2

00 0

2 2

0 0

33 ( ) ( )

2

( ( ) ( )) ( ( ) ( ))


n M M n M



1 2 3 4 5 6 7 8






1 0

1 0

1 0

( ) ( ) ( )

( )

( ) ( 1 2)

( )

k k k k k k k k k k

k k k k k

k k k k k

k k k k k

M b P k P f P

b P b P b

k P k P k k

f P f P f

+ + =

= +

= + = = +


Spring

elements

kk1 11732 Nmbar

kk0 68 Nm

Damping

elements

bk1 962 Nmsbar

bk0 085 Nms

Force

elements

fk1 9122 Nbar

fk0 473 N

0-100

6

0

100

55

200

Fo

rce

[N

]

Contraction length

[mm]

300

400

4

Pressure [bar]

500

103 2 151



Load

PAM 1 PAM 2Plate

Linear encoder


regulator valves

Card

PCI-1711

Card

PCI-QUAD04

Pressure signal

Control signal

Position signal

Pulley


1 2 3 4 5 6 7 8



Platform setup


Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle


Loadcell

Pulley


Proportional valve


Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping


PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


1 INTRODUCTION

Research Objectives

Propose Position

Tracking Controller


0102





Propose Finite-Time

Force Controller





05



1 2 3 4 5 6 7 8


CONTENTS







Introduction

7

3

4

5

6

C

H

A

P

T

E

R

1

7

8


1 2 3 4 5 6 7 8




n turns

nπD

bL

1 t

urn

s

D

2 2

2

3

4

dV dV d b LF P P P

dL dL d n














0 0( ) ( )x f x x g x u d= + +



( )2 2 2

00 0

2 2

0 0

33 ( ) ( )

2

( ( ) ( )) ( ( ) ( ))


n M M n M



1 2 3 4 5 6 7 8






1 0

1 0

1 0

( ) ( ) ( )

( )

( ) ( 1 2)

( )

k k k k k k k k k k

k k k k k

k k k k k

k k k k k

M b P k P f P

b P b P b

k P k P k k

f P f P f

+ + =

= +

= + = = +


Spring

elements

kk1 11732 Nmbar

kk0 68 Nm

Damping

elements

bk1 962 Nmsbar

bk0 085 Nms

Force

elements

fk1 9122 Nbar

fk0 473 N

0-100

6

0

100

55

200

Fo

rce

[N

]

Contraction length

[mm]

300

400

4

Pressure [bar]

500

103 2 151



Load

PAM 1 PAM 2Plate

Linear encoder


regulator valves

Card

PCI-1711

Card

PCI-QUAD04

Pressure signal

Control signal

Position signal

Pulley


1 2 3 4 5 6 7 8



Platform setup


Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle


Loadcell

Pulley


Proportional valve


Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping


PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


CONTENTS







Introduction

7

3

4

5

6

C

H

A

P

T

E

R

1

7

8


1 2 3 4 5 6 7 8




n turns

nπD

bL

1 t

urn

s

D

2 2

2

3

4

dV dV d b LF P P P

dL dL d n














0 0( ) ( )x f x x g x u d= + +



( )2 2 2

00 0

2 2

0 0

33 ( ) ( )

2

( ( ) ( )) ( ( ) ( ))


n M M n M



1 2 3 4 5 6 7 8






1 0

1 0

1 0

( ) ( ) ( )

( )

( ) ( 1 2)

( )

k k k k k k k k k k

k k k k k

k k k k k

k k k k k

M b P k P f P

b P b P b

k P k P k k

f P f P f

+ + =

= +

= + = = +


Spring

elements

kk1 11732 Nmbar

kk0 68 Nm

Damping

elements

bk1 962 Nmsbar

bk0 085 Nms

Force

elements

fk1 9122 Nbar

fk0 473 N

0-100

6

0

100

55

200

Fo

rce

[N

]

Contraction length

[mm]

300

400

4

Pressure [bar]

500

103 2 151



Load

PAM 1 PAM 2Plate

Linear encoder


regulator valves

Card

PCI-1711

Card

PCI-QUAD04

Pressure signal

Control signal

Position signal

Pulley


1 2 3 4 5 6 7 8



Platform setup


Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle


Loadcell

Pulley


Proportional valve


Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping


PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8




n turns

nπD

bL

1 t

urn

s

D

2 2

2

3

4

dV dV d b LF P P P

dL dL d n














0 0( ) ( )x f x x g x u d= + +



( )2 2 2

00 0

2 2

0 0

33 ( ) ( )

2

( ( ) ( )) ( ( ) ( ))


n M M n M



1 2 3 4 5 6 7 8






1 0

1 0

1 0

( ) ( ) ( )

( )

( ) ( 1 2)

( )

k k k k k k k k k k

k k k k k

k k k k k

k k k k k

M b P k P f P

b P b P b

k P k P k k

f P f P f

+ + =

= +

= + = = +


Spring

elements

kk1 11732 Nmbar

kk0 68 Nm

Damping

elements

bk1 962 Nmsbar

bk0 085 Nms

Force

elements

fk1 9122 Nbar

fk0 473 N

0-100

6

0

100

55

200

Fo

rce

[N

]

Contraction length

[mm]

300

400

4

Pressure [bar]

500

103 2 151



Load

PAM 1 PAM 2Plate

Linear encoder


regulator valves

Card

PCI-1711

Card

PCI-QUAD04

Pressure signal

Control signal

Position signal

Pulley


1 2 3 4 5 6 7 8



Platform setup


Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle


Loadcell

Pulley


Proportional valve


Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping


PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8






1 0

1 0

1 0

( ) ( ) ( )

( )

( ) ( 1 2)

( )

k k k k k k k k k k

k k k k k

k k k k k

k k k k k

M b P k P f P

b P b P b

k P k P k k

f P f P f

+ + =

= +

= + = = +


Spring

elements

kk1 11732 Nmbar

kk0 68 Nm

Damping

elements

bk1 962 Nmsbar

bk0 085 Nms

Force

elements

fk1 9122 Nbar

fk0 473 N

0-100

6

0

100

55

200

Fo

rce

[N

]

Contraction length

[mm]

300

400

4

Pressure [bar]

500

103 2 151



Load

PAM 1 PAM 2Plate

Linear encoder


regulator valves

Card

PCI-1711

Card

PCI-QUAD04

Pressure signal

Control signal

Position signal

Pulley


1 2 3 4 5 6 7 8



Platform setup


Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle


Loadcell

Pulley


Proportional valve


Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping


PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8



Platform setup


Proportional

pressure

regulator

valves

Optical

encoder

Control stick

Loadcells

Pneumatic

Muscle


Loadcell

Pulley


Proportional valve


Environment

APAM system

PAMPulley

PositionSignal

ControlSignal

StiffnessDamping


PCPCI Cards

Control box

(b)

1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8



Slave robot

Environment

Master robot

Control stick

Device Description

PAM




Max pressure 8 bar


Max pressure 8 bar

DAQ Card

ADVANTECH PCI-1711



Linear

encoder


Resolution 0005 mm



Platform setup

1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


CONTENTS






Introduction

7

4

5

6

C

H

A

P

T

E

R

1

7

8



1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8






2




1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8



2 1 1 2 1s e k e k e= + +

( ) ( ) (( ) ( ) )

0

1

0 1 2 2 1 2

0 1 2 0 1 1

t

t

n d

s t s t k e k e e

f x x g x u x d


+ + minus

( ) ( )( ) ( ) 31

0 1 3 3 4ˆ

su t g x sign

minus = minus + +

( )( ) ( )

( )3 3 0

3

ˆ ˆ1ˆ or

( )mt t

tsign t

= +=

+ =






1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

Control design

1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8



( ) ( ) ( )( )( )( ) ( )

2 0 1 2

0 1

ˆ

d d d

d d


g x t t u t t


minus minus minus

( ) ( )1

1ˆ


= + minus


where


Control design

1 1 1 1

2 2 1

de x x

x

= = minus = minus

( )( ) ( )1 2

1

0 1 1 1 0 1 2

1

1 1 1 1 1 2 2 2 2

n du g x x f x x

minus

minus

= minus minus




1 2d dx x

Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu

1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


Parameters



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 α = 911




Proposed control

PID control








1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8






L2[u] 09961 10270 09602


L2[u] 11284 11152 10900

MultistepL2[e] 08587 06846 06368

L2[u] 07670 07333 07247


with 2-kg-load

L2[e] 12578 08314 04779

L2[u] 11414 11506 11287


with 7-kg-load

L2[e] 15937 10827 05067

L2[u] 11613 11729 11584









1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8



Summary


Time-delay

Estimator

The PAM system

Control law

Robust

control signal

Adaptation

law

Nominal

control signal

Transfer


Sliding Surface

Terminal

Sliding Surface

Reference

Trajectory

d

The proposed

control approach

u

1 2x x

su

3ˆ

1 2x x

s

1

2

nu






1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

5

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8





2





static friction

1


1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8



( )[ ]

1 20

( ) ( ( )) = + +t

v

f f fe e sign e d

( ) ( )[ 1]5

5

4

ˆ

+

minus= minus

v

signsign

1 1 [ ]

1 2

[ ]

4 5

ˆ( ) ( ) ( )

ˆ ( )

v

c d f f

v

u t d t K e sign e

sign

minus minus= + + +

+ +




2

2

ˆ ˆ ˆ2ˆ

+ += e e e e e e

e

e


APAM system

Force based

Time-delay

Estimator

Integral Terminal


Adaptive law

Environment

Configuration

Reference

Trajectoryu

d

ˆe

d

+

+


approach

2k

Plant + Environment

Force Observer

e


Control design

1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8



for ITSMC η1 = 3 η2 = 15 η3 = 25 κ1 = κ3 = 10

κ2 = κ4 = 2 γ1 = γ2 = 57 k1 = 2 k2 = 55 v = 911




Proposed control

PID control





of 02(rads)


α3 = 5 α4 = 15


Parameters


1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8




Fig 44 Control gain


Sine 01 Hz 21050 12152 09440 05158

Sine 04 Hz 32054 18045 20632 19807

Multistep 26508 23548 20618 17685








Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

6

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8



( ( )) ( )r dZ e t F t= [ ]

1 2

( ) ( ) ( )

( ) ( )

= + +

minus minus minus +

r r r

v

e

Me t Ce t Ke t

sign F t

The force control





( ) ( ) ( )

= + f Pf f If f

t

x s K e t K e d


Precise position

control

Prescribed

Impedance Model

Environment

Configuration

APAM system

xx

d

Force controlx

rx

fe

r

Fe

Fd

ef

-

+ -+ -

+

RL-based

approximation

ϕ

Control design

1 2 3 4 5 6 7 8

( ) ( ) ( )r r de t Ae t BF t= +



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8



LQR solution



( ) ( )=d e rF t e t


( ) ( )( ) ( ) ( ) ( ) ( ) ( )

= minus +T T

s r d r r d dt



0+ + + =T T

eA PA S P A PB


( )1

1 minus

minus= minus +T T

e B PB R B PA



1 2 3 4 5 6 7 8

offline soliton








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8








( ) ( ) ( )min ( ) ( ) ( ) 0


d

t

s rtF

V e t r t r e dt


( )( ) ( )

2 ( ) ( ) ( ) ( ) 0

T T

r r

T T

r d d d

e t A P PA S P e t

e t PBF t F t RF t

+ + minus

+ + =



( ) ( ) ( ) ( ) ( )= = T


IRL approximation


( ) ( )

( )

( )( ) ( ) ( )

where ( ) ( ) ( ) ( ) ( ) ( )

t

s r r dt

T T

r d r r d d

V e t r e F e d

r e F e Se F RF

minus minus=

= +


1 2 3 4 5 6 7 8

( ) ( )=d e rF t e t










( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8








( )( ) ( )1

ˆ N

r d i r d i

i

Q e F e F =

=


11( ) exp

2

T


( )

( ) ( ) ( ) ( )

ˆ ( ) ( ) ( )1( ) ( )

( )

=

= minus +

r d

t t t t

Q e t F t tt t

t t

IRL approximation


( )

( )

( )( ) ( ) ( )

( ) ( )

t tt

r dt

t

r d

Q e t F t r e d

e Q e t t F t t

+

minus minus

minus

=

+ + +

( ) ( ) ( ) ( ) ( )T T




1 2 3 4 5 6 7 8

1

( )( )

( )

i r di r d N

i r di

e Fe F

e F

=

=

( ) ( )=d e rF t e t



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8



learning process



κe = 00025 S = 1 R = 10 T = 0001 A = minus4006

B = minus0002

υ = 09 γ = 09 δΔ = 065 ε = 01 ε-decay = 099

K = 10 β = 01

LQR solution

RL aproximation


ρ2 = 20 v = 911 α = 57


xd = 39917 mm xe = 35 mm

Parameters

Validation results

1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8





the experiment






IRL approximation



LQR solution

Validation results

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

C

H

A

P

T

E

R

1

7

8




1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8





2





demonstrated

1


1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8




( )1 20

ˆ ( )

ˆ (|| || )

= + + +

+ + minus

t

s h h h

q B h h

K e e e d

K Z Z K


1 2

1 2 1

2 1 2

ˆ ˆˆ ( ) ( )

ˆ ˆˆ ˆ ˆ || ||

ˆˆ ˆ ˆ( ) || ||

=


T T

T T T

h h h

T

h h

q W W q

W F F W q kF W

W Gq W kG W

2

h hm

d d d

K

M s C s K

=

+ +


impedance model

Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed

Control design

1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8




1

1

[ ]

minus

minus

= +

= minus

=

=

h h h h d

h e

h e p d

T T T

d d d

A B e

A k

B k

e e e

( )1

+=

+

p d

e

sG s

k s


0

0

= + = +

= minus

q d qd e

e

h h hh e

A e Be X AX Buu

B A u KX


Prescribed

Impedance Model

Haptic interface

based-APAM

Torque

control lawHuman

Operator

Optimal Law-based

IRL

Kh+-

NN functional

approximation

1

s

θ +

-

τ θ d θ m eh

τ h

ξ

ed


Control design

1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8




( )T T T

m d d d h h h e et

J e Q e Q u Ru d

= + +



0

0

q d qd

e

h h hh

e

A e BeX u

B A

u KX

= = +

= minus




To minimize

solve


Control design



1 2 3 4 5 6 7 8

( ) 1

( )

arg min ( ) ( )e

T

e eu t

u V t X t u R B PX

minus

= = minus

( )7 7( ( )) ( ) ( ) ( ( ))t T

T T T


+

= + minus +

( )7 ( ( )) ( ) ( )T T T T


= + =



( )( ) ( ) = ( ) ( )

( ) ( )

t tT T T T

t

T


X t t PX t t

+

+

+ + +






0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8





0 0

1= minuseu K X



1 1

1

+ minus= minusi T i

eu R B P X



Control design

1 2 3 4 5 6 7 8

( )( )( ) ( )= ( ) ( )

( ) ( )

t t TT i T i i T

t

T i


X t t P X t t

+

+

+ + +

Begin

1 2

1 2

1

( ) ( )

( ) ( ) ( )

( )

N i i i

Ti i N i

i T

t t t

K K K

p

minus

= = minus +

=

=



0

1 10 1 ( ) 0P i K A BK= = minus

Initialization

1 1i T i


Policy update

End

No

Yes

1i irarr +







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8







κe = 00025 S = 1 R = 10 T = 0001


υ = 09 γ = 09 δΔ = 065 ε = 01

ε-decay = 099 K = 10 β = 01

LQR solution

RL aproximation


= 100 κp = 20 κd = 10 and ke = 018 s

θd = 10sin(02πt) Md = 1 Kg Bd = 718

Nsmm-1 Kd =1576 Nmm-1 Kh = 0046

Parameters

Z

Simulation

Validation results

1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8







robot cooperation







Experiment

Validation results

Discussion

1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


CONTENTS





Introduction

7

4

5

6

C

H

A

P

T

E

R

1

8




1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8










1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8



( )1 2id = minus +

1 1

2

= minus + = minus +

id

T

i i id i

T

i ia = minus


[ ]



( )i i i is = +

[ ]

2

1 2

if 0 or 0 and ( )

( ) if 0 and

i

i i i i i i

i i

i i i i i i i i

s s

z z sign s

= =

+


where

where



Master Slave

Environment

APAM Slave

system

Slave

Controller

Estimator

Oper

ator

Master

Controller

APAM Master

system

Estimator

Impendence

filter

+

++

-

+

-

+

+md

m s

s

sd

ˆsdˆmd

mf

m

d

Control design

1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8




λi = 8 pi = 5 qi = 7 ςi = 10-4 νi = 911

εi = 01 ρ1i = 10 ρ2i = 2

Proposed control



for AFOB κi(0) = 01 λi = 10 ηi = 0 m = 01

a = 10



Parameters




Validation results

1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8









Validation results

1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8




tracking errors










Summary



1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8


CONTENTS





Introduction

4

5

6

C

H

A

P

T

E

R

1

7




1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8













Conclutions

1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8



Future works









1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8

20-May-21 | 53





1 2 3 4 5 6 7 8

a presentation of doctoral dissertation

Documents