impedance control of systems with joint flexibility

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IMPEDANCE CONTROL OF SYSTEMS WITH JOINT FLEXIBILITY

A dissertation submitted to the University of Manchester for the degree of Master of

Science in the Faculty of Engineering and Physical Sciences

2014

Marat Imanbayev

School of Electrical and Electronic Engineering


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List of Contents

List of Tables 4

List of Figures 5

List of Symbols 7

1 Introduction 13

1.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 Aims and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Literature review 17

2.1 Torque control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.1 Series elastic actuator . . . . . . . . . . . . . . . . . . . . . . 182.2 Impedance control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Impedance control architectures . . . . . . . . . . . . . . . . . 21

2.2.2 Passivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Series elastic joint dynamics 23

3.1 SEA system description . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 SEA system modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 State space representation . . . . . . . . . . . . . . . . . . . . 27

3.3 Open loop simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4 Bandwidth limitations of the torque controlled by PI controller. . . . 29

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Torque controller design 32

4.1 PI controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 LQR controller design . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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4.3 Comparison between PI and LQR controllers. . . . . . . . . . . . . . 38

4.3.1 Step response characteristics . . . . . . . . . . . . . . . . . . . 39

4.3.2 Stability margins . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.3 Torque ripples attenuation . . . . . . . . . . . . . . . . . . . . 40

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Impedance control 42

5.1 Approach and implementation . . . . . . . . . . . . . . . . . . . . . . 42

5.2 Control problem formulation . . . . . . . . . . . . . . . . . . . . . . . 43

5.3 Stability regions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.3.1 Stability regions with inner PI controller . . . . . . . . . . . . 45

5.3.2 Stability regions with inner LQR controller . . . . . . . . . . . 45

5.4 Passivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.4.1 Coupled stability . . . . . . . . . . . . . . . . . . . . . . . . . 485.5 Rejection of the torque ripple in the impedance . . . . . . . . . . . . 51

5.6 Impedance tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.6.1 Impedance tracking with inner PI controller . . . . . . . . . . 53

5.6.2 Impedance tracking with inner LQR controller . . . . . . . . . 55

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6 Conclusions and future work 58

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

References 60

A Open loop transfer functions and simulations 62

A.1 Open loop transfer functions . . . . . . . . . . . . . . . . . . . . . . . 62

A.2 Open loop simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 63

B Miscellaneous MATLAB codes 64

B.1 Open loop modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 64B.2 LQR design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

B.3 Outer loop stability regions . . . . . . . . . . . . . . . . . . . . . . . 66

Word count: 12553

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List of Tables

2.1 Comparison of various impedance control schemes . . . . . . . . . . . 22

3.1 Electrical and mechanical parameters for the SEA model . . . . . . . 26

4.1 PI controllers with corresponding closed loop characteristics . . . . . 33

4.2 LQR controllers with corresponding closed loop characteristics . . . . 37

5.1 Passivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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List of Figures

1.1 Block diagram of an impedance loop with an inner torque loop . . . . 14

2.1 Block diagram of Series elastic actuator. . . . . . . . . . . . . . . . . 18

3.1 SEA experimental setup. 1-motor pulley, 2-harmonic drive, 3-motor

position sensor, 4-DC motor, 5-cable drive, 6-link position sensor, 7-output pulley, 8-rigid strain gauge torque sensor, 9-output link . . . . 23

3.2 Schematic structure of SEA . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Block diagram of the SEA model . . . . . . . . . . . . . . . . . . . . 27

3.4 Open loop response of joint torque Ts . . . . . . . . . . . . . . . . . . 29

3.5 Root locus of C(s)G(s) around the origin computed for the model

with gravity (right) and without gravity (left). The open loop poles

and zeros are red, whereas the controller pole and zero are black . . . 30

4.1 Inner torque loop with PI controller . . . . . . . . . . . . . . . . . . . 334.2 Closed loop torque response (left) and link position response (right) to

step change in torque reference . . . . . . . . . . . . . . . . . . . . . 34

4.3 Torque ripple frequency response (0.013 N m). . . . . . . . . . . . . . 35

4.4 Inner torque loop with LQR controller . . . . . . . . . . . . . . . . . 36

4.5 Closed loop torque responses (left) and link position response (right)

to step change in torque reference . . . . . . . . . . . . . . . . . . . . 38

4.6 Torque ripple frequency response (0.013N m) . . . . . . . . . . . . . 38

4.7 Step response of torque controlled by PI and LQR controllers. . . . . 39

4.8 Torque ripple attenuation with PI and LQR controllers . . . . . . . . 40

4.9 Torque ripples effect on torque response: PI case (left) and LQR case

(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5.1 Block diagram of the system with inner torque loop (blue) and outer

impedance loop (green). . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Cascaded control system block diagram for ICT . . . . . . . . . . . . 44

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5.3 Stability regions varying the gain of PI controller (keepingz= 3) . 45

5.4 Stability regions varyingQmatrix of LQR controller . . . . . . . . . 46

5.5 Passivity regions (log scale): PI controllers . . . . . . . . . . . . . . . 48

5.6 Passivity regions (log scale): LQR controllers. . . . . . . . . . . . . . 48

5.7 The admittanceYcoupled to a spring . . . . . . . . . . . . . . . . . 495.8 Root locus ofKeY(s)/s. Open loop zeros are black . . . . . . . . . . 49

5.9 The admittanceYcoupled to a mass . . . . . . . . . . . . . . . . . . 50

5.10 Root locus ofJesY(s). Open loop zeros are black . . . . . . . . . . . 50

5.11 Bode diagram (magnitude) of the transfer functionTs

Trip(s). Inner PI

controllers have fixed zero at z= 3 . . . . . . . . . . . . . . . . . . 51

5.12 Bode diagram (magnitude) of the transfer functionTs

Trip(s) with LQR

controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.13 Bode diagram of the link impedance (PI case). Target impedance

(K = 40, D = 5) (red) and real impedance computed for k = 0.03

(blue),k = 1 (black), k = 2.7 (green) . . . . . . . . . . . . . . . . . . 54

5.14 Bode diagram of the link impedance (PI case). Target impedance

(K= 500, D = 30) (red) and real impedance computed for k = 0.03

(blue),k = 1 (black), k = 2.7 (green) . . . . . . . . . . . . . . . . . . 54

5.15 Bode diagram of the link impedance (LQR case). Target impedance

(K= 40, D= 5) is red . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.16 Bode diagram of the link impedance (LQR case). Target impedance(K= 500, D= 30) is red. . . . . . . . . . . . . . . . . . . . . . . . . 56

A.1 Open loop responses of SEA model. Small graphs inside each figure

show the responses for the first 0.6 s. . . . . . . . . . . . . . . . . . . 63

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List of Symbols

ql Link angular velocity

qm Motor angular velocity

Bl Link viscous friction constant

Bm Motor viscous friction constant

C(s) Transfer function of PI controller

D Desired damping

Dh Passive spring damping

G(s) Transfer function between input voltage and joint torque

H Link centre of mass

Je Passive environmental mass inertia

Jl Link inertia

Jm Rotor+harmonic drive inertia

K Desired stiffness

k Gain of PI controller in zero-pole-gain form

Kb Motor constant

Ke Passive environmental stiffness

Kg Stiffness due to gravity

Kh Passive spring stiffness

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Ki LQR controller state feedback gain

ki Integral gain of PI controller

kp Proportional gain of PI controller

Kt Torque constant

L Motor inductance

M Link mass

N Gear box ratio

Q State cost matrix in LQR control

ql Link angular position

qm Motor angular position

R Motor resistance

Rq Control cost matrix in LQR

Tc Compensation torque

Td Torque produced by spring and damping action

Tg Gravity torque

Tl Load disturbance torque

Tm Torque generated by the motor

Ts Measured joint torque

Tfric Meshing friction torque

Tref Reference torque

Trip Torque ripple

z Zero of PI controller in zero-pole-gain form

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Abstract

The aim of this dissertation is to investigate the performance of an impedance con-

trol strategy evaluated on a series elastic actuator. Impedance control improves the

interaction ability of the robotic systems due to controlling the interaction dynamics

rather than explicitly force or position. Impedance parameters, such as stiffness and

damping, vary within specified range of values.

In this thesis, the impedance control is implemented based on an inner torque

loop with two different torque controllers, namely PI and LQR. The effects of varying

controllers parameters on the closed loop bandwidth and disturbance attenuation

have been examined. Moreover, suitable control structure for LQR controller has

been proposed.

Having closed an outer impedance loop, the performance of the impedance control

in terms of disturbance attenuation and impedance emulation has been discussed.

Stability and passivity regions for varying values of stiffness and damping within a

specified range and for different inner torque controllers have been analyzed. As aresult, it has been determined that there exists a trade-off between maximizing the

regions of stability/passivity, and maximizing the frequency regions where the desired

impedance can be emulated.

The main contribution of this thesis is a thorough comparison between the im-

plementations of the impedance control strategy on the link side for two controllers,

namely PI and LQR.

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Declaration

No portion of the work referred to in the dissertation has been submitted in support

of an application for another degree or qualification of this or any other university or

other institute of learning.

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Intellectual property statement

i. The author of this dissertation (including any appendices and/or schedules to

this dissertation) owns certain copyright or related rights in it (the Copyright)

and s/he has given The University of Manchester certain rights to use such

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Acknowledgements

I would like to express my gratitude to my supervisor Dr Martin Brown for the useful

comments, remarks and engagement through the learning process of this master thesis.

Furthermore, I would like to thank Dr Gustavo Medrano-Cerda for introducing me

to the topic as well for the support on the way.

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Chapter 1

Introduction

This dissertation focuses on the control problems that arise in the systems interact-

ing with the environment. The control method that improves the interaction abil-ity of robotic systems that operate in unknown or unstructured environments is an

impedance control.

1.1 Motivation

Impedance control is an approach to control the dynamic interaction between a ma-

nipulator and its environment. An example of a robotic system that can gain benefit

from impedance control is a bipedal robot performing a locomotion task in an un-

known terrain. The research in this area has been conducted in a large amount

recently by the robotic community because biped robots can traverse rough terrains

more efficiently compared to wheeled robots. Moreover, there are considerably more

terrains suitable for humanoids rather than for wheeled vehicles, which makes them

suitable for many dangerous and hazardous tasks where human interaction is needed.

Search and rescue operations in unsafe environments (chemical gases, radiation) can

serve as examples. Furthermore, biped robots have been considered to act as assis-

tants to disabled or elderly people. However, the reason the biped robots are not that

common as the wheeled counterparts is the difficulty to control and design them.This thesis will investigate the performance of an impedance control strategy

evaluated on a series elastic actuator, which is vastly used in a robot locomotion

tasks. The impedance control problem is formulated as an impedance outer loop

with a specified range of impedance parameters, defined by stiffness and damping,

and an inner loop which has to be designed to meet the inner closed loop stability

requirements and to assess its effect on an entire closed loop system.

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Impedance control does not explicitly control force or position. Instead, it controls

the relationship between the force on one hand and the velocity on the other. This

relation is called an impedance. It requires a velocity as an input and a force as an

output. The inverse of the impedance is called admittance, and it is a relation from

the force to the velocity. In non-linear systems impedance and admittance may notbe reciprocals of each other, this condition is only guaranteed in a linear case.

In this project, the control architecture where an impedance controller is closed

around an inner torque loop has been used. Figure1.1shows this control architecture,

whereqrefandqlare the link desired (reference) and actual position, respectively. The

output of the impedance controller is fed as a reference torqueTrefto the inner torque

loop. Torque and position sensors feed back the joint torque (Ts) and link position

(ql) measurements. The output of the torque controller (u) is the command sent to

the actuator.

Torque

controller

Actuator

+

Link dynamics

lqrefq refT

sT

Torque loop

Impedance

Controller

Impedance loop

u

Figure 1.1: Block diagram of an impedance loop with an inner torque loop

First, consider the torque loop alone. The conventional way of controlling the

inner loop is to use a simple PI controller. Derivative term is avoided since it is

difficult to implement if the torque sensors have excessive noises. Whenever torque

is controlled by a conventional PI controller, limitations on the torque bandwidth

arise which depend on the dynamics of the load. The cause of this limitation is

the presence of a low frequency zero very close to the origin when the system is

unconstrained to move (or complex zeros very close to the imaginary axis when it

is constrained). In this situation, increasing the controller gain will not lead to animprovement in the tracking performance and may lead to instability. Thus, a more

sophisticated controller must be used to handle this limitation better. The controller

that uses more information about the system model is a state feedback controller,

namely LQR. Acting not only on the tracking error, but also on the states of a

dynamic system, it gives the developer more degrees of freedom to tune and allows

him to bypass the inherent torque bandwidth limitation, thus to improve the overall

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system performance.

There are few problems that arise when an outer impedance loop is closed around

the inner torque loop. Dynamics of the entire controlled system do not explicitly

depend on either impedance parameters or parameters of the inner controller. On

the contrary, there are mutual influences existing between inner and outer loop thatcannot be ignored. Neglecting these influences can lead to results and conclusions that

are incorrect and do not represent the reality. For example, increasing the bandwidth

of the inner loop may lead to a reduction in the range of impedance parameters

where the overall closed loop system will remain stable (stability regions). On the

other hand, increase in bandwidth may enlarge the frequency range where the real

impedance tracks the target one.

Another issue that should be addressed is the stability of the system when there

is a contact with the passive environment. The term passive environment defines

the type of environment that cannot inject energy during interaction. Most of the

standard environments are considered passive; thus, the interaction with passive envi-

ronments will be considered in this thesis. Indeed, a system that is stable can become

unstable whenever interacting with the environment. This property of stability is

called passivity and it is a more strict condition than just stability alone.

The work of this thesis will deal with the design of the inner torque controllers

that can deal with the above-mentioned issues. Moreover, the analysis of the stability

and passivity properties for the entire impedance control system will be performed.

1.2 Aims and objectives

The main goal of this thesis is to investigate the influence of the inner torque controller

on the impedance loop for different outer loop parameters. The key objectives that

should be addressed during the work are:

1. Derivation and investigation of an open loop dynamic model of a single link

series elastic actuator.

2. Design of a suitable torque controller to control the inner torque loop.

3. Analysis of the entire closed loop system after closing outer impedance loop.

4. Investigation of a stability of a controlled system coupled to passive environ-

ment.

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1.3 Contributions

The main contributions of this thesis are as follows:

SEA: model and simulation. The electromechanical model for the single

elastic joint has been formulated and simulated. Nonlinear effect of torque

ripples, produced by the harmonic drive, has been modeled to represent more

realistic case. Its impact on both torque and impedance loop performance has

been investigated.

Torque control: PI and LQR. Two approaches have been proposed to control

an inner torque loop: using a simple PI controller to control a voltage-to-torque

dynamic and LQR controller to control a state space system with torque as

an output. Inherent limitations which can occur in torque control loop has

been analyzed. The effect of PI controller gain and zero on the closed loop

bandwidth and disturbance attenuation has been examined. Furthermore, for

LQR controller, suitable control structure has been formulated. A methodology

to select state cost weighting matrix has been suggested. It has been shown

that LQR controller achieves a superior performance as oppose to PI controller

to regulate inner torque loop because LQR has more than just two degrees of

freedom to tune (gain and zero in a PI controller).

Impedance control: varying stiffness and damping. Impedance control

strategy on the link side has been implemented to control the impedance taking

into account the interaction with the environment that can occur during opera-

tion. Performance of the impedance control in terms of disturbance attenuation

and impedance emulation has been discussed. Stability and passivity regions for

varying values of stiffness and damping within specified range and for different

inner torque controllers have been analyzed.

Coupled stability: spring and masses. Simplest class of passive environ-

ments that interacts with the controlled systems has been determined. Such

passive environments are defined by simply masses and springs, and the analy-

sis has been based on two root loci of a system coupled with a spring or mass.

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Chapter 2

Literature review

This chapter will provide an overview of the most relevant contributions in torque

control and impedance control, which are the main topics of this dissertation.

2.1 Torque control

Robotic systems are often modeled as rigid body systems. Physical causality pre-

scribes that a rigid body inertia has a force as input and motion as an output (Focchi,

2013). Applying this to robots, they have torque as inputs to their link dynamics.

This is the motivation to implement torque control at the joints.

Being able to precisely control the joint torque has many advantages. Torque

control allows to implement various forms of impedance control, control of contact

forces, virtual model control, as well as model based controls, such as rigid body

dynamics based control (inverse dynamics, gravity compensation), and operational

space control (Boaventuraet al.,2012). Moreover, the main disturbances present in a

manipulator operation are all torque types (e.g. friction torque, gravity torque), which

naturally are best dealt within the torque/force domain by implementing low-level

torque control.

Research on robot torque control dates back to the 50s with industrial manipu-

lators. Torque control was initially developed to deal with non-linearity and frictioneffects presented in the actuators of industrial robots (Zinn et al., 2004). However,

stability problems emerged immediately due to the stiff mechanical interfaces between

the actuator and its load. The open loop gain of the actuator with stiff interface is

very high. Even a minor change in position of the load will produce considerable

forces affecting the actuator. Therefore, whenever contacting with the environment,

high gain feedback control would very likely result in destructive performance since

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both the high gain control and the environment impose position (Buchli, 2011).

To overcome this issues the solution was to deliberately introduce springs in series

with the actuator. This is the idea behind series elastic actuators (SEA), first intro-

duced by Pratt and Williamson (Prattet al., 1995). In this case, the environmental

shocks that occur during the interaction between the manipulator and the environ-ment are largely reduced. The following subsection will describe the SEA in more

details.

2.1.1 Series elastic actuator

Series elastic actuator (SEA) is one of the passive compliant actuators used in robotics

manipulators. The basic configuration of a series elastic actuator is shown in Figure

2.1(Prattet al., 1995).

Motor Gearbox Load

Series

Elasticity

Figure 2.1: Block diagram of Series elastic actuator

The introduction of springs helps to reduce the large impact forces and better

handle shocks from the environment because the output impedance is substantially

reduced (Focchi, 2013). As a result, some of the benefits of using the SEA include

greater shock tolerance, lower reflected inertia, and less damage to the environment

(Pratt et al., 1995). Moreover, springs act as an energy storage elements, and can

turn the force control problem into a position control problem, greatly improving force

accuracy. In SEA, the load force can be measured through the deflection of the spring

without necessity to equip the manipulator with torque sensors. In particular, the

output force is equal to the position difference across the series elasticity multiplied

by its spring constant.One of the drawbacks of SEA is that the amount of passive stiffness is fixed and

cannot be adjusted on the fly if the task requirements have changed. A possible

solution is to replace it with an active compliance. This type of compliance can be

tuned on-the-go to fit the task requirements based on the output information from

force or related sensors. However, the implementation of the active compliance is

computationally costly, and thus it may be unsuitable for many applications.

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The introduction of springs also limits the overall system bandwidth. The dynamic

characteristics of the robot with elastic actuators significantly limit the closed-loop

torque performance, making the closed-loop system unstable for high gains (Eppinger

& Seering,1987). Another limiting factor to the torque bandwidth when the torque

is transmitted through the spring is an intrinsic load velocity feedback that exists inthe torque dynamics and does not depend on the actuator. In this situation, there is

a zero in the torque transfer function that depends on load dynamics (damping and

inertia), and that can limit the achievable closed loop bandwidth (Focchi,2013). The

model based control strategy that overcomes the effect of this zero is called a velocity

compensation. This compensation deals with the load velocity feedback existing in

the load dynamics and performs a pole-zero cancellation, which makes it possible to

increase the controller gains, thus to achieve the desired torque bandwidth. Focchi

has designed and implemented this kind of compensation on hip abduction/adduction

(HAA) joint of the hydraulic quadruped HyQ robot and thoroughly analyzed its effect

on overall system performance in (Focchi, 2013). However, the concept of velocity

compensation can be generalized to any mechanical system where force/torque is

under control, and a compliant element is present, regardless of the type of actuator

in use (e.g. hydraulic/electric/pneumatic) (Boaventura et al.,2012).

Nevertheless, simple control strategies, such as PD control, are still used in the

most industrial robots. The bandwidth of the controllers has to be reduced until

robustness against highly non-linear dynamics is reached. If the controller can use

more extensive information about the model of the system, then the performance ofthe controlled system can be enhanced. Thus, a control structure in the form of a

joint state feedback controller was proposed in (Albu-Schaeffer & Hirzinger, 2001)

to control the DLR lightweight robot. Since it used both motor and link states, its

performance was proven to be superior to that of the simple PD controller.

2.2 Impedance control

Controlling the interaction of a robot manipulator with the environment is crucialfor accomplishing a variety of tasks in industrial applications, such as mechanical

part mating, object contour surface tracking, employment of tools for machining

mechanical part (Chiaverini et al., 1999), as well as in biped robot locomotion tasks,

such as walking, running, and climbing ladders. If the environment is structured

and well-known, high gain position control is usually used to maximize the tracking

accuracy. This high gain position control works well in industrial robots with low

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locomotion speed. Additionally, when it comes to mechanical interface, "the stiffer,

the better" is the traditional rule of good design for industrial manipulators (Pratt

et al., 1995).

Conversely, high gain control becomes unsuitable for dynamic locomotion or for

situations where contact with unstructured environment is involved. Whenever therobot operates in an unstructured environment, where the knowledge of it is imprecise

or when the motion is constrained (e.g. by the ground in the case of a legged robot),

the necessity to be compliant to these external impacts becomes a crucial requirement.

In this case, some interaction control is required.

Two methods of regulating the interaction forces using compliance were proposed:

passive and active. Passive method consists in the introduction of physical compliant

elements, such as spring and damping, between the motor and the link. On the other

hand, active method is based on virtual emulation of compliance using feedback sensormeasurements of joint forces. A major benefit of actively controlling the compliance

is its flexibility to change its dynamic characteristic (e.g. stiffness and damping) on-

the-fly (Focchi, 2013). This versatility is a great advantage versus an introduction of

passive compliant elements in the structure that have only a fixed value of stiffness

and damping. However, due to physical presence of the passive compliant element,

it can store and release energy, thus increasing the energy efficiency of the system

and reducing the work made by actuators. Nevertheless, the existing drawback is

additional dynamics introduced by springs that may induce unwanted oscillations.

However, compliant manipulator is still an optimal choice for the tasks involving

environment interactions.

There are many ways of controlling the interaction forces or the compliance:

impedance control (Hogan,1985), hybrid force control (Raibert & Craig, 1981), and

others. The distinction is in the approach to control the force. Direct force control

method is the one where the controller directly regulates the contact force to a desired

reference. Hybrid force control and other classical force control methods belong to this

category. Alternatively, indirect control method is the one where the force/torque is

controlled via motion control. Impedance control falls in this category. In particular,impedance control tries to regulate simultaneously both the mechanical impedance,

which is the dynamic relation between the torque and the velocity of the manipulator,

and the position of the manipulator. This dynamic characteristic can be defined at

different locations. In the case of SEA, the active impedance can be generated either

on the motor side or the link side. However, many aspects still create stability issues

on impedance control. For instance, the range of stable stiffness and damping values

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can be limited by actuator dynamics and inner loops bandwidth.

2.2.1 Impedance control architectures

Impedance control problem can be defined as an outer impedance controller com-

manding a reference to the inner loop. In practice, the inner control loop can be

a position loop (position based impedance control), a velocity loop (velocity based

impedance control), or a torque loop (torque based impedance control) (Mosadeghzad

et al.,2012).

Nowadays, torque sensing is available in many robots and IC based on torque

inner loop (ICT) has become widely adopted (Eppinger & Seering,1987;Prattet al.,

1995). One of the advantages of implementing ICT scheme is that the outer impedance

loop directly generates a reference command to be tracked by the inner torque loop.However, performance-wise the achievable torque bandwidth is limited when using a

simple PI controller. This problem arises when the link has a small value of viscous

damping.

One alternative for ICT scheme is to design inner velocity loop (ICV), but still

implementing torque feedback. In this way, the robustness and tolerance to non-linear

effects of friction can be increased. In this instance, the drive is rather the velocity

source than the torque one. The ICV scheme can be implemented in terms of either

joints or motors position. However, the significant drawback of this scheme is a

necessity to employ the torque derivative feedback. In practice, most of the torque

sensor signals are noisy, and calculations of the torque derivatives are not a trivial

problem. More details about the ICV scheme can be found in (Mosadeghzadet al.,

2012).

Another possibility to implement the IC is to use inner position loop (ICP). Using

the ICP scheme as the basis of the impedance control is quite common because most

of the current robots are equipped with the position control loops by default. Similar

to the ICV scheme, the ICP can be employed in terms of motor or link positions.

One drawback that arises implementing this type of the IC scheme is a requirementfor the link (or motor) position to be multiplied by some gain in the feedback path,

which depends on where the ICP scheme is implemented (link or motor side). The

errors in this gain value can cause high steady state errors in the reference response

of the impedance controller (Mosadeghzadet al.,2012).

The summary of the discussed impedance control schemes is presented in Table

2.1.

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Table 2.1: Comparison of various impedance control schemes

Scheme Advantages Disadvantages

ICT The torque generated by outer loop

can be used directly as a reference

Closed loop bandwidth is limited

using PI controller

ICV Increases robustness and decreases

non-linear effect of friction

Requires torque derivative to

implement

ICP Robotic devices are commonly

equipped by position control

systems

Sensible to errors in the gain value

2.2.2 Passivity

Another important specification which is necessary for a manipulator interacting with

the environment is called a coupled stability. In fact, whenever the manipulator is

in contact with the environment, the dynamics of the whole system depend both

on the dynamics of the manipulator and the environment. Since the environment is

considered passive in most cases, the practical solution is to design a controller such

that the manipulators system behaves passively at the interaction port (Colgate,

1986). In this case, the stable interaction with any passive environment is ensured.

Passivity property was defined and thoroughly investigated for mechanical systems by

(Colgate,1986). Later, Colgate and Hogan (Colgate & Hogan,1988) have derived the

necessary and sufficient conditions for stability in contact with passive objects based

on a Nyquist stability analysis. Similar to nominal stability, passivity is also influenced

by inner loop bandwidth. Comparison of how passivity property is preserved when

the impedance control loop is implemented with various nested inner loops (namely

velocity, position, and torque) can be found in the work done by (Mosadeghzadet al.,

2012).

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Chapter 3

Series elastic joint dynamics

In this chapter, the SEAs system description and modeling will be provided. An

open loop system will be simulated and analyzed. The torque transfer function willbe investigated in terms of poles and zeros. It will be shown that the open loop zeros

impose a limitation on the torque bandwidth.

3.1 SEA system description

The model of SEA, which is thoroughly used in this dissertation, is based on the

experimental setup utilized in (Mosadeghzadet al., 2012) and (Mosadeghzad et al.,

2013). This setup is shown in Figure 3.1. It consists of a brushless DC motor, aharmonic drive gearbox, a cable drive, and a load link. The system is equipped with

position and torque sensors.

Figure 3.1: SEA experimental setup. 1-motor pulley, 2-harmonic drive, 3-motorposition sensor, 4-DC motor, 5-cable drive, 6-link position sensor, 7-output pulley,8-rigid strain gauge torque sensor, 9-output link

23


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3.2 SEA system modeling

Generally, a single SEA consists of an electrical motor, a reduction gear, a passive

elastic element with a parallel intrinsic damping and an output link or load. The

schematic of the model is depicted in Figure3.2.

Motor Link

N

hD

hK

sT

mBl

B gK

mq

mT

lq

lT

Figure 3.2: Schematic structure of SEA

Using Euler-Lagrange equation, the dynamics of the link is given by:

Jlql+ Blql= Tl+ Td Tg (3.1)

where ql, qlare the link angular velocity and acceleration,JlandBlare the link inertia

and the link viscous friction,Tlis the load disturbance torque,Tdis the driving torque

and Tg is the gravity torque.

The torqueTd is produced by spring and damping, and it can be expressed by thedifference between motors and links positions and velocities:

Td= Kh

qmN ql

+ Dh

qmN ql

= Ts+ Dh

qmN ql

(3.2)

where qm, ql, qm and ql are the angular positions and velocities of motor and link,

Kh and Dh are the stiffness and the damping of the model, and N is the gear ratio

to increase the torque from the motor. The electrical dynamics of the motor can be

described by the following equation:

LIm+ RIm= Vm Kbqm (3.3)

whereLandR are the motor coil inductance and resistance, Imand Vmare the motor

current and voltage, and Kb is the back-emf motor constant. The torque inputTm,

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which is generated by the motor, is given by:

Tm= KtIm (3.4)

whereKt is a torque-current constant. Thus, the dynamic equation from the motor

side is given by:

Jmqm+ Bmqm= Tm TdN

(3.5)

whereJm is the total inertia of the electric rotor and the harmonic drive and Bm is

the viscous friction of the rotor and the harmonic drive.

As is known, a gravity torqueTgacting on the link can be expressed as a non-linearposition dependent torque:

Tg =MgHcos(ql) (3.6)

where g is the gravitational acceleration (9.81 m/s2). It is possible to model this

torque as a torsional spring Kg . Therefore, the linearized gravity torque becomes:

Tg =Kgql (3.7)

where computed value ofKg is 9.0341 Nm/rad, and is obtained after the linearization

around the equilibrium pointql=

2.

Rearranging the equations (3.1)-(3.7), the final set of equations describing the

SEA dynamics is the following (Mosadeghzadet al.,2012):

LIm+ RIm= Vm Kbqm

Jmqm+ Bmqm= KtIm Kh

N

qm

N ql

Dh

N

qm

N ql

1

N(Tfric+ Trip)

Jlql+ Blql= Kh

qm

N ql

+ Dh

qm

N ql

+ Tl Kgql

(3.8)

whereTfric and Trip are the meshing friction torque and the torque ripple produced

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by the harmonic drive, which have been added to show the non-linear effects existing

in the model. The system is driven in open loop by a voltage input signal. The

parameters for the model are given in Table 3.1.

Table 3.1: Electrical and mechanical parameters for the SEA model

L [H] Motor inductance 3.2 104

R [] Motor resistance 0.664

Kb[Nms/rad] Motor constant 4.10 102

Kt[Nm/A] Torque constant 4.10 102

Jm[kgm2] Rotor+harmonic drive inertia 1.3870 105

Bm[Nms/rad] Motor viscous friction constant 1.9958 105

Jl[kgm2] Link inertia 0.2518

M[kg] Link mass 3.60455

H[m] Link centre of mass 0.25548497

Bl[Nms/rad] Link viscous friction constant 2.0 103

Kh[Nm/rad] Passive spring stiffness 912

Dh[Nms/rad] Passive spring damping 1.1856

N Gear box ratio 150

It is worth mentioning that sometimes the actuators dynamic (the electric motor)

is neglected and is thus excluded from the dynamic model of SEA for the sake of

simplicity. In those cases, the motor is considered to be the ideal velocity source.

However, this simplification is justified only when those dynamics are very fast and

are constant up to certain frequency, which must be greater than operating one. In

this case, the electrical dynamics, the transfer function between motor current and

motor velocity is:

qm(s)

Im(s)=

Kt

Jms+ Bm(3.9)

and has a pole (s= BmJm

=1.44 rad/s) at low frequency. Due to small open loop

bandwidth the electric motor cannot be considered as an ideal velocity source, and

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thus its dynamic must be included in the model of the system. However, this is not

the limitation to the bandwidth of the closed loop system. The real limitation is

the voltage saturation of the actuator. Therefore, hereafter all the simulations will

be based on the situation where the actuator has a voltage limit of[30 V; 30 V] to

represent more realistic case.Using equations (3.8), one can express the model in the block diagram form, which

is shown in Figure3.3.

RLs

Kt

+ mm BsJ +

1

N

1

N

1

NK

D

h

h

hK s1

h

h

K

D

ll BsJ +

1

bK

mV mq& sT lq&

ripfric TT +

gK s1

lT

Figure 3.3: Block diagram of the SEA model

One drawback of using a harmonic drive gearbox is that it introduces torque rip-

ples that vary for different drives, assemblies, speeds, and loads. This torque has

detrimental effect on torque control and creates vibrations and wearings of the com-

ponents. However, a special characteristic of harmonic drives is that the dominating

component of torque ripple is periodic in nature and has fundamental component

with frequency twice the wave generator cycle (Mosadeghzadet al.,2012). Therefore,

in this case, the dominating fundamental torque ripple component has been modeled

as:

Trip = 2 cos(2qm) (3.10)

This term is added along with a motor torque in the equation (3.8), and it enters thesystem with the peak amplitude of

2

N0.013 N m.

3.2.1 State space representation

By manipulating equations (3.8), a continuous time linear system can be obtained in

state space form with measured joint torque as one of the states:

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x=

0; 1; 0; 0; 0

-Kh

JmN2 -

Kh

Jl; -

Bm

Jm-

Dh

JmN2 -

Dh

Jl;

KhKg

Jl;

KhBl

Jl-

KhBm

Jm;

KhKt

JmN

0; 0; 0; 1; 01

Jl;

Dh

JlKh; -

Kg

Jl; -

Bl

Jl; 0

0; -KbN

KhL; 0; -

KbN

L ; -

R

L

x +

0

0

0

0

1

L

u

y=

1 0 0 0 0

x

where state vector x, controlu, and outputy are as follows:

x=

Ts

Ts

ql

ql

Im

, u= Vm, y= Ts (3.11)

This representation of the system will be useful in designing a state feedback

controller in the subsequent chapter.

3.3 Open loop simulations

In SEA, the joint torque can be measured by the deflection of the elastic element:

Ts= Kh

qm

N ql

(3.12)

Therefore, from Figure3.3,the transfer function from the input voltageVmto the

measured torque Ts is (Tfric+ Trip = 0 and Tl= 0):

G(s) = 5.616 107s2 + 4.461 105s + 2.015 109

s5 + 2085s4 + 4.06 105s3 + 1.546 107s2 + 1.397 109s + 2.176 108 (3.13)

It has been computed using MATLAB code provided in Appendix B.The simu-

lations have been carried out using "step" command. Its open loop response to the

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input step change in the voltage (0-1 V) is shown in Figure3.4. Hereafter, the ini-

tial position of the link is set to zero. The poles of (3.13) are: 1872.3, 191.9,

10.3j61.4, and0.16. The pair of complex poles introduce oscillations with time

period of 2/61.4 0.1 s that settle down after 4/10.30.4 s. The first four poles can

be considered as "fast" poles because their effect is insignificant after 0.4 s. The lastmentioned pole, s = 0.16, is the dominant one, which causes the settling time to

be 4/0.16 25 s. This pole only appears when the gravity is taken into consideration.

The effect of this pole is seen better on the link and motor positions and velocities

responses because motors and links position and velocity transfer functions share

the same set of poles with torque transfer function. It can be seen that within 0.4 s

(before the dominant gravity pole) the position acts as an integrator of the velocity.

Since it was not the main object of interest, these transfer functions along with their

corresponding open loop step responses can be found in Appendix A.

0 10 20 30 402

0

2

4

6

8

10Joint torque

Ts

[Nm]

Time [sec]

0 0.52

0

2

Figure 3.4: Open loop response of joint torque Ts

The zeros of transfer function (3.13) will impose limitations on the torque closed

loop bandwidth. The next section will be devoted to investigation of this limitation,

which arises when the torque is controlled by a conventional PI controller.

3.4 Bandwidth limitations of the torque controlled

by PI controller

Consider the numerator of the transfer function (3.13). The roots of the numerator

ares = 0.0040j5.9898, which are low frequency zeros very close to imaginary axis

in continuous time. A simple PI controller is formulated as:

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C(s) =kp+ki

s =kp

s+ki

kps

(3.14)

wherekp andki are proportional and integral gains, respectively. The pole at origin,

introduced by the controller, will become the dominant pole of the closed loop system.

If the controller gain is increased, it will move towards one of the open loop zeros.

Increasing the controller gain infinitely does not increase the bandwidth of the closed

loop system, since the dominant pole of the controller is attracted towards one of

these complex zeros. The Figure3.5 (right) shows the root locus of the loop transfer

functionC(s)G(s)around the origin, where this phenomenon is clearly seen.

0,01 0,005 010

5

0

5

10

Root locus

Real axis

Im

aginaryaxis

4 2 010

5

0

5

10

Root locus

Real axis

Im

aginaryaxis

Figure 3.5: Root locus ofC(s)G(s)around the origin computed for the model withgravity (right) and without gravity (left). The open loop poles and zeros are red,whereas the controller pole and zero are black

Figure3.5(left) shows the root locus ofC(s)G(s), but for the system modeled

with no gravity included. In this case, there is a single open loop zero, s = 0.0079,

instead of two complex ones. Here, it is evident that the dominant pole ofC(s) at

the origin cannot go beyond the frequency of the open loop zero. Therefore, the same

problem arises, which is the limitation of the closed loop bandwidth by the frequency

of this zero.

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3.5 Summary

In this chapter, the derivation of the SEA model dynamics has been presented. It has

been shown that the electrical (the motor) dynamics cannot be considered as an ideal

velocity source and thus must be incorporated in the model due to the small open

loop bandwidth. The open loop simulations have been performed to get some insight

of the dynamics of the system. Finally, the origin of the torque bandwidth limitation

that is present when controlling the joint torque with a simple PI controller has been

explained.

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Chapter 4

Torque controller design

In the previous chapter, the model of SEA has been designed. In this chapter, the

formulation of the torque controller for SEA will be presented. This torque controlloop will be included in a nested loop architecture with an outer impedance loop as

described in details in Chapter 5. Two types of controllers, a simple PI and a Linear

Quadratic Regulator (LQR), will be proposed to regulate the joint torque measured by

the spring deflection. The performance will be evaluated via step response character-

istics and disturbance attenuation. Simulation results have proved that the designed

LQR controller has surpassed the performance of the PI controller.

4.1 PI controller designPID controllers are widely used in robotics. They are reasonably easy to tune and

robust in the presence of parameter uncertainty. Traditionally, in nested control archi-

tectures, the controller design is carried out with the aim to maximize the bandwidth

of the inner loop (torque loop in our case). However, as it will be shown further, this

approach is not always consistent with outer loop specifications (impedance parame-

ters in this case).

A block diagram of inner torque loop with a PID controller is shown in Figure

4.1. From figure, torque loop is subjected to a torque ripple disturbance Trip, whichhas been defined earlier in the previous chapter. Also, only proportional and integral

terms (PI) have been included in a controller structure. The reason why a derivative

term has not been included is that the sensors used in torque measurement provide

very noisy signals. The derivatives of these noisy signals are hard to compute and may

cause many problems. Hence, a continuous time implementation of the PI controller

has the following form:

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C(s) =kp+ki

s =kp

s+ki

kps

(4.1)

where proportional gainkp and integral gain ki are to be tuned to achieve loop gain

margin larger than6 dB and phase margin larger than30. Letz= kikp

andk = kp,

then the controller (4.1) can be expressed as the transfer function with gain k, zeroz

and pole at origin.

Link

dynamics

mV sTrefT + e

pk

iks

1

+

+

Electrical

dynamics

+

+

ripT

Figure 4.1: Inner torque loop with PI controller

The controller design procedure is as follows. First, nominal controller gain k

and zero zhave been found that meet the stability margin requirements. Next, gain

and zero of the controller has been varied taking into account the above-mentioned

margins demands. A set of controller parameters satisfying the given specificationswith corresponding response characteristics are shown in Table4.1:

Table 4.1: PI controllers with corresponding closed loop characteristics

# k z Bandwidth (3 dB)

[rad/s]

Rise time

[s]

Settling time

[s]

Overshoot

[%]

1 2.7 -7 4.89 0.0049 5.43 37.74

2 2.7 -3 4.22 0.0050 3.04 32.46

3 0.03 -3 0.53 3.57 38.28 40.99

4 1 -3 2.97 0.64 5.27 38.31

The closed loop step responses for proposed controllers are shown in Figure 4.2.

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It can be seen that reducing the gaink leads to increase in the rise time and settling

time. Also, moving the zero ztowards the origin reduces the amount of oscillations.

However, moving the zero too close to the origin or increasing the gain too much

deteriorates the stability margins. From the Table 4.1 and Figure 4.2, controller

#2 (purple) has the best response characteristics in terms of overshoot and settlingtime. The in-depth examination of this response in terms of poles and zeros will be

provided in Section4.3. Although the main interest is to control joint torque, other

intermediate signals, such as link position (see Figure 4.2right), are also important.

It can be seen that link position response is also oscillatory. Indeed, from (3.12) it

is evident that the joint torque depends on the position of the link, thus these two

responses will correlate with each other.

0 2 4 6

0

0.5

1

1.5

Time [s]

Ts

[N

m]

Torque step response

k=2.7; z=7

k=2.7; =3

k=0.03; z=3

k=1; z=3

0 2 4 6

0

0.05

0.1

0.15

0.2

Time [s]

ql

[rad]

Link position

Figure 4.2: Closed loop torque response (left) and link position response (right) tostep change in torque reference

The bode magnitude diagram for transfer function between the torque ripples Trip

and the joint torque Ts for different controllers is shown in Figure 4.3. As can be

seen, controllers #1 and #2 (blue and purple) provide the best ripple attenuation atlow and mid frequencies comparing to other controllers. The pronounced minimum

in every controllers magnitude, the anti-resonance at 6 rad/s, is due to the pair of

transmission zeros in the transfer function from torque ripple to the joint torque. At

this frequency, the ripples have almost zero effect on measured torque because the

joint torque produced by them is canceled by the torque produced by the motion of

the link.

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102

100

102

104

90

80

70

60

50

40

30

20

10

0

10

Frequency [rad/s]

Magnitude[dB]

k=2.7; z=7

k=2.7; z=3

k=0.03; z=3

k=1; z=3

Figure 4.3: Torque ripple frequency response (0.013 Nm)

4.2 LQR controller design

As has been mentioned earlier in Subsection 2.1.1, if the controller can use more

extensive information about the model of the system, then the performance of the

controlled system can be enhanced. The implementation of a state feedback controller

to control a robotic system, described in (Albu-Schaeffer & Hirzinger,2001), showed

a superior performance to that of the PD controller. However, the electrical dynamics

were not included in the model. In this work, the model of the system includes the

dynamics of the motor. Moreover, a methodology to select the state gain vector will

be provided with the aid of LQR control.

Linear Quadratic Regulator (LQR) is one of the milestones in control theory,

providing a possibility to find a solution for optimal control problem. In this case, for

the continuous time linear system described by (3.11) with a quadratic cost function

defined as:

J=

0

xTQx+ uTRu

dt (4.2)

the feedback control law that minimizes the value of the cost is given by:

u= K1Ts K2 Ts K3ql K4ql K5Im, where K=R1BTP (4.3)

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and P is found by solving the continuous time algebraic Riccati equation. To make

LQR track the reference signal and better handle disturbances, the systems state

equations have been modified by adding additional state for the integral of the error

(Ts Tref)dt. Moreover, the control law has been modified in the way to make

the gain K1 act on the error rather than on the torque Ts alone. This modificationintroduces a zero in a loop transfer function that will become useful when the outer

position loop is closed. Thus, expanding equation (4.3) and setting a tracking error

to be e = Ts Trefwill give the following equation for control law:

u= K1e K2Ts K3ql K4ql K5Im K6

e dt (4.4)

The reasons why derivative of the error e = Ts Tref has not been included

in the control law (4.4) are the same as in PID case. The rate of change of Ts is

available from the motor and link velocities measurements, whereas the derivative of

a reference signal, Tref, is very hard to compute and requires an approximation using

low-pass filters, which complicates already complex high-order system. Moreover,

if the gains K2, K3, K4 and K5 are set to zero, the resulted controller will be the

exact PI controller. Hence, it supports the idea that the proposed LQR controller

can achieve as good performance as the PI controller, or even superior one because it

has more degrees of freedom to tune. The block diagram of the proposed controlled

system is shown in Figure4.4.

refT

+

e1K

6Ks

1

5:2K

Link

dynamics

Electrical

dynamics

+

+

ripT

sT

Figure 4.4: Inner torque loop with LQR controller

The idea behind LQR control tuning is to find weighting matricesQ 0andR >0

and to compare the results with the specified design goals. Since the system has one

inputVm, the easiest way to tune LQR is setting R = 1 and changing only Q matrix

entries. The common practice is to set Q=diag [q1; q2; q2; q3; q4; q5; q6], where each

entry is the weighting factor for corresponding state. The proposed weighting with

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their effects on closed loop characteristics are provided in Table4.2.

Table 4.2: LQR controllers with corresponding closed loop characteristics

# q1 q4 q6 Bandwidth (3 dB)[rad/s]

Rise time[s]

Settling time[s]

Overshoot%

1 10 0 100 3.62 0.0097 1.25 8.77

2 100 0 100 4.85 0.0044 2.49 14.41

3 100 100 100 4.07 0.0044 1.56 11.81

4 500 0 500 5.46 0.0029 3.98 11.09

5 500 500 500 4.28 0.0029 1.56 13.74

The choice of these particular weightings is not an arbitrary one. Step response

performance characteristics and disturbance attenuation have been the tuning criteria.

However, more emphasis has been placed on friction (ripple) disturbance attenuation,

because the step performance of a system after closing outer loop is of much greater

importance. It has been found that increasing the weighting on tracking error (q1) sig-

nificantly improves the friction disturbance attenuation (see Figure4.6). Conversely,

increasing the weighting on Ts(q2) deteriorates disturbance rejection, but reduces the

overshoot. An increment in link position weighting (q3) worsens both settling time

and disturbance attenuation. On the other hand, the weighting on link velocity (q4)

noticeably improves settling time and reduces oscillations both of joint torque and

link position. The weighting on motor current (q5) increases the rise and settling

times. Finally, the weighting on the integral of the error (q6) increases the speed

of response in general. Having combined these observations, it has been decided to

operate onlyq1, q4and q6weightings, since they have more effects on the performance

of the system than the other ones. The closed loop step responses for proposed con-

trollers are shown in Figure4.5. The torque ripple disturbance attenuation frequencyresponses are shown in Figure4.6.

From Figures 4.5 and 4.6, it is seen that increasing q1 from 10 to 100 leads to

decreasing the magnitude of ripples from 22 dB to32 dB (blue and purple lines).

Also, adding weighting q4 helps to reduce the oscillations both in torque and link

position responses, yet not affecting much the ripple attenuation (red line). Controller

#4 (black) achieves peak ripple response of around 37 dB, which is the best result

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0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [s]

Ts

[Nm

]


q1=10; q

6=100

q1=100; q

6=100

q1=100; q

4=100; q

6=100

q1=500; q

6=500

q1=500; q

4=500; q

6=500

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

0.2

Time [s]

ql

[rad

]

Link position

Figure 4.5: Closed loop torque responses (left) and link position response (right) tostep change in torque reference

among other proposed controllers. Addingq4 weighting again reduces oscillations on

closed loop responses, yet keeps the similar friction attenuation.

102

100

102

104

100

90

80

70

60

50

40

30

20

Frequency [rad/s]

Magnit

ude[dB]

q1=10; q

6=100

q1=100; q

6=100

q1=100; q

4=100; q

6=100

q1=500; q

6=500

q1=500; q

4=500; q

6=500

Figure 4.6: Torque ripple frequency response (0.013N m)

4.3 Comparison between PI and LQR controllers

In this section, the comparison between PI and LQR controllers will be provided in

terms of step response characteristics, stability margins, and torque ripple attenua-

tion. The selected controllers have achieved suitable design requirements and will be

used only to show the difference in results that has been obtained using two different

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control design strategies. Thus, they are not the "best" controllers. The selected PI

controller has k = 2.7 and z = 3 (controller #2 in Table 4.1). The selected LQR

controller is defined by Q = diag [500; 0; 0; 500; 0; 500] and R = 1 (controller #5 in

Table4.2).

4.3.1 Step response characteristics

The response of two proposed controllers are shown in Figure 4.7. It is clearly seen

the response provided by the system controlled by LQR controller is much better than

by PI controller in terms of settling time, overshoot and rise time. The corresponding

poles of the closed loop system controlled by PI controller are: 1918.5, 77.6

j281, 8.7, 1.3 j3.1. There are two pairs of complex poles. First pair introduce

oscillations with a period of 2/281 0.02 s that settle down after 4/77.6 0.05 s. The

seconds pair are the dominant poles and the oscillations induced by them are with

the time period of 2/3.1 2.03 s that vanish after 4/1.3 3.08 s. These poles are the

ones that move towards open loop zeros with increasing controllers gain, thus limit

the torque bandwidth (see Section3.4).

The state feedback gain generated by LQR controller is Ki = [22; 0; 23; 17; 0; 22]

(values have been rounded towards nearest integer) that places the poles of closed

loop system at: 1904.3,553.9j593.7,2j5.6,1. Dominant pair of complex

poles induces noticeable oscillations with time period 2

/5.6

1.12 s that settle after4/2 2 s.

0 1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [s]

Ts

[N

m]


0 0.05 0.10

0.5

1

1.5

PI

LQR

Figure 4.7: Step response of torque controlled by PI and LQR controllers

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4.3.2 Stability margins

In theory, LQR must provide infinite gain margin and phase margin of at least 60.

In our case, LQR acts as a controller rather than as a regulator, since the integrator

has been added. Therefore, its stability margins must be checked and compared to

PI controller margins. The gain and phase margins for system controlled by the PI

controller are 14.6 dB and 32.5, respectively. As for the system controlled by the

LQR controller, its gain and phase margins are 16.6 dB and 63.3, respectively.

Indeed, the LQR controller provides the phase margin above60, which is twice as

great as in PI case. However, the gain margin of the LQR controlled system is 16dB,

but it is still more than the gain margin of PI controlled system. Therefore, proposed

LQR controller provides more robustness compared to a simple PI controller.

4.3.3 Torque ripples attenuation

The torque ripple frequency responses for two controllers are shown in Figure 4.8. It

is evident that LQR controller significantly improves ripples attenuation of the closed

loop system at low and mid frequencies by almost 20 dB.

102

100

102

104

100

90

80

70

60

50

40

30

20

10

Frequency [rad/s]

Magnitude

[dB]

PI

LQR

Figure 4.8: Torque ripple attenuation with PI and LQR controllers

To show the oscillatory behavior of the torque ripples, their effects have beendemonstrated using torque step responses. The joint torque responses of a system

affected by the torque ripples are shown in Figure 4.9.

The second order systems response has been used as a reference to reduce initial

spikes and to emphasize on ripples effect. It is no surprise that LQR provides superior

torque ripples attenuation compared to PI controller, due to better bode magnitude

diagram (see Figure4.8).

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0 1 2 3 4 5 60.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [s]

Ts[

N

m]

Torque response

Ref

Tsno ripples

Tswith ripples

0 1 2 3 4 5 60.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [s]

Ts[

N

m]

Torque response

Ref

Tsno ripples

Tswith ripples

Figure 4.9: Torque ripples effect on torque response: PI case (left) and LQR case(right)

4.4 Summary

In this chapter, two different controllers for the inner torque loop have been designed,

namely PI and LQR. In particular, the effect of the gain and the zero of a PI con-

troller on performance of closed loop system has been demonstrated via simulations.

Then, to surpass the performance of a PI controller, the LQR control strategy has

been introduced. The influence of a state weighting matrix on the closed loop sys-

tem performance has been analyzed. Via simulations, it has been proven that the

LQR controller shows better performance in terms of stability margins, step responsecharacteristics and disturbance attenuation compared to a conventional PI controller.

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Chapter 5

Impedance control

In the previous chapter, the implementation of a torque loop has been presented

to improve it in terms of bandwidth and disturbance rejection. In particular, it hasbeen shown that bandwidth and disturbance rejection can be increased by using LQR

controller instead of conventional PI controller. This is true for the torque loop alone;

however, when an outer impedance loop is closed, the dynamic of the controlled

system changes, thus this statement must be checked in order to select appropriate

controller. The results in this chapter show that the LQR controller provides larger

range of impedance parameters where the entire closed loop system remains stable

and passive.

5.1 Approach and implementation

To have desirable impedance on the link side, a spring-mass-damper system is being

modeled. Spring and mass are both energy storing elements, and damper is an energy

dissipating element. In other words, being able to control the impedance makes it

possible to control the energy exchange during interaction. Therefore, impedance

control is considered to be the interaction control (Buchli,2011). In this project, only

spring-damper system is being considered, since the mass (inertia) gain in impedance

controller requires the measurement of acceleration of the link, which in reality is veryhard to obtain and demands expensive sensors.

Now, considering the torque loop implemented in Chapter 4, an impedance con-

troller is added as an outer loop as shown in Figure5.1. The impedance gains Kand

Drepresent the desired stiffness and damping for the link joint. These gain units will

be expressed in SI units,Nm/radfor KandNms/radfor D, and henceforth will be

omitted for the sake of brevity. The output of the impedance controller provides the

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reference torqueTreffor the inner loop:

Tref=K(qref ql) Dql+ Tc (5.1)

where Tc is an external compensation torque than can be added to remove the

effect of gravity.

Controller

Actuator

+

Load dynamics

K

sD

lq

lq

refq refT cT

sT

+

fricTlTTorque loop

Figure 5.1: Block diagram of the system with inner torque loop (blue) and outerimpedance loop (green)

For our work, the range of values for impedance loop gains that is considered to be

sufficient isK(0 2000]and D (0 30] (Mosadeghzadet al.,2013). This range was

determined for the ankle joint of the humanoid robot CoMan to adjust the system

performance throughout the locomotion cycle.

5.2 Control problem formulation

Let us express the block diagram5.1in the block diagram form where each component

is represented via transfer function block. This block diagram is shown in Figure5.2.

Here,G1 is the voltage-to-torque transfer function, G2 is the torque-to-link position

transfer function, C1 is the inner torque controller, C2 is the impedance controller

parameters, and Gd is the transfer function from load disturbance to joint torque.

The design goal is to determine the controller C1 that will ensure the closed loop

stability and stability when coupled to passive environment given the fixed impedanceparametersC2. This is a non-trivial problem, and there may not be a solution even

if the coupled stability condition (passivity) is removed. If there is no solution, the

largest range of impedance parameters C2 and inner controller C1, which satisfy the

condition of stability and passivity, must be determined. However, there are no

analytical tools available for solving this problem. The numerical calculations have to

be carried out for each specific set of impedance parameters. An approach to solving

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C2

G1

C1

G2

GdlT

sT lqrefq

Figure 5.2: Cascaded control system block diagram for ICT

the control problem, proposed in (Mosadeghzadet al., 2013), can be summarized in

the following steps:

1. Design the inner torque controllerC1 that satisfies the inner closed loop spec-

ifications. In our case, these are in terms of inner closed loop bandwidth and

stability margins.

2. Adjust the controllerC1 to achieve overall closed loop stability for the largest

range of impedance parametersC2.

3. Repeat step 2, but now considering passivity.

Step 1 has been performed in Chapter 4, where PI and LQR controllers have been

designed considering the above-mentioned specifications. In the next section, step 2will be carried out to find the stability regions for the system with different inner

torque loop controllers. The notion of passivity will be introduced in Section5.4 and

step 3 will be presented.

5.3 Stability regions

In this section, the closed loop stability region will be determined for a given range of

impedance parametersKandD by varying the controllers parameters. The stabilityof the overall system is determined by computing the closed loop eigenvalues and

checking that they are less than zero (stability boundary in continuous time). First,

this condition will be checked for the system controlled by a PI controller, and then

for the system controlled by the LQR controller. In this section, the stability regions

will be plotted for the following values of impedance parameters: K (0 7000] and

D (0 40]. The reason for this is an interest in stability boundary, which can be seen

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only if plotted for the given large range. The distance to stability boundary can be

treated as an indicator of robustness of the system.

5.3.1 Stability regions with inner PI controller

The results are presented in Figure 5.3, where the red area corresponds to stable

region, whereas blue area is the unstable region. It has been determined that the

zero of a PI controller does not significantly affect the stability regions because it

does not noticeably affect the bandwidth of the inner loop. Therefore, the analysis is

carried out for a PI controller varying only its gain k. Increasing the gain k leads to

increasing the bandwidth of the inner torque loop (see Table 4.1). Figure5.3shows

that, as controller gain triples in value from 0.9 to 2.7, the stability region increases for

high stiffness, from 3000 to 6000 in stiffness units. However, if controller gain keeps

tripling from 2.7 to 7.29, stability region "moves" to the left, making high values ofdamping unstable. This clearly illustrates that increasing the bandwidth of the inner

loop is not always consistent with the requirements for the outer loop, because high

values of damping (D >20) are within the range of impedance interest.

D

K

x 103

10 20 30

2

4

6

(a)k = 0.9

D

K

x 103

10 20 30

2

4

6

(b)k = 2.7

D

K

x 103

10 20 30

2

4

6

(c)k = 7.29

Figure 5.3: Stability regions varying the gain of PI controller (keeping z= 3)

5.3.2 Stability regions with inner LQR controller

The results are presented in Figure 5.4, where the red area corresponds to stableregion, whereas blue area is the unstable region. As can be seen, even mediocre

LQR controller (a) provides larger stability region for high stiffness and damping, in

contrast to the PI controller. Increasing the bandwidth of the inner loop only increases

the stability region (see Figure (b) and (c)). The narrow region where stability fails

is for high stiffness and low damping only (beyond K > 2000). In general, it has

been proven that LQR controller provides larger ranges of impedance parameters

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that keep the system stable and that increasing the bandwidth of the torque loop

leads to enlargement of those ranges, as opposed to the conventional PI controller.

D

K

x 103

10 20 30

2

4

6

(a)Q = [10; 0;0; 0;0; 100]

D

K

x 103

10 20 30

2

4

6

(b)Q = [100; 0; 0;100;0; 100]

D

K

x 103

10 20 30

2

4

6

(c)Q = [500; 0;0; 500;0; 500]

Figure 5.4: Stability regions varying Q matrix of LQR controller

5.4 Passivity analysis

The term passive environment refers to the environment that cannot inject energy

during interaction; it can only exchange or dissipate energy. From (Colgate & Hogan,

1988) it is known that a strictly passive system, connected to any passive environ-

ment, is necessarily stable. This type of stability is ensured if the systems driving

port impedance / admittance is passive. Driving port impedance / admittance is the

impedance / admittance of the system at the point where the interaction with the

environment occurs. LetY(s)denote the transfer function of the driving port admit-tance. In our case, this is the transfer function from the load disturbance torqueTl

to the link velocity ql. Admittance is used in the analysis since the model (3.8) has

the load torque as an input and the link velocity as an output. Then, Y(s)is passive

if and only if:

1. Y(s)has no poles in (s)> 0;

2. the phase ofY(s)lies between -90 and 90 degrees.

In this section, the analysis of the passivity is as follows. First, the transfer functionY(s) has been computed for torque loop alone and for the whole system with the

outer impedance loop by setting low (K = 40, D = 5) and high (K = 500, D =

30) impedance parameters. Next, the analysis has been repeated by varying inner

torque loop controllers parameters. For each set of parameters, the analysis has been

performed first checking the stability of Y(s) and then verifying that the phase of

Y(s) lies in the range 90 +90. If these conditions are not satisfied, then Y(s)

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is not passive for the particular set of parameters. Finally, regions of Y(s) for the

controllers where passivity maintains for either low or high impedance parameters

will be visualized to obtain accurate range of allowed impedance parameters. The

results of this analysis are summarized in Table5.1. Only the controllers that satisfy

the stability margins requirements for the inner loop have been used (6dB and30

).

Table 5.1: Passivity

Torque

loop

(alone)

Impedance loopK= 40

D= 5

Impedance loopK= 500

D= 30

PI

k= 2.7z = 0.1 Yes No No

k= 2.7z = 3 Yes Yes No

k= 2.7z = 7 Yes No Nok= 0.03z = 3 Yes No No

k= 1z = 3 Yes Yes No

LQR

Q= [10; 0;0; 0;0; 100] Yes Yes No

Q= [100;0;0;0;0;100] Yes Yes No

Q= [100; 0; 0; 100; 0; 100] Yes Yes No

Q= [500;0;0;0;0;500] Yes Yes No

Q= [500; 0; 0; 500; 0; 500] Yes Yes No

As the results in the table show, the passivity is always maintained in the sys-

tem where only torque loop is closed, which means that it will remain stable when

interacting with the passive environment. On the other hand, the overall system

(torque loop + impedance loop) is never passive for high impedance parameters

(K= 500, D = 30). For low impedances (K = 40, D = 5) the system after closing

the impedance loop is passive for every proposed LQR controller, but fails to remain

passive for the PI controllers with the lowest bandwidth (k = 0.03) and the highest

bandwidth (k = 2.7 z = 7). This illustrates the fact that the inner closed loopbandwidth requirements is not always consistent with outer impedance loop specifi-

cations when using a PI controller. The Figures5.5and5.6show the passivity regions

(K(0 2000], D (0 30]) for the controllers where passivity maintains at least for

low impedance parameters, where white corresponds to passive stable, gray is non-

passive stable, and black is unstable system. In PI case (see Figure5.5), increasing

the controller gain from 1 to 2.7 decreases the range of damping where the system

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remains passive. For LQR (see Figure5.6), controllers (d) and (e) achieves passivity

for all values of the stiffnessKwhen the dampingD = 10. In general, LQR controller

provides larger passivity regions compared to PI controller.

D

K

100

101

100

102

(a)k = 1z = 3

D

K

100

101

100

102

(b)k = 2.7z = 3

Figure 5.5: Passivity regions (log scale): PI controllers

D

K

100

101

100

102

(a)Q = [10; 0;0; 0;0; 100]

D

K

100

101

100

102

(b)Q = [100;0;0;0;0;100]

D

K

100

101

100

102

(c)Q = [100; 0; 0;100;0; 100]

D

K

100

101

100

102

(d)Q = [500;0;0;0;0;500]

D

K

100

101

100

102

(e)Q = [500; 0;0; 500;0; 500]

Figure 5.6: Passivity regions (log scale): LQR controllers

5.4.1 Coupled stability

The purpose of this subsection is to determine the class of passive environments that

makes the system become non-passive and, as a result, unstable when interacting

with these environments. If a manipulator is stable when coupled to a particular, but

restricted, class of passive environments, the "worst" environment, the term intro-

duced by (Colgate,1986), then it is stable when coupled to all passive environments.

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Moreover, this restricted class of environments is easily param

impedance control of systems with joint flexibility

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