robust non-linear observer for a non-collocated flexible system mohsin waqar intelligent machine...

Robust Non-Linear Observer for a Non-collocated Flexible System

Mohsin Waqar

Intelligent Machine Dynamics Lab

Georgia Institute of Technology

December 12, 2007

http://www.gatech.edu/

Agenda

•Project Motivation and Goals

•Non-collocated Flexible System and Non-minimum Phase Behavior

•Control Overview

•Test-bed Overview

•Plant Model

•Optimal Observer – The Kalman Filter

•Robust Observer – Sliding Mode

•Project Roadmap

•Project Roadmap


Motivation for Research – Flexible Robotic Arms

1) Manipulators with very large workspaces (long reach):

Example - handling of nuclear waste.

2) Manipulators with constraint on mass:

Example – space manipulators.

3) Manipulators with constraint on cost:

Example – Camotion Depalletizer

4) Manipulators with Actuator/Sensor Non-collocation:

Collocation can be impossible.

Source: NASA.gov

Source: camotion.com


Problem Statement

•Contribute to the field of active vibration suppression in motion systems

•Examine the usefulness of the Sliding Mode Observer as part of a closed-loop system in the presence of non-collocation and model uncertainty.


What is a Flexible Robotic Arm?

•Robotic arm is subject to torsion, axial compression, bending.

•Structural stiffness, natural damping, natural frequencies and boundary conditions are important to consider.

Source: Shabana, A. A. Vibration of Discrete and Continuous Systems. 1997.

Note: pole = eigenvalue = mode = natural frequencyX


What is a Flexible Robotic Arm? - References

W.J. Book, “Modeling, Design, and Control of Flexible Manipulator Arms: A Tutorial Review,” Proceedings of the 29th Conference on Decision and Control, Dec. 1990.

W.J. Book, “Structural Flexibility of Motion Systems in Space Environment,” IEEE Transactions on Robotics and Automation, Vol. 9, No. 5, pp. 524-530, Oct. 1993.

W.J. Book, “Flexible Robot Arms,” Robotics and Automation Handbook., pp. 24.1-24.44, CRC Press, Boca Raton, FL, 2005. X


Non-Minimum Phase Behavior (in continuous time system)

Causes:

Combination of non-collocation of actuators and sensors and the flexible nature of robot links

Detection: •System transfer function has positive zeros.

Effects:

•Limited speed of response.

•Initial undershoot (only if odd number of pos. zeros).

•Multiple pos. zeros means multiple direction reversal in step response.

•PID control based on tip position fails.

Source: Cannon, R.H. and Schmitz, E. “Initial Experiments on the

End-Point Control of a Flexible One-Link Robot.” 1984.


Non-Minimum Phase Behavior(in continuous time system)

Effects:

•Limited gain margin (limited robustness of closed-loop system)

•Model inaccuracy (parameter variation) becomes more troubling (Zero-flipping).

X

X

X

Re

Im


Non-Minimum Phase Behavior - References

R.H. Cannon and D. E. Rosenthal, “Experiments in Control of Flexible Structures with Noncolocated Sensors and Actuators,” J. Guidance, Vol. 7, No. 5, Sept.-Oct. 1984.

R.H. Cannon and E. Schmitz, “Initial Experiments on the End-Point Control of a Flexible One-Link Robot,” International Journal of Robotics Research, 1984.

D.L. Girvin, “Numerical Analysis of Right-Half Plane Zeros for a Single-Link Manipulator,” M.S. Thesis, Georgia Institute of Technology, Mar. 1992.

J.B. Hoagg and D.S. Bernstein, “Nonminimum-Phase Zeros,” IEEE Control Systems Magazine, June 2007.


Control Overview

Design objective: Accuracy, repeatability and steadiness of the beam end point.

Linear Motor

Flexible Arm

Sensors

ObserverFeedback

Gain K

FeedforwardGain F

Commanded

Tip Position

Noise

V

+

-

yδFu

x


Test-Bed Overview

NI SCB-68Terminal

Board

Anorad EncoderReadhead

Anorad Interface Module

LS7084Quadrature

Clock Converter

PCB 352aAccelerometer

PCB Power Supply

Anorad DC ServoAmplifier

Linear Motor

LV Real Time 8.2Target PC

w/NI-6052E DAQ

Board

R

C

+-

+-

160VDC

PWM

-10 to +10VDC


Test-Bed Rigid Sub-system ID

2

( )

( )b mY s K

V s Ms bs

Starting Parameters: Km=8.17; % overall motor gain [N/V]M=9.6; % base mass [kg]b=50; % track-base viscous damping [N*s/m]

Using fmincon in Matlab with bounds:Km: +/- 25%M: +/- 10%b: (0,inf)

Final Parameters with Step Input:[Km,M,b] = [6.9, 10.52, 38.97]

With ramp input (0-5V over 2 sec):[Km,M,b] = [6.13, 10.56, 35.16]


Agenda



•Control Overview


•Plant Model



•Project Roadmap

•Project Roadmap


Flexible Arm Modeling

Lumped Parameter System (or Discrete System)

Distributed Parameter System (or Continuous System)

Mashner (2002) Beargie (2002)

•Finite degrees of freedom.•Described by one second-order ODE per degree/order of the system.

•Symbolic form retains infinite degrees of freedom and non-minimum phase characteristics.•Describes rigid body motion of link and elastic deflection of link. •Described by second order PDE.

Approaches: 1) Lagrangian: Obergfell (1999)2) Newton Euler: Girvin (1992)

Approximate methods:3) Transfer Matrix Method: Krauss (2006), Girvin (1992)4) Assumed Modes Method: Sangveraphunsiri (1984),

Huggins (1988), Lane (1996)


Flexible Arm Modeling – Assumed Modes Method

mw(x,t)

x

E, I, ρ, A, L

Assumptions:

•Uniform cross-section

•3 flexible modes + 1 rigid-body mode

•Undergoes flexure only (no axial or torsional displacement)

•Linear elastic material behavior

•Horizontal Plane (zero g)

•No static/dynamic friction at slider

•Light damping (ζ << 1)

F



'(0) 0i

Geometric Boundary Conditions:

For i = 1 to 4

Ritz Basis Functions:

'1 ( ) 1x '2 ( ) cos

2

xx

L

'3 ( ) cos

xx

L

'4

3( ) cos

2

xx

L

m

x

E, I, ρ, A, L

F w(x,t)

Source:

J.H. Ginsberg, “Mechanical and Structural Vibrations,” 2001



Mq Cq Kq Q

2 2

0

1 1( , )

2 2

L

mT Aw dx mw x t m

x

E, I, ρ, A, L

F w(x,t)

2

0

1

2

L

V EIw dx

22

20

L

dis

wP EI dx

x

,in FP Fw x t

2 21 0

31 0 1 02 1 2

00 0 0 02 3

2 1 6 1 0 1 00

3 2 5 0 0 0 02 6 1

03 5 2

M AL m

4 3

3 4 33

3 4

0 0 0 0

0 032 6

270

6 2 10

27 810 0

10 32

EIK

L

C K

1

1

1

1

Q F



Mq Cq Kq Q

0K M

1

2TM

m

x

E, I, ρ, A, L

F w(x,t)

4

EI

AL

q 2[ ( )]T TQ C diag

1 1

2 2

3 3

4 4

5 1

6 2

7 3

8 4

x

x

x

x

x

x

x

x

x Ax Bu

( 0, )

( , )

w x ty x Cx Du

w x L t


Length (m) .4

Width (m) 0.0412

Thickness (m) .0024

Material AISI 1018 Steel

Density (kg/m^3) 9838

Young’s Modulus (GPa)

205

Tip Mass (kg) .1375

First Mode 14.3 hz

Second Mode 81.5 hz

Third Mode 331.4 hz

First Mode 5.5 hz

Second Mode 49.5 hz

Third Mode 130.5 hz

Length (m) .707

Width (m) 0.0262

Thickness (m) .0037


Density (kg/m^3)

5903


205

Tip Mass (kg) 1.76

Length (m) .32

Width (m) 0.035 (1 3/8”)

Thickness (m) .003175 (1/8”)


Density (kg/m^3)

7870


205

Tip Mass (kg) .110

First Mode 5.83 hz

Second Mode 45.2 hz

Third Mode 186.5 hz

AMM Model with

Optimization Bounds:

(0,inf) Length and Tip Mass

+/- 25% All Others

Flexible Arm Model vs Experimental

AMM Model with

Optimization Bounds:

+/- 25% On All ParametersExperimental Data


Flexible Arm Model vs Experimental

System ID setup:

•Loop rate 1khz

•Closed-loop PID

•0-40hz Chirp Reference Signal with 0.1-0.5 cm p-p amplitude

•15 data sets averaged

( )

( )tipa s

F s


Flexible Arm Model – Root Locus

( )

( )tipa s

F s


Agenda



•Control Overview


•Plant Model



•Project Roadmap

•Project Roadmap


Performance Criteria for Observer Study

What is a useful observer anyway?

•Robust (works most of the time)

•Accuracy not far off from optimal estimates

•Not computationally intensive

•Straightforward design

•Uses simple rather than a complex plant model


A Hypothesis for Observer StudyE

stim

ate

Mea

n S

quar

e E

rror

(M

SE

)

-50% -25% 0 25% 50%

% Deviation in some Beam Parameter

SMO

Kalman Filter (KF)

WHOOPS…PARABOLAS SHOULD BE FACING UPPPP!!!!!!!


Overview of Steady State Kalman Filter

Why Use?

•Needed when internal states are not measurable directly (or costly).

•Sensors do not provide perfect and complete data due to noise.

•No system model is perfect

Notable Aspects:

•Designed off-line (constant gain matrix) and reduced computational burden

•Minimizes sum of squares of estimate error (optimal estimates)

•Predictor-Corrector Nature

Shortcomings:

Limited robustness to model parameter variation

Steady State KF gives sub-optimal estimates at best


How it works - Kalman Filter

Filter Parameters:

Noise Covariance Matrix Q – measure of uncertainty in plant. Directly tunable.

Noise Covariance Matrix R – measure of uncertainty in measurements. Fixed.

Error Covariance Matrix P – measure of uncertainty in state estimates. Depends on Q.

Kalman Gain Matrix K – determines how much to weight model prediction and fresh measurement. Depends on P.

Kalman Filter

Plant Dynamics

Measurement & State Relationships

Noise Statistics

Initial Conditions

State Estimates with minimum square of error


+

-

+

v

x

1/s

A

B C+

1/s

~A

B C+

L

+

K

F-

ur

x y

y

Filter Design:

1. Find R and Q

1a) For each measurement, find μ and σ2 to get R

1b) Set Q small, non-zero

2. Find P using Matlab CARE fcn

3. Find L=P*C'*inv(R)

4. Observer poles given by eig(~A-LC)

5. Tune Q as needed



ˆ~ˆ

0

A BK LC LC x BK L rx

BK A x BK vx

+-

+v

x

1/s

A

B C+

1/s

~A

B C+

L

+

K

F-

ur x y

y

0 0ˆ

ˆ 0 0

u K Kx r

y C DK DKx v

y DK C DK I

ˆˆ ˆ ˆ( )x Ax Bu L y y



Kalman Filter – LabVIEW Simulation

Observer model = Plant model


Kalman Filter – LabVIEW Simulation

Plant model: A + Δ A

Δ A may be from system wear and tear or change in tip mass


Kalman Filter – Testbed On-Line Estimation

Measured Tip Accel. vs Estimated

-12

-10

-8

-6

-4

-2

0

2

4

6

0 2 4 6 8 10

Time (sec)

Ac

ce

lera

tio

n (

m/s

^2

)

Measured

Estimated

Measured Base Position vs Estimate (Q=2e-6)

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0 2 4 6 8 10 12

Time (sec)

Po

sit

ion

(m

)

MeasuredEstimated

Tip Acceleration (m/s^2) Base Position (m)

Mean -6.607491857 1.61682E-06

Variance 0.000259078 3.86945E-10

Note: Accelerometer DC Bias of -0.67 volts or -6.61 m/s^2

3.86945E-10 0

0 0.000259078R


Agenda



•Control Overview


•Plant Model



•Project Roadmap

•Project Roadmap


Sliding Mode Observer – Lit. Review

•Slotine et al. (1987) – Suggests a general design procedure. Simulations shows superior robustness properties.

•Chalhoub and Kfoury (2004) – 4th order observer with single measurement. Adapts Slotine’s design approach with modifications to observer structure. Presents a unique method for selecting switching gains. Simulations of a single flexible link with observer in closed-loop. Shows KF unstable in presence of uncontrolled modes while SMO remains stable.

•Chalhoub and Kfoury (2006) – 6th order observer with 3 measurements. Same approach as earlier paper. Simulations of the rigid/flexible motion in IC engine.


Sliding Mode Observer – Lit. Review

Kim and Inman (2001) – SMO design based on Lyapunov equation. Unstable estimates by KF in presence of uncontrolled modes while SMO remains stable. Simulations and experimental results of closed-loop active vibration suppression of cantilevered beam (not a motion system).

Zaki et al. (2003) – 14th order observer with 3 measurements, with design based on Lyapunov equation. Experimental results (including parameter variation studies) from three flexible link testbed with PD control. Observer in open loop.

Elbeheiry and Elmaraghy (2003) – 8th order observer with two measurements, with design based on Lyapunov equation. Simulations and experimental results from 2 link flexible joint testbed with PI control. Observer in open loop.


• Sliding Surface – A line or hyperplane in state-space which is designed to accommodate a sliding motion.

• Sliding Mode – The behavior of a dynamic system while confined to the sliding surface.

• Signum function (Sgn(s)) if • Reaching phase – The initial phase of the closed loop

behaviour of the state variables as they are being driven towards the surface.

1

1

Sliding Mode Observer – Definitions

0

0

s

s


• Design a sliding surface. One surface per measurement.

• Design a sliding condition to reach the sliding surface in finite time.

• Design sliding observer gains to satisfy the sliding condition.

Sliding Mode Observer – 3 Basic Design Steps


Sliding Mode Observer – Overview

(0,0)

~

x

12,n y x

Error Vector Trajectory

Sliding Surface

1 1 1ˆs x x

x

0, 0x x

1 1 1s s s

Example:

Single Sliding Surface

Dynamics on Sliding Surface

Sliding Condition


Sliding Mode Observer Form

Example:

1 21

2

,x x

y xx f

2 1 11

2 12

ˆˆˆˆ

x L xx

f L xx

2 1 11

2 12

x L xx

f L xx

2 1 1 1 11

2 1 2 12

sgn( )

sgn( )

x L x k xx

f L x k xx

12,n y x

Luenberger Observer: Observer Error Dynamics:

f Is due to model imperfection. Has potential to destabilize observer error dynamics.

Sliding Mode Observer:


ˆˆ ˆ ˆ ˆ( ) (sgn( ))x Ax Bu L y y K y y

ˆ( ) ( ) ( ) (sgn( ))e t A LC e t K y y

TV e Pe

+ bounded nonlinear

perturbations

With proper selection of K which is based on some P, the Lyapunov function candidate can be used to show that is negative definite and so error dynamics are stable.

V

Sliding Mode Observer Form

General Form:

Error Dynamics:


Simplified Flexible Arm Model:

n=4, y=x1 (tip position)

Plant zeros: -0.7+i1.3e7, -0.7-i1.3e7,

2.75

Simulation Parameters:

Parameter Mismatch: 100% (spring constant)

deltaF for SMO Design: 20%

Eta for SMO Design: 0.05

Sliding Mode Observer LabVIEW Simulation

No K

No L

Just LMSE: 0.00076

L + KMSE: 0.0001


December 2007: Extend SMO to AMM Model and 2 measurements

January 2008: SMO with Closed Loop Control Simulation

KF with Closed Loop Control on Testbed

SMO with Closed Loop Control on Testbed

February 2008: Conduct Parameter Studies

Roadmap


December 2007: Extend SMO to AMM Model and 2 measurements

January 2008: SMO with Closed Loop Control Simulation

KF with Closed Loop Control on Testbed

SMO with Closed Loop Control on Testbed

February 2008: Conduct Parameter Studies

Roadmap

Questions?


robust non-linear observer for a non-collocated flexible system mohsin waqar intelligent machine...

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