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

1
Robust Non-Linear Observer for a Non-collocated
Flexible System
Mohsin Waqar Intelligent Machine Dynamics
Lab Georgia Institute of Technology December 12,
2007
2
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

3
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
4
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.

5
What is a Flexible Robotic Arm?
X
Source Shabana, A. A. Vibration of Discrete and
Continuous Systems. 1997.

• Robotic arm is subject to torsion, axial
  compression, bending.
• Structural stiffness, natural damping, natural
  frequencies and boundary conditions are important
  to consider.

Note pole eigenvalue mode natural frequency
6
What is a Flexible Robotic Arm? - References
X
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.
7
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.
8
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).

Im
X
X
X
Re
9
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.
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Control Overview
Noise V

Commanded Tip Position
y
F
u
d
Linear Motor
Flexible Arm
Sensors
Feedforward Gain F
-
Observer
Feedback Gain K
Design objective Accuracy, repeatability and
steadiness of the beam end point.
11
Test-Bed Overview
R
PCB 352a Accelerometer
PCB Power Supply
C
LS7084 Quadrature Clock Converter
Anorad Encoder Readhead
Anorad Interface Module

-
LV Real Time 8.2 Target PC w/ NI-6052E DAQ Board

NI SCB-68 Terminal Board
160VDC
Anorad DC Servo Amplifier
Linear Motor
-
PWM
-10 to 10VDC
12
Test-Bed Rigid Sub-system ID
Starting Parameters Km8.17 overall motor
gain N/V M9.6 base mass kg b50
track-base viscous damping Ns/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
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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

14
Flexible Arm Modeling
Lumped Parameter System (or Discrete
System) Distributed Parameter System (or
Continuous System)

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

Mashner (2002) Beargie (2002)

• 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)
15
Flexible Arm Modeling Assumed Modes Method
E, I, ?, A, L
m
F
w(x,t)
x

• 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 (? ltlt 1)

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Flexible Arm Modeling Assumed Modes Method
Geometric Boundary Conditions
For i 1 to 4
Ritz Basis Functions
Source J.H. Ginsberg, Mechanical and
Structural Vibrations, 2001
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Flexible Arm Modeling Assumed Modes Method
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Flexible Arm Modeling Assumed Modes Method
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Flexible Arm Model vs Experimental
AMM Model with Optimization Bounds (0,inf)
Length and Tip Mass /- 25 All Others
AMM Model with Optimization Bounds /- 25 On
All Parameters
Experimental Data
Length (m) .4
Width (m) 0.0412
Thickness (m) .0024
Material AISI 1018 Steel
Density (kg/m3) 9838
Youngs Modulus (GPa) 205
Tip Mass (kg) .1375
Length (m) .707
Width (m) 0.0262
Thickness (m) .0037
Material AISI 1018 Steel
Density (kg/m3) 5903
Youngs Modulus (GPa) 205
Tip Mass (kg) 1.76
Length (m) .32
Width (m) 0.035 (1 3/8)
Thickness (m) .003175 (1/8)
Material AISI 1018 Steel
Density (kg/m3) 7870
Youngs Modulus (GPa) 205
Tip Mass (kg) .110
First Mode 5.83 hz
Second Mode 45.2 hz
Third Mode 186.5 hz
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
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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

21
Flexible Arm Model Root Locus
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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

23
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

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A Hypothesis for Observer Study
WHOOPSPARABOLAS SHOULD BE FACING UPPPP!!!!!!!
SMO
Estimate Mean Square Error (MSE)
Kalman Filter (KF)
-50 -25 0 25 50
Deviation in some Beam Parameter
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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

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How it works - Kalman Filter
Plant Dynamics
Kalman Filter
State Estimates with minimum square of error
Measurement State Relationships
Noise Statistics
Initial Conditions
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.
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How it works - Kalman Filter

• Filter Design
• Find R and Q
• 1a) For each measurement, find µ and s2 to get
  R
• 1b) Set Q small, non-zero
• 2. Find P using Matlab CARE fcn
• Find LPC'inv(R)
• Observer poles given by eig(A-LC)
• 5. Tune Q as needed

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How it works - Kalman Filter
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Kalman Filter LabVIEW Simulation
Observer model Plant model
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Kalman Filter LabVIEW Simulation
Plant model A ? A
? A may be from system wear and tear or change in
tip mass
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Kalman Filter Testbed On-Line Estimation
  Tip Acceleration (m/s2) 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/s2
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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

33
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 Slotines 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.

34
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.
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Sliding Mode Observer Definitions

• 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.

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Sliding Mode Observer 3 Basic Design Steps

• 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.

37
Sliding Mode Observer Overview
Example
Sliding Surface
Single Sliding Surface Dynamics on Sliding
Surface Sliding Condition
Error Vector Trajectory
(0,0)
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Sliding Mode Observer Form
Example
Observer Error Dynamics
Luenberger Observer
Is due to model imperfection. Has potential to
destabilize observer error dynamics.
Sliding Mode Observer
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Sliding Mode Observer Form
General Form Error Dynamics
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.
40
Sliding Mode Observer LabVIEW Simulation
Simplified Flexible Arm Model n4, yx1 (tip
position) Plant zeros -0.7i1.3e7, -0.7-i1.3e7,
2.75 Simulation
Parameters Parameter Mismatch 100 (spring
constant) deltaF for SMO Design 20 Eta for SMO
Design 0.05
No K No L
L K MSE 0.0001
Just L MSE 0.00076
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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
42
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
Questions?