Robust Non-Linear Observer for
a Non-collocated Flexible System
Mohsin Waqar
Intelligent Machine Dynamics Lab
Georgia Institute of Technology
December 12, 2007
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
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.
Source: NASA.gov
3) Manipulators with constraint on cost:
Example – Camotion Depalletizer
4) Manipulators with Actuator/Sensor Non-collocation:
Collocation can be impossible.
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?
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
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.
Causes:
Non-Minimum Phase Behavior
(in continuous time system)
Combination of non-collocation of actuators and sensors and the flexible
nature of robot links
Detection:
•System transfer function has positive zeros.
Source: Cannon, R.H. and Schmitz, E. “Initial Experiments on the
Effects:
End-Point Control of a Flexible One-Link Robot.” 1984.
•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.
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 (Zeroflipping).
Im
X
X
X
Re
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
Noise
V
+
Commanded
Tip Position
u
Feedforward
Gain F
Linear
Motor
F
Flexible
Arm
δ
y
Sensors
-

Feedback
Gain K
x
Observer
Design objective: Accuracy, repeatability and steadiness of the beam end point.
Test-Bed Overview
R
PCB 352a
Accelerometer
PCB
Power Supply
Anorad Encoder
Readhead
Anorad
Interface Module
LV Real Time 8.2
Target PC
w/
NI-6052E DAQ
Board
C
LS7084
Quadrature
Clock Converter
NI SCB-68
Terminal
Board
+
+
-
Anorad DC Servo 160VDC
Amplifier
PWM
-10 to +10VDC
Linear
Motor
Test-Bed Rigid Sub-system ID
Yb ( s)
Km

V ( s) Ms 2  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
•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
Flexible Arm Modeling
Lumped Parameter System
(or Discrete System)
Mashner (2002)
Beargie (2002)
•Finite degrees of freedom.
•Described by one second-order ODE per degree/order
of the system.
Distributed Parameter System
(or Continuous 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
E, I, ρ, A, L
F
m
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 (ζ << 1)
Flexible Arm Modeling – Assumed Modes Method
E, I, ρ, A, L
Geometric Boundary Conditions:
 i (0)  0
'
For i = 1 to 4
F
m
w(x,t)
x
Ritz Basis Functions:
 1' ( x)  1
x 
 2 ( x)  cos  
 2L 
x 
'
 3 ( x)  cos  
 L 
 3 x 
'
 4 ( x)  cos 

 2L 
'
Source:
J.H. Ginsberg, “Mechanical
and Structural Vibrations,”
2001
Flexible Arm Modeling – Assumed Modes Method
L
1
1
T    Aw2 dx   mw( xm , t ) 2
20
2
E, I, ρ, A, L
m
F
L
1
V   EIw2 dx
20
x

 1

 2

M   AL 
 0

 2

 3
2
 2w 
Pdis    EI  2  dx
 x 
0
L
Pin   Fw  xF , t 
1
1
QF 
1

1
C K
2

0
1
2
2
3
2
3
1
2
6
5
0 0

4
0 

32
EI 
K 3
3
L 0

6

0 0

0
0
3
6
4
2
27 3
10
Mq  Cq  Kq  Q
2 
3 

1
0 
0



m

 1
6 

5 
0

1

2 
0 

0 

3
27 
10 
81 4 
32 
0 1
0 0
0 1
0 0
0
0 
0

0
w(x,t)
Flexible Arm Modeling – Assumed Modes Method
Mq  Cq  Kq  Q
E, I, ρ, A, L
F
 K  M   0

q  
  T Q  T C  [diag ( 2 )]


T
w(x,t)
x
EI

 AL4

m
M 
1
2
x  Ax  Bu
 w( x  0, t ) 
y
   x    Cx  Du

 w( x  L, t ) 
 x1  1 
 x   
 2  2
 x3  3 
   
 x4    4 
 x5  1 
   
 x6   2 
 x   
 7  3
 x8   4 
Flexible Arm Model vs Experimental
AMM Model with
Experimental Data
AMM Model with
Optimization Bounds:
Optimization Bounds:
(0,inf) Length and Tip Mass
+/- 25% On All Parameters
+/- 25% All Others
Length (m)
.32
Length (m)
.4
Length (m)
.707
Width (m)
0.035 (1 3/8”)
Width (m)
0.0412
Width (m)
0.0262
Thickness (m)
.003175 (1/8”)
Thickness (m)
.0024
Thickness (m)
.0037
Material
AISI 1018 Steel
Material
AISI 1018 Steel
Material
AISI 1018 Steel
Density
(kg/m^3)
7870
Density (kg/m^3)
9838
Density
(kg/m^3)
5903
Young’s
Modulus (GPa)
205
Young’s
Modulus (GPa)
205
Young’s
Modulus (GPa)
205
Tip Mass (kg)
.110
Tip Mass (kg)
.1375
Tip Mass (kg)
1.76
First Mode
5.5 hz
First Mode
14.3 hz
First Mode
5.83 hz
Second Mode
49.5 hz
Second Mode
81.5 hz
Second Mode
45.2 hz
Third Mode
130.5 hz
Third Mode
331.4 hz
Third Mode
186.5 hz
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
atip ( s)
F ( s)
Flexible Arm Model – Root Locus
atip ( s)
F ( s)
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
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
Estimate Mean
Square Error (MSE)
A Hypothesis for Observer Study
WHOOPS…PARABOLAS SHOULD BE
FACING UPPPP!!!!!!!
SMO
Kalman Filter
(KF)
-50%
-25%
0
25%
50%
% Deviation in some Beam Parameter
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
Plant Dynamics
Measurement & State Relationships
Noise Statistics
Kalman
Filter
State Estimates
with minimum
square of error
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.
How it works - Kalman Filter
v
r
u
F
-
x
B
+
1/s
y
+
C
Filter Design:
A
1. Find R and Q
1a) For each measurement,
find μ and σ2 to get R
+
1b) Set Q small, non-zero
L
2. Find P using Matlab CARE fcn
3. Find L=P*C'*inv(R)
4. Observer poles given by
eig(~A-LC)
x̂
+
B
+
-
1/s
~A
5. Tune Q as needed
K
ŷ
C
How it works - Kalman Filter
r
F
-
u
B
ˆ ˆ  Bu  L( yˆ  y )
xˆ  Ax
u    K
 yˆ   C  DK
  
 y    DK
LC   xˆ   BK




A   x   BK
0
 K
ˆ
 x
0      DK
x

 DK
C 
L  r 
0   v 
0
r 
0  
v

I 
xC
v
y +
A
L
B
 xˆ   ~ A  BK  LC
 
 BK
 x 
+
1/s
+
+
1/s
~A
K
x̂
C
ŷ
+
-
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
Tip Acceleration (m/s^2)
Base Position (m)
Mean
-6.607491857
1.61682E-06
Variance
0.000259078
3.86945E-10
0
3.86945E-10

R
0
0.000259078

Note: Accelerometer DC Bias of -0.67 volts or -6.61 m/s^2
Measured Tip Accel. vs Estimated
Measured Base Position vs Estimate (Q=2e-6)
0.03
6
0.02
2
0.01
0
-2
0
2
4
6
8
Measured
Estimated
-4
-6
10
Position (m)
Acceleration (m/s^2)
4
0
0
2
4
6
8
-0.01
-0.02
-8
Measured
Estimated
-0.03
-10
-12
-0.04
Time (sec)
Time (sec)
10
12
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
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 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.
1
• Signum function (Sgn(s))   
1
s  0
if 

s  0
• Reaching phase – The initial phase of the closed loop
behaviour of the state variables as they are being
driven towards the surface.
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.
Sliding Mode Observer – Overview
Example:
n  2, y  x1
Single Sliding Surface
Sliding
Surface
s1  xˆ1  x1
Error
Vector
Trajectory
Dynamics on Sliding
Surface
x   x  0,   0
Sliding Condition
s1s1   s1
x
~
x
(0,0)
Sliding Mode Observer Form
Example:
n  2, y  x1
 x1   x2 
 x    f  , y  x1
 2  
Luenberger Observer:
 xˆ1   xˆ2  L1 x1 
  ˆ

 xˆ2   f  L2 x1 
f
Observer Error Dynamics:
 x1   x2  L1 x1 
 


f

L
x
2 1
 x2  
Is due to model imperfection. Has potential to destabilize observer error
dynamics.
Sliding Mode Observer:
 x1   x2  L1 x1  k1 sgn( x1 ) 
 


f

L
x

k
sgn(
x
)
2 1
2
1 
 x2  
Sliding Mode Observer Form
General Form:
ˆ ˆ  Bu  L( yˆ  y )  K (sgn( yˆ  y ))
xˆ  Ax
Error Dynamics:
e(t )  ( A  LC )e(t )  K (sgn( yˆ  y ))
+ bounded nonlinear
perturbations
With proper selection of K which is based on some P, the Lyapunov
function candidate V  eT Pe can be used to show that V is negative
definite and so error dynamics are stable.
Sliding Mode Observer LabVIEW Simulation
Simplified Flexible Arm Model:
n=4, y=x1 (tip position)
Plant zeros: -0.7+i1.3e7, -0.7-i1.3e7,
2.75
No K
No L
Simulation Parameters:
Parameter Mismatch: 100% (spring constant)
deltaF for SMO Design: 20%
Eta for SMO Design: 0.05
Just L
MSE: 0.00076
L+K
MSE: 0.0001
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
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?