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Transcript of CSPA 2008 Presentation
FRICTION MODELING AND COMPENSATION IN MOTION CONTROL SYSTEM USING SVR
Tijani, I.B., Wahyudi M., and Talib H.H.
Presentation @ CSPA 2009
BY
Tijani, I.B.
PRESENTATION OVERVIEW
INTRODUCTION
METHODOLOGY OF SVR OVERVIEW
DEVELOPMENT OF SVR-MODEL
IMPLEMENTATION FOR COMPESATION
RESULTS
CONCLUSION
REFERENCES
INTRODUCTION
Background: The interest in the study of friction in control engineering has been driven by the need for precise motion control in
most of the industrial applications such as machine tools, robot systems, semiconductor manufacturing systems and Mechatronic systems.
Effects of Friction in motion control system: (Armstrong, 1994) makes the motion of a positioning system slow
causes steady state error or limit cycles near the reference position
Generally, Friction: is inherently present in all machines/systems incorporating parts with relative motion is characterized with complex nonlinear behaviors: stiction, Stribeck, friction lag, dwell time and depends
on factors such as: Temperature, Contact geometry, Surface materials, Presence and type of lubrication, and
Relative motion
INTRODUCTION
The need for its compensation and precise modeling: Model-Based Approach, and problem of model selection!
System Dynamics:
Control Effort:
INTRODUCTION
Problem Statement:
For Model-based friction, there is not yet a universally accepted model for friction.
Hence selection of model thus remains problem-dependent,
and selecting appropriate and accurate model from pool of available models (Tustin, Lorentzian,Gaussian,Polynomial,seven parameter, Dahl, Lugre, Luven, GMS, etc) for a particular application is challenging (in terms of time, computational efforts etc) due to complexity of parametric modeling of the friction nonlinearities.
RESEARCH OBJECTIVE/JUSTIFICATION
A non-parametric friction model based on Artificial Intelligent using einsensitivity support vector regression ( -SVR) is proposed and developed in this work for the identification and compensation of friction in motion control system.
The work has been necessitated by the need for simple and yet efficient model-free representation of friction.
the choice of SVR has been motivated by its unique qualities in approximating nonlinear function among AI-paradigms, and
Also, SVR has not been extensively explored compared to ANN in friction modeling as indicated in the literatures reviewed.
Plant Modeling Overview
Experimental Plant
Linear Model
Non-Linear(Friction)
+
-
u w1/s
thetha
Qss
A
su
s
2)(
)( Linear Model
Friction Model
Friction Experiment.
Experimental Set-up: With MATLAB Xpc-Target Interface
Break-away experiment : to yield friction torque @ v=0
Steady state Friction-Velocity experiment: measuring armature current for several steady state velocities in the range and
Friction Experiments: for friction-velocity data
20 02
THEORY OF SVR
Generally, given a set of N input/output data such that
and the goal of SVR learning theory is to find a function which minimizes the expected risk:
(1)
Where is loss function is unknown probability distribution function
Since function P is unknown, expression (1) can not be directly computed, hence unlike traditional Empirical risk minimization principle that minimizes only the empirical risk(training error),statistical learning theory seeks to obtain a small risk in terms of both training error and model complexity by minimizing the regularized risk function (structural risk function);
(2)
INTRODUCTION CONT.’
Where is the regularization term(or complexity penalizer) used to find the flattest function with sufficient approximation qualities, and is empiric risk defined as:
(3)
Using e-insensitivity loss function proposed by Vapnik (1995) ,[1]:
(4)
the goal of the function estimation in -SVR is thus to minimizes;
(5)
2
2
1w
][ fRemp
METHODOLOGY OF SVR
Mapping the input space to High dimensional
space using the Kernel trick
Subject to
Formulation of the Constrained
Optimization problem in the primal weight space
Using
METHODOLOGY OF SVR CONT.’
Formulating the Lagrangian
Applying the conditions for optimal
solution
Solve the Dual Optimization
Problem with QP Subject to
DEVELOPMENT OF SVR-FRICTION MODEL
MODELING STEPS
Kernel Selection
Parameters
Combinations(rho,C,
and e)
Start
Computing the
Lagrange multipliers,nsv,and bias term
Computing the
Decision Function(
SVR model)
Model Validatio
n
Model Selection(RMSE,and nsv)
END
Input Data Partitionin
gFriction-Velocity Data Pairs
DEVELOPMENT OF SVR-FRICTION MODEL
The steps was implemented with MATLAB codes written with reference to the original SVM MATLAB Toolbox codes by Gun[2].
After 50 and 30 parameters combinations for positive and negative directions respectively, the combinations with least RMSE and at the same time smallest number of support vectors (nsv) were selected as reported below:
C nsv RMSE
Positive 2.5 550 0.0005 16(29%) 0.00047403
Negative 1.5 70000 0.00025 32(58%/) 0.00070062
C
•SVR Friction Model Learning for Negative Motion
PositiveNegative
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0-0.01
-0.009
-0.008
-0.007
-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0
Velocity(rad/s)
Fric
tion(
Nm
)
SVR-FRICTION MODEL RESULTs
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.82
3
4
5
6
7
8
9
10x 10
-3
Velocity(rad/s)
Fric
tion(
Nm
)
SVR MODEL RESULTS CONT.’
Combine SVR-Model with Validation Data Set
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.01
-0.008
-0.006
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
Velocity(rad/s)
Fric
tion
Forc
e(N
m)
IMPLEMENTATION FOR COMPENSATION
For MATLAB Real-time implementation of the Developed SVR model, the computed Lagrange multipliers and bias were integrated in an Embedded Matlab function:
Input Kernel Computation Output computation
Input v
Predicted ,f
IMPLEMENTATION FOR COMPENSATION
Experimental Set-up For Position Control:
Combine SVR-Models
RESULTS: PTP Positioning Control
1 Degree Step Input 0.5 Degree Step Input
0 0.05 0.1 0.150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Posit
ion(
degr
ee)
Time(sec.)
SVRPD Only
0 0.05 0.1 0.150
0.2
0.4
0.6
0.8
1
1.2
1.4
Time(sec.)
Posit
ion
(deg
ree)
SVRPD Only
RESULTS: PTP Positioning Control
Friction Compensators
STEP INPUTS
Positive Inputs Negative Inputs
0.5-deg. 1-deg. -0.5-deg. -1-deg.
ess(%) Tr(sec.) ess(%) Tr(sec.) ess(%) Tr(sec.) ess(%) Tr(sec.)
No Compensator 37.6 Inf. 7.6 0.017 44.26 inf. 21 0.017
v-SVR 0.8 0.01 0.4 0.015 0.8 0.013 0.4 0.013
% Reduction in steady state error
97.8 94.7 98.2 98.1
RESULTS: Tracking Positioning Control
0.5 Degree,1Hz Sine Reference Position error comparison
0.0 0.5 1.0 1.5-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Po
sitio
n (
de
gre
e)
Time (sec.)
Reference PD Only with SVR
0.00 0.25 0.50 0.75 1.00 1.25 1.50-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Pos
itio
n er
ror
Time (sec.)
PDOnly vSVR
RESULTS: Tracking Positioning Control
1 Degree,1Hz Sine Reference
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Posi
tion (
degre
e)
Time (sec.)
PDOnly vSVR SVR
0.0 0.5 1.0 1.5-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Po
siti
on
err
or
Time (sec.)
PDOnly vSVR
Friction Compensators Root Mean Square Errors (RMSE)
0.5-deg 1-deg.
No Compensator 0.0656 0.0874
v-SVR Model 0.0322 0.0530
% reductionin RMSE 50.9 39.35
Position error comparison
CONCLUSION
SVM based friction model with exponential kernel function was successfully developed and implemented for friction compensation in PTP and Tracking motion control.
The results obtained for modeling and compensation show SVR as a viable and efficient technique in representing and compensating frictional effects in motion control system.
However, the non-smoothness in the tracking responses especially at low reference input was attributed to the effect of velocity estimation and imperfection of the sensor used in the compensation scheme. This could be improved upon with the use of more efficient position sensor and/or observer based estimation or other more sophisticated velocity estimation scheme.
SELECTED REFRENCES
1. Armstrong-Helouvry B., Control of Machines with Friction, Boston, MA, Kluwer, 1991
2. Armstrong-Helouvry B., Dupont P. and De Wit C., “A survey of models, analysis tools and compensation method for the control of machines with friction”, Automatica, Vol. 30, No. 7 (1994) pp. 1083-1138.
3. V. Vapnik. The Nature of Statistical Learning Theory. Springer, New York, 1995.
4. S. R. Gunn. Support Vector Machines for Classification and Regression. Technical Report, Image Speech and Intelligent Systems Research Group, University of Southampton, 1997.
“Had however this friction really existed, in the many centuries that these heavens have revolved they would
have been consumed by their own immense speed of everyday…”
Leonardo da Vinci (1452-1519)
The Notebooks 56 V