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)

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