ELECO 2011 Prensentation

7/31/2019 ELECO 2011 Prensentation

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Istanbul Technical University

Department of Control Engineering

Istanbul, Turkey

Emre SARIYILDIZ & Hakan TEMELTA

ELECO 2011

A Comparison Study of the NumericalIntegration Methods in the Trajectory TrackingApplication of Redundant Robot Manipulators

December 20111


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CONTENTS

Introduction

Differential Kinematics Model

Numerical Integration of the Joint Velocities Explicit Euler Integration Method

Runge-Kutta 4 Method

Euler Trapezoidal Predictor & Corrector Method

Adams-Moulton Method (Fourth Order)

Trajectory Tracking Application

Simulation Results

Conclusions

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INTRODUCTION

Kinematic Problem: The problem of kinematic isto describe the motion of the manipulator withoutconsideration of the forces and torques causing the

motion.

Kinematic

Forward Kinematics Inverse Kinematics

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INTRODUCTION

Redundancy:If the dimension of the joint space is greater than thedimension of the work space then the robot iskinematically redundant

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DIFFERENTIAL KINEMATIC MODEL

Differential Kinematics:

generalized inverse of the Jacobian matrix

gJ tipq q V

0 0

t tg

dt J dt tipq q q V

gJ q

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DIFFERENTIAL KINEMATIC MODEL

Advantages:

It can be easily implemented any kind of mechanism.(Structure Free)

Easy to implement

Disadvantages

Locally linearized approximation of the inversekinematic problem

Heavy computational calculation and bigcomputational time

It requires numerical integration which suffersfrom numerical errors

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NUMERICAL INTEGRATION

Numerical Integration Methods:

Single-step numerical integration methods

Explicit Euler Integration Runge-Kutta 4

Multi-step numerical integration methods

Predictor & Corrector Adams-Moulton

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Explicit Euler Integration Runge-Kutta 4

where where

in which,

1k k kt t t t q q q

gk k kt J t t tipq q V

11

2 26

k k k k k k t t t t t t 1 2 3 4q q q q q q

gk k kt J t t 1 tipq q V 12

gk k

kt J t t

2 1 tipq q V

12

gk k

kt J t t

3 2 tipq q V 1gk k kt J t t 4 3 tipq q V

12

gk k k k

tt t J t t

tipq q q V

1 12

2

gk k k

k

tt t J t t

2 tipq q q V

2 12g

k k k kt t t J t t

3 tipq q q V

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Predictor & Corrector

where is predicted joint velocities in which,

1 12

k k k k

tt t t t

q q q q

1 k k kt t t t q q q 1 g

k k k k t t J t t t tipq q q V

ktq

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Adams & MoultenPredictor

If then

If then

If then

If then

Corrector

If then

If thenIf then

1kt t 1 k k kt t t t q q q

2kt t

3kt t

4kt t

1 1 32

k k k k

tt t t t

q q q q

1 1 2 23 16 512

k k k k k

tt t t t t

q q q q q

1 1 2 3 55 59 37 924

k k k k k k

tt t t t t t

q q q q q q

1kt t

2kt t3kt t

1 12

k k k k

tt t t t

q q q q

1 1 15 8

12k k k k k

t

t t t t t

q q q q q

1 1 1 29 19 524

k k k k k k

tt t t t t t

q q q q q q

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TRAJECTORY TRACKINGAPPLICATION

qdot qNI

NumericalIntegration

Vt

error

qdotJ

Jacobian

q poFK

ForwardKinematics

t

Vt

po

DT

DesiredTrajectory

Clock

Vt

Po

qdotTTP

Trajectory TrackingAlgorithm (Predictor)

Vt

po

qdotprdct

po1TT

TrajectoryTracking Algorithmt

Vt

po

DT

DesiredTrajectory

Clock

Explicit Numerical Integration

Euler Integration and Runge-Kutta 4

Implicit Numerical Integration

Predictor&Corrector and Adams Moulten

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SIMULATION RESULTS

Matlab, Virtual Reality Toolbox (VRML)

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SIMULATION RESULTS

Computational Efficiency

Simulation times of the numerical integration methods algorithms (second)

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SIMULATION RESULTS

Accuracy

0 5 100

2

4

6

8x 10

-4

Time (s)

(a)

Errors(rad&m)

or. error

psn. error

0 5 100

2

4

6

8x 10

-6

Time (s)

(b)

Errors(rad&m)

or. error

psn. error

0 5 100

0.2

0.4

0.6

0.8

1x 10

-8

Time (s)(a)

Errors(rad&m)

or. error

psn. error

0 5 100

0.2

0.4

0.6

0.8

1

1.2x 10

-13

Errors(rad&m)

Time (s)(b)

or. error

psn. error

Explicit Euler Integration(a) ms(b) ms

Runge Kutta 4(a) ms(b) ms

100t 10t

100t 10t

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SIMULATION RESULTS

Accuracy

Predictor&Corrector(a) ms(b) ms

Adams Moulten(a) ms(b) ms

100t 10t

100t 10t

0 5 100

1

2

3

45

6x 10

-4

Time (s)

(a)

Errors(rad&m)

0 5 10

0

1

2

3

45

6x 10

-6

Errors(rad&m)

Time (s)

(b)

or. error

psn. error

or. error

psn. error

0 5 100

0.05

0.1

0.15

0.2

Time (s)

(a)

Errors(rad&m)

0 5 10

0

1

2

x 10-4

Time (s)

(b)

Errors(rad&m)

or. error

psn. error

or. error

psn. error

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SIMULATION RESULTS

Stability

a) Explicit Euler Integrationb) Runge-Kutta 4

sec1t

0 5 100.01

0.02

0.03

0.04

0.05

0.06

Time (s)

(a)

Errors(rad&m)

or. error

psn. error

0 5 100

0.02

0.04

0.06

0.08

0.1

Time (s)

(b)

Errors(rad&m)

or. error

psn. error

a) Predictor&Correctorb) Adams Moulten

sec1t

0 5 100

0.05

0.1

0.15

0.2

Time (s)(a)

Errors(rad&m)

0 5 10

0

1

2

x 10-4

Time (s)(b)

Errors(rad&m)

or. error

psn. error

or. error

psn. error

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SIMULATION RESULTS

Around the Singularity Points

17

Total orientation and position errors (radian and meter) of the end effectoraround the singular configurations of robot arm for the Explicit Euler, Runge-

Kutta 2 and Runge-Kutta 4 numerical integration methods.

0 20 40 600

0.2

0.4

0.6

0.8

Time (sec)

Orie

ntationErrorofEulerInt.

0 20 40 600

0.02

0.04

0.06

Time (sec)Orient

ationErrorofRungeKutta2

0 20 40 600

1

2

3

4

5

6x 10

-5

Time (sec)OrientationErrorofRungeKutta4

0 20 40 600

0.05

0.1

0.15

0.2

Time (sec)

Positio

nErrorofEulerInt.

0 20 40 600

0.002

0.004

0.006

0.008

0.01

Time (sec)PositionE

rrorofRungeKutta2

0 20 40 600

0.5

1

1.5

2

2.5x 10

-5

Time (sec)PositionErrorofRungeKutta4

10t ms


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SIMULATION RESULTS

Around the Singularity Points

18

Total orientation and position errors (radian and meter) of the end effectoraround the singular configurations of robot arm for the Predictor & Corrector,

Adams-Bashforth and Adams-Moulton numerical integration methods.

0 20 40 600

0.2

0.4

0.6

0.8

Orienta

tionErrorofPre&Cor.

Time (sec)

0 20 40 600

0.02

0.04

0.06

0.08

Time (sec)Orienta

tionErrorofAd.

Moult.

0 20 40 600

0.02

0.04

0.06

0.08

0.1

Time (sec)Orienta

tionErrorofAd.

Bash.

0 20 40 600

0.05

0.1

0.15

0.2

Time (sec)PositionErrorofPre&Cor

0 20 40 600

0.005

0.01

0.015

Time (sec)PositionErrorofAd.

Moult.

0 20 40 600

0.01

0.02

0.03

0.04

Time (sec)

Position

ErrorofAd.B

ash.

10t ms


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CONCLUSIONS

The performance of the trajectory tracking algorithm is drasticallyaffected by the chosen numerical integration method.

More accurate and more computationally efficient trajectory tracking

algorithms can be obtained by changing the numerical integration methods.Even, the trajectory tracking algorithm may become unstable because of thechosen numerical integration method.

Runge-Kutta and Adams-Moulton numerical integration methods givesatisfactory results in the trajectory tracking application.

Runge-Kutta is the most accurate but least computationally efficientmethod.

Euler integration is the most computationally efficient but least accuratemethod.

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CONCLUSIONS

Note:

In the trajectory tracking application, Runge-Kutta based algorithm gives

quite satisfactory results when the sampling rates are high. As thesampling rates increase, computational load of the trajectory tracking

algorithm decreases. However Runge-Kutta based algorithms require

extra computations and they have high computational load, the

satisfactory results at high sampling rates may reduce even eliminate

this disadvantage.

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Thank you for your kindattention !

Questions ?

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