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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NUMERICAL INTEGRATION
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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NUMERICAL INTEGRATION
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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NUMERICAL INTEGRATION
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
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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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