Uzair Ansari

Saqib Alam

Syed Minhaj un Nabi Jafri

SUPARCO, PakistanICS 2014

Inverse Dynamics Control with

Sliding Mode Neural Network

compensator for Space

Manipulators

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Motivation

• Mathematical modeling of Robotic Manipulator.

• Design of high precision control system subject to

model uncertainties and external disturbances.

Sequence of Presentation

OBJECTIVES

AUTOPILOT DESIGN

Simulation Results

MANIPULATOR DYNAMICS

Objectives

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ESOBJECTIVES

• Development of Two-Link Manipulator SimulationModel in MATLAB/Simulink®.

• Design of Inverse dynamic control explicitly based onmathematical model

• Design of compensator using Adaptive Sliding modebased Radial Basis Function Neural Network.

• Efficacy proof of designed controller by simulationresults and its comparison with Computed TorqueControl.

Modeling of Two Link

Planar Robotic Arm

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SMathematical ModelingThe mathematical model of Two-Link Robotic Manipulator

consists of two types of modeling

A. Kinematic Modelinga. Forward Kinematics

b. Inverse Kinematics

B. Dynamic Modelinga. Newton–Euler

b. Euler-Lagrange formulations

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SKinematic Modeling

Forward Kinematics: It gives position of its end effector with

the knowledge of all the joint angles.

Inverse Kinematics: It gives all the Joint angles with the

knowledge of position of its end effector.

For computing Forward and Inverse Kinematics, Denavit–

Hartenberg (DH) is frequently used technique for Kinematics

modeling.

( )K q

( )K q

j Theta d a Alpha

1 q1 0 1 0

2 q2 0 1 0

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SDynamic ModelingThe dynamic equation of n-link manipulator is written as

,d

M q q C q q q F q G q T Q

2 2 2

1 2 1 2 2 2 1 2 2 2 2 2 1 2 2

2 2

2 2 2 1 2 2 2 2

1 1 1 1 1cos cos

4 4 2 4 2

1 1 1cos

4 2 4

m m l m l m l l q m l m l l q

M q

m l m l l q m l

1 2 1 1 2 2 1 2

2 2 1 2

1 1cos cos

2 2

1cos

2

m m gl q m gl q q

G q

m gl q q

2

2 1 2 1 2 2 2

2

2 1 2 1 2

12 sin

2, &

1sin

2

m l l q q q q

C q q q

m l l q q

1 1

2 2

s

s

f ign qF q

f ign q

Autopilot Design

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NChallenges

Highly nonlinearity

High degree of coupling

Model uncertainty and time-variant

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NIDSNN Control Algorithm

To precisely follow the desired trajectory, the proposed control

architecture consists of:

Feed-forward term: based on Inverse Dynamics of manipulator

which computes the control moment or required torque which is

explicitly based on the modeled dynamics of the system.

Feedback term: based on sliding mode based neural network

compensator is employed in order to compensate for unmodeled

dynamics and to guarantee desired closed-loop behavior

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NIDSNN Control Algorithm

Inverse Dynamic Control

The dynamics equation of manipulator is written as:

Recursive Newton-Euler algorithm is applied to derive the

equation of inverse dynamics of manipulator

,d

M q q C q q q F q G q T Q

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NIDSNN Control Algorithm

Shortcoming! (Inverse Dynamic Control)

Exact Dynamic Model

Structured uncertainties

Unstructured uncertainties.

It is necessary to devise a control strategy that can handle these

nonlinear factors and uncertainties.

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NIDSNN Control Algorithm

Sliding Mode based NN Compensator

Sliding Mode based Radial Basis Neural Network compensator

is used to compensate the errors due to unmodeled dynamics and

disturbances.

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NIDSNN Control Algorithm

Sliding Mode based NN Compensator

The input to the RBF NN compensator is the sliding surface,

based on joint error and its rate denoted by:

The neuron in hidden layer is defined by Radial basis function

i.e.

The output of Sliding mode based RBF NN compensator is

written as

qnS Ce e

2

2( ) exp , 1,2,...5.

2

i

i

i

s ch S i

ˆ '*nj i iU w h

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NIDSNN Control Algorithm

Sliding Mode based NN Compensator

Lyapunov synthesis approach is used to develop an adaptive

control algorithm in which the weights of the RBF neural

network are adjusted adaptively online.

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NIDSNN Control Algorithm

Lyapunov Stability

Lyapunov function is defined as:

Stability Criteria

The derivation is written as:

21 1

2 2

TL s w w

TL ss w w

0LL

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NIDSNN Control Algorithm

Problem formulation

Let the nonlinear system is defined as:

We know that

Placing u we can write:

( , )x f x x gu dt

s ce e

( )d ds e ce x x ce x fx gu dt ce ksign s

1( ( ))du x fx dt ce ksign s

g

( )s fx dt ksign s

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NIDSNN Control Algorithm

Lyapunov Stability

The derivation is written as:

The adaptive law is selected as

TL ss w w

ˆ( ( ) ) ( ( ))TL w sh s w s dt ksign s

1ˆ ( )w sh s

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NIDSNN Control Algorithm

Lyapunov Stability

Now we can write:

Since approximation error is sufficiently small and

We get

( ( )) ( )L s dt ksign s s dt k s

k dt

0LL

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NIDSNN Control Algorithm

Block Diagram

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RSimulink Model

Simulation Results

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NSimulation results

To evaluate the performance of proposed

controller, a reference trajectory is generated in

XY co-ordinate.

Initial pose: (1.5m, -0.5m)

Final pose: (-1.5m, 0.8m)

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Simulation results

Trajectory tracking using Inverse Dynamic

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

20

40

60

80

100

120

140Joint 1 trajectory

Time [Sec]

q1 [

deg]

Reference

ID

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-100

-50

0

50

100Joint 2 trajectory

Time [Sec]

q2 [

deg]

Reference

ID

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NSimulation results

• To evaluate the robustness, 20%-30%

variations in system dynamics are included.

• The following slides will show the performance

of Inverse Dynamics control

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Simulation results

Trajectory tracking using Inverse Dynamic

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

20

40

60

80

100

120

140Joint 1 trajectory

Time [Sec]

q1 [

deg]

Reference

ID

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-100

-50

0

50

100

150Joint 2 trajectory

Time [Sec]

q2 [

deg]

Reference

ID

SIM

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LTS

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

20

40

60

80

100

120

140Joint 1 trajectory

Time [Sec]

q1 [

Deg]

Reference

ID

IDSNN

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3

-2

-1

0

1

2

3Joint 1 Torque

Time [Sec]

Q [

Nm

]

IDSNN

ID

Simulation results

Comparison of IDSNN with ID

SIM

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-100

-50

0

50

100

150Joint 2 trajectory

Time [Sec]

q2 [

Deg]

Reference

ID

IDSNN

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6Joint 2 Torque

Time [Sec]

Q [

Nm

]

IDSNN

ID

Simulation results

Comparison of IDSNN with ID

SIM

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

20

40

60

80

100

120

140Joint 1 trajectory

Time [Sec]

q1 [

Deg]

Reference

CTC

IDSNN

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.6

-0.4

-0.2

0

0.2

0.4Joint 1 Error

Time [Sec]

Err

or

[deg]

CTC

IDSNN

Simulation results

Comparison of IDSNN with CTC

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Manipulator Dynamic Simulator

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Conclusion

• The Mathematical modeling of two-link planar robotic manipulator

and its control design using IDSNN is successfully implemented in

Simulink/MATLAB.

• Sliding Mode based Neural Network feedback term is added to

compensate for unmodeled dynamics and to guarantee stability.

• The weights of the neural networks adjusts adaptively based on

Lyapunov principle to make controller robust against parametric

variations

• The results also depict that the feedback control using SNN

effectively deals with the imperfections remaining in calculating the

inverse dynamics of the manipulator.

• A comparison of the proposed algorithm has also been carried out

with conventional computed torque controller that indicates the superior

performance of the proposed design.

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Future Work

• Development of mathematical model of free-floating

system using Virtual Manipulator technique in order to conserve

fuel which permits the spacecraft to move freely in response to

manipulator motions

• The development of image based visual servoing algorithm

? Questions?