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Uzair Ansari
Saqib Alam
Syed Minhaj un Nabi Jafri
SUPARCO, PakistanICS 2014
Inverse Dynamics Control with
Sliding Mode Neural Network
compensator for Space
Manipulators
MO
TIV
ATI
ON
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
OB
JEC
TIV
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
MA
NIP
ULA
TOR
DY
NA
MIC
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
MA
NIP
ULA
TOR
DY
NA
MIC
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
MA
NIP
ULA
TOR
DY
NA
MIC
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
AU
TOP
ILO
T D
ESIG
NChallenges
Highly nonlinearity
High degree of coupling
Model uncertainty and time-variant
AU
TOP
ILO
T D
ESIG
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
AU
TOP
ILO
T D
ESIG
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
AU
TOP
ILO
T D
ESIG
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.
AU
TOP
ILO
T D
ESIG
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.
AU
TOP
ILO
T D
ESIG
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
AU
TOP
ILO
T D
ESIG
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.
AU
TOP
ILO
T D
ESIG
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
AU
TOP
ILO
T D
ESIG
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
AU
TOP
ILO
T D
ESIG
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
AU
TOP
ILO
T D
ESIG
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
AU
TOP
ILO
T D
ESIG
NIDSNN Control Algorithm
Block Diagram
MO
DEL
ING
AN
D C
ON
TRO
L O
F Q
UA
DR
OTO
RSimulink Model
Simulation Results
AU
TOP
ILO
T D
ESIG
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)
SIM
ULA
TIO
N R
ESU
LTS
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
AU
TOP
ILO
T D
ESIG
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
SIM
ULA
TIO
N R
ESU
LTS
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
ULA
TIO
N R
ESU
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
ULA
TIO
N R
ESU
LTS
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
ULA
TIO
N R
ESU
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
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
SIM
ULA
TIO
N R
ESU
LTS
Manipulator Dynamic Simulator
CO
NC
LUSI
ON
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.
CO
NC
LUSI
ON
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?
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