Adaptive Neural Network Sliding Mode Control For Electrically-Driven Robot [626504]

CEAI, Vol.14, No.4, pp. 27-32, 2012 Printed in Romania

Adaptive Neural Network Sliding Mode Control For Electrically-Driven Robot
Manipulators

S. Sefriti*, J. Boumhidi**, R. Naoual*, I. Boumhidi*

*LESSI Laboratory, Department Of Physics, Faculty Of Sciences,
Sidi Mohammed Ben Abdellah University, Fez, Morocco
e-mail: sefriti.salma@ hotmail.com
**L IIAN Laboratory, Department of Informatics, Faculty of Sciences,
Sidi Mohammed Ben Abdellah University, Fez, Morocco
e-mail: [anonimizat]

Abstract: In this paper a method for neural network sliding mode control design (NNS) is proposed for
the robust tracking control of the electrically-driven two-links robot manipulators. The aim of this study
is to overcome some shortcomings of the standard sliding mode controller (SMC) such as the produced higher amplitude of chattering, du e to the higher switching gain re quired in the pr esence of large
uncertainties. In the proposed NNS, the sliding mode control with a boundary layer approach is combined
with the neural network ( NN) to control the electrically-driven two-links robot. The NN is used for the
prediction of the model unknown parts and hence it enables a lower switching gain to be used in the
presence of large uncertainties. The stability is shown by the Lyapunov Theory and the control action
used did not exhibit any chattering behavior. As a result, a high-precision position tracking performance is obtained without any oscillatory behavior. The effectiveness of the designed NNS is illustrated by
simulations.
Keywords : Adaptive Control, Neural Network, Robot Manipulators, Sliding Mode Control.
1. INTRODUCTION
The robot manipulator is a complex nonlinear system, whose
dynamic parameters are difficult to forecast precisely. In fact,
it is almost impossible to obtain exact dynamic models as the
system is described by a nominal model with large
uncertainties. To deal with parameters uncertainties, various
methods have been proposed, including the Sliding Mode
Control (Slotine, 1984;Utkin, 1992), and the neural network based controls (Patino et al, 2002; Hussain and Ho, 2004; Liu
et al, 2003). The SMC is a nonlinear control strategy that is
well known for its strong robustness and accuracy. The main feature of this method is to drive the system states on a user-
specified surface in the state space (switching surface), and to
maintain the states on the su rface for all subsequent time.
However, in the presence of larg e uncertainties, the controller
has a higher switching gain and produces higher amplitude of
chattering. As a result, it is im possible to achieve in practical
systems. One possible method to eliminate this chattering
problem is based on the boundary layer solution (Slotine and
Sastry, 1983; Slotine, 1984). Though, this method can resolve the problem for systems with small uncertainties only.
The NN-based controls (Ciliz, 2005; Sun et al, 2011; Sun et
al, 2000) have been closely scoped out in the NN applications in robot tracking control. Most of the papers get the results
that the tracking errors can be uniformly ultimately bounded
as in (Sun et al, 2000) or asymptotically converge to zero as
in (Ciliz, 2005; Sun et al, 2011). However, the considered
uncertainties are small or some gains parameters are
sufficiently large in the case of large uncertainties, which
lead to the oscillatory behaviour. In this paper, a neural
network structure is proposed to estimate the unknown parts
of the two-links robot model, so that the system uncertainties
can be kept small and hence enable a lower switching gain to
be used. The network weights are adjusted during the online implementation by using the gradient descent method (GD)
(Rumelhart et al, 1986). The proposed control consists of the
so-called equivalent control added to robust control term, the NN predicted terms are incorporat ed in the equivalent control
component, enabling the robust component to be used with a
small gain which is responsible of compensating only the network errors prediction. As a result, the responses will be
fast and smooth without any oscillatory behaviour. The
stability is shown by using the Lyapunov theory. The rest of the paper is divided into five sections. In Section
2, the system model is presented. In Section 3, the proposed
neural network sliding mode controller is shown. Section 4 presents the simulations results. Finally, a conclusion is given
in section 5
.
2. SYSTEM MODEL OF THE ELECTRICALLY-DRIVEN
TWO-LINK S ROBOT
The dynamic model of the electrically-driven two-links robot
control may be expressed as follows (Dawson et al, 1992):

28 C ONTROL ENGINEERING AND APPLIED INFORMATICS


 
qE BJiqGqqqCqqM
 
 )( ),( )(
(1)
where qqq,, denote respectively the joints position, which is
the controlled output of the sy stem, velocity, and acceleration
vectors.
 is the torque and the considered control law is the current
i applied to the servo motors.





2212
2111)(
MM
MM
qM : is an inertia matrix that is symmetric
and positive definite where:
)2cos(21242
1242 1 11q llm lm I I M 
),2cos(21222 12q llm I M 
),2cos(21222 21q llm I M  and2 22I M .
with Tqq q2 1 as the positions, 2,1ll as the
lengths,2,1mm are the masses and 2,1II as the respective
inertias of the first and second segment.





22 2112 11),(C CC C
qqC : represents the cen trifugal forces
where:
)2sin(2212211q qllm C 
)2sin()2 1(212212q q qllm C  
)2sin(1212221q qllm C   and 022 C
)(qG is the coriolis matrix given as:





)2 1sin(22)1sin(11)2 1sin(22)(q q glmq glm q q glm
qG
J, B and E are constant, positive definite and diagonal
matrices.
The system model can be written as the following state-space
form:




  
,t)x(ξ)u,x(nh xx xx x,t)x(ξ)u,x(nh xx xx x
2 2 66 55 41 1 33 22 1

(2) where: ),(1uxnh and ),(2uxnh are the nominal representation
of the system with respectively the unknown parts ),(1tx
and ),(2tx .
2)4,1(12 1)4,1(11)(1),,(1uxxng uxxngxnf uxnh  
2)4,1(22 1)4,1(21)(2),,(2uxxng uxxngxnf uxnh  
Tuu ui2 1 is the input contro l of the system
Txx q4 1 : is the controlled output position
Tx x q5 2 ,Tx x q6 3 ,
 Tx x x x xxx6 5 4 3 2 1
 
   BGGqE BCCq BMC M MTxnf xnf xnf
 
  1)(2)(1)(
and 


)(22)(21)(12)(11)( )(1
xngxngxng xng
xngJx M .
3. NEURAL NETWORK SLIDING MODES CONTROL
DESIGN
3.1 Controller Design
Let’s define some variables as:

 Ttx tx tx ),(2),(1),(  (3)

This represents the unknown parts of the system.

 T
dx xdx xdqqe4 4 1 1  (4)
is the output tr acking error with:
T
dxdxdq4 1 : is the desired output.
The control problem is to find a current control law so that
the state )(tq can track the desired trajectorydq.
The relative degree 3r , then the sliding variable can be
defined as:

e e eS  (5)

, are diagonal matrices defined as follows:

CONTROL ENGINEERING AND APPLIED INFORMATICS 29






220011

 , 




220011



 and  are selected such that the roots of the following
characteristic polynomial are specifi ed in the open left half of
the complex plane :

0)1(
22)2(
22)3(0)1(
11)2(
11)3(
    
s s ss s s
  
(6)

The sliding variable derivative is:
e e
dxdx
gutx f S 
   



63),( (7)
To ensure that a sliding mode exists on a switching surface
and that this switching surface can be reached in finite time,
the condition given below has to be satisfied:
0STS (8)
The control law that satisfies (8 ) is given by (Alaoui et al,
2007):
)(63)( )(1
S signke edxdx
xnf xngu







 (9)

where sign(.) is the sign function given by:



0 if10 if00 if1
)(
SSS
S sign
The positive switching gain to compensate the uncertainties
is : k which is designed as:
kB (10)
with Bas the upper bound of the uncertainties given by:
B tx),( (11)
To eliminate the chatteri ng effect caused by the
discontinuous control law, the boundary layer approach can be used. The control becomes as follows:
)(63)( )(1
Ssatke edxdx
xnf xngu







 (12)
Where sat is the saturation function, given by:
 

otherwise) sgn(if /
)(
SS S
Ssat 
(13)
with is the boundary layer thickness.
This method can resolve the problem for systems with small
uncertainties. For systems w ith large uncertainties, we
propose in this study the use neural networks to model the
unknown parts of the two-links robot nonlinear functions
given in (3), so that the system uncertainties can be kept small.
Let’s denote the prediction of the unknown non linear
functions parts as:
 Ttx tx tx ),(2ˆ),(1ˆ ),(ˆ  (14)
where )(1ˆx , )(2ˆx are the expressions for the network
outputs given in a later section.

),(ˆ),( ),( tx tx tx   (15)
And *),(tx (16)
where *is the upper bound of the network error prediction.
Theorem: Consider the robot manipulator modelled by (2) in
the presence of larg e uncertainties. If the system control is
designed as:
sueuui  ˆ
e edxdx
tx xnf xngeu








63)),(ˆ)(()(1ˆ

)()(1Ssatxngksu
with k*
The trajectory tracking errors will conv erge, in finite time, to
the vicinity of S = 0 as S , with is the small
boundary layer thickness.
Proof. Consider the candidate Lyapunov function:
STS V
21 then STS V
Replacing the expression of Sgiven in (7) we get:
)
63),( ( e e
dxdx
ugtx fTS V 
   




By replacing the expression of ugiven in the theorem we
get:
))( ),(ˆ),(( S ksattx txTS V   
)( ),( )( ),( SsatTkS txTS SsatTkStxTS      

30 C ONTROL ENGINEERING AND APPLIED INFORMATICS

)(*SsatTkSTS 
By choosing k* , with k as a small gain, which is
responsible only for compensating the network errors
prediction, we get:
For any small 0 , if S , )( )( S sign Ssat , the
function 0 )*(  Sk V . However, in a small -
vicinty of the origin (boundary layer),
SSsat)( is
continuous, the system trajectories are confined to a boundary
layer of a sliding mode manifold 0S , then the high
tracking precision S is obtained .

Fig. 1. NNS controller scheme.
3.2 Neural Network Design
In this paper, we consider a NN with two layers of adjustable
weights (Lewis et al, 1999) (Fig. 1). x: is the state input
variables and the output variables are:
),(1ˆ
1tx y , ),(2ˆ
2tx y

Fig. 2. The architecture of a multilayer neural network for the prediction of uncertain parts.

) ( )( xW WxyT
jT
k k 2,1k (17)
where:
(.) represents the hidden-layer activation function
considered as a sigmoid function given by:
ses
11)( (18)
 T
kN k k kw w w W …2 1 and

 T
jN j j jw w w W …2 1 are respectively the
interconnection weights between the hidden and the output
layers and between the input and the hidden layers.
The actual output )(xydk (desired output which is the
difference between the actual and nominal functions) is:
)( )( )( x xy xyk k dk  (19)
Where: )(xk is the NN approximation error.
Remark : Before incorporating the networks into the
proposed sliding mode control strategy, the networks were
trained offline. The objective of offline training is to let the
networks learn the functional nonlinearities to a certain degree of accuracy before impl ementing into the controller,
and thus can give faster online adaptation as needed. After
the pre-training step, we would have reasonably good initial
values of the network weights.
The network weights are ad justed during the online
implementation. The method used is based on the gradient
descent method (GD), which is a simple and fast method for
online adaptation.
The essence of the GD consists of iteratively adjusting the
weights in the direction opposite to the gradient of E, so as to
reduce the discrepancy according to:

kjkkj
wE
tw
 (20)
Where 0k is the usual learning rate. The gradient terms
kjwE
 can be derived using the backpropagation algorithm
(Rumelhart et al, 1986). The cost function E is defined as the
error index and the least square error criterion is often chosen as follows:

2
12
21
kk E (21)
4. SIMULATION RESULTS

In this section, we test the proposed control algorithm on a
two-links robot described by the model (2). The control objective is to maintain the system in order to track the
desired angle trajectory:

) cos()3/(1tdx and ) sin()3/(2/4tdx 

CONTROL ENGINEERING AND APPLIED INFORMATICS 31

The parameters are considered to be 6.01m and
4.02m .
The considered sampling period is 0.01s.
The considered uncertainties are a vector random noise with
the magnitude equal to unity.

5005E , 1000 10B and 10000 100J

The switching functions coefficients are defined as:
422 11 22 11  .

Fig. 3. Two-links robot manipulator.

From figures 4 and 7, it can be seen that the tracking
performance is obtained without any oscillatory behaviour
even in the presence of large uncertainties. The
corresponding control current signals are given in Fig. 5 and
8. The figures 6 and 9 show the adjusted torques of joints 1
and 2.

Fig. 4. Angle response 1xand desired trajectory dx1.

Fig. 5. Control 1u(input current of join actuator 1).

Fig. 6. Torque of joint 1 (N-m).

Fig. 7. Angle response 4xand desired trajectory dx4.

32 C ONTROL ENGINEERING AND APPLIED INFORMATICS

Fig. 8. Control 2u(input current of join actuator 2).

Fig. 9. Torque of joint 2 (N-m).
5. CONCLUSIONS
This paper addressed the robust trajectory tracking problem
for a robot manipulator in the presence of large uncertainties
without any chattering behaviour. The designed method is a
combination of the sliding mode control with a boundary
layer approach and the neural network employed to approximate the nonlinear model functions unknown parts
with online adaptation of parameters. Simulation results have
shown a good performance of the proposed method to track
the desired trajectory without any oscillatory behaviour.
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