International Journal of Computers, Communications & Control
Vol. II (2007), No. 4, pp. 328-339
Neural Networks-based Adaptive State Feedback Control of Robot
Manipulators
Ghania Debbache, Abdelhak Bennia, Noureddine Goléa
Abstract: This paper proposes an adaptive control suitable for motion control of
robot manipulators with structured and unstructured uncertainties. In order to design
an adaptive robust controller, with the ability to compensate these uncertainties, we
use neural networks (NN) that have the capability to approximate any nonlinear func-
tion over a compact space. In the proposed control scheme, we need not derive the
linear formulation of robot dynamic equation and tune the parameters. To reduce the
NNs complexity, we consider the properties of robot dynamics and the decomposition
of the uncertainties terms. The proposed controller is robust against uncertainties and
external disturbance. The validity of the control scheme is demonstrated by computer
simulations on a two-link robot manipulator.
Keywords: Robot manipulator, neural networks, adaptive control, stability.
1 Introduction
Robot manipulators are multivariable nonlinear systems and are frequently subjected to structured
and unstructured uncertainties even in a well-structured setting for industrial use. Structured uncertain-
ties are mainly caused by imprecision in the manipulator link properties, unknown loads, and so on.
Unstructured uncertainties are caused by unmodeled dynamics, such as, nonlinear friction, disturbances,
and high-frequency dynamics. As a result, it is difficult to obtain an accurate mathematical model so
that computed torque controllers [1-6] or other model-based controllers [7-8] can be accurately applied.
Although adaptive controllers [1-5, 7] can achieve fine control and compensate for partially unknown
manipulator dynamics (i.e., structured uncertainties), they often suffer from incapacity to deal with un-
structured uncertainties. Hence, there is a need for model-free adaptive control strategies.
The application of neural networks to robots dynamic control is not new [9-11]. Though the proposed
methods have been practically successful, it has proved extremely difficult to develop a general analysis
and design theory for early NNs control systems. During the last few years, a number of papers have been
presented to deal with the problem of robot adaptive control [12-14]. The basic idea of these methods
is to design the feedback controller based on the computed torque principle, and to use an adaptive NN
to approximate the robot nonlinearities needed in the control input design. However, most of the above
designs present the drawback that, robot dynamic model is presented as single nonlinearity approximated
by a single NN with the robot real and desired positions and velocities as inputs, which results in large
NN with lot of parameters to be tuned.
In this paper, our goal is to develop a method for designing an adaptive NN control for rigid robot
manipulators. A structured or partitioned NN structure, that simplifies the controller design and makes
for faster weight tuning online, is designed to ensure the closed loop stability. Robust update laws
are used to tune the NNs parameters, and to ensure their boundedness. Lyapunov stability theory is
used to drive the stability conditions, and to show the robustness against uncertainties and disturbances.
Simulation tests for a two-link robot, under uncertainties, disturbance and parameters variations, show
the accuracy and the robustness of the proposed adaptive scheme.
Copyright © 2006-2007 by CCC Publications
Neural Networks-based Adaptive State Feedback Control of Robot Manipulators 329
2 Robot control problem
The Lagrange–Euler formulation, the dynamic equation of an n-joint robot arm can be expressed as
M(q)
..
q + c(q,
.
q) + g(q) + τc(q,
.
q) + τd (q,
.
q) = u (1)
where M(q) ∈ Rn×n bounded positive definite inertia matrix; c(q, .q) ∈ Rn vector representing centrifugal
and Coriolis effects; g(q) ∈ Rn vector representing gravitational torques; τc(q,
.
q) ∈ Rn, τd (q,
.
q) ∈ Rn
vectors representing the dynamic effects as nonlinear frictions, small joint and link elasticities, backlash
and bounded torque disturbances. Here the uncertainties effect is decomposed as continuous part τc(q,
.
q)
and discontinuous part τd (q,
.
q). u ∈ Rn vector of joint torques supplied by the actuators; q ∈ Rn vector of
joint positions;
.
q ∈ Rn vector of joint velocities and ..q ∈ Rn vector of joint accelerations.
Taking as state vector xT =
[
xT1 ... x
T
n
]
with xTi =
[
qi
.
qi
]
, the robot model (1) can be rewrit-
ten as
.
x = Ax + B
[
F(q,
.
q) + G(q)u + d(q,
.
q)
]
(2)
where
F(q,
.
q) =
f1
(
q,
.
q
)
...
fn(q,
.
q)
:= −M−1(q)
[
c
(
q,
.
q
)
+ g(q) + τc
(
q,
.
q
)]
G(q) =
g11 (q) . . . g1n (q)
...
. . .
...
gn1 (q) . . . gnn (q)
:= M−1(q)
d(q,
.
q) =
d1
(
q,
.
q
)
...
dn(q,
.
q)
:= −M−1(q)τd
(
q,
.
q
)
and A =diag[A1, .., An], B =diag[b1, .., bn] with
Ai =
[
0 1
0 0
]
, bi =
[
0
1
]
, i = 1..n
The control problem can be stated as: for a given bounded reference trajectories qr,
.
qr and
..
qr ∈ Rn design
the control input torques u such as the robot’s states follow their references, with all involved signals in
closed loop remain bounded.
3 Neural networks
The general function of one hidden layer feedforward neural network can be described as in (3) as
the weighted combination of N activation functions. Here the input vector x and ϕi (.) represents the ith
activation function (with its parameters vector θi) connected to the output by weight wi.
y =
N
∑
i=1
ϕi (x, θi) wi (3)
The numbers of the input and output layers coincide with the dimension of the input vector and output
information number, respectively. Since the above neural networks will be trained on line to achieve the
330 Ghania Debbache, Abdelhak Bennia, Noureddine Goléa
control task, and in order to reduce computation load, we will assume that the activation functions pa-
rameters θi are fixed, i.e., their number and shape is a priori determined. The only adjustable parameters
are the wights wi. Then, (3) can be rewritten in the compact form
y = wT φ (x) (4)
where φ T (x) =
[
ϕ1 (x) ... ϕN (x)
]
and wT =
[
w1 ... wN
]
.
It is known from NNs approximation theory [15-19] that the modeling error can be reduced arbitrarily
by increasing the number N, i.e., the number of the linear independent activation functions in the network.
That is, a smooth function f (x) , x ∈ Ωx ⊂ Rn can be written as
f (x) = w∗T φ (x) + ε (x) (5)
where ε (x) is the network inherent approximation error, and w∗ is an optimal weight vector.
Various well-known results, see e.g. [16-19], for various activation functions ϕi(.), based, e.g. on the
Stone-Weierstrass theorem, say that any sufficiently smooth function can be approximated by a suitably
large NN [5-8]. The functional range of NN (4) is said to be dense, if for any f (x) and a constant ε∗ > 0
there exist finite N and w∗ such that (5) holds with |ε (x)| < ε∗. The rang of activation functions include
for instance the step, the ramp, the sigmoid and radial basis functions. Several algorithms are proposed
in the literature to select the structure and parameters for those kind of NNs, see e.g. [20-21].
4 Neural state feedback
Due to approximation property (5), we can assume that the nonlinear terms in (2) can be approxi-
mated as
fi(q,
.
q) = θ∗Tfi φi(q,
.
q) + εi(q,
.
q)
gii(q) = θ ∗Tgi ψi(q) + εi(q)
i = 1..n (6)
where θ ∗Tfi φi(q,
.
q) and θ∗Tgi ψi(q) are NNs of the from (4), and εi(q,
.
q), εi(q) are the inherent approxima-
tion errors due to the finite size of the NNs. The optimal weights θ∗fi and θ
∗
gi defined above are quantities
required only for analytical purpose. Typically θ ∗fi and θ
∗
gi are chosen to minimize εi(q,
.
q) and εi(q) over
the compact regions Ω f and Ωg respectively, that is
θ ∗fi = arg minθ fi
{
sup
q,
.
q∈Ω f
∣∣ fi(q,
.
q)−θ Tfi φi(q,
.
q)
∣∣
}
θ∗gi = arg minθgi
{
sup
q∈Ωg
∣∣gii(q)−θ Tgi ψi(q)
∣∣
}
Assumption 1: The neural networks approximation errors are bounded by
∣∣εi(q,
.
q)
∣∣ ≤ ε0i and |εi(q)|≤ ε0i,
i = 1..n, for some constants ε0i and ε0i.
Assumption 1 results from the universal approximation property of neural networks, that can approx-
imate any well-defined function over a compact space with finite approximation error.
Using (6) in (2), the robot dynamic can be written as
.
x = Ax + B
[
Θ∗f Φ(q,
.
q) + Θ∗gΨ(q)u + H(q)u + ω(q,
.
q)
]
(7)
where Φ(q,
.
q) =block-diag
[
φ1(q,
.
q), .., φn(q,
.
q)
]
, Ψ(q,
.
q) =block-diag
[
ψ1(q,
.
q), .., ψn(q,
.
q)
]
, Θ∗f =block-
Neural Networks-based Adaptive State Feedback Control of Robot Manipulators 331
diag
[
θ ∗Tf1 , .., θ
∗T
fn
]
, Θ∗g =block-diag
[
θ∗Tg1 , .., θ
∗T
gn
]
, ω(q,
.
q) = ε + d(q,
.
q), with ε T =
[
ε1 ... εn
]
, and
H(q) =
ε1 g12 (q) . . . g1n (q)
g21 (q)
. . .
...
...
. . . g(n−1)n (q)
gn1 (q) . . . gn(n−1) (q) εn
Based on (7), the control inputs are defined as
u = [ΘgΨ(q)]
−1 [−Θ f Φ(q,
.
q) +
..
qr + Ke
]
(8)
where eT =
[
(qr −q)T
( .
qr −
.
q
)T ] is the tracking error vector, Θg, Θ f are the estimated neural net-
works parameters, and K =diag[K1, ..., Kn] with Ki ∈ R2 is PD gain vector, chosen such as the matrix
Ac = A−BK is Hurwitz.
Then, introducing the control input (8) in (7) yields
.
e = Ace−B
[
Θ̃ f Φ(q,
.
q) + Θ̃gΨ(q)u + H(q)u + ω(q,
.
q)
]
(9)
where Θ̃ f = Θ∗f −Θ f and Θ̃g = Θ∗g −Θg are the parameters estimation errors.
From (9), it can be seen that the tracking error vector is driven by the coupling terms and the finite
approximation accuracy effects reflected by H(q) and the uncertainty term ω(q,
.
q).
To design the neural networks parameters update laws and to ensure boundedness of the involved
signals in the closed loop robot control, the following assumptions are used:
Assumption 2: The diagonal elements of G(q) are bounded such as gm ≤diag[g11(q), .., gnn(q)] ≤ gM ,
the matrix H(q) is bounded by |H(q)| ≤ h0, and the disturbance term ω(q,
.
q) is bounded by
∣∣ω(q, .q)
∣∣ ≤
ω0.
Assumption 3: The neural networks parameters are bounded by the constraint sets Ω f and Ωg such
that: Ω f =
{
Θ f |
∣∣Θ f
∣∣ ≤ fM
}
and Ωg = {Θg | gm ≤ |Θg| ≤ gM}, respectively, where fM , gm, and gM are
some known constants.
The first part of assumption 2 follows from the fact that M(q) is bounded positive definite matrix,
the second part follows from the boundedness of M(q) and εi(q). Finally, the third part follows from
boundedness of M(q), τd and εi(q,
.
q). The bounds used in assumption 3 result from the assumptions 1-2
and are used to ensure the boundedness of the neural networks outputs.
In order to constraint the parameters Θ f and Θg within the sets Ω f and Ωg, respectively, we use the
following parameter projection algorithm [22]:
.
Θ f =
−γ1BT PeΦT (q,
.
q)
if
∣∣Θ f
∣∣ < fM or (
∣∣Θ f
∣∣ = fM
and tr
[
BT PeΦT (q,
.
q)ΘTf
]
≥ 0)
−γ1BT PeΦT (q,
.
q)
+γ1tr
[
BT PeΦT (q,
.
q)Θ̂Tf
] (
1+|Θ f|
fM
)2
Θ f
if
∣∣Θ f
∣∣ = fM and
tr
[
BT PeΦT (q,
.
q)ΘTf
]
< 0
(10)
and
.
Θg =
−γ2BT PeuT ΨT (q)
if
∣∣∣Θ̂g
∣∣∣ < gM or (
∣∣∣Θ̂g
∣∣∣ = gM
and tr
[
BT PeuT ΨT (q)Θ̂Tg
]
≥ 0)
−γ2BT PeuT ΨT (q)
+γ2tr
[
BT PeuT ΨT (q)ΘTg
] ( 1+|Θg|
gM
)2
Θg
if |Θg| = gM and
tr
[
BT PeuT ΨT (q)ΘTg
]
≤ 0
(11)
332 Ghania Debbache, Abdelhak Bennia, Noureddine Goléa
where γ1, γ2 > 0 are design parameters, and P = PT > 0 is the solution, for a given Q = QT > 0, of the
Lyapunov equation
ATc P + PAc = −Q (12)
Moreover, in order to guarantee |Θg| ≥ gm such that an inverse of ΘgΨ(q) always exists, we use the
following law to adjust the parameter Θg.
1. Whenever any element [Θg]i j = gm use
[ .
Θg
]
i j
=
{
−γ2
[
BT PeuT ΨT (q)
]
i j if
[
BT PeuT ΨT (q)
]
i j < 0
0 if
[
BT PeuT ΨT (q)
]
i j ≥ 0
(13)
2. Otherwise, use (11).
where [A]i j stands for the i jth element of the matrix A.
The stability properties of the proposed NNs adaptive state feedback are summarized by the following
theorem.
Theorem 1: The robot adaptive control composed by the robot dynamic (2), the control input (8), the
update laws (10)-(11) and (13) verifying assumptions 1-3, guarantees the following:
1.
∣∣Θ f
∣∣ ≤ fM and gm ≤ |Θg| ≤ gM
2. |e| ∈ L∞
3. |u| ∈ L∞
Proof :
1. To prove |Θg| ≤ gM , let Vg = 12γ2 tr
[
ΘTg Θg
]
, then
.
V g = 1γ2 tr
[
.
Θ
T
g Θg
]
. If the first line of (11) is true,
we have either |Θg| < gM or
.
V g = −tr
[(
BT PeuT ΨT (q)
)T Θg
]
≤ 0 when |Θg| = gM , that is, we get
always |Θg| ≤ gM . If the second line of (11) is true, we have |Θg| = gM and
.
V g = tr
[
−
(
BT PeuT ΨT (q)
)T
Θg + tr
[
BT PeuT ΨT (q) ΘTg
] ( 1 +|Θg|
gM
)2
ΘTg Θg
]
= −tr
[(
BT PeuT ΨT (q)
)T
Θg
]
+ tr
[
BT PeuT ΨT (q) ΘTg
] ( 1 +|Θg|
gM
)2
tr
[
ΘTg Θg
]
= −tr
[(
BT PeuT ΨT (q)
)T
Θg
]
+ tr
[
BT PeuT ΨT (q) ΘTg
] ( 1 +|Θg|
gM
)
|Θg|2 (14)
since |Θg| = gM, we get
.
V g = tr
[
BT PeuT ΨT (q) ΘTg
]
g2M ≤ 0 (15)
that is, |Θg| ≤ gM . Therefore, we have |Θg| ≤ gM , ∀t ≥ 0.
From (13) we see that if |Θg|i j = gm, then
[ .
Θg
]
i j
≥ 0; that is, we have that |Θg|i j ≥ gm.
Using the same analysis, we can prove that
∣∣Θ f
∣∣ ≤ fM , ∀t ≥ 0.
Neural Networks-based Adaptive State Feedback Control of Robot Manipulators 333
2. Consider the Lyapunov function
V =
1
2
eT Pe +
1
2γ1
tr
[
Θ̃Tf Θ̃ f
]
+
1
2γ2
tr
[
Θ̃Tg Θ̃g
]
(16)
The differentiation of (16) along (9) yields
.
V = −1
2
eT Qe−eT PB
[
Θ̃ f Φ(q,
.
q) + Θ̃gΨ(q)u + H(q)u + ω(q,
.
q)
]
+
1
γ1
tr
[ .
Θ̃
T
f Θ̃ f
]
+
1
γ2
tr
[ .
Θ̃
T
g Θ̃g
]
(17)
which can be arranged as
.
V = −1
2
eT Qe−eT PB
[
H(q)u + ω(q,
.
q)
]
+
1
γ1
tr
[( .
Θ̃
T
f −γ1Φ(q,
.
q)eT PB
)
Θ̃ f
]
+
1
γ2
tr
[( .
Θ̃
T
g −γ2Ψ(q)ueT PB
)
Θ̃g
]
(18)
Then, using (10)-(11) and the fact that
.
Θ̃ f = −
.
Θ f (
.
Θ̃g = −
.
Θg), one can show that the third and
fourth terms in (18) are always ≤ 0. Therefore, (18) can be written as
.
V ≤ −1
2
eT Qe−eT PB
[
H(q)u + ω(q,
.
q)
]
(19)
Further, (19) can be upper bounded by
.
V ≤ −1
2
λmin (Q)|e|2 +|e||PB|
[
|H(q)||u|+
∣∣ω(q, .q)
∣∣] (20)
Then, using (8) and the result in 1, the input torques can be upper bounded by
|u| ≤
∣∣∣[ΘgΨ(q)]−1
∣∣∣
[∣∣Θ f Φ(q,
.
q)
∣∣ +
∣∣..qr
∣∣ +|K||e|
]
≤ 1
gm
( fM + q0 +|K||e|) (21)
where q0 is an upper bound on the desired accelerations
..
qr.
Then, using (21) and the assumptions 1-3 in (20), becomes
.
V ≤ −1
2
λmin (Q)|e|2 + κ1 |e|2 + κ2 |e| (22)
where κ1 = h0gm |PBK| and κ2 = |PB|
(
h0
gm
( fM + q0) + ω0
)
.
Hence, (22) can be arranged as
.
V ≤ −1
2
(λmin (Q)−2κ1)|e|2 + κ2 |e| (23)
If the matrix Q is chosen such as λmin (Q) > 2κ1, then
.
V ≤ 0 whenever the tracking error is outside
the region given by
|e| ≤ 2κ2
(λmin (Q)−2κ1)
(24)
which implies that |e| ∈ L∞.
334 Ghania Debbache, Abdelhak Bennia, Noureddine Goléa
3. Using the result (24) in (21) yields
|u| ≤ 1
gm
(
fM + q0 +|K|
2κ2
(λmin (Q)−2κ1)
)
(25)
this implies that |u| ∈ L∞.
Remarks:
1. Only the diagonal elements of G (q) are estimated and used in the control inputs design. By doing
this, we avoid the estimation of the coupling terms (considered here as disturbances) and the need
to compute the inverse of the estimation of G (q).
2. Although, the control torques (8) are presented in vector form, they can be, in practice, computed
independently, since ΘgΨ(q) and K are diagonal matrices and no information is needed from the
other input torques.
3. From (25), it can be seen that constraints on the control inputs, i.e., |ui| ≤ ui max can be meet by
tuning the the PD gain Ki and the desired accelerations magnitudes q0.
5 Simulation Results
To test the proposed adaptive neural control, we consider the two-link manipulator (fig. 1) whose
dynamic equations of motion (1) are:
M (q) =
[
(m1 + m2) l21 m2l1l2 (s1s2 + c1c2)
m2l1l2 (s1s2 + c1c2) m2l22
]
c
(
q,
.
q
)
=
[
0 −m2l1l2 (c1s2 −s1c2)
.
q2
−m2l1l2 (c1s2 −s1c2)
.
q1 0
]
g(q) =
[
−g (m1 + m2) s1l1
−gm2l2s2
]
where c1 = cos(q1), c2 = cos(q2), s1 = sin(q1) and s2 = sin(q2). The robot physical parameters are:
l1 = l2 = 1m, m1 = m2 = 1kg, and g = 9.81 m/s2.
The uncertainties terms in (1) are given by:
τc =
[ .
q1 + sin (3q1)
1.2
.
q2 + 0.5 sin (2q2)
]
, τd =
[
0.2sign
( .
q1
)
0.1sign
( .
q2
)
]
The NNs adaptive control design for the two-link robot is as follows:
1. In order to construct the NNs approximators, each variable q1, q2 ∈ [−π, π] and
.
q1,
.
q2 ∈ [−2π, 2π]
range is devised into 3 sub domains, which yields four RBF networks to approximate f1
(
q,
.
q2
)
, f2
(
q,
.
q1
)
, g11 (q)
and g22 (q), with qT =
[
q1 q2
]
, with, respectively, 27, 27, 9 and 9 RBFs. The designed RBF
networks take the following compact forms:
f̂1
(
q,
.
q2
)
= θ Tf1 φ1
(
q,
.
q2
)
(26)
f̂2
(
q,
.
q1
)
= θ Tf2 φ2
(
q,
.
q1
)
(27)
ĝ11 (q) = θ Tg1 ψ1 (q) (28)
ĝ22 (q) = θ Tg2 ψ2 (q) (29)
Neural Networks-based Adaptive State Feedback Control of Robot Manipulators 335
m2
l2
q2
m1
q1
l1
Figure 1: Two link robot
where θ f1 , θ f2 ∈ R27×1 and θg1 , θg2 ∈ R9×1 are the NNs adjustable parameters. This approach is far
from being optimal, but it has the merit to reduce the number of parameters to be learned and thus
to reduce the update algorithm complexity and execution time.
2. The control input is designed as in (8) with the adaptive NNs defined in (26)-(29). The PD gain is
defined as K =diag[K1, K2] with K1 = K2 =
[
16 8
]
.
3. For the choice of Q = 100I4 and the solutions of (12) we get P.
4. By analyzing the dynamic of the robot, the following bounds are fixed gM = 2.5, gm = 0.5 and
fM = 20. The adjustable parameters are updated using (10)-(11) with γ1 = 0.01 and γ2 = 0.001.
In the simulation, the NNs parameters are initialized as |θg1 (0)| = |θg2 (0)| = 0.5 and
∣∣θ f1 (0)
∣∣ =∣∣θ f2 (0)
∣∣ = 0. The initial states, in all simulations, are xT (0) =
[
0.5 2 −0.5 1
]
.
The first simulation test concerns the regulation of the joint positions under nominal conditions, i.e.,
no parameters changes and no disturbances. As depicted in figure 2.a, the joint positions exhibit a good
transient performance, and no error is remarked in steady state regime. Figure 2.b shows the regulation
performance under uncertainties effects. From this figure it is seen that these uncertainties affect little the
regulation performance and small steady state error is introduced and the NNs adaptive control achieves
good compensation of the uncertainties effects. In figure 2.c, in addition to the uncertainties effects, a
payload change is introduced at t = 15s when m2 passes from 1kg to 3kg. This situation is rapidly taken
in account by the NNs control and it’s effect is compensated.
336 Ghania Debbache, Abdelhak Bennia, Noureddine Goléa
0 5 10 15 20
0
2
4
x
1
0 5 10 15 20
0
2
4
x
2
0 5 10 15 20
−20
0
20
u
1
t (sec)
0 5 10 15 20
0
2
4
x
3
0 5 10 15 20
0
2
4
x
4
0 5 10 15 20
−20
0
20
u
2
t (sec)
a) nominal case
0 5 10 15 20
0
2
4
x
1
0 5 10 15 20
0
2
4
x
2
0 5 10 15 20
−20
0
20
40
u
1
t (sec)
0 5 10 15 20
0
2
4
x
3
0 5 10 15 20
0
2
4
x
4
0 5 10 15 20
−20
0
20
40
u
2
t (sec)
b) τc + τd uncertainties effects
0 5 10 15 20
0
2
4
x
1
0 5 10 15 20
0
2
4
x
2
0 5 10 15 20
−20
0
20
40
u
1
t (sec)
0 5 10 15 20
0
2
4
x
3
0 5 10 15 20
0
2
4
x
4
0 5 10 15 20
−20
0
20
40
u
2
t (sec)
c) τc + τd + m2 variation effects
Figure 2: Regulation performance
0 10 20 30 40
−2
0
2
x
1
0 10 20 30 40
−5
0
5
x
2
0 10 20 30 40
−20
0
20
u
1
t (sec)
0 10 20 30 40
−2
0
2
x
3
0 10 20 30 40
−5
0
5
x
4
0 10 20 30 40
−20
0
20
u
2
t (sec)
a) nominal case
0 10 20 30 40
−2
0
2
x
1
0 10 20 30 40
−5
0
5
x
2
0 10 20 30 40
−20
0
20
u
1
t (sec)
0 10 20 30 40
−2
0
2
x
3
0 10 20 30 40
−5
0
5
x
4
0 10 20 30 40
−20
0
20
u
2
t (sec)
b) τc + τd uncertainties effects
0 10 20 30 40
−2
0
2
x
1
0 10 20 30 40
−5
0
5
x
2
0 10 20 30 40
−40
−20
0
20
u
1
t (sec)
0 10 20 30 40
−2
0
2
x
3
0 10 20 30 40
−5
0
5
x
4
0 10 20 30 40
−40
−20
0
20
u
2
t (sec)
c) τc + τd + m2 variation effects
Figure 3: Tracking performance
The second simulation test concerns the tracking performance for the desired trajectories qr1 =
sin(t) + sin(2t) and qr2 = cos(t) + cos(2t). Figure 3.a presents the tracking performance in the nominal
case. As depicted, the joint positions exhibit a good transient performance, and small error is remarked
in steady state regime. Figure 3.b shows the tracking performance under uncertainties effects. It is clear
that these uncertainties introduce acceptable tracking error, and the NNs control inputs compensate the
uncertainties with a little effort. In figure 3.c, we add to the uncertainties effects, a payload change is
introduced at t = 20s when m2 passes from 1kg to 3kg. This variation affects essentially the developed
torques to compensate the additional mass, and small error is remarked.
6 Conclusion
In this paper, an adaptive NNs control scheme for rigid robot control, was proposed. The adaptive ca-
pability of handling modeling errors and external disturbances was demonstrated. The error convergence
rate with the NNs adaptive approach was found to be fast. Asymptotic stability of the control system is
established using the Lyapunov approach. Simulation studies for a two-link robot verify the flexibility,
adaptation and tracking performance of the proposed approach. The major contributions of the paper
are as follows: reduction of the NNs complexity and no robustifying control is required to achieve the
stability or to enhance the control performance.
Neural Networks-based Adaptive State Feedback Control of Robot Manipulators 337
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Ghania Debbache
Electrical Engineering Institute, Oum El-Bouaghi University, 04000 Oum
El-Bouaghi, Algeria
gdebbache@yahoo.fr
Abdelhak Bennia
Electronic Departement, Constantine University
25000 Constantine, Algeria
abdelhak.bennia@laposte.net
Noureddine Goléa
Electrical Engineering Institute, Oum El-Bouaghi University, 04000 Oum
El-Bouaghi, Algeria
n.golea@lycos.com
Received: March 15, 2007
Neural Networks-based Adaptive State Feedback Control of Robot Manipulators 339
Ghania Debbache was born in Constantine, Algeria, in 1974.
She received the B.Sc. and M.Sc. degrees in electronics from
the Constantine University, Algeria, in 1997 and 2000, respec-
tively. She is currently pursuing the Ph.D. degree on the intel-
ligent control at the University Constantine. Currently, she is a
Teaching Assistant at the Technologic Sciences Institute of Oum
El Bouaghi University, Algeria.
Abdelhak Bennia received his D.E.S. degree in 1983 in physics
from the University of Constantine, Algeria. From 1984 to 1990
he attended the graduate school at Virginia Tech, majoring in
electrical engineering. He received the M.Sc. degree in 1986
and the Ph.D. degree in 1990. Since 1990, he has been with
the electronics department of the University of Constantine. His
current research interests are deconvolution, system identification
and neural networks applied to character recognition, control sys-
tems, and signal processing.
Noureddine Goléa was born in Batna, Algeria, in 1967. He
received the Engineer and Master grades from Sétif University,
Algeria, and the Doctorat from Batna University, Algeria, all in
industrial control, in 1991, 1994, and 2000, respectively. From
1991 to 1994, he was with Electronics Institute at Sétif Univer-
sity, and from 1994 to 1996, he was with the Electronics Institute
at Batna University. Currently, he is Professor of electrical engi-
neering in the Technologic Sciences Institute at Oum El-Bouaghi
University, Algeria. His research interests are nonlinear and adap-
tive control, and intelligent control applied to motion control.