INT J COMPUT COMMUN, ISSN 1841-9836
Vol.7 (2012), No. 3 (September), pp. 518-529

An Adaptive Iterative Learning Control for Robot Manipulator in Task
Space

T. Ngo, Y. Wang, T.L. Mai, J. Ge, M.H. Nguyen, S.N. Wei

ThanhQuyen Ngo, M. Hung Nguyen
HCM City University of Industry
College of Electrical Engineering
HCM City, Vietnam
E-mail: thanhquyenngo2000@yahoo.com,
nmhung289@yahoo.com

YaoNan Wang, Ji Ge, ShuNing Wei
Hunan University
College of Electrical and Information Engineering
Changsha, Hunan Province 410082, P.R.China
E-mail: yaonan@hnu.cn, gechunxi@126.com,
weishuning@sina.com

T. Long Mai
HCM City University of Industry
College of Electronic Engineering
HCM City, Vietnam
E-mail: mailongtk@gmail.com

Abstract: In this paper, adaptive iterative learning control (AILC) of uncertain robot
manipulators in task space is considered for trajectory tracking in an iterative opera-
tion mode. The control scheme incluces a PD controller with a gain switching tech-
nique plus a learning feedforward term, is exploited to predict the desired actuator
torque. By using Lyapunov method, an adaptive iterative learning control scheme is
presented for robotic system with both structured and unstructured uncertainty, and
the overall stability of the closed-loop system in the iterative domain is established.
The validity of the scheme is confirmed through a numerical simulation.
Keywords: PD Control; Learning control; robot manipulator.

1 Introduction

In general, robotic manipulators have to face various uncertainties in their dynamics, such as fric-
tion, and external disturbance. It is difficult to establish exactly mathematical model for the design of a
model-based control system. In order to deal with this problem, the branches of current control theories
are broad include classical control, robust control, adaptive control, optimal control, nonlinear control,
neural network control, fuzzy logic and intelligent control. However, many adaptive control approaches
are rejected as being too computationally intensive because of the required of real-time parameter iden-
tification and control design

Owing to its simplicity and robustness to modelling uncertainties, are usually used for repetitive
tasks. In this case, the reference trajectory is repeated over a definite operation time. So, iterative
learning control (ILC) for robot manipulator has attracted considerable attention in the recent years [1]-
[3], the ideal of learning control is to improve the tracking performance from iteration to iteration for
applications of robotics in industry which a single repetitive task [4], [5].

ILC is a relatively recent but well-established area of study in control theory. ILC, which can be
categorized as an intelligent control methodology, is an approach for improving the transient perfor-
mance of system that operates repetitively over a fixed time interval [4]. Starting from the classical

Copyright c⃝ 2006-2012 by CCC Publications



An Adaptive Iterative Learning Control for Robot Manipulator in Task Space 519

Arimoto-type ILC algorithm, we can develop a PID-like update law can be given in [6]. So far some
of robot manipulators control in the published literature [7]-[9] and etc, proposed an adaptive ILC to
deal with parameter uncertainties, such as the link length, mass inertia, and friction nonlinearity, with a
self-organizing capability.

In this paper, a new method is given based on a combination of the advantages of a several control
methods into a hybrid one. In particular, it is further extended to the task space or the so-called Cartesian
space. To apply robot manipulators to a wide class of tasks, it will be necessary to control not only the
position of the end-effector, but also the force exerted by the end-effector on the object. By designing the
control law in task space, force control can be easily formulated.

This paper is organized as follows. Section 2 described a dynamic model of an n-link robot ma-
nipulator in task space. Section 3 presents AILC and its features are discussed. By using Lyapunov
method to prove the asymptotic convergence of proposed controller. Numerical simulation results of a
two-link robot manipulator in task space under the possible occurrence of uncertainties are provided to
demonstrate the tracking control performance of the proposed AILC system in section 4. Conclusions
are drawn in section 5.

2 Robotic Dynamic Model In Task Space

In general, the dynamic of an n-link robot manipulator may be expressed in the Lagrange form [10]
as:

D(qi(t))q̈i(t) + C(qi(t), q̇i(t))q̇i(t) + G(qi(t), q̇i(t)) +τa(t) = τ
i(t) (1)

With t ∈ [0, t f ] denotes the time and i ∈ N denotes the iteration, qi(t) ∈ Rn , q̇i(t) ∈ Rn, q̈i(t) ∈ Rn are
the joint position, joint velocity, and joint acceleration variables vector, respectively. D(qi(t)) ∈ Rn×n is
the inertia matrix, C(qi(t), q̇i(t)) ∈ Rn is the coriolis-ccentripetal matrix, is the gravity vector plus friction
force vector. Bounded unknown disturbances are denoted by τa(t) ∈ Rn and the control input torque is
denoted by τi(t) ∈ Rn. For convenience, a two-link robot manipulator, as shown in Fig.1, is utilized to
verify dynamic properties are given in section 4.

Figure 1: Architecture of two-link robot manipulator.

Usually, the manipulator task specification is given relative to the end-effector. Thus, it is natural to
attempt to derive the control algorithm directly in task space, better than in the joint space. Denote the
end-effector position and orientation in task space by x ∈ Rn . The task space dynamic can be rewritten
as follow:

Dx(q
i(t))ẍi(t) + Cx(q

i(t), q̇i(t))ẋi(t) + Gx(q
i(t), q̇i(t)) = τa(t) + Fx

i(t) (2)

Where
Dx(q

i(t)) = J−T (qi(t))D(qi(t))J−1(qi(t))



520 T. Ngo, Y. Wang, T.L. Mai, J. Ge, M.H. Nguyen, S.N. Wei

Cx(q
i(t), q̇i(t)) = J−T (qi(t))(C(qi(t), q̇i(t)) − D(qi(t))J−1(qi(t))J̇(qi(t)))J−1(qi(t))

Gx(q
i(t), q̇i(t)) = J−T (qi(t))G(qi(t), q̇i(t)), Fix = J

−T (qi(t))τi(t)

and J(qi(t)) ∈ Rn×n is the configuration-dependent Jacobian matrix, which is assumed as non-singular in
the finite work space Ω. The above dynamic equation has the following useful structural properties [11],
which can be exploited to facilitate the controller design in the next section.

Property 1: The inertia matrix Dx(qi(t)) is symmetric and positive definite. It is also bounded as a
function of q: m1 I ≤ Dx

(
qi(t)

)
≤ m2 I, where m1,m2 > 0.

Property 2: Ḋx(qi(t)) − 2Cx(qi(t), q̇i(t)) is a skew symmetric matrix. Therefore, yT [Ḋx(qi(t)) −
2Cx(qi(t), q̇i(t))]y= 0, where y is a n × 1 nonzero vector.

Assumption 1: The given desired joint trajectory xd (t) belongs to C2[0, t f ], where C2[0, t f ] denotes
2nd -order continuously differentiable functions on t ∈ [0, t f ].

Assumption 2: Initial condition xi(0) = xd (0), ẋi(0) = ẋd (0) and for all i ≥ 1.

3 AILC Design

Linearizing the system (2) along the desired trajectory xd(t), ẋd (t), ẍd (t) at the ith iterative, we obtain
the following linear time-varying system according [8]:

D(t)ëi(t) + [C(t) + C1(t)]ė
i(t) + R(t)ei(t) + n(ëi, ėi,ei, t) −τa(t) = S (t) − Fi(t) (3)

Where:
D(t) = Dx(xd (t)), C(t) = Cx(xd (t), ẋd (t))

R(t) =
∂Dx
∂x

∣∣∣∣∣
xd (t)

ẍd (t) +
∂Cx
∂x

∣∣∣∣∣
xd (t),ẋd (t)

ẋd (t) +
∂Gx
∂x

∣∣∣∣∣
xd (t)

S (t) = Dx(xd (t))ẍd (t) + Cx(xd (t), ẋd(t))xd (t) + Gx(xd (t)) +τa(t)

e(t) = xd (t) − x(t)
The term n(ëi, ėi,ei, t) contains the higher order terms ëi(t), ėi(t) and ei(t) and it can be negligible. The

control problem is to find a control law so that the end-effector position x(t) can track specific commands
xd (t). We construct controller as follows:

Fi = Fie + H
i (4)

Where the first term Fie = K
i
p(e

i(t)) + Kid (ė
i(t)) is feedback PD control law with the following gain

switching rule in [8]:

Ki+1p = β(i)K
i
p, K

i+1
d = β(i)K

i
d, β(i + 1) >β(i), i = 0,1,2, ..., N (5)

With Kip, K
i
d are the initial proportional and derivative control gain matrices that diagonal positive

definite, Ki+1p , are the control gains of the ith iterative. β(i) > 1 is the gain switching factor. The
gains adaptive law in (5) are used to adjust the PD gains from iterative to iterative. And Hi is the ini-
tial predicted feedforward control input to be computed at each iterative by a learning rule. According
demonstrated in [8], the feedback PD control law with the gain switching factor in (5) plus the feed-
forward learning control law with the input force profile, the convergence of system (3) is guaranteed.
However, in order to, the trajectory tracking convergence fast in some initial iterative, we cannot increase
the switching factor arbitrarily large because actuator forces are limited, especially when the system has
modelling errors or nonlinearity. Hence, according [12] to deal this problem, we propose feedforward
control input Hi(t) with a learning rule so that Hi(t) converges to R(t) for all t ∈ [0, t f ] as follow:

Hi+1 = Hi +αFie (6)



An Adaptive Iterative Learning Control for Robot Manipulator in Task Space 521

Figure 2: Schematic diagram of the learning control scheme.

At the initial stage of learning, the Hi(t) are set to zero. α is a positive constant often called a training
factor. Therefore, for the ith and (i + 1)th iterations, applying the input (4), (6) to system (3), we obtain
an error equation as follows:

D(t)ëi(t) + [C(t) + C1(t)]ė
i(t) + R(t)ei(t) = S (t) − Fie − H

i (7)

D(t)ëi+1(t) + [C(t) + C1(t)]ė
i+1(t) + R(t)ei+1(t) = S (t) − Fi+1e − H

i −αFie (8)

To simplicity the proof of stability, let Kip = aK
i
d for the initial iteration, and define the filter errors

as follow:
x̃i(t) = ėi(t) + aei(t) (9)

Also, define δx̃i = x̃i+1 − x̃i and δei = ei+1 − ei. Then, from (9):

δx̃i = δėi + aδei (10)

From (5)-(9) and (10), one can obtain the following equation:

Dδ ˙̃xi + (C + C1 − aD + Ki+1d )δx̃
i
+ (R − a(C + C1 − aD))δei =−(Ki+1d + (α− 1)K

i
d )x̃

i (11)

The following theorem can be proved.
Theorem: Suppose robot system (2) satisfies property (1, 2) and assumption (1, 2). With the control

input (4), the gain switching rule (5) and learning rule (6). The following should hold for all t ∈ [0, t f ].

xi(t) → xd (t), ẋi(t) → ẋd (t), i →∞

If the controller gains are selected so that the following relationships hold:

lp = λmin((2 −α)Kid + 2C1 − 2aD) > 0 (12)

lr = λmin((2 −α)Kid + 2C + 2R/a − 2Ċ1/a) > 0 (13)

lplr ≥ ∥R/a − (C + C1 − aD)∥2max (14)



522 T. Ngo, Y. Wang, T.L. Mai, J. Ge, M.H. Nguyen, S.N. Wei

Where λmin(A) is the minimum eigenvalue of matrix A, and ∥M∥max = max∥M(t)∥ for t ∈ [0, t f ].
Here, ∥M∥ represents the Euclidean norm of M.

Proof: We select a performance index V i(t) as follow:

V i =

t∫
0

e−ρτ(x̃i)T Qx̃idτ≥ 0 (15)

Thus, β(i) > 1 according (5) so we have Ki+1e > K
i
d and α is a positive constant, so Q = K

i+1
d + (α−

1)Kid) > 0. From the definition of V
j, for the (i + 1)th iteration, one can get:

V i+1 =

t∫
0

e−ρτ(x̃i+1)T Qx̃i+1dτ (16)

Let ∆V i = V i+1 − V i then from (15), (16) and (11), we obtain:

∆V i =

t∫
0

e−ρτ((δx̃i)T Qδx̃i + 2(δx̃i)T Qx̃i)dτ

=

t∫
0

e−ρτ((δx̃i)T Qδx̃i + 2(δx̃i)T Qx̃i)dτ

=

t∫
0

e−ρτ(δx̃i)T Qδx̃idτ− 2
t∫

0

e−ρτ(δx̃i)T Dδ ˙̃xidτ

− 2
t∫

0

e−ρτ(δx̃i)T ((C + C1 − aD + Ki+1d )δx̃
i
+ (R − a(C + C1 − aD))δei)dτ

(17)

Applying the partial integration, from assumption 2, and property 2, we have:
t∫

0

e−ρτ(δx̃i)T Dδ ˙̃xidτ= e−ρτ(δx̃i)
T

D(δx̃i)
∣∣∣∣t
0
−

t∫
0

(e−ρτ(δx̃i)T D)′δx̃idτ

= e−ρτ(δx̃i)T (t)D(t)δx̃i(t) +ρ

t∫
0

e−ρτ(δx̃i)T Dδx̃idτ

−
t∫

0

e−ρτ(δx̃i)T Dδ ˙̃xidτ− 2
t∫

0

e−ρτ(δx̃i)T Cδx̃idτ

(18)

Substituting (18) into (17), and Q = Ki+1d + (α− 1)K
i
d yields:

∆V i =−e−ρτ(δx̃i)T (t)D(t)δx̃i(t) −ρ
t∫

0

e−ρτ(δx̃i)T Dδx̃idτ

− 2
t∫

0

e−ρτ(δx̃i)T (R − a(C + C1 − aD))δeidτ

−
t∫

0

e−ρτ(δx̃i)T (Ki+1d − (α− 1)K
i
d + 2C1 − 2aD)δx̃

idτ

(19)



An Adaptive Iterative Learning Control for Robot Manipulator in Task Space 523

From (5), we have:

t∫
0

e−ρτ(δx̃i)T Ki+1d δx̃
idτ= β(i + 1)

t∫
0

e−ρτ(δx̃i)T Kidδx̃
idτ ≥

t∫
0

e−ρτ(δx̃i)T Kidδx̃
idτ (20)

Substituting (10) into (19) and noticing (20), we obtain:

∆V i ≤− e−ρτ(δx̃i)T (t)D(t)δx̃i(t) −ρ
t∫

0

e−ρτ(δx̃i)T Dδx̃idτ

−
t∫

0

e−ρτ(δėi)T ((2 −α)Kid + 2C1 − 2aD)δė
idτ

− 2a
t∫

0

e−ρτ(δei)T ((2 −α)Kid + 2C1 − 2aD)δė
idτ

− 2
t∫

0

e−ρτ(δėi)T (R − a(C + C1 − aD))δeidτ

− a2
t∫

0

e−ρτ(δei)T ((2 −α)Kid + 2C1 − 2aD)δe
idτ

− 2a
t∫

0

e−ρτ(δei)T (R − a(C + C1 − aD))δeidτ

(21)

Applying the partial integration again gives:

t∫
0

e−ρτ(δei)T ((2 −α)Kid + 2C1 − 2aD)δė
idτ= e−ρτ(δei)

T
((2 −α)Kid + 2C1 − 2aD)(δe

i)
∣∣∣∣t
0

+ρ

t∫
0

e−ρτ(δei)T ((2 −α)Kid + 2C1 − 2aD)δe
idτ

−
t∫

0

e−ρτ(δėi)T ((2 −α)Kid + 2C1 − 2aD)δe
idτ

+ 2

t∫
0

e−ρτ(δei)T (aḊ − Ċ1)δeidτ

(22)



524 T. Ngo, Y. Wang, T.L. Mai, J. Ge, M.H. Nguyen, S.N. Wei

Therefore

∆V i ≤− e−ρτ(δx̃i)T Dδx̃i −ρ
t∫

0

e−ρτ(δx̃i)T Dδx̃idτ

− ae−ρτ(δei)T ((2 −α)Kid + 2C1 − 2aD)δe
i

−ρa
t∫

0

e−ρτ(δei)T ((2 −α)Kid + 2C1 − 2aD)δe
idτ−

t∫
0

e−ρτwdτ

≤− e−ρτ(δx̃i)T Dδx̃i − ae−ρτ(δei)T lpδei −ρ
t∫

0

e−ρτ(δx̃i)T Dδx̃idτ

−ρa
t∫

0

e−ρτ(δei)T lpδe
idτ−

t∫
0

e−ρτwdτ

(23)

Where

w=(δėi)T ((2 −α)Kid + 2C1 − 2aD)δė
i
+ 2a(δėi)T (R/a − (C + C1 − aD))δei

+ a2(δei)T ((2 −α)Kid + 2R/a + 2C − 2Ċ1/a)δe
i

(24)

Let P = R/a − (C + C1 − aD). Then from (12) and (13), we obtain

w≥ lp∥δė∥2 + 2aδėT Pδe + a2lr∥δe∥2 (25)

Applying the Cauchy-Schwartz inequality, we obtain:

δėT Pδe ≥−∥δė∥∥P∥max ∥δe∥ (26)

From (12)-(14)

w= lp∥δė∥2 − 2a∥δė∥∥P∥max ∥δe∥+ a2lr∥δe∥2

= lp(∥δė∥−
a
lp
∥P∥max ∥δe∥)2 + a2(lp −

1
lr
∥P∥2max)∥δe∥

2

≥ 0

(27)

According property 1 and (27), based on (23), it can be ensured that ∆V i ≤ 0, therefore V i+1 ≤ V i.
From the definition, Kid is a positive definite matrix. From the definition of V

i, V i ≥ 0 and V i is bounded.
As a result, yi(t) → ∞ when i → ∞. Because ei(t) and ėi(t) are two independent variables, and a is
a positive constant. Thus, if i → ∞, ei(t) → 0 and ėi(t) → 0 for t ∈ [0, t f ]. Finally, the following
conclusions hold for t ∈ [0, t f ].

xi(t) → xd (t), ẋi(t) → ẋd (t), i →∞

From the above analysis it can be seen that the adaptive PD control method can guarantee that the
tracking errors converge arbitrarily close to zero as the number of iterations increases. The following
case studies based on simulation will demonstrate this conclusion.



An Adaptive Iterative Learning Control for Robot Manipulator in Task Space 525

4 Numerical Simulation

A two-link robot manipulator as shown in Fig.1 is utilized in this paper to verify the effectiveness of
the proposed control scheme. The dynamic equation of the robot manipulator is adopted in [13].

D(q) =
m1 + m2 + 2m3 cos(q2) m2 + m3 cos(q2)

m2 + m3 cos(q2) m2


C(q, q̇) =

−m3q̇2 sin(q2) −m3(q̇1 + q̇2) sin(q2)
m3q̇1 sin(q2) 0


G(q) =

m4gcos(q1) + m5gcos(q1 + q2)
m5gcos(q1 + q2)


and mi are the parameters of interest given by M = P + pl L with

M =
[
m1 m2 m3 m4 m5

]T
P =

[
p1 p2 p3 p4 p5

]T
L =

[
l21 l

2
2 l1l2 l1 l2

]T
and pl is the payload, ł1 = 1(m) and ł2 = 1(m) are the lengths of link 1 and link 2, respectively, P is the
parameter vector of the robot itself. g = 9.8 m/s2. The jacobian matrix is known as:

J(q) =
−l1 sin(q1) − l2 sin(q1 + q2) −l2 sin(q1 + q2)

l1 cos(q1) + l2 cos(q1 + q2) l2 cos(q1 + q2)


For the convenience of the simulation, the nominal parameters of the robotic system are given as:

P =
[
1.66 0.42 0.63 3.75 1.25

]T
kg.m2

The desired reference trajectories in the Cartesian space are:

xd1 = 1.0 + 0.2 cos(πt), xd2 = 1.0 + 0.2 sin(πt).

Which represent a circle of radius 0.2m and its centres is located at (x1, x2) = (1.0,1.0)m. The robot
is initially rested with its end-effector positioned at the center of the circle. With initial condition are
x1(0) = xd1(0) = 1.2, x2(0) = xd2(0) = 1.0, ẋ1(0) = ẋd1(0) = 1.0, ẋ2(0) = ẋd2(0) = 1.2 and t = [0,2].

The most important parameters that effect the control performance of the robotic system are the
external disturbance τa(t), the friction term f (q̇), in simulation, payload variation situation and external
disturbance situation occurring at fifth the iteration are considered. The payload variation situation is
that pl = 0(kg) from first the iteration to fourth the iteration, and then it was put on pl = 3 after the fifth
iteration. The disturbance situation is that external forces are injected into the robotic system, and their
shapes are expressed as follows:

τa (t) =
[
5 sin(5t) 5 sin(5t)

]T
(28)

In addition, friction forces are also considered in this simulation and given as:

f (q̇) =
[
2q̇1 + 0.8sgn(q̇1) 4q̇2 + 0.1sgn(q̇2)

]T
(29)



526 T. Ngo, Y. Wang, T.L. Mai, J. Ge, M.H. Nguyen, S.N. Wei

0.6 0.8 1 1.2 1.41.4
0.4

0.6

0.8

1

1.2

1.4

 Position X1 (m)

P
os

it
io

n 
X

2 
(m

)

 

 
Actual Desired

0.6 0.8 1 1.2 1.41.4
0.6

0.8

1

1.2

1.41.4

 Position X1 (m)

P
os

it
io

n 
X

2 
(m

)

 

 
Actual Desired

0 5 10 15
0

0.2

0.4

0.6

0.8

1

Iteration Number

M
ax

im
um

 E
rr

or
s 

(m
)

 

 
RMS of X2 RMS of X1

Figure 3: Reference tracking and actual trajectory for end-effector at different iterative under AILC is
adopted in [11]. (a) Position tracking of end-effector at first iteration, (b) Position tracking of end-effector
after fifteenth iteration and (c) profile of rms position errors payload and disturbance variation after the
fifth iteration.

In order to exhibit the superior control performance of AILC system, two extra control systems
including an adaptive switching learning PD control system (ASL-PD) is represented in [8], an iterative
learning control (ILC) is represented in [12].

In three simulation situations, The PD control gain was set to be the same as [8]:

Kip = K
i
d = diag {30,30} , K

i+1
p = 2iK

i
p, K

i+1
d = 2iK

i
d, β= 0.75

The simulated results of ILC system, the responses of end-effector position at first and fifteenth
iteration and tracking error from iteration to iteration to be depicted Fig. 3(a), (b) and (c) respectively.
From the simulated results, we can be seen that the tracking performance was not acceptable because the
errors were too large compared to ASL-PD and proposed controllers, especially at the fifth iteration poor
tracking errors responses are resulted due to the occurrence of payload variation situation and external
disturbance. In the ASL-PD system are depicted in [8]. The end-effector position responses, tracking
error from iteration to iteration to be depicted in Fig. 4(a), 4(b), and 4(c), respectively. The end-effector
tracking performance is obvious under the occurrence of payload variation and external disturbance. The
convergence rate increased greatly compared with the ILC control method.

Now, the AILC system depicted in Fig. 2 is applied to control the robot manipulator for comparison.
The simulated results of end-effector position responses and tracking error from iteration to iteration
are depicted in Fig. 5(a), 5(b) and 5(c), respectively. Table 1 shows tracking performance of proposed
system from the initial iteration to fifteenth are obvious. Therefore, the comparison of their method and
our method demonstrated a bit fast convergence rate with the proposed control method. Specially, control



An Adaptive Iterative Learning Control for Robot Manipulator in Task Space 527

Iterative 1 5 10 15

ILC
max |ei1|(m) 0.0822 0.0773 0.0168 0.0140
max |ei2|(m) 0.1998 0.5634 0.0130 0.0105

ASL-PD
max |ei1|(m) 0.0398 0.0076 0.0012 0.0010
max |ei2|(m) 0.0646 0.0593 0.0008 0.0006

AILC
max |ei1|(m) 0.0377 0.0076 0.0011 0.0009
max |ei2|(m) 0.1741 0.0590 0.0008 0.0006

Table 1: Maximum tracking errors from iteration to iteration.

Iterative 1 5 10 15

ASL-PD
max |τi1|(N.m) 90 930 3330 7230
max |τi2|(N.m) 51.45 531.6 1903 3618

AILC
max |τi1|(N.m) 82.50 772.5 2647 5647
max |τi2|(N.m) 47.14 441.6 1513 3228

Table 2: Maximum control forces from iteration to iteration.

torques of proposed system is much smaller than the ASL-PD controller. Detail results are depicted in
table 2.

5 Conclusions

This paper has successfully implemented an AILC scheme to control the position of end-effector
in task space for achieving desired position control. All the system dynamics may be unknown. A
new control method is a combination of advantages some other method into a hybrid one as explained
above. By using Lyapunov theorem, the asymptotic convergence of the closed-loop control system can be
ensured whether or not the uncertainties occur. Simulation results of a two link robot manipulator in task
space via various existing control methods including ASL-PD and ILC control were also applied in this
paper to compare and display the manipulative performance of the proposed control system. According
to the result as depict in Figs. 3-5 and table 1-2, the desired position tracking and tracking errors response
of the AILC scheme decrease with the increase of the iteration number under wide range of payload and
external disturbance.

The main of the paper is to construct a simple scheme, easy implementation, fast convergence. Es-
pecially, in this paper is applied not only to control the position of the end-effector, but also the force
exerted by the end-effector on the object. By designing the control law in task space, force control can
be easily formulated.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (60775047; 60835004),
the National High Technology Research and Development Program of China (863 Program) (2007AA04Z244;
2008AA04Z214).

The authors would like to thank the associate editor and the reviewers for their valuable comments.



528 T. Ngo, Y. Wang, T.L. Mai, J. Ge, M.H. Nguyen, S.N. Wei

0.6 0.8 1 1.2 1.41.4
0.6

0.8

1

1.2

1.41.4

 Position X1 (m)

P
os

it
io

n 
X

2 
(m

)

 

 
Actual Desired

0.6 0.8 1 1.2 1.41.4
0.6

0.8

1

1.2

1.4

 Position X1 (m)

P
os

it
io

n 
X

2 
(m

)

 

 
Actual Desired

0.6 0.8 1 1.2 1.41.4

0.7

0.8

0.9

1

1.1

1.2

1.3

 Position X1 (m)

P
os

it
io

n 
X

2 
(m

)

 

 
Actual Desired

0.6 0.8 1 1.2 1.41.4

0.8

1

1.21.2

 Position X1 (m)

P
os

it
io

n 
X

2 
(m

)

 

 
Actual Desired

0 5 10 15
0

0.2

0.4

0.6

0.8

1

Iteration Number

M
ax

im
um

 E
rr

or
s 

(m
)

 

 
RMS of X2 RMS of X1

Figure 4: Reference tracking and actual trajectory
for end-effector at different iterative under ASL-
PD is adopted in [6]. (a) Position tracking of end-
effector at first iteration, (b) Position tracking of
end-effector after fifteenth iteration and (c) pro-
file of rms position errors payload and disturbance
variation after the fifth iteration.

0 5 10 15
0

0.2

0.4

0.6

0.8

1

Iteration Number

M
ax

im
um

 E
rr

or
s 

(m
)

 

 
RMS of X2 RMS of X1

Figure 5: Reference tracking and actual trajectory
for end-effector at different iterative under AILC
controller. (a) Position tracking of end-effector at
first iteration, (b) Position tracking of end-effector
after fifteenth iteration and (c) profile of rms posi-
tion errors payload and disturbance variation after
the fifth iteration.



An Adaptive Iterative Learning Control for Robot Manipulator in Task Space 529

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