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J Control Theory Appl 2010 8 (2) 160–165

DOI 10.1007/s11768-010-0010-2

Adaptive learning tracking control of robotic

manipulators with uncertainties

Rui YAN, Keng Peng TEE, Haizhou LI

(

Institute for Infocomm Research, 1 Fusionopolis Way, #21-01 Connexis, Singapore 138632)

Abstract: An adaptive learning tracking control scheme is developed for robotic manipulators by a synthesis of

adaptive control and learning control approaches. The proposed controller possesses both adaptive and learning properties

and thereby is able to handle robotic systems with both time-varying periodic uncertainties and time invariant parame-

ters. Theoretical proofs are established to show that proposed controllers ensure asymptotical tracking performance. The

effectiveness of the proposed approaches is validated through extensive numerical simulation results.

Keywords: Adaptive control; Learning control; Robotic dynamic systems; Uncertainties

1 Introduction

Robotic manipulators are required to perform various

tasks, and hence the robotic system has many uncertain-

ties, such as external disturbances, parameter uncertainty

and sensor errors. It has been shown in [1] and [2] that the

simple controllers such as the PD and PID feedback are ef-

fective for setpoint control despite the nonlinearity and un-

certainty of the robotic system. Moreover, to deal with the

trajectory tracking problem of robotic systems in the pres-

ence of uncertainties, many kinds of control schemes in-

cluding adaptive control and robust control have been pro-

posed [3∼9].

By exploring the physical properties of the robotic sys-

tem, adaptive control of robotic manipulators has been ex-

tensively studied by using Lyapunov synthesis as the main

tool to handle the constant uncertainty [3∼5]. Robust con-

trol strategies are well used for the control of highly non-

linear and uncertain systems [6]. In most cases, it is enough

to know the upper bounds of the uncertainties, regardless

of whether they are constant parameters, time-varying dis-

turbances or non-linear functions of system state variables.

Furthermore, for non-linear uncertainties of system state

variables, adaptive neural-network control has also been ex-

tensively investigated by using the approximation capabil-

ity of neural networks without the knowledge of the upper

bounds of the uncertainties in [7] and [8].

Learning control aims at improving the system perfor-

mance via directly updating the control input, either repeat-

edly over a ﬁxed ﬁnite time interval, or repetitively (cycli-

cally) over an inﬁnite time interval. Recently, there have

been some works in combining adaptive control and learn-

ing control techniques to solve time-varying uncertainties

in the robotic system model [9∼11]. However, the above

researches have assumed that the robotic systems perform

the same tasks over a speciﬁed time interval. In order to re-

ject periodic uncertainties, a robust learning control scheme

was developed to deal with the repeated trajectory or peri-

odic trajectory in [12]. In this work, a new controller is pro-

posed to deal with periodic uncertainties for any trajectory.

The classical adaptive updating law and the periodic learn-

ing law are used jointly for robotic systems with both time-

varying and time invariant parameters. Generally speaking,

the classical adaptive updating law does not work for time

varying parameters. The periodic learning control law, on

the other hand, does not perform as well as the classical

adaptive updating law for time invariant parameters due to

smoothness problem.

A Lyapunov-Krasovskii functional is constructed to

achieve the classical adaptive updating mechanism and the

periodic learning mechanism of multiple periods. This pa-

per is organized as follows. Section 2 gives robot dynam-

ics and kinematics. The adaptive learning tracking control

scheme is presented in Section 3. Section 4 illustrates a sim-

ulation example. The conclusion is given in Section 5.

2 Robot dynamics and kinematics

2.1 Robot dynamics

The dynamics of rigid-body robots with n degrees of

freedom can be expressed by the following general second-

order differential equation:

D(q)¨q + C(q, ˙q)˙q + G(q)+τ

= τ, (1)

where q ∈ R

is the vector of generalized coordinates,

D(q) ∈ R

n×n

is the inertia matrix, C(q, ˙q) ∈ R

n×n

the matrix of Coriolis and centrifugal force, G(q) ∈ R

the gravity vector, τ ∈ R

is the vector of input torques and

∈ R

is the vector of unknown bounded disturbances.

Several important properties of the dynamic system de-

scribed by (1) are given as follows [2, 4].

Property 1 The inertia matrix D(q) is symmetric and

uniformly positive deﬁnite for all q ∈ R

Property 2 The matrix

D(q) − 2C(q, ˙q) is skew-

symmetric so that ν

(

D(q) − 2C(q, ˙q))ν =0for all

ν ∈ R

Property 3 The dynamic model as described by (1) is

linear-in-the parameters, is written as [3, 4, 7]

D(q)¨q + C(q, ˙q)˙q + G(q)=ξ(q, ˙q, ¨q)θ, (2)

Received 12 January 2010.

 South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2010

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