Page 1

Published in IET Control Theory and Applications

Received on 12th July 2010

Revised on 11th May 2011

doi:10.1049/iet-cta.2010.0389

ISSN 1751-8644

Friction and uncertainty compensation of robot

manipulator using optimal recurrent cerebellar

model articulation controller and elasto-plastic

friction observer

S.I. Han J.M. Lee

School of Electrical Engineering, Pusan National University, Busan, Republic of Korea

E-mail: jmlee@pusan.ac.kr

Abstract: A model-free control scheme with the elasto-plastic friction observer is presented for robust and high-precision

positioning of a robot manipulator. The traditional model-based adaptive controller requires information on the robotic

dynamics in advance and thus undergoes robustness problem because of complex dynamics and non-linear frictions of a

robot system. This problem is overcome by an employed model-free recurrent cerebellar model articulation controller

(RCMAC) system and friction estimator for friction and uncertainty compensation of a robot manipulator. The adaptive laws

of the RCMAC networks to approximate an ideal equivalent sliding mode control law and adaptive friction estimation laws

based on the elasto-plastic friction model are derived based on the Lyapunov stability analysis. To guarantee stability and

increase convergence speed of the RCMAC network, the optimal learning rates are obtained by the fully informed particle

swarm (FIPS) algorithm. The robust positioning performance of the proposed control scheme is verified by simulation and

experiment for the Scorbot robot in the presence of the joint dynamic friction and uncertainty.

1 Introduction

In recent years, the compensation of the joint friction and

uncertainty has become important issue to maintain precise

low-speed motion of a robot manipulator and adaptability of

the control scheme for several advanced applications, from

industrial robots to medical robots. Friction modelling and

compensation of it are particularly important issues in a

robotic manipulator because of the influence of friction on

system performance and stability. Hence, without resorting to

high control gains, friction forces need to be identified and

compensated to improve the transient performance and

reduce the steady-state tracking errors. However, an effective

compensation of friction still remains as a challenging

research subject, especially for very slow motions because

friction often reveals complex behaviours by several working

conditions. Thus, it is not easy work to build an elaborate

model that can embody all such friction characteristics.

To develop further advanced friction model which can

describereal physicalfriction

several elaborate friction models have been studied, such as

Dahl’s model [1], the LuGre model [2], the elasto-plastic

model [3, 4], the Leuven model [5] and the generalised

Maxwell-Slip (GMS) model [6]. The Leuven and GMS

models provide relatively accurate reproducibility of friction

behaviour, especially in the pre-sliding region than other

friction models.However,

considerable identification effort because of their complex

phenomenaaccurately,

thesemodelsmaypay

expressions. Furthermore, the control schemes based on

these models are even more complicated at the design level

and during implementation. On the other hand, the LuGre

friction model is an extension of Dahl’s model that can

provide representations of Stribeck friction, viscous friction,

rising static friction and friction memory during slip. It has

a single-state mathematical structure, so it can be easily

implemented intomany

limitation of this model, however, is that it exhibits drift

phenomenon subjected to an arbitrarily small bias force and

arbitrarilysmall vibration.

experimental results show that this drift is spurious. It is

shown that drift is due to the fact that pre-sliding

displacement in the LuGre model always includes a plastic

component. To minimise drift, a class of single-state models

are defined in which pre-sliding is elasto-plastic, that is,

under loading the displacement is first purely elastic before

transitioning to plastic.

The piecewise continuous function defined in [3, 4] to

achieve elasto-plastic property plays a significant role in the

elasto-plastic friction model. In real situations, the accurate

estimation of this function’s value is generally difficult and

laborious and repetitive experiments for identifying it may

be required. To the best of our knowledge, the friction

observers based on the elasto-plastic model have never

reported except for the discrete-time estimation method [4].

Thus, in this paper, we will employ the estimation laws for

friction state and parameters based on the elasto-plastic

applications.A significant

Mostofthe practical

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IET Control Theory Appl., 2011, Vol. 5, Iss. 18, pp. 2120–2141

doi:10.1049/iet-cta.2010.0389

www.ietdl.org

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model. For the LuGre friction model, to estimate the friction

parameters, several friction observers [7–10] are developed,

but these observershave

implementation. A developed sliding mode-based friction

parameter observer in this paper has a simple structure and

no limitation when compared with the previous observers

proposed by Canudas de Wit and Lischinsky [7] and Xie [10].

Friction compensation of robots has been reviewed in

[11–15]. The decomposition-based control approach was

proposed using the Stribeck friction model in [11, 12].

The robust compliant motion control was presented with

consideration of Coulomb friction and stiction in [13]. The

robust fuzzy friction compensation with a local friction

observer was studied in [14]. Friction identification and

compensation for the joint friction control of robots using

the LuGre model were carried out by simple control

schemes[15].Most ofthese

schemes are based on a model-based adaptive control

method or the LuGre model, which has weakness to the

uncertaintiesof robotsbecause

modelling of manipulator dynamics, friction and external

unknown disturbances as well as drift of the LuGre

friction model.

Recently, artificial intelligent methods such as neural

networks, fuzzy logic and genetic algorithm have been

developed as model-free controls since, in general, the

conventional model-based robust control often fails to

achieve satisfactory performance for a system with system

dynamics that is difficult to model or have severe time-

varying non-linearities [16–18]. These artificial intelligent

methods utilise human brain learning and logic processes

and approximate the unknown system dynamics or the

ideal controller after learning to certain accuracy without

the need for information about the system dynamics. The

cerebellarmodel articulation

classified asa non-fully

associative memory network with overlapping receptive

fields and have been developed in the literature [19–21].

The CMAC to approximate a non-linear function over a

domain of interest to any desired accuracy has been

already validated. To utilise dynamic mapping, recurrent

CMAC (RCMAC) has been developed [22–26], which

includesthe delayedself-recurrent

memory space. To achieve highly accurate approximation,

the conventionalRCMAC [22–26]

adding feedforward weighing path to them.

It is well known that the SMC is a powerful robust scheme

to control non-linear systems with uncertainties [27, 28]. The

representative features of the SMC are its insensitivity to

system parameter variations, external disturbances and fast

dynamic responses. On the other hand, the SMC has

additional favourable features such as the relatively simple

controller structure and easier implementation than other

control schemes. Based on these features, the sliding mode-

based RCMAC (SRCMAC) system, which is developed in

this paper, combines the advantages of the SMC with

robust characteristics and the RCMAC with on-line learning

ability. Hence, the stability, convergence and robustness of

the control system can be enhanced.

The learning rates of the RCMAC network are the

important factor that influences the performance of the

SRCMAC system. At a low learning rate, convergence is

easily guaranteed but the learning speed is apt to be much

slower. On the contrary, an excessively high learning rate

may cause a stability problem. To optimise the learning

rates of the RCMAC network, particle swarm optimisation

some limitationsfor

frictioncompensation

of their incomplete

controller

connected

(CMAC)

perceptron-like

are

unit association

aremodified by

(PSO) [29] is employed. The PSO was developed based

on the simulation of animal social behaviours such as bird

flocking and fish schooling. Unlike a genetic algorithm,

the PSO has no complicated evolutionary operators, such

as thecrossover andmutation.

particularly useful for solving optimisation problems with

a large number of variables. From several PSO algorithms,

the fully informed particle swarm (FIPS) algorithm [30] is

employed to optimise the learning rates of the RCMAC

network.

Summarising the above statements, in this paper, we will

develop an improved SRCMAC control system combined

with a friction observer and FIPS algorithm (SRCFO_FIPS)

for precise joint tracking performance to the joint friction

and uncertainty of the robot manipulator. Some simulation

and experiment for the Scorbot robot system with non-

linear friction and uncertainty were carried out to evaluate

the proposed control scheme.

ThePSO can be

2

non-linear dynamic friction

Modelling of the robot manipulator and

2.1Robot manipulator dynamics

The dynamic equation of an n rigid-link robot system

with frictional joints is described as the following Lagrange

form

M(q)¨ q + C(q, ˙ q)˙ q + G(q) + Tf(q, ˙ q) = t

(1)

where q, ˙ q, ¨ q [ Rn×1denote the joint position, velocity and

acceleration vectors, respectively, the moment of inertia

matrix M(q) [ Rn×nis a positive-definite symmetric matrix,

C(q, ˙ q) [ Rn×n

isthe centripetal

˙M(q) − 2C(q, ˙ q) is a skew-symmetric matrix, G(q) [ Rn×1

is the gravity vector, Tf(q, ˙ q) [ Rn×1is the non-linear

friction torque vector and t [ Rn×1is the control torque

vector by the joint actuators. Considering the modelling

uncertainties and external disturbance, the robot system in

(1) can be reformulated as

Coriolismatrix,

¨ q = fn(q, ˙ q) + gn(q)[u − Tfn+ Tu] (2)

where the subscript n represents the system parameters in the

nominal condition,fn(q, ˙ q) = −gn(q)[Cn(q, ˙ q) + Gn(q)],

gn(q) = M−1

as Tu= DM(q)¨ q + DC(q, ˙ q) + DG(q) + DTf− TL, DM(q),

DC(q, ˙ q),

DG(q) and

DTf

uncertainties of M(q), C(q, ˙ q), G(q) and Tf, respectively,

and TL[ Rn×1is the disturbance vector. The uncertainties

of DM(q), DC(q, ˙ q), DG(q) and DTfare bounded by some

positive constants ri(i ¼ m, c, g, f ) such that ?DM? ≤ rm,

?DC? ≤ rc, ?DG? ≤ rgand ?DTf? ≤ rf. The disturbance is

assumed such that TL[ L2[0, T], ∀T [ [0, 1), and TLis

bounded by some positive constant rd: ?TL? ≤ rd. Thus,

the lumped uncertainty is assumed to be bounded with a

finitevalue.To guarantee

performance, an elaborate non-linear friction model should

be considered.

n(q); u ¼ t, Tuis a lumped uncertainty defined

representtheunknown

more improvedcontrol

IET Control Theory Appl., 2011, Vol. 5, Iss. 18, pp. 2120–2141

doi:10.1049/iet-cta.2010.0389

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2.2Elasto-plastic dynamic friction model

The friction state equation of the LuGre friction model is

˙ z =

˙ q1− s01h1(˙ q1)z1

⎢

⎢

...

˙ qi− s0ihi(˙ qi)zi

...

˙ qn− s0nhn(˙ qn)zn

⎡

⎢

⎢

⎢

⎣

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎥

(3)

where ziis the average defection vector of the elastic bristle,

s0i. 0 is the stiffness of elastic bristle and

hi(˙ qi) =

|˙ qi|

[Tci+ (Tsi− Tci)exp(−(˙ qi/˙ qsi)2)]

(4)

In general, the function hi(˙ q) . 0 depends on many factors

such as the material properties, lubrication and temperature,

and Tci is Coulomb friction, Tsi is stiction level, ˙ qsiis

Stribeck angular velocity. The dynamic friction torque

vector is described by

Tfn=

s01z1+ s11˙ z1+ s21˙ q1

...

s0izi+ s1i˙ zi+ s2i˙ qi

...

s0nzn+ s1n˙ zn+ s2n˙ qn

⎡

⎢

⎢

⎢

⎢

⎢

⎣

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎥

(5)

where s1i. 0 is the damping coefficient in elastic range and

s2i. 0 is the viscous friction coefficient. The LuGre friction

model captures many aspects of complex friction, but allows

no further modelling of the pre-sliding domain. In Figs. 1b

and d, it is shown that, when the LuGre friction model, a

drift is always produced when arbitrarily small forces or

torques are applied as shown in Fig. 1a. This behaviour is

not physically consistent, since in reality an object remains

situated in the presence of small loads as shown in Fig. 1f,

where this is the author’s previous experimental result

[31] for the torque input shown in Fig. 1a. Although the

LuGre friction model assumes that plastic and elastic

displacements are always combined, the elasto-plastic

friction model enables precise modelling of the pre-sliding

domain and the dynamic transition to sliding domain, where

the following formulation is used for the bristle displacement

˙ z =

˙ q1− a1(z1, ˙ q1)s01h1(˙ q1)z1

⎢

⎢

...

˙ qi− ai(zi, ˙ qi)s0ihi(˙ qi)zi

...

˙ qn− an(zn, ˙ qn)s0nhn(˙ qn)zn

⎡

⎢

⎢

⎢

⎣

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎥

(6)

The main difference with the LuGre friction model is

represented by the function ai(zi, ˙ qi): this is an adhesion

map that controls the rate of change of zi; specially, when

the bristle displacement zi is smaller than a given

breakaway displacement zbai, then ai(zi, ˙ qi) is set to zero

and consequently (6) reduces to ˙ zi= ˙ qi. This means that in

the range |zi| ≤ zbai, the model shows purely elastic pre-

sliding regime, and therefore does not exhibit drift as

shown in Figs. 1c and e. These results correspond to the

experimental result as shown in Fig. 1f. Note that the

LuGre friction model is a special case of (6) where

ai(zi, ˙ qi) = 1. The function ai(zi, ˙ qi) is parameterised as

⎧

⎪⎪⎩

ai(zi, ˙ qi)

0,

ami,

1,

0,

|zi| , zbai,

zbai, |zi| , zssi(˙ qi),

|zi| , zssi(˙ qi),

sgn(˙ qi) = sgn(zi)

sgn(˙ qi) = sgn(zi)

sgn(˙ qi) = sgn(zi)

sgn(˙ qi) = sgn(zi)

⎪⎪⎨

(7)

where zbai(0 , zbai, zssi(˙ qi)) is the breakaway displacement

below which the pre-sliding is purely elastic. ami(˙ zi, ˙ qi) is

indeed a function of ˙ qi and zi, which describes the

transition between elastic and plastic behaviour, as

ami(zi, ˙ qi) =1

2

1 + sin pzi− (zssi(˙ qi) + zbai)/2

zssi(˙ qi) − zbai

????

(8)

Following [3], zssi(˙ qi) is defined as

zssi(˙ qi) =sgn(˙ qi)

s0i

[Tci+ (Tsi− Tci)e−(˙ qi/qsi)2] (9)

Since the friction torque of the elasto-plastic model is equal to

that of the LuGre model, from (5) and (6), Tfncan be rewritten

as

Tfn=

s01z1− a1s31h1(˙ q1)z1+ s41˙ q1

...

s0izi− ais3ihi(˙ qi)zi+ s4i˙ qi

...

s0nzn− ans3nhn(˙ qn)zn+ s4n˙ qn

⎡

⎢

⎢

⎢

⎢

⎢

⎣

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎥

(10)

where s3i¼ s0i.s1i, s4i¼ s1i+ s2i. Since, however, the

statevariable

zi

cannot

ai(zi, ˙ qi) (0 ≤ ai(zi, ˙ qi) ≤ 1) is an important factor in the

elasto-plastic model, we suggest the friction state observer

to estimate zias follows

bemeasureddirectly and

˙ˆ z =

˙ q1− ˆ a1ˆ s01h1(˙ q1)ˆ z1

⎢

⎢

...

˙ qi− ˆ aiˆ s0ihi(˙ qi)ˆ zi

...

˙ qn− ˆ anˆ s0nhn(˙ qn)ˆ zn

⎡

⎢

⎢

⎢

⎣

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎥

(11)

where ˆ ziis the estimation of zi, ˆ ai(zi, ˙ qi) is the estimation of

ai(zi, ˙ qi) and

ˆ s0i is the estimation of s0i. Then, the

estimation error of the friction state variable is given from

(6) and (11) as follows

˙˜ z =

−a1s01h1(˙ q1)˜ z1− a1h1(˙ q1)ˆ z1˜ s01− ˆ s01h1(˙ q1)ˆ z1˜ a1

...

−ais0ihi(˙ qi)˜ zi− aihi(˙ qi)ˆ zi˜ s0i− ˆ s0ihi(˙ qi)ˆ zi˜ ai

...

−ans0nhn(˙ qn)˜ zn− anhn(˙ qn)ˆ zn˜ s0n− ˆ s0nhn(˙ qn)ˆ zn˜ an

⎡

⎢

⎢

⎢

⎢

⎢

⎣

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎥

(12)

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IET Control Theory Appl., 2011, Vol. 5, Iss. 18, pp. 2120–2141

doi:10.1049/iet-cta.2010.0389

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Fig. 1

a Applied input torque

b Angular displacement predicted by the LuGre model

c Angular displacement predicted by the elasto-plastic model

d Torque against displacement for the LuGre model

e Torque against displacement for the elasto-plastic model

f Experimental result in ball-screw system: torque against displacement

Simulated results of the LuGre and the elasto-plastic friction model and real experimental result

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doi:10.1049/iet-cta.2010.0389

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where ˜ zi= zi− ˆ zi, ˜ ai=ai− ˆ aiand ˜ s0i= s0i− ˆ s0i. Thus,

the estimation of friction Tfnis given as

ˆTfn=

ˆ s01ˆ z1− ˆ a1ˆ s31h1(˙ q1)ˆ z1+ ˆ s41˙ q1

⎢

⎢

...

ˆ s0iˆ zi− ˆ aiˆ s3ihi(˙ qi)ˆ zi+ ˆ s4i˙ qi

...

ˆ s0nˆ zn− ˆ anˆ s3nhn(˙ qn)ˆ zn+ ˆ s4n˙ qn

⎡

⎢

⎢

⎢

⎣

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎥

(13)

where ˆ s3iand ˆ s4iare estimations of s3iand s4i, respectively.

In the elasto-plastic model, ai(zi, ˙ qi) is a very important

factor, but determining it experimentally is difficult.

Therefore the estimation procedure of it as well as other

friction parameters should be considered.

2.3Sliding mode control

The control objective for a robot manipulator is to determine

an intelligent adaptive control system such that the system

output vector q can track a desired joint trajectory qd, which

is a given continuously differentiable and uniformly

bounded in joint space, and the tracking error e ¼ qd2 q

should be kept as small as possible.

To achieve our control objective, we define a sliding

surface in the space of the error vector

S =

s1(e1)

...

si(ei)

...

sn(en)

⎡

⎢

⎢

⎢

⎢

⎢

⎣

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎥

=

c1e1+ ˙ e1

...

ciei+ ˙ ei

...

cnen+ ˙ en

⎡

⎢

⎢

⎢

⎢

⎢

⎣

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎥

= LTE (14)

where ciare positive constants, ei¼ qdi2 qi(i ¼ 1, ... , n),

L ¼ [C I]T, E = [e

˙ e]Tand

C =

c1

...

0

...

0

···

..

···

···

···

0

...

ci

···

···

···

..

···

0

...

0

...

cn

.

···

0

.

⎡

⎢

⎢

⎢

⎢

⎢

⎣

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎥

Thus

˙E = VE + B[¨ qd− fn(q, ˙ q) − gn(q)(u − Tfn+ Tu)] (15)

where V =

reaching phase since the function S ¼ 0 when t ¼ 0. The

tracking control problem is to choose a control law u such

that the system state is driven to the sliding surface S ¼ 0 for

all t . 0 regardless of friction and the lumped uncertainty. In

order to design this sliding mode controller (SMC), we first

derive the equivalent control law ueq, which will determine

the dynamic of the system behaviour on the sliding surface.

0

0

I

0

??

, B = 0

I

??T. Note that there is no

The equivalent control law is derived by setting

˙S|u=ueq=

˙ s1

...

˙ si

...

˙ sn

⎡

⎢

⎢

⎢

⎢

⎢

⎣

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎥

= 0

(16)

which leads to

ueq= g−1

n(q)[¨ qd− fn(q, ˙ q) − gn(q)Tu+ LTVE] (17)

We consider the friction and unified smooth control law to

avoid excessive chattering in control input and dispel the

lumped system uncertainty as

us= g−1

uf= Tfn

n(q)gS (18)

(19)

where g ¼ diag(g1, ...,gn) . 0 is a diagonal constant matrix.

Then, the ideal control law is given as

uid= ueq+ us+ uf

(20)

Taking the time derivative of both sides of (14) and using (14)

and (20), we obtain

˙S = LTVE + LTB[¨ qd− fn(q, ˙ q) − gn(q)(u − Tfn+ Tu)]

= −gS (21)

We define a Lyapunov function candidate

V1=1

2STS (22)

Considering (21), its time derivative with (22) is

˙V1= ST˙S

= −g?S?2≤ 0(23)

Thus, the SMC control input introduced in (20) can guarantee

the stability of the control system. By properly choosing the

values of g, we can easily design the system performances

such as rise time, overshoot and settling time. However,

if the system parameters are unknown or perturbed, the

equivalent control of (17) cannot guarantee the performance

and stability of the controlled system. Considering the above

estimation of the friction, the ideal equivalent control law can

be changed as

uid= g−1

n(q)[¨ qd− fn(q, ˙ q) − gn(q)Tu+ KTAE]

+ g−1

n(q)gS +ˆTfn= ueq+ us+ ˆ uf

(24)

where ˆ uf=ˆTfn. Substituting (24) into (2) yields the following

result

˙S = −gS −˜Tfn

(25)

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where

˜ s0i− aihi(˙ qi)ˆ zi˜ s3+ ˙ qi˜ s4i− ˆ s3ihi(˙ qi)ˆ zi˜ ai,

s0i− ais3ihi(˙ qi). Then, assuming that the friction torque is

well estimated, that is, limt?1˜Tfn= 0 in (26) by using some

friction estimation, then the stability condition of (23) is

satisfied.

˜Tfn= Tfn−ˆTfn, ˜Tfni= Tfni−ˆTfni= Gi(˙ qi)˜ zi+ ˆ zi

Gi(˙ qi) and =

3

parameter estimation

RCMAC-based approximation and friction

However, the ideal controller in (24) cannot be implemented

for practical applications since f(q, ˙ q) and g(q) cannot be

exactly obtained in general and the lumped uncertainty Tu

is unknown in advance. Thus, the stability of (23) cannot

be really satisfied. In this paper, we propose the intelligent

controller system toachieve

performance robust to uncertainty. This type of controller

consists of an RCMAC network, a friction observer and

robust compensator suggested by

thedesired tracking

usrc= urc+ us+ ˆ uf+ ur

(26)

where urc is used to imitate controller ueq, the robust

compensatorur

isdesigned

approximated error. In this section, the laws of the

intelligent approximation and estimation of the friction

parameter will be derived on the basis of the approximation

theory and the Lyapunov stability theory to achieve the

robust position tracking performance in the closed system.

to compensatefor the

3.1 RCMAC system

An RCMAC is presented in Fig. 2 and composed of input

space with recurrent units, association space with recurrent

units, receptive-field space, weight memory space and

output space. The signal propagation and the basic function

in each space of the RCMAC are introduced as follows.

1. Input space I: For a given x = [x1, x2, ..., xna]T[ Rna,

each input state variable is quantised into discrete regions

(called elements) according to given control space. The

number of elements, na, is termed as a resolution.

2. Association memory space A: Several elements can be

accumulated as a block, the number of blocks, nb, is

usually greater than or equal to two. The A denotes an

associationmemory space

components. In this space, each block performs a receptive-

field basis function, and a mother wavelet is selected as the

receptive-field basis function, which can be represented as

with

nc

(nc¼ na× nb)

Fjk= exp −(xi+ wjkFjk(t − T) − mjk)2

(vjk)2

??

,

for k = 1, 2, ..., nb

(27)

where Fjkrepresents the output of the kth block receptive-

field basis function for the jth input xjwith mean mjkand

variance vjk, wjk represents the recurrent weight and

Fjk(t 2 T) denotes the value of Fjkthrough delay time T.

Fig. 2

Proposed RCMAC network architecture

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2125

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Fig. 3 shows the schematic diagram of a 2-D RCMAC with

na¼ 5 and nt¼ 4 (nt is the number of elements in a

complete block). In Fig. 3, x1is divided into blocks A and

B, and x2is divided into blocks a and b. By shifting each

variable an element, different blocks will be obtained. For

example, blocks C and D for x1and blocks c and d for x2

are possible shifted elements. Each block in this space has

three adjustable parameters mjk, sjkand wjk.

3. Receptive-field space T: Areas formed by blocks, named

as Ab and Bb are called ‘receptive fields’. In this study, the

number of receptive fields, nc, is equal to nb. We define the

following kth multidimensional receptive-fields function

bk(xi, mk, vk, wk)

=

?

k = 1, 2, ..., nc

na

j=1

Fjk= fjk

?

na

j=1

−(xij+ wjkFjk(t − T) − mjk)2

(vjk)2

??

,

(28)

where mk=[mk1, mk2,..., mkna]T[ Rna, vk=[vk1, vk2, ...,

vkna]T[ Rnaand wk= [wk1, wk2, ...,wkna]T[ Rna. The

multidimensional receptive-field function can be expressed

in a vector notation as

Q(xi, wi, m, v, w) = [b1, b2, ..., bnc]T

(29)

where

m = [mT

v = [vT

wj= [wT

1, ..., mT

k, ..., mT

nc]T[ Rnanc,

1, ..., vT

k, ..., vT

nc]T[ Rnanc

and

1, ···, wT

k, ..., wT

nc]T[ Rnanc

4. Weight memory space W: Each location of T to a particular

adjustable value in the weight memory space with nc

components can be expressed as

Ww= [ww1, ..., wwi, ..., wwno]

ww11

...

wwk1

...

wwnc

=

···

..

ww1i

...

wwki

...

wwnci

···

...

ww1no

...

wwkno

...

wwncno

.

··· ···

..

···

···

.

···

⎡

⎢

⎢

⎢

⎢

⎣

⎢

⎢

⎢

⎢

⎤

⎥

⎥

⎥

⎥

⎦

⎥

⎥

⎥

⎥

(30)

where wwi= [ww1i, ..., wwki, ..., wwnci]T[ Rncand wwki

denotes connectingweight

associated with the kth receptive field.

5. Output space O: The output of the RCMAC is the

algebraic summation of the activated weights in the weight

memory by adding the feedforward activation weights from

valueof the

ithoutput

Fig. 3

Two-dimensional RCMAC with na¼ 5 and nt¼ 4

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Page 8

the input space

Oi= wT

wiQ + wfixi=

?

nc

k=1

wwkpbk(xi, mk, vk, wj) + wfixi

(31)

where the connecting weight wfiis the output action strength

of the ith input associated with the output of the RCMAC.

The output of the RCMAC can be expressed in a vector

form as

O = [O1, ..., Oi, ..., On]T= WT

wQ + wT

fx (32)

where Wo= [WT

Rna, Qo= [Q x]T[ Rndand nd¼ na+ nc.

w

wT

f]T[ Rnd, wf= [wf 1, ..., wfna]T[

3.2

approximation of the RCMAC

Design of controller and observer based on

An RCMAC-based controller is the output of the RCMAC

that can be represented in a vector form

urc= O =

O1

...

Oi

...

On

⎡

⎢

⎢

⎢

⎢

⎢

⎣

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎥

= WT

oQo

(33)

From the approximation theory [18], the optimal RCMAC-

based controller u∗

rcwill be designed to mimic the controller

ueqin (24) such that

ueq= u∗

rc= W∗T

oQo+ 1

(34)

where W∗

approximation error vector 1 is assumed to be bounded by

?1? ≤ ?r?, in which r is a positive constant vector. The

control law of the SRCMAC system is assumed to take the

following form

ois the optimal parameter vectors of Woand the

usrc= urc+ us+ ˆ uf+ ur

(35)

where urc= WT

approximator u∗

to reduce residual approximation error. Substituting (35)

into (2) yields

oQo

rc, and the robust compensator uris designed

utilises to approximate the ideal

˙S = −gS −˜Tfn+ gn(q)(u∗

rc− urc− ur) (36)

Substituting (34) and (35) into (36) yields

˙S =

˙ s1

...

˙ si

...

˙ s1

−g1s1−˜Tfn+ gn1(q1)(˜ wT

⎢

⎢

⎡

⎢

⎢

⎡

⎢

⎢

⎢

⎢

⎢

⎣

⎢

⎤

⎥

⎥

⎥

⎥

⎥

⎦

⎥

=

w1Q + ˜ wf 1x1+ 11− ur1)

...

wiQ + ˜ wfixi+ 1i− uri)

...

wnQ + ˜ wfnxn+ 1n− urn)

−gisi−˜Tfni+ gni(qi)(˜ wT

−gnsn−˜Tfnn+ gnn(qn)(˜ wT

⎢

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

(37)

where ˜ wwi= ˜ w∗

We define a Lyapunov function candidate

wi− ˜ wiand ˜ wfi= wfi− ˆ wfi.

V2=

?

n

i=1

V2i

(38)

where

V2i=

1

2gni(˙ qi)s2

i+

1

2ho

˜ wT

wi˜ wwi+

1

2hf

˜ w2

fi+1

2˜ z2

i

+

1

2h0

˜ s2

0i+

1

2h3

˜ s2

3i+

1

2h4

˜ s2

4i+

1

2ha

˜ a2

i

(39)

where hi≥ 0 (i ¼ o, f, 0, 3, 4, a) are design parameters.

Differentiating with respect to time of (39) yields

˙V2i=

1

gni(qi)si˙ si

1

ho

˙˜ wT

wi˜ wwi+1

hf

˜ wfi˙˜ wfi

+ ˜ zi˙˜ zi+1

h0

˜ s0i˙˜ s0i+1

h3

˜ s3i˙˜ s3i+1

h4

˜ s4i˙˜ s4i+1

ha

˜ ai˙˜ ai

(40)

In order to estimate the friction parameters of the elasto-

plastic model, the friction parameter adaptation laws are

employed as

˙˜ s0i= −˙ˆ s0i= −h0ˆ zi˜Tfni

(41)

˙˜ s3i= −˙ˆ s3i= h3hi(˙ qi)ˆ zi˜Tfni

(42)

˙˜ s4i= −˙ˆ s4i= −h4˙ qih(˙ qi)˜Tfni

(43)

˙˜ ai= −˙ˆ ai= haˆ ziˆ s3h(˙ qi)˜Tfni

(44)

Substituting (41)–(44) into (40) with (37), (40) can be

IET Control Theory Appl., 2011, Vol. 5, Iss. 18, pp. 2120–2141

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Page 9

rearranged as

˙V2i= ˜ wT

wisiQ −1

ho

˙ˆ wwi

??

+ ˜ wfisixi−1

hi

˙ˆ wfi

??

+ si1i− siuri−

gis2

gni(qi)

i

−

1

gni(qi)[Gi(˙ qi)˜ zisi+ ˆ zi˜ s0isi− aihi(˙ qi)ˆ zi˜ s3isi

− ˙ qi˜ s4isi− ˆ s3ihi(˙ qi)ˆ zi˜ aisi] − ˆ ziGi(˙ qi)˜ s0i˜ zi− ˆ z2

+ aihi(˙ qi)ˆ z2

+ ˆ z2

− ˆ z2

− h2

− ˙ q2

− aih2

+ ˆ ziˆ s3ihi(˙ qi)Gi(˙ qi)˜ ai˜ zi− ˆ z2ˆ s2

− aihi(˙ qi)ˆ zi˜ zi˜ s0i− ais0ihi(˙ qi)˜ z2

i˜ s2

0i

i˜ s0i˜ s3i− ˙ qiˆ zi˜ s0iˆ s4i+ ˆ s3ih(˙ qi)ˆ z2˜ s0i˜ ai

ihi(˙ qi)˜ s3i˜ s0i− aih2

iˆ s3ihi(˙ qi)˜ ai˜ s0i+ hi(˙ qi)ˆ ziGi(˙ qi)˜ s3i˜ zi

i(˙ qi)ˆ z2

i(˙ qi)ˆ z2

i˜ s2

3i+ ˙ qihi(˙ qi)ˆ zi˜ s3i˜ s4i

i˜ s3i˜ ai− ˆ zi˙ qihi(˙ qi)˜ s4i˜ s0i− aiˆ zihi(˙ qi)˜ s4i˜ s3i

4i− hi(˙ qi)˙ qiGi(˙ qi)˜ s4i˜ zi+ ˆ s3ihi(˙ qi)ˆ zi˜ s4iai

i(˙ qi)ˆ z2

is2

iˆ s3i˜ ai˜ s3i− ˆ ziˆ s3ihi(˙ qi)˙ qi˜ ai˜ s4i

3ih2

i(˙ qi)˜ a2

2

i− ˆ aihi(˙ qi)ˆ zi˜ ai˜ zi

(45)

We choose the learning and adaptive laws in order to achieve

˙V2i≤ 0 as

˙ˆ wwi= hosiQ

(46)

˙ˆ wfi= hfsixi

uri=(r2

(47)

i+ 1)

2r2

i

si

(48)

Substituting (46)–(48) into (45), it follows that

˙V2i= si1i−(r2

i+ 1)

2r2

i

s2

i−

gis2

gni(qi)−

i

1

gni(qi)[Gi(˙ qi)˜ zisi+ ˆ zi˜ s0isi

− aihi(˙ qi)ˆ zi˜ s3isi− ˙ qi˜ s4isi− ˆ s3ihi(˙ qi)ˆ zi˜ aisi]

− ˆ ziGi(˙ qi)˜ s0i˜ zi− ˆ z2

+ ˆ s3ih(˙ qi)ˆ z2˜ s0i˜ ai+ ˆ z2

+ ˙ qihi(˙ qi)ˆ zi˜ s3i˜ s4i+ hi(˙ qi)ˆ ziGi(˙ qi)˜ s3i˜ zi− h2

− ˆ zi˙ qihi(˙ qi)˜ s4i˜ s0i− aiˆ zihi(˙ qi)˜ s4i˜ s3i− ˙ q2

− hi(˙ qi)˙ qiGi(˙ qi)˜ s4i˜ zi+ ˆ s3ihi(˙ qi)ˆ zi˜ s4iai

− aihi(˙ qi)ˆ zi˜ zi˜ s0i− ais0ihi(˙ qi)˜ z2

− ˆ z2

i˜ s2

0i+ aihi(˙ qi)ˆ z2

ihi(˙ qi)˜ s3i˜ s0i− aih2

i˜ s0i˜ s3i− ˙ qiˆ zi˜ s0iˆ s4i

i(˙ qi)ˆ z2

i˜ s2

3i

i(˙ qi)ˆ z2

i˜ s3i˜ ai

is2

4i

i− ˆ aihi(˙ qi)ˆ zi˜ ai˜ zi

i(˙ qi)ˆ z2

iˆ s3ihi(˙ qi)˜ ai˜ s0i− aih2

iˆ s3i˜ ai˜ s3i

− ˆ ziˆ s3ihi(˙ qi)˙ qi˜ ai˜ s4i+ ˆ ziˆ s3ihi(˙ qi)Gi(˙ qi)˜ ai˜ zi− ˆ z2ˆ s2

iMiJi+ si1i−(r2

2r2

3ih2

i(˙ qi)˜ a2

2

= −JT

i+ 1)

i

s2

i

−

gis2

gni(qi)−

i

1

gni(qi)[Gi(˙ qi)˜ zisi+ ˆ zi˜ s0isi

− aihi(˙ qi)ˆ zi˜ s3isi− ˙ qi˜ s4isi− ˆ s3ihi(˙ qi)ˆ zi˜ aisi]

?

matrixMi

canbe

manipulations, (49) is

(49)

where Ji=

Therefore since the positive semi-definiteness of the

proved

˜ s0i

˜ s3i

˜ s4i

˜ zi

˜ ai

?T(see (50))

fromsomealgebraic

˙V2i≤−1

2s2

i−−1

2

s2

r2

i

i

−2si

ri

ri1i+r2

i12

i

??

+1

2r2

i12

i−

gis2

gni(qi)

i

−

1

gni(qi)[Gi(˙ qi)˜ zisi+ˆ zi˜ s0isi−aihi(˙ qi)ˆ zi˜ s3isi

− ˙ qi˜ s4isi− ˆ s3ihi(˙ qi)ˆ zi˜ aisi]

=−1

2

ri

?

?

?

5

4gi2

+ ˙ q2

gi1

gni(qi)s2

+a2

2s2

i−1

si

−ri1i

?

?

?

?2

Gi(˙ qi)˜ zi

gi2

+1

2r2

?2

i12

i−

?

?2

?2

gi1

gn(qi)s2

i

−

gi2

5gni(qi)

si+5

2

+ si+5

2

ˆ zi˜ s0i

gi2

?2

?

−

gi2

5gni(qi)

si−5

2

aihi(˙ qi)ˆ zi˜ s3i

gi2

+ si−5

2

˙ qi˜ s4i

gi2

??2

?

−

gi2

5gni(qi)

si−5

2

ˆ s3ihi(˙ qi)ˆ zi˜ ai

gi2

+

[Gi(˙ qi)2˜ z2

i+ˆ z2

i˜ s2

0i+a2

ih2

i(˙ qi)ˆ z2

i˜ s2

3i

i˜ s2

4i+ ˆ s2

3ih2

i(˙ qi)ˆ z2

i˜ a2

i]

5

4gi2

≤−

i+1

2r2

i12

i+

[Gi(˙ qi)2˜ z2

i+ˆ z2

i˜ s2

0i

ih2

i(˙ qi)ˆ z2

i˜ s2

3i+ ˙ q2

i˜ s2

4i+ ˆ s2

3ih2

i(˙ qi)ˆ z2

i˜ a2

i](51)

where gi¼ gi1+ gi2, gi1 and gi2 are positive constant

vectors. Integrating both sides of (51), we have

V2i(T) − V2i(0)

gi1

gni(qi)

?T

?T

≤ −

?T

0

s2

idt +

5

4gi2

Gi(˙ qi)2

?T

0

˜ z2

idt

?

+ ˆ z2

i

×

0

˜ s2

0idt + a2

ih2

i(˙ qi)ˆ z2

i

?T

?T

0

˜ s2

3idt + ˙ q2

?

i

×

0

˜ s2

4idt + ˆ s2

3ih2

i(˙ qi)ˆ z2

i

0

˜ a2

idt

+1

2r2

i

?T

0

12

idt

(52)

Mi=

ˆ z2

ihi(˙ qi)

ˆ zi˙ qihi(˙ qi)

aihi(˙ qi)ˆ zi

ˆ z2

iˆ s3ihi(˙ qi)

i

−aihi(˙ qi)ˆ z2

aih2

aihi(˙ qi)ˆ zi

0

aih2

i

˙ qiˆ zi

ˆ ziGi(˙ qi)

−hi(˙ qi)ˆ ziGi(˙ qi)

hi(˙ q)˙ qiGi(˙ qi)

ais0ihi(˙ qi)

−ˆ ziˆ s3ihi(˙ qi)Gi(˙ qi)

−ˆ s3ihi(˙ qi)ˆ z2

ˆ s3ih2

−ˆ s3ihi(˙ qi)ˆ zi

ˆ aihi(˙ qi)ˆ zi

ˆ z2

i

−ˆ z2

i(˙ qi)ˆ z2

i

−˙ qiˆ zihi(˙ q)

˙ q2

ihi(˙ q)

0

ˆ ziˆ s3ihi(˙ qi)˙ qi

i(˙ qi)ˆ z2

i

i(˙ qi)ˆ z2

iˆ s3i

iˆ s2

3ih2

i(˙ qi)

⎡

⎢

⎢

⎣

⎢

⎢

⎤

⎥

⎥

⎦

⎥

⎥

(50)

2128

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doi:10.1049/iet-cta.2010.0389

www.ietdl.org

Page 10

Since V2i(t) ≥ 0, the inequality in (52) implies the following

inequality

g1i

?T

0

s2

idt ≤1

2g−1

ni(qi)s2

i(0)+

1

2ho

˜ wT

wi(0)˜ wwi(0)

+

1

2hf

w2

fi(0)+˜ z2

i(0)+1

h0

˜ s2

0i(0)

+1

h3

˜ s2

3i(0)+1

?

h4

?T

˜ s2

4i(0)+1

ha

?T

˜ a2

i(0)

+

5

4gi2

?T

G2

i(˙ qi)

0

?T

˜ z2

idt

+ˆ z2

i

0

˜ s2

0idt+a2

ih2

i(˙ qi)ˆ z2

i

×

0

˜ s2

3idt+ ˙ q2

i

0

˜ s2

4idt+˙ q2

?

i

?T

?T

0

˜ s2

4idt

+ ˆ s2

3ih2

i(˙ qi)ˆ z2

i

?T

0

˜ a2

idt

+1

2r2

i

0

12

idt

(53)

If the system starts with initial conditions

wfi(0) =0, ˜ zi(0)=0˜ sji(0)= 0 (j =0, 3, 4)

3, 4) and ˜ ai(0)=0, (53) is rewritten as

?T

?T

+1

0

˜ wwi(0)=0,

˜ sji(0) =0 ( j = 0,

gi1

0

s2

idt ≤az

?T

0

˜ z2

idt+a0

?T

0

˜ s2

0idt+a3

?T

?T

0

˜ s2

3idt

+a4

0

?T

˜ s2

4idt+aa

0

˜ a2

idt

2r2

i

12

idt

(54)

where

az=

5

4gi2

G2

i(˙ qi)(55)

a0=

5

4gi2

(qi)ˆ z2

i

(56)

a3=

5

4gi2

a2

ih2

i(˙ qi)ˆ z2

i

(57)

a4=

5

4gi2

˙ q2

i

(58)

aa=

5

4gi2

ˆ s2

3ih2

i(˙ qi)ˆ z2

i

(59)

Since ˜ zi, ˜ sji(j =0, 3,4) and ˜ aiare square integrable, that is,

?1

t ? 1 by Barbalat’s Lemma [28]. We also have si? 0 as

t ? 1, thus ei? 0 as t ? 1.

0˜ z2

4, a) and riare finite, then, ˜ zi?0, ˜ sji?0 and ˜ ai?0 as

idt ,1,?1

0˜ s2

jidt ,1, respectively, and ai(i ¼ z, 0, 3,

3.3Online learning of the RCMAC parameters

The performance of the RCMAC will be significantly affected

by selection of the connective weight, recurrent weight, mean

and variance. For achieving effective on-line learning, the

gradient descent method is introduced. The adaptation law

of (46)

˙ˆ wwi= −ho

∂V2i

∂wwi

= −ho

∂V2i

∂Oi

∂Oi

∂wwi

= −ho

∂V2i

∂Oi

Q

= hosiQ

(60)

From the above equation, the Jacobian of the controlled

system is obtained as

∂V2i

∂Oi

= −si

(61)

The learning algorithms based on the gradient descent method

for the receptive-field basis functions and recurrent weights in

association memory space and input space are derived as

follows

˙ mjk= −hm

∂V2i

∂mjk

= −hm

∂V2i

∂Oi

∂Oi

∂bk

∂bk

∂Fjk

∂Fjk

∂mjk

= hmsiwwkbk

2(mjk− xi− wjkFjk(t − T))

v2

jk

(62)

˙ vjk= −hv

∂V2i

∂vjk

= −hv

∂V2i

∂Oi

∂Oi

∂bk

∂bk

∂Fjk

∂Fjk

∂vjk

= hvsiwwkbk

2(xi+ wjkFjk(t − T) − mjk)2

v3

jk

(63)

˙ wjk= −hjk

∂V2i

∂wjk

= −hjk

∂V2i

∂Oi

∂Oi

∂bk

∂bk

∂Fjk

∂Fjk

∂wjk

= hjksiwwkbk

2(xi− wjkFjk(t − T) − mjk)

v2

jk

Fjk(t − T)

(64)

where hm, hvand hjkare the learning rates related to the

mean, variation and recurrent weights, respectively.

4

learning rates of the RCMAC

FIPS approach for the improved on-line

The learning rates of (46), (47) and (62)–(64) should be

selected to obtain a desirable control performance. Thus, we

consider FIPS algorithm to decide the optimal learning rates

of each parameter of the RCMAC. The FIPS is a kind of

evolutionary algorithm to search for the best solution by

simulating themovement

collection or swarm of particles are defined, where each

particle is assigned a random position in the N-dimensional

problem space. These particles fly with a certain velocity

and find the global best position after some iteration. Each

particle can adjust its velocity vector, based on its

momentum and the influence of its best position (pbest) as

well as the best position of its neighbours (gbest), and then

calculate a new position that the particle is to fly to as

shown below

andflockingofbirds.A

? vi= x ? vi+

?

Ni

n=1

?U(0, w) ⊗ (? pnbh(n)−? xi)

Ni

??

(65)

? xi= ? xi+? vi

(66)

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where ? vithe velocity of particle i;?U(0, w) [ RNa vector of

uniformly distributed random values between 0 and fi; ⊗

component-wise multiplication; nbh(n) the particle’s nth

neighbour; Nithe number of neighbours of the particle; ? xi

the current position of particle i; ? pithe best position in

history of particle i; and N the dimensional vectors of the

search space. The constant x is called the constriction

coefficient, which is necessary to keep velocities under

control, as they would quickly increase to unacceptable

levels within a few iterations and defined as follows

x =

2

?

w − 2 +

?????????

w2− 4w

(67)

where w . 4 is a design parameter.

The FIPS algorithm is adopted to improve the on-line

learning ability of the RCMAC by tuning the on-line

learning rates ho, hf, hm, hvand hjk. The total number of

the particles was set to N ¼ 50 and the dimension of the

particle was also set to D ¼ 5. The desired position is set to

? xid

= [ho

randomly generated as follows

hm

hv

hjk

hi]

andtheelementsare

lbhi≤ ? xi≤ ubhi

(68)

where lbhiand ubhidesignate the outcome of a uniformly

distributed random variable ranging over the given lower-

and upper-bounded values of the learning rate hi. The

global best of the swarm is updated using the equation as a

minimisation problem

? gnbh(i)= arg min

? pi

[f (? pi)](69)

where f (.) is a function that evaluates the fitness value for a

given position. In this study, the fitness is selected as

Fitness = d1?e? + d2max{|e|}(70)

where d1and d2are weight coefficients. The configuration of

the proposed SRCMAC combined with friction observer and

FIPS algorithm (SRCFO_FIPS) is depicted in Fig. 4.

5

manipulator with friction and uncertainty

Simulation and experiment for the robot

This section describes the simulated and experimental

evaluation of the proposed control scheme applied to the

Scorbot robot system. The photograph and structure of the

Scorbot robot control system are represented in Figs. 5 and

6. We select only two links (upper arm ¼ link1 and

forearm ¼ link2) among four links of the Scorbot robot

manipulator to simplify the verification process of our

Fig. 4

Block diagram of the proposed SRCFO_FIPS control system

Fig. 5

Photograph of the Scrobot robot control system

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position control. From (1), the dynamic equation for two DOF

(degree-of-freedom) links of the Scorbot robot manipulator is

described as

M(q)¨ q + C(q, ˙ q)˙ q + G(q) + Tf(q, ˙ q) = t

(71)

where (see equations at the bottom of the page)

Chosen values of link parameters are m1¼ 12.1 kg,

m2¼ 3.59 kg, L1¼ 0.3 m, L2¼ 0.41 m and g ¼ 9.806 m/s2.

The friction parameters from the experiment are selected

as

Ts1¼ 0.42 N m,

Tc2¼ 0.335 N m, s01¼ 5400 N m/rad, s02¼ 8700 N m/rad,

s11¼ 5.4 N m s/rad, s12¼ 6.2 N m s/rad, s21¼ 1.22 N m s/

rad, s22¼ 1.85 N m s/rad. The transient function values of

the elasto-plastic model are chosen as a1¼ 0.8 and a2¼ 1.

The direct kinematics for a circle trajectory in task space is

given by

Ts2¼ 0.351 N m,

Tc1¼ 0.412 N m,

h(q) =

L1sin(q1) + L2sin(q1+ q2)

L1cos(q1) + L2cos(q1+ q2)

??

(72)

Then, the desired end-effector trajectory of the manipulator is

Yd(t) =

xd

yd

??

=

xc+ R cos(v × t)

yc+ R sin(v × t)

??

(73)

where xc¼ yc¼ 20.1 m, R ¼ 0.0025 m and v ¼ 0.45 rad/s.

This trajectory makes the manipulator tip trace a circle in the

x02 y0 plane having a radius R ¼ 0.0025 m. The desired

trajectory Ydwas translated to the corresponding joint space

desired position trajectory qdvia the inverse kinematics of the

simulated two DOF links manipulator

qd= h−1(Yd) =

⎡

⎢

qd1

qd2

?

??

=

tan−1 yd/xd− tan−1(L2sin(qd2))

L1+ L2cos(qd2)

tan−1 1 − (x2

(x2

?

d+ y2

d+ y2

d− L2

d− L2

1− L2

1− L2

2)2/(2L1L2)2

2)/(2L1L2)

??

⎢

⎢

⎣

⎤

⎥

⎥

⎥

⎦

(74)

5.1 Simulation

All simulated tests were carried out by the Matlab package

under the initial conditions q1(0) ¼ 24.6886 rad, q2(0) ¼

2.8956 rad,

˙ q1(0) = 0rad/s and ˙ q2(0) = 0 rad/s.

parameters of the sliding-mode surface are chosen as

c1¼ c2¼ 1000, g1¼ g2¼ 4.75. The learning rates of the

RCMACnetwork areselected

ho¼ hf¼ hm¼ hv¼ hjk¼ 0.8, to guarantee stability of

the RCMAC network. High learning rates may cause the

RCMAC network to produce unstable output, although the

convergence speed would become faster. The design gains

of the approximated error compensator are r1¼ r2¼ 3. In

the simulation, we designed four controllers to evaluate the

performance of the proposed control system: SMC, sliding-

mode-based RCMAC controller (SRC), sliding-mode-based

RCMAC controller with the friction parameter observer

(SRCFO) and sliding-mode-based RCMAC controller with

the friction parameter observer and FIPS (SRCFO_FIPS).

Moreover, the following two cases were considered in the

simulation process to illustrate the effectiveness of the

proposed control systems:

The

as lowvalues,

Case I: All parameters are nominal condition, and TL¼ 0.

Case II: TL1¼ 0, TL2¼ 100 N m and m2¼ 10 × m2nact on

link 2 at 9 and 19 s, respectively,

where the subscript n means a nominal value of each

parameter.

The simulated results of the SMC for Case I is depicted

in Fig. 7, where the joint position tracking errors of the

SMC system were much high because of friction effect

of each joint.Thus,the

performance could not be obtained. Next, simulated results

using the RCMAC for the position tracking are given in

Fig. 8,wherethe RCMAC

improved tracking performance than that of the SMC by

virtue of robustness of the RCMAC network. Thus, the

SRC system gives improved tracking performance than the

SMC system. However, since the size of the tracking

errors of the SRC system was still high because of

friction, friction needs to be compensated to obtain precise

satisfactory circletracking

systemprovidedmuch

Fig. 6

link 2 ¼ forearm

Scorbot robot arm and joint: link 1 ¼ upper arm and

M(q) =

(m1+ m2)L2

1+ m2L2

m2L2

2+ 2m2L1L2cos(q2)

2+ L1L2m2cos(q2)

−m2L1L2sin(q2)˙ q2

m2L1L2sin(q2)˙ q2

m2L2

2+ L1L2m2cos(q2)

m2L2

2

?

?

?

?

?

C(q, ˙ q) =

2− 2m2L1L2sin(q2)˙ q1˙ q2

1

?

?

G(q) =

m2L2cos(q1+ q2) + (m1+ m2)L1g cos(q1)

m2L2cos(q1+ q2)

?

Tf(q, ˙ q) =

s01z1+ s11˙ z1+ s21˙ q1

s02z2+ s12˙ z2+ s22˙ q2

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tracking performance. The simulated results of the SRCFO

system for Case I are presented in Fig. 9, where the

magnitudesof the tracking

reduced by the proposed friction compensation of the

friction observers as shown in Fig. 9. The introduced four

errorsweresignificantly

friction parameter observers well estimated the given

friction parameters s0, s3, s4 and a as shown in

Figs. 9f–i. Thus, the friction state z and torque Tfof each

joint were also well estimated, as shown in Figs. 9j and k,

respectively.

Fig. 7

a Position tracking result of the joint 1 (desired: solid; tracked output: dashed)

b Position tracking result of the joint 2 (desired: solid; tracked output: dashed)

c Position tracking errors of the joints (link 1: solid; link 2: dashed)

d Control input of the joint (link 1: solid; link 2: dashed)

e Cartesian reference tracking result of endpoint in the X–Y space (actual: solid; desired: dashed)

Simulated results of the SMC system

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Next, forCase II,simulation wascarriedout to test robustness

to mass variation and external disturbance. As shown in

Figs. 10a–c, the tracking errors of the SRCFO system were

significantly increased than that of the SRCFO_FIPS system.

In the SRCFO_FIPS system, when an external disturbance was

applied to link 2, the learning rates of the RCMAC network

automatically adjusted to seek their optimal values to minimise

the tracking error and guarantee stability by FIPS as shown in

Figs. 10d. Fig. 10e shows the fitness value for d1¼ 0.5 and

d2¼ 1. Therefore the simulated results show robustness of the

Fig. 8

a Position tracking result of the joint 1 (desired: solid; tracked output: dashed)

b Position tracking result of the joint 2 (desired: solid; tracked output: dashed)

c Position tracking errors of the joints (link 1: solid; link 2: dashed)

d Control input of the joint (link 1: solid; link 2: dashed)

e Cartesian reference tracking result of endpoint in the X–Y space (actual: solid, desired: dashed)

Simulated results of the SRC system

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Fig. 9

a Position tracking result of the joint 1 (desired: solid; tracked: dashed)

b Position tracking result of the joint 2 (desired: solid; tracked: dashed)

c Position tracking error of the joint (joint 1: solid; joint 2: dashed)

d Control input (joint 1: solid; joint 2: dashed)

e Cartesian reference tracking result of endpoint in the X–Y space (actual: solid; desired: dashed)

f

ˆ s01(solid) and ˆ s02(dashed)

g ˆ s31and ˆ s32(dashed)

h ˆ s41(solid) and ˆ s42(dashed)

i

ˆ a1and ˆ a2(dashed)

j Friction state ziand ˆ zi

k Friction torque TfiandˆTfi

Simulated results of the SRCFO system

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proposed SRCFO_FIPS system to the uncertainty of the robot

manipulator as well as non-linear friction.

5.2 Experiment

The experiment for the proposed control scheme was

performed to confirm simulated results. In the experiment,

the SMC, SRC_FIPS, SRCFO_FIPS control systems were

designed to investigate the usefulness of the proposed

control system. The designed controllers generated in the

computer are implemented by the Matlab RTI system using

MF624 board of the Humusoft company [32]. The control

signals were transferred into the DC servo motor of the

Scorbot robot through the servo drive. The generated motor

Fig. 9

Continued

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torques in the robot joint were calculated by

tmi=nikti

Rmi

Vi+

niktikbi

Rmi

+ fmi

??

˙ui

(i = 1, 2) (75)

where V is the control input voltage, n ¼ 127 is the gear ratio

of the harmonic drive gear, Rm¼ 0.8294 V is the motor

resistance, kt¼ 1.82 × 1022N m/A is the torque constant,

kb¼ 1.82 × 1022V/rad/s is the back emf constant and

fm¼ 0.0001 N m/rad/s is the motor friction torque. The

Fig. 10

increased

Simulated results of the SRCFO and SRCFO_FIPS system when the external disturbance is applied to link 2 and mass of endpoint is

a Tracking errors of the joint 1 (SRCFO: solid, SRCFO_FIPS: dashed)

b Tracking errors of the joint 2 (SRCFO: solid, SRCFO_FIPS: dashed)

c Cartesian reference tracking result of endpoint in the X–Y space (desired: solid; SRCFO: dashed; SRCFO_FIPS: dotted)

d Learning rates trained by FIPS of the SRCFO_FIPS system

e Fitness value

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selected values of the motor parameters are same in both

motors attached in joints. The sample time was selected as

250 Hz.

The experimental results of the SMC system are

represented in Fig. 11. In Figs. 11a–c, it can be shown that

the position tracking performance of each link was still

Fig. 11

a Position tracking result of the joint 1 (desired: solid; tracked output: dashed)

b Position tracking result of the joint 2 (desired: solid; tracked output: dashed)

c Position tracking errors of the joints (link 1: solid; link 2: dashed)

d Control input of the joint (link 1: solid; link 2: dashed)

e Cartesian reference tracking result of endpoint in the X–Y space (actual: solid; desired: dashed)

Experimental results of the SMC system

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poor because of non-linearities of the robot system. Hence,

the circle tracking result was also worse as shown in

Fig. 11e. Next, the experimental results of the SRC_FIPS

system are represented in Fig. 12. Here, the learning rates

of the RCMAC were selected as the fixed values based on

the results of the FIPS optimisation to fasten the real-time

Fig. 12

a Position tracking result of the joint 1 (desired: solid; tracked output: dashed)

b Position tracking result of the joint 2 (desired: solid; tracked output: dashed)

c Position tracking errors of the joints (link 1: solid; link 2: dashed)

d Control input of the joint (link 1: solid; link 2: dashed)

e Cartesian reference tracking result of endpoint in the X–Y space (actual: solid; desired: dashed)

Experimental results of the SRC_FIPS system

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Fig. 13

a Position tracking result of the joint 1 (desired: solid, tracked: dashed)

b Position tracking result of the joint 2 (desired: solid, tracked: dashed)

c Position tracking error of the joint (joint 1: solid; joint 2: dashed)

d Control input (joint 1: solid; joint 2: dashed)

e Cartesian reference tracking result of endpoint in the X–Y space (actual: solid; desired: dashed)

f

ˆ s01(solid) and ˆ s02(dashed)

g ˆ s31(solid) and ˆ s32(dashed)

h ˆ s41(solid) and ˆ s42(dashed)

i

ˆ a1and ˆ a2(dashed)

j Friction state ziand ˆ zi

k Friction torque TfiandˆTfi

Experimental results of the SRCFO_FIPS system

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computation speed of the control system. Then, the position

tracking performance was more improved than that of the

SMC system by virtue of the RCMAC with FIPS, but the

satisfactory performance could not still be obtained because

of joint frictions. Finally, in Fig. 13, it can be seen that the

magnitude of the position tracking error of the SRCF_FIPS

control system were smaller than that of other two control

systems because of compensation for the non-linear friction

and uncertainty.

6 Conclusions

In this paper, the robust model-free control scheme was

developed to provide much enhanced position tracking

performance of the uncertain robot manipulator with non-

linear dynamic joint friction and system uncertainty. To

eliminate the drawback of the LuGre friction model, the

elasto-plastic model-based friction parameter observer was

first studied to estimate the friction parameters of the joint

Fig. 13

Continued

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frictions. For approximation for an ideal sliding mode control,

the improved RCMAC network than the conventional

RCMAC with the sliding mode control was designed. From

the Lyapunov stability theorem, adaptive laws for the

controller and friction parameter observer were derived. The

FIPS algorithm was also employed to obtain optimal

learningrates guaranteeing

convergence speed in the RCMAC network when the

uncertainty is applied. The favourable position tracking

performance of the proposed control scheme could be

validated from some simulation and experiment for the

Scorbot robot manipulator in the presence of joint friction

by effective compensating friction and uncertainty.

stabilityand suitable

7 Acknowledgments

This research was supported by the MKE(The Ministry of

Knowledge Economy), Korea, under the Specialized Field

Navigation/Localization Technology Research Center support

programme supervised by the NIPA (National IT Industry

Promotion Agency) (NIPA-2010-(C7000-1001-0004)).

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