Robotics and Autonomous Systems 61 (2013) 555–564

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Robotics and Autonomous Systems

journal homepage: www.elsevier.com/locate/robot

Adaptive variable structure controller of redundant robots with mobile/fixedobstacles avoidanceTarek Madani a, Boubaker Daachi b,∗, Abdelaziz Benallegue a

a Laboratoire d’Ingénierie des Systèmes de Versailles (LISV), 10–12, avenue de l’Europe, 78140 Vélizy, Franceb Laboratoire Image, Signaux & Systèmes Intelligents (LISSI), 122–124, rue Paul Armangot, 92400 Vitry sur Seine, France

h i g h l i g h t s

• Adaptive controller without knowledge of the dynamical model of the robot.• The stability is proved theoretically using the Lyapunov principle.• The controller is designed directly in the task space.• No inversion of models is required.• No collision with moving obstacles is possible.

a r t i c l e i n f o

Article history:Received 5 March 2012Received in revised form11 February 2013Accepted 22 February 2013Available online 15 March 2013

Keywords:Adaptive controlReference modelVariable structure systemRedundant robotsMobile obstacle avoidance

a b s t r a c t

In this paper, a variable structure adaptive controller is proposed for redundant robot manipulatorsconstrained by moving obstacles. The main objective of the controller is to force the model states ofthe robot to track those of a chosen reference model. In addition, the controller is designed directly inCartesian space and no knowledge on the dynamic model is needed, except its structure. The parametersof the controller are adapted using adaptive laws obtained via Lyapunov stability analysis of the closedloop. The performances of the proposed controller are evaluated using a 3DOF robotmanipulator evolvingin a vertical plane constrained by amobile obstacle. The obtained results show its effectiveness comparedto other tested variable structure controllers.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

The previous versions of robot manipulators were confined tomake simple and repetitive tasks in known and static environ-ments. Today’s robots are assigned to tasks that aremore andmorecomplex like ubiquitous robots that contribute more in the lifeof a human in order to recover it. Therefore, they are becomingfaster and operate in dynamic environments. Hence, the challengeis to address the problem of self-tuning and adaptation regardingthe environment and possible real time changes. Using redundantrobot manipulators [1–4] can be the solution to this problem. Infact, they are dexterous and able to avoid obstacles even if they aremobile (see Fig. 1). In order to accomplish its task, the robot has tobe controlled in a way that it has to avoid collision with obstaclesand to ensure their security, for instance in the automotive field the

∗ Corresponding author. Tel.: +33 141807324.E-mail address: daachi@u-pec.fr (B. Daachi).

0921-8890/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.robot.2013.02.010

manipulating robots are not immune to the passage of humans andgears. In general, the required task is specified in Cartesian space,also called the operational space of the robot. Therefore, it is morepractical to synthesize control laws directly in this space whichrepresents the evolving space of the robot. Thus, the controller syn-thesis of robot manipulators is transformed into a problem of tra-jectory planning of joint space from Cartesian space constrainedbymobile obstacles. Each obstacle can bewrapped in a spherewithgreater radius in order to increase the securitymargin. The passageof a space to another is carried out via geometrical and kinemati-cal transformations [5]. These transformations are based on directand inverse models describing the relation between the Cartesianpositions according to the joint positions (geometrical model) andCartesian velocities according to the joint velocities (kinematicsmodel). If the robot is redundant, there is are infinite solutions rep-resenting the various configurations of the robot for a given pointin Cartesian space. Thus, the problem to be solved is to determinethe optimal joint positions and velocities corresponding to the de-sired trajectories expressed in task space under mobile obstaclesconstraints.

556 T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564

Several works addressing, in general, the problem of adaptivecontrol of redundant robots have been proposed in the literature.The work presented in [6] deals with the robust adaptive controltracking of a 6 degree of freedom parallel robot, called C5 parallelrobot. The proposed approach is based on the coupling of slidingmodes and multi-layers perceptron neural networks (MLP-NNs).It does not require the inverse dynamic model for deriving thecontrol law. In our previous work [7], we used adaptive neural-networks to control redundant robots in task space without thepresence of mobile obstacles. In [8], neural network-based nonlin-ear dynamical control of kinematically redundant robot manipula-tors is also proposed; it is based on several assumptions that arenot always satisfied. In [9], the authors present an adaptive controlscheme for a redundant robot subject to spatial constraints. Thefunctions of the dynamic model are partially known. It means thatthe dynamical model can be written as a known regressor multi-plied by unknown vector of parameters [10]. Themain drawback ofthe use of neural-networks approach to control redundant robotsis a great number of parameters [11,12].

The interest of this paper is in the synthesis of a variable struc-ture adaptive controller [13–20]. We use an optimization strategybased on energy criterion, allowing the robot to use optimal jointconfigurations and avoid any collision with possible mobile obsta-cles. We combine the variable structure systems, adaptive controland reference model, to propose a robust and self-adjusting con-trol law in the Cartesian space of the robot manipulator. It shouldbe noted that the proposed control law does not require inversionof geometric and kinematic models. The knowledge of the robotdynamic model is also not required.

The paper is organized as follows: Section 2 describes the dy-namics of the robot manipulator, its structural properties and theprinciple of avoidance of moving obstacles. Section 3 presents theproposed controller with the analysis of stability of the system inclosed loop. Simulation results on a robot planewith 3DOF are pre-sented and discussed in Section 4. Finally, the conclusion is pre-sented in Section 5.

2. Problem formulation

Let us consider a n-DOF robot manipulator whose dynamicmodel is expressed in extended Cartesian space as given in [11]:

M∗(q, xo)X + C∗(xo, xo, q, q)X + H∗(q, q, xo, xo, xo) = u∗ (1)

u = JTu∗ (2)

where:

• q ∈ ℜn is the vector of joint positions

• q ∈ ℜn is the vector of joint velocities

• q ∈ ℜn is the vector of joint accelerations

• M∗ is the inertia matrix• C∗ is the matrix of Coriolis terms• H∗ is the vector of gravitational forces, frictions and all other

dynamics• J is the extended Jacobian matrix• u is the vector of torques acting on the joints• xo, xo and xo represent respectively, the positions, the velocities

and the accelerations of obstacles.• X, X and X are given by:

X =

xh

X =

xh

X =

xh

with x, x and x representing respectively, the positions, the veloc-ities and the accelerations in the Cartesian space of the effector(their dimension ism). h, h and h of dimension (n−m), are vectors

Fig. 1. The problem of obstacles avoidance.

that represent a set of variables depending on the obstacle move-ment [11]. The choice of the variable h is given by:

h(q, xo) = NT ∂Ω(q, xo)∂q

(3)

where the matrix N represents the null space of the Jacobian ma-trix J andΩ(q, xo) represents a criterion to be optimized [21].

The problem of the controller is to track any desired trajectoryand avoids any possible collision with mobile obstacles. To over-come this problem and in the same time to avoid any knowledgeon the dynamic model of the system (the matrices M∗, C∗ and thevector H∗ are unknown), we propose to use sliding modes tech-nique and reference model.

3. Variable structure control law synthesis

The control problem of redundant robot manipulators havinguncertainties in the dynamic model and often disturbed by fixedor mobile obstacles in their environment is a real challenge. Con-siderable research has focused on developing advanced manipula-tor control systems, and some techniques have emerged such asthe computed torque method, which requires precise knowledgeof the manipulator parameters and dynamic model [22]. In thiscase, the good performance of the control system depends on thehigh fidelity of the mathematical model used to describe the be-havior dynamics. For this purpose, an adaptive variable structurecontroller with reference model has been proposed in this section.The control lawwhich does not require any prior knowledge of thedynamical model is directly developed in task space.

3.1. Robot dynamic and reference model in state space

The proposed approach consists in synthesis of adaptive controllaw without using any a priori knowledge concerning the robotdynamics. The principal goal is to realize tracking of a givenreference model in the augmented task space.

Let xs ∈ ℜ2n be the state vector defined by:

xs =

XX

. (4)

T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564 557

The Eq. (1) can be represented by the following state equation:

xs =

0n×n In×nA1 A2

A(q,q,xo,xo)

xs +

0n×nB1

B(q,xo)

u∗+

0nH1

H(q,q,xo,xo,xo)

(5)

where In×n ∈ ℜn×n is the identity matrix and:

A1 = 0n×n

A2(q, q, xo, xo) = −M∗−1(q, xo)C∗(q, q, xo, xo) ∈ ℜn×n (6)

B1(q, xo) = M∗−1(q, xo) ∈ ℜn×n

H1(q, q, xo, xo, xo) = −M∗−1(q, xo)H∗(q, q, xo, xo, xo) ∈ ℜn.

The controller objective is to force the model states (5) to trackthose of the reference model characterized by the state vectorxm ∈ ℜ

2n given by:

xm =

Xm

Xm

(7)

and:

xm =

0n×n In×nAm1 Am2

Am

xm +

0nBm1

Bm

r (8)

where Am ∈ ℜ2n×2n is the evolution matrix, Bm ∈ ℜ

2n×n is theinput vector and r ∈ ℜ

n is the input of the reference model.Let the tracking error be denoted by:

e = xm − xs. (9)

The problem is as follows: for a given vector r , determine astable sliding surface s = 0n so that, in ideal sliding regime, thetracking error e tends to 0n when t tends to ∞. Finally we have tosynthesize the control law u∗ ensuring the attractiveness to thesliding surface s = 0n.

Let s ∈ ℜn the vector of switching laws be defined by:

s = Λe (10)

where Λ = [Λ1, In×n] ∈ ℜn×2n and Λ1 is a square matrix of

dimensions (n × n).Consider e =

ϵT , ϵT

T with ϵ = (Xm − X) ∈ ℜn. In ideal

sliding regime, the switching law (10) is close to zero and the errordynamics is given by ϵ = −Λ1ϵ. The sliding surface s = 0n is stableif and only if the matrixΛ1 is positive definite.

Before presenting our variable structure control, it is necessaryto define some basic functions.

Definition 1. The function ∥·∥p for p ≥ 1 represents the Höldernorm. It is defined by:

∥·∥p : ℜn

→ ℜ+ (11)x1

...xn

→|x1|p + · · · + |xn|p

1p .

Definition 2. For all matrix A ∈ ℜm×n the norm ∥·∥p is given by:

∥·∥p : ℜm×n

→ ℜ+ (12)

A → sup∥x∥p=0

∥Ax∥p

∥x∥pwith x ∈ ℜ

n.

3.2. Control law synthesis

For the control law, we propose:

u∗= K1 + K2 · xs + K3 · e + K4 · r (13)

with:K1 = K1 +∆K1 ∈ ℜ

n

K2 = K2 +∆K2 sgn(xs)T ∈ ℜn×2n

K3 = K3 +∆K3 sgn(e)T ∈ ℜn×2n

K4 = K4 +∆K4 sgn(r)T ∈ ℜn×n.

(14)

Ki are constant terms and∆Ki ∈ ℜn are variable terms given by:

∆Ki =

κi0 (αi0s + sgn(s))+

2j=1

κijαijs + sgn(s)

∥e∥j

2

for i = 1, 2κi (αis + sgn(s)) for i = 3, 4

(15)

where α· ∈ ℜ∗+ are positive constants and κ· ∈ ℜ

+ are positivegains with:

·

κ10 = w10(∥s∥1 − β10κ10 ∥s∥22)

·

κ1j = w1j(∥s∥1 − β1jκ1j ∥s∥22) ∥e∥

j2 for j = 1, 2

·

κ20 = w20(∥s∥1 − β20κ20 ∥s∥22) ∥x∥1

·

κ2j = w2j(∥s∥1 − β2jκ2j ∥s∥22) ∥e∥

j2 ∥x∥1 for j = 1, 2

·

κ3 = w3(∥s∥1 − β3κ3 ∥s∥22) ∥e∥1

·

κ4 = w4(∥s∥1 − β4κ4 ∥s∥22) ∥r∥1

(16)

w· ∈ ℜ∗+ are arbitrary positive adaptation gains and β· ∈ ℜ

∗+

are positive constants given by:

αind

βind≥ ξind > 0, ∀ ind ∈ 10, 11, 12, 20, 21, 22, 3, 4 (17)

ξind represents constants defined later (Eq. (24)).The control (13) has a variable structure. It contains four

changes of structure:

• a variable structure K1,

• a feedback variable state K2 · x,• a feedback variable tracking error K3 · e,• anticipation of desired reference variable K4 · r.

In joint space, the torques are obtained using Eq. (2).The proposed controller is shown in Fig. 2.Let A0 ∈ ℜ

2n×2n be a matrix defined by:

A0 =

0n×n −In×nΛ1 In×n +Λ1

. (18)

The matrix A0 is chosen to verify:

ΛA0 = Λ. (19)

One can notice that (I2n×2n − A0) represents the null space of thematrixΛ. It meansΛ(I2n×2n − A0) = 0n×2n.

Consider B =0n×n, B−1

1

∈ ℜ

n×2n the pseudo-inverse of B (BB= In×n). So:

H = BBH(Am − A) = BB(Am − A)(Am + A0) = BB(Am + A0)

Bm = BBBm.

(20)

558 T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564

Fig. 2. Principle scheme of the variable structure controller.

In our controller we consider the following assumptions:

Assumption 1. The matrix B1 is not a zero matrix, bounded andpositive definite. One can consider the following inequalities:

0 < ρ1 ≤ ∥B1∥2 ≤ ρ2 and ρ1 ≤ λmin(B1)B(Am + A0)− K3

2

≤ δ3BBm − K4

2

≤ δ4

(21)

where ρ1,2 and δ3,4 are unknown bounded constants that arestrictly positive and λmin(B1) is the smallest eigenvalue of thematrix B1.

Assumption 2. The obstacle positions, velocities and accelerationsare assumed known and bounded. The matrix B and the vector Hsatisfy:BH + K1

2

≤ δ10 +

2j=1

δ1j ∥e∥j2 (22)

where δ1· are unknown constants bounded and strictly positive.

Assumption 3. Matrices B and A satisfy:B(Am − A)− K2

2

≤ δ20 +

2j=1

δ2j ∥e∥j2 (23)

where δ2· are unknown constants bounded and strictly positive.

Remark 1. Due to the mechanical and physical characteristics ofrobot manipulators (1), the stability of the referencemodel (8) andthe boundedness of obstacles motion, the Assumptions 1–3 arevalid. So, the constants ξind defined in (17) can be fixed such as:

ξind ≥ δindρ2

ρ1, ∀ ind ∈ 10, 11, 12, 20, 21, 22, 3, 4 . (24)

Remark 2. The matrix B1 is positive definite and matrix theoryshows that B1 ≥ λmin(B1)In×n. Therefore, using Assumption 1:

sTB1s ≥ λmin(B1) sT s∥s∥22

≥ ρ1 ∥s∥22 (25)

sTB1 sgn(s) ≥ λmin(B1) sT sgn(s) ∥s∥1

≥ ρ1 ∥s∥1 . (26)

3.3. Stability study

Let us consider the system (1) and suppose that the Assump-tions 1–3 hold true. The controller defined by Eqs. (13)–(15) withadaptation laws (16), guarantees boundedness of all the controlsignals and attractiveness to the sliding surface s = 0n.

Proof. Consider the Lyapunov function:

V =12

sT s + ρ1θ

TW−1θ

(27)

where W = diag [w10, w11, w12, w20, w21, w22, w3, w4] and θ =

(θ − θ )with:

θ =κ10, κ11, κ12, κ20, κ21, κ22, κ3, κ4

T (28)

represents the estimation of the vector:

θ = [κ10, κ11, κ12, κ20, κ21, κ22, κ3, κ4]T (29)αind

βind≥ κind ≥ δind

ρ2

ρ1,

∀ ind ∈ 10, 11, 12, 20, 21, 22, 3, 4 . (30)

The derivative of V with respect to the time:

V = sT s − ρ1θTW−1

·

θ . (31)

Using Eqs. (13), (19) and (20), we obtain:

s = Λe= ΛAmxm + Bmr − Ax − Bu − H

= Λ−A0e − H + (Am − A)x + (Am + A0)e + Bmr − Bu

= −ΛA0e s

+Λ

−BBH + BB(Am − A)x + BB(Am + A0)e

+ BBBmr − B (K1 + K2x + K3e + K4r)

= −s + ΛBB1

−[BH + K1] + [B(Am − A)− K2]x

+ [B(Am + A0)− K3]e + [BBm − K4]r. (32)

The expansion of (31) using the derivative of the switching law(32), the Eqs. (13)–(16) and the properties (20), gives:

V = −sT s + V1 + V2 + V3 + V4 (33)

T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564 559

where:

V1 = sTB1−[BH + K1] + [B(Am − A)− K2]x

+ [B(Am + A0)− K3]e + [BBm − K4]r (34)

V2 = −sTB1s

κ10α10 +

2j=1

κ1jα1j ∥e∥j2

+

κ20α20 +

2j=1

κ2jα2j ∥e∥j2

∥x∥1

+ κ3α3 ∥e∥1 + κ4α4 ∥r∥1

(35)

V3 = −sTB1 sgn(s)

κ10 +

2j=1

κ1j ∥e∥j2

+

κ20 +

2j=1

κ2j ∥e∥j2

∥x∥1 + κ3 ∥e∥1 + κ4 ∥r∥1

(36)

V4 = −ρ1θTW−1

·

θ (37)

= −ρ1 ∥s∥1

κ10 +

2j=1

κ1j ∥e∥j2

+

κ20 +

2j=1

κ2j ∥e∥j2

∥x∥1 + κ3 ∥e∥1 + κ4 ∥r∥1

+ ρ1 ∥s∥22

κ10β10κ10 +

2j=1

κ1jβ1jκ1j ∥e∥j2

+

κ20β20κ20 +

2j=1

κ2jβ2jκ2j ∥e∥j2

∥x∥1

+ κ3β3κ3 ∥e∥1 + κ4β4κ4 ∥r∥1

. (38)

Using the Assumptions 1–3, the two inequalities (25) and (26),and the property ∥·∥1 ≥ ∥·∥2, we have:

V1 ≤ ∥s∥2 ∥B1∥2

BH + K1

2+

B(Am − A)− K2

2∥x∥2

+

B(Am + A0)− K3

2∥e∥2 +

BBm − K4

2∥r∥2

≤ ρ2 ∥s∥1

δ10 +

2j=1

δ1j ∥e∥j2

+

δ20 +

2j=1

δ2j ∥e∥j2

∥x∥1 + δ3 ∥e∥1 + δ4 ∥r∥1

(39)

V2 ≤ −ρ1 ∥s∥22

κ10α10 +

2j=1

κ1jα1j ∥e∥j2

+

κ20α20 +

2j=1

κ2jα2j ∥e∥j2

∥x∥1

+ κ3α3 ∥e∥1 + κ4α4 ∥r∥1

(40)

V3 ≤ −ρ1 ∥s∥1

κ10 +

2j=1

κ1j ∥e∥j2

+

κ20 +

2j=1

κ2j ∥e∥j2

∥x∥1 + κ3 ∥e∥1 + κ4 ∥r∥1

(41)

V4 ≤ V41 + V42 + V43 (42)

V41 = −ρ1 ∥s∥1

κ10 +

2j=1

κ1j ∥e∥j2

+

κ20 +

2j=1

κ2j ∥e∥j2

∥x∥1 + κ3 ∥e∥1 + κ4 ∥r∥1

V42 = ρ1 ∥s∥1

κ10 +

2j=1

κ1j ∥e∥j2

+

κ20 +

2j=1

κ2j ∥e∥j2

∥x∥1 + κ3 ∥e∥1 + κ4 ∥r∥1

V43 = ρ1 ∥s∥22

κ10β10κ10 +

2j=1

κ1jβ1jκ1j ∥e∥j2

+

κ20β20κ20 +

2j=1

κ2jβ2jκ2j ∥e∥j2

∥x∥1

+ κ3β3κ3 ∥e∥1 + κ4β4κ4 ∥r∥1

.

From (33) and using the sum of (39)–(42), it becomes:

V ≤ −sT s − P1 ∥s∥1 − P2 ∥s∥22 (43)

with:

P1 = ψ1 + ψ2 ∥x∥1 + ψ3 ∥e∥1 + ψ4 ∥r∥1 (44)

where:

ψi =

(ρ1κi0 − ρ2δi0)+

2j=1

(ρ1κij − ρ2δij) ∥e∥j2

for i = 1, 2(ρ1κi − ρ2δi) for i = 3, 4

(45)

and:

P2 = ρ1 (ϕ1 + ϕ2 ∥x∥1 + ϕ3 ∥e∥1 + ϕ4 ∥r∥1) (46)

where:

ϕi =

(αi0 − βi0κi0)κi0 +

2j=1

(αij − βijκij)κij ∥e∥j2

for i = 1, 2(αi − βiκi)κi for i = 3, 4.

(47)

According to (30) we can say that the two quantities P1 andP2 are positive because:ρ1κind − ρ2δind ≥ 0αind − βindκind ≥ 0 ,

∀ ind ∈ 10, 11, 12, 20, 21, 22, 3, 4 . (48)

Consequently, the Eq. (43) implies V ≤ −sT s. Hence if s = 0n wehave V < 0 and if s = 0n we have V = 0. The switching law s andall estimation errors θ are bounded. Invoking Barbalat’s Lemma,wecan say that s goes to zero because V is checked bounded. Finally,the control law (13) ensures the attraction to the sliding surfaces = 0n.

560 T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564

Fig. 3. 3DOF redundant robot constrained with one mobile obstacle.

Fig. 4. Desired eight-shaped trajectory.

Remark 3. The constant terms K· in (14) can be chosen in a man-ner to reduce the bounds of δ· in (21)–(23). This permits to re-duce the inferior bounds of κ· in (30). Consequently the adaptive

variable terms∆K· in (15) are minimized and the control energyof (2) will be optimized.

4. Simulation results

In this section, we consider a 3DOF robot manipulator evolvedin the vertical plan. The first segment is constrained by a mobileobstacle (Fig. 3). The radius of the circle enclosing the movingobstacle is directly integrated into our calculations (radius =

0.05mand center xo = (x1o, x2o)). Variable or fixed, the calculationof the distance between the robot parts and the obstacle isconducted online. It means that we do not ask if the radius haschanged or will change. The motion of the obstacle is carried outaccording to the following equations:x1o = 0.1x2o = 0.2 × (1 − |sin(π fot)|)fo = 0.5 Hz.

(49)

The matrices of the reference model are given as follows:

Am1 = Am2 = Bm1 = I3×3.

The reference trajectory r is:

r =

1 + 0.1 sin(4π fot)+ 0.4π fo cos(4π fot)− 1.6π2f 2o sin(4π fot)0.1 cos(2π fot)− 0.2π fo sin(2π fot)− 0.4π2f 2o cos(2π fot)

0

.These equations imply that the desired trajectory represents

an eight-shaped trajectory (Fig. 4). To ensure the avoidance of theobstacle, the desired trajectories hd, hd and hd are defined by:

hd = hd = hd = 0. (50)

The parameters of the sliding mode controller are:

α3 = α4 = α1i = α2i = 20 (i = 1, 2, 3)w3 = w4 = w1j = w2j = 5 (j = 1, 2, 3)β3 = β4 = β1k = β2k = 5 (k = 1, 2, 3)

K4 = 2 × ones(3, 1) K2 = K3 = (C C)

K4 = C C = 2 × ones(3, 3)

The initial configurations of the robot and the obstacle are givenin Fig. 5. One can notice that this configuration is singular butour approach permits to overcome it using the redundancy ofthe robot. Figs. 6 and 7 show the effectiveness of the proposedcontroller in both trajectory tracking and obstacle avoidance. Likeonly the first part of the robot is constrained by the obstacle, therobot uses its redundancy and relies more on its other parts.

Fig. 5. Initial positions of the mobile obstacle and the robot.

T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564 561

Fig. 6. Sliding mode tracking for several instants of simulation.

Fig. 7. Tracking results within all the period of simulation (in solid line: desired trajectory and in dashed line: trajectory obtained by the proposed approach).

Fig. 8. Tracking results within all the period of simulation (in solid line: desired trajectory and in dashed line: trajectory obtained by the proposed approach).

The results of tracking in directions x and y given in Fig. 8 aresatisfactory and confirm the good quality of tracking presented intask space. Moreover, one can easily notice on the curves of theFig. 9 an optimization of joint displacements and consequently the

use of a minimal energy. Also it can be concluded that having agood trajectory tracking in task space does not imply obtainingjoint trajectories closed to joint trajectories relative to the desiredtask.

562 T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564

Fig. 9. Joint displacements.

Fig. 10. Some configurations of the robot.

For more illustrations, we have chosen four configurations ofthe robot within their corresponding obstacle positions (Fig. 10).We can easily see that the robot uses its redundancy to realize theassigned task and at the same time avoids any collision of its firstpart with the mobile obstacle. As a comparison, we have consid-ered the following control laws:

u∗= Kp · e + Kv · e + ϱ · sign(e) (51)

u∗= Kp · e + Kv · e + υ · sign(e + ζ · e) (52)

u∗= K1 + K2 · xs + K3 · e (53)

u∗= K1 + K2 · xs + K3 · e + K4 · r (54)

where:

• Kp = ϱ = 200• Kv = υ = 20• ζ = 10.

In the following Figs. 11–13, we can observe clearly the differ-ence between our proposed approach and other controllers. Themain conclusion that we infer by these additional results, is thatevery time the adaptation part of our sliding mode approach is in-cluded in the controller, the results are satisfactory. In all these re-sults, the moving obstacle avoidance is a priority. This means thatthe robot avoids the obstacle even if it does not perform its task(Figs. 11–13).When the sliding part is not present in the controller,which means that there is no change adaptation dynamics causedby the sudden arrival of the obstacle, the robot is completely unbal-anced and essentially tries to avoid the obstacle in the detriment

of the task. However, when the adaptive sliding mode is a part ofthe controller, even alone, it offers the robot, the adaptation capac-ity so that it can both perform the task and avoid collision with theobstacle. Of course, if themoving obstaclemakes the task unattain-able by the effector of the robot, no controllermay exceed the robotworkspace that is defined by its physical characteristics.

For more illustrations, we give another kind of trajectory (seeFig. 14).

5. Conclusion

Collision avoidance of moving obstacles is only limited by thebandwidth of the robot. In our developments, there is no impact,whether the frequency is high or not. Our proposition concerns thecontrol of redundant serial robots constrained by moving obsta-cles. The considered a priori knowledge is only the model struc-ture of robot manipulators. No on line inversion of the matricesis considered. The control is calculated in one step and directly inthe operational space. Also no inversion model is specified in ourapproach. The problem of mobile obstacles collision avoidance istransformed into optimization problem. The theoretical study con-ducted andpresented is general and valid for any robot constrainedby obstacles. For the simulation results we considered an eight-shaped trajectory to have more excitation about Coriolis dynam-ics. The obstacle hinders the first part of the robot motion and hasa variable speed. Regarding the comparison with other controllers,we can conclude the effectiveness of the proposed approach.

T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564 563

Fig. 11. Tracking results after adaptation period.

Fig. 12. Tracking results after adaptation period.

Fig. 13. Distance between the obstacle and the first part of the robot.

564 T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564

Fig. 14. Additional trajectory.

References

[1] S. Yahya, M. Moghawemi, H.A.F. Mohamed, Geometrical approach of pla-nar hyper-redundant manipulators: inverse kinematics, path planning andworkspace, Simulation Modelling Practice and Theory 19 (1) (2011) 406–422.

[2] A. Benallegue, B. Daachi, N.K. M’Sirdi, Stable neural network adaptive controlof redundant robot manipulators, in: The 10th International Conference onAdvanced Robotics, ICAR’2001, Budapest, Hungary, August 22–25, 2001.

[3] J. Bailleul, Kinematic programming alternatives for redundant manipulators,in: Proc. IEEE Int. Conf. On Robotics and Automation, St. Louis, 1985.

[4] C.A. Klein, C.H. Huang, Review of pseudoinverse control for use with kine-matically redundant manipulators, IEEE Transactions on Systems Man andCybernetics (1983) 245–250.

[5] R. Koker, C. Oz, T. Cakar, H. Ekiz, A study of neural network based inverse kine-matics solution, Robotics and Autonomous Systems 49 (3–4) (2004) 227–234.

[6] B. Achili, B. Daachi, Y. Amirat, A. Ali-Cherif, Robust adaptive control for aparallel robot, International Journal of Control 83 (10) (2010) 2107–2119.

[7] B. Daachi, A. Benallegue, A neural network adaptive controller for end-effectortracking of redundant robot manipulators, Journal of Intelligent and RoboticSystems (Springer) 46 (3) (2006) 245–262.

[8] Naveen Kumar, Vikas Panwarb, N. Sukavanamc, S.P. Sharmac, J.H. Borma,Neural network-based nonlinear tracking control of kinematically redundantrobot manipulators, Mathematical and ComputerModelling Journal 53 (2011)1889–1901.

[9] Brice Le Boudec, Maarouf Saad, Vahé Nerguizian, Modeling and adaptivecontrol of redundant robots, Journal of Mathematics and Computers inSimulation 71 (2006) 395–403.

[10] Rui YAN, Keng Peng TEE, Haizhou LI, Adaptive learning tracking controlof robotic manipulators with uncertainties, Journal of Control Theory andApplications 8 (2) (2010) 160–165.

[11] B. Daachi, T. Madani, A. Benallegue, Adaptive neural controller for redun-dant robot manipulators and collision avoidance with mobile obstacles,Neurocomputing 79 (2012) 50–60.

[12] H.P. Singh, N. Sukavanam,Neural network based control scheme for redundantrobotmanipulators subject tomultiple self-motion criteria, Mathematical andComputer Modelling (2011) http://dx.doi.org/10.1016/j.mcm.2011.10.007.

[13] A.V. Topalov, O. Kaynak, Neuro-adaptive SM tracking control of robot manip-ulators, International Journal of Adaptive Control and Signal Processing 21(2007) 674–691.

[14] T. Madani, A. Benallegue, Backstepping sliding mode control applied to aminiature quadrotor flying robot, in: IECON—32nd Annual Conference onIEEE Industrial Electronics, 2006, pp. 700–705.

[15] D. Boukhetala, F. Boudjema, T. Madani, M.S. Boucherit, N.K. M’Sirdi, A newdecentralized variable structure control for robot manipulators, InternationalJournal of Robotics and Automation 18 (1) (2003) 28–40.

[16] M. Ertugrul, O. Kaynak, Neuro sliding mode control of robotic manipulators,Mechatronics 10 (2000) 239–263.

[17] T. Ahmed-Ali, F. Lamnabhi-Lagarrigue, Sliding observer–controller designfor uncertain triangular nonlinear systems, IEEE Transactions on AutomaticControl 44 (1999) 1244–1249.

[18] T.P. Leung, Q.J. Zhou, Chun Yi Su, An adaptive variable structure model follow-ing control design for robot manipulators, IEEE Transactions on AutomaticControl AC-36 (1991) 347–353.

[19] V.I. Utkin, Sliding Modes and their Application in Variable Structure Systems,MIR, Moscow, 1978.

[20] S.V. Emelianov, Variable Structure Control Systems, Nauka, Moscow, 1967.[21] A. Ramdane-Cherif, Inversion Des Modèles Géométrique et Cinématique

d’un Robot Redondant: Une Solution Neuronale Adaptative, Ph.D. Thesis,Université Pierre et Marie Curie, 1998.

[22] J.J.E. Slotine, W. Li, Adaptive manipulator control: a case study, IEEE Transac-tions on Automatic Control 33 (1988) 995–1003.

Dr. Tarek Madani received the engineering and Magisterdegrees in automatic control for electrical engineeringfrom the Ecole Nationale Polytechnique, Algiers, Algeria,in 1997 and 2000, respectively, and the Ph.D. degree inautomatic control and robotics from the University ofVersailles in 2005. He was an Assistant Professor at theUniversity of Versailles from 2005 to 2007. He joined theLISV laboratory (in French, Laboratoire d’Ingénierie desSystémes de Versailles) where he involved in differentprojects. His main research interests artificial neuralnetworks control, adaptive control, backstepping control,

decentralized control, variable structure control and sliding mode observer appliedto robotics and nonlinear systems.

Boubaker Daachi is Associate Professor of robotics andcomputer science since 2003. He received his Ph.D. inRobotics from the University of Versailles in 2000, hisM.S. in robotics from the University of Paris 6 in 1998and his B.S in computer science from the University ofSétif (Algeria) in 1995. His research interests includeadaptive control of complex systems, soft computingand ambient intelligence. Application fields are robotics,pervasive and distributed systems. He has publishedmorethan 40 papers in scientific journals, books and conferenceproceedings. He has been involved in the organizing

committees of some national and international events.

Abdelaziz Benallegue received his B.Sc. in Electronicsfrom Ecole Nationale Polytechnique d’Alger, Algeria in1986, his M.Sc. (DEA) in Robotics and Ph.D. in Roboticsand Automatic Control from Université Pierre & MarieCurie, Paris, France, respectively in 1987 and 1991. Hewas an associate professor of Automatic control androbotics at the university of Pierre and Marie Curie, Paris6, France from 1992–2002. He joined the University ofVersailles St Quentin in September 2002 as Professor ofAutomatic Control and Robotics. His research interestsare mainly related to linear and nonlinear control theory

including adaptive control, robust control andneural network learning control,withapplications to robot manipulators and aerial vehicles.