- Email: [email protected]
fixed obstacles avoidance
Robotics and Autonomous Systems 61 (2013) 555–564
Contents lists available at SciVerse ScienceDirect
Robotics and Autonomous Systems journal homepage: www.elsevier.com/locate/robot
Adaptive variable structure controller of redundant robots with mobile/fixed obstacles avoidance Tarek 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, France
b
Laboratoire Image, Signaux & Systèmes Intelligents (LISSI), 122–124, rue Paul Armangot, 92400 Vitry sur Seine, France
highlights • • • • •
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.
article
info
Article history: Received 5 March 2012 Received in revised form 11 February 2013 Accepted 22 February 2013 Available online 15 March 2013 Keywords: Adaptive control Reference model Variable structure system Redundant robots Mobile obstacle avoidance
abstract In this paper, a variable structure adaptive controller is proposed for redundant robot manipulators constrained by moving obstacles. The main objective of the controller is to force the model states of the robot to track those of a chosen reference model. In addition, the controller is designed directly in Cartesian space and no knowledge on the dynamic model is needed, except its structure. The parameters of the controller are adapted using adaptive laws obtained via Lyapunov stability analysis of the closed loop. The performances of the proposed controller are evaluated using a 3 DOF robot manipulator evolving in a vertical plane constrained by a mobile obstacle. The obtained results show its effectiveness compared to other tested variable structure controllers. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The previous versions of robot manipulators were confined to make simple and repetitive tasks in known and static environments. Today’s robots are assigned to tasks that are more and more complex like ubiquitous robots that contribute more in the life of a human in order to recover it. Therefore, they are becoming faster and operate in dynamic environments. Hence, the challenge is to address the problem of self-tuning and adaptation regarding the environment and possible real time changes. Using redundant robot manipulators [1–4] can be the solution to this problem. In fact, they are dexterous and able to avoid obstacles even if they are mobile (see Fig. 1). In order to accomplish its task, the robot has to be controlled in a way that it has to avoid collision with obstacles and to ensure their security, for instance in the automotive field the
∗
Corresponding author. Tel.: +33 141807324. E-mail address: [email protected] (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 and gears. In general, the required task is specified in Cartesian space, also called the operational space of the robot. Therefore, it is more practical to synthesize control laws directly in this space which represents the evolving space of the robot. Thus, the controller synthesis of robot manipulators is transformed into a problem of trajectory planning of joint space from Cartesian space constrained by mobile obstacles. Each obstacle can be wrapped in a sphere with greater radius in order to increase the security margin. The passage of a space to another is carried out via geometrical and kinematical transformations [5]. These transformations are based on direct and inverse models describing the relation between the Cartesian positions according to the joint positions (geometrical model) and Cartesian velocities according to the joint velocities (kinematics model). If the robot is redundant, there is are infinite solutions representing the various configurations of the robot for a given point in Cartesian space. Thus, the problem to be solved is to determine the optimal joint positions and velocities corresponding to the desired trajectories expressed in task space under mobile obstacles constraints.
556
T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564
Several works addressing, in general, the problem of adaptive control of redundant robots have been proposed in the literature. The work presented in [6] deals with the robust adaptive control tracking of a 6 degree of freedom parallel robot, called C5 parallel robot. The proposed approach is based on the coupling of sliding modes and multi-layers perceptron neural networks (MLP-NNs). It does not require the inverse dynamic model for deriving the control law. In our previous work [7], we used adaptive neuralnetworks to control redundant robots in task space without the presence of mobile obstacles. In [8], neural network-based nonlinear dynamical control of kinematically redundant robot manipulators is also proposed; it is based on several assumptions that are not always satisfied. In [9], the authors present an adaptive control scheme for a redundant robot subject to spatial constraints. The functions of the dynamic model are partially known. It means that the dynamical model can be written as a known regressor multiplied by unknown vector of parameters [10]. The main drawback of the use of neural-networks approach to control redundant robots is a great number of parameters [11,12]. The interest of this paper is in the synthesis of a variable structure adaptive controller [13–20]. We use an optimization strategy based on energy criterion, allowing the robot to use optimal joint configurations and avoid any collision with possible mobile obstacles. We combine the variable structure systems, adaptive control and reference model, to propose a robust and self-adjusting control law in the Cartesian space of the robot manipulator. It should be noted that the proposed control law does not require inversion of geometric and kinematic models. The knowledge of the robot dynamic model is also not required. The paper is organized as follows: Section 2 describes the dynamics of the robot manipulator, its structural properties and the principle of avoidance of moving obstacles. Section 3 presents the proposed controller with the analysis of stability of the system in closed loop. Simulation results on a robot plane with 3 DOF are presented and discussed in Section 4. Finally, the conclusion is presented in Section 5.
Fig. 1. The problem of obstacles avoidance.
that represent a set of variables depending on the obstacle movement [11]. The choice of the variable h is given by: h(q, xo ) = N T
∂ Ω (q, xo ) ∂q
(3)
where the matrix N represents the null space of the Jacobian matrix J and Ω (q, xo ) represents a criterion to be optimized [21]. The problem of the controller is to track any desired trajectory and avoids any possible collision with mobile obstacles. To overcome this problem and in the same time to avoid any knowledge on the dynamic model of the system (the matrices M ∗ , C ∗ and the vector H ∗ are unknown), we propose to use sliding modes technique and reference model.
2. Problem formulation Let us consider a n-DOF robot manipulator whose dynamic model is expressed in extended Cartesian space as given in [11]: M ∗ (q, xo )X¨ + C ∗ (xo , x˙ o , q, q˙ )X˙ + H ∗ (q, q˙ , xo , x˙ o , x¨ o ) = u∗
(1)
u = J T u∗
(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 , x˙ o and x¨ o represent respectively, the positions, the velocities and the accelerations of obstacles. X , X˙ and X¨ are given by:
X =
x h
X˙ =
x˙ h˙
X¨ =
x¨ h¨
with x, x˙ and x¨ representing respectively, the positions, the velocities and the accelerations in the Cartesian space of the effector (their dimension is m). h, h˙ and h¨ of dimension (n − m), are vectors
3. Variable structure control law synthesis The control problem of redundant robot manipulators having uncertainties in the dynamic model and often disturbed by fixed or mobile obstacles in their environment is a real challenge. Considerable research has focused on developing advanced manipulator control systems, and some techniques have emerged such as the computed torque method, which requires precise knowledge of the manipulator parameters and dynamic model [22]. In this case, the good performance of the control system depends on the high fidelity of the mathematical model used to describe the behavior dynamics. For this purpose, an adaptive variable structure controller with reference model has been proposed in this section. The control law which does not require any prior knowledge of the dynamical 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 control law without using any a priori knowledge concerning the robot dynamics. The principal goal is to realize tracking of a given reference model in the augmented task space. Let xs ∈ ℜ2n be the state vector defined by:
xs =
X . X˙
(4)
T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564
The Eq. (1) can be represented by the following state equation:
x˙ s =
0n×n A1
In×n A2
A(q,˙q,xo ,˙xo )
xs +
0n×n B1
u∗ +
0n H1
3.2. Control law synthesis
For the control law, we propose:
(5)
u∗ = K1 + K2 · xs + K3 · e + K4 · r
B(q,xo )
557
H (q,˙q,xo ,˙xo ,¨xo )
(13)
with:
where In×n ∈ ℜn×n is the identity matrix and:
B1 (q, xo ) = M ∗−1 (q, xo ) ∈ ℜn×n
K = K¯ 1 + ∆K1 ∈ ℜn 1 K2 = K¯ 2 + ∆K2 sgn(xs )T ∈ ℜn×2n K = K¯ 3 + ∆K3 sgn(e)T ∈ ℜn×2n 3 K4 = K¯ 4 + ∆K4 sgn(r )T ∈ ℜn×n .
H1 (q, q˙ , xo , x˙ o , x¨ o ) = −M ∗−1 (q, xo )H ∗ (q, q˙ , xo , x˙ o , x¨ o ) ∈ ℜn .
K¯ i are constant terms and ∆Ki ∈ ℜn are variable terms given by:
A1 = 0n×n A2 (q, q˙ , xo , x˙ o ) = −M ∗−1 (q, xo )C ∗ (q, q˙ , xo , x˙ o ) ∈ ℜn×n
(6)
The controller objective is to force the model states (5) to track those of the reference model characterized by the state vector xm ∈ ℜ2n given by:
Xm X˙ m
xm =
(7)
2
for i = 1, 2 κˆ i (αi s + sgn(s))
where α{·} ∈ ℜ gains with:
0n×n Am1
2 κˆ i0 (αi0 s + sgn(s)) + κˆ ij αij s + sgn(s) ∥e∥j
∗+
and: x˙ m =
∆ Ki =
In × n Am2
Am
0n Bm1
xm +
r
(8)
Bm
where Am ∈ ℜ2n×2n is the evolution matrix, Bm ∈ ℜ2n×n is the input vector and r ∈ ℜn is the input of the reference model. Let the tracking error be denoted by: e = xm − xs .
(9)
(14)
j=1
(15)
for i = 3, 4
are positive constants and κˆ {·} ∈ ℜ+ are positive
· κˆ 10 = w10 (∥s∥1 − β10 κˆ 10 ∥s∥22 ) · j 2 κˆ 1j = w1j (∥s∥1 − β1j κˆ 1j ∥s∥2 ) ∥e∥2 · κˆ 20 = w20 (∥s∥1 − β20 κˆ 20 ∥s∥22 ) ∥x∥1 · κˆ 2j = w2j (∥s∥1 − β2j κˆ 2j ∥s∥22 ) ∥e∥j2 ∥x∥1 · κˆ 3 = w3 (∥s∥1 − β3 κˆ 3 ∥s∥22 ) ∥e∥1 · κˆ 4 = w4 (∥s∥1 − β4 κˆ 4 ∥s∥22 ) ∥r ∥1
for j = 1, 2 for j = 1, 2
(16)
The problem is as follows: for a given vector r, determine a stable sliding surface s = 0n so that, in ideal sliding regime, the tracking error e tends to 0n when t tends to ∞. Finally we have to synthesize the control law u∗ ensuring the attractiveness to the sliding surface s = 0n . Let s ∈ ℜn the vector of switching laws be defined by:
w{·} ∈ ℜ∗+ are arbitrary positive adaptation gains and β{·} ∈ ℜ∗+
s = Λe
The control (13) has a variable structure. It contains four changes of structure:
(10)
where Λ = [Λ1 , In×n ] ∈ ℜn×2n and Λ1 is a square matrix of dimensions (n × n). T T
Consider e = ϵ , ϵ˙ with ϵ = (Xm − X ) ∈ ℜ . In ideal sliding regime, the switching law (10) is close to zero and the error dynamics is given by ϵ˙ = −Λ1 ϵ . The sliding surface s = 0n is stable if and only if the matrix Λ1 is positive definite. Before presenting our variable structure control, it is necessary to define some basic functions.
T
n
Definition 1. The function ∥·∥p for p ≥ 1 represents the Hölder norm. It is defined by:
∥·∥p : ℜn → ℜ+ x1 p 1 .. p . → |x1 | + · · · + |xn | p .
(11)
αind ≥ ξind > 0, βind
∀ ind ∈ {10, 11, 12, 20, 21, 22, 3, 4}
Definition 2. For all matrix A ∈ ℜm×n the norm ∥·∥p is given by: (12)
(17)
ξind represents constants defined later (Eq. (24)).
• • • •
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 ∈ ℜ2n×2n be a matrix defined by:
A0 =
0n×n
Λ1
−I n × n . In×n + Λ1
(18)
The matrix A0 is chosen to verify:
ΛA0 = Λ.
xn
∥·∥p : ℜm×n → ℜ+ ∥Ax∥p A → sup with x ∈ ℜn . ∥x∥p ̸=0 ∥x∥p
are positive constants given by:
(19)
One can notice that (I2n×2n − A0 ) represents the null space of the matrix Λ. It means Λ(I2n×2n − A0 ) = 0n×2n . 1 ´ Consider B´ = 0n×n , B− ∈ ℜn×2n the pseudo-inverse of B (BB 1 = In×n ). So:
´ H = BBH (Am − A) = BB´ (Am − A) (A + A0 ) = BB´ (Am + A0 ) m ´ m. Bm = BBB
(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:
3.3. Stability study
Assumption 1. The matrix B1 is not a zero matrix, bounded and positive definite. One can consider the following inequalities: 0 < ρ1 ≤ ∥B1 ∥2 ≤ ρ2
and ρ1 ≤ λmin (B1 )
´ B(Am + A0 ) − K¯ 3 ≤ δ3 2 ´ BBm − K¯ 4 ≤ δ4
(21)
2
where ρ{1,2} and δ{3,4} are unknown bounded constants that are strictly positive and λmin (B1 ) is the smallest eigenvalue of the matrix B1 . Assumption 2. The obstacle positions, velocities and accelerations are assumed known and bounded. The matrix B´ and the vector H satisfy: 2 ´ δ1j ∥e∥j2 BH + K¯ 1 ≤ δ10 + 2
(22)
j=1
where δ1{·} are unknown constants bounded and strictly positive. Assumption 3. Matrices B´ and A satisfy: 2 ´ δ2j ∥e∥j2 B(Am − A) − K¯ 2 ≤ δ20 + 2
1 T s s + ρ1 θ˜ T W −1 θ˜ 2
(27)
where W = diag [w10 , w11 , w12 , w20 , w21 , w22 , w3 , w4 ] and θ˜ = (θ − θˆ ) with: T θˆ = κˆ 10 , κˆ 11 , κˆ 12 , κˆ 20 , κˆ 21 , κˆ 22 , κˆ 3 , κˆ 4 (28) represents the estimation of the vector:
θ = [κ10 , κ11 , κ12 , κ20 , κ21 , κ22 , κ3 , κ4 ]T ρ2 αind ≥ κind ≥ δind , βind ρ1 ∀ ind ∈ {10, 11, 12, 20, 21, 22, 3, 4} .
(29)
(30)
V˙ = sT s˙ − ρ1 θ˜ T W −1 θˆ .
(31)
j =1
∀ ind ∈ {10, 11, 12, 20, 21, 22, 3, 4} .
(24)
Remark 2. The matrix B1 is positive definite and matrix theory shows that B1 ≥ λmin (B1 )In×n . Therefore, using Assumption 1: sT B1 s ≥ λmin (B1 ) sT s ≥ ρ1 ∥s∥22
(25)
∥s∥22
s B1 sgn(s) ≥ λmin (B1 ) s sgn(s) ≥ ρ1 ∥s∥1 . T
V =
·
(23)
Remark 1. Due to the mechanical and physical characteristics of robot manipulators (1), the stability of the reference model (8) and the boundedness of obstacles motion, the Assumptions 1–3 are valid. So, the constants ξind defined in (17) can be fixed such as:
ρ2 , ρ1
Proof. Consider the Lyapunov function:
The derivative of V with respect to the time:
where δ2{·} are unknown constants bounded and strictly positive.
ξind ≥ δind
Let us consider the system (1) and suppose that the Assumptions 1–3 hold true. The controller defined by Eqs. (13)–(15) with adaptation laws (16), guarantees boundedness of all the control signals and attractiveness to the sliding surface s = 0n .
T
∥s∥1
(26)
Using Eqs. (13), (19) and (20), we obtain: s˙ = Λe˙
= Λ{Am xm + Bm r − Ax − Bu − H } = Λ{−A0 e − H + (Am − A)x + (Am + A0 )e + Bm r − Bu} ´ + BB´ (Am − A)x + BB´ (Am + A0 )e = − ΛA0 e +Λ −BBH s ´ m r − B (K1 + K2 x + K3 e + K4 r ) + BBB ´ + K1 ] + [B´ (Am − A) − K2 ]x = −s + ΛB {−[BH B1
´ m − K4 ]r }. + [B´ (Am + A0 ) − K3 ]e + [BB
(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 + V˙ 1 + V˙ 2 + V˙ 3 + V˙ 4
(33)
T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564
where:
κˆ 10 +
V˙ 3 ≤ −ρ1 ∥s∥1
´ + K¯ 1 ] + [B´ (Am − A) − K¯ 2 ]x V˙ 1 = sT B1 {−[BH
´ m − K¯ 4 ]r } + [B´ (Am + A0 ) − K¯ 3 ]e + [BB 2 j ˙V2 = −sT B1 s κˆ 10 α10 + κˆ 1j α1j ∥e∥2
+
(34)
κˆ 20 +
2
κˆ 20 α20 +
+
κˆ 2j α2j ∥e∥
j 2
κ20 +
(35)
κˆ 20 +
+
V˙ 42 = ρ1 ∥s∥1
κˆ 1j ∥e∥
+ (36)
(37) 2
κ˜ 20 +
+
2
+
∥x∥1 + κˆ 3 ∥e∥1 + κˆ 4 ∥r ∥1
κ20 β20 κˆ 20 +
2 j =1
2
κ1j β1j κˆ 1j ∥e∥
j 2
κ2j β2j κˆ 2j ∥e∥j2 ∥x∥1 .
From (33) and using the sum of (39)–(42), it becomes:
V˙ ≤ −sT s − P1 ∥s∥1 − P2 ∥s∥22
2
P1 = ψ1 + ψ2 ∥x∥1 + ψ3 ∥e∥1 + ψ4 ∥r ∥1
j =1
j =1
ψi =
.
(44)
where:
κ˜ 2j β2j κˆ 2j ∥e∥j2 ∥x∥1
+ κ˜ 3 β3 κˆ 3 ∥e∥1 + κ˜ 4 β4 κˆ 4 ∥r ∥1
(43)
with:
κ˜ 1j β1j κˆ 1j ∥e∥j2
2
κ˜ 2j ∥e∥j2 ∥x∥1 + κ˜ 3 ∥e∥1 + κ˜ 4 ∥r ∥1
κ˜ 20 β20 κˆ 20 +
j 2
j 2
+ κ3 β3 κˆ 3 ∥e∥1 + κ4 β4 κˆ 4 ∥r ∥1
∥x∥1 + κ3 ∥e∥1 + κ4 ∥r ∥1
κˆ 1j ∥e∥
κ10 β10 κˆ 10 +
j =1
κ˜ 1j ∥e∥j2
κ˜ 10 β10 κˆ 10 +
κˆ 2j ∥e∥
j =1
+ ρ1 ∥s∥22
2
V˙ 43 = ρ1 ∥s∥22
j =1
j 2
j=1
·
V˙ 4 = −ρ1 θ˜ T W −1 θˆ
κ˜ 10 +
κˆ 20 +
2
j =1
= −ρ1 ∥s∥1
(41)
κ1j ∥e∥j2 j=1
κ2j ∥e∥
κˆ 10 +
κˆ 2j ∥e∥j2 ∥x∥1 + κˆ 3 ∥e∥1 + κˆ 4 ∥r ∥1
∥x∥1 + κˆ 3 ∥e∥1 + κˆ 4 ∥r ∥1
2
j=1
j 2
2
j =1 2
j=1
+ κˆ 3 α3 ∥e∥1 + κˆ 4 α4 ∥r ∥1 2
j 2
(42)
∥x∥1 +
κˆ 2j ∥e∥
κ10 +
V˙ 41 = −ρ1 ∥s∥1
κˆ 10 +
κˆ 1j ∥e∥j2 j =1
j=1
V˙ 3 = −sT B1 sgn(s)
2
V˙ 4 ≤ V˙ 41 + V˙ 42 + V˙ 43
2
j =1
j =1
+
559
(38)
2 (ρ1 κi0 − ρ2 δi0 ) + (ρ1 κij − ρ2 δij ) ∥e∥j2 j =1
for i = 1, 2 (ρ1 κi − ρ2 δi ) for i = 3, 4
(45)
and: Using the Assumptions 1–3, the two inequalities (25) and (26), and the property ∥·∥1 ≥ ∥·∥2 , we have:
P2 = ρ1 (ϕ1 + ϕ2 ∥x∥1 + ϕ3 ∥e∥1 + ϕ4 ∥r ∥1 )
(46)
where:
´ V˙ 1 ≤ ∥s∥2 ∥B1 ∥2 BH + K¯ 1 + B´ (Am − A) − K¯ 2 ∥x∥2 2 2 ´ ¯ + B´ (Am + A0 ) − K¯ 3 ∥e∥2 + BB m − K4 ∥r ∥2 2 2 2 j ≤ ρ 2 ∥s ∥1 δ10 + δ1j ∥e∥2 j =1 2 j + δ20 + δ2j ∥e∥2 ∥x∥1 + δ3 ∥e∥1 + δ4 ∥r ∥1
ϕi =
(39)
j =1
V˙ 2 ≤ −ρ1 ∥s∥22
κˆ 10 α10 +
2 j =1
+
κˆ 20 α20 +
2 j =1
j=1
for i = 1, 2 (αi − βi κi )κˆ i for i = 3, 4.
(47)
According to (30) we can say that the two quantities P1 and P2 are positive because:
ρ1 κind − ρ2 δind ≥ 0 , αind − βind κind ≥ 0
∀ ind ∈ {10, 11, 12, 20, 21, 22, 3, 4} . (48) T Consequently, the Eq. (43) implies V˙ ≤ −s s. Hence if s ̸= 0n we have V˙ < 0 and if s = 0n we have V˙ = 0. The switching law s and all estimation errors θ˜ are bounded. Invoking Barbalat’s Lemma, we can say that s goes to zero because V¨ is checked bounded. Finally,
κˆ 1j α1j ∥e∥j2
κˆ 2j α2j ∥e∥j2 ∥x∥1
+ κˆ 3 α3 ∥e∥1 + κˆ 4 α4 ∥r ∥1
2 (αi0 − βi0 κi0 )κˆ i0 + (αij − βij κij )κˆ ij ∥e∥j2
(40)
the control law (13) ensures the attraction to the sliding surface s = 0n .
560
T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564
variable terms ∆K{·} in (15) are minimized and the control energy of (2) will be optimized. 4. Simulation results In this section, we consider a 3DOF robot manipulator evolved in the vertical plan. The first segment is constrained by a mobile obstacle (Fig. 3). The radius of the circle enclosing the moving obstacle is directly integrated into our calculations (radius = 0.05 m and center xo = (x1o , x2o )). Variable or fixed, the calculation of the distance between the robot parts and the obstacle is conducted online. It means that we do not ask if the radius has changed or will change. The motion of the obstacle is carried out according to the following equations:
x1o = 0.1 x2o = 0.2 × (1 − |sin(π fo t )|) fo = 0.5 Hz.
(49)
The matrices of the reference model are given as follows: Am1 = Am2 = Bm1 = I3×3 . Fig. 3. 3DOF redundant robot constrained with one mobile obstacle.
The reference trajectory r is: 1 + 0.1 sin(4π fo t ) + 0.4π fo cos(4π fo t ) − 1.6π 2 fo2 sin(4π fo t ) r = 0.1 cos(2π fo t ) − 0.2π fo sin(2π fo t ) − 0.4π 2 fo2 cos(2π fo t ) . 0
These equations imply that the desired trajectory represents an eight-shaped trajectory (Fig. 4). To ensure the avoidance of the obstacle, the desired trajectories hd , h˙ d and h¨ d are defined by: hd = h˙ d = h¨ d = 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) K¯ 4 = 2 × ones(3, 1) K¯ 2 = K¯ 3 = (C C ) K¯ 4 = C Fig. 4. Desired eight-shaped trajectory.
Remark 3. The constant terms K¯ {·} in (14) can be chosen in a manner to reduce the bounds of δ{·} in (21)–(23). This permits to reduce the inferior bounds of κ{·} in (30). Consequently the adaptive
C = 2 × ones(3, 3)
The initial configurations of the robot and the obstacle are given in Fig. 5. One can notice that this configuration is singular but our approach permits to overcome it using the redundancy of the robot. Figs. 6 and 7 show the effectiveness of the proposed controller in both trajectory tracking and obstacle avoidance. Like only the first part of the robot is constrained by the obstacle, the robot 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 are satisfactory and confirm the good quality of tracking presented in task space. Moreover, one can easily notice on the curves of the Fig. 9 an optimization of joint displacements and consequently the
use of a minimal energy. Also it can be concluded that having a good trajectory tracking in task space does not imply obtaining joint trajectories closed to joint trajectories relative to the desired task.
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 of the robot within their corresponding obstacle positions (Fig. 10). We can easily see that the robot uses its redundancy to realize the assigned task and at the same time avoids any collision of its first part with the mobile obstacle. As a comparison, we have considered the following control laws: u∗ = Kp · e + Kv · e˙ + ϱ · sign(e)
(51)
∗
u = Kp · e + Kv · e˙ + υ · sign(˙e + ζ · e)
(52)
∗
(53)
∗
(54)
u = K1 + K2 · xs + K3 · e u = K1 + K2 · xs + K3 · e + K4 · r where:
• Kp = ϱ = 200 • Kv = υ = 20 • ζ = 10. In the following Figs. 11–13, we can observe clearly the difference between our proposed approach and other controllers. The main conclusion that we infer by these additional results, is that every time the adaptation part of our sliding mode approach is included in the controller, the results are satisfactory. In all these results, the moving obstacle avoidance is a priority. This means that the 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 caused by the sudden arrival of the obstacle, the robot is completely unbalanced and essentially tries to avoid the obstacle in the detriment
of the task. However, when the adaptive sliding mode is a part of the controller, even alone, it offers the robot, the adaptation capacity so that it can both perform the task and avoid collision with the obstacle. Of course, if the moving obstacle makes the task unattainable by the effector of the robot, no controller may exceed the robot workspace that is defined by its physical characteristics. For more illustrations, we give another kind of trajectory (see Fig. 14). 5. Conclusion Collision avoidance of moving obstacles is only limited by the bandwidth of the robot. In our developments, there is no impact, whether the frequency is high or not. Our proposition concerns the control of redundant serial robots constrained by moving obstacles. The considered a priori knowledge is only the model structure of robot manipulators. No on line inversion of the matrices is considered. The control is calculated in one step and directly in the operational space. Also no inversion model is specified in our approach. The problem of mobile obstacles collision avoidance is transformed into optimization problem. The theoretical study conducted and presented is general and valid for any robot constrained by obstacles. For the simulation results we considered an eightshaped trajectory to have more excitation about Coriolis dynamics. The obstacle hinders the first part of the robot motion and has a 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
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.
563
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 planar hyper-redundant manipulators: inverse kinematics, path planning and workspace, Simulation Modelling Practice and Theory 19 (1) (2011) 406–422. [2] A. Benallegue, B. Daachi, N.K. M’Sirdi, Stable neural network adaptive control of redundant robot manipulators, in: The 10th International Conference on Advanced 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 kinematically redundant manipulators, IEEE Transactions on Systems Man and Cybernetics (1983) 245–250. [5] R. Koker, C. Oz, T. Cakar, H. Ekiz, A study of neural network based inverse kinematics 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 a parallel robot, International Journal of Control 83 (10) (2010) 2107–2119. [7] B. Daachi, A. Benallegue, A neural network adaptive controller for end-effector tracking of redundant robot manipulators, Journal of Intelligent and Robotic Systems (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 redundant robot manipulators, Mathematical and Computer Modelling Journal 53 (2011) 1889–1901. [9] Brice Le Boudec, Maarouf Saad, Vahé Nerguizian, Modeling and adaptive control of redundant robots, Journal of Mathematics and Computers in Simulation 71 (2006) 395–403. [10] Rui YAN, Keng Peng TEE, Haizhou LI, Adaptive learning tracking control of robotic manipulators with uncertainties, Journal of Control Theory and Applications 8 (2) (2010) 160–165. [11] B. Daachi, T. Madani, A. Benallegue, Adaptive neural controller for redundant 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 redundant robot manipulators subject to multiple self-motion criteria, Mathematical and Computer 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 manipulators, International Journal of Adaptive Control and Signal Processing 21 (2007) 674–691. [14] T. Madani, A. Benallegue, Backstepping sliding mode control applied to a miniature quadrotor flying robot, in: IECON—32nd Annual Conference on IEEE Industrial Electronics, 2006, pp. 700–705. [15] D. Boukhetala, F. Boudjema, T. Madani, M.S. Boucherit, N.K. M’Sirdi, A new decentralized variable structure control for robot manipulators, International Journal 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 design for uncertain triangular nonlinear systems, IEEE Transactions on Automatic Control 44 (1999) 1244–1249. [18] T.P. Leung, Q.J. Zhou, Chun Yi Su, An adaptive variable structure model following control design for robot manipulators, IEEE Transactions on Automatic Control 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 Transactions on Automatic Control 33 (1988) 995–1003.
Dr. Tarek Madani received the engineering and Magister degrees in automatic control for electrical engineering from the Ecole Nationale Polytechnique, Algiers, Algeria, in 1997 and 2000, respectively, and the Ph.D. degree in automatic control and robotics from the University of Versailles in 2005. He was an Assistant Professor at the University of Versailles from 2005 to 2007. He joined the LISV laboratory (in French, Laboratoire d’Ingénierie des Systémes de Versailles) where he involved in different projects. His main research interests artificial neural networks control, adaptive control, backstepping control, decentralized control, variable structure control and sliding mode observer applied to robotics and nonlinear systems.
Boubaker Daachi is Associate Professor of robotics and computer science since 2003. He received his Ph.D. in Robotics from the University of Versailles in 2000, his M.S. in robotics from the University of Paris 6 in 1998 and his B.S in computer science from the University of Sétif (Algeria) in 1995. His research interests include adaptive control of complex systems, soft computing and ambient intelligence. Application fields are robotics, pervasive and distributed systems. He has published more than 40 papers in scientific journals, books and conference proceedings. He has been involved in the organizing committees of some national and international events.
Abdelaziz Benallegue received his B.Sc. in Electronics from Ecole Nationale Polytechnique d’Alger, Algeria in 1986, his M.Sc. (DEA) in Robotics and Ph.D. in Robotics and Automatic Control from Université Pierre & Marie Curie, Paris, France, respectively in 1987 and 1991. He was an associate professor of Automatic control and robotics at the university of Pierre and Marie Curie, Paris 6, France from 1992–2002. He joined the University of Versailles St Quentin in September 2002 as Professor of Automatic Control and Robotics. His research interests are mainly related to linear and nonlinear control theory including adaptive control, robust control and neural network learning control, with applications to robot manipulators and aerial vehicles.
Contents lists available at SciVerse ScienceDirect
Robotics and Autonomous Systems journal homepage: www.elsevier.com/locate/robot
Adaptive variable structure controller of redundant robots with mobile/fixed obstacles avoidance Tarek 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, France
b
Laboratoire Image, Signaux & Systèmes Intelligents (LISSI), 122–124, rue Paul Armangot, 92400 Vitry sur Seine, France
highlights • • • • •
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.
article
info
Article history: Received 5 March 2012 Received in revised form 11 February 2013 Accepted 22 February 2013 Available online 15 March 2013 Keywords: Adaptive control Reference model Variable structure system Redundant robots Mobile obstacle avoidance
abstract In this paper, a variable structure adaptive controller is proposed for redundant robot manipulators constrained by moving obstacles. The main objective of the controller is to force the model states of the robot to track those of a chosen reference model. In addition, the controller is designed directly in Cartesian space and no knowledge on the dynamic model is needed, except its structure. The parameters of the controller are adapted using adaptive laws obtained via Lyapunov stability analysis of the closed loop. The performances of the proposed controller are evaluated using a 3 DOF robot manipulator evolving in a vertical plane constrained by a mobile obstacle. The obtained results show its effectiveness compared to other tested variable structure controllers. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The previous versions of robot manipulators were confined to make simple and repetitive tasks in known and static environments. Today’s robots are assigned to tasks that are more and more complex like ubiquitous robots that contribute more in the life of a human in order to recover it. Therefore, they are becoming faster and operate in dynamic environments. Hence, the challenge is to address the problem of self-tuning and adaptation regarding the environment and possible real time changes. Using redundant robot manipulators [1–4] can be the solution to this problem. In fact, they are dexterous and able to avoid obstacles even if they are mobile (see Fig. 1). In order to accomplish its task, the robot has to be controlled in a way that it has to avoid collision with obstacles and to ensure their security, for instance in the automotive field the
∗
Corresponding author. Tel.: +33 141807324. E-mail address: [email protected] (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 and gears. In general, the required task is specified in Cartesian space, also called the operational space of the robot. Therefore, it is more practical to synthesize control laws directly in this space which represents the evolving space of the robot. Thus, the controller synthesis of robot manipulators is transformed into a problem of trajectory planning of joint space from Cartesian space constrained by mobile obstacles. Each obstacle can be wrapped in a sphere with greater radius in order to increase the security margin. The passage of a space to another is carried out via geometrical and kinematical transformations [5]. These transformations are based on direct and inverse models describing the relation between the Cartesian positions according to the joint positions (geometrical model) and Cartesian velocities according to the joint velocities (kinematics model). If the robot is redundant, there is are infinite solutions representing the various configurations of the robot for a given point in Cartesian space. Thus, the problem to be solved is to determine the optimal joint positions and velocities corresponding to the desired trajectories expressed in task space under mobile obstacles constraints.
556
T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564
Several works addressing, in general, the problem of adaptive control of redundant robots have been proposed in the literature. The work presented in [6] deals with the robust adaptive control tracking of a 6 degree of freedom parallel robot, called C5 parallel robot. The proposed approach is based on the coupling of sliding modes and multi-layers perceptron neural networks (MLP-NNs). It does not require the inverse dynamic model for deriving the control law. In our previous work [7], we used adaptive neuralnetworks to control redundant robots in task space without the presence of mobile obstacles. In [8], neural network-based nonlinear dynamical control of kinematically redundant robot manipulators is also proposed; it is based on several assumptions that are not always satisfied. In [9], the authors present an adaptive control scheme for a redundant robot subject to spatial constraints. The functions of the dynamic model are partially known. It means that the dynamical model can be written as a known regressor multiplied by unknown vector of parameters [10]. The main drawback of the use of neural-networks approach to control redundant robots is a great number of parameters [11,12]. The interest of this paper is in the synthesis of a variable structure adaptive controller [13–20]. We use an optimization strategy based on energy criterion, allowing the robot to use optimal joint configurations and avoid any collision with possible mobile obstacles. We combine the variable structure systems, adaptive control and reference model, to propose a robust and self-adjusting control law in the Cartesian space of the robot manipulator. It should be noted that the proposed control law does not require inversion of geometric and kinematic models. The knowledge of the robot dynamic model is also not required. The paper is organized as follows: Section 2 describes the dynamics of the robot manipulator, its structural properties and the principle of avoidance of moving obstacles. Section 3 presents the proposed controller with the analysis of stability of the system in closed loop. Simulation results on a robot plane with 3 DOF are presented and discussed in Section 4. Finally, the conclusion is presented in Section 5.
Fig. 1. The problem of obstacles avoidance.
that represent a set of variables depending on the obstacle movement [11]. The choice of the variable h is given by: h(q, xo ) = N T
∂ Ω (q, xo ) ∂q
(3)
where the matrix N represents the null space of the Jacobian matrix J and Ω (q, xo ) represents a criterion to be optimized [21]. The problem of the controller is to track any desired trajectory and avoids any possible collision with mobile obstacles. To overcome this problem and in the same time to avoid any knowledge on the dynamic model of the system (the matrices M ∗ , C ∗ and the vector H ∗ are unknown), we propose to use sliding modes technique and reference model.
2. Problem formulation Let us consider a n-DOF robot manipulator whose dynamic model is expressed in extended Cartesian space as given in [11]: M ∗ (q, xo )X¨ + C ∗ (xo , x˙ o , q, q˙ )X˙ + H ∗ (q, q˙ , xo , x˙ o , x¨ o ) = u∗
(1)
u = J T u∗
(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 , x˙ o and x¨ o represent respectively, the positions, the velocities and the accelerations of obstacles. X , X˙ and X¨ are given by:
X =
x h
X˙ =
x˙ h˙
X¨ =
x¨ h¨
with x, x˙ and x¨ representing respectively, the positions, the velocities and the accelerations in the Cartesian space of the effector (their dimension is m). h, h˙ and h¨ of dimension (n − m), are vectors
3. Variable structure control law synthesis The control problem of redundant robot manipulators having uncertainties in the dynamic model and often disturbed by fixed or mobile obstacles in their environment is a real challenge. Considerable research has focused on developing advanced manipulator control systems, and some techniques have emerged such as the computed torque method, which requires precise knowledge of the manipulator parameters and dynamic model [22]. In this case, the good performance of the control system depends on the high fidelity of the mathematical model used to describe the behavior dynamics. For this purpose, an adaptive variable structure controller with reference model has been proposed in this section. The control law which does not require any prior knowledge of the dynamical 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 control law without using any a priori knowledge concerning the robot dynamics. The principal goal is to realize tracking of a given reference model in the augmented task space. Let xs ∈ ℜ2n be the state vector defined by:
xs =
X . X˙
(4)
T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564
The Eq. (1) can be represented by the following state equation:
x˙ s =
0n×n A1
In×n A2
A(q,˙q,xo ,˙xo )
xs +
0n×n B1
u∗ +
0n H1
3.2. Control law synthesis
For the control law, we propose:
(5)
u∗ = K1 + K2 · xs + K3 · e + K4 · r
B(q,xo )
557
H (q,˙q,xo ,˙xo ,¨xo )
(13)
with:
where In×n ∈ ℜn×n is the identity matrix and:
B1 (q, xo ) = M ∗−1 (q, xo ) ∈ ℜn×n
K = K¯ 1 + ∆K1 ∈ ℜn 1 K2 = K¯ 2 + ∆K2 sgn(xs )T ∈ ℜn×2n K = K¯ 3 + ∆K3 sgn(e)T ∈ ℜn×2n 3 K4 = K¯ 4 + ∆K4 sgn(r )T ∈ ℜn×n .
H1 (q, q˙ , xo , x˙ o , x¨ o ) = −M ∗−1 (q, xo )H ∗ (q, q˙ , xo , x˙ o , x¨ o ) ∈ ℜn .
K¯ i are constant terms and ∆Ki ∈ ℜn are variable terms given by:
A1 = 0n×n A2 (q, q˙ , xo , x˙ o ) = −M ∗−1 (q, xo )C ∗ (q, q˙ , xo , x˙ o ) ∈ ℜn×n
(6)
The controller objective is to force the model states (5) to track those of the reference model characterized by the state vector xm ∈ ℜ2n given by:
Xm X˙ m
xm =
(7)
2
for i = 1, 2 κˆ i (αi s + sgn(s))
where α{·} ∈ ℜ gains with:
0n×n Am1
2 κˆ i0 (αi0 s + sgn(s)) + κˆ ij αij s + sgn(s) ∥e∥j
∗+
and: x˙ m =
∆ Ki =
In × n Am2
Am
0n Bm1
xm +
r
(8)
Bm
where Am ∈ ℜ2n×2n is the evolution matrix, Bm ∈ ℜ2n×n is the input vector and r ∈ ℜn is the input of the reference model. Let the tracking error be denoted by: e = xm − xs .
(9)
(14)
j=1
(15)
for i = 3, 4
are positive constants and κˆ {·} ∈ ℜ+ are positive
· κˆ 10 = w10 (∥s∥1 − β10 κˆ 10 ∥s∥22 ) · j 2 κˆ 1j = w1j (∥s∥1 − β1j κˆ 1j ∥s∥2 ) ∥e∥2 · κˆ 20 = w20 (∥s∥1 − β20 κˆ 20 ∥s∥22 ) ∥x∥1 · κˆ 2j = w2j (∥s∥1 − β2j κˆ 2j ∥s∥22 ) ∥e∥j2 ∥x∥1 · κˆ 3 = w3 (∥s∥1 − β3 κˆ 3 ∥s∥22 ) ∥e∥1 · κˆ 4 = w4 (∥s∥1 − β4 κˆ 4 ∥s∥22 ) ∥r ∥1
for j = 1, 2 for j = 1, 2
(16)
The problem is as follows: for a given vector r, determine a stable sliding surface s = 0n so that, in ideal sliding regime, the tracking error e tends to 0n when t tends to ∞. Finally we have to synthesize the control law u∗ ensuring the attractiveness to the sliding surface s = 0n . Let s ∈ ℜn the vector of switching laws be defined by:
w{·} ∈ ℜ∗+ are arbitrary positive adaptation gains and β{·} ∈ ℜ∗+
s = Λe
The control (13) has a variable structure. It contains four changes of structure:
(10)
where Λ = [Λ1 , In×n ] ∈ ℜn×2n and Λ1 is a square matrix of dimensions (n × n). T T
Consider e = ϵ , ϵ˙ with ϵ = (Xm − X ) ∈ ℜ . In ideal sliding regime, the switching law (10) is close to zero and the error dynamics is given by ϵ˙ = −Λ1 ϵ . The sliding surface s = 0n is stable if and only if the matrix Λ1 is positive definite. Before presenting our variable structure control, it is necessary to define some basic functions.
T
n
Definition 1. The function ∥·∥p for p ≥ 1 represents the Hölder norm. It is defined by:
∥·∥p : ℜn → ℜ+ x1 p 1 .. p . → |x1 | + · · · + |xn | p .
(11)
αind ≥ ξind > 0, βind
∀ ind ∈ {10, 11, 12, 20, 21, 22, 3, 4}
Definition 2. For all matrix A ∈ ℜm×n the norm ∥·∥p is given by: (12)
(17)
ξind represents constants defined later (Eq. (24)).
• • • •
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 ∈ ℜ2n×2n be a matrix defined by:
A0 =
0n×n
Λ1
−I n × n . In×n + Λ1
(18)
The matrix A0 is chosen to verify:
ΛA0 = Λ.
xn
∥·∥p : ℜm×n → ℜ+ ∥Ax∥p A → sup with x ∈ ℜn . ∥x∥p ̸=0 ∥x∥p
are positive constants given by:
(19)
One can notice that (I2n×2n − A0 ) represents the null space of the matrix Λ. It means Λ(I2n×2n − A0 ) = 0n×2n . 1 ´ Consider B´ = 0n×n , B− ∈ ℜn×2n the pseudo-inverse of B (BB 1 = In×n ). So:
´ H = BBH (Am − A) = BB´ (Am − A) (A + A0 ) = BB´ (Am + A0 ) m ´ m. Bm = BBB
(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:
3.3. Stability study
Assumption 1. The matrix B1 is not a zero matrix, bounded and positive definite. One can consider the following inequalities: 0 < ρ1 ≤ ∥B1 ∥2 ≤ ρ2
and ρ1 ≤ λmin (B1 )
´ B(Am + A0 ) − K¯ 3 ≤ δ3 2 ´ BBm − K¯ 4 ≤ δ4
(21)
2
where ρ{1,2} and δ{3,4} are unknown bounded constants that are strictly positive and λmin (B1 ) is the smallest eigenvalue of the matrix B1 . Assumption 2. The obstacle positions, velocities and accelerations are assumed known and bounded. The matrix B´ and the vector H satisfy: 2 ´ δ1j ∥e∥j2 BH + K¯ 1 ≤ δ10 + 2
(22)
j=1
where δ1{·} are unknown constants bounded and strictly positive. Assumption 3. Matrices B´ and A satisfy: 2 ´ δ2j ∥e∥j2 B(Am − A) − K¯ 2 ≤ δ20 + 2
1 T s s + ρ1 θ˜ T W −1 θ˜ 2
(27)
where W = diag [w10 , w11 , w12 , w20 , w21 , w22 , w3 , w4 ] and θ˜ = (θ − θˆ ) with: T θˆ = κˆ 10 , κˆ 11 , κˆ 12 , κˆ 20 , κˆ 21 , κˆ 22 , κˆ 3 , κˆ 4 (28) represents the estimation of the vector:
θ = [κ10 , κ11 , κ12 , κ20 , κ21 , κ22 , κ3 , κ4 ]T ρ2 αind ≥ κind ≥ δind , βind ρ1 ∀ ind ∈ {10, 11, 12, 20, 21, 22, 3, 4} .
(29)
(30)
V˙ = sT s˙ − ρ1 θ˜ T W −1 θˆ .
(31)
j =1
∀ ind ∈ {10, 11, 12, 20, 21, 22, 3, 4} .
(24)
Remark 2. The matrix B1 is positive definite and matrix theory shows that B1 ≥ λmin (B1 )In×n . Therefore, using Assumption 1: sT B1 s ≥ λmin (B1 ) sT s ≥ ρ1 ∥s∥22
(25)
∥s∥22
s B1 sgn(s) ≥ λmin (B1 ) s sgn(s) ≥ ρ1 ∥s∥1 . T
V =
·
(23)
Remark 1. Due to the mechanical and physical characteristics of robot manipulators (1), the stability of the reference model (8) and the boundedness of obstacles motion, the Assumptions 1–3 are valid. So, the constants ξind defined in (17) can be fixed such as:
ρ2 , ρ1
Proof. Consider the Lyapunov function:
The derivative of V with respect to the time:
where δ2{·} are unknown constants bounded and strictly positive.
ξind ≥ δind
Let us consider the system (1) and suppose that the Assumptions 1–3 hold true. The controller defined by Eqs. (13)–(15) with adaptation laws (16), guarantees boundedness of all the control signals and attractiveness to the sliding surface s = 0n .
T
∥s∥1
(26)
Using Eqs. (13), (19) and (20), we obtain: s˙ = Λe˙
= Λ{Am xm + Bm r − Ax − Bu − H } = Λ{−A0 e − H + (Am − A)x + (Am + A0 )e + Bm r − Bu} ´ + BB´ (Am − A)x + BB´ (Am + A0 )e = − ΛA0 e +Λ −BBH s ´ m r − B (K1 + K2 x + K3 e + K4 r ) + BBB ´ + K1 ] + [B´ (Am − A) − K2 ]x = −s + ΛB {−[BH B1
´ m − K4 ]r }. + [B´ (Am + A0 ) − K3 ]e + [BB
(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 + V˙ 1 + V˙ 2 + V˙ 3 + V˙ 4
(33)
T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564
where:
κˆ 10 +
V˙ 3 ≤ −ρ1 ∥s∥1
´ + K¯ 1 ] + [B´ (Am − A) − K¯ 2 ]x V˙ 1 = sT B1 {−[BH
´ m − K¯ 4 ]r } + [B´ (Am + A0 ) − K¯ 3 ]e + [BB 2 j ˙V2 = −sT B1 s κˆ 10 α10 + κˆ 1j α1j ∥e∥2
+
(34)
κˆ 20 +
2
κˆ 20 α20 +
+
κˆ 2j α2j ∥e∥
j 2
κ20 +
(35)
κˆ 20 +
+
V˙ 42 = ρ1 ∥s∥1
κˆ 1j ∥e∥
+ (36)
(37) 2
κ˜ 20 +
+
2
+
∥x∥1 + κˆ 3 ∥e∥1 + κˆ 4 ∥r ∥1
κ20 β20 κˆ 20 +
2 j =1
2
κ1j β1j κˆ 1j ∥e∥
j 2
κ2j β2j κˆ 2j ∥e∥j2 ∥x∥1 .
From (33) and using the sum of (39)–(42), it becomes:
V˙ ≤ −sT s − P1 ∥s∥1 − P2 ∥s∥22
2
P1 = ψ1 + ψ2 ∥x∥1 + ψ3 ∥e∥1 + ψ4 ∥r ∥1
j =1
j =1
ψi =
.
(44)
where:
κ˜ 2j β2j κˆ 2j ∥e∥j2 ∥x∥1
+ κ˜ 3 β3 κˆ 3 ∥e∥1 + κ˜ 4 β4 κˆ 4 ∥r ∥1
(43)
with:
κ˜ 1j β1j κˆ 1j ∥e∥j2
2
κ˜ 2j ∥e∥j2 ∥x∥1 + κ˜ 3 ∥e∥1 + κ˜ 4 ∥r ∥1
κ˜ 20 β20 κˆ 20 +
j 2
j 2
+ κ3 β3 κˆ 3 ∥e∥1 + κ4 β4 κˆ 4 ∥r ∥1
∥x∥1 + κ3 ∥e∥1 + κ4 ∥r ∥1
κˆ 1j ∥e∥
κ10 β10 κˆ 10 +
j =1
κ˜ 1j ∥e∥j2
κ˜ 10 β10 κˆ 10 +
κˆ 2j ∥e∥
j =1
+ ρ1 ∥s∥22
2
V˙ 43 = ρ1 ∥s∥22
j =1
j 2
j=1
·
V˙ 4 = −ρ1 θ˜ T W −1 θˆ
κ˜ 10 +
κˆ 20 +
2
j =1
= −ρ1 ∥s∥1
(41)
κ1j ∥e∥j2 j=1
κ2j ∥e∥
κˆ 10 +
κˆ 2j ∥e∥j2 ∥x∥1 + κˆ 3 ∥e∥1 + κˆ 4 ∥r ∥1
∥x∥1 + κˆ 3 ∥e∥1 + κˆ 4 ∥r ∥1
2
j=1
j 2
2
j =1 2
j=1
+ κˆ 3 α3 ∥e∥1 + κˆ 4 α4 ∥r ∥1 2
j 2
(42)
∥x∥1 +
κˆ 2j ∥e∥
κ10 +
V˙ 41 = −ρ1 ∥s∥1
κˆ 10 +
κˆ 1j ∥e∥j2 j =1
j=1
V˙ 3 = −sT B1 sgn(s)
2
V˙ 4 ≤ V˙ 41 + V˙ 42 + V˙ 43
2
j =1
j =1
+
559
(38)
2 (ρ1 κi0 − ρ2 δi0 ) + (ρ1 κij − ρ2 δij ) ∥e∥j2 j =1
for i = 1, 2 (ρ1 κi − ρ2 δi ) for i = 3, 4
(45)
and: Using the Assumptions 1–3, the two inequalities (25) and (26), and the property ∥·∥1 ≥ ∥·∥2 , we have:
P2 = ρ1 (ϕ1 + ϕ2 ∥x∥1 + ϕ3 ∥e∥1 + ϕ4 ∥r ∥1 )
(46)
where:
´ V˙ 1 ≤ ∥s∥2 ∥B1 ∥2 BH + K¯ 1 + B´ (Am − A) − K¯ 2 ∥x∥2 2 2 ´ ¯ + B´ (Am + A0 ) − K¯ 3 ∥e∥2 + BB m − K4 ∥r ∥2 2 2 2 j ≤ ρ 2 ∥s ∥1 δ10 + δ1j ∥e∥2 j =1 2 j + δ20 + δ2j ∥e∥2 ∥x∥1 + δ3 ∥e∥1 + δ4 ∥r ∥1
ϕi =
(39)
j =1
V˙ 2 ≤ −ρ1 ∥s∥22
κˆ 10 α10 +
2 j =1
+
κˆ 20 α20 +
2 j =1
j=1
for i = 1, 2 (αi − βi κi )κˆ i for i = 3, 4.
(47)
According to (30) we can say that the two quantities P1 and P2 are positive because:
ρ1 κind − ρ2 δind ≥ 0 , αind − βind κind ≥ 0
∀ ind ∈ {10, 11, 12, 20, 21, 22, 3, 4} . (48) T Consequently, the Eq. (43) implies V˙ ≤ −s s. Hence if s ̸= 0n we have V˙ < 0 and if s = 0n we have V˙ = 0. The switching law s and all estimation errors θ˜ are bounded. Invoking Barbalat’s Lemma, we can say that s goes to zero because V¨ is checked bounded. Finally,
κˆ 1j α1j ∥e∥j2
κˆ 2j α2j ∥e∥j2 ∥x∥1
+ κˆ 3 α3 ∥e∥1 + κˆ 4 α4 ∥r ∥1
2 (αi0 − βi0 κi0 )κˆ i0 + (αij − βij κij )κˆ ij ∥e∥j2
(40)
the control law (13) ensures the attraction to the sliding surface s = 0n .
560
T. Madani et al. / Robotics and Autonomous Systems 61 (2013) 555–564
variable terms ∆K{·} in (15) are minimized and the control energy of (2) will be optimized. 4. Simulation results In this section, we consider a 3DOF robot manipulator evolved in the vertical plan. The first segment is constrained by a mobile obstacle (Fig. 3). The radius of the circle enclosing the moving obstacle is directly integrated into our calculations (radius = 0.05 m and center xo = (x1o , x2o )). Variable or fixed, the calculation of the distance between the robot parts and the obstacle is conducted online. It means that we do not ask if the radius has changed or will change. The motion of the obstacle is carried out according to the following equations:
x1o = 0.1 x2o = 0.2 × (1 − |sin(π fo t )|) fo = 0.5 Hz.
(49)
The matrices of the reference model are given as follows: Am1 = Am2 = Bm1 = I3×3 . Fig. 3. 3DOF redundant robot constrained with one mobile obstacle.
The reference trajectory r is: 1 + 0.1 sin(4π fo t ) + 0.4π fo cos(4π fo t ) − 1.6π 2 fo2 sin(4π fo t ) r = 0.1 cos(2π fo t ) − 0.2π fo sin(2π fo t ) − 0.4π 2 fo2 cos(2π fo t ) . 0
These equations imply that the desired trajectory represents an eight-shaped trajectory (Fig. 4). To ensure the avoidance of the obstacle, the desired trajectories hd , h˙ d and h¨ d are defined by: hd = h˙ d = h¨ d = 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) K¯ 4 = 2 × ones(3, 1) K¯ 2 = K¯ 3 = (C C ) K¯ 4 = C Fig. 4. Desired eight-shaped trajectory.
Remark 3. The constant terms K¯ {·} in (14) can be chosen in a manner to reduce the bounds of δ{·} in (21)–(23). This permits to reduce the inferior bounds of κ{·} in (30). Consequently the adaptive
C = 2 × ones(3, 3)
The initial configurations of the robot and the obstacle are given in Fig. 5. One can notice that this configuration is singular but our approach permits to overcome it using the redundancy of the robot. Figs. 6 and 7 show the effectiveness of the proposed controller in both trajectory tracking and obstacle avoidance. Like only the first part of the robot is constrained by the obstacle, the robot 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 are satisfactory and confirm the good quality of tracking presented in task space. Moreover, one can easily notice on the curves of the Fig. 9 an optimization of joint displacements and consequently the
use of a minimal energy. Also it can be concluded that having a good trajectory tracking in task space does not imply obtaining joint trajectories closed to joint trajectories relative to the desired task.
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 of the robot within their corresponding obstacle positions (Fig. 10). We can easily see that the robot uses its redundancy to realize the assigned task and at the same time avoids any collision of its first part with the mobile obstacle. As a comparison, we have considered the following control laws: u∗ = Kp · e + Kv · e˙ + ϱ · sign(e)
(51)
∗
u = Kp · e + Kv · e˙ + υ · sign(˙e + ζ · e)
(52)
∗
(53)
∗
(54)
u = K1 + K2 · xs + K3 · e u = K1 + K2 · xs + K3 · e + K4 · r where:
• Kp = ϱ = 200 • Kv = υ = 20 • ζ = 10. In the following Figs. 11–13, we can observe clearly the difference between our proposed approach and other controllers. The main conclusion that we infer by these additional results, is that every time the adaptation part of our sliding mode approach is included in the controller, the results are satisfactory. In all these results, the moving obstacle avoidance is a priority. This means that the 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 caused by the sudden arrival of the obstacle, the robot is completely unbalanced and essentially tries to avoid the obstacle in the detriment
of the task. However, when the adaptive sliding mode is a part of the controller, even alone, it offers the robot, the adaptation capacity so that it can both perform the task and avoid collision with the obstacle. Of course, if the moving obstacle makes the task unattainable by the effector of the robot, no controller may exceed the robot workspace that is defined by its physical characteristics. For more illustrations, we give another kind of trajectory (see Fig. 14). 5. Conclusion Collision avoidance of moving obstacles is only limited by the bandwidth of the robot. In our developments, there is no impact, whether the frequency is high or not. Our proposition concerns the control of redundant serial robots constrained by moving obstacles. The considered a priori knowledge is only the model structure of robot manipulators. No on line inversion of the matrices is considered. The control is calculated in one step and directly in the operational space. Also no inversion model is specified in our approach. The problem of mobile obstacles collision avoidance is transformed into optimization problem. The theoretical study conducted and presented is general and valid for any robot constrained by obstacles. For the simulation results we considered an eightshaped trajectory to have more excitation about Coriolis dynamics. The obstacle hinders the first part of the robot motion and has a 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
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.
563
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 planar hyper-redundant manipulators: inverse kinematics, path planning and workspace, Simulation Modelling Practice and Theory 19 (1) (2011) 406–422. [2] A. Benallegue, B. Daachi, N.K. M’Sirdi, Stable neural network adaptive control of redundant robot manipulators, in: The 10th International Conference on Advanced 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 kinematically redundant manipulators, IEEE Transactions on Systems Man and Cybernetics (1983) 245–250. [5] R. Koker, C. Oz, T. Cakar, H. Ekiz, A study of neural network based inverse kinematics 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 a parallel robot, International Journal of Control 83 (10) (2010) 2107–2119. [7] B. Daachi, A. Benallegue, A neural network adaptive controller for end-effector tracking of redundant robot manipulators, Journal of Intelligent and Robotic Systems (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 redundant robot manipulators, Mathematical and Computer Modelling Journal 53 (2011) 1889–1901. [9] Brice Le Boudec, Maarouf Saad, Vahé Nerguizian, Modeling and adaptive control of redundant robots, Journal of Mathematics and Computers in Simulation 71 (2006) 395–403. [10] Rui YAN, Keng Peng TEE, Haizhou LI, Adaptive learning tracking control of robotic manipulators with uncertainties, Journal of Control Theory and Applications 8 (2) (2010) 160–165. [11] B. Daachi, T. Madani, A. Benallegue, Adaptive neural controller for redundant 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 redundant robot manipulators subject to multiple self-motion criteria, Mathematical and Computer 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 manipulators, International Journal of Adaptive Control and Signal Processing 21 (2007) 674–691. [14] T. Madani, A. Benallegue, Backstepping sliding mode control applied to a miniature quadrotor flying robot, in: IECON—32nd Annual Conference on IEEE Industrial Electronics, 2006, pp. 700–705. [15] D. Boukhetala, F. Boudjema, T. Madani, M.S. Boucherit, N.K. M’Sirdi, A new decentralized variable structure control for robot manipulators, International Journal 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 design for uncertain triangular nonlinear systems, IEEE Transactions on Automatic Control 44 (1999) 1244–1249. [18] T.P. Leung, Q.J. Zhou, Chun Yi Su, An adaptive variable structure model following control design for robot manipulators, IEEE Transactions on Automatic Control 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 Transactions on Automatic Control 33 (1988) 995–1003.
Dr. Tarek Madani received the engineering and Magister degrees in automatic control for electrical engineering from the Ecole Nationale Polytechnique, Algiers, Algeria, in 1997 and 2000, respectively, and the Ph.D. degree in automatic control and robotics from the University of Versailles in 2005. He was an Assistant Professor at the University of Versailles from 2005 to 2007. He joined the LISV laboratory (in French, Laboratoire d’Ingénierie des Systémes de Versailles) where he involved in different projects. His main research interests artificial neural networks control, adaptive control, backstepping control, decentralized control, variable structure control and sliding mode observer applied to robotics and nonlinear systems.
Boubaker Daachi is Associate Professor of robotics and computer science since 2003. He received his Ph.D. in Robotics from the University of Versailles in 2000, his M.S. in robotics from the University of Paris 6 in 1998 and his B.S in computer science from the University of Sétif (Algeria) in 1995. His research interests include adaptive control of complex systems, soft computing and ambient intelligence. Application fields are robotics, pervasive and distributed systems. He has published more than 40 papers in scientific journals, books and conference proceedings. He has been involved in the organizing committees of some national and international events.
Abdelaziz Benallegue received his B.Sc. in Electronics from Ecole Nationale Polytechnique d’Alger, Algeria in 1986, his M.Sc. (DEA) in Robotics and Ph.D. in Robotics and Automatic Control from Université Pierre & Marie Curie, Paris, France, respectively in 1987 and 1991. He was an associate professor of Automatic control and robotics at the university of Pierre and Marie Curie, Paris 6, France from 1992–2002. He joined the University of Versailles St Quentin in September 2002 as Professor of Automatic Control and Robotics. His research interests are mainly related to linear and nonlinear control theory including adaptive control, robust control and neural network learning control, with applications to robot manipulators and aerial vehicles.