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Procedía Engineering 96 (2014) 111 - 125

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Modelling of Mechanical and Mechatronic Systems MMaMS 2014

Constrained finite-time control of mobile manipulators

Miroslaw Galicki*

Faculty of Mechanical Engineering, University of Zielona Gora, 65-516 Zielona Gora, Podgorna 50, Poland

Abstract

This work offers the solution at the control feed-back level of the accurate positioning in a finite time of the end-effector whose mobile manipulator is subject to control and complex state constraints (both holonomic singularity and collision avoidance). Based on the Lyapunov stability theory, a suitably defined extended task error and exterior penalty function approach, a class of simple non-linear controllers converging in a finite time, which fulfil control and state constraints, is proposed. The numerical simulation results carried out for a mobile manipulator consisting of a nonholonomic differentially steered wheeled mobile platform and a holonomic manipulator of two revolute kinematic pairs, operating both in a two-dimensional unconstrained work space and work space including the obstacles, illustrate performance of the proposed controllers.

© 2014TheAuthors. Published byElsevier Ltd. Thisis an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Peer-review under responsibility of organizing committee of the Modelling of Mechanical and Mechatronic Systems MMaMS 2014 Keywords: Mobile manipulator task; collision avoidance; Lyapunov stability.

1. Introduction

Mobile manipulators have been recently applied to accurate end-effector positioning tasks (e.g. precise parts assembly in a work space including many (unknown) obstacles). The aim of the positioning task is to find a control which moves the mobile manipulator from its initial configuration and velocity expressed in generalized coordinates (relative rotations and/or translations of the kinematic pairs) to a desired end-effector location expressed in the task (work) space coordinates. As is well known, relationship between generalized coordinates and the task ones (mobile manipulator kinematic equations) is strongly non-linear. The platform of the mobile manipulator, which is far from the desired end-effector location, has to shift to a preferable (not specified) posture, at which the end-effector attains desired location. As is known, no smooth time invariant state feed-back controllers exist for the platform

* Corresponding author. Tel.: +48-68-3282614; fax: +48-68-3282617. E-mail address: m.galicki@ibem.uz.zgora.pl

1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Peer-review under responsibility of organizing committee of the Modelling of Mechanical and Mechatronic Systems MMaMS 2014 doi:10.1016/j.proeng.2014.12.120

accomplishing a point-to-point task [1], [2]. However, by combining the mobility of the non-holonomic platform with the manipulability of the holonomic manipulator, performance characteristics of such a kinematically redundant mechanism (mobile manipulator) are improved which enable it to accomplish complicated tasks, e.g. accurate end-effector positioning in work spaces including many known and/or unknown obstacles. In order to eliminate undesirable vibrations of the end-effector at the desired location caused by the kinematic redundancy of the mobile manipulator, its velocity and in particular the platform velocity, has to be reduced to zero. Moreover, to enable the end-effector to accomplish precisely fine motions from the desired location, subject in particular, to the immobile platform, the holonomic manipulability measure, introduced by Yoshikawa [3], should be large at this location. Due to imposed aforementioned technological constraints, the mobile manipulator is required to accurately move in a finite time from the initial configuration and velocity to the desired end-effector location. The mobile manipulator velocity should be reduced at the desired location to zero also in a finite time. Even a small end-effector displacement from the desired location, caused e.g. by interruption (after a finite time) of only asymptotically convergent positioning regulator, may then prevent the end-effector from precise accomplishment of the fine motions. Furthermore, when moving in the work space with obstacles, the mobile manipulator and the end-effector should avoid collisions.

Several approaches to controlling the mobile manipulators subject to both kinematic and dynamic equations were proposed in the literature. Nevertheless, all of them provide at most asymptotically stable solutions. The first approach utilises pseudo-inverse techniques to resolve mobile manipulator redundancy [13]-[22], [41]. The main disadvantage related with most of known pseudo-inverse techniques applied to fulfil state inequality constraints (collision avoidance of both mobile manipulator and end-effector with obstacles), is lack of obstacle influence on the end-effector movement. The second approach, developed in works [23]-[27], involves input-output decoupling controllers and both kinematic and dynamic equations. The algorithms from [23]-[27] require additional output functions, inverse of the so-called extended Jacobian matrix and they are not suitable to collision avoidance problems. From the literature survey, it follows that all the aforementioned control algorithms are not able to shift the mobile manipulator in a finite-time in such a way as to fulfil control and state equality (both zero position error and zero final mobile manipulator velocity) and inequality (both collision avoidance of the whole mobile manipulator with obstacles and holonomic singularity avoidance at the desired end-effector location) constraints subject to kinematic and dynamic equations.

The present work addresses the problem of finite-time controlling the mobile manipulators subject to control and state variable constraints. Several approaches which partilly relate to a class of finite-time control tasks may be distinguished [4]-[6], [8]-[12]. However, all those approaches deal only with stationary robotic manipulators, neglect their kinematic equations and can not be applied to tasks with control and/or state inequality constraints. We first propose new forms of various terminal sliding modes (TSM's) which result from the access to kinematic redundancy of the non-holonomic mechanical system and are defined by useful tasks to be accomplished. Then, all the TSM's are simultaneously applied in both the reaching phase and the sliding phase, resulting in a new continuous TSM control for mobile manipulators with finite-time stability provided that some practically reasonable assumptions are fulfilled. Our constrained controller may be directly used in a real-time to accomplish complicated mobile manipulator tasks provided that parameters of dynamic equations are known with sufficient accuracy. Kinematic and dynamic parameters of the mobile manipulator can be obtained with sufficient accuracy by means of the calibration and identification techniques given e.g. in [28], [29]. If those parameters are not accurately known, then proposed non-trivial control law generates admissible reference trajectories (both non-holonomic reference platform and reference holonomic manipulator trajectories) in an on-line mode which are required as inputs to adaptive feed-back controllers (offered e.g. in work [30]) tackling uncertain dynamics of the non-holonomic mechanical system. Consequently, our work is not focused on constructing adaptive control techniques which are known from the literature. Instead, non-trivial control strategies providing admissible reference trajectories (which are time consuming and hard to numerically determine in classic approaches) as being the inputs to adaptive feed-back controllers are, in fact, subject of our interest. The remainder of the paper is organised as follows. Section 2 formulates the finite-time end-effector positioning task subject to collision avoidance with obstacles, holonomic singularity avoidance and limits on torques/forces, as a constrained control problem. Section 3 sets up a class of controllers solving the positioning task in a finite time, subject to control and state dependent constraints. Section 4 presents computer examples of the task accomplishment for a mobile manipulator consisting of a non-holonomic platform and a holonomic manipulator of two revolute

kinematic pairs, operating in both an unconstrained two-dimensional work space and a work space with obstacles. Finally, some concluding remarks are drawn in Section 5.

2. Problem formulation

Consider a mobile manipulator composed of a nonholonomic platform. It is described by the vector of generalized coordinates x £ , (platform posture xlc, x2fi, & and angles of driving wheels (p1, 02-see Fig. 1., where 8 is the orientation angle of the platform with respect to a global coordinate system Ox1x2; xl c, x2c stand for coordinates of the platform centre; W denotes one half of the distance between platform wheels; 2L is the platform length; R stands for the wheel radius; (a, b) denotes the point at which the holonomic manipulator base is fasten to the platform), where I > 1 and joint coordinates y£ Rn of a holonomic manipulator mounted on the platform, where n is the number of its kinematic pairs. The platform motion is usually subject to 1 < k < I nonholonomic constraints in the so-called Pfaffian form A(x)x = 0, where A(x) stands for the kxl matrix of the full rank (that is, rank(A(x)) = k) depending on x analytically. Suppose that Ker(A(x)) is spanned by vector fields a1(x),..., a;_k(x). Then, the Pfaffian constraint can be equivalently expressed by an analytic drift-less dynamic system

x = N{x)A, (1)

where N(x) = [a1(x),..., at_k(x)] ; rank(N{x)) = I — k and vector <A = (a1( ...,ai_k)T denotes auxiliary velocities of the platform (introduced in [32]). The dynamics of a mobile manipulator is given by the following equation [31]:

M'(q)q + F'(q, q) + [A(x)Okxn]TA = B'v, (2)

where q = (xTyT)T is the vector of generalized coordinates and represents the configuration of the mobile manipulator; (,qT,qT)T denotes the state vector of the mobile manipulator; M\q) stands for the (n + Z) X (n + Z) inertia matrix; F'{q,q) = C'{q,q)q + D'{q); C\q,q)q is the (n + V) -dimensional vector representing centrifugal and Coriolis forces; q denotes the mobile manipulator velocity; D'(q) stands for the (n + Z) -dimensional vector of generalized gravity forces; Okxn denotes the k Xn zero matrix; X is the vector of Lagrange multipliers corresponding to non-holonomic Pfaffian constraints; B' stands for the (n + Z) X (n X I — k) matrix describing which state variables of the platform and holonomic manipulator are directly driven by the actuators and Rn+i_k 3 v is the vector of controls (torques/forces). Applying the transformation

v = B~1{Mu + F), (3)

where u is a new control vector, M = CTM'C; C = . . ; F = CT(M'Cs + F'J; s = ^ I is the reduced

0 lnY ^ J \yj

velocity vector of the mobile manipulator; In stands for the nxn identity matrix and B = CTB', we obtain a simplified form of the mobile manipulator dynamic equations (a dynamic system with drift) expressed in all the configuration and reduced velocity variables [20]-[22]

q = Cu + Cs. (4)

Due to physical actuator limits, control vector v = {vt,..., vn+i_k)T is subject to the following constraints:

vmin,i <vt< vmaXji, i = l,...,l + n-k, (5)

where vmin i, vmax i denote the lower and upper limits, respectively on control vt.

The position and orientation p of the end-effector with respect to an absolute coordinate system is described by a mobile manipulator kinematic equations

p = fio), (6)

where p = (p1(... ,pm)T; f(q) = (A(qO,... ,/m(qO) denotes the m-dimensional (in general, nonlinear with respect to x and y) mapping and m stands for the dimension of the task space.

A task accomplished by a mobile manipulator consists in shifting the end-effector to a desired position and orientation pd £ in a finite time T, which may be given either explicitly or implicitly depending on a task specificity. Furthermore, the end-effector velocity should equal zero at pd = (pitd,... ,Pm,d) . Moreover, a stable accomplishment of the fine motions (precise assembly of parts, e.g., insertion of the shaft into the hole of the bearing) requires damping the end-effector vibrations which could potentially arize from the non-zero mobile manipulator velocity. Hence, it is natural to reduce this velocity to zero at the desired location pd. By introducing the task error e defined as e = {et,... ,em)T = p — pd = f{q) — pd, we may formally express the regulation aim by means of the following equations (state equality constraints):

lim e(t) = 0, lim e(t) = 0, lim s(t) = 0, (7)

t^T t^T t^T

where 0 < 7<ro denotes a finite performance time of the mobile manipulator task. It is also practically important to attain pd with a desirably large manipulability measure (defined as a valuation of difficulty of the holonomic manipulator operation, proposed by Yoshikawa [3]. If a manipulability value were small, then a reconfiguration of the whole mobile manipulator at location pd would be necessary to efficiently accomplish the fine motions. Thus, the following (state inequality) constraint is imposed on the final (unknown) configuration of the holonomic part:

D(q(D)>p', (8)

where D (■) is the manipulability measure of the holonomic manipulator; p' denotes a positive scalar coefficient (user specified). During the mobile manipulator movement, collision-avoidance (state inequality) constraints resulting from the existence of obstacles in the work space, are induced. The general form of these constraints can be written in the following manner:

{qj,(q(t)) > 0}, (ci(p(t)) > 0}, y = l: N0, t e [0,7], (9)

where c,^ denotes either a distance function [33] between mobile manipulator (without the end-effector) and an obstacle or an analytic description of an obstacle [34]; c1e is a distance function between the end-effector and an obstacle or analytic description of the obstacle; N0 stands for the total number of collision-avoidance constraints and p = f(q) denotes current end-effector location in the task space. For purpose of further analysis, we explicitly separate end-effector collision avoidance constraints from the rest. We postulate further on that q(0) together with its small neighbourhood does not cause a collision, i.e., > 0, cJe (p0 = /(<7(0))^ > 0. Moreover, functions cJe, c^

from (9) are assumed to belong to a class of smooth mappings with bounded derivatives with respect to any p and q. Thus, the mobile manipulator regulation aim may now be reformulated as follows: find a control u which moves the mobile manipulator from initial configuration and velocity <7(0), ¿(0) (or equivalently ^(0) such that equalities e = 0, e = 0 and s = 0 are fulfilled in a finite time 7. Moreover, u should steer the mobile manipulator in such a way as to fulfil control constraints (5) and avoid both holonomic singular configuration at t = T and collisions with obstacles for t £ [0,7].

3. Constrained control of mobile manipulator

In order to involve state inequality constraints (8)-(9) in the mobile manipulator control, suitable exterior penalty functions are introduced. To be more precise, we introduce the following exterior penalty functions to satisfy both inequality (8) and collision avoidance constraints (9):

Us(q) = c'Es(q) (10)

tfe(p) = ceJEe(cJe), Um{q) = c"Em{q), (11) where Es{q) = (D(q) - p')4 for D{q) < p' and Es(q) = 0 otherwise; Ee(cJe) = (c'e - p")4 for cJe < p" and

Ee(cJe) = 0 otherwise; Em(q) = Y!jl=oCm,jEm{c^n) ; Em{c]m) = (c^ - p") for c4 < p" and Em{c]m) = 0 , otherwise, pj' stands for a given threshold value which activates the 7-th inequality constraint after exceeding this value; N' < N0 is the number of only active constraints (9) (i.e. such that c1e — p" < 0 or c'm— p" < 0; c', cej, cmj and c" denote positive, constant coefficients (strengths of penalty). Let us note that configuration set {p = fiq)'. 0 < Cg < pj'} u{q\0<c^< pj'} determines an interacting or a safety zone around the j -th active obstacle, j = 1\N'. Without loss of generality, we further assume that s(0) = 0. Moreover, based on the assumption made in Section 2, we have Ue(p0) = Um(q(0)) = 0.

Let us note that minimization of the following cost function (constructed from penalty functions Us and Um):

U = Us + Um (12)

leads to both collision-free movements of the mobile manipulator and its holonomic singularity-free final configuration, i.e., to fulfilment of state inequality constraints (8)-(9). Similarly, in order to avoid collisions of the end-effector with obstacles and to attain desired location pd, the following modified criterion is introduced:

Ve=i||e||2+Ue(p)|p=/(i?). (13)

Solutions to the problem of the mobile manipulator (and end-effector) motion in the presence of obstacles can be generally divided into the two classes: global and local methods. Global methods are (by their nature) computationally expensive. Furthermore, they are, in fact, inapplicable when there are unmodeled or moving obstacles in the work space. Because of these limitations, global collision avoidance is implemented in an off-line mode. Since our aim is to avoid collisions of the mobile manipulator with obstacles (and not to globally plan collision-free trajectory) during its movement, we are interested in utilizing local methods, which are less computationally involved than global techniques. Moreover, local methods can be applied for real-time sensor based configuration modifications in a neighbourhood of an obstacle. These attributes make them implementable in an online collision avoidance. In order to both escape from the obstacle interacting zones and to attain desired location pd, a new extended task error £ is introduced below

£=(£l.....£m)T = d-^. (14)

Let us observe that £ = 0 (minimum of function Ve) implies equality

e+^=0. (15)

Assuming linear independence of vectors e and ^p, from (15) one obtains e = 0 and ^ = irrespectively for the end-

effector. By differentiating (14) with respect to time, we have

¿=Jeq= J sS,

where ]£ = ^ is the mx (I + n) extended Jacobian matrix; j£ = ]£C. Matrix j£ is further assumed to be of the full

rank. To involve state constraints (8)-(9) into control law, generalized vector sliding variables Z and Zt, respectively will be introduced as follows

Z = {Zi.....Zm)T = £+ A0£a, (17)

coefficient; i = 1, ... ,m, and

where a = p1; p2 are odd numbers; p1 < p2 < 2p1; A0 = diag(A01,... ,A0 m); A0^ > 0 denotes constant gain

Zi = (^1,1.....zu_k+ny = P£(s + CT^), (18)

where P£ = Ii-k+n — jf J^. Based on (17), (18), we propose the following (simple) controller to tackle both state equality and inequality constraints (5), (7-9):

= ¡K-JeCs-hq - A0a£a"1£ - A^) + P£ (-A2zf -P£(s + - (19)

where ft is defined similarly as a ; £a_1 = diag{£'^~1, ...,££,_1) ; A-l = diag(Alil,... ,Alm) ; A2 = diag(A2?i, ■■■,A2i_k+n); A1^,A2^ > 0 are constant gain coefficients. The closed-loop error dynamics is obtained by inserting the right-hand side of (19) into (4)

q = CUg'i; + Cs. (20)

We now give the following new result.

Theorem 1. If J£ is a full rank matrix in a (closed) region of the task space, vectors e and are linearly independent, CT is linearly independent on rows of j£ and A0, A1( A2 > 0 then control law (19) guarantees stable

convergence in a finite time of task errors (e, e, s) to the origin (e, e, s) = (0,0,0). Proof: Consider a Lyapunov function candidate

Vge=±(Z,Z)+±(Z1,Z1). (21)

Differentiating Vge with respect to time results in the following expression:

Vge = (Z, }£q + }£q + A0aE^E) + (Zu P£ (s + CT + (Zv P£ (us/e + £ (CT (22)

Substituting Ug'H, q for the right-hand sides of (19) and (20), respectively, results after simple algebra in the relation

vge = -2r=iAi,^f+1 - -^=i+nA2j < 0. (23)

Based on (22), (23), we can see that Z,ZX evolve according to the following equations:

Z = -A tZP, Zi = -A2zf. (24)

Eression (23) may be further transformed into the form

%e < -min{AM,A2j}

Applying inequality from [12] to component

we obtain

£+1 £+1

%e < -mi.n{Au,A2j}2 2 (Vge) 2 .

Inequality (25) leads to the following two TSM's: Z = 0 and Z1 = 0 for t > T2,

IV (0) 2

where T2' <-—-Based on (21) and (25), we have for t > T2'.

min{A1,i,A2,i}(l-^)2 i,j

¿+A0£a = 0

Differential equation (26) implies stable convergence of £ and £ to 0 in a finite time t > 7\', where

((£(o),£(o))\

7\' < ■

min{Aoi}(l-a)2

Hence,

£ = 0, £ = J£s = 0

for t > Tl + T2'. Due to linear independence of vectors e and , we conclude that

e = 0, ¿ = 0, ^=0 3p

for t > Tl + T2. It is also worth to note that the end-effector collision-freely (minimisation of Ve) attains desired location pd. Moreover, from (21) and (25), it follows that

for t > T2. Hence, taking into account lower equation of (27) and (29), we have

j4+^(crS)) = 0, p*(s+p*(cr£)) = o (30)

for t > Tl + T2. Since [Jg P,c]r is (by assumption) full rank matrix, we obtain

s = —

Relation (31) shows that mobile manipulator moves in the null space of J£ for e = 0. Differentiating U with respect to time, one obtains

U = {CT — ,s).

On account of (31), we have

¿ = -<P£(C^),P£(C^)><O (33)

which means both holonomic singularity and collision avoidance of the mobile manipulator with obstacles at desired end-effector location pd. By assumption, CT^is linearly independent on rows of j£, i.e., P£ ^ 0 for ^ 0.

Hence, based on (33), we obtain equality U = 0 provided that

CT—= 0. (34)

dq V '

From (31)-(34), we finally conclude that s = 0. It is worth to stress out that the time needed to fulfil equality (34) strongly depends on the geometry of obstacles. Since we are interested only in local collision avoidance with simple obstacles (and not to globally plan a collision-free trajectory in a cluttered environment), this time is also assumed to be finite. Consequently, control law (19) implies stable convergence in a finite time of task errors (e, e, s) to the origin (e, e, s) = (0,0,0). In addition, assuming that — <£ ker(CT), from (34), it follows that

- = 0, (35)

dq V '

i.e., mobile manipulator fulfils state inequality constraints at pd.

We know from the proof of Theorem 1, that the end-effector collision-freely (minimisation of Ve) attains desired location pd. Furthermore, the final mobile manipulator configuration both does not collide with obstacles and is holonomic singularity-free (see eqn (35)). Nevertheless, it remains to be shown that controller (19) generates collision-free trajectory of the whole mobile manipulator (and not only of the end-effector) in the whole time horizon of movement, i.e. also for e ^ 0. For this purpose, we first prove boundedness of ||£||. By introducing, an auxiliary Lyapunov function candidate VA = i ||£||2, we easily deduce that ||£(t)|| < ||£(0) ||, t > 0. Moreover, from (21) and (23), one obtains

||£+A0£a||2<2l^(0). (36)

Inequality (36) implies boundedness of £. By assumption (see Section II),

Z1(0) = P£(s(0) + Cr-^| ) = 0. (37)

From (24), it follows that

Z1(t) = P£(s(t) + CTd-£\ ) = 0 (38)

for t > 0, i.e., controller (19) maintains the second TSM Zt = 0 all the time. Inserting the right-hand side of identity s = P£s + Jf J£s = P£s + f£t into (32), we obtain

¿ = <C^,P£s> + <C^,J*£>. (39)

From (10)-(12), (38) and (39), it follows that

Let, e.g., c' be equal to c' = c"c"', where c"' > 0. Hence, (40) may be rewritten as follows

Ù = -(c")2

On account of the fact that CT and jf£ are bounded (j£ is non-singular and £ is bounded), it is always possible to choose sufficiently large coefficient c" from (41) is (by assumption) linearly independent on rows

of j£, i.e., P^ 0 for CT^ 0) such that Ù is non-positive, i.e.,

which means both holonomic singularity and collision avoidance of the whole mobile manipulator with obstacles when moving in the work space. Moreover we can show boundedness of mobile manipulator velocity s. This remark follows from the definition of Vge and the facts that ||Z||, HZjJI are bounded monotonically decreasing functions of time. In such a case, we can easily derive the following inequality:

||J£s||2 + ||P£S||2<21' (0) + ||A0£a||

■ dU

2UZJ 11^(^)11 +2||Z||||A0m. (43)

Since matrix has (by assumption) full rank, from (43) we can easily obtain un upper estimation on ||s||.

Furthermore, it is not difficult to show that there exist suitable gain coefficients A0, Aj, A2, c' and c" for which vector v = B~1(Musg'H + CTM'Cs + CTC'Cs + CTD') fulfils control limits (5). As we know from formula (19)

and Theorem 1., the convergence of task errors (e, e, s) to zero may be ensured provided that vectors e ^ 0 and ^^ are linearly independent, CT is linearly independent on rows of J£. If this is not the case, the mobile manipulator

stops before the desired location pd is attained and thus global methods must be utilized to escape from a local minimum (at which s = s = 0) and to attain pd (see, for example, work [40] for a real-time version of A* used to a redundant manipulator). Let us also note, that it is difficult to obtain a result for global stability in the presence of obstacles. Involving a Filipov solution [37], [38], [39] results in discontinuous right hand side of motion equations, i.e. discontinuity of manipulator velocity, which induces the undesirable effect of chattering. Furthermore, the difficulty of obtaining solutions in the case of [38], [39] is related with necessity of transforming the original control problem into free configuration space with assigned potential value of navigation function at each point of this space. On the other hand, the main advantage of the (local) solution proposed here is the continuity of manipulator control. It is also worth emphasising that the knowledge of obstacles shapes is not required by generating the mobile manipulator control. Therefore, the control scheme (19) may be applicable to unknown environments.

4. Computer example

Based on a selected mobile manipulator task, this section demonstrates the performance of controller given by equation (19). For this purpose, a mobile manipulator, schematically shown in Fig. 1, is considered. In all numerical simulations, the SI units are used. The holonomic part (a SCARA type stationary manipulator) with two revolute kinematic pairs n = 2 operating for simplicity of computations in two-dimensional task space m = 2, is mounted on the platform which is assumed to be physically driven by two wheels of angular velocities (0i02)r.

The kinematic equations of the mobile manipulator equal

fr ce(a + lici + ^12) ~se{b + l1s1 + l2s12) + xljC \

q ~ \se(a + l1c1 + l2cy12) + ce(b + + Z2s12) + x2,c/

where ce = cos 0 ; se = sin 9 ; q = cosy1; s1 = sin y1; c12 = cos^ + y2) ; s12 = sin(y! + y2) ; l1 = l2 = 0.4 stand for the link lengths of its holonomic part; a = 0.85, b = 0.2; yx, y2 denote joint coordinates of the holonomic

manipulator. Hence, vector of generalized coordinates q takes the form q = (xl c x2fi tp1 <fi2 y1 y2) . Matrix C is equal to

HV(x) ^

0 I2J,

s = (a1 a2 y1 y2)T and u = (ux u2 u3 u4)T. Parameters R and W are equal to R = 0.2, W = 0.2. The performance time T is not specified in all the simulations. The components of the dynamic equations of the mobile manipulator take the following values: platform mass mp = 94; wheel mass mw = 5;platform moment of inertia Ip = 6.609; the masses of the links of the holonomic manipulator equal m1 = m2 = 4,respectively and D'(q) = 0. The lower vmin and upper vmax control limits are chosen as follows vmin = (—45 — 45 —2 — 2)T and vmin = (45 45 2 2)T, respectively. Controller gains, where i = 0,1,2 take thefollowing scalar values A0 = 0.2, A-l = 2, A2 = 3, a = 3/5 and P = 5 / 7. The task of the mobile manipulator is to make the end-effector attain the desired location pd. On account of the fact that I + n — (k — m) = 2,the mobile manipulator becomes redundant. Let us introduce the following task errors:

respectively, to evaluate the performance of controller (19).

Fig. 1. Kinematic scheme of the mobile manipulator and the positioning task to be accomplished.

The initial configuration and velocity of the mobile manipulator equal ^(0) = (00^000 0)r, ^(0) = (0 0 0 0 0 0 0)r, respectively and desired location pd is equal to pd = (6,8)r. The mobile manipulator was used to solve the position regulation problem by the assumption that there are two obstacles (circles), schematically presented in Fig. 1, in the work space. The boundaries of obstacles take the form (x1 — 3.5)2 + {x2 — 3.5)2 — 0.452 = 0 and (x1 — 0.5)2 + (x2 — 5.5)2 — 2.052 = 0. In order to accomplish both the holonomic singularity free and collision avoidance task, controller (19) was applied in this experiment. The length of nonholonomic platform equals 2L = 1.8. For simplicity of simulation, we do not take into account platform wheels in the collision avoidance. Hence, the total number of active collision avoidance constraints N', which are assumed herein to be distance functions, fulfils inequality N' < 2 (c,^, c1e stand for the distances between the mobile manipulator, the end-effector, respectively, and the 7-th obstacle, j = 1.2). In order to calculate numerically the values of functions c,^, c1e each link of the holonomic part was discretized into 3 points. The same discretization was carried out for each side of the nonholonomic platform. The end-effector was represented by one point. Following the ideas presented in [21], we introduced for the platform point xp (see Fig. 1), the additional penalty function U'(xp) = £/=1 , where cj is a positive, constant

coefficient (strength of penalty). The role of the penalty terms Um and U' is to ensure the escape of the mobile manipulator from the interaction (safety) zones of the circle obstacles. On account of the fact that inequality (41) from the last section provides for c" conservative estimation which would result in an impetuous movement in the safety zones, coefficients c", cmj, ce j and cj were chosen experimentally. Consequently, the settings include c" = 7.2 ■ 10"2, cm, 1 = 8, cm 2 = 7, ce l = 0.015, ce 2 = 0.06, c{ = = 80. The threshold values p" and taken for computations are equal to p" = 0.8 and p2 = 0.7. The chosen values of cmj, cej and Cj result in deep penetration of safety zones but provide mild mobile manipulator movement which is a desirable property. The results of computer simulations for controller (19) are presented in Figures 2-7.

Fig.2. Task errors ea, e2 for controller (19) in work space with obstacles.

Fig.3. Norm ||s|| of the mobile manipulator velocities for controller (19) in work space with obstacles.

Fig.4. Holonomic manipulability measure D for controller (19) in task space with obstacles.

Fig.5. Distance dmbm_ 1 between the mobile manipulator and centre of the 1-st obstacle for controller (19).

Fig.6. Distance dmbm_2 between the mobile manipulator and centre of the 2-nd obstacle for controller (19).

Fig.7. Control vector v corresponding to controller (19) in work space with obstacles.

As is seen from Fig. 2, this controller stably and in the finite time horizon T = 45 attains desired end-effector location pd. Figure 3 presents the Euclidean norm of the mobile manipulator velocities which also stably attains zero at T = 45. From Fig. 4, it follows that the controller (19) provides holonomic singularity-free configuration for c' = 500 and p' = 0.16. Moreover, Figures 5-6 present distances dmbm_t, ¿ = 1:2 between the mobile manipulator and the centres of obstacles 1 and 2, respectively. As can be seen from Figs 5-6, the manipulator singularity- and collision-freely penetrates the safety zone of the 1-st obstacle for t belonging approximately to interval [8, 15] and the safety zone of the second obstacle is singularity- and collision-freely penetrated for t £ [6, 17], respectively. Figure 7 presents mobile manipulator controls computed based on (3). As is seen from Fig. 7, vector v as being the mapping of UgH does not violate control limits (5) in the whole time horizon of the movement.

Conclusion

In this paper, the mobile manipulator positioning task subject to control and state equality and inequality constraints, has been discussed. Using the Lyapunov stability theory, a class of nonlinear controllers both generating the mobile manipulator trajectory fulfilling the control and state constraints and eliminating the undesirable end-effector vibrations, has been derived. The control scheme proposed in this paper generates continuous controls (even

near boundaries of obstacles) which is a desirable property in an on-line control. Moreover, the approach presented forces the mobile manipulator trajectory to escape from both holonomic singular configurations and obstacle neighbourhoods which makes our control algorithm practically useful. Furthermore, the generation scheme proposed provides the user with the capability to vary the level of information needed by controller (19) depending on the form of functions c4, c1e. That is, the approach presented is equally applicable to analytical descriptions of obstacles in the work space or distances (provided by the robot sensors) between the mobile manipulator and obstacles. Numerical simulations carried out on an exemplary mobile manipulator consisting of a differentially steered wheeled mobile robot and a holonomic manipulator of two revolute kinematic pairs have confirmed theoretical results obtained in Section 3. The advantage of using the method proposed is the possibility to implement it in an on-line control with singularity and collision avoidance provided that the parameters of dynamic equations are given with sufficient accuracy. Alternatively, our control strategies provide reference trajectories in real time which may serve as inputs to adaptive control algorithms. Our future research will focus on application of controller (19) to singularity and collision avoidance tasks with moving obstacles.

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