Digital control of nonholonomic systems two case studies

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DIGITALCONTROL OF NONHOLONOMIC SYSTEMS : T W O CASE STUDIES * A. Chelouaht, P. Di GiamberardinoS, S . Monaco$, and D. Normand-Cyrott SDipartimento di Informatica e Sistemistica Universith di Roma “La Sapienza” via Eudossiana 18, 00184 &ma, Italy

tLaboratoire des Signaux 0 Systemes CNRS, SUPELEC, Plateau d e Moulon 91190 Gif sur Yvette France

Abstract

an exact sampled version on the basis of which control in velocity and acceleration is proposed. In section 4, simulation results concerning exact steering and path following are presented.

This paper deals with digital control of nonholonomic systems. I t is shown, through two classical examples, t h a t multirate control can be very effective for solving t h e motion planning problem. T h e theory is discussed for t h e car with one trailer and t h e hopping robot examples. Simulation results concerning exact steering and path following are presented.

1

2.1

The kinematic model

For simplicity, we discuss hereafter t h e case of

a car with one trailer but t h e same control method can be applied t o t h e case with n trailers M illustrated in [21]. Let the kinematic equations

Introduction

Control and stabilization of dynamic systems satisfying nonholonomic constraints have been widely investigated in recent years. This growing interest is justified by t h e theoretical issues in pure mathematics and mechanics involved for t h e study of such dynamics which moreover apply to many physical examples in robotics, space and multibody systems. Several control strategies have been proposed for exact steering and stabilization. Let us recall t h a t maximally nonholonomic systems satisfy the controllability rank condition ([6],[171) but t h a t classical continuous time state feedback laws d o not apply ([2]). Among others, various solutions a r e proposed in terms of sinusoids ([18]), piecewise smooth ([3, I]), piecewise constants ([8, 7, 11, ZO]), time varying control schemes ([16, 4, 14, 131) or more recently in terms of dynamic feedbacks and flat systems ([5, 151). Our interest for t h e thematic initiates noticing t h a t difficult continuous control problems may benefit of a preliminary sampling procedure of the dynamics. This happens for example when dealing with systems including delays ([lo]) or non minimum phase systems. In fact, it is noted in [ll] that a quite large class of nonholonomic constrained systems admit exact sampled models (polynomial state equations). Among them, one finds t h e chained form ([12]) systems which can be associated with many mechanical systems. On t h e basis of these exact discrete time dynamics, nonlinear digital feedback laws can be set to achieve exact steering between two states configurations. Then step after step, a previously planned trajectory can b e followed enabling obstacles avoidance for example. In [ll], the usual unicycle example is treated. In [ZO], a comparison is done between multirate control and sinusoids on the basis of t h e fire truck example which counts three controls. In contrast t o other approaches, it can b e said t h a t exact steering between arbitrary s t a t e configurations with any requirement concerning body orientation is possible and t h a t t h e control computation is extremely simple; in general matrices inversions only are required. T h e object of t h e present paper is to discuss the benefit of multirate digital control method on t h e basis of two examples, t h e car with one trailer and t h e hopping robot. Control in velocity is designed and a more realistic control in acceleration is introduced and illustrated. In fact, in this second situation, the s t a t e space models exhibit a non zero drift term but t h e method still applies without any major

y t

Figure 1: T h e kinematic car with one trailer where BO is the angle of the car body with respect t o t h e z-axis (horizontal), q5 t h e steering angle with respect to t h e car body, and 01 the angle of t h e trailer with respect to t h e z-axis. T h e two controls u1 and u~ are the driving velocity of t h e rear wheels of t h e car and the steering velocity of t h e front wheels of t h e car respectively. T h e system (2.1) can be transformed under coordinates change and feedback transformation, into a canonical one-chained form ([12]) by firstly setting w l = u1 cos(@o) and w2 = u 2 . However, the singularity which occurs for 80 = f ( 2 k 1)d brings t o unfeasible controls when one has t o follow a corner trajectory for example. In what follows, a solution to overcome this difficulty is proposed by rewriting the equations (2.1) according t o a @-rotationof the 2-y axes, where t h e amplitude 8 is fixed in t h e control algorithm to render admissible t h e planned trajectory. More precisely, the 8-rotation given by t h e relation

+

difficulty. I t has to be noted t h a t these two examples satisfy t h e conditions re-

quired for admitting, under coordinates change and feedback transformation, chained forms in velocities. However, due t o singularities occuring into these transformations or due to others physical constraints, t h e control algorithm cannot always be designed directly on the chained form. In t h e first example a rotation is performed so that any kind of trajectory can b e f o d w e d so as a corner. In the second example, t h e kinematic equations are transformed under coordinates and input changes into a special form, which is no more a chained form, but can exactly b e sampled. T h e paper is organized as follows: in section 2, on the basis of the model of t h e car with one trailer transformed into a one-chained form, digital control schemes are proposed both in velocity and acceleration. Section 3 is devoted t o the hopping robot example the continuous time model is transformed into a special form admiking

transforms t h e system (2.1) into

+$

= cos(po)ul = s4po)ui

4 = v z

-

bo

bi

where po = 00 0 and p1 = 81 Setting now t h e input changes

*This work is supported in parts by grants from MEN in Eance and AS1 study contract Rs-157-91in Italy

OiQi-22i6/93/$3.00CQ 1993 IEEE

The car with one trailer

2

= =

pll(d)Ul

; isin(p0

- p1)ul

(2.3)

- B.

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the equations (2.3) can be rewritten in the form of a driftleu linear analytic system

i = i7l(C)Wl + !do*

(2.5)

where { = (6, gl 4, po, p ~ and ) the ~ smooth vector flelds g1 and g2 are deduced from (2.3). It clearly appeam that the singularity po = *(2k l)f, due.to the input change8 (2.4) can be overcame by adequately c h w i n g

u # eo

+

* (zk+ 113.

It can be checked t h a t the sufficient conditionr for t h e existence of a change of coordinate8 and a feedback transformation putting the system (2.6) into t h e canonical onc-chained form ([12]) an ratiafled. So,defining = a({), as

€2

= 2 = L3,,h(O

€3

= L:,h(C)

€1

€4

= L,,h(C)

€5

= h(C)

(2.6) It has t o b e noted that the neunpled dynamlcr 2.12a, 2.12b) is exact and of order 4 at most with rempect to 6. horeover, due to the choice of t h e multirate orden, the flmt equation (2.12a) is linear with respect to ul(k) and the remaining dynamicn of dimension 4 (2.12b) is linear with r a p & t o ua(k). The com utation of t h e multirate controls is greatly simplified by this s t a t e fecompoaition and ir achieved by fintly computing the control u1 on t h e basis of (2.12a) and then u2 on the bmir of (2.12b) after r u k t i t u t i n g the previourly computed value of u1. Given a pair of states { x " , xf}, with io' ioo, and ita corresponding pair {€*,(I}, with ({ Cf, according to t b coordinate change (2.6), one easily a o l m the rteering problem by aolving, with respect to u1 and U?, the net of equations

with.

2 = r({)(vl, ) ~

and setting ( u l , ~

y\)T

/ (U

one can verify that e({) is a coordinatea change and r(()an input change which transform the system (2.3) into the one-chained form

= =

=

€2Ul (2.9) €3~1 €5 (4ul Remarks: The singularitien 6 = and p o p1 = *f represent natural physical limitations in the sense that it is not pouible for the trailer (U well M for t h e front wheels t o be perpendicular to t h e car body. (1

€2

U1 ua

(3 €4

= =

*+,

€{ =

-

ua

the s t a t e vector of (2.1), the exact steering problem can be stated M follows: given an initial state xo = (c",y',4",8~,0~)T and a d a i r e d final rtate xf = (cJ,yf, #,e{, flnd a digital control strategy rteering the system (2.1) f"t h e initial condition x ( k ) = X" to t h e flnal condition ~ ( k1) = ,gf. In [ll], it har been shown that very appealing propertier of chained systems is t h a t they maintain t h e controllability property for almost all sampling period and admit a flnite exact dincretisation. It results that exact rteering under mu!tirate control can be achieved. Denoting by 6 > 0 the sampling perrod, one ha8

C{

ti(k

+ +) = ti(&+ 9)+

~

1

f@)(k 1

+ 9 )repmntr = (i + *)a,

€i(k+1) [(k+ 1) with

=

+ v), (2.11)

€i(k)+6ui(k)

-

v2i

1

=

L3 h ( C ) ( w e- u l ~ t , h ( ~ ) ) i =1 , . . . , 4

sa 91

(2.19)

with desired car and trailer orientations is achieved.

U-rotation angle can be set to zero. In t h b case,systems (2.1) and (2.3) coincide.

k y r without changin(jlthe s-rd'mate. When a parallel parking in c o h d e n d we plan t e path in two st-. An intermediate point in ch-n which b halfway bett h e initial urd final value8 in all the coordinatea except io and t h e intermediate s value is chosen to be t h e initial c i n c d by the d i f k e n c e between t h e initial and

flnal y valuen.

i

the one obtains for ( = (€1

= u(6,.f(k),ul(k))

(2.18)

(I) It ha# t o be noted that, according to this strategy, exact steering

i=: 1,-.-,4 (2.10)

8= ~,i=l,...,s,and1=1,...,4 where e!"(, computed at time t sampled model

-0(6,{~,~1))

(U) If a corner trajectory ir not planed, namely U{ # k f , then the

+ f6[

Of Order

(ff

Rsmarhr

In fact, applying such a procedure to t h e present case, one nets

+

A-'(6,t41)

The controls u1 and u2 being piecewise conatants, the real controls v1 and u2 are piecewise continuous with dincontinuity at the sampling instants.

D

fort E k6, (k+ 1)6[ for t E \k6 9 6 , k6

=

T h e nonaingularity of t h e matrix A is ensured by the condition # o or equivalently u1 # 0. T h e controls u1 and u2, applied t o t h e real system (2.1), are given

1

Proposition 1 ([II]) Given a pair of rtatsr (xo, X I }ratirhing d - co # 0, then there exirtr a digital control rtrategy rteering esactlv xo to xf in one rtep of amplitude 6 with u1 = and

= ul(k) = uli(k)

(2.16b)

by

+

ul(t) ul(t)

(2.16a)

~ ( 6 , $ , ~ ( 1+)A ( ~ , U ? ) U ~

(2.17)

Digital control in velocity Exact Stee~IiI~g. Denoting by x = (E,#, 4, eOsel)T

{

=

[f

€;+6~1

thus obtaining

2.2

a multirate control of order 4 on u2.

+

+

Of

,oTthe (2-12a)

+ A(8,ul(k))u2(k) (2.12b)

PJ-tory traang. Due to the dMign of the digital control law on t h e basis of the state repracntation (2.12s and (2.12b), the control strategy is nothing elno but a multirate )!back one over r t e p of amplitude 6. Such a feedback rtrategy would give the usual advantages Ur o b u s t n a and the poribility to track, step after step a given t*cctOW preclcly~ dcftnl?W a quence Of points: satidying p i E] f , % [ and ' s # c", one can combine multirate feedback and path planning in d e r to ateer oae point from another

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step after step. In t h e case of a corner trajectory or a trajectory with l ) f , one must set B in such a way t h a t po the angle 0: = f ( 2 k be different of f ( 2 k 1 ) 3 . For example, if we want t o follow t h e trajectory described by t h e figure 2, one can set B = f .

+ +

Y

IL I:

A

B

Figure 2: Trajectory composed with a circle segment and straight lines. Starting from A (resp. C),.the point B ('esp. can be directly steered. A certam number of intermdate points on the segment BC can be taken.

gets a n exact sampled model of finite order with respect t o 6 which satisfies t h e controllability rank condition ([lq) for almost all 6 > 0. Arguing as previously, the exact steering problem can be stated as follows: given a n initial s t a t e = (to, yo, 9", Ob, B y , U;, ~ 5 and) a d m i r e d final statexf = ( t f , y f , 9 J , B ~ , B : , v : , v ~ ) * , find adigital control strategy steering R" to RI. In fact, applying such a procedure to t h e present case, one sets

xo

{

a1(t)

=

iiz(t)

=

1,)

t ++

t i , l ( k ) fort E 156, k6 $S[ a l z ( k ) for t E k6 $6, k6 6 [ forte [k6+?6,k6+;6[

~

+

i=l,-.. (2.25)

Denoting by ( € 1 , € 6 ) by ~ h and (€~,€3,€4,€5,Cr)~ by (2, t h e resulting multirate sampled system is given, for & = f , by the equations

9)

2.3

Digital control in acceleration

Piecewise continuous velocities not being feasible in practice, it is more realistic t o design the multirate control no more on t h e basis of t h e system (2.3), but o n t h e basis of its dynamic extension so enabling a control in acceleration. This extended dynamics corresponds t o t h e system (2.3) completed with

(2.20) Let R has

= (2, y, 9, Bo, 81,

~

2

Remarks:

be) t h e~ extended state vector, one

+

+

(2.21) = Go (RI GI(R)el Gz ( R k z where Go, GI,and Gz are deduced from (2.3) and (2.20). T h e dynamics (2.21) is linear analytic with a nonzero drift term. Extending t h e change of coordinates (2.6) and t h e feedback transformation (2.8) defined for t h e control in velocity, we can similary define a change of coordinates and a feedback transforming t h e system (2.21) into t h e dynamic extension of (2.9). For, let us set $. = a@), as

(i) I t has to b e noted t h a t A1 and A2 a r e always nonsingular for 6 > 0 except, regarding A2, for specific values of (€;, €{) and

(l).

(ti, More precisely, t h e determinant of A2 satisfies a polynomial in I,€,; €{,€$, €,'. A pair of states (go,R') will b e said t o be admissible if it corresponds to a nonzero determinant and if po €] - $, $[. In case of singularity, one has simply to plan t h e steering in two or more paths.

(ii) T h e configuration of a l ( t ) given in (2.25) is arbitrarily chosen

(5 being not a multiple of 2). Other possibilities can be tested as fil(t) = a l l ( k ) for f E [k6,k6 #6[ and fil(t) = i i l z ( k ) for f € [k6 #4(k 1)6[.

+

(2.22)

+

+

According to these remarks, one sets

x'},

of states {no, then there ezists a digital control strategy steering ezactly 2" to ' 2 in one step of amplitude 6 with a multirate of order 2 on ii1 and of order 5 on ilz. 0

Proposition 1 Given an admissible pair

3 3.1

The hopping robot The kinematic model

Following t h e litterature ([12]), we consider a very simple model of t h e hopping robot, considering it as a body, with mass mb, with an actuated leg, of mass mi supposed concentrated a t the foot, t h a t can rotate and extend, and analiaing it only during the free flight between two consecutive jumps. If we suppose t o control t h e rate of change of the leg angle and its extension, t h e first two equations of t h e system can be set as

41 €1

&

= =

€6

€3

€7

€4

€5

= = bb = t4b

€6

€7

= Gl

= iil

(2.24)

which represents t h e dynamic extension of (2.9). T h e system (2.24) is no more into onachained form but still satisfies a kind of triangular form, M) t h a t applying t h e sampling procedure described by t h e relation (2.11) for 8 = f and 1 = 1,... , 5 , one

= v1 = v2

From t h e conservation of angular momentum, setting for simplicity mbd: = 1, we have

e + m1(l+

10)2(8

+ 9) = 0

(3.2)

which can be written as

(3-3)

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3.2

Tkansformation into special kinematic forms

T h e solution of such a system has t h e general expression

Chained Po-

It can b e eamly verified t h a t a system like (3.1) and (3.3)can b e transformed into onechained form of dimension 3. On t h e basis of this three dimensional one-chained form, it is easily understood t h a t digital control in velocity and acceleration can b e performed arguing as in proposition 1 and 2. However, t h e state transformation bring8 to unrealistic driving control v when [+lo 5 0. This is illustrated by simulation results not reported here for reapon of convqnience. Such a situati0.n occurs because .no mathemptical constraint on t h e model associated to t h e physical constrarnt I lo > 0 is taken into account. In order to overcome this difficulty we propome herebfter another s t a t e and feedback transformation so t h a t t h e r e ~ u l t i n gsystem does not satisfy any more a onachained form but still admits a finite order exact sampled model.

+-

'Rdormation into a mpecial kind of discretisable form I t is poesible to find, for (3.1,3.3),another transformation which brings to systems admitting exact discretization. For, one sets

€=

(I; )

=

(&'e+

*

lo)

)

(3.10) Remark: T h e condition = €5 brings t o u2 = 0 in (3.8),such a situation is t h e same as previously discussed in term of parallel parking for a car with one trailer. Here, it corresponds t o require t h e same initial and final leg extensions. In such a case, one has to divide t h e procedure in two steps.

(3.4)

3.4 so

obtaining

'

/

€1 €2 €3

= (1 +€;)U1 = U2 =

Digital control in acceleration

In a more realistic situation the model (3.1J.3)has t o be extended with t h e introduction of two other dynamical equations deflning t h e rate of change of t h e U,, i = 1 , 2 as follows (3.6)

-€$a1

which still admits a finite discretization. Precisely, according t o t h e sampling procedure recalled in Section 1 for v i and vz constant over time intervals of amplitude 6, one obtains

+ 1) = el@) + 6 (1 + €$(k))Ul(k) + g (2€Z(k)U2(k)Ul(k)) + g (2ui(k)u1 (k)) + 6uz(k) €2(k + 1) = + $ (-2€2(k)~l(k)ua(k)) e3(k + 1) = b ( k ) + 6 + g (-2uz(k)u1 i-($(k)ui(k)) (k))

€l(k

where el and e2 a r e t h e driving accelerations. Also for this system, obtained by adding one level of integrators t o the input of a system t h a t can b e transformed into a chained form, it is possible to determine a s t a t e space transformation and a s t a t e feedback which produce a finite sampled system. In fact, using t h e transformations

€2(k)

(3:7) which is exact, linear with respect to t h e control q and of finite order with w p e c t to up. Hereafter, it is shown t h a t digital control in velocity and in acceleration for exact steering is still solvable on t h e basis of (3.6).

3.3

Digital control in velocity

Considering a multirate control strategy of order one on on u1,precisely setting

u2

and two

t h e model is transformed into €1