An introduction to motion planning under multirate digital c - Decision ...

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TM2 IS00 AN INTRODUCTION TO MOTION PLANNING UNDER MULTIRATE DIGITAL CONTROL S. Monaco* and D.Normand-Cyrot**

* Dipartimento di lnfonnatica b Sistemistica, Universitk di Roma "LaSapienza", via Eudossiana, 18.001 84, Roma, Italy.

** Labomtoire des Signaux et SysGmes, CNRS-BE, Plateau de Moulon, 91 192 Gif-sur-Yvette. Cedex, France.

In fact, multirate digital control is currently investigated in

a linear context ([l, 41) when digital control of continuous

Abstract

systems is investigated. Such methods allow us to achieve, at sampling times, regulation, stabilization, decoupling or tracking objectives set in a continuous-timedomain. Moreover, multirate digital control can be applied to controllable continuous-time systems, even to those which are nonminimum phase, to achieve stabilization. Typical discrete-time requirements, such as dead beat or minimum time responses, can also be required. In a nonlinear context, it has recently been shown that multirate digital control can be proposed as the digital analogue of linearizing feedback laws for systems with a relative degree which is larger than one ((151). Roughly speaking, if the sampling period is equal to 8, multirate digital control consists in designing piecewise constant controls over fractions of this sampling period (say 3 = 6 / p where p is an integer 2 0). Such a procedure which causes an increasing number of freedom degrees on the control, enables to satisfy several objectives. For example, when linearization under sampling is discussed, it has been proved that a multirate procedure of order equal to the relative degree enables to maintain, at sampled instants, the linearizing or &coupling properties of the continuous input/output response. Under controllability assumption of the continuous-time system it can also be shown that full linearization under multirate of order n can be achieved. Other critical continuous-time situations can also take benefit from such control procedures, such as time delayed or non minimum phase systerhs ( [ 171). Considering the steering problem which essentially requires the complete controllability assumption, it seems interesting aying to apply digital methods. Even if many questions are still open concerning the existence of solutions and the "right" choice of multirate order, the strict purpose of this paper is to show that digital control can be a possible solution in a number of practical examples. The benefits essentially concern the simplicity of the designed trajectories, the number of paths required for steering (exact steering for chained systems), the computing time. The paper is organized as follows. The introduction recalls the terminology and the approaches which inspired the present work. Section 2 recalls some basic properties of multirate sampled systems. In fact, exponential representations of the sampled dynamics introduced in I161 are here used to compute the steering control. Steering under digital control is outlined in Section 3 for both systems with drift and without drift tern. In Section 4 chained systems are considered, in this case an exact solution is explicitly designed. The section ends with a classical example from mobile robotics.

In this paper we propose digital control methods for steering real analytic controllable systems between arbitrary state configurations. The main idea is to achieve a multirate sampled procedure to perform motions in all the directions of controllability under piecewise constant controls. When it is applied to nonholonomic control systems without drift, the procedure simplifies. In particular, it results in exact steering on chained systems recently introduced in the motion planning literature. A classical example is reported.

1. Introduction Let us consider a continuous-time nonlinear system: i ( f ) = A x ) + u1g1(x) + ... + U&&) (1.1) where thef and gi are real analytic vector fields defined on a connected smooth manifold M of R" and assumed complete. Given two state vectors xo and xj. The set problem consists in designing a control procedure steering xo to XF For nonholonomic driftless systems (f= 0), this problem has widely been investigated over the last few years, especially in literature on motion planning in robotics and several efficient control algorithms have been proposed. Restricting ourselves to the more recent papers which inspire our work, we refer to [6, 8, 9, 10, 11, 18, 19, 21, 231 where a more complete biblio raphy can be found. kssuming with m < n the complete controllability of the system (1.1) to guarantee the existence of a solution ([7,23]), the problem is to design a reasonable one (with respect to computing time, complexity, convergence to the solutions, number of paths, control constraints and others). Among the various proposed approaches, it appears that much can be said about nilpotent or nilpotentizable systems (under coordinate changes and feedback) ([6, 7, 14, 151). In particular in [18, 191 special classes of systems, which can be steered using sinusoidal inputs, are introduced. These systems, referred to as chained systems, have convenient triangular forms for which such a control reveals many advantages (optimality, small number of paths). Other techniques are based on exponential representations of the trajectories and decomposition in suitable bases of the control Lie algebra associated to (1.1) ([S, 10, 61). Some attempts to extend these methods to systems with drift can be found in [S. 231. Let us recall that techniques of linearization cannot be applied to nonholonomic systems which are generally not fully feedback linearizable, and that linear techniques fail since the controllability rank condition of a driftless dynamics cannot be preserved under linearization. With regard to these difficulties, it is not too surprising that digital strategies could lead to efficient solutions.

CH3229-2/92/000O-1780$1.OO Q 1992 IEEE

2. Nonlinear Multirate Sampling Consider a linear analytic continuous-time dynamics

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where x E M (a smooth manifold of JR"), LE JR" ,f and g, are analytic vector fields on M , which are assumed complete. Denote by E the control Lie algebra associated to (2.1) (i.e., the smallest Lie algebra which is invariant under f, g; and containingf , g;) and EOthe Lie ideal of L generated by the g; The distribution A = span ( c ,c E 2)is said to be nonsingular on M if there exists an integer d such that:

where the F'i = F';(E:, ..., E : ) are homogeneous Lie polynomial of degree i in the vectorfields E;' where E: E E; fori21. Precisely, for the first terms one obtains

"?(E:, E:, E:) =E: + 3 1 [E:, E:],

X is said to be nilpotent of order p ( Zp= 0) if the following

decreasing sequence stops:

L'=E,

P2(E;,E:) =E:

P1(E:)= E:

V X EM

dim A(x) = d

(2.7)

...

with

E 2 =[ E ,XI,...) si= [ E ,Li-l]

In the sequelf and g indifferently denote the vector fields and their associated Lie derivatives. At the same time g] denotes the Lie bracket of vector fields

62

1 - 6adf + 2ad2/ - e-Sadf

Through the paper 60 will denote a sampling period small enough to guarantee the convergence of the series exp'ansions manipulated.

E: :=

Usual sampling

where W denotes the shuffle product ([201).

Given a sampling period 6 (6 E IO, 60[) , let us now assume in (2.1), m = I (gl = g ) and that the control U,(() = U(() is constant over time intervals of amplitude 6:

Composing the successive exponentials one obtains ([ 161):

U(()

= u(k) for k 6 1 t < (k + 1)s

Definition 2.1 ( [ I S ] ) A nonlinear sampled dynamics of the

form

x(k

+ 1 ) = FS(x(k), u(k)) &[?

represeMation:

6 €10, SO[, (2.3) admits the exponential

(9

6'

+ 1 ) =x(k) + i2 I -~

(1)

x(k)

Xi

4 . 9

P

c

U?P

- 1) Pf i!

e

where

a " l ' c f , g ) = e i j - f , s " d / ( F ' ~ ( f g= ) ) q(E';, ..., E ; )

with

E; = eiJj-fJs"dfEl)

i!

'1

___. ,e

l 2

I

x(0)

(2.9)

for i 2 1 and j 2 1. (2.10)

Multirate sampling

su

Assume now the CO 01 U(() constant over fraction of the time intend 5 (say 3 = - , for p 2 1) and denote u;(k)the constant value of u(t) over &+(i - 1) 5 , k 6 + i 8[, (i = 1, ..., p ) . Definition 2.2 A nonlinear sampled dynamics with p inputs

Remark 2.1 For computing purposes, it is interesting to note that, according to (2.4), the dynamics (2.3) can be written ;is x(k

E

x(p) =ephf

(2.3)

where F S : M X B" + B" is analyric for 6 E] 0, is said the sampled analogue of (2.1) iffor the same initialization x(O), the state evolutions of (2.1) and (2.3) coincide at sampled instants t = k6; namely ifx(t = k6) = x(k) for k > 0. Proposition 2.1 For

W $ad2g(6g)

Proposition 2.3 For S E ] 0 , 601, x(0) E B " and a sequence of piecewise constant inputs U (O),...,U (p - l ) , the state x(p),reached at time t = p6 under the sampled dynamics action (2.4). is given by:

(2.2)

k2O

@ad?

of the form

(2.5)

x(k

+ 1 ) = F'(x(k),

ul(k), ..., up(k))

(2.1 1)

where F6 :M X 8 p -+ 8"is analytic for 6 E J , &I,is said the multirate sampled analogue of(2.1)of order p if, for the same initialization. the state evolutions of (2.1) and (2.1 I ) coincide at sampled instants t = k6, namely ifx(t = kS) = x(k) for k 2 I .

where x ( k ) represents the i-th time derivative of x ( t ) computed at time t = k6. Recalling the Baker-Campbell-Hausdorff formula ([ 121) and related combinatoric expansions it has been shown in [16] that the sampled dynamics (2.4) can be regrouped according to the successive powers in U as follows.

6

8 = - , the multirate sampled dynamics of order p , associated to (8.1). admits the eqonential representation:

Proposition 2.4 For 6 € 1 0 , 601and

Proposition 2.2 ([I61 ) For 6 E ] 0 , 601. (2.4) satisfies the

expansion :

(2.6)

The proof is deduced from (2.9) replacing 6 by 6 and u(i) by u ; + 1 ;(i = 0, . . . , p - 1).

P

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&[.Then compute the "extended equation" defined in [9, 10,6, 81 as follows:

E ] 0,

Remarks (i) Local controllability properties of the sampled dynamics (2.4 can be expressed in terms of the Lie algebra generated by for for i ,j 2 1 ([14, 221).

2

n - 1

in - 1) for satisfying other constraints such as optimality. 2421.

Remark When Z is nilpotent, the h;(6,U) are polynomials. Proposition 3.1 Given a single input continuous-time dynamics ( 2 . 1 ) satisfying ( a ) , ( b ) , (c), given a pair of "admissible states" { X O , xf}, there exists a multirate digital control of order n - 1 or u steering xo to xf in one step of amplitude 6 ifthe set of n equations:

admits a solution

(6,U I ,. . . ,Un -1

).

The proof is achieved by equating the exponents of (3.6) and (3.7).

Remarks (i) In practice, the solution, which is an 11 n-uple (6, ..., U,- 1) is approximated by means of formal series inversion (see Grobner [SI for example) making use of formal computin languages ([2,31). (ii) this non zero drift case, the fact that the steering time 6 is not free but computed as the control from series reversion represents a strong restriction.

8,

Under the usual assumptions set in the motion planning literature, it has been shown in Section 3 that it is possible to steer under multirate digital control the system in an "admissible" state configuration, at least in a approximated meaning. The remaining difficulties are the choice of the order of the multirate on the various inputs channels depending on the desired objective and the control computations. An interesting situation occurs when triangular systems are considered, in particular when chained systems are investigated, In this case the previous study is highly simplified. When we look at the literature it clearly appears that a great number of practical examples verify such particular forms. Moreover, if this is not the case, it often occurs that such forms can be obtained under coordinate changes and feedback [9,10,11,18,19] and sufficient conditions to convert systems into chained forms have been given in [ 18,191. This considerably enlarges the interest to discuss such forms. In this section we will consider two inputs single chained systems and show with an example how the control algorithm is simplified in this case and achieves exact steering. Consider a two input system transformed under coordinates and input changes into the chained form :

Motion planning for systems without drift term From G I , define the sequence of distributions with

G; = Gi [GI, G,

~

11

-

1 + [GI, Gj- 11

= span { [g, h ] . g

E

Motion planning of nonholonomic chained systems

4.

VI,

G I , h E Gi- I ) , i 2 1.

System (2.1) is said to be maximally nonholonomic if there exists an integer p < n such that rankG;= rankG, + 1 for all i 2 p + 1 and rankc, + ~ ( x=) n on M.Consider now a maximally nonholonomic driftless dynamics with two inputs, that is, (2.1) with m = 2 a n d f i 0 satisfying conditions (a-b-c). Choose arbitrarily a multirate control of order ( n - 1) on the control u2. and one on u1.

Lemma 3.2 Given the dynamics (2.1) with m = 2 and f - 0 , for 6 E ] 0. &[, the state reachedfrom xo in one slep under the multirate sampled dynamics of order (n - I)on u2 and one on ul can be written as: x( 6 ) = ~ ' ( x o~, 1 . ~ 2 1..., , ~2n-1) 7 ; 1 ( f i , u 1 ~ ~ 2+ ) g l + hn(&ul,u)gn

=e where the &(8,ul

..e

x3 = X 2 U I

b o ) (3.9)

x, = x,

) are series in their arguments.

(4.1)

- 1241

and assume that it is controllable on M. In this case it can easily be verified that:

Remark When dE is nilpotent, then the h;(6,u1,&2 ) are polynomial. Arguing as previously, one obtains: Proposition 3.2 Given the dynamics ( 2 . 1 ) with m = 2 and

k

adglg2 = (-ilk

f = 0 satisfying ( a ) ,(b),( c ) , given a pair of admissible states {xo. xf} ifthere exists a solution to the set ofequations:

and then the system can be steered in one step of amplitude Gfrom ~0 to xf. using a single rate digital control on U! and a multirate control of order n - I on u2.

a

fork = 0, xk+2 adgig2 = 0 for i 2 n - 1

adg~adgigz= 0

..., n - 2;

for i 2 0

thus obtaining with 8 := (-1)'-2adgF2g2= := span (gi,gz. g3,

a for i ax;

= 3,

.., n.

..., gn I

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In order to illustrate the control procedure, let us consider the model of steering a unicycle ([ 6,10,19]). The equations

According to the definitions (2.8), the multirate sampled system of order one on u1 and n - 1 on u2 will be characterized by the vector fields, obtained by substituting ulgl tof and g2 to g . One finds with

arc

6 3 =n -l

(4.9) where x and y represent the coordinates of the centre of the unicycle and Oexpress the rotational angle with respect to x.

E?' = e-jiduigi(E:) ,

for

Let us transform the system into a chained form, for it is classical ([ 6,10,19]) to consider the input changes w1 = cosOel, and w2 = e2 and the coordinate changes (cosO# 0), X I = x, x2 = tge, x3 = y , one finds:

j = 1 , ..., n - 2 (4.3)

E{=O and [ E' { , E1 , ] = O f o r i l 2 , j l l , 1 2 1 .

(4.4) (4.10)

4,

Simple computations, inspired from [ 161, show that

with U ] = W I and u2 = c0s2Bw2. Setting 8 = one obtains the sampled version of system (4.10) under single rate on u1 and multirate of order 2 on u2, that is:

LenJma 4.1 If ul z 0 then the n - 1 vector fields E: ( x ) , .. ., E:- (x) are linearly independent on M. Lemma 4.2 Given any initial stale xo, the state reachedgt time Gunder the action of u1 over [O, 6[ and u2i over [(i -I) 6, i 6 u o r i = I , ..., n - 1 is given by:

XI(@

= XOl + S U I

6

x2(@ = x02 +y( U 2 1 + u22) Xd@

which can be written because of (4.3) as

= x03 + 6 X 2 O U 1 +

SL 8 (3u21 + U 2 2 ) U l

Given a pair of states { X O . XI), setting XJ= xo + 6a, one has to solve the three equations

S U I = 6al 6 T ( ~ 2 +1 ~ 2 2=6a2 )

Ix

(The bar indicates the evaluation at x of the function inside the parentheses)

B 8 (3u21+ uz)u1=6a3 - 6 ~ 2 0 ~ 1

Because of the Jtriangularity of the chained form, the exponential expansion (4.6) stops and one easily obtains exactly

setting with a1 f 0 the exact solution:

(-46))' = (X0)I + 6 U l

(x(@)2= (x0)2+

3 ( ~ 2 1 +...+~ 2 n - 1 )

(4.7) .., n

(x(@>; =(e"lgl(xo) + u z l ~ - (xo)+ l . . . + u z n - I ~ f ( x O )i) i=; 3,

Setting for example6 = 1, xo = (0,0,-0.6) and XJ = (3,3,0), Figs. 1-2 illustrates the steering action from xo to XJ in one path of amplitude 6 = 1. The example tested illustrates the benefits of the digital control proposed with respect to the number of paths required, the simplicity of the control for exact steering, its limited amplitude and the state trajectories (Figs. 12). It has to be note that, after reversing the input changes, the control u1 and u2 (Fig. 4) become continuous in the sense that they are simulated at the frequency for integrating system (4.9) (Fig. 3).

Exact steering is thus achieved solving the n equalities:

-

&I=

&ll

6(u21+ ...+ U Z n - l ) = 6a2

( u 2 1 ~ -(xo)+ 1

(4.8)

...+ u 2 n - l ~ : ( x O ) ) =(& i -e&lgI(xo) -xo)i

;

i = 3, ...,n Proposition 4.1 Given a pair of admissible states { X O , XJ] if a1 f 0, then there exists a solution U ] , u21 , ... ,uzn - 1 , to the set of algebraic equations (4.8) and the system (4.1) is exactly steered in one step from xo to xf , with U! = a1 and a multirate control of order n -I on u2.

REFERENCES Astrom K.J. and B. Wittenmark, Computer Controlled Systems - Theory and Practice, Englewood Cliffs, NJ : Prentice Hall, 1984. Barbot J.P., Computer aided dy?ign for sampling nonlinear analytic systems, in Applied algebraic algorithms and error correcting codes", Lect. Notes in Comp. Sc. 357 (T. Mora Ed.), 1989, pp. 74-88.

Remarks (i) When u1 = 0 but u2 # 0, one proceeds in the same way with a multirate of order n - 1 on u1 anone on 242. (ii) When u 1 = 0 =u2 one has to proceed in two steps. Simulated example 1784


figurc I : y position (state versus x)

Barbot J.P.,A forward accessibility algorithm for nonlinear discrete-time systems, Lect. Notes in Cont. and Info. Sci. (A. Bensoussan and J.L. Lions, eds.), 1990, pp. 314-323. Franklin G.F. and J.D.Powell, Digital Control of Dynamic Systems, Reading, MA: Addison-Wesley, 1977. Grobner W.. Serie di Lie e Loro Applicazioni, eds. Cremonese. Roma, 1973. Guyon C., G. Jacob, M. Petitol Motion planning and symbolic computation, preliminary report, 1991. Isidori A., Nonlinear Control Systems, Springer-Verlag, 2"d edition, 1989. Jacob G., Lyndon discretisation and exact motion planning, hoc. EEC-91, Hem&, 1991, pp. 1507-1513. Laffeniere G., A general strategy for computing geering controls of systems without drift, Proc.of the 30 CDC, Briehton. 1991. DD. 1115-1120. [lo] Lafferrieie G. &id H.J. Sussmann, Motion planning for controllable systems without drift: A preliminary report "Rutgers University, Report S YCON-90-04. 1990. [ 11) Laumond JP., Nonholonomic motion planning versus controllability via the multibody car system example, Report STANCS-90, 1345, Stanford University, 1990. [12] Mielnik B. and J. Plebanski, Combinatorial approach to Baker-Campbell-Hausdorficxponants. Ann. Inst. Hcnri PoincarC, Vol. XII, 3, 1970, pp. 215-250. [ 131 MOMCOS.and D. Normqd-Cyrot, On the sampling of a linear control system, 24 IEEE-CDC, Fort Lauerdale, USA, 1985, pp. 1457-1462. (141 Monaco S. and D. Normand-Cyrot, Invariant Distributions under Sampling, in Theory and Applications of Nonlinear Control Systems, (C.I. Bymes and A. Lindquist eds.) Elsevier Science Publishers, North-Holland, 1986. pp. [15] MOMCOS. and D. Normand-Cyrot, Multirate digital Control, Rap. D.I.S., University of Rome "La Sapienza", 1988. [16] MOMCOS. and D. Normand-Cyrot, A combinatorial approach to the nonlinear sampling problem, in Lect. Notes in Cont. and Info. Sci. (A. Bensoussan and J.L. Lions, eds.), 144, Springer-Verlag. 1990, pp. 788-797. [ 171 MOMCOS. and D. Normand-Cyrot, On the sampling of time delayed systems, Proc. ECC'91, Hermks, 1991, pp.556-560 [I81 Murray R.M. and S.S. Sastry, Steering nonholonomic systems in chained form, hoc. of the 30Lh CDC. Brighton 1991, 1121-1126. [19] Murray R.M. and S.S. Sastry. Nonholonomic motion planning: steering using sinusoids, Memorandum nr. UCBERL M91/45. Univ. of California. Berkeley, 1991. [20] Ree R.. Lie elements and algebra associated with shuffles, Ann. of Maths, 66,1958, pp. 210-220. [21] Sasuy S.S and 2. Li, Robot motion planning with nonholonomic constraints, h o c . 281h IEEE, CDC, Tampa, 1989,211-216. (221 Sontag E.D., Orbit theorems and sampling (M. Fliess and M. Hazewinkel eds.) in Algebraic and Geometric Methods in Nonlinear Control Theory, D. Reider Company, 1986, pp. 441-483. [23] Sussmann H.J., Orbits of families of vector fields and integrability of distributions, Trans.Amer. Math. Soc. 180 (1973). pp. 171-188. [241 Sussmann HJ., A product expansion for the Chen series in Theory and Applications of Nonlinear Control Systems, C. Byrnes and A. Lindquist Eds., NorthHolland, 1986, pp. 323-335. [25] Sussmann H.J., Local controllability and motion planning for some classes of synems with drift, hoc. 301h CDC, Brighton, 1991, 11 10-1113.

figure 2

B position (state versus x)

:

os

0

I S

I

figure 3 : Continuous

0

01

02

03

I

2s

2

3

time controls el(-), ez(-.)

04

OS

06

07

O t

09

TlME

- - - - - -.- - - - - L2I

6

1~

U1

U1

I

1785
