Definitions. Basic problems

To sum up, we shall be intereste'd in control systems of the form

m

= ~ uiXi(x), x E M, (Z)

i=l

where the configuration space M of the system is a C °O manifold, X 1 , . . . , Xm are Coo vector fields on M, and the control function u(t) = ( u l ( t ) , . . . , ut(t)) takes values in a fixed compact convex K of R TM, with nonempty interior, and

Geometry of Nonholonomic Systems 57

symmetric

with respect to the origin. Such systems are called

symmetric

(or

driftless).

One also says t h a t controls enter

linearly

in (Z).

For any choice of u as a measurable function defined on some interval [0,

T],

with value in K , equation (Z) becomes a differential equation

ic = ~ ui(t)X~(x).

(1)

i = 1

Given any point xo on M, we can integrate (1), taking

x(0)=x0 (2)

as an initial condition. For the sake of simplicity, we shall suppose t h a t this equation has a well-defined solution on [0, T] for all choices of u (this is guaran- teed if M is compact or if M = R n, and vector fields X~ are bounded). Call this solution x~. One says t h a t xu is the path with initial point x0 and controlled by u. We shall mainly be interested in its final value

x~(T).

Classically, points in M are called the

states

of the system. One says for example t h a t the system is

steered

from state Xo to state

xu(T)

by means of the control function u.

One also says t h a t xu (T) is

accessible,

or

reachable,

in time T from Xo. We shall denote by

A(x, T)

the set of points of M accessible from x in time T (or in time < T, it is the same thing for symmetric systems), and by

A(x)

the set of points accessible from x, t h a t is

A(x) = U A(x,T).

T > 0

Basic problems of Control Theory are:

- determine the accessible set A(x);

- given a point

y,

accessible from x, find control functions steering the system from x to y;

- do the preceding in minimal time;

- more generally, find control function u ensuring any given property of xu (t), the path controlled by u.

Given xo, the control function

u(t)

is considered as the

input

of the system, and

xu(t)

as the

output.

In a more general setting, the output is only some function

h(x)

of the state x, h being called the observation: the state is only partially known. Here we will take as observation h = Id, and call indifferently x the state or the output.

We can now state another basic problem:

58 A. Bella'iche, F. Jean and J.-J. Risler

- can one find a map k : M - ~ / ( such that the differential equation

= f ( x , k(x)) (3)

has a determined behaviour, for example, has a given point xo as an at- tractor?

Since in this problem, the output is reused as an input, such a map k is called a feedback control law, or a closed-loop control. If (3) has xo as an attractor, one says t h a t k is a stabilizing feedback at x0.

1.3 T h e c o n t r o l distance

R e t u r n to the control system (Z). For x, y E M, define d(x, y) as the infimum of times T such t h a t y is accessible from x in time T, so d(x, y) = +co if y is not accessible from x. It is immediate to prove that d(x, y) is a distance [distance function] on M. Of course, this is the case only because we supposed t h a t K is symmetric with respect to the origin in R m.

Distance d will be called the control distance.

We can define d in a different way. First, observe t h a t since K is convex, symmetric, with nonempty interior, we can associate to it a norm I1' IlK on R TM, such that K is the unit ball llu[[g ___ 1. Now, for a controlled path c = x~ : [a, b] ~ M obtained by means of a control function u e Ll([a, b], Rm), we set

length(e)

⁼

IJu(t) Hg dt. (4)

If c carl be obtained in such a way from several different u's, we take the infimum of the corresponding integrals. Then, d(x, y) is the infimum of the lengths of controlled paths joining x to y (and, of course, this is intended in the definition of an infimum, + c o if no such path exists).

A slightly variant construction may be useful. ~5:ansfer the function I1" IlK to T~M, by setting

IlVllK = inf{ II(ul, . . . , Um)NK I V ---- Ul X I (X) ~" " " + u m X m ( x ) }.

We get in this way a function on T x M which is a norm on s p a n ( X l ( X ) , . . . , Xm(x)) and takes the value + c o for vector not in this subspace. We can now define the length of any absolutely continuous path e : [a, b] -¢ M as

b

length(c) = ]a lla(t)ilK dt

and the distance d(x, y) as the infimum of length of paths joining x and y.

Geometry of Nonholonomic Systems 59 Note that distances corresponding to different K , say K1 and/(2, are equiv- alent: there exists some positive constants A and B such that

Adl (x, y) <_ d2 (x, y) < Bdl (x, y).

The most convenient version of the control distance is obtained by taking for K the unit ball of R m, which gives

2 i/2

Ilull = (u~ + - . . + Urn) •

In this case, the distance d is called the sub-Riemannian distance attached to the system of vector fields X1,..., Xm. As a justification for this name, observe that, locally, any Riemannian distance can be recovered in such a way by taking m = n, and as X l ( x ) , . . . , Xn(x) an orthonormal basis, depending on x, of the tangent space TxM. A more general, more abstract, definition of sub-Riemannian metrics can be given, but we shall not use it in this book.

Now, observe that d(x, y) < oo if and only if x and y are reachable from one another, that A(x, T) is nothing else that the ball of center x and radius T (for d), and that controlled paths joining x to y in minimal time are simply minimizing geodesics.

Many problems of control theory, or path planning, get in this way a geo- metric interpretation. For another example, one could think to obtain a feed- back law k(x) stabilizing the system at x = Xo by choosing k so as to ensure

](x,k(x)) to be the gradient of d(x, xo). Unfortunately, this does not work, even if we take the good version of the gradient, i.e., the sub-Riemannian one:

g r a d . f = + . . . + (x .f)xm.

and take k(x) = (Xl.f,... ,Xmf) for that purpose. But studying the reasons of this failure is very instructive. Such a geometric interpretation, using the sub-Riemaniann distance, really brings a new insight in theory, and it will in several occasions be very useful to us.

1.4 Accessibility. T h e t h e o r e m s o f C h o w a n d S u s s m a n n

We shall deduce-the classical theorem of Chow (Chow [7], Rashevskii [28]) from a more precise result by Sussmann. Sussmann's theorem will be proved using L 1 controls. However, it can be shown that the results obtained are, to a great extend, independent of the class of control used (see Bella'iche [2]).

Consider a symmetric control system, as described above,

m

= E u ' X i ( x ) ' x E M , u E K . (Z)

i = 1

60 A. Bella~che, F. Jean and J.-J. Risler

Recall the configuration space M is a C ~ manifold, X 1 , . . . , Xm are C ~ vector fields on M, and K, the control set, or parameter set, is a fixed compact convex of R m, with nonempty interior, symmetric with respect to the origin.

In all this section, we fix a point x0 E M, the initial point, and a positive time T. Set

n T = L 1 ([0, T], am).

We shall call this space the space o] controls. It may be considered as a normed space by setting

//

Ilull = Ilu(t)llK dt.

Given u E 7/T, we consider the differential equation { ~ = ~i~=1 ui(t)Xi(x), 0 < t < T

x(0) = x0 (5)

Under suitable hypotheses, the differential equation (5) has a well defined so- lution x~(t). We will denote by

Endxo,T : 7/T -+ M

the mapping wich maps u to x~(T). We will call End~o,T, or End for short, the end-point map.

Now, the accessible set A(xo) (the set of points accessible from Xo for the system ~ , regardless of time) is exactly the image of Endxo,T. Indeed, every controlled path c : [0, T'] -~ M, defined by the control u : [0, T t] ~ M may be reparametrized by [0, T]. Conversely, if u E 7/, and L = length(x~), the control function

u(¢(t)) 0 < t < L,

v ( t ) =

where ¢ is defined as a right inverse to the mapping

//

s ~+ []U(T)llK dT

from [0, T] to [0, L] takes its values in K, and defines the same geometric path

a s u .

T h e o r e m 1.1 ( S u s s m a n n [36], S t e f a n [35]). The set A(xo) of points ac- cessible from a given point Xo in M is an immersed submanifold.

We shall prove this theorem using arguments from differential calculus in Banach spaces, taking advantage from the fact that the end-point map is a differentiable mapping from 7i to M, a finite dimensional manifold.

Geometry of Nonholonomic Systems 61 Recall the rank of a differentiable mapping at a given point is by definition the rank of its differential at that point. The theorem of the constant rank asserts that the image of a differential map with constant rank is an immersed submanifold (for more details about this part of the proof, see Bella'iche [2]).

D e f i n i t i o n . Let p the maximal rank of the end-point map Endz0,T : 7-/T --+ M. infinitesimal variations ~xw(3T) obtained from infinitesimal variations 6w of w. We can consider special variations of w, namely variations of the first part of w only, leaving the two other parts unchanged. In other words, we consider tile control functions mor'pl~ism of M, these variations of x,v(3T) form a subspace of dimension p of the tangent space TyM. The space formed by variations of the xw(3T) caused by unrestricted variations of the control w has thus dimension > p, an so has dimension p. This proves that w is normal.

Of course, the fact that w is in ~f~3T instead of being in "~T is harmless. •

62 A. Bellai'che, F. Jean and J.-J. Risler set. Whence it results that the accessibility components are also the connected component of M for the topology defined by d.

When the Chow's Condition holds, one says that system (Z) is controllable.

The reciprocal of Chow's theorem, that is, if (Z:) is controllable, the Xi's and

Geometry of Nonholonomic Systems 63 1.5 T h e s h a p e o f t h e a c c e s s i b l e set in t i m e 6

The purpose of this section is to study the geometric structure of

A(x, ~)

^for

small ~. Let us recall t h a t

A(x, E)

denotes the set of points accessible from x in time ~ (or in time < ~, it is the same thing) by means of control ui such t h a t ~ u~ < 1. In other words,

A(x, s)

is equal to

B(x, ~),

the sub-Riemaniann closed ball centered at x with radius e.

We suppose in the sequel t h a t Chow's condition is satisfied for the control system (~). Choosing some chart in a neighbourhood of x0, we may write (1)

a s

m

i----1

The differential equation (1) thus appear as a perturbation of the trivial equa- tion

iTS

= (7)

Classical arguments on perturbation of differential equations show t h a t the solution of (6) is given by

x(T) = x(O) + ui(t) dt Xi(O) + O(llull2),

(8)

where, for u, we use the L 1 norm. Thus, with a linear change of coordinates, the set of points accessible from x(0) = 0 in time T < ~ satisfies, for small

A(x,

e) C C[-~, e]~' x [-E 2, ~2]~-~1,

where nl is the rank of the family X I ( 0 ) , . . . ,Xm(0). As a first step, the set

A(x, E)

is then included in a flat pancake.

The expression (8) implies also t h a t the differential of the end-point map- ping at the origin in 7 / i s the linear map

u ~-~ ~=l (~oTu~(t) dt) X~(O) •

Since, typically, we suppose m < n, this linear map has rank nl < n and the end-point mapping is not a submersion at 0 E 7-/. Following our definition, this means t h a t the constant path at Xo is an abnormal path. This result has a lot of consequences.

64 A. Bella'/che, F. Jean and J.-J. Risler

Given a neighbourhood U of xo, there may not exist a

smooth

^mapping

x ~ u ~ of U into 7/ such that the control u ~ steers xo to x, or, as well, x to xo. A stronger result is Brockett's theorem asserting the non-existence of a continuous feedback law, stabilizing system (Z) at a given point Xo, when

m<n.

To go further in the description of the set

A(x, ~)

we can use the so-called

iterated integrals.

For example, the system Xl = Ul

X2 = u2 (9)

X3 ~ Xl~t2 --

X2Ul

x l ( 0 ) = x (0) = 3(0) = o is solved by

T

xi(T)= fo ul(t) dt T

x2(T) = ~o

u2(t) dt ^(I0)

x 3 ( T ) =

]i T (~otl Ui (t2) dt2) u2(tl) dti - ~0 T ( fot' U2(t2) dt2) ui (ti) dtl

This scheme works for chained or triangular systems, that is, 2j depends only on the controls and x i , . . . , x j - i . But we shall see that it can be put to work for any system. To begin with, let us rewrite (9) as

.T ---- U l X l ( x ) -{-

u2X2(x), x(O) : O.

T h e n (10) can be read as

x(T) = x(O) + ul(t)dt

X I ( O ) +

u~(t)dt

X 2 ( O ) +

P u t this way, the formula for

x(T)

can readily be generalized to any control system of the form

x" = E uiXi(x).

i < ~ < m

Geometry of Nonholonomic Systems 65 One gets (the proof is not hard, cf. Brockett [5])

(L" )

z(r) = z(O) + ~ u~(t) dt + O(llnll ~) X~(O) +

i_<i<m

loT (/otlUj(t2) dt2) ui(tx)dtl) [Xi, Xj](O)-~'O(llul,3),

or, written in a more civilized manner

x(T) = x(O) + ~ (AT(u) + O(llull2))X~(O) +

l < i < r a

AT (u)[Xi,X~]( 0 ) +

O(llull 3)

l <i<j<rn

which can, for given T, be considered as a limited expansion of order 2 of the end-point mapping about 0 in 7-/. Observe t h a t AT(u) is a linear function with respect to u E 7-/, and AT(u) is a quadratic function on 7-/. This expansion generalizes the expansion (8) and the set A(x, e) satisfies now

A(x,e) C C[-a,e] m × [-e2,e2] "2-"1 x [-ea,ez] '~-n2. (11) Having shown t h a t A(x, ~) is contained in some box, one can ask whether it contains some other box of the same kind. Of course, before this question can be taken seriously, one has to replace inclusion (11) by

A(x, e) C C[-e, el"' x [ - e 2, e2] n=-"l x [ - e a, ca] "a-"= x . . . where the integers n l , n2, na, . . . are the best possible.

Now, except for the case n2 = n which can be dealt with directly, the proof of an estimate like

C'[-a,e] m x [-e2,e2] nz-nl x [-ea,e3] ha-ha x . . . c A(x,e)

requires new techniques and special sets of coordinates. Instead of computing limited expansion up to order r, we will compute an expansion to order 1 only, but by assigning weights to the coordinates. This will be done in §§1.6-1.8.

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