The car with <span class="mathjax-formula">$N$</span> trailers : characterisation of the singular configurations

THE CAR WITH N TRAILERS: CHARACTERISATION OF

THE SINGULAR CONFIGURATIONS

FREDERIC JEAN

Abstract. In this paper we study the problem of the car withⁿtrailers.

It was proved in previous works (9], 12]) that when each trailer is perpendicular with the previous one the degree of nonholonomy is^Fn⁺³

(the (ⁿ+ 3)-th term of the Fibonacci's sequence) and that when no two consecutive trailers are perpendicular this degree isⁿ+ 2. We compute here by induction the degree of non holonomy in every state and obtain a partition of the singular set by this degree of non-holonomy. We give also for each area a set of vector elds in the Lie Algebra of the control system wich makes a basis of the tangent space.

1. Introduction

A car with

n

trailers is a nonholonomic system it is, indeed, subject to non integrable constraints, the rolling without sliding of the wheels. The conguration of the system is given by two positions coordinates and

n

+ 1 angles. There are only two inputs, namely one tangential velocity and one angular velocity which represent the action on the steering wheel and on the accelerator of the car.

The problem of nding control laws was intensively treated in many pa- pers throughout the literature: for instance by using sinusoids (see the works of Murray, Sastry and alii 11], 15]) or from the point of view of dieren- tially at systems (introduced by Fliess and alii 3]).

In general the study of such systems (to prove controllability, to nd con- trol laws,

:::

) involves tools from nonlinear control theory and dierential geometry. In particular an important concept for such problems is the de- gree of nonholonomy, which expresses the level of Lie-bracketing needed to generate the tangent space at each conguration. This degree comes up for instance in estimation of the complexity required to steer the system from a point to another (see 7], 2]).

Laumond (6]) has presented a kinematic model for the car with

n

trailers in 1991 and has proved the controllability for this model. He has also proved that the degree of nonholonomy of the system is bounded toward the top by 2ⁿ⁺¹. Srdalen has afterwards proved in 12] that, when no two consec- utive trailers are perpendicular, this degree is equal to

n

+ 2. The system

Institut de Mathematiques, Universite Pierre et Marie Curie, Case 247, 4, place Jussieu, F-75 252 Paris Cedex 05. E-mail: jean@mathp6.jussieu.fr.

Received by the journal February 23, 1996. Accepted for publication August 5, 1996.

c Societe de Mathematiques Appliquees et Industrielles.

0

n-1

n n

vn

x y

(x,y)

ω θ θ

θ

Figure 1. Model of the car with

n

trailers.

congurations corresponding to such cases are called the regular points of the conguration space (see 1], 8]).

More recently, it has been proved that the degree of nonholonomy is bounded by the

n

+ 3-th Fibonacci number (13]) and that this bound is a maximum (9],10]) which is reached if and only if each trailer (exept the last one) is perpendicular to the previous one.

To close denitively the problem, we still have to study the non regular points for which the maximum degree of nonholonomy is not reached. For the car with 2, 3 and 4 trailers, a complete classication of the singularities has already been done in 4]. The goal of our paper is to extend this classi- cation to any number of trailers. Let us note that some results given here have already been presented without proof in 5].

In Section 2 of this report, we are going to write equations, give deni- tions and notations and construct an induction procedure. Section 3 groups together the main result of this paper, Theorem 3.1, and some conclusion on the form of the singular locus and on the degree of nonholonomy. Section 4 is devoted to the demonstration of Theorem 3.1, but the proof of some technical lemmas are relegated to the appendix.

2. Equations and notations 2.1. Control system

In this paper we are going to use the same representation as Fliess 3]

and Srdalen 12] for the car with

n

trailers. A car in this context will be represented by two driving wheels connected by an axle. The state is parametrised by

q

= (

xy

⁰

:::

n)^T where:

(

xy

) are the coordinates of the last trailer,

n is the orientation angle of the car with respect to the

x

-axis,

i, for 0

i

n

^;1, is the orientation angle of the trailer (

n

^;

i

) with respect to the

x

-axis.

Esaim: Cocv, October 1996, Vol. 1, pp. 241-266

The kinematic model of a car with two degrees of freedom pulling

n

trailers can be given by:

x

_ = cos

⁰

v

⁰

y

_ = sin

⁰

v

⁰

_⁰ = _R¹1sin(

^1;

⁰)

v

¹

_ ...

i = _Ri¹+1sin(

i^+1;

i)

v

i⁺¹

_ ...

n^;1 = _R¹_nsin(

n^;

n^;1)

v

n

_n =

!

n

:

where

R

i is the distance from the trailer (

n

^;

i

) to the trailer (

n

^;

i

+ 1),

!

n is the angular velocity of the car and

v

n is the tangential velocity of the car.

v

n and

!

n are the two inputs of the system.

The tangential velocity

v

i of trailer

n

^;

i

is given by:

v

i= ^Yn

j⁼i⁺¹cos(

j^;

j^;1)

v

n

:

Let us denote:

f

_ni=^Qn_j⁼_i⁺¹cos(

j^;

j^;1)

v

i =

f

_ni

v

n

i

= 0

n

^;1

:

The motion of the system is then characterized by the equation:

q

_=

!

n

X

¹ⁿ(

q

) +

v

n

X

²ⁿ(

q

) with

8

>

>

<

>

>

:

X

¹ⁿ = _@^@_n

X

²ⁿ = cos

⁰

f

⁰ⁿ _@@x+ sin

⁰

f

⁰ⁿ_@@y + ^sin(_R¹1^;0)

f

¹ⁿ_@^@0 +

+ ^sin(n_R^;_n^n;1)_@n^@;1

(2.1) We will suppose that the distance

R

i doesn't depend on

i

and to simplify we shall, from now on, consider it equal to 1 (we will come back to this hypothesis in Subsection 3.4).

2.2. Characterization of the singular locus

We are going now to dene the singular locus of the control system

f

X

¹ⁿ

X

^2ng, and give a characterization of this locus easy to use.

In this section

n

is xed and we write

X

¹ and

X

² instead of

X

¹ⁿand

X

²ⁿ. Let^L1(

X

¹

X

²) be the set of linear combinations with real coecients of

X

¹ and

X

². We dene recursively the distribution ^Lk =^Lk(

X

¹

X

²) by:

Lk =^Lk^;1+ ^X

i⁺j⁼k^Li

^Lj]

where ^Li

^Lj] denotes the set of all brackets

VW

] for

V

^2Li and

W

^2Lj.

Esaim: Cocv, October 1996, Vol. 1, pp. 241-266

Thus ^Lk is the set of linear combinations of iterated Lie brackets of

X

¹ and

X

² of length

k

. The union ^L of all ^Lk is a Lie subalgebra of the Lie algebra of vector elds on^R2(^S1)ⁿ⁺¹. It is called the Control Lie Algebra of the system^f

X

¹

X

^2g.

Let us introduce some notations. For

s

1 denote by

i

= (

i

¹

:::i

s) a sequence of

s

elements in ^f1

2^gand by ^As the sets of these sequences such that

i

¹= 1 if

s >

1, that is:

A

1 =^f(1)

(2)^g

As=^f

i

= (1

i

²

:::i

s) ^j

i

j = 1 or 2^g if

s >

1

:

^(2.2) The functions

u

(

i

) and

d

(

i

) indicate respectively the number of occurences of 1 and the number of occurences of 2 in the sequence

i

= (

i

¹

:::i

s), and the length of the sequence is^j

i

^j=

s

. Obviously we have

u

(

i

) +

d

(

i

) =^j

i

^j.

The vector eld

::: X

i¹

X

i²]

:::X

is^;1]

X

is] will be denoted by

X

i] or

X

i¹

:::X

is] and its value in

q

by

X

i]q. By using the Jacobi identity, we can write a bracket of length

k

(i.e. belonging to ^Lk) as a sum of

X

i] with^j

i

^j

k

.

Moreover, because of skew symetry of Lie bracket,

X

i] = 0 if

i

¹ =

i

² and

X

²

X

i¹

:::X

is] = ^;

X

i¹

X

²

:::X

is]. Then ^Lk is generated by the brackets

X

i] such that

i

^2As, for 1

s

k

.

For a given state

q

, let

L

k(

q

) be the subspace of

T

q(^R2(^S1)ⁿ⁺¹) wich consists of the values at

q

taken by the vector elds belonging to ^Lk. We have an increasing sequence of dimensions:

2 = dim

L

¹(

q

)dim

L

k(

q

)

n

+ 3

:

(2.3) If this sequence stays the same in an open neighbourhood of

q

, the state

q

is called a regular point of the control system otherwise,

q

is called a sin- gular point of the control system (see 1]). Thus the sequence (2.3) at any state

q

allows to characterize regular and singular points, i.e., the singular locus.

To determinate the sequence (2.3), we dene, for

i

^2f1

n

+ 3^g:

_ni(

q

) = min^f

k

^j dim

L

k(

q

)

i

^g

d

_ni(

q

) = min^f

d

^j dimspan^h

X

j]q

^j

j

^j

_ni(

q

)

d

(

j

)

d

ⁱ

i

^{g (2.4)} In other words, the fact that

k

=

_ni(

q

) is equivalent to:

dim

L

k(

q

)

i

dim

L

k^;1(

q

)

< i

^(2.5) The sequence (2.3) can be deduced from the

_ni(

q

)'s,

i

= 1

:::n

+3, by:

if⁹

i

^2f1

n

+3^gsuch that

k

=

_ni(

q

), then dim

L

k(

q

) is strictly greater than dim

L

k^;1(

q

) and equal to the greatest

j

such that

_nj(

q

) =

k

, otherwise dim

L

k(

q

) = dim

L

k^;1(

q

).

Thus the functions

_ni(

q

),

i

= 1

:::n

+ 3, characterize completely the singular locus. Hence the paper is devoted to the calculation of these func- tions. Notice that the functions

d

_ni(

q

) are not useful in the characterization

Esaim: Cocv, October 1996, Vol. 1, pp. 241-266

of the singular locus, they are only a tool for the computation of the func- tions

_ni(

q

).

To simplify we will omitt the dependence on

q

in

_ni(

q

) and

d

_ni(

q

). Accord- ing to its denition,

_niincreases with respect to

i

, for

i

lesser than dim

L

(

q

) (when

i

is strictly greater than this dimension,

_ni is equal to ^;1). We will prove in this paper (in Theorem 3.1) that this sequence is strictly in- creasing with respect to

i

for 2

i

n

+ 3, which means that, for any

k

, dim

L

k(

q

)^;dim

L

k^;1(

q

) 1. In other words, we will prove that, for 2

i

n

+ 3,

k

=

_ni is equivalent to (compare with (2.5)):

dim

L

k(

q

) =

i

dim

L

k^;1(

q

) =

i

^;1 (2.6)

We can yet calculate the rst values of these sequences.

-

L

¹(

q

) is two dimensionnal for all

q

, and

¹ⁿ and

ⁿ² are equal to 1. To span a two dimensionnal linear space we need both

X

¹and

X

²whereas for a one dimensionnal linear space

X

¹ is sucient. Then

d

ⁿ¹ = 0 and

d

ⁿ² = 1.

-

L

²(

q

) is generated by the family

X

¹

X

²

X

¹

X

²] which is three di- mensional for all

q

(it is clear from Formula (2.1)), so

³ⁿ= 2. Moreover it is not possible to nd another three dimensionnal family of vector elds which contains \a fewer number of 2", then

d

ⁿ³ = 1.

Finally, for all state

q

:

¹ⁿ= 1

d

ⁿ¹ = 0

²ⁿ= 1

d

ⁿ² = 1

³ⁿ= 2

d

ⁿ³ = 1

:

^(2.7)

2.3. Induction procedure

For

q

^{2 R2}(^S1)ⁿ⁺¹ and 1

p < n

, we will denote by

q

^p the projec- tion of

q

on the rst (

n

+3^;

p

) coordinates, that is

q

^p = (

xy

⁰

:::

n^;p)^T. Let us consider now the system of a car with

n

^;

p

trailers. The states

q

⁰ belong to^R2(^S1)^n;p+1 and the control system^f

X

^1n;p

X

^2n;pgis given by Formulas (2.1) with

n

^;

p

instead of

n

, that is:

(

X

^1n;p = _@^@_n^;_p

X

^2n;p = cos

⁰

f

^0n;p_@x^@ + sin

⁰

f

^0n;p_@y^@ +

s

¹

f

^1n;p_@^@0 ++

s

n^;p @@n^;p^;1

where

c

m = cos(

m^;

m^;1),

s

m = sin(

m^;

m^;1) and

f

_i^n;p =

c

i⁺¹

c

n^;p. Hence for any

q

⁰ we have the sequences

_j^n;p(

q

⁰) and

d

ⁿ_j^;p(

q

⁰),

j

= 1

:::n

^;

p

+ 3. The dimensions of the spaces

L

k(

X

^1n;p

X

^2n;p)(

q

⁰),

k

1, are characterized by the sequence

_j^n;p(

q

⁰).

On the other hand,

X

^1n;p and

X

^2n;p can be seen as vector elds on ^R2 (^S1)ⁿ⁺¹ whose last

p

coordinates are zero and which values at

q

depends only on the projection

q

^p. We can then consider ^L(

X

^1n;p

X

^2n;p) as a subalgebra of^L(

X

¹ⁿ

X

²ⁿ).

Esaim: Cocv, October 1996, Vol. 1, pp. 241-266

Remark 2.1. The motion of a point

q

^{0 2R2} (^S1)^n;p+1 is characterized by an equation _

q

⁰=

u

¹

X

^1n;p(

q

⁰) +

u

²

X

^2n;p(

q

⁰), and so depends only on the state

q

⁰. On the other hand, the motion of the point

q

^p is given by the projection on the rst

n

^;

p

coordinates of the motion equation of

q

: then the motion of

q

^p depends on

q

and not only on

q

^p.

According to Formulas (2.1), we can write

X

²ⁿas:

X

²ⁿ=

s

n

X

^1n;1+

c

n

s

n^;1

X

^1n;2++

c

n

c

n^;p⁺²

s

n^;p⁺¹

X

^1n;p+ (2.8) +

c

n

c

n^;p⁺²

c

n^;p⁺¹

X

^2n;p where

c

m= cos(

m^;

m^;1) and

s

m= sin(

m^;

m^;1).

With this relation, for 1

p

n

^;1, we will be able to express a vector eld in ^L(

X

¹ⁿ

X

²ⁿ) in function of

X

¹ⁿ

:::X

^1n;p+1 and of vector elds in

L(

X

^1n;p

X

^2n;p).

For instance, for

p

= 1, we have:

X

²ⁿ =

s

n

X

^1n;1+

c

n

X

^2n;1

X

¹ⁿ

X

²ⁿ] =

c

n

X

^1n;1;

s

n

X

^2n;1

X

¹ⁿ

X

²ⁿ

X

²ⁿ] = ^;

X

^1n;1+

X

^1n;1

X

^2n;1]

:

^(2.9) Hence^L(

X

¹ⁿ

X

²ⁿ) is equal to ^L(

X

^1n;1

X

^2n;1)^h

X

¹ⁿⁱ, where ^h

X

¹ⁿⁱ is the subalgebra generated by

X

¹ⁿ. Formula (2.8) allows to describe the projec- tion of ^L(

X

¹ⁿ

X

²ⁿ) on ^L(

X

^1n;1

X

^2n;1). We will see for instance that the projection of^Lk(

X

¹ⁿ

X

²ⁿ) is^Lk^;1(

X

^1n;1

X

^2n;1).

The induction will be done in the following way: we will assume that the functions

_j^n;p(

q

^p) (

j

= 1

:::n

^;

p

+3) are known for any

p < n

and, by us- ing the relation (2.8), we will calculate the dimensions of the

L

k(

X

¹ⁿ

X

²ⁿ)(

q

), and so the

_ni(

q

)'s (

i

= 1

:::n

+ 3), in function of the

_j^n;p(

q

^p)'s.

From now on the dependence on

q

or

q

^p will be omitted if there is no possible confusion for example we will write

_ni instead of

_ni(

q

) and

_j^n;p instead of

_j^n;p(

q

^p).

3. Singular configurations 3.1. Exposition of the result

In this chapter we present the main result of this paper, Theorem 3.1, which gives the recursion formulas satised by the sequence of functions

_ni. The proof of the theorem is given in Section 4.

Let us introduce a sequence

a

p by:

a

¹ = ²

a

p = arctansin

a

p^;1

:

^(3.1) This sequence is clearly positive and decreasing, that is: 0

< a

p

<

² for

p >

1. Notice that the recursion relationship is odd, that is if we dene an

Esaim: Cocv, October 1996, Vol. 1, pp. 241-266

other sequence

a

_p by the same recursion relationship and the initial value

a

^;1 =^;2, we have

a

^;_p =^;

a

p.

Let us dene also some brackets

A

ⁿ(

d

)] (

> d

or

=

d

= 1) by:

8

>

>

<

>

>

:

A

ⁿ(1

0)] =

X

¹ⁿ

A

ⁿ(1

1)] =

X

²ⁿ

A

ⁿ(

d

)] =

X

¹ⁿ

X

²ⁿ

:::X

²ⁿ

| {z }

d

X

¹ⁿ

:::X

¹ⁿ

| {z }

^;d^;1 ]

for >

2 (3.2) From this denition we can see that the bracket

A

ⁿ(

d

)] is of length

and that

X

²ⁿ occurs

d

times in it.

With these notations, we have:

Theorem 3.1.

8

q

^{2 R2} (^S1)ⁿ⁺¹, for 2

i

n

+ 3,

_ni is streactly increasing with respect to

i

, and we have

d

_ni=

ⁿ_i^;1;1.

We can calculate the functions

_ni(

q

) by the following induction formulas, for

i

^2f3

n

+ 3^g:

1. If

n^;

n^;1 =², then:

_ni=

_i^n;1;1+

d

ⁿ_i^;1;1

:

2. If ⁹

p

² 1

n

^;2] and = 1 such that

k ^;

k^;1 =

a

k^;p for every

k

^2f

p

+ 1

n

^g, then:

_ni= 2

_i^n;1;1;

d

ⁿ_i^;1;1

:

3. Otherwise,

_ni=

_i^n;1;1+ 1

:

Moreover, a basis ^Bn = ^f

B

_ni

i

= 1

:::n

+ 3^g of

T

q(^R2 (^S1)ⁿ⁺¹) is given by:

B

_ni=

A

ⁿ(

_ni

d

_ni)]q (3.3) 3.2. Form of the singular locus

Let us study the sequence

ⁿ = (

_ni)i⁼²:::n⁺³ (we remove

ⁿ¹ because it is always equal to

²ⁿ). The level sets of this sequence give a partition of the conguration space. For example, Figure 2 shows us the partition obtained for

n

= 3. Since each area is a cylinder with respect to the di- rection

^1;

⁰, we have just shown the projection of these cells on a plane

^1;

⁰ =

constant

. The complement of the four lines are the regular points of the system and corresponds to the values (1

2

3

4

5) of the sequence

³. For

n

3, let

q

²be the projection of

q

on the rst

n

+1 coordinates. Theo- rem 3.1 allows us to calculate the values of

ⁿ(

q

) in function of

^n;2(

q

²). We illustrate it in Figure 3, where we represent the set of points

q

= (

q

²

n^;1

n) wich have the same projection

q

².

Esaim: Cocv, October 1996, Vol. 1, pp. 241-266

Let us consider now a point

q

such that

k^;

k^{;1 6}=² for

k

= 2

:::n

. It is clear (for instance by looking at Figures 2 and 3) that there exists a neighbourhood of

q

in which the sequence

ⁿis constant. Conversaly, if there is an integer

k

, 2

k

n

, such that

k^;

k^;1 =², such a neighbourhood doesn't exist. Then we have determined the regular and singular points (see also 12] and 1]).

π/2

π/2 π/

- 2

π/

- 2

π/4

π/4 -

θ3- θ2

(1,2,3,5,7) (1,2,3,5,8)

(1,2,3,4,5) (1,2,3,4,7)

(1,2,3,4,6)

0 θ θ2- 1

Figure 2. Partition of the conguration space (

n

= 3) by the values of the sequence (

³²

³³

⁴³

⁵³

⁶³).

θn-2

θn-1-

θ θn- n-1

π/2 π/2

π/2 θn-1- θn-2

π/

- 2 - π/2

π/

- 2 - π/2

π/4

- - π/4

π/4 π/4

π/2 B

0 E

D C D C

E G B

A

A

0 F

H a

a

p+2 p+1

θ θn- n-1

q

² in case 3

q

² in case 1 or 2

A :

_ni=

_i^n;2;2+ 2 B :

_ni=

_i^n;2;2+

d

ⁿ_i^;2;2+ 1 C :

_ni= 2

_i^n;2;2+ 1 D :

_ni= 2

_i^n;2;2+

d

ⁿ_i^;2;2 E :

_ni=

_i^n;2;2+ 2

d

ⁿ_i^;2;2 F :

_ni= 2

_i^n;2;2;

d

ⁿ_i^;2;2+ 1 G :

_ni= 3

_i^n;2;2;

d

ⁿ_i^;2;2 H :

_ni= 3

ⁿ_i^;2;2;2

d

ⁿ_i^;2;2

Figure 3. Cells of the subset (

q

²

n^;1

n).

Esaim: Cocv, October 1996, Vol. 1, pp. 241-266

Theorem 3.2. The singular locus of the system is the set of the points for which there exists

k

²2

n

] such that

k^;

k^;1 =².

3.3. Application to the degree of nonholonomy

The degree of nonholonomy of the system at a point

q

is the degree from wich the sequence (2.3) is constant, that is the degree

r

such that:

L

r^;1(

q

)

6=

L

r(

q

) =

L

r⁺¹(

q

) = =

L

(

q

)

:

With our notations, this degree

r

is given by the greatest

_ni, for

i

n

+3.

Thus Theorem 3.1 implies that the degree of nonholonomy of the system is equal to

_nn⁺³ at any point

q

, and then the rank of the Control Lie Algebra at any point is

n

+ 3.

Let us recall the Chow theorem (also called the Lie Algebra Rank Con- dition): if the rank of the Control Lie Algebra at any point

q

of the cong- uration space is equal to the dimension of the tangent space in this point, the system is controllable (see for instance 14]).

This condition is satised here, therefore the system is controllable. We are meeting a classic result, wich was rst proved by Laumond in 1990 (6]).

By using Theorem 3.1, we can study the function

_nn⁺³ and nd some other results about the degree of nonholonomy (these results were already proved in 12],13], and 10]):

Theorem 3.3.

(i) At a regular point, that is a point such that

k ^;

k^{;1 6}=² for every

k

²2

n

], the degree of nonholonomy of the system is

n

+ 2.

(ii) The maximum of the degree of nonholonomy is the (

n

+3)-th Fibonacci number

F

n⁺³ (recall that the Fibonacci sequence is dened by

F

⁰ = 0,

F

¹ = 1,

F

n⁺² =

F

n⁺¹ +

F

n), and this maximum is obtained if and only if all the trailers are perpendicular (except the last one), that is if

k ^;

k^;1 =² for every

k

²2

n

].

Proof.

This theorem is obtained by applying the recursion formulas of Theorem 3.1 and by using the values for

n

= 0 given by Formula (2.7):

³ⁿ= 2 =

F

³ and

d

ⁿ³ = 1 =

F

².

Hence we see that

n

+ 2

_nn⁺³

F

n⁺³. Moreover, for

n

4,

_nn⁺³ can take all the values from

n

+2 to

F

n⁺³, but this property is no more true for

n >

4.

3.4. Case where the distances between the trailers are not all equals

We have assumed (see Subsection 2.1) that the distance

R

i between the trailer (

n

^;

i

) and the trailer (

n

^;

i

+ 1) is independent on

i

and equal to 1.

If we remove this hypothesis, the result is the same as Theorem 3.1, except

Esaim: Cocv, October 1996, Vol. 1, pp. 241-266

that we have to replace the case 2 by:

2 bis.If ⁹

p

² 1

n

^;1] such that

k ^;

k^;1 =

a

k^;p(

p

) for every

k

²

f

p

+ 1

n

^g, then:

_ni= 2

_i^n;1;1;

d

ⁿ_i^;1;1 where the sequence

a

k^;p(

p

) is dened by:

(

a

¹(

p

) = ²

a

k^;p(

p

) = arctan(_R^R_k;1^k sin

a

k^;p^;1(

p

))

for k > p

+ 1

:

The proof of this result is similar than the one of Theorem 3.1, but requires more notations. Therefore we don't give it in this paper. The dierence with the case

R

i = 1 is that the sequence of angles wich gives singularities de- pends on

p

. For instance, if

R

p^{+2 6}=

R

p⁺¹ =

R

p,

a

²(

p

^;1) is equal to ⁴ whereas

a

²(

p

) is not.

4. Proof of Theorem 3.1

In this chapter,

n

1 is xed. The proof is organized as follows: in a rst time, we study the relationships between the Lie Algebra for the

n

-trailers system and the Lie Algebras for the systems with less than

n

trailers. In a second time we use these relationships to establish the induction formulas for the functions

_ni(

q

). The main point of the proof is the rst part, that is Lemma 4.1. This kind of proof is inspired by 10].

4.1. Preliminary result

We have seen in Subsection 2.3 that, for 2

< m < n

, a vector eld in

L(

X

¹ⁿ

X

²ⁿ) can be decomposed in a linear combination (with functions as coecients) of

X

¹ⁿ

:::X

^1m+1 and of vector elds in ^L(

X

^1m

X

^2m). Lemma 4.1 gives such a decomposition and allows to conclude, in some particular cases, on the nullity or non nullity of the decomposition coecients.

We will denote, for

m

n

:

'

m =

m^;

m^;1

c

m = cos

'

m

s

m = sin

'

m

t

m = sin

'

m^;cos

'

msin

'

m^;1

:

^(4.1)

Lemma 4.1. Let

p

, 1

p

n

^;1 and

i

^{2 Aj}_i^j,

i

⁶= (1) (the sets ^As are dened in Formula 2.2). Then there exist functions

h

k(

'

n^;p⁺¹

:::'

n),

n

^;

p

+ 1

k

n

^;1 and

f

l(

'

n^;p⁺¹

:::'

n) in

C

¹(^Sp) depending on

i

such that:

X

_ni] = ^nX;1

k⁼n^;p

h

k

X

^1k+^Xd

s⁼¹

X

l^2As

f

l

X

_l^n;p] where

d

= max^f1

d

(

i

)^;

p

+ 1^g.

Moreover, if^j

i

^j

p

+ 1, then:

Esaim: Cocv, October 1996, Vol. 1, pp. 241-266

1. we have:

f

l =^X

b²Il(

c

n^;p⁺¹)^b0;b1(

t

n^;p⁺²)^b1;b2(

t

n)^bp;1;bp

g

bl (4.2) where the functions

g

bl(

'

n^;p⁺¹

:::'

n)²

C

¹(^Sp) depend on

bl

and

i

and the set

I

l^Zp⁺¹ satises:

I

l^f

b

= (

b

⁰

:::b

p)^j

d

(

l

) =

b

⁰

b

p0

^Xp

j⁼¹

b

j ^j

i

^j;j

l

^jg 2. if

b

²

I

l is such that

b

p= 0, then ^Pp_j^=1;1

b

j

< d

(

i

)^;j

l

^j

3. if we denote by

I

_l⁺ the following subset of

I

l:

I

_l⁺=^f

b

²

I

l^j Xp

j⁼¹

b

j=^j

i

^j;j

l

^jg

for every

b

²

I

_l⁺, there exist an integer

b

>

0 and a function

G

(

'

n^;p⁺¹,

:::'

n) which depends only on ^j

i

^j, ^j

l

^j,

d

(

i

),

d

(

l

) and

b

such that:

g

bl=

b

G

_b^j_i^jj_l^j_d⁽_i⁾_d⁽_l⁾

4. if

X

_ni] =

A

ⁿ(

+ (

p

^;1)

+

r

+ (

p

^;1)

)], with

>

r

1, and if

X

_l^n;p] is such that

l

^{2 A} and

d

(

l

) =

, the sequence(

:::r

) belongs to

I

_l⁺(the denition of the bracket

A

ⁿ(

d

)] is given by (3.2)) 5. if

p

= 1,^j

i

^j3, and

d

(

i

) = ^j

i

^j;1, then the coecient of

X

_l^n;1] such

that

l

^2A_d⁽_i⁾ and

d

(

l

) =

d

(

i

)^;1 is:

f

l(

'

n) = (

c

n)^d(i);2

:

Proof. The proof is quite long and technical, so it is done in the appendix (where the lemma is divided in four parts: Lemmae 5.1, 5.2, 5.3 and 5.4).

The point 3 implies that functions

G

(

'

n^;p⁺¹

:::'

n) doesn't depend on the sequences

i

and

l

but only on the length ^j

i

^j,^j

l

^jand on the \number of 2" in these sequences (namely

d

(

i

) and

d

(

l

)). The form of the sequences

i

and

l

acts only on the integer

b, and then not on the sign of

g

bl. The exact form of

G

(

'

n^;p⁺¹

:::'

n) is given in Lemma 5.3 but it is not useful here.

Remark 4.2. It appears from this lemma that the terms

c

n^;p⁺¹,

t

n^;p⁺²,

:::

,

t

nhave a particular part in the decomposition (4.2) (it will be conrmed in what follows). Thus it is interesting to notice that all of these quantities are zero if there exists =1 such that

k^;

k^;1=

a

k^;p for

k

=

p

+1

:::n

. In this case, the function

f

l can be non zero only if there exists

b

²

I

l such that

b

⁰ ==

b

p (we set 0⁰= 1).

4.2. Proof

4.2.1. Plan of the proof. We are not going to prove directly Theorem 3.1 but the following proposition, which implies the theorem.

Proposition 4.3. Let

n

1. Then, for every state

q

we have:

_ni increases strictly with respect to

i

(for

i >

1),

d

_ni=

_i^n;1;1,

Esaim: Cocv, October 1996, Vol. 1, pp. 241-266

- if⁹1

p

n

^;1

and

=1 such that

'

n^;p⁺¹ =

a

¹

:::'

n=

a

p

(

a

p is dened by (3.1)), then:

_ni=

ⁿ_i^;1;1+ 1

for i

= 3

:::p

+ 2

_ni=

ⁿ_i^;;_p^p+

p d

ⁿ_i^;;_p^p

for i

=

p

+ 2

:::n

+ 3 - otherwise,

_ni=

ⁿ_i^;1;1+ 1 for

i

= 3

:::n

+ 3,

f

B

_ni =

A

ⁿ(

_ni

d

_ni)]q

i

= 1

:::n

+ 3^g is a basis of

T

q(^R2(^S1)ⁿ⁺¹) (the denition of the brackets

A

ⁿ(

d

)] is given by (3.2)),

every vector

X

_ni] such that

i

^{2 A}_ni and

d

(

i

) =

d

_ni has a positive coordinate on the basis vector

B

_ni (recall that, if

i

^{2 A}s and

s >

1, then

i

¹= 1 (see (2.2)).

The rst four points of this proposition are equivalent to Theorem 3.1 (the induction formulas for

_ni are the same but expressed in a dierent way). The last point of the proposition is an induction hypothesis required for the proof and then is omitted in the theorem.

The proof will be done by induction on

n

. We assume that Proposition 4.3 is true for every

m < n

, and we will prove that it is true also for

n

by proceeding as follows:

- for each

q

and

i

we have "candidates" for

_ni and

d

_ni (given by the proposition)

- in Lemma 4.5 and Corollary 4.8, we prove that these "candidates" are less than

_ni and

d

_ni

- with Lemma 4.9, we establish that there exists a basis of

T

q(^R2 (^S1)ⁿ⁺¹) formed by vectors the length (and number of

X

²ⁿ) of which are equal to the "candidates", for

i

= 1 to

n

+ 3

- by using Lemma 4.7 we prove that

_ni and

d

_ni are indeed equal to the

"candidates"

- Lemma 4.10 allows us to establish the last point of Proposition 4.3 and so to conclude.

The form of Proposition 4.3 implies that we have to distinguish several possibilities for the state

q

:

9 =1 such that

'

n=

a

¹,

9

p

2

and

= 1 such that

'

n^;p⁺¹ =

a

¹

:::'

n^;1 =

a

p^;1 and

'

n=

a

p,

9

p

2

and

= 1 such that

'

n^;p⁺¹ =

a

¹

:::'

n^;1 =

a

p^;1 and

'

n⁶=

a

p and ⁶=

a

¹,

such a

p

2 doesn't exist and

'

n⁶=

a

¹. We can resume these possibilities in two cases:

q

²(

a

) if ⁹

p

^2f1

:::n

^;1^gand, if

p >

1

⁹ =1 such that

'

n⁶=

a

¹ and, if

p >

1

'

n^;p⁺¹=

a

¹

:::'

n^;1=

a

p^;1

'

n⁶=

a

p

q

²(

b

) if ⁹

p

1

p

n

^;1

and =1 such that

'

n^;p⁺¹=

a

¹

:::'

n=

a

p

:

(4.3) For instance the generic case is

q

²(

a

) and

p

= 1. In this case,

'

n⁶=² (since

a

¹ = ² from (3.1)) and there is no sequence

'

n^;p⁺¹

:::'

n^;1 equal

Esaim: Cocv, October 1996, Vol. 1, pp. 241-266