Mathematics

In the first half of the 90's, through the work of Laumond, Sordalen, Risler, Jean and others it had become clear that the kinematic model of a car drawing a given number l of attached trailers represents a rank--2 distribution D (on the configuration space \Sigma = R^2 \times (S^1)^{l+1}) having the property that the Lie square [D, D] of D is a rank--3 distribution, the Lie square of [D, D] is a rank--4 distribution, and so on. Distributions satisfying this condition are called Goursat. Every Goursat distribution around a generic point behaves (locally) in a unique way best visualised by chained model known in control theory. For trailer systems this behaviour happens around positions where NONE of the angles between: the car and 1st trailer, the 1st and 2nd trailer, ... , the (l-2)th and (l-1)th, is right. All other positions in \Sigma are singular. I will speak about: 1) a fundamental stratification of \Sigma by Jean '96 separating different singular behaviours of the system and drawing attention to specific values of the angles. This includes the computation of the growth vector at a point, hence -- also the nonholonomy degree of any given position of the system. 2) how Jean's classification prompts a basic geometric classification (if only rough; the local classification problem for Goursat objects is a lot more hard) of singularities of Goursat objects. 3) whether the car with l trailers is a universal model for local behaviours of Goursat distributions on manifolds of dimension l+3 (i.e., whether any Goursat germ is equivalent to the kinematic model at certain point of \Sigma). For l \le 5 the affirmative answer was known to me in '97. Pasillas--Lepine informed me in '98 that a general answer was also yes. In '99 this result was elegantly shown by Montgomery & Zhitomirskii in their big work on Goursat. It is a byproduct of performing -- over Goursat distributions -- so-called Cartan's prolongations as described by Bryant & Hsu in their '93 paper. I will recall that construction and explain the resulting universality of the trailer systems. 4) [time permitting; briefly and only giving references] which evolutions of the trailer systems are locally C^1--rigid.