In this paper we address the general problem of approximating, in a certain optimal way, non admissible motions of a kinematic system with nonholonomic constraints. Since this kind of problems falls into the general subriemannian geometric setting, it is natural to consider optimality in the sense of approximating by means of subriemannian geodesics. We consider systems modeled by a subriemanni...

The theme of this paper is the generalization of Goursat normal forms for Pfaffian systems with co-dimension greater than two. There are necessary and sufficient conditions for the existence of coordinates that transform a Pfaffian system with co-dimension greater than two into an extended Goursat normal form, which is the dual of the multiple-chain, single-generator chained form mentioned in o...

Let V be a vector field distribution on manifold M . We give an efficient algorithm for the construction of local coordinates on M such that V may be locally expressed as some partial prolongation of the contact distribution C (1) q , on the 1 st order jet bundle of maps from R to R, q ≥ 1. It is proven that if V is locally equivalent to a partial prolongation of C (1) q then the explicit const...

We show that the category of regular epimorphisms in a Barr exact Goursat category is almost Barr exact in the sense that (it is a regular category and) every regular epimorphism in it is an effective descent morphism.

In this paper, we develop the machinery of exterior diierential forms, more particularly the Goursat normal form for a Pfaaan system, for solving nonholonomic motion planning problems, i.e. planning problems with non-integrable velocity constraints. We apply this technique to solving the problem of steering a mobile robot with n trailers. We present an algorithm for nding a family of transforma...

We investigate 3-permutability, in the sense of universal algebra, in an abstract categorical setting which unifies the pointed and the non-pointed contexts in categorical algebra. This leads to a unified treatment of regular subtractive categories and of regular Goursat categories, as well as of E-subtractive varieties (where E is the set of constants in a variety) recently introduced by the f...

The solution to the biharmonic equation in a simply connected region in the plane is computed in terms of the Goursat functions. The boundary conditions are conformally transplanted to the disk with a numerical conformal map. A linear system is obtained for the Taylor coeecients of the Goursat functions. The coeecient matrix of the linear system can be put in the form I + K where K is the discr...

In Book 1, Proposition 7, Problem 2 of his 1687 Philosophiae Naturalis Principia Mathematica, Isaac Newton poses and answers the following question: Let orbit a particle moving in central force field be an off-center circle. How does magnitude depend on position onthat circle? this article, we identify potential that can produce such force, only at zero energy. We further map zero-energy orbits...