Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems

This paper develops a constructive method for finding a nilpotent basis for a special class of smooth nonholonomic distributions. The main tool is the use of the Goursat normal form theorem which arises in the study of exterior differential systems. The results are applied to the problem of finding a set of nilpotent input vector fields for a nonholonomic control system, which can then used to construct explicit trajectories to drive the system between any two points. A kinematic model of a rolling penny is used to illustrate this approach. The methods presented here extend previous work using "chained form" and cast that work into a coordinate-free setting. This work is motivated by the recent interest in trajectory generation for mechanical systems with nonholonomic constraints. Consider the problem of steering a mechanical system with n-dimensional configuration space M from an initial configuration xO to a final configuration xl, subject to a set of independent kinematic constraints ha,ving the form The wj's are a basis for a codistribution on M which restricts the velocity of the system to be zero in certain directions. We assume the wj 's are smooth and linearly independent over the ring of smooth functions. Such constraints can arise when two surfaces roll against each other, such as the rolling between a wheel and the road, or in space-based systems where the total angular momentum of the system Research supported in part by a grant from the Powell Foundation..