Controllability of a system of two symmetric rigid bodies in three space

Abstract

Imposing the constraint that a mechanical system having two completely symmetric 3-D rigid bodies have zero total angular momentum, the angular velocities of these bodies are negatives of one another, and the transfer of this sytem from one position to another is nonholonomic. While Chow's theorem establishes the fixed-endpoint controllability of the system, this result does not explicitly exhibit any motions between a given set of endpoints. In the present work it is shown how to explicitly construct simple motions of this system on (0, 1) with arbitrary endpoints in SO(3)/sup 2/. In particular, using normalized quaternions to describe rotations, the author constructs, in terms of elementary functions, a continuous motion with the given endpoints; it consists of at most three successive motions during each of which the bodies rotate on fixed axes. It is assumed that the rigid bodies have coincident mass centers, that each has a completely symmetric mass distribution about its mass center, and that the inertia tensor of each body is the identity matrix. >