New results on control of multibody systems which conserve angular momentum

Abstract

A planar system of rigid bodies interconnected by one degree of freedom rotational joints is considered. This multibody system is referred to as a multilink, and the rigid bodies are referred to as links. The angular momentum of the multilink is conserved but is not necessarily zero. We show that if the number of links is at least four, then periodic joint motions can make the absolute orientation of a specified base link track exactly a specified function of time whose time derivative is periodic. This result on the use of periodic joint motions for orientation tracking extends previous work [15], [20], [22] on using periodic joint motions for rest-to-rest reorientation. It has interesting physical consequences. Specifically, in the case of non-zero angular momentum periodic joint motions can maintain the orientation of the base link constant. In the case of zero angular momentum periodic joint motions can change the orientation of the base link at a specified angular rate. We also demonstrate that if the multilink consists of at least three links, then for any value of the angular momentum joint motions can reorient the multilink arbitrarily over an arbitrary time interval. This result extends similar results in [15] for zero angular momentum and in [20] that apply for nonzero angular momentum but not for an arbitrary time interval. In terms of their control-theoretic aspects, the problems treated in the paper can be viewed as controllability problems for a class of nonlinear control system with time-dependent drift.