Minimum Perturbation Coordinates on SO(3)

Abstract

Systems that use internal shape changes to control their orientation in space have interested the geometric controls community for some time. Examples include the classic problem of a falling cat and the more applied attitude control of satellites. The dynamics of these systems are dominated by conservation of angular momentum, which induces a set of constraints between changes in shape and spatial orientation. This relationship can be combined with Lie bracket theory to identify shape changes that produce desired net rotations. The major weakness of the Lie bracket approach is that it only works for relatively small-amplitude motions; these methods depend on a local linearization of the system dynamics, which breaks down as larger motions are considered. Recent work on a related problem, planar locomotion, has shown that this breakdown can be mitigated by identifying a set of minimum perturbation coordinates for the system; application of Lie bracket theory in the minimum perturbation coordinates allows these methods to be applied to a broader and more interesting class of shape changes. In this work, we bring the derivation of minimum perturbation coordinates to the space of three-dimensional rotations. We show that as a result, we are able to derive visual tools that provide the control designer intuition into selecting cyclic controllers for inertial systems in free flight. These tools are demonstrated on a minimal satellite model taken from the literature.