Analysis of Fully Reversed Sequences of Rotations of a Free Rigid Body

Abstract

A fully reversed (FR) sequence of rotations is defined as a series of rotations of a free rigid body about its body-fixed axes such that the rotation about each axis is fully reversed at the end of the sequence. Due to the non-commutative property of finite rigid body rotations, an FR sequence can effect non-zero changes in orientation of the rigid body even though the net rotation about each axis is zero. The FR sequences are useful for attitude maneuvers of miniature spacecraft that use elastic deformation-based microactuators and for other airborne or neutrally buoyant underwater vehicles where actuations effecting orientation change are restricted. This paper considers the kinematics of six-rotation FR sequences and discusses if each of them is capable of producing any desired change in orientation. It is concluded that out of a total of 24 six-rotation sequences, 12 are capable of this but not the remaining 12. The results are proved using mathematical formalism and are also interpreted using numerical computations and graphical visualization.