Inverse Kinematics of an Untethered Rigid Body Undergoing a Sequence of Forward and Reverse Rotations

A sequence of rotations considered in this paper is a series of rotations of an untethered 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 noncommutative property of finite rigid body rotations, such a sequence can effect nonzero changes in the orientation of the rigid body even though the net rotation about each axis is zero. These sequences are useful for attitude maneuvers of miniature spacecraft that use elastic deformation-based microactuators, or of other airborne or neutrally buoyant underwater vehicles. This paper considers the inverse kinematics problem of determining the angles in a given sequence to achieve a desired change in the orientation. Two types of problems are addressed. For the first problem, where four-rotation sequences are used, an analytical solution is presented and it is shown that a pointing vector attached to the rigid-body can be arbitrarily oriented. In the second problem, six-rotation sequences are used to control all three of the orientation freedoms of the rigid body. Some of the six-rotation sequences can provide any change in orientation while others are limited in their capabilities. A general numerical solution for all types, and a closed-form analytical solution for one type are presented along with the numerical examples and graphical visualization.