Projective Geometry of Attitude Parameterizations with Applications to Control

Abstract

This paper examines the projective geometry of three-parameter attitude representations that are constructed by projecting a four-parameter unit quaternion representation from its unit hypersphere onto a three-dimensional hyperplane. Using this geometrical perspective, the paper demonstrates how kinematics of relative attitude motion characteristic of attitude tracking problems follow naturally from comparing projections from two different reference directions. The paper also demonstrates that among a continuum of possible projection pole placements there exist optimal distances for which resulting projected parameterizations yield magnitudes that accurately approximate all practical rotation angles. These parameterizations referred to in this paper as proxy-rotation vectors can be custom tuned for any expected range of rotation angles. They and their kinematics are free from trigonometric functions and do not require special handling if the rotation angle approaches zero. It is shown that this computational simplicity of the proxy-rotation vectors can be advantageous for linear feedback attitude controls and for certain classes of time-efficient attitude steering laws, where they can replace the more computationally cumbersome true rotation vector. The paper studies how kinematic singularities affect closed-loop stability and demonstrates that redesigning control laws to ensure closed-loop convergence toward the nearest equilibrium is equivalent to augmenting attitude parameterizations with their shadow counterparts.