An inverse solution for torque-free axisymmetric rigid body dynamics, with a test for non-precessional motion

Abstract

The rotational motion of a torque-free axisymmetric rigid body is precession. This motion has been expressed analytically in the literature given the body's initial orientation and rotational dynamics parameters, i.e. inertia ratio and initial angular velocities or precession parameters. The inverse problem of deriving these dynamics parameters given orientation in time has been implemented numerically but has not yet been solved analytically. If a rigid body is precessing, and its orientation with respect to an arbitrary inertial frame is provided at three equally spaced points in time such that the rotational motion is not undersampled, an analytical inverse solution is presented for the precession rate, relative spin rate, coning angle and angular velocities; if the precessional motion is due to inertial axisymmetry and torque-free motion, the inertia ratio is also derived. Additionally, an analytical methodology is presented to test for non-precessional motion. These techniques are applicable to various problems in space science and astronomy, where non-precessional motion or the rotational dynamics parameters of this type of rigid body must be accurately derived from its orientation or relative orientation in time.