| Abstract | The rotational attitude dynamics of a rigid platform incorporating internal momentum exchange devices, such as reaction wheels, control moment gyroscopes, or spinning rotors embedded in the body frame, constitute a canonical class of gyrostatic systems in which the total angular momentum departs from the classical Eulerian prediction by a constant body-fixed bias vector. This study presents a systematic semi-analytic framework for determining the attitude of such gyrostatic rigid bodies under constant applied torques, without restriction to infinitesimal angular excursions. The framework rests on three mathematical pillars: a 3-1-2 Euler angle kinematic representation in which the precession angle is recovered by direct quadrature after determining the nutation and spin angles; an exact Riccati differential equation governing the stereographic projection of the direction cosine matrix onto the complex plane, valid for arbitrarily large attitude excursions; and a complexvariable formulation of the modified Euler equations that naturally decomposes the solution into Fresnel integrals
and closed-form elementary exponential functions. Four dimensionless parameters completely characterize all system behavior: the nutation frequency coefficient, the normalized external transverse torque
amplitude, the gyrostatic linear forcing strength from transverse gyrostatic momentum components, and the precession frequency modulation parameter from the axial gyrostatic momentum component. A particularly
consequential analytical result is that the transverse gyrostatic forcing integral reduces exactly to elementary exponential functions, eliminating numerical quadrature and reducing computational effort by approximately
40% relative to full special-function evaluation. Asymptotic expansions for high spin rates and Taylor series for near-zero conditions provide efficient and accurate evaluations across the complete operational envelope. Numerical validation against direct adaptive Runge-Kutta integration of the full nonlinear modified Euler equations demonstrates excellent agreement, with transverse angular velocity errors below 10?? 2 rad/s and relative errors below 15% at nutation oscillation peaks over extended time intervals. The formulation finds immediate application
in momentum-biased spacecraft, dual-spin satellites, and spin-stabilized projectiles, where rapid and accurate attitude prediction is essential for control design and mission planning. |