An analytical solution for the rotational dynamics of axisymmetric rigid bodies under time-varying body-fixed torques

This article presents a general analytical solution concerning the attitude motion of axisymmetric rigid bodies subject to time-varying body-fixed torques. Leveraging the axisymmetry feature of the body, it is demonstrated that the system becomes linear and time-variant, allowing the use of a state space approach. This leads to two key contributions: (i) A new closed-form analytical solution is derived by applying a single series expansion to the system’s zero-state response. Central to this approach is a novel recursive formulation that streamlines the computation of high-order derivatives, enabling a unified global approximation of system dynamics. This eliminates the need for separate expansions of each torque component and supports efficient offline construction of high-precision analytical representations. (ii) The closed-form structure enables analytical gradients, making the approach highly suitable for embedded optimal control and embedded machine learning applications under constrained computational resources. Simulation results demonstrate both the accuracy and computational effectiveness of the proposed solution.