Optimization of asymmetric gyrostatic satellite kinematics in a resistive medium: A novel elliptic function solution

In this study, a new approach for solving the dynamic motion of an asymmetric satellite with optimally controlled torques is investigated. The satellite moves in an atmospheric resistance medium and is impacted by gyrostatic torque (GT). The Euler equations were used to formulate the governing satellite equations. A small parameter is introduced through the assumption of a small control torque magnitude. Using elliptic function theory, explicit expressions for the angular velocities in the unperturbed regime are derived, extending the classical Euler-Poinsot solutions to include coupled gyrostatic and control effects. The semi-optimal control law is derived from a minimum-energy objective functional and is shown to preserve the integrability structure of the system. Perturbation analysis yields evolution equations for the angular momentum and kinetic energy in the presence of resistive torques. Numerical validation outcomes with < 10-6 relative error over 100 rotation periods. Parametric studies reveal distinct operational regimes: gyrostatic amplification enhances momentum capacity while maintaining stability; medium resistance provides stabilization but increases compensatory energy consumption; and control axes exhibit specialized roles, with b2 serving as the primary momentum driver and b3 exhibiting inverse energy relationships. The analytical framework provides a 100x computational speedup for mission design optimization compared to direct numerical integration, with applications to the attitude control of asymmetric satellites in low Earth orbit. The findings directly apply to low-Earth-orbit satellites experiencing atmospheric drag, where optimal power management is crucial for mission longevity. Earth observation satellites, communication satellites, and space telescopes with complex, nonsymmetric geometries can benefit from the developed control torque optimization strategies, especially the discovery that different control axes serve specialized roles in energy management and attitude stability. The study’s insights into GT effects are particularly valuable for spacecraft with large spinning components, such as solar arrays or antennas, whereas elliptic modulus analysis provides essential guidance for mission planning and attitude determination algorithms.