New Solution to Euler’s Equations for Asymmetric Bodies: Applications to Spacecraft Reorientation

Abstract

This work derives the kinematic and dynamic solution to Euler’s equations of motion for an asymmetric rigid body subject to a torque aligned parallel or antiparallel to the instantaneous angular momentum vector. These are the first complete solutions to Euler’s equations for an asymmetric rigid body subject to nonaxial torques. The antiparallel solution is shown to be the optimal trajectory for spacecraft detumbling subject to a bound on the Euclidean norm of the applied torque. Comparisons to a numerically integrated optimal control law validate the solution for highly asymmetric spacecraft and long maneuver times. Combining the parallel and antiparallel torque solutions allows for the analytical expression of a rest-to-rest reorientation maneuver for any asymmetric spacecraft. This maneuver is completely parameterized by a single body frame angular momentum vector, and is thus termed a momentum-axis maneuver. Comprehensive trade studies indicate that the momentum-axis maneuver is superior in terms of maneuver time to time-optimal eigenaxis maneuvers for large-angle rotations, and generally within of numerically computed true optimum solutions. As these solutions are both analytic and computationally cheaper than either the eigenaxis or true optimum solutions, this approach could have widespread applications for spacecraft attitude control.