Exact solutions to Euler's equations for rigid body motion with application to detumbling satellites

TL;DR

This paper derives exact analytical solutions to Euler's equations for asymmetric rigid bodies under torque, and applies these solutions to model and visualize satellite detumbling in both body-fixed and inertial frames.

Contribution

It presents novel exact solutions using Jacobi elliptic functions for asymmetric bodies under torque and demonstrates their application to satellite detumbling.

Findings

Analytical solutions agree with numerical simulations.

Solutions enable visualization of satellite motion in inertial frame.

Method applicable to arbitrary initial conditions.

Abstract

Exact solutions are found for Euler's equations of rigid body motion for general asymmetrical bodies under the influence of torque by using Jacobi elliptic functions. Differential equations are determined for the amplitudes and the parameters of the elliptic functions. The solution is then applied to the detumbling of a satellite with arbitrary initial rotation rates where numerical solutions are seen to be in agreement with the analytical solution. The body fixed frame solution is then transformed to the inertial frame by use of a quaternion rotation matrix to depict the motion in figures and in animations within a Mathematica notebook which is openly published on the Wolfram community.