Accurate discretizations of torqued rigid body dynamics

Abstract

This paper investigates the solution of the free rigid body equations of motion, as well as of the equations governing the torqued rigid body. We will consider two semi-exact methods for the solution of the free rigid body equations, and we discuss the use of both rotation matrices and quaternions to describe the motion of the body; our focus is on the quaternion formulation. The approach to which we give the most attention is based on the Magnus series expansion, and we derive numerical methods of order 2, 4, 6, and 8, which are optimal as they require a minimal number of commutators. The other approach uses Gaussian quadrature to approximate an elliptic integral of the third kind. Both methods rely on the exact solution of the Euler equation which involves the exact computation of the elliptic integral of the first kind. For the solution of the torqued rigid body equations, we divide the equations into two systems where one of them is the free rigid body equations; the solutions of these two systems are then combined in the Störmer-Verlet splitting scheme. We use these methods to solve the so-called marine vessel equations. Our numerical experiments suggest that the methods we present are robust and accurate numerical integrators of both the free and the torqued rigid body.