Rigid body trajectories in different 6D spaces

Abstract

The objective of this paper is to show that the group SE3 with an imposed Lie-Poisson structurecan be used to determine the trajectory in a spatial frame of a rigid body in Euclidean space.Identical results for the trajectory are obtained in spherical and hyperbolic space by scaling thelinear displacements appropriately since the influence of the moments of inertia on the trajectoriestends to zero as the scaling factor increases. The semidirect product of the linear and rotationalmotions gives the trajectory from a body frame perspective. It is shown that this cannot be used todetermine the trajectory in the spatial frame. The body frame trajectory is thus independent of thevelocity coupling. In addition, it is shown that the analysis can be greatly simplified by aligningthe axes of the spatial frame with the axis of symmetry which is unchanging for a natural systemwith no forces and rotation about an axis of symmetry