Invariant Description of Rigid Body Motion Trajectories

This paper presents a minimal invariant coordinate-free description of rigid body motion trajectories. Based on a motion model for the instantaneous screw axis, a time-based coordinate-free description consisting of six scalar functions of time is defined. Analytical formulas are presented to obtain these functions from the pose or twist coordinates of a motion trajectory. The time-based functions are then stripped from their temporal information yielding five independent geometric functions together with a scalar motion profile. The geometric functions are shown to be invariant with respect to time scale, linear and angular scale, motion profile, reference frame, and reference point on the rigid body used to express the translational components of the motion. An algorithm is given to reconstruct a coordinate representation of a motion trajectory from its coordinate-free description. A numerical example illustrates the validity of the approach.