Fine-tuning geometrically constrained planar motions

This paper presents a method for fine-tuning a geometrically constrained planar motion in the context of motion approximation. It builds on the recent work that seeks to identify and extract point trajectories of an explicitly given planar motion. Once two point trajectories are obtained, the remaining issue is to determine the length of the "coupler link" that connects the two point trajectories such that the resulting motion best approximates the original motion. In this paper, the concept of standard deviation in statistics and probability theory is used to define the "distance" between two planar motions. This distance definition is bi-invariant with respect to the choice of both moving and fixed reference frames. Furthermore, the concept of kinetic energy is also used for combining translation with rotation when calculating the distance between two planar displacements. A simple, direct search method for obtaining the optimum length of the coupler link is presented that minimizes the standard deviation of the motion error in terms of the kinetic energy based distance measure for planar displacements.