An Average-Distance Minimizing Motion Sweep for Planar Bounded Objects.

Abstract

This paper studies the problem of how to construct an interpolating planar motion between two positions of a bounded object as opposed to an infinitely large moving plane. Central to this investigation is the question of metrics, i.e., how to characterize the spatial separation or "distance" between two positions of the bounded object. The concept of shape dependent object norms proposed by Kazerounian and Rastegar [5] and refined by Chirikjian and Zhou [1] were used to compute the average distance between two positions for all points of the bounded body. The "ideal" interpolating motion of a bounded object, called motion sweep in this paper, is the one such that, at every intermediate position along the motion, the sum of the average distances from this intermediate position to each of the end positions is minimized. It is found that the resulting motion sweep is not the commonly known motion that linearly interpolates both translation and rotation parts independently but a new type of straight-line motion such that the translational part is coupled to the rotation part via sinusoidal functions $\text{sin}\left(\left(1-t\right)\Delta \theta \right)$ and $\text{sin}\left(t\Delta \theta \right)$ , where $\Delta \theta$ is the range of rotation angle, instead of the usual $\left(1-t\right)$ and $t$ without including $\Delta \theta$ .