An Average-Distance Minimizing Motion Sweep for Bounded Spatial Objects and Its Application in Bézier-Like Freeform Motion Generation

Abstract

This paper uses the ellipsoidal parameters associated with volume moments of inertia of a bounded solid object to construct a motion sweep joining two poses of the solid object, in contrast to earlier works on motion interpolation in SE(3) without taking into account the shape of the moving object. The paper borrows the concept of shape-dependent object norms introduced by Kazerounian and Rastegar [1] and refined by Chirikjian and Zhou [2] to compute as a metric the average of the squared distances (or ASD) among all homologous points of the bounded body between two given poses and seeks to obtain an optimal interpolating motion that minimizes a combination of two ASD distances from each intermediate pose to the two given poses. It is found that the ASD minimizing motion sweep is a novel straight-line motion such that while the centroid of the object follows a straight line, the orientation of the object is constrained so that the ASD metric is minimized. Furthermore, the rotational component can be determined by polar decomposition of the linearly interpolated rotation matrices, scaled by the object's inertial parameters. As an illustration of one of its applications, this motion sweep is repeatedly applied using the de Casteljau algorithm to generate Bézier-like freeform motions, whose paths are in general dependent on the shape of the inertia ellipsoid.