Lower Bounds of the Allowable Motions of One N-Dimensional Ellipsoid Contained in Another

Abstract

This paper studies the representations of a subset of the allowable motions for an N-dimensional ellipsoid inside another slightly larger ellipsoid without collision based on the idea of the Kinematics of Containment. As an extension to the previous work on the closed-form lower bounds, this paper proposes another two lower bounds based on the first-order algebraic condition of containment and the closed-form Minkowski difference between two ellipsoids respectively. Querying processes for a specific configuration of the moving ellipsoid and the calculations of the volume of the proposed lower bounds in configuration space (C-space) are introduced. Examples for the proposed lower bounds in 2D and 3D Euclidean space are implemented and the corresponding motion volumes in C-space are compared with different shapes of the ellipsoids. Finally a case study of the application on automated assembly is introduced.