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research-article

The Kinematics of Containment for N-Dimensional Ellipsoids

[+] Author and Article Information
Sipu Ruan

3400 N Charles St Hackerman Hall 136 Baltimore, MD 21218-2680 ruansp@jhu.edu

Jianzhong Ding

Beihang Univesity, New Main Building, Room B312 Haidian District Beijing, Beijing 100191 China jianzhongd@buaa.edu.cn

Qianli Ma

Aptiv Automotive Pittsburgh, PA 15238 qianli.b.ma@aptiv.com

Gregory S. Chirikjian

3400 North Charles Street Baltimore, MD 21218 gchirik1@jhu.edu

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanisms and Robotics. Manuscript received October 30, 2018; final manuscript received April 4, 2019; published online xx xx, xxxx. Assoc. Editor: Xianwen Kong.

ASME doi:10.1115/1.4043458 History: Received October 30, 2018; Accepted April 04, 2019

Abstract

Knowing the set of allowable motions of a convex body moving inside a slightly larger one is useful in applications such as automated assembly mechanisms, robot motion planning, etc. The theory behind this is called the “Kinematics of Containment (KC)”. In this article, we show that when the convex bodies are ellipsoids, lower bounds of the KC volume can be constructed using simple convex constraint equations. In particular, we study a subset of the allowable motions for an n-dimensional ellipsoid being fully contained in another. The problem is addressed in both algebraic and geometric ways, and two lower bounds of the allowable motions are proposed. Containment checking 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 volumes in C-space are compared with different shapes of the ellipsoids. Practical applications using the proposed theories in motion planning problems and parts-handling mechanisms are then discussed.

Copyright © 2019 by ASME
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