How to avoid collisions in 3D-realizations for moving graphs

• Published in: Mechanism and machine theory Vol. 162; p. 104337
• Main Author: Qi, Jiayue
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.08.2021
• ISSN: 0094-114X; 1873-3999
• DOI: 10.1016/j.mechmachtheory.2021.104337

•Parameterizing the positions of vertices of a graph gives us a moving graph.•Assigning different heights to the edges of a moving graph gives us an L-model.•The 3D-realization of an L-model is called an L-linkage.•A property of the moving graph guarantees the collision-freeness of its L-linkage.•An algorithm for building this collision-free linkage is provided. If we parameterize the positions of all vertices of a given graph in the plane such that distances between adjacent vertices are fixed, we obtain a moving graph. An L-linkage is a realization of a moving graph in 3D-space, by representing edges using horizontal bars and vertices by vertical sticks. Vertical sticks are parallel revolute joints, while horizontal bars are links connecting them. We give a sufficient condition for a moving graph to have a collision-free L-linkage. Furthermore, we provide an algorithm guiding the construction of such a linkage when the moving graph fulfills the sufficient condition, via computing a height function for the edges (horizontal bars). In particular, we prove that any Dixon-1 moving graph has a collision-free L-linkage and no Dixon-2 moving graphs have collision-free L-linkages, where Dixon-1 and Dixon-2 moving graphs are two classic families of moving graphs.