Continuous Unfolding of Polyhedra - a Motion Planning Approach

Zhonghua Xi and Jyh-Ming Lien

Abstract

Cut along the surface of a polyhedron and unfold it to a planar structure without overlapping is known as Unfolding Polyhedra problem which has been extensively studied in the mathematics literature for centuries. However, whether there exists a continuous unfolding motion such that the polyhedron can be continuously transformed to its unfolding has not been well studied. Recently, researchers started to recognize continuous unfolding as a key step in designing and implementation of self-folding robots. In this paper, we model the unfolding of a polyhedron as multi-link tree-structure articulated robot, and address this problem using motion planning techniques. Instead of sampling in continuous domain what are traditional motion planners do, we propose to sample only in the discrete domain. Our experimental results show that sampling in the discrete domain is efficient and effective for finding feasible unfolding paths.

Method

Sampling in Discrete Domain

With the increase of the degree of freedom (DOF), sampling in continuous domain become harder and harder to find even one valid configuration. Instead, we sample in discrete domain, the folding angle of each crease line is sampled only from {0, target-folding-angle}, we show that this strategy has a much higher probability to find a valid configuration. And the configurations found by this strategy are also important since many folding motions can be achieved by only one intermediate configuration.

Folding process of a Bunny model, live demo

Continuous Blooming of Convex Polyhedra

We found all unfoldings of convex polyhedra generated by Steepest Edge Unfolding are like flowers and have a linear path that directly connect start and goal. Thus, we can such unfolding process as continuous blooming.

Continous unfolding of a 161 DOF sphere, live demo

Publication

Continuous Unfolding of Polyhedra - a Motion Planning Approach, Zhonghua Xi and Jyh-Ming Lien, 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Sep. 2015
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