Efficient Planar Caging Test Using Space Mapping

TL;DR: An algorithm to solve a geometric problem—find whether a given formation of planar points can constrain (or cage) a planar shape and offers robustness to avoid uncertainty in the tasks.

Abstract: This paper presents an efficient algorithm to test whether a planar object can be caged by a formation of point agents (point fingertips or point mobile robots). The algorithm is based on a space mapping between the 2-D work space ( $\mathcal {W}$ space) and the 3-D configuration space ( $\mathcal {C}$ space) of the given agent formation. When performing caging test on a planar object, the algorithm looks up the space mapping to recover the $\mathcal {C}$ space of the given agent formation, labels the recovered $\mathcal {C}$ space, and counts the number of labeled surfaces to judge the success of caging. The algorithm is able to work with various planar shapes, including objects with convex boundaries, concave boundaries, or holes. It can also respond quickly to varying agent formations and different object shapes. Experiments and analysis on different objects and fingertip formations demonstrate the completeness, robustness, and efficiency of our proposal. Note to Practitioners —This paper proposes an algorithm to solve a geometric problem—find whether a given formation of planar points can constrain (or cage) a planar shape. Users can use the proposed algorithm to actuate a formation of robotic fingertips to perform caging-based grasping tasks or use the proposed algorithm to actuate a formation of mobile robots to perform cooperative transportation tasks. The algorithm inherits the merits of caging and helps users to avoid explicit force analysis. It offers robustness to avoid uncertainty in the tasks. The code of our work is in the supplementary material.