Two-Finger Squeezing Caging of Polygonal and Polyhedral Object

The problem of object caging is defined as a problem of designing a formation of fingers to restrict an object within a bounded space. Assuming two pointed fingers and a rigid polygonal or polyhedral object, this paper addresses the problem of two-finger squeezing caging, i.e., to characterize all possible formations of the fingers that are capable of caging the object via limiting their separation distance. Our study is done entirely in the object´s frame allowing the object to be considered as a static obstacle so that the analysis can be performed in terms of the finger motion. Our solution is based on partitioning the configuration space of the problem into finite subsets called nodes. A graph of these nodes can then be constructed to represent all possible finger motion where a search based method can be applied to solve the caging problem. The partitioning of the configuration is based on convex decomposition of the free space. Let m be the number of convex subsets from the decomposition, our proposed algorithm reports all squeezing cage sets in O(n² + nm + m² log m) for a polygonal input with n vertices and O(nN³ + n² + nm + m² log m) for a polyhedron with n vertices and having TV edges exhibiting a reflex angle. After reporting all squeezing cages, the proposed algorithm can answer whether a given finger placement can cage the object within a logarithmic time.