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(n 2 +nm+m 2 log m) for a polygonal input with n vertices and O(nN 3 + n 2 + nm + m 2 log m) for a polyhedron with n vertices and having N 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.
The object caging problem focuses on designing a formation of fingers that keeps an object within a bounded space without immobilizing it. This paper addresses the problem of designing such formation for object represented by a polytope in any finite dimensional workspace and for any specified number of pointed finger. Our goal is to characterize all the caging sets, each of which corresponds to a largest connected set of initial formations of fingers guaranteed to cage the object, up to maintaining a certain class of real-valued measurement induced by the whole fingers' formation below a critical value. In our previous works, such measurement is simply the distance between two fingers (the formation). We found that it is possible to apply the framework based on graph search from the previous works to broader classes of measurements. In this paper, we introduce two of measurements, called dispersion and concentration and propose a generalized approach to query and to report all caging sets with respect to a given dispersion or concentration.
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