Caging rigid polytopes via finger dispersion control

Abstract

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.