Abstract
We study a robot finger model in the framework of the theory of expansions in non-integer bases. We investigate the reachable set and its closure. A control policy to get approximate reachability is also proposed.
Highlights
Aim of this paper is to give a model of a robot’s finger in the framework of the theory of expansions in non-integer bases and to use the methods of this theory to study the reachability set and its closure
Neither the reachable set nor the approximate reachable set are explicitly known, we show that the iteration Fρ,ω on the convex hull of the approximate reachable set yields approximations of the approximate reachable set whose accuracy is monotonically increasing with the number of iterations
We study some convexity issues and we get the following results: the approximate reachable set is not a convex set, but if we restrict ourselves to the closure of the full-rotation configurations, a simple condition characterizes its convexity
Summary
Aim of this paper is to give a model of a robot’s finger in the framework of the theory of expansions in non-integer bases and to use the methods of this theory to study the reachability set and its closure. For every point x belonging to the approximate reachable set, we define the expansion of x, namely a particular couple of control sequences ensuring the finger to reach an arbitrarily small neighborhood of x. We give an explicit description of the convex hull of the reachable points in the full-rotation, full-extension and general cases.
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