Abstract
We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle R, and outputs an explicit system of piecewise linear motion planners for R. The algorithm is designed in such a way that the cardinality of the output is probabilistically close (with parameters chosen by the user) to the minimal possible cardinality.This yields the first automated solution for robust-to-noise robot motion planning in terms of simplicial complexity (SC) techniques, a discretization of Farber’s topological complexity TC. Besides its relevance toward technological applications, our work reveals that, unlike other discrete approaches to TC, the SC model can recast Farber’s invariant without having to introduce costly subdivisions. We develop and implement our algorithm by actually discretizing Macías-Virgós and Mosquera-Lois’ notion of homotopic distance, thus encompassing computer estimations of other sectional category invariants as well, such as the Lusternik-Schnirelmann category of polyhedra.
Highlights
In its more simplified form, a motion planning algorithm A for a given mechanical system S is a function which assigns, to any pair of initial-final states of S, a motion of S starting and ending at the given states
We find somehow surprisingly that the subdivision ingredient implicit in (1) can be unnecessary in a simplicial complexity (SC)-recasting of Farber’s topological complexity (TC), nullifying the critical compromise noted in Remark 1
We describe and implement a randomized algorithm based on Farber’s topological complexity model for robust-to-noise motion planning of an automated guided vehicle (AGV)
Summary
In its more simplified form, a motion planning algorithm A for a given mechanical system S is a function which assigns, to any pair of initial-final states of S , a motion of S starting and ending at the given states. The ultimate goal for A is to allow S to function in an autonomous regime Such a task is fundamental in modern technological problems. The notion of topological complexity (TC), introduced by Michael Farber in [4], is a mathematical model measuring the continuity instabilities in the motion planning problem of robots. Farber’s topological complexity of X, TC( X ), is the sectional category of the end-point evaluation map e : P( X ) → X × X, i.e., the fibration taking a free path γ ∈ P( X ) to the pair (γ(0), γ(1)). TC( X ) + 1 is the smallest cardinality of open covers {Ui }i of X × X so that e admits a continuous section σi on each Ui. Note that we use reduced terms, so that a contractible space has zero topological complexity. In view of the continuity requirement on local rules, an optimal motion planner minimizes the possibility of accidents in the performance of a robot moving in a noisy environment
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