Simplicial complexity: piecewise linear motion planning in robotics

摘要

Using the notion of contiguity of simplicial maps, and its relation (via iterated subdivisions) to the notion of homotopy between continuous maps, we adapt Farber's topological complexity to the realm of simplicial complexes. We show that, for a finite simplicial complex K, our discretized concept recovers the topological complexity of the realization ∥K∥. Our approach lays the theoretical grounds for designing and implementing algorithms that search for optimal motion planners for autonomous systems in real-life applications.