Simplicial Complexity: piecewise linear motion planning in robotics

Using the notion of contiguity of simplicial 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 $\mathrm{\parallel }K\mathrm{\parallel }$. Our approach is well suited for designing and implementing algorithms that search for optimal motion planners for autonomous systems in real-life applications.