Geometric and Topological Techniques in Graph Search-based Robot Planning

3 Work done so far 4 3.1 Planning with Homotopy class constraints . . . . . . . . . . . . . . . . . . . . . . 4 3.1.1 The problem in 2 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1.2 The problem in 3 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.3 Generalization and extension to higher dimensions . . . . . . . . . . . . . 6 3.2 Metric information using graph search techniques – Voronoi Tessellation in Nonconvex and non-Euclidean metric spaces Using Graph Search . . . . . . . . . . . . 8 3.3 Metric Transformation for Efficient Optimal Planning in Environments with Nonuniform Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3.1 Problem definition in 2 dimensions with isotropic metric, and its partial solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3.2 An Illustrative Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3.3 Existence of f for a given m . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.4 Dimensional Decomposition – Distributed Optimization using Separable Optimal Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12