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Note that μH satisfies all properties of a metric [9]. For our fixed path set generated by Green-Kelly, we precomputed each pairwise path metric value of (2) and stored them in a lookup table for rapid online access. Alg. 4 performs path classification on a set of paths that have already tested collision-free at runtime. We form a graph G = (V, E) in which node vi ∈ V corresponds to path pi . Edge ei j ∈ E joins nodes vi and v j when this relation holds: μH (pi , p j ) ≤ d, (3) where d is the diameter of the robot.
In: Algorithmic Foundation of Robotics VII, pp. : Computing minimum length paths of a given homotopy class. : Polytime algorithm for the shortest path in a homotopy class amidst semi-algebraic obstacles in the plane. In: ISSAC 1998: Proceedings of the 1998 Int. Symp. on Symbolic and algebraic computation, pp. : Search-based Path Planning with Homotopy Class Constraints. In: Proceedings of the Conference on Artificial Intelligence, AAAI (2010) 18 T. Igarashi and M. : A global geometric framework for nonlinear dimensionality reduction.
Durrant-Whyte Fig. 1 A car (small rectangle) is following a previously planned path (solid line) to the goal (dot), but got off track due to disturbances. The left image shows a pure pursuit controller trying to get back on track as quickly as possible, resulting in unnecessary turns (dashed line). A more natural solution is shown in the second picture where the available space is used to elegantly correct the path during the upcoming turn. Mechanical systems frequently exhibit symmetries that can be represented as Lie groups of translation or rotation [8, 14, 17].