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Additional resources for Algorithmic Foundation of Robotics VII: Selected Contributions of the Seventh International Workshop on the Algorithmic Foundations of Robotics
Jaillet and T. Sim´eon b a Fig. 16. Path Deformation Roadmaps for 2D environments: (a) a labyrinth with many homotopy classes. (b) an indoor environment with a complex geometry. The third experiment (Figure 17) involves a narrow passage problem for a squared robot with 3-dof (two translations and one rotation). The robot has four ways to go through the narrow passage, depending on its orientation. Therefore the narrow passage corresponds to four homotopy classes in the conﬁguration space. a c b 3Π 2 Π y θ y θ x x Π 2 x y y 0 θ x Fig.
It represents the connectivity of FW as an Workspace-Based Connectivity Oracle W 1 2 39 C 3 4 1 Lf 2 3 4 Fig. 1. A partition of W induces a partition of C. Obstacles are shaded in dark color. A workspace channel and its lifted version are shaded in light color. In general, Lf is a one to many mapping. It maps a region of W to several regions of C. adjacency graph GT for the cell decomposition T . Each node of GT represents a cell in T , and two nodes are connected by an edge if the corresponding cells are adjacent.
2 Homotopy Versus Useful Roadmap Paths First we informally discuss the relation between homotopy and the representative path varieties that it would be desirable to store in the roadmap. The capture of the homotopy classes of Cf ree corresponds to a stronger property Path Deformation Roadmaps 21 than connectivity. 1). Homotopy deﬁnes an equivalence relation on the set of all paths of Cf ree . A roadmap capturing the homotopy classes means that every valid path (even cyclic paths) can be continuously deformed into a path of the roadmap.