Search-Based Path Planning with Homotopy Class Constraints

Homotopy classes of trajectories, arising due to the presence of obstacles, are defined by the set of trajectories joining same start and end points which can be smoothly deformed into one another by gradual bending and stretching without colliding with obstacles. Despite being mostly an uncharted research area, homotopy class constraints often appear in path planning problems. For example, in multi-agent planning problems (Zhang, Kumar, and Ostrowski 1998; Karabakal and Bean 1995), the trajectories often need to satisfy certain proximity or resource constraints or constraints arising due to tasks allocated to agents, which translates into restricting the solution trajectories to certain homotopy classes. In exploration and mapping problems (Bourgault et al. 2002), agents often need to plan trajectories based on their mission or part of the environment they are assigned for mapping or exploration, and hence restrict their trajectories to certain homotopy classes. Other examples of path planning with homotopy class constraints include predicting paths for dynamic entities and computing heuristics for path planning with dynamic constraints. Motion planning with homotopy class constraints have been studied in the past using geometric approaches (Grigoriev and Slissenko 1998; Hershberger and Snoeyink 1991) and probabilistic road-map construction (Schmitzberger et al. 2002) techniques. Such techniques suffer from complexity of representation of homotopy classes and are not immediately integrable with standard graph search techniques. While comparing trajectories in different homotopy classes and finding the different homotopy classes in an environment is possible using such techniques, optimal path planning with homotopy class constraints is not achievable in an efficient way. The abstract triangle graph in triangulationbased path planning (Demyen and Buro 2006) technique can be used to represent various homotopy classes in an environment. However the construction of the graph and planning in it relies on the assumptions that the obstacles in the environ-