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Many robotics applications benefit from being able to compute multiple geodesic paths in a given configuration space. Existing paradigm is to use topological path planning, which can compute optimal paths in distinct topological classes. However, these methods usually require nontrivial geometric constructions, which are prohibitively expensive in 3-D, and are unable to distinguish between distinct topologically equivalent geodesics that are created due to high-cost/curvature regions or prismatic obstacles in 3-D. In this article, we propose an approach to compute k geodesic paths using the concept of a novel neighborhood-augmented graph, on which graph search algorithms can compute multiple optimal paths that are topo-geometrically distinct. Our approach does not require complex geometric constructions, and the resulting paths are not restricted to distinct topological classes, making the algorithm suitable for problems where finding and distinguishing between geodesic paths are of interest. We demonstrate the application of our algorithm to planning shortest traversible paths for a tethered robot in 3-D with cable-length constraint. 

Abstract We consider the problem of multi-robot path planning in a complex, cluttered environment with the aim of reducing overall congestion in the environment, while avoiding any inter-robot communication or coordination. Such limitations may exist due to lack of communication or due to privacy restrictions (for example, autonomous vehicles may not want to share their locations or intents with other vehicles or even to a central server). The key insight that allows us to solve this problem is to stochastically distribute the robots across different routes in the environment by assigning them paths in different topologically distinct classes, so as to lower congestion and the overall travel time for all robots in the environment. We outline the computation of topologically distinct paths in a spatio-temporal configuration space and propose methods for the stochastic assignment of paths to the robots. A fast replanning algorithm and a potential field based controller allow robots to avoid collision with nearby agents while following the assigned path. Our simulation and experiment results show a significant advantage over shortest path following under such a coordination-free setup.