Search-Based Path Planning with Homotopy Class Constraints in 3D

Homotopy classes of trajectories, arising due to the presence of obstacles, are defined as sets of trajectories that can be transformed into each other by gradual bending and stretching without colliding with obstacles. The problem of exploring/finding the different homotopy classes in an environment and the problem of finding least-cost paths restricted to a specific homotopy class (or not belonging to certain homotopy classes) arises frequently in such applications as predicting paths for unpredictable entities and deployment of multiple agents for efficient exploration of an environment. In (Bhattacharya, Kumar, and Likhachev 2010) we have shown how homotopy classes of trajectories on a two-dimensional plane with obstacles can be classified and identified using the Cauchy Integral Theorem and the Residue Theorem from Complex Analysis. In more recent work (Bhattacharya, Likhachev, and Kumar 2011) we extended this representation to threedimensional spaces by exploiting certain laws from the Theory of Electromagnetism (Biot-Savart law and Ampere’s Law) for representing and identifying homotopy classes in three dimensions in an efficient way. Using such a representation, we showed that homotopy class constraints can be seamlessly weaved into graph search techniques for determining optimal path constrained to certain homotopy classes or forbidden from others, as well as for exploring different homotopy classes in an environment. 1 Homotopy Classes and Homology Classes of Trajectories Two trajectories τ1 and τ2 connecting the same start and end coordinates, xs and xg respectively, are called homotopic iff one can be continuously deformed into the other without intersecting any obstacle (Figure 1). Sets of homotopic trajectories form homotopy classes. Give two trajectories, one may naively attempt to check if indeed one can be deformed into the other. However, such Copyright © 2012, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. This is a condensed, non-technical overview of work previously published in the proceedings of Robotics: Science and Systems, 2011 conference (Bhattacharya, Likhachev, and Kumar 2011). Figure 1: Illustration of homotopy and homology equivalences in 2 dimensions. In this example τ1 and τ2 are both homotopic (because of the existence of the sequence of trajectories shown by the dashed curves) as well as homologous (because of the presence of the area shown by blue hashing). But τ3 is not homotopic nor homologous to either. a process is highly non-trivial and may be extremely difficult to automate. Even if one is able to check, using such a method, whether of not two trajectories are homotopic, it is extremely difficult to incorporate the method in searchbased planning algorithms to plan trajectories that are constrained to or avoid certain homotopy classes. Thus, what one desires is to construct a functional of the trajectories, H(τ) (which we will call the H-signature of τ ), such that its value will uniquely identify the homotopy class of the trajectory (i.e. a complete invariant of homotopy classes of trajectories). We also desire thatH be of the form of an integration, i.e., H(τ) = ∫ τ dh (where dh is some differential 1-form – a quantity that can be integrated along a curve). This will let us compute least-cost paths in non trivial configuration spaces with topological constraints using graph search-based planning algorithms. It is possible to find such desired 1-forms, dh, as we did in our previous work for 2-dimensional configuration space (Bhattacharya, Kumar, and Likhachev 2010), where we exploited some theorems from complex analysis. However, it can be shown that by virtue of computing such integrals, what we end up obtaining from H(τ) are complete invariants for homology classes rather than homotopy classes. Homology, although a close relative of homotopy and similar Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence