1. Trajectories in same homotopy classses Trajectories in different homotopy classses Definition of Homotopy Class Set of trajectories joining same start and goal points that can be smoothly deformed into one another without intersecting obstacles Deploying multiple agents: Searching/exploring the map Pursuing an agent with uncertain paths Motivational Examples initial final ? ? ? ? start goal Predicting: Possible paths of an agent with uncertainty in behaviors Possible paths taken by an agent when only start and final positions are known Constraints: Avoid high-risk regions and homotopy classes Follow a known homotopy class in order to perform certain tasks Geometric approach [Hershberger et al.; Grigoriev et al.] -Not well-suited for graph representation -Inefficient for planning with homotopy class constraints Triangulation based method [Demyen et al.] -Not suitable for non-Euclidean cost functions -Requires triangulation-based discretization schemes. -Complexity increases significantly if environment contains many small obstacles. -Cannot be easily used with an arbitrary graph search and arbitrarily discretization. Homotopy class in literature Our Approach – Exploit Theorems from Complex Analysis To plan least-cost paths for arbitrary cost functions (not necessarily Euclidean distances) within a particular homotopy class or while avoiding certain homotopy classes. To develop efficient representation of homotopy classes that supports planning using arbitrary discretizations and graph representations, (uniform discretization, unstructured discretization, triangulation, visibility graph, etc.) and using any standard graph search algorithm. (Dijkstra’s, A*, D*, ARA*, etc.) Goal: Homotopy class constraint Basic Principle: Re Im Represent the X-Y plane by a complex plane i.e. A point (x,y) is represented as z = x + iy ζ1ζ1 ζ2ζ2 ζ3ζ3 Place “representative points”, ζ i, inside significant obstacles (we can ignore small obstacles which we don’t want to contribute towards homotopy classes) Define an Obstacle Marker function such that it is Complex Analytic everywhere, except for having poles (singularities) at the representative points f 0, for example, can be any arbitrary polynomial in z Complex Analytic Function ≡ Complex Differentiable F (z) ≡ F (x + iy) ≡ u(x, y) + i v(x, y) Equivalently, F ( ) = ( ) with u, v following certain properties ( 2 u = 2 v = 0) which are guaranteed when x & y are implicitly used within z in construction of F. x y u(x,y) v(x,y) ζ1ζ1 ζ2ζ2 ζ3ζ3 τ1τ1 τ2τ2 τ3τ3 τ1τ1 τ2τ2 τ3τ3 = ≠ A direct consequence of Cauchy Integral Theorem and Residue Theorem The value of uniquely defines the homotopy class of a trajectory τ Computing L(e) for any straight line segment e (e.g. edge of a graph laid down on the environment) z1z1 z2z2 Can be computed numerically by further discretizing e If e is “small”, and f 0 an order- N-1 polynomial, this can be computed analytically in a fast and efficient way. k l = argmin k l L -augmented graph Augment each node, z, with distinct L-values of trajectories leading to it from start. Integrating L-values along paths by adding up L-values of the edges Cost function remains same Given the graph laid upon the environment, we construct G L by augmenting each state z with L-value of trajectory leading to it from start coordinate Insight into graph topology Experimental Analysis Homotopy class exploration “Visibility” constraint translates to homotopy class constraint Non-Euclidean cost function Planning in X-Y-Time No homotopy class constraint: A homotopy class blocked: Exploring 20 homotopy classes in a 1000x1000 uniformly discritized environment Implementation on a Visibility Graph (polygonal obstacles) Conclusions: Developed a compact and efficient representation of homotopy classes, using which homotopy class constraints can be imposed on existing graph search-based planning methods. Acknowledgements We gratefully acknowledge support from ONR grant no. N00014-09-1-1052, NSF grant no. IIS-0427313, ARO grant no. W911NF-05-1-0219, ONR grants no. N00014-07-1-0829 and N00014-08-1-0696, and ARL grant no. W911NF-08-2-0004. A – set of allowed homotopy classes B – set of blocked homotopy classes Search-based Path Planning with Homotopy Class Constraints Subhrajit Bhattacharya | Vijay Kumar | Maxim Likhachev GRASP L ABORATORY University of Pennsylvania e zszs zgzg z1z1 z2z2 ζ1ζ1 unique goal statestart {z s, 0+0i} ζ1ζ1 start e1e1 e2e2 e3e3 e4e4 {z 2, L(e 1 )} {z g, L(e 1 )+L(e 3 )} e1e1 e2e2 e3e3 e4e4 (z 1, L(e 2 )) {z g, L(e 2 )+L(e 4 )} ≠ G G L Goal states are distinguished based on the path taken to reach it z in G {z, L(z s →z)} in G L Illustration of the effect of augmenting L-values with state coordinates:

2. Addendum For the simple cases in 2-dimensions we have not distinguished between homotopy and homology. The distinction however does exist even in 2-d. See our more recent [AURO 2012] paper or [RSS 2011] paper for a comprehensive discussion on the distinction between homotopy and homology, examples illustrating the distinction, and its implications in robot planning problems. [AURO 2012] Subhrajit Bhattacharya, Maxim Likhachev and Vijay Kumar (2012) "Topological Constraints in Search-based Robot Path Planning". Autonomous Robots, 33(3):273-290, October, Springer Netherlands. DOI: 10.1007/s10514-012- 9304-1. [RSS 2011] Subhrajit Bhattacharya, Maxim Likhachev and Vijay Kumar (2011) "Identification and Representation of Homotopy Classes of Trajectories for Search-based Path Planning in 3D". [Original title: "Identifying Homotopy Classes of Trajectories for Robot Exploration and Path Planning"]. In Proceedings of Robotics: Science and Systems. 27-30 June.