Path homotopy invariants and their application to optimal trajectory planning

We consider the problem of optimal path planning in different homotopy classes in a given environment. Though important in robotics applications, path-planning with reasoning about homotopy classes of trajectories has typically focused on subsets of the Euclidean plane in the robotics literature. The problem of finding optimal trajectories in different homotopy classes in more general configuration spaces (or even characterizing the homotopy classes of such trajectories) can be difficult. In this paper we propose automated solutions to this problem in several general classes of configuration spaces by constructing presentations of fundamental groups and giving algorithms for solving the word problem in such groups. We present explicit results that apply to knot and link complements in 3-space, discuss how to extend to cylindrically-deleted coordination spaces of arbitrary dimension, and also present results in the coordination space of robots navigating on an Euclidean plane.

[1]  Didier Wolf,et al.  Capture of homotopy classes with probabilistic road map , 2002, IEEE/RSJ International Conference on Intelligent Robots and Systems.

[2]  Subhrajit Bhattacharya,et al.  Path Homotopy Invariants and their Application to Optimal Trajectory Planning , 2015 .

[3]  Daniel C. Cohen,et al.  Motion planning in tori , 2008 .

[4]  Vijay Kumar,et al.  Topological constraints in search-based robot path planning , 2012, Auton. Robots.

[5]  Vijay Kumar,et al.  Path planning for a tethered mobile robot , 2014, 2014 IEEE International Conference on Robotics and Automation (ICRA).

[6]  William Fulton,et al.  The Van Kampen Theorem , 1995 .

[7]  Joan S. Birman,et al.  Braids, Links, and Mapping Class Groups. (AM-82) , 1975 .

[8]  J. Birman.  Braids, Links, and Mapping Class Groups. , 1975 .

[9]  J. Müller,et al.  Group Theory , 2019, Physical Mathematics.

[10]  R. Ho.  Algebraic Topology , 2022 .

[11]  C. Weinbaum,et al.  The word and conjugacy problems for the knot group of any tame, prime, alternating knot , 1971 .

[12]  Gaurav S. Sukhatme,et al.  A topological approach to using cables to separate and manipulate sets of objects , 2013, Int. J. Robotics Res..

[13]  Fred Cohen,et al.  Sensor Beams, Obstacles, and Possible Paths , 2008, WAFR.

[14]  John F. Canny,et al.  New lower bound techniques for robot motion planning problems , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[15]  Graham A. Niblo,et al.  The geometry of cube complexes and the complexity of their fundamental groups , 1998 .

[16]  Daniel E. Koditschek,et al.  Coordinated Robot Navigation via Hierarchical Clustering , 2015, IEEE Transactions on Robotics.

[17]  Charles F. Miller,et al.  Combinatorial Group Theory , 2002 .

[18]  Ronald L. Rivest,et al.  Introduction to Algorithms , 1990 .

[19]  Martin Greendlinger,et al.  On Dehn presentations and Dehn algorithms , 1986 .

[20]  John Hershberger,et al.  Computing Minimum Length Paths of a Given Homotopy Class (Extended Abstract) , 1991, WADS.

[21]  L. Kauffman.  An Introduction to Knot Theory , 2001 .

[22]  Steven M. LaValle,et al.  Nonpositive Curvature and Pareto Optimal Coordination of Robots , 2006, SIAM J. Control. Optim..

[23]  Vijay Kumar,et al.  Human-Robot Collaborative Topological Exploration for Search and Rescue Applications , 2014, DARS.

[24]  Michael Farber.  Topological Complexity of Motion Planning , 2003, Discret. Comput. Geom..

[25]  David B. A. Epstein,et al.  Word processing in groups , 1992 .

[26]  Michael Buro,et al.  Efficient Triangulation-Based Pathfinding , 2006, AAAI.

[27]  Armindo Costa,et al.  MOTION PLANNING IN SPACES WITH SMALL FUNDAMENTAL GROUPS , 2008, 0806.4113.

[28]  Joseph S. B. Mitchell,et al.  New results on shortest paths in three dimensions , 2004, SCG '04.

[29]  Dima Grigoriev,et al.  Polytime algorithm for the shortest path in a homotopy class amidst semi-algebraic obstacles in the plane , 1998, ISSAC '98.

[30]  Michael Farber,et al.  Topological complexity of collision free motion planning algorithms in the presence of multiple moving obstacles , 2006 .

[31]  Vijay Kumar,et al.  Invariants for homology classes with application to optimal search and planning problem in robotics , 2012, Annals of Mathematics and Artificial Intelligence.