Optimal Multirobot Path Planning on Graphs: Complete Algorithms and Effective Heuristics

We study optimal multirobot path planning on graphs (MPP) over four minimization objectives: the makespan (last arrival time), the maximum (single-robot traveled) distance, the total arrival time, and the total distance. Having established previously that these objectives are distinct and NP-hard to optimize, in this paper, we focus on efficient algorithmic solutions for solving these optimal MPP problems. Toward this goal, we first establish a one-to-one solution mapping between MPP and a special type of multiflow network. Based on this equivalence and integer linear programming (ILP), we design novel and complete algorithms for optimizing over each of the four objectives. In particular, our exact algorithm for computing optimal makespan solutions is a first that is capable of solving extremely challenging problems with robot-vertex ratios as high as 100%. Then, we further improve the computational performance of these exact algorithms through the introduction of principled heuristics, at the expense of slight optimality loss. The combination of ILP model based algorithms and the heuristics proves to be highly effective, allowing the computation of 1.x-optimal solutions for problems containing hundreds of robots, densely populated in the environment, often in just seconds.

[1]  Seth Hutchinson,et al.  Path planning for permutation-invariant multirobot formations , 2005, IEEE Transactions on Robotics.

[2]  Vijay Kumar,et al.  Leader-to-formation stability , 2004, IEEE Transactions on Robotics and Automation.

[3]  Mark H. Overmars,et al.  Prioritized motion planning for multiple robots , 2005, 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[4]  Jay E. Aronson,et al.  A survey of dynamic network flows , 1989 .

[5]  Tucker R. Balch,et al.  Behavior-based formation control for multirobot teams , 1998, IEEE Trans. Robotics Autom..

[6]  Daniela Rus,et al.  An Effective Algorithmic Framework for Near Optimal Multi-robot Path Planning , 2015, ISRR.

[7]  G. Swaminathan.  Robot Motion Planning , 2006 .

[8]  Bartholomew O. Nnaji.  Theory of automatic robot assembly and programming , 1992 .

[9]  SharonGuni,et al.  Conflict-based search for optimal multi-agent pathfinding , 2015 .

[10]  Dan Halperin,et al.  k-color multi-robot motion planning , 2014, Int. J. Robotics Res..

[11]  Bruce Randall Donald,et al.  Moving furniture with teams of autonomous robots , 1995, Proceedings 1995 IEEE/RSJ International Conference on Intelligent Robots and Systems. Human Robot Interaction and Cooperative Robots.

[12]  David Tolpin,et al.  ICBS: The Improved Conflict-Based Search Algorithm for Multi-Agent Pathfinding , 2015, SOCS.

[13]  Nancy M. Amato,et al.  Behavior-based evacuation planning , 2010, 2010 IEEE International Conference on Robotics and Automation.

[14]  T. Murphey,et al.  Switching Rules for Decentralized Control with Simple Control Laws , 2007, 2007 American Control Conference.

[15]  Steven M. LaValle,et al.  Planning algorithms , 2006 .

[16]  Steven M. LaValle,et al.  Optimal Multi-Robot Path Planning on Graphs: Structure and Computational Complexity , 2015, ArXiv.

[17]  Dinesh Manocha,et al.  Centralized path planning for multiple robots: Optimal decoupling into sequential plans , 2009, Robotics: Science and Systems.

[18]  Richard E. Korf,et al.  Compressed Pattern Databases , 2007, J. Artif. Intell. Res..

[19]  Steven M. LaValle,et al.  Multi-agent Path Planning and Network Flow , 2012, WAFR.

[20]  D A N I E L R A T N E R A N D M A N F R E D W A R M,et al.  The ( n 2-1 )-Puzzle and Related Relocation Problems , 2008 .

[21]  Samuel Loyd,et al.  Mathematical Puzzles of Sam Loyd , 1959 .

[22]  Alexander Zelinsky,et al.  A mobile robot exploration algorithm , 1992, IEEE Trans. Robotics Autom..

[23]  Kostas E. Bekris,et al.  Multi-Agent Pathfinding with Simultaneous Execution of Single-Agent Primitives , 2021, SOCS.

[24]  Martin Nilsson,et al.  Cooperative multi-robot box-pushing , 1995, Proceedings 1995 IEEE/RSJ International Conference on Intelligent Robots and Systems. Human Robot Interaction and Cooperative Robots.

[25]  Dinesh Manocha,et al.  Reciprocal Velocity Obstacles for real-time multi-agent navigation , 2008, 2008 IEEE International Conference on Robotics and Automation.

[26]  John McPhee,et al.  A Complete and Scalable Strategy for Coordinating Multiple Robots Within Roadmaps , 2008, IEEE Transactions on Robotics.

[27]  Ailsa H. Land,et al.  An Automatic Method of Solving Discrete Programming Problems , 1960 .

[28]  Howie Choset,et al.  Principles of Robot Motion: Theory, Algorithms, and Implementation ERRATA!!!! 1 , 2007 .

[29]  Steven M. LaValle,et al.  Optimal motion planning for multiple robots having independent goals , 1996, Proceedings of IEEE International Conference on Robotics and Automation.

[30]  Raffaello D'Andrea,et al.  Coordinating Hundreds of Cooperative, Autonomous Vehicles in Warehouses , 2007, AI Mag..

[31]  Manfred K. Warmuth,et al.  NxN Puzzle and Related Relocation Problem , 1990, J. Symb. Comput..

[32]  Nathan R. Sturtevant,et al.  Conflict-based search for optimal multi-agent pathfinding , 2012, Artif. Intell..

[33]  Dinesh Manocha,et al.  Reciprocal collision avoidance with acceleration-velocity obstacles , 2011, 2011 IEEE International Conference on Robotics and Automation.

[34]  Dan Halperin,et al.  Finding a needle in an exponential haystack: Discrete RRT for exploration of implicit roadmaps in multi-robot motion planning , 2016, Int. J. Robotics Res..

[35]  Tomás Lozano-Pérez,et al.  On multiple moving objects , 1986, Proceedings. 1986 IEEE International Conference on Robotics and Automation.

[36]  Pavel Surynek,et al.  A novel approach to path planning for multiple robots in bi-connected graphs , 2009, 2009 IEEE International Conference on Robotics and Automation.

[37]  Kostas E. Bekris,et al.  Towards Using Discrete Multiagent Pathfinding to Address Continuous Problems , 2012, MAPF@AAAI.

[38]  Richard M. Wilson,et al.  Graph puzzles, homotopy, and the alternating group☆ , 1974 .

[39]  Magnus Egerstedt,et al.  Automatic Generation of Persistent Formations for Multi-agent Networks Under Range Constraints , 2009, Mob. Networks Appl..

[40]  Daniela Rus,et al.  Pebble Motion on Graphs with Rotations: Efficient Feasibility Tests and Planning Algorithms , 2012, WAFR.

[41]  Roni Stern,et al.  The Increasing Cost Tree Search for Optimal Multi-Agent Pathfinding , 2011, IJCAI.

[42]  Jie Ding,et al.  Scheduling of microfluidic operations for reconfigurabletwo-dimensional electrowetting arrays , 2001, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[43]  Wolfram Burgard,et al.  A Probabilistic Approach to Collaborative Multi-Robot Localization , 2000, Auton. Robots.

[44]  Cees Witteveen,et al.  Push and Rotate: a Complete Multi-agent Pathfinding Algorithm , 2014, J. Artif. Intell. Res..

[45]  Malcolm Ross Kinsella Ryan.  Exploiting Subgraph Structure in Multi-Robot Path Planning , 2008, J. Artif. Intell. Res..

[46]  Nathan R. Sturtevant,et al.  A new approach to cooperative pathfinding , 2008, AAMAS.

[47]  Kostas E. Bekris,et al.  Push and Swap: Fast Cooperative Path-Finding with Completeness Guarantees , 2011, IJCAI.

[48]  S. Zucker,et al.  Toward Efficient Trajectory Planning: The Path-Velocity Decomposition , 1986 .

[49]  Gaurav S. Sukhatme,et al.  Constrained coverage for mobile sensor networks , 2004, IEEE International Conference on Robotics and Automation, 2004. Proceedings. ICRA '04. 2004.

[50]  Steven M. LaValle,et al.  Fast, Near-Optimal Computation for Multi-Robot Path Planning on Graphs , 2013, AAAI.

[51]  Neil Robertson,et al.  Graph Minors .XIII. The Disjoint Paths Problem , 1995, J. Comb. Theory B.

[52]  Howie Choset,et al.  M*: A complete multirobot path planning algorithm with performance bounds , 2011, 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[53]  Roland Geraerts,et al.  Space-Time Group Motion Planning , 2012, WAFR.

[54]  Jean-Claude Latombe,et al.  A General Framework for Assembly Planning: The Motion Space Approach , 2000, Algorithmica.

[55]  Srinivas Akella,et al.  Coordinating Multiple Droplets in Planar Array Digital Microfluidic Systems , 2005, Int. J. Robotics Res..

[56]  Pavel Surynek,et al.  Towards Optimal Cooperative Path Planning in Hard Setups through Satisfiability Solving , 2012, PRICAI.

[57]  Vijay Kumar,et al.  Capt: Concurrent assignment and planning of trajectories for multiple robots , 2014, Int. J. Robotics Res..

[58]  Ravindra K. Ahuja,et al.  Network Flows: Theory, Algorithms, and Applications , 1993 .

[59]  Paul G. Spirakis,et al.  Coordinating Pebble Motion on Graphs, the Diameter of Permutation Groups, and Applications , 2015, FOCS.

[60]  G. Whelan,et al.  Cooperative search and rescue with a team of mobile robots , 1997, 1997 8th International Conference on Advanced Robotics. Proceedings. ICAR'97.

[61]  D. R. Fulkerson,et al.  Constructing Maximal Dynamic Flows from Static Flows , 1958 .

[62]  Lisa Zhang,et al.  Logarithmic hardness of the undirected edge-disjoint paths problem , 2006, JACM.

[63]  Richard E. Korf,et al.  Complete Algorithms for Cooperative Pathfinding Problems , 2011, IJCAI.

[64]  Trevor Scott Standley.  Finding Optimal Solutions to Cooperative Pathfinding Problems , 2010, AAAI.

[65]  David Silver,et al.  Cooperative Pathfinding , 2005, AIIDE.

[66]  Dan Halperin,et al.  Motion Planning for Unlabeled Discs with Optimality Guarantees , 2015, Robotics: Science and Systems.

[67]  A. Land,et al.  An Automatic Method for Solving Discrete Programming Problems , 1960, 50 Years of Integer Programming.

[68]  Steven M. LaValle,et al.  Planning optimal paths for multiple robots on graphs , 2012, 2013 IEEE International Conference on Robotics and Automation.