Mathematical programming for Multi-Vehicle Motion Planning problems

Real world Multi-Vehicle Motion Planning (MVMP) problems require the optimization of suitable performance measures under an array of complex and challenging constraints involving kinematics, dynamics, communication connectivity, target tracking, and collision avoidance. The general MVMP problem can thus be formulated as a mathematical program (MP). In this paper we present a mathematical programming (MP) framework that captures the salient features of the general MVMP problem. To demonstrate the use of this framework for the formulation and solution of MVMP problems, we examine in detail four representative works and summarize several other related works. As MP solution algorithms and associated numerical solvers continue to develop, we anticipate that MP solution techniques will be applied to an increasing number of MVMP problems and that the framework and formulations presented in this paper may serve as a guide for future MVMP research.

[1]  H.T. Friis,et al.  A Note on a Simple Transmission Formula , 1946, Proceedings of the IRE.

[2]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2002, SIAM J. Optim..

[3]  Jonathan P. How,et al.  Aircraft trajectory planning with collision avoidance using mixed integer linear programming , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[4]  Micha Sharir,et al.  Planning, geometry, and complexity of robot motion , 1986 .

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

[6]  Sven Leyffer,et al.  Integrating SQP and Branch-and-Bound for Mixed Integer Nonlinear Programming , 2001, Comput. Optim. Appl..

[7]  R. Vanderbei.  LOQO user's manual — version 3.10 , 1999 .

[8]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[9]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[10]  P. Abichandani,et al.  Multi-vehicle path coordination under communication constraints , 2008, 2008 American Control Conference.

[11]  Mehran Mesbahi,et al.  On maximizing the second smallest eigenvalue of a state-dependent graph Laplacian , 2006, IEEE Transactions on Automatic Control.

[12]  Srinivas Akella,et al.  Coordinating Multiple Double Integrator Robots on a Roadmap: Convexity and Global Optimality , 2005, Proceedings of the 2005 IEEE International Conference on Robotics and Automation.

[13]  Yoshiaki Kuwata,et al.  Experimental demonstrations of real-time MILP control , 2003 .

[14]  J. Meditch,et al.  Applied optimal control , 1972, IEEE Transactions on Automatic Control.

[15]  Antonio Bicchi,et al.  Conflict resolution problems for air traffic management systems solved with mixed integer programming , 2002, IEEE Trans. Intell. Transp. Syst..

[16]  Francesco Borrelli,et al.  Decentralized Receding Horizon Control and Coordination of Autonomous Vehicle Formations , 2008, IEEE Transactions on Control Systems Technology.

[17]  Johan Efberg,et al.  YALMIP : A toolbox for modeling and optimization in MATLAB , 2004 .

[18]  Ahmad A. Masoud,et al.  Kinodynamic Motion Planning , 2010, IEEE Robotics & Automation Magazine.

[19]  Srinivas Akella,et al.  Coordinating Multiple Robots with Kinodynamic Constraints Along Specified Paths , 2005, Int. J. Robotics Res..

[20]  C. Tomlin,et al.  Decentralized optimization, with application to multiple aircraft coordination , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[21]  John H. Reif,et al.  Complexity of the mover's problem and generalizations , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[22]  Eric Feron,et al.  Decentralized Cooperative Trajectory Planning of Multiple Aircraft with Hard Safety Guarantees , 2004 .

[23]  Alberto Bemporad,et al.  Control of systems integrating logic, dynamics, and constraints , 1999, Autom..

[24]  M. Ani Hsieh,et al.  An Optimal Approach to Collaborative Target Tracking with Performance Guarantees , 2009, J. Intell. Robotic Syst..

[25]  Roger Fletcher,et al.  Practical methods of optimization; (2nd ed.) , 1987 .

[26]  Dinesh Manocha,et al.  Reciprocal n-Body Collision Avoidance , 2011, ISRR.

[27]  F. Borrelli,et al.  MILP and NLP Techniques for centralized trajectory planning of multiple unmanned air vehicles , 2006, 2006 American Control Conference.

[28]  J. Gower.  Properties of Euclidean and non-Euclidean distance matrices , 1985 .

[29]  L. Singh,et al.  Trajectory generation for a UAV in urban terrain, using nonlinear MPC , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

[30]  David Kendrick,et al.  GAMS, a user's guide , 1988, SGNM.

[31]  Jorge Nocedal,et al.  Knitro: An Integrated Package for Nonlinear Optimization , 2006 .

[32]  E. J.,et al.  ON THE COMPLEXITY OF MOTION PLANNING FOR MULTIPLE INDEPENDENT OBJECTS ; PSPACE HARDNESS OF THE " WAREHOUSEMAN ' S PROBLEM " . * * ) , 2022 .

[33]  Brian W. Kernighan,et al.  AMPL: A Modeling Language for Mathematical Programming , 1993 .

[34]  Stephen P. Boyd,et al.  Automatic code generation for real-time convex optimization , 2010, Convex Optimization in Signal Processing and Communications.

[35]  Moshe Kam,et al.  Multi-vehicle path coordination in support of communication , 2009, 2009 IEEE International Conference on Robotics and Automation.

[36]  Lorenz T. Biegler,et al.  On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming , 2006, Math. Program..

[37]  Stephen P. Boyd,et al.  Fast Model Predictive Control Using Online Optimization , 2010, IEEE Transactions on Control Systems Technology.

[38]  Michael R. Bussieck,et al.  MINLP Solver Software , 2011 .

[39]  F. Borrelli,et al.  A study on decentralized receding horizon control for decoupled systems , 2004, Proceedings of the 2004 American Control Conference.

[40]  Tom Schouwenaars,et al.  Safe Trajectory Planning of Autonomous Vehicles , 2006 .

[41]  I. Garcia,et al.  Trajectory optimization for satellite reconfiguration maneuvers with position and attitude constraints , 2005, Proceedings of the 2005, American Control Conference, 2005..

[42]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[43]  J. How,et al.  Two-stage path planning approach for solving multiple spacecraft reconfiguration maneuvers , 2008 .

[44]  John E. Hopcroft,et al.  Movement Problems for 2-Dimensional Linkages , 1984, SIAM J. Comput..

[45]  E. Yaz.  Linear Matrix Inequalities In System And Control Theory , 1998, Proceedings of the IEEE.