Mixed-integer programming in motion planning

Abstract This paper presents a review of past and present results and approaches in the area of motion planning using MIP (Mixed-integer Programming). Although in the early 2000s MIP was still seen with reluctance as method for solving motion planning-related problems, nowadays, due to increases in computational power and theoretical advances, its extensive modeling capabilities and versatility are coming to the fore and enjoy increased application and appreciation. This class of control problems involves, essentially, either a selection from a limited number of alternatives or a constrained optimization problem over a non-convex domain. In both situations, MIP has proven to be an efficient modeling technique as it will be shown in the present review paper. Furthermore, an emphasis is laid on the existing alternatives for implementation and on various experimental validations documented in the literature.

[1]  Rong Su,et al.  Traffic light scheduling for pedestrians and vehicles , 2017, 2017 IEEE Conference on Control Technology and Applications (CCTA).

[2]  YangQuan Chen,et al.  Formation control: a review and a new consideration , 2005, 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[3]  Steven L. Waslander,et al.  Tunnel-MILP: Path Planning with Sequential Convex Polytopes , 2008, AIAA Guidance, Navigation and Control Conference and Exhibit.

[4]  Charalampos P. Bechlioulis,et al.  Robot Navigation in Complex Workspaces Using Harmonic Maps , 2018, 2018 IEEE International Conference on Robotics and Automation (ICRA).

[5]  Luke Fletcher,et al.  A perception‐driven autonomous urban vehicle , 2008, J. Field Robotics.

[6]  David L. Woodruff,et al.  Pyomo — Optimization Modeling in Python , 2012, Springer Optimization and Its Applications.

[7]  Jianqiao Yu,et al.  UAV path planning using artificial potential field method updated by optimal control theory , 2016, Int. J. Syst. Sci..

[8]  Ionela Prodan,et al.  Exact and overapproximated guarantees for corner cutting avoidance in a multiobstacle environment , 2018, International Journal of Robust and Nonlinear Control.

[9]  Jonathan P. How,et al.  Coordination and control of multiple UAVs with timing constraints and loitering , 2003, Proceedings of the 2003 American Control Conference, 2003..

[10]  Goele Pipeleers,et al.  Online distributed motion planning for multi-vehicle systems , 2016, 2016 European Control Conference (ECC).

[11]  Ramjee Prasad,et al.  Data collection using miniature aerial vehicles in wireless sensor networks , 2016, IET Wirel. Sens. Syst..

[12]  Iain Dunning,et al.  JuMP: A Modeling Language for Mathematical Optimization , 2015, SIAM Rev..

[13]  Yoshihiro Kanno,et al.  Alternating direction method of multipliers as a simple effective heuristic for mixed-integer nonlinear optimization , 2018 .

[14]  G. Ziegler.  Lectures on Polytopes , 1994 .

[15]  J. How,et al.  Mixed-integer programming for control , 2005, Proceedings of the 2005, American Control Conference, 2005..

[16]  L. Ljung,et al.  Identification of hybrid systems via mixed-integer programming , 2001 .

[17]  Donald E. Kirk,et al.  Optimal control theory : an introduction , 1970 .

[18]  Lydia E. Kavraki,et al.  Measure theoretic analysis of probabilistic path planning , 2004, IEEE Transactions on Robotics and Automation.

[19]  George L. Nemhauser,et al.  Modeling disjunctive constraints with a logarithmic number of binary variables and constraints , 2011, Math. Program..

[20]  Karl Berntorp,et al.  Positive Invariant Sets for Safe Integrated Vehicle Motion Planning and Control , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[21]  Laurie G. Cuthbert,et al.  Energy-Efficient UAV Communication with Multiple GTs Based on Trajectory Optimization , 2018, Mob. Inf. Syst..

[22]  Jonathan P. How,et al.  Receding horizon control of autonomous aerial vehicles , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[23]  Leonardo Taccari,et al.  Integer programming formulations for the elementary shortest path problem , 2016, Eur. J. Oper. Res..

[24]  Marwan Bikdash,et al.  Hybrid mixed-logical linear programming algorithm for collision-free optimal path planning , 2007 .

[25]  Jizhong Xiao,et al.  A literature review of UAV 3D path planning , 2014, Proceeding of the 11th World Congress on Intelligent Control and Automation.

[26]  M. Jünger,et al.  50 Years of Integer Programming 1958-2008 - From the Early Years to the State-of-the-Art , 2010 .

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

[28]  Behçet Açikmese,et al.  Trajectory optimization with inter-sample obstacle avoidance via successive convexification , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[29]  Carl D. Laird,et al.  A stochastic programming approach for gas detector placement using CFD-based dispersion simulations , 2012, Comput. Chem. Eng..

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

[31]  Ionela Prodan,et al.  Hyperplane arrangements in mixed-integer programming techniques. Collision avoidance application with zonotopic sets , 2013, 2013 European Control Conference (ECC).

[32]  Robert R. Bitmead,et al.  Persistently Exciting Model Predictive Control for SISO systems , 2012 .

[33]  Tomás Lozano-Pérez,et al.  An algorithm for planning collision-free paths among polyhedral obstacles , 1979, CACM.

[34]  Tobias Achterberg,et al.  SCIP: solving constraint integer programs , 2009, Math. Program. Comput..

[35]  Stephen P. Boyd,et al.  Embedded Mixed-Integer Quadratic optimization Using the OSQP Solver , 2018, 2018 European Control Conference (ECC).

[36]  Manfred Morari,et al.  Multi-Parametric Toolbox 3.0 , 2013, 2013 European Control Conference (ECC).

[37]  George B. Dantzig,et al.  Solution of a Large-Scale Traveling-Salesman Problem , 1954, Oper. Res..

[38]  Fred W. Glover,et al.  The feasibility pump , 2005, Math. Program..

[39]  B. Moor,et al.  Mixed integer programming for multi-vehicle path planning , 2001, 2001 European Control Conference (ECC).

[40]  Huei Peng,et al.  Speed trajectory planning at signalized intersections using sequential convex optimization , 2017, 2017 American Control Conference (ACC).

[41]  James K. Kuchar,et al.  A review of conflict detection and resolution modeling methods , 2000, IEEE Trans. Intell. Transp. Syst..

[42]  Thorsten Koch,et al.  Solving mixed integer linear and nonlinear problems using the SCIP Optimization Suite , 2012 .

[43]  Marios M. Polycarpou,et al.  Distributed Traffic Signal Control Using the Cell Transmission Model via the Alternating Direction Method of Multipliers , 2015, IEEE Transactions on Intelligent Transportation Systems.

[44]  Madan Lal Mittal,et al.  Traveling Salesman Problem: an Overview of Applications, Formulations, and Solution Approaches , 2010 .

[45]  Detlef Stolten,et al.  Spatio-temporal optimization of a future energy system for power-to-hydrogen applications in Germany , 2018, Energy.

[46]  Hans D. Mittelmann,et al.  Mixed-integer nonlinear programming formulation of a UAV path optimization problem , 2017, 2017 American Control Conference (ACC).

[47]  Carleton Coffrin,et al.  The QC Relaxation: A Theoretical and Computational Study on Optimal Power Flow , 2017, IEEE Transactions on Power Systems.

[48]  Steven M. LaValle,et al.  Randomized Kinodynamic Planning , 1999, Proceedings 1999 IEEE International Conference on Robotics and Automation (Cat. No.99CH36288C).

[49]  Jonathan P. How,et al.  Cooperative path planning for multiple UAVs in dynamic and uncertain environments , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[50]  Walter Lucia,et al.  The obstacle avoidance motion planning problem for autonomous vehicles: A low-demanding receding horizon control scheme , 2015, Syst. Control. Lett..

[51]  Calin Belta,et al.  Robotic swarm control from spatio-temporal specifications , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[52]  Arthur Richards,et al.  Merging Vehicles at Junctions using Mixed-Integer Model Predictive Control , 2018, 2018 European Control Conference (ECC).

[53]  Hai Lin,et al.  Communication-aware motion planning for multi-agent systems from signal temporal logic specifications , 2017, 2017 American Control Conference (ACC).

[54]  Sanjay Mehrotra,et al.  A branch-and-cut method for 0-1 mixed convex programming , 1999, Math. Program..

[55]  Arthur Richards,et al.  Inter‐sample avoidance in trajectory optimizers using mixed‐integer linear programming , 2015 .

[56]  R. Raman,et al.  Modelling and computational techniques for logic based integer programming , 1994 .

[57]  Robin Deits,et al.  Efficient mixed-integer planning for UAVs in cluttered environments , 2015, 2015 IEEE International Conference on Robotics and Automation (ICRA).

[58]  R. Murray,et al.  Flat Systems , 1997 .

[59]  Alberto Bemporad,et al.  Identification of piecewise affine systems via mixed-integer programming , 2004, Autom..

[60]  Brian W. Kernighan,et al.  AMPL: a mathematical programming language , 1989 .

[61]  Francois Defay,et al.  Collision-Free Rendezvous Maneuvers for Formations of Unmanned Aerial Vehicles , 2017 .

[62]  Rosdiadee Nordin,et al.  Energy-Efficient Wireless Sensor Networks for Precision Agriculture: A Review , 2017, Sensors.

[63]  Jonathan P. How,et al.  Performance evaluation of rendezvous using model predictive control , 2003 .

[64]  Jean-Claude Latombe,et al.  On the Probabilistic Foundations of Probabilistic Roadmap Planning , 2006, Int. J. Robotics Res..

[65]  Mo Chen,et al.  Multi-vehicle collision avoidance via hamilton-jacobi reachability and mixed integer programming , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[66]  Ionela Prodan,et al.  On the Tight Formation for Multi-agent Dynamical Systems , 2012, KES-AMSTA.

[67]  Behçet Açikmese,et al.  Lossless convexification of non-convex optimal control problems for state constrained linear systems , 2014, Autom..

[68]  Roberto Kawakami Harrop Galvão,et al.  On the use of mixed‐integer linear programming for predictive control with avoidance constraints , 2009 .

[69]  Lydia E. Kavraki,et al.  A Random Sampling Scheme for Path Planning , 1997, Int. J. Robotics Res..

[70]  M. Pereira,et al.  A Mixed Integer Disjunctive Model for Transmission Network Expansion , 2001, IEEE Power Engineering Review.

[71]  Christos G. Cassandras,et al.  Optimal dynamic formation control of multi-agent systems in environments with obstacles , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[72]  Dimos V. Dimarogonas,et al.  Decentralized Control of Uncertain Multi-Agent Systems with Connectivity Maintenance and Collision Avoidance , 2017, 2018 European Control Conference (ECC).

[73]  Rubens Junqueira Magalhães Afonso,et al.  Task allocation and trajectory planning for multiple agents in the presence of obstacle and connectivity constraints with mixed‐integer linear programming , 2020 .

[74]  A. Bemporad.  Solving Mixed-Integer Quadratic Programs via Nonnegative Least Squares , 2015 .

[75]  Z. Qu,et al.  Cooperative Control of Dynamical Systems: Applications to Autonomous Vehicles , 2009 .

[76]  Behçet Açikmese,et al.  Successive convexification of non-convex optimal control problems and its convergence properties , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[77]  Florent Altché,et al.  Partitioning of the free space-time for on-road navigation of autonomous ground vehicles , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[78]  Rudy R. Negenborn,et al.  Distributed Model Predictive Control Made Easy , 2013 .

[79]  Xu Wang,et al.  Collision avoidance of mobile robots by using initial time delays , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[80]  Stephen P. Boyd,et al.  CVXPY: A Python-Embedded Modeling Language for Convex Optimization , 2016, J. Mach. Learn. Res..

[81]  Mérouane Debbah,et al.  Distributed caching in 5G networks: An Alternating Direction Method of Multipliers approach , 2015, 2015 IEEE 16th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC).

[82]  John N. Hooker,et al.  Integrated methods for optimization , 2011, International series in operations research and management science.

[83]  Renaud Deborne,et al.  Collision-free trajectory planning for overtaking on highways , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[84]  Roberto Kawakami Harrop Galvão,et al.  Waypoint trajectory planning in the presence of obstacles with a tunnel-MILP approach , 2013, 2013 European Control Conference (ECC).

[85]  André Luckow,et al.  Optimal scheduling of autonomous vehicle arrivals at intelligent intersections via MILP , 2017, 2017 American Control Conference (ACC).

[86]  Jonathan P. How,et al.  UAV Task Assignment , 2008, IEEE Robotics & Automation Magazine.

[87]  Matteo Fischetti,et al.  Feasibility pump 2.0 , 2009, Math. Program. Comput..

[88]  A. R. Meyer,et al.  Handbook of Theoretical Computer Science: Algorithms and Complexity , 1990 .

[89]  Luigi del Re,et al.  Efficient mixed integer programming for autonomous overtaking , 2017, 2017 American Control Conference (ACC).

[90]  Ping Lu,et al.  Solving Nonconvex Optimal Control Problems by Convex Optimization , 2014 .

[91]  Arthur Richards,et al.  Inter‐sample avoidance in trajectory optimizers using mixed‐integer linear programming , 2013 .

[92]  Emilio Frazzoli,et al.  Sampling-based algorithms for optimal motion planning , 2011, Int. J. Robotics Res..

[93]  M. Morari,et al.  An efficient branch and bound algorithm for state estimation and control of hybrid systems , 1999, 1999 European Control Conference (ECC).

[94]  Florin Ilarion Miertoiu,et al.  Feasibility Pump Algorithm for Sparse Representation under Laplacian Noise , 2019 .

[95]  Alessandro Rucco,et al.  A finite-time cutting plane algorithm for distributed mixed integer linear programming , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[96]  Scott A. Bortoff,et al.  Energy-Efficient Trajectory Planning for a Mobile Agent by Using a Two-Stage Decomposition Approach , 2014 .

[97]  Christian Kirches,et al.  Mixed-integer nonlinear optimization*† , 2013, Acta Numerica.

[98]  Masayuki Fujita,et al.  Model predictive control with a mixed integer programming for merging path generation on motor way , 2017, 2017 IEEE Conference on Control Technology and Applications (CCTA).

[99]  David Q. Mayne,et al.  Constrained model predictive control: Stability and optimality , 2000, Autom..

[100]  Vihangkumar V. Naik,et al.  Embedded Mixed-Integer Quadratic Optimization using Accelerated Dual Gradient Projection , 2017 .

[101]  Michal Kvasnica,et al.  Trajectory planning and following for UAVs with nonlinear dynamics , 2017, 2017 21st International Conference on Process Control (PC).

[102]  Jean-Claude Latombe,et al.  Robot motion planning , 1970, The Kluwer international series in engineering and computer science.

[103]  Charalampos P. Bechlioulis,et al.  Prescribed Time Scale Robot Navigation , 2018, IEEE Robotics and Automation Letters.

[104]  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).

[105]  Bernhard Rinner,et al.  Networked UAVs as aerial sensor network for disaster management applications , 2010, Elektrotech. Informationstechnik.

[106]  Robin Deits,et al.  Computing Large Convex Regions of Obstacle-Free Space Through Semidefinite Programming , 2014, WAFR.

[107]  Alberto Bemporad,et al.  Optimal and receding horizon drift counteraction control: Linear programming approaches , 2017, 2017 American Control Conference (ACC).

[108]  Stefano Di Cairano,et al.  A Structure Exploiting Branch-and-Bound Algorithm for Mixed-Integer Model Predictive Control , 2019, 2019 18th European Control Conference (ECC).

[109]  Wu Yue,et al.  Path Planning for UAV to Collect Sensors Data Based on Spiral Decomposition , 2018 .

[110]  Stefan Ulbrich,et al.  Adaptive Observation Strategy for Dispersion Process Estimation Using Cooperating Mobile Sensors , 2014 .

[111]  R. D'Andrea,et al.  Modeling and control of a multi-agent system using mixed integer linear programming , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[112]  T. H. Feiroz Khan,et al.  Mobile Collector Aided Energy Reduced (MCER) Data Collection in Agricultural Wireless Sensor Networks , 2016, 2016 IEEE 6th International Conference on Advanced Computing (IACC).

[113]  Karl Berntorp,et al.  Motion planning with invariant set trees , 2017, 2017 IEEE Conference on Control Technology and Applications (CCTA).

[114]  Benoît Chachuat,et al.  Branch-and-lift algorithm for obstacle avoidance control , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[115]  Jonathan P. How,et al.  Plume Avoidance Maneuver Planning Using Mixed Integer Linear Programming , 2001 .

[116]  Alberto Bemporad,et al.  Receding horizon drift counteraction and its application to spacecraft attitude control , 2017 .

[117]  Hassan L. Hijazi,et al.  Complex number formulation and convex relaxations for aircraft conflict resolution , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[118]  Ionela Prodan,et al.  Mixed-Integer Representations in Control Design: Mathematical Foundations and Applications , 2015 .

[119]  Alberto Bemporad,et al.  A simple effective heuristic for embedded mixed-integer quadratic programming , 2020, Int. J. Control.

[120]  William W. Hager,et al.  Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method , 2011, Comput. Optim. Appl..

[121]  Sajal K. Das,et al.  Efficient Aerial Data Collection with UAV in Large-Scale Wireless Sensor Networks , 2015, Int. J. Distributed Sens. Networks.

[122]  Manfred Morari,et al.  A decomposition method for large scale MILPs, with performance guarantees and a power system application , 2014, Autom..

[123]  François Soumis,et al.  An integrated aircraft routing, crew scheduling and flight retiming model , 2005, Comput. Oper. Res..

[124]  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).

[125]  Michal Kvasnica,et al.  OPTIPLAN: A Matlab Toolbox for Model Predictive Control with Obstacle Avoidance , 2017 .

[126]  Juan Pablo Vielma,et al.  Mixed Integer Linear Programming Formulation Techniques , 2015, SIAM Rev..

[127]  Ionela Prodan,et al.  Complexity Bounds for Obstacle Avoidance within a Zonotopic Framework , 2019, 2019 American Control Conference (ACC).

[128]  I. Grossmann,et al.  New algorithms for nonlinear generalized disjunctive programming , 2000 .

[129]  Wolfgang Kuehn,et al.  Rigorously computed orbits of dynamical systems without the wrapping effect , 1998, Computing.

[130]  Rong Su,et al.  Modelling and traffic signal control of a heterogeneous traffic network with signalized and non-signalized intersections , 2017, 2017 IEEE Conference on Control Technology and Applications (CCTA).

[131]  Raffaello D'Andrea,et al.  Iterative MILP methods for vehicle-control problems , 2005, IEEE Transactions on Robotics.

[132]  Z. Taşkın.  Tutorial Guide to Mixed-Integer Programming Models and Solution Techniques , 2008 .

[133]  H. P. Williams,et al.  Model Building in Mathematical Programming , 1979 .

[134]  Daniel E. Koditschek,et al.  Exact robot navigation using artificial potential functions , 1992, IEEE Trans. Robotics Autom..

[135]  José A. De Doná,et al.  Reference governor design for tracking problems with fault detection guarantees , 2012 .

[136]  J. Christopher Beck,et al.  Scheduling a Dynamic Aircraft Repair Shop with Limited Repair Resources , 2013, J. Artif. Intell. Res..

[137]  Wenwu Yu,et al.  An Overview of Recent Progress in the Study of Distributed Multi-Agent Coordination , 2012, IEEE Transactions on Industrial Informatics.

[138]  Roberto Kawakami Harrop Galvão,et al.  Reduction in the number of binary variables for inter‐sample avoidance in trajectory optimizers using mixed‐integer linear programming , 2016 .

[139]  Yuanqi Mao,et al.  Successive Convexification of Non-Convex Optimal Control Problems with State Constraints , 2017, 1701.00558.

[140]  Francesco Borrelli,et al.  An auto-generated nonlinear MPC algorithm for real-time obstacle avoidance of ground vehicles , 2013, 2013 European Control Conference (ECC).

[141]  Russell Bent,et al.  Polyhedral approximation in mixed-integer convex optimization , 2016, Math. Program..

[142]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[143]  Timothy W. McLain,et al.  Small Unmanned Aircraft: Theory and Practice , 2012 .