Chance-Constrained Optimal Path Planning With Obstacles

Autonomous vehicles need to plan trajectories to a specified goal that avoid obstacles. For robust execution, we must take into account uncertainty, which arises due to uncertain localization, modeling errors, and disturbances. Prior work handled the case of set-bounded uncertainty. We present here a chance-constrained approach, which uses instead a probabilistic representation of uncertainty. The new approach plans the future probabilistic distribution of the vehicle state so that the probability of failure is below a specified threshold. Failure occurs when the vehicle collides with an obstacle or leaves an operator-specified region. The key idea behind the approach is to use bounds on the probability of collision to show that, for linear-Gaussian systems, we can approximate the nonconvex chance-constrained optimization problem as a disjunctive convex program. This can be solved to global optimality using branch-and-bound techniques. In order to improve computation time, we introduce a customized solution method that returns almost-optimal solutions along with a hard bound on the level of suboptimality. We present an empirical validation with an aircraft obstacle avoidance example.

[1]  E. L. Lawler,et al.  Branch-and-Bound Methods: A Survey , 1966, Oper. Res..

[2]  Dagfinn Gangsaas,et al.  Wind models for flight simulator certification of landing and approach guidance and control systems , 1974 .

[3]  Lennart Ljung,et al.  System Identification: Theory for the User , 1987 .

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

[5]  Ramon E. Moore.  Global optimization to prescribed accuracy , 1991 .

[6]  P. Gács,et al.  Algorithms , 1992 .

[7]  S. LaValle,et al.  Motion Planning , 2008, Springer Handbook of Robotics.

[8]  Manfred Morari,et al.  Robust constrained model predictive control using linear matrix inequalities , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[9]  B. Faverjon,et al.  Probabilistic Roadmaps for Path Planning in High-Dimensional Con(cid:12)guration Spaces , 1996 .

[10]  Basil Kouvaritakis,et al.  Stable generalized predictive control with constraints and bounded disturbances , 1997, Autom..

[11]  J Figueira,et al.  Stochastic Programming , 1998, J. Oper. Res. Soc..

[12]  Yinyu Ye,et al.  Interior point algorithms: theory and analysis , 1997 .

[13]  Michael Nikolaou,et al.  Chance‐constrained model predictive control , 1999 .

[14]  Nadine Le Fort-Piat,et al.  Safe Task Planning Integrating Uncertainties and Local Maps Federations , 2000, Int. J. Robotics Res..

[15]  Martin W. P. Savelsbergh,et al.  Progress in Linear Programming-Based Algorithms for Integer Programming: An Exposition , 2000, INFORMS J. Comput..

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

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

[18]  C. Scherer,et al.  LMI-based closed-loop economic optimization of stochastic process operation under state and input constraints , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[19]  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).

[20]  Pu Li,et al.  A probabilistically constrained model predictive controller , 2002, Autom..

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

[22]  O. Bosgra,et al.  Closed-loop stochastic dynamic process optimization under input and state constraints , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[23]  O. Bosgra,et al.  A full solution to the constrained stochastic closed-loop MPC problem via state and innovations feedback and its receding horizon implementation , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[24]  Dominique Gruyer,et al.  Safe path planning in an uncertain-configuration space , 2003, 2003 IEEE International Conference on Robotics and Automation (Cat. No.03CH37422).

[25]  R. D'Andrea,et al.  Low observability path planning for an unmanned air vehicle using mixed integer linear programming , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[26]  Jonathan P. How,et al.  Hybrid Model for Trajectory Planning of Agile Autonomous Vehicles , 2004, J. Aerosp. Comput. Inf. Commun..

[27]  Aachen,et al.  Stochastic Inequality Constrained Closed-loop Model Predictive Control: With Application To Chemical Process Operation , 2004 .

[28]  I Ivo Batina,et al.  Model predictive control for stochastic systems by randomized algorithms , 2004 .

[29]  Juris Vagners,et al.  PARALLEL EVOLUTIONARY ALGORITHMS FOR UAV PATH PLANNING , 2004 .

[30]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[31]  Hui Li,et al.  Generalized Conflict Learning for Hybrid Discrete/Linear Optimization , 2005, CP.

[32]  D. Q. Mayne,et al.  A simple tube controller for efficient robust model predictive control of constrained linear discret , 2005 .

[33]  Arthur G. Richards,et al.  Robust constrained model predictive control , 2005 .

[34]  Brian C. Williams,et al.  Coordinating Agile Systems through the Model-based Execution of Temporal Plans , 2005, AAAI.

[35]  Lakhmi C. Jain,et al.  Path Planning and Obstacle Avoidance for Autonomous Mobile Robots: A Review , 2006, KES.

[36]  Giuseppe Carlo Calafiore,et al.  The scenario approach to robust control design , 2006, IEEE Transactions on Automatic Control.

[37]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[38]  L. Blackmore.  A Probabilistic Particle Control Approach to Optimal, Robust Predictive Control , 2006 .

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

[40]  Ali A. Jalali,et al.  A Survey on Robust Model Predictive Control from 1999-2006 , 2006, 2006 International Conference on Computational Inteligence for Modelling Control and Automation and International Conference on Intelligent Agents Web Technologies and International Commerce (CIMCA'06).

[41]  Behçet Açikmese,et al.  A nonlinear model predictive control algorithm with proven robustness and resolvability , 2006, 2006 American Control Conference.

[42]  Alain Lambert,et al.  Safe Path Planning in an Uncertain-Configuration Space using RRT , 2006, 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[43]  Hui X. Li,et al.  A probabilistic approach to optimal robust path planning with obstacles , 2006, 2006 American Control Conference.

[44]  Giuseppe Carlo Calafiore,et al.  New results on the scenario design approach , 2007, 2007 46th IEEE Conference on Decision and Control.

[45]  Giuseppe Carlo Calafiore,et al.  Linear Programming with Probability Constraints - Part 2 , 2007, 2007 American Control Conference.

[46]  Masahiro Ono,et al.  Robust, Optimal Predictive Control of Jump Markov Linear Systems Using Particles , 2007, HSCC.

[47]  Masahiro Ono,et al.  An Efficient Motion Planning Algorithm for Stochastic Dynamic Systems with Constraints on Probability of Failure , 2008, AAAI.

[48]  L. Blackmore.  Robust Path Planning and Feedback Design Under Stochastic Uncertainty , 2008 .

[49]  L. Blackmore,et al.  Convex Chance Constrained Predictive Control without Sampling , 2009 .

[50]  M Ono,et al.  Chance constrained finite horizon optimal control with nonconvex constraints , 2010, Proceedings of the 2010 American Control Conference.

[51]  Masahiro Ono,et al.  A Probabilistic Particle-Control Approximation of Chance-Constrained Stochastic Predictive Control , 2010, IEEE Transactions on Robotics.