LQR-trees: Feedback Motion Planning via Sums-of-Squares Verification

Advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of attraction for smooth non-linear systems. Here we present a feedback motion-planning algorithm which uses rigorously computed stability regions to build a sparse tree of LQR-stabilized trajectories. The region of attraction of this non-linear feedback policy “probabilistically covers” the entire controllable subset of state space, verifying that all initial conditions that are capable of reaching the goal will reach the goal. We numerically investigate the properties of this systematic non-linear feedback design algorithm on simple non-linear systems, prove the property of probabilistic coverage, and discuss extensions and implementation details of the basic algorithm.

[1]  Matthew T. Mason,et al.  The mechanics of manipulation , 1985, Proceedings. 1985 IEEE International Conference on Robotics and Automation.

[2]  Anil V. Rao,et al.  Practical Methods for Optimal Control Using Nonlinear Programming , 1987 .

[3]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[4]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .

[5]  Yinyu Ye.  Interior-Point Polynomial Algorithms in Convex Programming (Y. Nesterov and A. Nemirovskii) , 1994, SIAM Rev..

[6]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

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

[8]  A. Rantzer,et al.  System analysis via integral quadratic constraints , 1997, IEEE Trans. Autom. Control..

[9]  Martin Berz,et al.  Computation and Application of Taylor Polynomials with Interval Remainder Bounds , 1998, Reliab. Comput..

[10]  Natan Peterfreund,et al.  Convergence analysis of nonlinear dynamical systems by nested Lyapunov functions , 1998, IEEE Trans. Autom. Control..

[11]  Daniel E. Koditschek,et al.  Sequential Composition of Dynamically Dexterous Robot Behaviors , 1999, Int. J. Robotics Res..

[12]  Steven M. LaValle,et al.  Rapidly-Exploring Random Trees: Progress and Prospects , 2000 .

[13]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[14]  Tor Arne Johansen,et al.  Computation of Lyapunov functions for smooth nonlinear systems using convex optimization , 2000, Autom..

[15]  Masayuki Inaba,et al.  Motion planning for humanoid robots under obstacle and dynamic balance constraints , 2001, Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No.01CH37164).

[16]  A. Rantzer.  A dual to Lyapunov's stability theorem , 2001 .

[17]  E. Feron,et al.  Real-time motion planning for agile autonomous vehicles , 2000, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

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

[19]  Peter J Seiler,et al.  SOSTOOLS: Sum of squares optimization toolbox for MATLAB , 2002 .

[20]  A. Papachristodoulou,et al.  On the construction of Lyapunov functions using the sum of squares decomposition , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[21]  Michael S. Branicky,et al.  Nonlinear and Hybrid Control Via RRTs , 2002 .

[22]  A. Megretski.  Positivity of trigonometric polynomials , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[23]  Pablo A. Parrilo,et al.  Nonlinear control synthesis by convex optimization , 2004, IEEE Transactions on Automatic Control.

[24]  Alexandre M. Bayen,et al.  A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games , 2005, IEEE Transactions on Automatic Control.

[25]  Ian M. Mitchell,et al.  A Toolbox of Hamilton-Jacobi Solvers for Analysis of Nondeterministic Continuous and Hybrid Systems , 2005, HSCC.

[26]  Michael A. Saunders,et al.  SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization , 2005, SIAM Rev..

[27]  Alexandre M. Bayen,et al.  Computational Techniques for the Verification and Control of Hybrid Systems , 2005 .

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

[29]  Rafal Goebel,et al.  Smooth patchy control Lyapunov functions , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[30]  L. Rodrigues,et al.  Control of Large Angle Attitude Maneuvers for Rigid Bodies Using Sum of Squares , 2007, 2007 American Control Conference.

[31]  Anders Rantzer,et al.  Convex Programs for Temporal Verification of Nonlinear Dynamical Systems , 2007, SIAM J. Control. Optim..

[32]  Christopher G. Atkeson,et al.  Random Sampling of States in Dynamic Programming , 2007, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[33]  Andrew Packard,et al.  Stability Region Analysis Using Polynomial and Composite Polynomial Lyapunov Functions and Sum-of-Squares Programming , 2008, IEEE Transactions on Automatic Control.

[34]  Ufuk Topcu,et al.  Local stability analysis using simulations and sum-of-squares programming , 2008, Autom..

[35]  Graziano Chesi,et al.  Estimating the domain of attraction for non-polynomial systems via LMI optimizations , 2009, Autom..

[36]  G. Franzé,et al.  An Off-Line MPC Strategy for Nonlinear Systems Based on SOS Programming , 2008 .

[37]  Russ Tedrake,et al.  LQR-trees: Feedback motion planning on sparse randomized trees , 2009, Robotics: Science and Systems.

[38]  Russ Tedrake,et al.  Path planning in 1000+ dimensions using a task-space Voronoi bias , 2009, 2009 IEEE International Conference on Robotics and Automation.

[39]  Antonis Papachristodoulou,et al.  Analysis of Polynomial Systems With Time Delays via the Sum of Squares Decomposition , 2009, IEEE Transactions on Automatic Control.

[40]  Russ Tedrake,et al.  Simulation-based LQR-trees with input and state constraints , 2010, 2010 IEEE International Conference on Robotics and Automation.