Safe Feedback Motion Planning: A Contraction Theory and ℒ1-Adaptive Control Based Approach

Autonomous robots that are capable of operating safely in the presence of imperfect model knowledge or external disturbances are vital in safety-critical applications. In this paper, we present a planner-agnostic framework to design and certify safe tubes around desired trajectories that the robot is always guaranteed to remain inside. By leveraging recent results in contraction analysis and ℒ1-adaptive control we synthesize an architecture that induces safe tubes for nonlinear systems with state and time-varying uncertainties. We demonstrate with a few illustrative examples1 how contraction theory-based ℒ1-adaptive control can be used in conjunction with traditional motion planning algorithms to obtain provably safe trajectories.

[1]  R. Schneider.  Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[2]  Hanmin Lee,et al.  Optimized L1 Adaptive Controller for Trajectory Tracking of an Indoor Quadrotor , 2017 .

[3]  Jean-Jacques E. Slotine,et al.  Contraction Metrics in Adaptive Nonlinear Control , 2019, ArXiv.

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

[5]  Chengyu Cao,et al.  Stability Margins of ${\cal L}_{1}$ Adaptive Control Architecture , 2010, IEEE Transactions on Automatic Control.

[6]  Jonathan P. How,et al.  Dynamic Tube MPC for Nonlinear Systems , 2019, 2019 American Control Conference (ACC).

[7]  Ian R. Manchester,et al.  Unifying Robot Trajectory Tracking with Control Contraction Metrics , 2015, ISRR.

[8]  Arjan van der Schaft,et al.  A receding-horizon approach to the nonlinear Hinfinity control problem , 2001, Autom..

[9]  Tristan Perez,et al.  Trajectory tracking passivity-based control for marine vehicles subject to disturbances , 2017, J. Frankl. Inst..

[10]  William S. Levine,et al.  Elastic tube model predictive control , 2016, 2016 American Control Conference (ACC).

[11]  P. Olver.  Nonlinear Systems , 2013 .

[12]  F. Allgöwer,et al.  Robust model predictive control with disturbance invariant sets , 2010, Proceedings of the 2010 American Control Conference.

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

[14]  A. Schaft,et al.  A receding-horizon approach to the nonlinear H ∞ control problem , 2000 .

[15]  Xiaofeng Wang,et al.  L1 adaptive controller for nonlinear time-varying reference systems , 2012, Syst. Control. Lett..

[16]  J. Slotine,et al.  Robust feedback motion planning via contraction theory , 2023, Int. J. Robotics Res..

[17]  Marco Pavone,et al.  Robust online motion planning via contraction theory and convex optimization , 2017, 2017 IEEE International Conference on Robotics and Automation (ICRA).

[18]  Jean-Jacques Slotine,et al.  Wasserstein Contraction of Stochastic Nonlinear Systems , 2019, 1902.08567.

[19]  L. Valavani.  Review of "L1 Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation" , 2011 .

[20]  Travis E. Gibson,et al.  Projection Operator in Adaptive Systems , 2011, ArXiv.

[21]  R. Mahony,et al.  Robust trajectory tracking for a scale model autonomous helicopter , 2004 .

[22]  Ian R. Manchester,et al.  Nonlinear stabilization via Control Contraction Metrics: A pseudospectral approach for computing geodesics , 2016, 2017 American Control Conference (ACC).

[23]  S. Raković.  Set Theoretic Methods in Model Predictive Control , 2009 .

[24]  Eduardo F. Camacho,et al.  Min-max Model Predictive Control of Nonlinear Systems: A Unifying Overview on Stability , 2009, Eur. J. Control.

[25]  A. D. Lewis,et al.  Geometric control of mechanical systems : modeling, analysis, and design for simple mechanical control systems , 2005 .

[26]  T. Apostol.  One-variable calculus, with an introduction to linear algebra , 1967 .

[27]  Naira Hovakimyan,et al.  L1 Adaptive Control Theory - Guaranteed Robustness with Fast Adaptation , 2010, Advances in design and control.

[28]  Xiaofeng Wang,et al.  Adaptive model predictive control of nonlinear systems with state‐dependent uncertainties , 2017 .

[29]  Dongkyoung Chwa,et al.  Coupled Multiple Sliding-Mode Control for Robust Trajectory Tracking of Hovercraft With External Disturbances , 2018, IEEE Transactions on Industrial Electronics.

[30]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[31]  James M. Rehg,et al.  Robust Sampling Based Model Predictive Control with Sparse Objective Information , 2018, Robotics: Science and Systems.

[32]  John M. Lee.  Riemannian Manifolds: An Introduction to Curvature , 1997 .

[33]  Irene M. Gregory,et al.  Flight Test of an L(sub 1) Adaptive Controller on the NASA AirSTAR Flight Test Vehicle , 2010 .

[34]  Peter I. Corke,et al.  Multirotor Aerial Vehicles: Modeling, Estimation, and Control of Quadrotor , 2012, IEEE Robotics & Automation Magazine.

[35]  Ian R. Manchester,et al.  An Amendment to "Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design" , 2017, ArXiv.

[36]  L. Praly,et al.  Adaptive nonlinear regulation: estimation from the Lyapunov equation , 1992 .

[37]  Naira Hovakimyan,et al.  Path Following for Unmanned Aerial Vehicles Using L1 Adaptive Augmentation of Commercial Autopilots , 2010 .

[38]  Winfried Stefan Lohmiller,et al.  Contraction analysis of nonlinear systems , 1999 .

[39]  Nasser M. Nasrabadi,et al.  Pattern Recognition and Machine Learning , 2006, Technometrics.

[40]  Naira Hovakimyan,et al.  Evaluation of an L1 Adaptive Flight Control Law on Calspan’s Variable-Stability Learjet , 2017 .