Moment-Based Exact Uncertainty Propagation Through Nonlinear Stochastic Autonomous Systems

In this paper, we address the problem of uncertainty propagation through nonlinear stochastic dynamical systems. More precisely, given a discrete-time continuous-state probabilistic nonlinear dynamical system, we aim at finding the sequence of the moments of the probability distributions of the system states up to any desired order over the given planning horizon. Moments of uncertain states can be used in estimation, planning, control, and safety analysis of stochastic dynamical systems. Existing approaches to address moment propagation problems provide approximate descriptions of the moments and are mainly limited to particular set of uncertainties, e.g., Gaussian disturbances. In this paper, to describe the moments of uncertain states, we introduce trigonometric and also mixed-trigonometric-polynomial moments. Such moments allow us to obtain closed deterministic dynamical systems that describe the exact time evolution of the moments of uncertain states of an important class of autonomous and robotic systems including underwater, ground, and aerial vehicles, robotic arms and walking robots. Such obtained deterministic dynamical systems can be used, in a receding horizon fashion, to propagate the uncertainties over the planning horizon in real-time. To illustrate the performance of the proposed method, we benchmark our method against existing approaches including linear, unscented transformation, and sampling based uncertainty propagation methods that are widely used in estimation, prediction, planning, and control problems.

[1]  Morteza Lahijanian,et al.  Online Mapping and Motion Planning Under Uncertainty for Safe Navigation in Unknown Environments , 2020, IEEE Transactions on Automation Science and Engineering.

[2]  Zhong Liu,et al.  Automated proof of mixed trigonometric-polynomial inequalities , 2020, J. Symb. Comput..

[3]  Stefan Streif,et al.  PoCET: a Polynomial Chaos Expansion Toolbox for Matlab , 2020, IFAC-PapersOnLine.

[4]  Xin Huang,et al.  Fast Risk Assessment for Autonomous Vehicles Using Learned Models of Agent Futures , 2020, Robotics: Science and Systems.

[5]  Allen Wang,et al.  Moment State Dynamical Systems for Nonlinear Chance-Constrained Motion Planning , 2020, ArXiv.

[6]  M. McHenry,et al.  A ROS-based Simulator for Testing the Enhanced Autonomous Navigation of the Mars 2020 Rover , 2020, 2020 IEEE Aerospace Conference.

[7]  Non-Gaussian Chance-Constrained Trajectory Planning for Autonomous Vehicles in the Presence of Uncertain Agents , 2020, ArXiv.

[8]  Ali-akbar Agha-mohammadi,et al.  Deep Learning Tubes for Tube MPC , 2020, Robotics: Science and Systems.

[9]  Zachary Manchester,et al.  (Preprint) AAS 20-071 SAMPLE-BASED ROBUST UNCERTAINTY PROPAGATION FOR ENTRY VEHICLES , 2020 .

[10]  Ashkan Jasour,et al.  Sequential Chance Optimization For Flow-Tube Based Control Of Probabilistic Nonlinear Systems , 2019, 2019 IEEE 58th Conference on Decision and Control (CDC).

[11]  Sasinee Pruekprasert,et al.  Moment Propagation of Discrete-Time Stochastic Polynomial Systems using Truncated Carleman Linearization , 2019, ArXiv.

[12]  Jeongeun Son,et al.  Probabilistic surrogate models for uncertainty analysis: Dimension reduction‐based polynomial chaos expansion , 2019, International Journal for Numerical Methods in Engineering.

[13]  Soon-Jo Chung,et al.  Trajectory Optimization for Chance-Constrained Nonlinear Stochastic Systems , 2019, 2019 IEEE 58th Conference on Decision and Control (CDC).

[14]  Brian C. Williams,et al.  Risk Contours Map for Risk Bounded Motion Planning under Perception Uncertainties , 2019, Robotics: Science and Systems.

[15]  Jean B. Lasserre.  Volume of sub-level sets of polynomials , 2019, 2019 18th European Control Conference (ECC).

[16]  Jean Lasserre.  Volume of Sublevel Sets of Homogeneous Polynomials , 2019, SIAM J. Appl. Algebra Geom..

[17]  Xin Huang,et al.  Hybrid Risk-Aware Conditional Planning with Applications in Autonomous Vehicles , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[18]  Brian C. Williams,et al.  Moment-Sum-of-Squares Approach for Fast Risk Estimation in Uncertain Environments , 2018, 2018 IEEE Conference on Decision and Control (CDC).

[19]  Tom Dhaene,et al.  Review of Polynomial Chaos-Based Methods for Uncertainty Quantification in Modern Integrated Circuits , 2018 .

[20]  Timothy D. Barfoot,et al.  State Estimation for Robotics , 2017 .

[21]  Sairaj V. Dhople,et al.  Approximate moment dynamics for polynomial and trigonometric stochastic systems , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[22]  Cristinel Mortici,et al.  The natural algorithmic approach of mixed trigonometric-polynomial problems , 2017, Journal of inequalities and applications.

[23]  Shankar Mohan,et al.  Convex estimation of the α-confidence reachable set for systems with parametric uncertainty , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[24]  Constantino M. Lagoa,et al.  Convex Chance Constrained Model Predictive Control , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[25]  Bert Debusschere,et al.  Intrusive Polynomial ChaosMethods for Forward Uncertainty Propagation , 2016 .

[26]  Constantino M. Lagoa,et al.  Semidefinite Programming For Chance Constrained Optimization Over Semialgebraic Sets , 2014, SIAM J. Optim..

[27]  Constantino Lagoa,et al.  Reconstruction of support of a measure from its moments , 2014, 53rd IEEE Conference on Decision and Control.

[28]  Mohammad Farrokhi,et al.  Adaptive neuro‐predictive control for redundant robot manipulators in presence of static and dynamic obstacles: A Lyapunov‐based approach , 2014 .

[29]  Constantino M. Lagoa,et al.  Convex relaxations of a probabilistically robust control design problem , 2013, 52nd IEEE Conference on Decision and Control.

[30]  Constantino M. Lagoa,et al.  Semidefinite relaxations of chance constrained algebraic problems , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[31]  Didier Henrion,et al.  Mean Squared Error Minimization for Inverse Moment Problems , 2012 .

[32]  O. Ernst,et al.  ON THE CONVERGENCE OF GENERALIZED POLYNOMIAL CHAOS EXPANSIONS , 2011 .

[33]  Nicholas Roy,et al.  Finite-Time Regional Verification of Stochastic Nonlinear Systems , 2012 .

[34]  Masahiro Ono,et al.  Chance-Constrained Optimal Path Planning With Obstacles , 2011, IEEE Transactions on Robotics.

[35]  Pieter Abbeel,et al.  LQG-MP: Optimized path planning for robots with motion uncertainty and imperfect state information , 2010, Int. J. Robotics Res..

[36]  Mohammad Farrokhi,et al.  Fuzzy improved adaptive neuro-NMPC for online path tracking and obstacle avoidance of redundant robotic manipulators , 2010, Int. J. Autom. Control..

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

[38]  Katie Byl,et al.  Metastable Walking Machines , 2009, Int. J. Robotics Res..

[39]  D. Xiu.  Fast numerical methods for stochastic computations: A review , 2009 .

[40]  Teresa A. Vidal-Calleja,et al.  Unscented Transformation of Vehicle States in SLAM , 2005, Proceedings of the 2005 IEEE International Conference on Robotics and Automation.

[41]  Jeffrey K. Uhlmann,et al.  Unscented filtering and nonlinear estimation , 2004, Proceedings of the IEEE.

[42]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[43]  Sebastian Thrun,et al.  Probabilistic robotics , 2002, CACM.

[44]  Rudolph van der Merwe,et al.  The unscented Kalman filter for nonlinear estimation , 2000, Proceedings of the IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (Cat. No.00EX373).

[45]  S. Resnick.  A Probability Path , 1999 .

[46]  Jeffrey K. Uhlmann,et al.  New extension of the Kalman filter to nonlinear systems , 1997, Defense, Security, and Sensing.

[47]  W. Steeb,et al.  Nonlinear dynamical systems and Carleman linearization , 1991 .

[48]  Tad McGeer,et al.  Passive Dynamic Walking , 1990, Int. J. Robotics Res..

[49]  G. Pólya,et al.  Polynomials and Trigonometric Polynomials , 1976 .