Guaranteed computation of robot trajectories

This paper proposes a new method for guaranteed integration of state equations. Within this framework, the variables of interest are trajectories submitted to both arithmetic and differential equations. The approach consists in formalizing a problem thanks to a constraint network and then apply these constraints to sets of trajectories. The contribution of the paper is to provide a reliable framework to enclose the solutions of these differential equations. Its use is shown to be simple, more general and more competitive than existing approaches dealing with guaranteed integration, especially when applied to mobile robotics. The flexibility of the developed framework allows to deal with non-linear differential equations or even differential inclusions built from datasets, while considering observations of the states of interest. An illustration of this method is given over several examples with mobile robots. A contractor-based approach is proposed for guaranteed integration of state equations.The framework is based on the use of tubes as envelopes of feasible trajectories.A dedicated differential contractor is provided to deal with dynamical systems.The use of tubes is well suited for multi-differential and non-linear equations.An illustration of this approach is given over several mobile robotics examples.

[1]  C. Combastel.  A State Bounding Observer for Uncertain Non-linear Continuous-time Systems based on Zonotopes , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[2]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[3]  Ramon E. Moore.  Methods and applications of interval analysis , 1979, SIAM studies in applied mathematics.

[4]  Luc Jaulin.  Mobile Robotics , 2019 .

[5]  Alexandre Goldsztejn,et al.  Tinkerbell Is Chaotic , 2011, SIAM J. Appl. Dyn. Syst..

[6]  Thor I. Fossen,et al.  Guidance and control of ocean vehicles , 1994 .

[7]  Alex M. Andrew,et al.  Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics , 2002 .

[8]  Nedialko S. Nedialkov,et al.  Validated solutions of initial value problems for ordinary differential equations , 1999, Appl. Math. Comput..

[9]  Luc Jaulin,et al.  Set-membership state estimation with fleeting data , 2012, Autom..

[10]  D. Bertsekas,et al.  Recursive state estimation for a set-membership description of uncertainty , 1971 .

[11]  E. Walter,et al.  Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics , 2001 .

[12]  Luc Jaulin,et al.  Loop detection of mobile robots using interval analysis , 2013, Autom..

[13]  Philippe Bonnifait,et al.  Localization Confidence Domains via Set Inversion on Short-Term Trajectory , 2013, IEEE Transactions on Robotics.

[14]  Christian Bessiere,et al.  Constraint Propagation , 2006, Handbook of Constraint Programming.

[15]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[16]  Alexandre Goldsztejn,et al.  The Reach-and-Evolve Algorithm for Reachability Analysis of Nonlinear Dynamical Systems , 2008, Electron. Notes Theor. Comput. Sci..

[17]  Eric Goubault,et al.  Inner approximated reachability analysis , 2014, HSCC.

[18]  Lyudmila Mihaylova,et al.  Non-linear state estimation using imprecise samples , 2013, Proceedings of the 16th International Conference on Information Fusion.

[19]  Sandor M. Veres,et al.  Outliers in bound-based state estimation and identification , 1993, 1993 IEEE International Symposium on Circuits and Systems.

[20]  Marcia Kilchenman O'Malley,et al.  Acumen: An Open-Source Testbed for Cyber-Physical Systems Research , 2015, IoT 360.

[21]  Walid Taha,et al.  Enclosing the behavior of a hybrid system up to and beyond a Zeno point , 2013, 2013 IEEE 1st International Conference on Cyber-Physical Systems, Networks, and Applications (CPSNA).

[22]  Luc Jaulin,et al.  Contractor programming , 2009, Artif. Intell..

[23]  Pascal Van Hentenryck,et al.  Helios: A Modeling Language for Global Optimization and its Implementation in Newton , 1997, Theor. Comput. Sci..

[24]  Luc Jaulin,et al.  Nonlinear bounded-error state estimation of continuous-time systems , 2002, Autom..

[25]  D. Wilczak,et al.  Cr-Lohner algorithm , 2011 .

[26]  Luc Jaulin,et al.  Solving Non-Linear Constraint Satisfaction Problems Involving Time-Dependant Functions , 2014, Math. Comput. Sci..

[27]  Alan K. Mackworth.  Consistency in Networks of Relations , 1977, Artif. Intell..

[28]  Gilles Trombettoni,et al.  A Contractor Based on Convex Interval Taylor , 2012, CPAIOR.

[29]  Martin Berz,et al.  Computational differentiation : techniques, applications, and tools , 1996 .

[30]  Vincent Drevelle,et al.  VIBes: A Visualizer for Intervals and Boxes , 2014, Math. Comput. Sci..

[31]  Nathalie Revol,et al.  Taylor models and floating-point arithmetic: proof that arithmetic operations are validated in COSY , 2005, J. Log. Algebraic Methods Program..

[32]  Pascal Van Hentenryck,et al.  Numerica: A Modeling Language for Global Optimization , 1997, IJCAI.

[33]  J. Norton,et al.  Bounding Approaches to System Identification , 1996 .

[34]  S. Veres.  Error Control in Polytope Computations , 2002 .

[35]  W. Tucker.  The Lorenz attractor exists , 1999 .

[36]  Nedialko S. Nedialkov,et al.  Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint-propagation techniques , 2011 .

[37]  L. Dubins.  On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents , 1957 .

[38]  A. Kurzhanski,et al.  Ellipsoidal Calculus for Estimation and Control , 1996 .

[39]  Julien Alexandre Dit Sandretto,et al.  Validated Explicit and Implicit Runge-Kutta Methods , 2016 .

[40]  Marco Nehmeier,et al.  filib++ , Expression Templates and the Coming Interval Standard , 2011, Reliab. Comput..