Learning stable dynamical systems using contraction theory

This paper discusses the learning of robot point-to-point motions via non-linear dynamical systems and Gaussian Mixture Regression (GMR). The novelty of the proposed approach consists in guaranteeing the stability of a learned dynamical system via Contraction theory. A contraction analysis is performed to derive sufficient conditions for the global stability of a dynamical system represented by GMR. The results of this analysis are exploited to automatically compute a control input which stabilizes the learned system on-line. Simple and effective solutions are proposed to generate motion trajectories close to the demonstrated ones, without affecting the stability of the overall system. The proposed approach is evaluated on a public benchmark of point-to-point motions and compared with state-of-the-art algorithms based on Lyapunov stability theory.

[1]  Matteo Saveriano,et al.  Human-aware motion reshaping using dynamical systems , 2017, Pattern Recognit. Lett..

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

[3]  A. Billard,et al.  Learning Stable Nonlinear Dynamical Systems With Gaussian Mixture Models , 2011, IEEE Transactions on Robotics.

[4]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[5]  Dongheui Lee,et al.  Incremental kinesthetic teaching of motion primitives using the motion refinement tube , 2011, Auton. Robots.

[6]  Aude Billard,et al.  Learning control Lyapunov function to ensure stability of dynamical system-based robot reaching motions , 2014, Robotics Auton. Syst..

[7]  Aude Billard,et al.  BM: An iterative algorithm to learn stable non-linear dynamical systems with Gaussian mixture models , 2010, 2010 IEEE International Conference on Robotics and Automation.

[8]  P. Sanguansat.  Multiple Multidimensional Sequence Alignment Using Generalized Dynamic Time Warping , 2012 .

[9]  Matteo Saveriano,et al.  Point cloud based dynamical system modulation for reactive avoidance of convex and concave obstacles , 2013, 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[10]  Jun Nakanishi,et al.  Dynamical Movement Primitives: Learning Attractor Models for Motor Behaviors , 2013, Neural Computation.

[11]  Matteo Saveriano,et al.  Distance based dynamical system modulation for reactive avoidance of moving obstacles , 2014, 2014 IEEE International Conference on Robotics and Automation (ICRA).

[12]  Stefan Schaal,et al.  Robot Programming by Demonstration , 2009, Springer Handbook of Robotics.

[13]  Klaus Neumann,et al.  Learning robot motions with stable dynamical systems under diffeomorphic transformations , 2015, Robotics Auton. Syst..

[14]  Aude Billard,et al.  On Learning, Representing, and Generalizing a Task in a Humanoid Robot , 2007, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[15]  Nicolas Perrin,et al.  Fast diffeomorphic matching to learn globally asymptotically stable nonlinear dynamical systems , 2016, Syst. Control. Lett..

[16]  Chee Kheong Siew,et al.  Extreme learning machine: Theory and applications , 2006, Neurocomputing.

[17]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Nonlinear Systems Analyzing stability differentially leads to a new perspective on nonlinear dynamic systems , 1999 .

[18]  Ashwin P. Dani,et al.  Learning Contracting Nonlinear Dynamics From Human Demonstration for Robot Motion Planning , 2015, HRI 2015.

[19]  Jean-Jacques E. Slotine,et al.  On partial contraction analysis for coupled nonlinear oscillators , 2004, Biological Cybernetics.

[20]  Aude Billard,et al.  Learning Stable Nonlinear Dynamical Systems With Gaussian Mixture Models , 2011, IEEE Transactions on Robotics.

[21]  Matteo Saveriano,et al.  Incremental kinesthetic teaching of end-effector and null-space motion primitives , 2015, 2015 IEEE International Conference on Robotics and Automation (ICRA).

[22]  Klaus Neumann,et al.  Neural learning of vector fields for encoding stable dynamical systems , 2014, Neurocomputing.