A Physically-Consistent Bayesian Non-Parametric Mixture Model for Dynamical System Learning

Abstract: We propose a physically-consistent Bayesian non-parametric approach for fitting Gaussian Mixture Models (GMM) to trajectory data. Physicalconsistency of the GMM is ensured by imposing a prior on the component assignments biased by a novel similarity metric that leverages locality and directionality. The resulting GMM is then used to learn globally asymptotically stable Dynamical Systems (DS) via a Linear Parameter Varying (LPV) re-formulation. The proposed DS learning scheme accurately encodes challenging nonlinear motions automatically. Finally, a data-efficient incremental learning framework is introduced that encodes a DS from batches of trajectories, while preserving global stability. Our contributions are validated on 2D datasets and a variety of tasks that involve single-target complex motions with a KUKA LWR 4+ robot arm.

[1]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[2]  Peter I. Frazier,et al.  Distance dependent Chinese restaurant processes , 2009, ICML.

[3]  Pierre Apkarian,et al.  Self-scheduled H∞ control of linear parameter-varying systems: a design example , 1995, Autom..

[4]  Ashwin P. Dani,et al.  Learning Partially Contracting Dynamical Systems from Demonstrations , 2017, CoRL.

[5]  Michael Stingl,et al.  PENLAB: A MATLAB solver for nonlinear semidefinite optimization , 2013 .

[6]  Seyed Sina,et al.  Compliant control of Uni/ Multi- robotic arms with dynamical systems , 2018 .

[7]  Aude Billard,et al.  Learning Stable Task Sequences from Demonstration with Linear Parameter Varying Systems and Hidden Markov Models , 2017, CoRL.

[8]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

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

[10]  Caroline Blocher,et al.  Learning stable dynamical systems using contraction theory , 2017, 2017 14th International Conference on Ubiquitous Robots and Ambient Intelligence (URAI).

[11]  Aude Billard,et al.  A Dynamical System Approach for Softly Catching a Flying Object: Theory and Experiment , 2016, IEEE Transactions on Robotics.

[12]  Brett Browning,et al.  A survey of robot learning from demonstration , 2009, Robotics Auton. Syst..

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

[14]  Oussama Khatib,et al.  Learning potential functions from human demonstrations with encapsulated dynamic and compliant behaviors , 2017, Auton. Robots.

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

[16]  Johan Löfberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004 .

[17]  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).

[18]  Aude Billard,et al.  Learning Augmented Joint-Space Task-Oriented Dynamical Systems: A Linear Parameter Varying and Synergetic Control Approach , 2018, IEEE Robotics and Automation Letters.

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

[20]  Dana Kulic,et al.  Learning Action Primitives , 2011, Visual Analysis of Humans.

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

[22]  Oliver Brock,et al.  Patterns for Learning with Side Information , 2015, 1511.06429.

[23]  A. Billard,et al.  Learning the Nonlinear Multivariate Dynamics of Motion of Robotic Manipulators , 2009 .

[24]  Rajesh P. N. Rao,et al.  Robotic imitation from human motion capture using Gaussian processes , 2005, 5th IEEE-RAS International Conference on Humanoid Robots, 2005..

[25]  Stefan Schaal,et al.  Scalable Techniques from Nonparametric Statistics for Real Time Robot Learning , 2002, Applied Intelligence.

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

[27]  Hongbin Wang,et al.  Highly efficient incremental estimation of Gaussian mixture models for online data stream clustering , 2005, SPIE Defense + Commercial Sensing.