Learning dynamical systems with bifurcations

Abstract Trajectory planning through dynamical systems (DS) provides robust control for robots and has found numerous applications from locomotion to manipulation. However, to date, DS for controlling rhythmic patterns are distinct from DS used to control point to point motion and current approaches switch at run time across these to enable multiple behaviors. This switching can be brittle and subject to instabilities. We present an approach to embed cyclic and point to point dynamics in a single DS. We offer a method to learn the parameters of complete DS through a two-step optimization. By exploiting Hopf bifurcations, we can explicitly and smoothly transit across periodic and non-periodic phases, linear and nonlinear limit cycles, and non-periodic phases, in addition to changing the equilibrium’s location and the limit cycle’s amplitude. We use diffeomorphism and learn a mapping to modify the learned limit cycle to generate nonlinear limit cycles. The approach is validated with a real 7 DOF KUKA LWR 4+ manipulator to control wiping and with a humanoid robot in simulation.

[1]  Pei Yu,et al.  Hopf Bifurcation and Normal Form Computation , 2012 .

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

[3]  A. Ijspeert,et al.  A general family of morphed nonlinear phase oscillators with arbitrary limit cycle shape , 2013 .

[4]  Taishin Nomura,et al.  Time series-based bifurcation diagram reconstruction , 1999 .

[5]  Aude Billard,et al.  Passive Interaction Control With Dynamical Systems , 2016, IEEE Robotics and Automation Letters.

[6]  Jan Peters,et al.  Learning motor primitives for robotics , 2009, 2009 IEEE International Conference on Robotics and Automation.

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

[8]  Jun Tani,et al.  Dynamic and interactive generation of object handling behaviors by a small humanoid robot using a dynamic neural network model , 2006, Neural Networks.

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

[10]  Kevin P. Murphy,et al.  Machine learning - a probabilistic perspective , 2012, Adaptive computation and machine learning series.

[11]  Ryuji Tokunaga,et al.  Reconstructing bifurcation diagrams only from time-waveforms , 1994 .

[12]  Aude Billard,et al.  From Human Physical Interaction To Online Motion Adaptation Using Parameterized Dynamical Systems , 2018, 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).

[13]  J N Kutz,et al.  Model selection for dynamical systems via sparse regression and information criteria , 2017, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[14]  Henri Poincaré,et al.  méthodes nouvelles de la mécanique céleste , 1892 .

[15]  Jun Morimoto,et al.  Accelerating synchronization of movement primitives: Dual-arm discrete-periodic motion of a humanoid robot , 2015, 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).

[16]  Oliver Kroemer,et al.  A Review of Robot Learning for Manipulation: Challenges, Representations, and Algorithms , 2019, J. Mach. Learn. Res..

[17]  Byron Boots,et al.  Euclideanizing Flows: Diffeomorphic Reduction for Learning Stable Dynamical Systems , 2020, L4DC.

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

[19]  Ricardo Campos,et al.  Hexapod locomotion: A nonlinear dynamical systems approach , 2010, IECON 2010 - 36th Annual Conference on IEEE Industrial Electronics Society.

[20]  Henk Nijmeijer,et al.  Robot Programming by Demonstration , 2010, SIMPAR.

[21]  Stefan Schaal,et al.  Encoding of periodic and their transient motions by a single dynamic movement primitive , 2012, 2012 12th IEEE-RAS International Conference on Humanoid Robots (Humanoids 2012).

[22]  Yasuhiro Fukuoka,et al.  Adaptive Dynamic Walking of a Quadruped Robot on Irregular Terrain Based on Biological Concepts , 2003, Int. J. Robotics Res..

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

[24]  S. Brunton,et al.  Discovering governing equations from data by sparse identification of nonlinear dynamical systems , 2015, Proceedings of the National Academy of Sciences.

[25]  G. Metta,et al.  A modular bio-inspired architecture for movement generation for the infant-like robot iCub , 2008, 2008 2nd IEEE RAS & EMBS International Conference on Biomedical Robotics and Biomechatronics.

[26]  Cong Wang,et al.  Dissipativity-based switching adaptive control , 2011, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.