BM: An iterative algorithm to learn stable non-linear dynamical systems with Gaussian mixture models

By Seyed Mohammad Khansari-Zadeh, Aude Gemma Billard. 2010

73 citations recorded by the source corpus.

We model the dynamics of non-linear point-to-point robot motions as a time-independent system described by an autonomous dynamical system (DS). We propose an iterative algorithm to estimate the form of the DS through a mixture of Gaussian distributions. We prove that the resulting model is asymptotically stable at the target. We validate the accuracy of the model on a library of 2D human motions and to learn a control policy through human demonstrations for two multi-degrees of freedom robots. We show the real-time adaptation to perturbations of the learned model when controlling the two kinematically-driven robots.

Subfields: Artificial Intelligence, Control and Systems Engineering, Probabilistic & Bayesian Methods

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Citations in the corpus

End-to-End Training of Deep Visuomotor Policies (2015, background): “The Gaussian mixture model provides a good approximation for such piecewise linear systems, with each mixture element corresponding to a different linear mode (Khansari-Zadeh and Billard, 2010).”
An Incremental Self-Organizing Architecture for Sensorimotor Learning and Prediction (2017, background): “Therefore, Khansari-Zadeh and Billard [37] suggested a learning procedure capable of modeling demonstrated motion sequences”
Learning contact-rich manipulation skills with guided policy search (2015, background): “As discussed in prior work [29], GMMs are well suited for modeling piecewise linear dynamics, which are a good approximation for articulated systems (such as robots) that contact objects in the environment.”
Neural learning of stable dynamical systems based on data-driven Lyapunov candidates (2013, background): “Recently, several studies emphasized that stability plays an important role for such tasks besides the undoubted modeling power of dynamical systems [7].”
Imitation Learning of Positional and Force Skills Demonstrated via Kinesthetic Teaching and Haptic Input (2011, background): “The superposition of basis vector fields is determined in (1) by an implicit time dependency, but other approaches using spatial and/or sequential information could also be used [23, 24].”
Learning Stable Nonlinear Dynamical Systems With Gaussian Mixture Models (2011, background): “The results from both approaches will be evaluated and compared in Section VI-A.”
Imitation learning of globally stable non-linear point-to-point robot motions using nonlinear programming (2010, methodology): “We also further examine SEDS in a library of 20 different human handwriting motions recorded using a Tablet PC, and compare it against our previous method Binary Merging (BM) [9] and those of four alternative methods GMR, LWPR, GPR, and DMP6.”
Learning and Reproduction of Gestures by Imitation (2010, background): “Analysis and solutions to the problem of stabilizing the first order system can be found in [25].”
Learning Neural Network Policies with Guided Policy Search under Unknown Dynamics (2014, background): “The dynamics of a robot can be reasonably approximated with such piecewise linear functions [9], and they are well suited for contacts, which are approximately piecewise linear with a hard boundary.”
Learning control Lyapunov function to ensure stability of dynamical system-based robot reaching motions (2014, methodology): “time-invariant) DS, in [50] we proposed an alternative approach, called Binary Merging (BM), to construct a locally stable estimate of nonlinear autonomous DS as a mixture of Gaussian functions.”
Robot motor skill coordination with EM-based Reinforcement Learning (2010, background): “The superposition of basis force fields is determined in (1) by an implicit time dependency, but other approaches using spatial and/or sequential information could also be used [25], [26].”
Learning robot motions with stable dynamical systems under diffeomorphic transformations (2015, background): “The stable estimator of dynamical systems (SEDS) [13] is a advanced version of the binary merging algorithm [9] and learns a dynamical system by means of a Gaussian mixture model”
Optimism-driven exploration for nonlinear systems (2015, methodology): “other probabilistic models are also suitable for use with our method, though the DPMM provides a good balance of flexibility and efficiency, particularly for physical systems that are well approximated by the piecewise linear dynamics that are induced by a Gaussian mixture [20].”