On-line learning and modulation of periodic movements with nonlinear dynamical systems

Abstract

The paper presents a two-layered system for (1) learning and encoding a periodic signal without any knowledge on its frequency and waveform, and (2) modulating the learned periodic trajectory in response to external events. The system is used to learn periodic tasks on a humanoid HOAP-2 robot. The first layer of the system is a dynamical system responsible for extracting the fundamental frequency of the input signal, based on adaptive frequency oscillators. The second layer is a dynamical system responsible for learning of the waveform based on a built-in learning algorithm. By combining the two dynamical systems into one system we can rapidly teach new trajectories to robots without any knowledge of the frequency of the demonstration signal. The system extracts and learns only one period of the demonstration signal. Furthermore, the trajectories are robust to perturbations and can be modulated to cope with a dynamic environment. The system is computationally inexpensive, works on-line for any periodic signal, requires no additional signal processing to determine the frequency of the input signal and can be applied in parallel to multiple dimensions. Additionally, it can adapt to changes in A. Gams ( ) · J. Lenarčič "Jožef Stefan" Institute, frequency and shape, e.g. to non-stationary signals, such as hand-generated signals and human demonstrations.

Performing Periodic Tasks: On-Line Learning, Adaptation and Synchronization with External Signals

“…The authors modify non-linear oscillators or pseudo-oscillators with a learning rule, which allows the modified oscillators to learn the frequency of the input signal. The approach works for different oscillators, from a simple phase oscillator (Gams et al, 2009), the Hopf oscillator, the Fitzhugh-Nagumo oscillator, etc. .…”

“…. Combining several adaptive frequency oscillators in a feedback loop allows extraction of several frequency components (Buchli et al, 2008;Gams et al, 2009). Applications vary from bipedal walking to frequency tuning of a hopping robot (Buchli et al, 2005).…”

“…Such feedback structures can be used as a whole imitation system that both extracts the frequency and learns the waveform of the input signal. Not many approaches exist that combine both frequency extraction and waveform learning in imitation systems (Gams et al, 2009;Ijspeert, 2008b). One of them is a two-layered imitation system, which can be used for extracting the frequency of the input signal in the first layer and learning its waveform in the second layer, which is the basis for this chapter.…”

Adaptive oscillators with human-in-the-loop: Proof of concept for assistance and rehabilitation

“…without delay with respect to the measured output. In counterpart, this filtering oscillator is designed to work only for quasi-sinusoidal signals, and would not work properly with other profiles, although adaptations can be found in [17], [18].…”

“…In the present paper, we used µ = 1 (such that the oscillator's intrinsic amplitude equals one) and γ = 8 (like in [13], [18]). The learning parameter ν determines the speed of the phase synchronization with respect to F(t).…”