Frequency analysis with coupled nonlinear oscillators

We present a method to obtain the frequency spectrum of a signal with a nonlinear dynamical system. The dynamical system is composed of a pool of adaptive frequency oscillators with negative mean-field coupling. For the frequency analysis, the synchronization and adaptation properties of the component oscillators are exploited. The frequency spectrum of the signal is reflected in the statistics of the intrinsic frequencies of the oscillators. The frequency analysis is completely embedded in the dynamics of the system. Thus, no pre-processing or additional parameters, such as time windows, are needed. Representative results of the numerical integration of the system are presented. It is shown, that the oscillators tune to the correct frequencies for both discrete and continuous spectra. Due to its dynamic nature the system is also capable to track non-stationary spectra. Further, we show that the system can be modeled in a probabilistic manner by means of a nonlinear Fokker–Planck equation. The probabilistic treatment is in good agreement with the numerical results, and provides a useful tool to understand the underlying mechanisms leading to convergence.

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Jonas Buchli, Auke Ijspeert, Ludovic Righetti
2008
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https://infoscience.epfl.ch/record/118459?ln=fr

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Concepts associés (16)

Concepts associés

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Publications associées (3)

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Adaptive Frequency Oscillators and Applications

Jonas Buchli, Auke Ijspeert, Ludovic Righetti

In this contribution we present a generic mechanism to transform an oscillator into an adaptive frequency oscillator, which can then dynamically adapt its parameters to learn the frequency of any periodic driving signal. Adaptation is done in a dynamic way: it is part of the dynamical system and not an offline process. This mechanism goes beyond entrainment since it works for any initial frequencies and the learned frequency stays encoded in the system even if the driving signal disappears. Interestingly, this mechanism can easily be applied to a large class of oscillators from harmonic oscillators to relaxation types and strange attractors. Several practical applications of this mechanism are then presented, ranging from adaptive control of compliant robots to frequency analysis of signals and construction of limit cycles of arbitrary shape.

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Dynamic hebbian learning in adaptive frequency oscillators

Jonas Buchli, Auke Ijspeert, Ludovic Righetti

Nonlinear oscillators are widely used in biology, physics and engineering for modeling and control. They are interesting because of their synchronization properties when coupled to other dynamical systems. In this paper, we propose a learning rule for oscillators which adapts their frequency to the frequency of any periodic or pseudo-periodic input signal. Learning is done in a dynamic way: it is part of the dynamical system and not an offline process. An interesting property of our model is that it is easily generalizable to a large class of oscillators, from phase oscillators to relaxation oscillators and strange attractors with a generic learning rule. One major feature of our learning rule is that the oscillators constructed can adapt their frequency without any signal processing or the need to specify a time window or similar free parameters. All the processing is embedded in the dynamics of the adaptive oscillator. The convergence of the learning is proved for the Hopf oscillator, then numerical experiments are carried out to explore the learning capabilities of the system. Finally, we generalize the learning rule to non-harmonic oscillators like relaxation oscillators and strange attractors.

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CPG-based prostheses control

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In this thesis we describe a strategy to control robotic knees and ankles. A dynamical system is used to generate a position trajectory to control a servo motor replacing the missing joint. The dynamical system consists in a pool of coupled oscillators modeling a central pattern generator (CPG). As a first step, anthropometric trajectories of the knee and ankle are learned by the system through the convergence of the oscillators to the specific frequencies, corresponding amplitudes and phase relations. The same system is then used to play back these trajectories. As a sensory feedback to trigger the playback we use one adaptive frequency oscillator to synchronized with the acceleration from the thigh. We use a bipedal model in a physics-based robot simulation environment to test the proposed system. Finally we present a simple hardware implementation of this system on the Agonist-Antagonist Active Knee prototype.

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