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The aim of this paper is to describe a coupled oscillator model for Bayesian inference. The coupled oscillator model comprises of a large number of oscillators with meanfield coupling. The collective dynamics of the oscillators are used to solve an inference problem: the empirical distribution of the population encodes a 'belief state' (posterior distribution) that is continuously updated based on noisy measurements. In effect, the coupled oscillator model works as a particle filter.The framework is described here with the aid of a model problem involving estimation of a walking gait cycle. For this problem, the coupled oscillator particle filter is developed, and demonstrated on experimental data taken from an Ankle-foot Orthosis (AFO) device.

This paper presents a methodology for state estimation of coupled oscillators from noisy observations. The methodology is comprised of two parts: modeling and estimation. The objective of the modeling is to express dynamics in terms of the so-called phase variables. For nonlinear estimation, a coupled-oscillator feedback particle filter is introduced.The filter is based on the construction of a large population of oscillators with mean-field coupling. The empirical distribution of the population encodes the posterior distribution of the phase variables. The methodology is illustrated with two numerical examples. I. INTRODUCTIONThis paper is concerned with the problem of estimating phases of coupled oscillators from noisy observation data. The motivation comes from biology where rhythmicity underlies several important behaviors -e.g., periodic gaits exhibited by animals during locomotion (walking, hopping, running etc); cf., [5]. The importance of the phase estimation problem is discussed in [13] and in several references therein. An important motivation for us comes from the problem of controlling rhythmic behaviors -e.g., by using feedback control to stabilize walking gaits [4]. For such applications, estimation is seen as an intermediate step towards control.The methodology proposed here comprises of two parts: 1. Modeling: The objective of the modeling is to obtain a reduced order model of the dynamics. The reduced order model comprises of weakly coupled oscillators, expressed in their phase variables; cf., [7], [8], [1].As an example, consider a dynamical system with two weakly coupled oscillators: The reduced order model has two phase variables, θ 1 (t) ∈ [0, 2π) and θ 2 (t) ∈ [0, 2π), representing the instantaneous phase of the two oscillators at time t. The dynamics evolve according to,

This paper presents a bio-inspired central pattern generator (CPG) architecture for feedback control of rhythmic behavior. The CPG circuit is realized as a coupled oscillator feedback particle filter. The collective dynamics of the filter are used to approximate a posterior distribution that is used to construct the optimal control input. The architecture is illustrated with the aid of a model problem involving pumping up of a swing, modeled as a parametrically forced pendulum. For this problem, the coupled oscillator particle filter is designed and its control performance demonstrated in a simulation environment.