Multi-dimensional feedback particle filter for coupled oscillators

Abstract

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,

Control with rhythms: A CPG architecture for pumping a swing

“…The controlled form of the feedback particle filter is given by, (14) where U t is the common control input (the Stratonovich notation is used to simplify the presentation). To adapt the methodology described in Sec.…”

“…The optimal control is shown to be a function of certain averages taken over the population, {Θ i t : 1 ≤ i ≤ N}. The methodology of this paper is a synthesis of several papers from our research group at the University of Illinois: the feedback particle filtering methodology for diffusion appears in [17], [16], [18]; its application to the problem of phase estimation is described in [13], [14]; a feedback particle filter-based approach to optimal control of partially observed diffusions is the subject of [11]. In the present paper, we bring together these research threads to propose a CPG architecture for optimal control of rhythmic behavior.…”

Q-learning for POMDP: An application to learning locomotion gaits

“…Coupled oscillator feedback particle filter: We construct a nonlinear filtering algorithm to approximate the posterior distribution of the phase variable θ (t), given time history of sensory measurements. The coupled oscillator feedback particle filter (FPF) is comprised of a system of N coupled oscillators [23], [21]. The empirical distribution of the oscillators is used to approximate the posterior distribution.…”

A coupled oscillators-based control architecture for locomotory gaits

“…The methodology of this paper is a synthesis of several papers from our research group at the University of Illinois: the feedback particle filtering methodology for diffusion appears in [30], [29], [28]; its application to the problem of phase estimation is described in [24], [26]; a feedback particle filter-based approach to optimal control of partially observed diffusions is the subject of [21], [25]. In the present paper, we bring together these research threads to propose a CPG architecture for control of locomotion.…”

“…Estimation using CPG: The estimation problem is to construct a nonlinear filter for estimating the instantaneous phase θ (t), given a time-history of noisy measurements. The problem is solved numerically by constructing a coupled oscillator feedback particle filter; cf., [24], [26]. The coupled oscillator filter comprises of N stochastic processes {θ i (t) : 1 ≤ i ≤ N}, where the value θ i (t) ∈ [0, 2π] is the state of the i th oscillator at time t. For large N, the empirical distribution of the population approximates the posterior distribution of the θ (t).…”