Feedback particle filter on matrix lie groups

Abstract

This paper is concerned with the problem of continuous-time nonlinear filtering for stochastic processes on a connected matrix Lie group. The main contribution of this paper is to derive the feedback particle filter (FPF) algorithm for this problem. In its general form, the FPF is shown to provide a coordinate-free description of the filter that automatically satisfies the geometric constraints of the manifold. The particle dynamics are encapsulated in a Stratonovich stochastic differential equation that retains the feedback structure of the original (Euclidean) FPF. The implementation of the filter requires a solution of a Poisson equation on the Lie group, and two numerical algorithms are described for this purpose. As an example, the FPF is applied to the problem of attitude estimation -a nonlinear filtering problem on the Lie group SO(3). The formulae of the filter are described using both the rotation matrix and the quaternion coordinates. Comparisons are also provided between the FPF and some popular algorithms for attitude estimation, namely the multiplicative EKF, the unscented quaternion estimator, the left invariant EKF, and the invariant ensemble Kalman filter. Numerical simulations are presented to illustrate the comparisons.

Kalman Filter and Its Modern Extensions for the Continuous-Time Nonlinear Filtering Problem

“…For sdes on a manifold, it is well known that the Stratonovich form is invariant to coordinate transformations (i.e., intrinsic) while the Ito form is not. A more in-depth discussion of the FPF for Lie groups appears in [30]. Table 5 includes a list of symbols used for the FPF.…”

“…In terms of the solution 1 The extension to multi-valued observation is straightforward and appears in [31]. 2 Although this paper is limited to R d , the proposed algorithm is applicable to nonlinear filtering problems on differential manifolds, e.g., matrix Lie groups (For an intrinsic form of the Poisson equation, see [30]). For domains with boundary, the pde is accompanied by a Neumann boundary φ (x) of (10), the gain function at time t is given by…”

Interacting particle filters for simultaneous state and parameter estimation

“…The feedback particle filter (FPF) [20,21] is one of these, so called, particle flow filters, which is characterized by the modified SDE…”

Attitude estimation with feedback particle filter

“…The FPF algorithm was originally proposed in the Euclidean setting of R n [42]. In a recent paper from our group, the FPF was extended to filtering on compact matrix Lie groups [45]. The FPF is an intrinsic algorithm: The particle dynamics, expressed in their Stratonovich form, respect the geometric constraints of the manifold.…”