This paper is concerned with the problem of continuous-time nonlinear filtering of stochastic processes evolving on connected Riemannian manifolds without boundary. 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 an intrinsic 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 manifold, and a numerical algorithm is described for this purpose. For the special case when the manifold is a matrix Lie group, explicit formulae for the filter are derived, using the matrix coordinates. Filters for two example problems are presented: the attitude estimation problem on SO(3) and the robot localization problem in SE(3). Comparisons are also provided between the FPF and popular algorithms for attitude estimation, namely the multiplicative extended Kalman filter (EKF), the invariant EKF, the unscented quaternion estimator, the invariant ensemble Kalman filter, and the bootstrap particle filter. Numerical simulations are presented to illustrate these comparisons.

中文翻译：

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黎曼流形和矩阵李群上的反馈粒子滤波器


本文研究的是在无边界连通黎曼流形上演化的随机过程的连续时间非线性滤波问题。本文的主要贡献是推导了针对该问题的反馈粒子滤波器（FPF）算法。在其一般形式中，FPF 提供了滤波器的内在描述，自动满足流形的几何约束。粒子动力学被封装在 Stratonovich 随机微分方程中，该方程保留了原始（欧几里得）FPF 的反馈结构。滤波器的实现需要求解流形上的泊松方程，为此描述了一种数值算法。对于流形是矩阵李群的特殊情况，使用矩阵坐标导出滤波器的显式公式。提出了两个示例问题的滤波器：SO(3) 上的姿态估计问题和 SE(3) 上的机器人定位问题。还提供了 FPF 与流行的姿态估计算法之间的比较，即乘法扩展卡尔曼滤波器 (EKF)、不变 EKF、无迹四元数估计器、不变集成卡尔曼滤波器和自举粒子滤波器。通过数值模拟来说明这些比较。