This paper develops a new filtering approach for state estimation in polynomial systems corrupted by arbitrary noise, which commonly arise in robotics. We first consider a batch setup where we perform state estimation using all data collected from the initial to the current time. We formulate the batch state estimation problem as a Polynomial Optimization Problem (POP) and relax the assumption of Gaussian noise by specifying a finite number of moments of the noise. We solve the resulting POP using a moment relaxation and prove that under suitable conditions on the rank of the relaxation, (i) we can extract a provably optimal estimate from the moment relaxation, and (ii) we can obtain a belief representation from the dual (sum-of-squares) relaxation. We then turn our attention to the filtering setup and apply similar insights to develop a GMKF for recursive state estimation in polynomial systems with arbitrary noise. The GMKF formulates the prediction and update steps as POPs and solves them using moment relaxations, carrying over a possibly non-Gaussian belief. In the linear-Gaussian case, GMKF reduces to the standard Kalman Filter. We demonstrate that GMKF performs well under highly non-Gaussian noise and outperforms common alternatives, including the Extended and Unscented Kalman Filter, and their variants on matrix Lie group.

翻译：摘要：本文针对机器人领域常见的受任意噪声污染的多项式系统，提出了一种新的状态估计滤波方法。我们首先考虑批处理框架，利用从初始时刻到当前时刻采集的所有数据进行状态估计。通过将批处理状态估计问题形式化为多项式优化问题（POP），并利用噪声的有限阶矩代替高斯噪声假设，我们放松了传统的高斯噪声约束。采用矩松弛法求解该多项式优化问题，并证明在秩条件满足时：(i) 可从矩松弛中提取可证明最优的估计值；(ii) 可从对偶（平方和）松弛中获得置信度表示。随后，我们将研究重点转向滤波框架，基于类似思路提出广义矩卡尔曼滤波器（GMKF），用于实现任意噪声多项式系统的递归状态估计。GMKF将预测与更新步骤建模为多项式优化问题，通过矩松弛进行求解，并能够传递非高斯置信度。在线性高斯情形下，GMKF退化为标准卡尔曼滤波器。实验表明，GMKF在强非高斯噪声环境下表现优异，性能优于扩展卡尔曼滤波器、无迹卡尔曼滤波器及其矩阵李群变体等常用替代方法。