Bayesian estimation is a vital tool in robotics as it allows systems to update the belief of the robot state using incomplete information from noisy sensors. To render the state estimation problem tractable, many systems assume that the motion and measurement noise, as well as the state distribution, are all unimodal and Gaussian. However, there are numerous scenarios and systems that do not comply with these assumptions. Existing non-parametric filters that are used to model multimodal distributions have drawbacks that limit their ability to represent a diverse set of distributions. In this paper, we introduce a novel approach to nonparametric Bayesian filtering to cope with multimodal distributions using harmonic exponential distributions. This approach leverages two key insights of harmonic exponential distributions: a) the product of two distributions can be expressed as the element-wise addition of their log-likelihood Fourier coefficients, and b) the convolution of two distributions can be efficiently computed as the tensor product of their Fourier coefficients. These observations enable the development of an efficient and exact solution to the Bayes filter up to the band limit of a Fourier transform. We demonstrate our filter's superior performance compared with established nonparametric filtering methods across a range of simulated and real-world localization tasks.

翻译：贝叶斯估计是机器人学中的关键工具，它使系统能够利用来自噪声传感器的不完整信息更新机器人状态的置信度。为使状态估计问题易于处理，许多系统假设运动和测量噪声以及状态分布均为单峰高斯分布。然而，存在大量场景和系统不满足这些假设。现有用于建模多峰分布的非参数滤波器存在局限性，制约了其表征多样化分布的能力。本文提出一种处理多峰分布的非参数贝叶斯滤波新方法，该方法基于谐波指数分布。本方法利用了谐波指数分布的两个关键特性：a) 两个分布的乘积可表示为它们对数似然傅里叶系数的逐元加法；b) 两个分布的卷积可通过其傅里叶系数的张量积高效计算。这些特性使得在傅里叶变换的带宽限制内，能够推导出贝叶斯滤波器的高效精确解。我们通过一系列仿真和实际定位任务证明，相较于现有非参数滤波方法，本滤波器具有更优越的性能。