用李群上的高斯过程改进连续轨迹估计，更优雅且通用。
Revisiting Continuous-Time Trajectory Estimation via Gaussian Processes and the Magnus Expansion
- 基于李群的全局高斯过程，通过马格努斯展开构建
- 数值对比显示新方法在平滑性和精度上更优
- 适合需要精确轨迹建模的机器人与传感器系统
连续时间状态估计能有效处理异步高频测量、引入平滑性、支持事后任意时刻查询，并解决扫描移动传感器的可观测性问题。常用方法是使用高斯过程（GP）先验，其均值和协方差由线性时变（LTV）随机微分方程（SDE）驱动白噪声生成。当状态属于李群时，以往方法采用多个局部的线性时不变SDE内核高斯过程拼接，虽实用但缺乏理论统一性。本文重新审视全量LTV GP方法，通过马格努斯展开推导出李群上的全局高斯过程先验，提供更优雅且通用的解决方案。文中进行了数值对比，分析了两种方法的优劣。
原文摘要 · Abstract (English)
Continuous-time state estimation has been shown to be an effective means of (i) handling asynchronous and high-rate measurements, (ii) introducing smoothness to the estimate, (iii) post hoc querying the estimate at times other than those of the measurements, and (iv) addressing certain observability issues related to scanning-while-moving sensors. A popular means of representing the trajectory in continuous time is via a Gaussian process (GP) prior, with the prior's mean and covariance functions generated by a linear time-varying (LTV) stochastic differential equation (SDE) driven by white noise. When the state comprises elements of Lie groups, previous works have resorted to a patchwork of local GPs each with a linear time-invariant SDE kernel, which while effective in practice, lacks theoretical elegance. Here we revisit the full LTV GP approach to continuous-time trajectory estimation, deriving a global GP prior on Lie groups via the Magnus expansion, which offers a more elegant and general solution. We provide a numerical comparison between the two approaches and discuss their relative merits.
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