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.

翻译：连续时间状态估计已被证明是一种有效手段，其优势体现在：(i) 处理异步与高频测量数据，(ii) 为估计结果引入平滑性，(iii) 支持在测量时刻之外的时间点进行事后查询，以及 (iv) 解决与“边扫描边移动”传感器相关的特定可观测性问题。一种常用的连续时间轨迹表示方法是采用高斯过程先验，该先验的均值与协方差函数由白噪声驱动的线性时变随机微分方程生成。当状态包含李群元素时，先前的研究通常采用由多个局部高斯过程拼凑而成的方法，每个局部过程使用线性时不变随机微分方程核；这种方法虽在实践中有效，但缺乏理论上的优雅性。本文重新审视了基于完整线性时变高斯过程的连续时间轨迹估计方法，通过马格努斯展开推导出李群上的全局高斯过程先验，从而提供了一种更为优雅且通用的解决方案。我们对两种方法进行了数值比较，并讨论了它们各自的优缺点。