Revisiting Continuous-Time Trajectory Estimation via Gaussian Processes and the Magnus Expansion

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.