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

2026-01-06

arXiv.org

AI Summary

This work addresses the lack of a globally consistent and theoretically rigorous Gaussian process (GP) framework for continuous-time trajectory estimation in Lie group state spaces. It introduces, for the first time, the Magnus expansion to construct a global linear time-varying Gaussian process (LTV-GP) prior on Lie groups. By integrating the geometric structure of Lie groups, linear time-varying stochastic differential equations, and Magnus series, the proposed approach enables a unified and smooth representation of continuous trajectories under asynchronous, high-frequency observations. In contrast to conventional methods that piece together local linear time-invariant GP approximations, this framework offers superior theoretical coherence and expressive power. Numerical experiments demonstrate its significant advantages in both estimation accuracy and robustness.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

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.

Problem

• continuous-time trajectory estimation
• Gaussian processes
• Lie groups
• linear time-varying SDE
• Magnus expansion

Innovation

• Gaussian Processes
• Magnus Expansion
• Continuous-Time Trajectory Estimation
• Lie Groups
• Linear Time-Varying Systems