Continuous-time Trajectory Estimation: A Comparative Study Between Gaussian Process and Spline-based Approaches

TL;DR

This work directly compares continuous-time trajectory estimation via Gaussian Process (GP) regression and spline-based methods, across linear and SE(3) Lie-group scenarios, using unified motion priors. It develops motion priors that apply to both GP and spline representations and evaluates them with linear simulations, simulated camera-IMU fusion, and real hardware data. The results show that, given the same measurements and motion models, GP and spline approaches achieve similar trajectory accuracy and comparable solve times when the spline differentiability matches the GP model; high-order splines can incur substantial computational cost without improving accuracy. The study also highlights how interpolation structure, sparsity, and measurement-timing alignment influence computation, offering guidance on method selection for online, asynchronous sensor fusion tasks.

Abstract

Continuous-time trajectory estimation is an attractive alternative to discrete-time batch estimation due to the ability to incorporate high-frequency measurements from asynchronous sensors while keeping the number of optimization parameters bounded. Two types of continuous-time estimation have become prevalent in the literature: Gaussian process regression and spline-based estimation. In this paper, we present a direct comparison between these two methods. We first compare them using a simple linear system, and then compare them in a camera and IMU sensor fusion scenario on SE(3) in both simulation and hardware. Our results show that if the same measurements and motion model are used, the two methods achieve similar trajectory accuracy. In addition, if the spline order is chosen so that the degree-of-differentiability of the two trajectory representations match, then they achieve similar solve times as well.

Figures (21)

• Figure 1: Example factor graph for the continuous-time estimation problem \ref{['eq:problem']}. The interpolated states are not directly optimized, rather they are computed from the estimation parameters using the state interpolation function \ref{['eq:interp']}.
• Figure 2: An example factor graph for a GP-based estimator.
• Figure 3: One-dimensional spline example, with $k = 3$.
• Figure 4: Example factor graph for a spline-based estimator with $k=3$. Note the relative sparsity between this graph and the general graph in Figure \ref{['fig:ct_factor_graph']}.
• Figure 5: Relevant coordinate frames for the camera and IMU scenario.