Continuous-Time State Estimation Methods in Robotics: A Survey

TL;DR

This survey clarifies how continuous-time state estimation reframes robotics estimation by modeling the state as a time-continuous function, enabling querying at arbitrary times and seamless fusion of asynchronous sensors. It systematically categorizes CT methods into interpolation/integration (LI and numerical), temporal splines, and temporal Gaussian processes (TGP), detailing mathematical foundations, Lie-group extensions, and optimization considerations. The work highlights the most influential literature up to 2024, compares CT approaches, and outlines open problems—knot and state-time selection, certifiability, handling complex dynamics, and benchmarking—while mapping CT methods to a broad set of robotic and cross-domain applications. The findings underscore CT methods’ potential to improve estimation accuracy, robustness, and planning/control integration, suggesting a path toward broader adoption in real-time, resource-constrained platforms. Overall, the survey provides a comprehensive, technically detailed reference that can guide researchers in selecting CT representations and in addressing remaining theoretical and practical challenges.

Abstract

Accurate, efficient, and robust state estimation is more important than ever in robotics as the variety of platforms and complexity of tasks continue to grow. Historically, discrete-time filters and smoothers have been the dominant approach, in which the estimated variables are states at discrete sample times. The paradigm of continuous-time state estimation proposes an alternative strategy by estimating variables that express the state as a continuous function of time, which can be evaluated at any query time. Not only can this benefit downstream tasks such as planning and control, but it also significantly increases estimator performance and flexibility, as well as reduces sensor preprocessing and interfacing complexity. Despite this, continuous-time methods remain underutilized, potentially because they are less well-known within robotics. To remedy this, this work presents a unifying formulation of these methods and the most exhaustive literature review to date, systematically categorizing prior work by methodology, application, state variables, historical context, and theoretical contribution to the field. By surveying splines and Gaussian processes together and contextualizing works from other research domains, this work identifies and analyzes open problems in continuous-time state estimation and suggests new research directions.

Figures (2)

• Figure 1: Example of 2D localization from noisy accelerometer, gyroscope, and range-bearing measurements (shown projected from interpolated poses) using two popular CT methods; B-splines (left), and 'exactly sparse' Gaussian processes (right). The Laplace approximation for the posterior (\ref{['eqn:laplace_approximation']}) is obtained via batch optimization. For the uniform cubic B-spline, the $SE(2)$ control points (diamonds) are estimated, while for the uniform 'constant-jerk' Gaussian process, $SE(2)$ states on the trajectory are estimated. The $3\sigma$ uncertainties for the estimated variables (pink) and the interpolated $SE(2)$ trajectories (blue) are shown. Since the latter method supports efficient covariance interpolation in manifold spaces, a $3\sigma$ uncertainty envelope is shown around the trajectory.
• Figure 2: Uniform vector-space B-splines of different orders. The control points, which are also exactly the points of the degenerate $k=1$ spline, are shown as diamonds. The linear $k = 2$ spline (thin solid line) is the connection between these control points. The quadratic $k = 3$ spline (dashed line) and cubic $k = 4$ spline (solid line) are also shown. Each segment is colored to correspond with the first of the $k$ control points used in its interpolation.