[Paper Review] Continuous-Time State Estimation Methods in Robotics: A Survey

William Henry Fox Talbot, Julian Nubert+6 more|arXiv (Cornell University)|Nov 6, 2024

Fault Detection and Control Systems4 citations

TL;DR

This survey presents a comprehensive unification and systematic review of continuous-time (CT) state estimation methods in robotics, focusing on splines and Gaussian processes to model robot states as continuous functions of time. It demonstrates that CT methods improve estimator performance, reduce preprocessing complexity, and enhance flexibility for planning and control by enabling interpolation at any query time, with a detailed analysis of open problems and future research directions.

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.

Motivation & Objective

• To address the underutilization of continuous-time (CT) state estimation methods in robotics despite their advantages over traditional discrete-time approaches.
• To provide a unified theoretical formulation that integrates splines and Gaussian processes within a common framework for CT state estimation.
• To systematically categorize prior work by methodology, application, state variables, historical context, and theoretical contributions.
• To identify open problems and suggest new research directions for advancing CT estimation in complex robotic systems.
• To support broader adoption of CT methods by clarifying their benefits in performance, robustness, and downstream task integration.

Proposed method

• Formulates continuous-time state estimation as an optimization problem where the state is represented as a smooth function of time, parameterized by control points or hyperparameters.
• Employs temporal splines (e.g., B-splines) and Gaussian processes (GPs) to model continuous trajectories, enabling interpolation at arbitrary time points.
• Uses batch optimization with Laplace approximation to compute the posterior distribution over continuous states, supporting uncertainty quantification.
• Introduces a sensor-agnostic formulation that decouples optimization variables from measurement sampling times, reducing dependency on discrete sensor triggers.
• Supports manifold-valued state representations (e.g., SE(2), SE(3)) with exact covariance interpolation, enabling uncertainty propagation on non-Euclidean spaces.
• Incorporates sensor-specific models (e.g., IMU preintegration, LiDAR deskewing, rolling shutter correction) within the CT framework to reduce motion distortion and preprocessing errors.

Experimental results

Research questions

• RQ1 How can continuous-time state estimation improve performance and robustness compared to discrete-time methods in robotic systems?
• RQ2 What are the key methodological differences and similarities between spline-based and Gaussian process-based continuous-time estimation?
• RQ3 In what ways do CT methods reduce sensor preprocessing complexity and improve time synchronization in multi-sensor robotic systems?
• RQ4 What are the open problems and limitations in current CT estimation frameworks, particularly regarding computational efficiency and scalability?
• RQ5 How can CT estimation be extended to support complex, non-Euclidean state spaces such as SE(2) and SE(3) with accurate uncertainty propagation?

Key findings

• Continuous-time methods significantly reduce motion distortion by modeling state evolution continuously, avoiding the need for motion compensation techniques like LiDAR deskewing.
• The use of B-splines and Gaussian processes enables accurate interpolation of states at any time, including measurement times, without requiring explicit state variables at each sample.
• CT estimation with Gaussian processes supports exact covariance interpolation on manifold spaces (e.g., SE(2)), allowing for meaningful uncertainty envelopes along trajectories.
• The framework reduces the number of optimization variables by decoupling state representation from sensor sampling times, improving computational efficiency and flexibility.
• The survey identifies that sensor calibration parameters (e.g., time offsets, biases, extrinsics) are naturally integrated into the CT formulation, enhancing robustness.
• Despite their advantages, CT methods remain underutilized due to limited awareness and a lack of standardized tooling, highlighting a need for broader adoption and implementation support.