Smoothing Out the Edges: Continuous-Time Estimation with Gaussian Process Motion Priors on Factor Graphs

Robust and efficient state estimation is crucial for perception, navigation, and control in robotics. State estimation problems are conveniently modeled using the factor-graph framework as enabled by modern software packages such as GTSAM or g2o. However, the standard solvers included in such frameworks are local and may converge to poor local minima, posing significant safety concerns. Conversely, techniques based on convex relaxations have been shown to provide a means of globally solving or certifying many state estimation problems. However, these relaxations 1) often require substantial effort to formulate, and 2) may incur significantly higher cost compared to efficient local solvers, as they require solving a large semidefinite program (SDP). In this work, we address both shortcomings by 1) creating a new procedure within the GTSAM framework for automatically constructing convex SDP relaxations for any factor graphs with common factor and variable types, and by 2) exploiting the Bayes tree constructions native to GTSAM to decompose the SDP problem, leading to significant speedup in solver time for chordally sparse problems. We demonstrate the favorable scaling of this structure-exploiting global estimator compared to standard local solvers for two case studies: A 3D pose-graph SLAM problem with a ring factor graph and a 2D localization problem with a chain factor graph. The software framework is available at https://github.com/borglab/gtsam.

Continuous-time SLAM provides a principled framework for fusing heterogeneous sensors while estimating smooth trajectories, and is particularly well-suited for handling heterogeneous, asynchronous sensor streams with non-uniform readout patterns, such as rolling shutter cameras, LiDAR scanners, radar sweeps, or event-based sensors. In this work, we introduce G-solver, a fully Gaussian and distributed framework that combines Gaussian Belief Propagation (GBP) with Gaussian Process (GP) motion priors for continuous-time trajectory estimation. Our GP model provides a probabilistic representation of the trajectory, enabling consistent interpolation and the use of data-driven hyperparameters, while GBP offers a scalable message-passing formulation well-suited for decentralized settings. The resulting solver naturally extends to multi-camera scenarios without specialized synchronization or engineering effort. We evaluate the approach on synthetic and real data, including rolling shutter and distributed multi-camera optimization, demonstrating accurate and stable estimation with runtimes comparable to existing continuous-time methods. An open-source implementation is released.

Reconstructing the shape of continuum manipulators from sparse, noisy sensor data is a challenging task, owing to the infinite-dimensional nature of such systems. Existing approaches broadly trade off between parametric methods that yield compact state representations but lack probabilistic structure, and Cosserat rod inference on factor graphs, which provides principled uncertainty quantification at the cost of a state dimension that grows with the spatial discretization. This letter combines the strength of both paradigms by estimating the coefficients of a low-dimensional Geometric Variable Strain (GVS) parameterization within a factor graph framework. A novel kinematic factor, derived from the Magnus expansion of the strain field, encodes the closed-form rod geometry as a prior constraint linking the GVS strain coefficients to the backbone pose variables. The resulting formulation yields a compact state vector directly amenable to model-based control, while retaining the modularity, probabilistic treatment and computational efficiency of factor graph inference. The proposed method is evaluated in simulation on a 0.4 m long tendon-driven continuum robot under three measurement configurations, achieving mean position errors below 2 mm for all three scenarios and demonstrating a sixfold reduction in orientation error compared to a Gaussian process regression baseline when only position measurements are available.