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arxiv: 2605.09073 · v1 · submitted 2026-05-09 · 💻 cs.RO

Recognition: no theorem link

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

Connor Holmes , Sven Lilge , Zi Cong Guo , Frank Dellaert , Timothy D. Barfoot
Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:49 UTC · model grok-4.3

classification 💻 cs.RO
keywords continuous-time estimationGaussian processesfactor graphsmotion priorsstate estimationroboticsGTSAM
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The pith

Factor graphs let practitioners adopt Gaussian process continuous-time estimation using familiar robotics tools.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Continuous-time state estimation with Gaussian processes yields smooth trajectories and naturally accommodates asynchronous sensors, yet remains underused compared to parametric alternatives. The paper shows how to express Gaussian process motion priors directly as factors in a factor graph, so that existing solvers and libraries can optimize the continuous-time problem without new mathematics. Three concrete GTSAM implementations are supplied to demonstrate the translation. If the reformulation works as intended, robotics teams already working with factor graphs can add continuous-time capabilities without switching frameworks.

Core claim

Gaussian process motion priors can be encoded as factors on a factor graph, allowing the full continuous-time estimation problem to be solved with standard factor-graph optimizers while retaining the nonparametric smoothness and interpolation properties of the Gaussian process.

What carries the argument

The Gaussian process motion prior factor, which connects state variables at arbitrary times through the chosen GP kernel and covariance function.

If this is right

  • Standard factor-graph solvers can optimize trajectories at any continuous time without separate interpolation steps.
  • Asynchronous measurements from different sensors can be added at their exact timestamps.
  • Users gain the ability to query the estimated state at any intermediate time while preserving consistency with the motion prior.
  • Existing discrete-time factor graphs can be extended incrementally by inserting GP factors between states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same factor-graph encoding could be applied to other nonparametric priors beyond Gaussian processes.
  • Custom motion models could be created simply by substituting different kernels inside the factor definition.
  • Mixed discrete-continuous problems become feasible by combining GP factors with conventional measurement factors.

Load-bearing premise

That casting Gaussian process priors in factor-graph language will lower the barrier enough for existing users of libraries such as GTSAM to adopt continuous-time methods.

What would settle it

If the three provided GTSAM examples produce trajectories whose smoothness or accuracy is statistically indistinguishable from discrete-time baselines on standard robot datasets, the claimed practical advantage would not hold.

Figures

Figures reproduced from arXiv: 2605.09073 by Connor Holmes, Frank Dellaert, Sven Lilge, Timothy D. Barfoot, Zi Cong Guo.

Figure 1.1
Figure 1.1. Figure 1.1: We consider a trajectory to be a continuous function of time. Measure￾ments yk occur (possibly asynchronously) at discrete times tk yet we would like to be able to query (estimate) the trajectory state x(τ ) at any time of interest, τ . the use of continuous-time estimation in a practical setting, in￾cluding how to handle high-rate and asynchronous measurements. • We release the code and the datasets use… view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: In GP regression, we begin with some prior over trajectories, including mean and covariance functions. After incorporating measurements, we obtain a posterior distribution over trajectories that can be queried at any time. We consider that we want to query the state at a number of times (τ0 < τ1 < . . . < τJ ) that may or may not be different from the measurement times (t0 < t1 < . . . < tK). The joint d… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: A simple factor graph for a classic estimation problem. The unary factor on x0 represents the prior knowledge about the initial state, the binary factors represent the motion model, and the remaining unary factors represent the measurements. In practice, we can have much more complicated factor graphs with loops. where x = (x0, . . . , xK) is the state at times t0, . . . , tK, v = (xˇ0, v1,0, . . . , vK,… view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Simple example of the elimination algorithm. The left shows the factor graph before elimination and the right shows the result of eliminating x from the factor graph, resulting in a new factor and a conditional density for x. We can visually understand the factors by assembling them into a factor graph, which is a bipartite graph with two types of nodes: variable nodes and factor nodes. The variable node… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: A continuous-time estimation problem with measurements at discrete times and a single query time. We refer to this as ‘continuous-time’ because the query(s) could be at any time along the trajectory. In the rest of this chapter, we will show that our insights from the previous chapter follows naturally when continuous-time estimation is cast into the factor graph framework. We first focus on how GP-based… view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: The same factor graph as [PITH_FULL_IMAGE:figures/full_fig_p026_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Elimination of the query state, xτ , from the factor graph in [PITH_FULL_IMAGE:figures/full_fig_p028_3_5.png] view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: The ground-truth sinusoidal trajectory is shown in green. Noisy position￾only measurements are shown in red. Our continuous-time solver first computes the solution at the measurement times (blue dots) and then interpolates (both mean and covariance) at several query times between the measurements times (blue line). We see that the solution is smooth and continuous, as expected, following the ground truth… view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: When eliminating an interpolated state with a measurement, we can exactly decompose the residual factor into two factors: a measurement factor ϕy(xk−1, xk) and a motion-prior factor ϕv(xk−1, xk). Assuming a motion prior of the type discussed in this article, it turns out that the second term can be refactored as p(xk, xτ |xk−1, v) = p(xτ |xk, xk−1, v)p(xk|xk−1, vk,k−1), (3.19) The residual factor, ϕ(xk−1… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: To make use of GPs on a Lie group, we define local variables in the Lie algebra around each measurement time and apply linear GP models to these local variables. model in the Lie algebra. For example, a WNOA model can be written down as ¨ξ(t) = w(t), (4.8) where w(t) is a white-noise process. As a first-order system, this can be written as " ˙ξ(t) ¨ξ(t) # = " 0 I 0 0# "ξ(t) ˙ξ(t) # + " 0 I # w(t), (4.9) … view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: General form of a motion prior defined over a Lie group trajectory. Between each pair of measurement times, we use a linear motion model in the Lie algebra. The variables for which we are actually optimizing live in the Lie group. this, we assume that we can carry out perturbations of the Lie group element and the generalized velocity and its derivatives as follows: Tk = Top,kExp(ϵk), ϖk = ϖop,k + ηk , .… view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Linearized Lie-group motion prior. The factor graph is now expressed over the perturbation states rather than the original Lie group variables. We can use our linear system results to solve this factor graph, then iterate to improve the linearization. 4.4 After-Main-Solve Querying After the main solve, we can query the states at any time by interpolating between the measurement times. We will do this in … view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: To query the mean of the state at time tk−1 < τ < tk, we start with the situation on the left. Notice that the factors are expressed in terms of the local GP variable, γk−1 . We then eliminate γk−1 (τ ) resulting in the situation on the right. τ : p(ετ |εk−1, εk) = N (µτ , Στ ). (4.21) Since this is only the conditional density for the perturbation variable, we need to convolve this with the marginal pos… view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: To query the covariance of the state at time tk−1 < τ < tk, we start with the situation on the left. Notice that the factors are expressed in terms of the perturbation variables. We then eliminate ετ resulting in the situation on the right. 4.5 Example Figures 4.6 shows a simple example of estimating a trajectory in SE(3) using a WNOA motion prior. The red frames are noisy pose measure￾ments, the dark bl… view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Example of SE(3) state estimation with a WNOA motion prior. The red frames are noisy pose measurements. The dark blue frames and covariance ellipsoids are computed in the main solve. The light blue frames and covariance ellipsoids are the interpolated states. The green frames are the ground truth. 0.0 2.5 5.0 7.5 10.0 t [s] 1.2 1.0 0.8 vx [m/s] 0.0 2.5 5.0 7.5 10.0 t [s] 0.04 0.02 0.00 0.02 0.04 vy [m/s]… view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: Generalized velocities associated with [PITH_FULL_IMAGE:figures/full_fig_p044_4_7.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: A simple SE(2) interpolation example demonstrating a full factor graph solve including all 60 states (green) compared to a reduced solve where only three of the 60 states are actually solved (blue). A unary factor is placed on each of the 60 states. The top row shows the estimated states in each case (with covariances). The bottom two rows show two different methods of interpolating the covariance of the… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: A mobile robot driving between rails. It is equipped with an odometer to measure translational speed and a laser rangefinder to measure the range to the large white cylinder. circular shape. This eliminates the need to process the raw laser scans directly, allowing the use of a simplified range measurement instead. Given the known location of the cylinder, the range measurement can be treated as a noisy … view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Robot state estimate over a subset of its trajectory for different measurement frequencies. Position and velocity measurements are provided only every 2.5 seconds, 5 seconds or 7.5 seconds, respectively. Between these times, the estimate relies solely on the Gaussian process prior. The plots show the mean estimates with 3-σ uncertainty envelopes. Ground-truth velocity is obtained by numerically different… view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Comparison estimating discrete states in the factor graph at every 0.1 s (left) versus only every 5 s when measurements are available (right). In the latter case, intermediate states are recovered using our post-query interpolation scheme. For this one-dimensional linear example, both approaches yield identical solutions. with relatively sparse data, leveraging the motion prior to produce a continuous es… view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Dataset setup and ground-truth trajectory. The left image shows the setup of the robot and the plastic cylinders that serve as the landmark in a room with 10 camera Vicon system used to obtain the ground-truth positions and orientations of the robot throughout the dataset. The plot on the right shows the full trajectory of the robot throughout the dataset. The start and end positions are marked with a gr… view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: GTSAM localization solution of a segment (100 seconds to 250 seconds) of the Lost in the Woods dataset with bearing-range measurements limited to 1 m and WNOA motion model factors between adjacent states (no odometry). The ground-truth trajectory is shown in green, while the estimated trajectory is shown in blue. Covariance ellipsoids based on three standard deviations are plotted in blue. In regions wit… view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: GTSAM SLAM solution with interpolation for the robot trajectory between 100 seconds and 250 seconds. The main solve only optimizes every 30th state in the trajectory (i.e., every 3 seconds), with the remainder of the states being interpolated. Ground truth is shown in green, while the estimates are shown in blue. Covariance ellipsoids are plotted based on three standard deviations. Bearing-range measurem… view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: Quantitative comparison of solving the full factor graph by including a state at every measurement time versus relying on interpolation with varying state inclusion frequencies (one state corresponds to 0.1 seconds). We report on accuracy and computation time. We also compare interpolation with and without the approximations applied to the derived noise model for measurements at interpolated times, illus… view at source ↗
Figure 5.9
Figure 5.9. Figure 5.9: Comparison of the estimate’s consistency when solving the problem using interpolation with and without the approximations applied to the derived noise model for measurements at interpolated times. interpolation scheme becoming inadequate to represent the true trajec￾tory. However, the accuracy of the estimated states degrades more when the simplified model is used. Intuitively, we attribute this effect t… view at source ↗
Figure 5.10
Figure 5.10. Figure 5.10: Starry Night dataset collection setup. A rigid sensor head is moved around in dim environment while collecting IMU data and stereo measurements to Vicon markers on the ground (top row). The ground-truth poses of the sensor head are collected using a Vicon motion capture system. We now turn to a 3D dataset to demonstrate the above concepts in SE(3). The setup consists of a sensor head that contains an in… view at source ↗
Figure 5.11
Figure 5.11. Figure 5.11: Comparing GTSAM localization solutions with and without inter￾polation between 112 seconds and 152 seconds of the Starry Night dataset. In the magenta trajectory, all states were included in the main solve. In the blue trajectory, every 5 th state was included in the main solve, with the remainder of the states being interpolated (cyan). We see that the two solutions are fairly close, despite that using… view at source ↗
Figure 5.12
Figure 5.12. Figure 5.12: Pose error plots for the GTSAM localization solutions in [PITH_FULL_IMAGE:figures/full_fig_p067_5_12.png] view at source ↗
read the original abstract

Continuous-time state estimation is gaining in popularity due to its abilities to provide smooth solutions, handle asynchronous sensors, and interpolate between data points. While there are two main paradigms, parametric (e.g., temporal basis functions, splines) and nonparametric (Gaussian processes), the latter has seen less adoption despite its technical advantages and relative ease of implementation. In this article, we seek to rectify this situation by providing a new simplified explanation of GP continuous-time estimation rooted in the language of factor graphs, which have become the de facto estimation paradigm in much of robotics. To simplify onboarding, we also provide three working examples implemented in the popular GTSAM estimation framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper presents a factor-graph formulation of Gaussian process (GP) continuous-time state estimation. It argues that GP methods offer advantages including trajectory smoothness, support for asynchronous sensors, and natural interpolation, yet have seen limited adoption; the work provides a simplified explanation in factor-graph language together with three working GTSAM implementations to lower the barrier for practitioners already using such libraries.

Significance. If the formulation and examples are correct, the paper has moderate significance as a bridge between nonparametric continuous-time estimation and the dominant factor-graph paradigm in robotics. The concrete, working GTSAM code constitutes direct evidence of implementability and reproducibility, which strengthens the claim of relative ease of onboarding and could aid wider adoption among existing GTSAM users.

minor comments (3)
  1. Abstract: the assertion that GP methods possess 'relative ease of implementation' is stated without a brief supporting comparison to parametric alternatives (e.g., splines); adding one sentence would strengthen the motivation.
  2. The manuscript would benefit from a short table or bullet list in the examples section that summarizes the three GTSAM implementations by sensor type, state dimension, and key GP kernel used.
  3. Conclusion: the final paragraph could explicitly note any remaining limitations of the factor-graph GP formulation (e.g., kernel choice or scaling with trajectory length) to give readers a balanced view.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive review and recommendation to accept. We are pleased that the contribution of recasting Gaussian-process continuous-time estimation in factor-graph language, together with the concrete GTSAM implementations, has been recognized as a useful bridge for practitioners.

Circularity Check

0 steps flagged

Explanatory reformulation of prior GP continuous-time methods into factor graphs, with concrete implementations

full rationale

The paper reformulates existing Gaussian-process motion priors using factor-graph language and supplies three working GTSAM examples to demonstrate onboarding ease. No derivation, prediction, or uniqueness claim reduces by construction to a parameter or result defined inside the present manuscript. Self-citations to foundational GP work (e.g., Barfoot et al.) provide background context but are not load-bearing for the central contribution, which rests on the supplied implementations rather than any closed mathematical loop. The work is self-contained as an expository bridge between estimation paradigms.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work is explanatory and relies on standard assumptions from the Gaussian-process and factor-graph literature without introducing new free parameters or invented entities.

axioms (2)
  • domain assumption Gaussian processes provide a suitable nonparametric prior for continuous-time motion
    Invoked when stating that the nonparametric paradigm has technical advantages.
  • domain assumption Factor graphs are the de-facto estimation paradigm in robotics
    Used to justify the choice of language for the simplified explanation.

pith-pipeline@v0.9.0 · 5417 in / 1224 out tokens · 35200 ms · 2026-05-12T01:49:29.777350+00:00 · methodology

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