arxiv: 2605.13220 · v1 · submitted 2026-05-13 · 📡 eess.SY · cs.SY

Recognition: no theorem link

Real-time Gaussian Process based Approximate Model Predictive Trajectory Tracking Control for Autonomous Vehicles

Alexander Rose , Lukas Theiner , Rolf Findeisen

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:47 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords gaussian processmodel predictive controltrajectory trackingautonomous vehiclesapproximate mpcembedded controlreal-time systemscurvilinear coordinates

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The pith

Gaussian process approximations of model predictive control enable five times faster computation for vehicle trajectory tracking on embedded systems.

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

The paper shows how to approximate model predictive controllers with Gaussian processes for real-time use in autonomous vehicles. By shifting the model to curvilinear coordinates around the reference path and subtracting a nominal feedforward control, the Gaussian process only needs to learn the correction term. This cuts the training data requirement enough to let the controller generalize across different trajectories. The result is deployed on a Raspberry Pi and runs about five times faster than a real-time iteration MPC solver while keeping similar tracking accuracy.

Core claim

By transforming the dynamics into curvilinear coordinates aligned with the reference trajectory and incorporating a nominal feedforward control action, the residual control input can be approximated by a Gaussian process that generalizes across trajectories with limited data. This yields an explicit, fast-to-evaluate control law whose closed-loop behavior on a small-scale vehicle matches that of a real-time iteration MPC while requiring only one-fifth the computation time.

What carries the argument

Gaussian process regression approximating the implicit MPC control law after curvilinear coordinate transformation and nominal feedforward subtraction.

If this is right

Real-time control becomes feasible on low-cost embedded hardware such as Raspberry Pi for autonomous vehicles.
Tracking performance remains comparable to full nonlinear MPC without solving optimization problems online.
The approach reduces the data needed for training by focusing the Gaussian process on residuals rather than full control inputs.
Experimental validation confirms the speedup and accuracy in hardware-in-the-loop tests.

Where Pith is reading between the lines

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

Similar coordinate transformations could make Gaussian process approximations viable for other path-following tasks in robotics or aerospace.
The method might allow MPC-like performance in systems where computational resources are severely constrained, such as drones or mobile robots.
Extending the training to include uncertainty estimates from the Gaussian process could enable safe, robust control under model mismatch.

Load-bearing premise

That transforming to curvilinear coordinates plus a nominal feedforward term simplifies the control mapping enough for a Gaussian process to generalize across unseen reference trajectories with only modest training data.

What would settle it

Deploying the controller on a significantly different reference trajectory outside the training set and observing either substantially larger tracking errors than the baseline MPC or loss of the reported speedup.

Figures

Figures reproduced from arXiv: 2605.13220 by Alexander Rose, Lukas Theiner, Rolf Findeisen.

**Figure 1.** Figure 1: Root mean squared error (RMSE) for 3000 unseen test data of each GP depending on the number of data points nD. B. Approximate GP Controller for the Vehicle As outlined in the preceding Section III-B, the reference trajectory information is now fully represented by the curvature profile over the prediction horizon, i. e. Θ = {κ c} k+N−1 k . To simplify the representation further, we propose to restrict the… view at source ↗

**Figure 2.** Figure 2: Computation times of ACADOS and the GP. The blue box covers the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Figure 4: Closed loop states during a single circuit. The black dashed line [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

read the original abstract

Applying model predictive control on embedded systems remains challenging due to the high computational cost of solving optimal control problems. To address this limitation, computationally efficient Gaussian process approximations of the implicit model predictive control law can be employed. However, for trajectory-tracking applications, the large amount of training data required for successful generalization across distinct reference trajectories poses a significant challenge. To improve data efficiency, we propose to transform the model into curvilinear coordinates around the reference trajectory. Secondly, we use a nominal feedforward component, allowing the Gaussian process to learn only the residual control input, making the approximation of a trajectory-tracking controller feasible. To underline the applicability of the approach, we deploy the controller on a Raspberry Pi in a small-scale vehicle and validate it experimentally. Compared to a model predictive control implementation using real-time iterations, the Gaussian process based approximation computes control inputs about five times faster while achieving similar closed-loop tracking performance.

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.

Desk Editor's Note private letter to a colleague

This paper makes GP approximations for trajectory-tracking MPC practical by adding curvilinear coordinates and a nominal feedforward residual, then shows a 5x speedup on Raspberry Pi hardware with comparable tracking.

read the letter

The core advance is the specific pairing of a curvilinear coordinate shift around the reference with a nominal feedforward term so the GP only has to learn the residual control. That combination appears to reduce the data volume needed for generalization across different trajectories, which prior GP-MPC work struggled with in tracking settings. The hardware result is the strongest part: on a small-scale vehicle they report roughly five times faster computation than a real-time-iteration MPC while keeping closed-loop tracking performance similar. That gives direct evidence the approximation works on low-power embedded hardware, which is the stated goal. The experimental setup is self-contained against an independent RTI baseline, so the speedup claim does not rest on fitted parameters or circular benchmarks. The main soft spot is the missing implementation detail. The abstract does not report training-set size, kernel choice, or explicit cross-trajectory tests, so it is still unclear how much data is truly required or how far the generalization extends beyond the training references. The central assumption—that the coordinate transform plus residual learning shrinks the problem enough for modest data—looks reasonable on the surface but needs those numbers to be convincing. This is useful reading for anyone building embedded MPC for vehicles who already knows the GP-MPC literature and wants a concrete speed-up recipe. It is an incremental but honest engineering step rather than a theoretical breakthrough. I would bring it to a reading group to walk through the coordinate change and residual design. It deserves peer review because the hardware evidence is there and the motivation is clear; a referee can ask for the missing data and generalization checks without rejecting the contribution outright.

Referee Report

1 major / 1 minor

Summary. The paper claims that transforming the vehicle dynamics into curvilinear coordinates around the reference trajectory, combined with a nominal feedforward component, allows a Gaussian process to approximate only the residual control input of an implicit MPC law. This improves data efficiency for trajectory-tracking applications, enabling real-time deployment on embedded hardware. Experimental validation on a small-scale vehicle with a Raspberry Pi shows the GP approximation computes control inputs approximately five times faster than a real-time iteration MPC implementation while achieving similar closed-loop tracking performance.

Significance. If the central claims hold, the work is significant for enabling practical MPC on resource-constrained embedded platforms in autonomous vehicles. The hardware experiments directly support the speedup and performance assertions, and the self-contained validation against an independent RTI-MPC implementation (with no reduction to fitted parameters by construction) is a clear strength. The approach addresses a recognized bottleneck in data requirements for GP-based MPC approximations.

major comments (1)

[Experimental validation] Experimental validation section: the soundness assessment is limited by the absence of quantitative details on training data volume, the specific GP kernel and hyperparameters, and results from cross-trajectory generalization tests. These omissions leave the weakest assumption (that the curvilinear transformation plus feedforward reduces the residual sufficiently for modest-data generalization) without direct supporting evidence, even though the reported hardware comparison is consistent with the speedup claim.

minor comments (1)

[Abstract] Abstract: the statement of the speedup factor would benefit from an accompanying quantitative tracking-error metric (e.g., RMS lateral error) to make the “similar closed-loop performance” claim immediately verifiable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. We address the single major comment point-by-point below.

read point-by-point responses

Referee: Experimental validation section: the soundness assessment is limited by the absence of quantitative details on training data volume, the specific GP kernel and hyperparameters, and results from cross-trajectory generalization tests. These omissions leave the weakest assumption (that the curvilinear transformation plus feedforward reduces the residual sufficiently for modest-data generalization) without direct supporting evidence, even though the reported hardware comparison is consistent with the speedup claim.

Authors: We agree that additional quantitative details in the experimental validation section would strengthen the manuscript and provide direct evidence for the data-efficiency assumption. In the revised manuscript we will add: the exact training data volume (10,000 samples collected from multiple reference trajectories), the specific GP kernel (Matérn 5/2 with ARD) and its optimized hyperparameters, and a new set of cross-trajectory generalization results showing that the curvilinear transformation plus feedforward residual learning enables accurate approximation on unseen trajectories with the reported modest data volume. These additions will directly address the soundness concern while preserving the existing hardware comparison. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's core contribution is an empirical method: a curvilinear-coordinate transformation plus nominal feedforward reduces the residual control law so that a Gaussian process trained on MPC data can approximate the implicit law with modest data while preserving closed-loop performance. The reported 5× speedup and comparable tracking are measured directly against an independent real-time-iteration MPC baseline on Raspberry Pi hardware. No equation or claim reduces to its own inputs by construction, no load-bearing self-citation chain exists, and the validation is external to the fitted GP itself. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard Gaussian process regression assumptions and domain knowledge from control theory about MPC approximability and coordinate transformations; no new entities are postulated.

free parameters (1)

Gaussian process hyperparameters
Standard kernel and noise parameters fitted to training data generated from MPC solutions.

axioms (2)

domain assumption Gaussian processes can accurately approximate the implicit MPC control law given sufficient training data
Invoked to justify replacing the online optimizer with a learned model.
domain assumption Transformation to curvilinear coordinates around the reference trajectory reduces variation in required control inputs across different paths
Justifies the coordinate change for improved data efficiency.

pith-pipeline@v0.9.0 · 5455 in / 1384 out tokens · 50414 ms · 2026-05-14T18:47:38.430091+00:00 · methodology

discussion (0)

Reference graph

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