pith. sign in

arxiv: 2604.19452 · v1 · submitted 2026-04-21 · 📡 eess.SY · cs.SY

Robust Nonlinear Trajectory Tracking Control for Autonomous Racing on Three-Dimensional Tracks

Joscha F. Bongard , Georg Jank , Simon Sagmeister , Boris Lohmann This is my paper

Pith reviewed 2026-05-10 01:59 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords nonlinear model predictive controltrajectory trackingautonomous racingthree-dimensional trackssingle-track modelrobust controlvehicle dynamicsconstraint tightening
0
0 comments X

The pith

Robust nonlinear MPC with a 3D single-track model tracks trajectories accurately on non-planar racing tracks.

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

This work presents a model predictive controller designed for autonomous race cars that must stay on a prescribed path at the limits of handling even when the road is not flat. The dynamics are derived from first principles as a three-dimensional single-track model, with only low-impact terms removed to keep the math fast enough to solve in real time. Vertical load changes caused by road shape are included in the prediction, and extra safety margins are added around the tire force limits to handle uncertainty in grip. Simulation tests on a detailed model of an actual racing circuit show that the car stays closer to the desired line than it would with a simpler flat-road model, without increasing the time needed to compute each control move. A reader should care because real circuits contain hills, dips, and banks that affect tire grip and vehicle balance in ways ignored by most current controllers.

Core claim

We derive a three-dimensional dynamic single-track model from first principles, selectively omitting negligible terms to preserve real-time capability. The resulting MPC integrates terrain-induced vertical loads and other 3D effects into its predictions. An uncertainty-aware constraint tightening scheme adds margins to keep the vehicle stable despite tire-road uncertainties. High-fidelity simulations on a real circuit model demonstrate improved trajectory-tracking accuracy at low computation times.

What carries the argument

Three-dimensional dynamic single-track vehicle model inside a nonlinear model predictive controller, combined with uncertainty-aware tightening of state and input constraints.

If this is right

  • The vehicle tracks reference trajectories more closely on tracks with elevation and banking.
  • Prediction accuracy improves because vertical load variations from road shape are modeled directly.
  • Uncertainty margins allow stable operation despite variations in tire friction on sloped surfaces.
  • Overall computation remains suitable for real-time implementation on embedded hardware.

Where Pith is reading between the lines

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

  • This selective model reduction might allow similar controllers for other dynamic systems where full equations are too slow.
  • Future work could test whether the same 3D model helps with path planning rather than just tracking.
  • Integration with tire force observers could shrink the uncertainty margins and permit even higher speeds.

Load-bearing premise

The assumption that terms omitted for computational speed do not reduce the model's ability to predict forces accurately enough for the tightened constraints to guarantee controllability on 3D tracks.

What would settle it

Running the controller on a track section with large elevation changes or banking angles and observing whether the vehicle departs from the planned trajectory or violates stability limits in high-fidelity simulation.

Figures

Figures reproduced from arXiv: 2604.19452 by Boris Lohmann, Georg Jank, Joscha F. Bongard, Simon Sagmeister.

Figure 1
Figure 1. Figure 1: Autonomous vehicles Dallara AV24 of TUM Au￾tonomous Motorsport and Unimore in a banked turn at the IAC. surfaces are often neglected in the design of tracking control modules, their explicit consideration has proven advanta￾geous in related AD modules, such as trajectory planning [5] and state estimation [6]. Therefore, this paper studies the potential benefits of integrating dynamic 3D effects in the pred… view at source ↗
Figure 2
Figure 2. Figure 2: Vehicle on banked and slope racetrack with relevant [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Side-by-side illustration of progressive tightening of [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Layout of the Autodromo Nazionale di Monza circuit overlaid on a 500 m grid. Track elevation is represented by a color gradient. The curves Lesmo I & II, used later, are highlighted. the uncertainty in order to tightly cover the set of uncertain trajectories and therefore induce the right amount of caution while avoiding unnecessary conservatism. In practice, the tube dynamics (25) grow during aggressive m… view at source ↗
Figure 6
Figure 6. Figure 6: Path deviation d and vertical loads FzF , FzR in the turns Lesmo I and II for the scenario at 0.98 planner acceleration limit scale. The solid lines in the second and third plots show the loads assumed by the MPC prediction model. The dashed lines show the simulation ground-truth axle load values, calculated as the sum of the left and right tire forces [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Overlaid histograms of MPC solve times for different [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
read the original abstract

We propose a robust nonlinear model predictive control (MPC) scheme for trajectory-tracking control of autonomous vehicles at the limits of handling on non-planar road surfaces. We derive the dynamics from first principles and selectively omit terms with negligible dynamic influence to maintain real-time capability. The resulting MPC with a three-dimensional (3D) dynamic single-track model integrates relevant dynamic effects directly into the prediction model and leverages them to improve prediction accuracy and therefore control performance. Even if the influence of terrain-induced vertical loads on the total acceleration potential is modeled, tire-road interactions are subject to uncertainty and disturbance. The uncertainty-aware constraint tightening scheme introduces a margin to constraint bounds to keep the vehicle controllable and stable in this environment. To validate our proposed approach, we perform high-fidelity dynamic double-track vehicle dynamics simulations on a model of a real circuit. We find that our algorithm can improve trajectory-tracking accuracy while maintaining low computation times.

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

3 major / 2 minor

Summary. The paper proposes a robust nonlinear MPC scheme for autonomous vehicle trajectory tracking at handling limits on non-planar 3D tracks. It derives a 3D dynamic single-track model from first principles, selectively omits terms deemed to have negligible dynamic influence to preserve real-time capability, incorporates terrain-induced vertical loads, and applies uncertainty-aware constraint tightening to handle tire-road uncertainties. Validation is performed via high-fidelity double-track simulations on a real-circuit model, with the central claim being improved tracking accuracy at low computation times.

Significance. If the modeling assumptions hold, the work would meaningfully advance MPC-based autonomous racing by directly embedding 3D effects (vertical load variation, non-planar geometry) into the prediction model rather than treating them as disturbances. The first-principles derivation combined with simulation-based validation on a realistic track model is a positive feature; reproducible high-fidelity simulation results would strengthen the contribution if the omitted-term justification is made rigorous.

major comments (3)
  1. [§3] §3 (model derivation): The selective omission of terms with 'negligible dynamic influence' is stated without quantitative error bounds, sensitivity analysis with respect to road curvature/banking, or verification that residual dynamics remain inside the uncertainty set used for constraint tightening. On non-planar surfaces these omitted couplings (load transfer, slip-angle interactions) directly modulate the friction ellipse; their absence risks prediction bias precisely where the tightening margin is smallest, undermining the controllability claim.
  2. [§4.3] §4.3 (uncertainty-aware tightening): The scheme assumes the reduced-order 3D model supplies predictions accurate enough for the tightened constraints to keep the vehicle inside the feasible set, yet no a-priori guarantee, post-hoc prediction-error quantification, or comparison against the full-order model is provided. This is load-bearing for the robustness claim.
  3. [§5] §5 (simulation results): The reported accuracy improvements lack error bars, multiple randomized runs, or ablation studies isolating the effect of the 3D model versus planar baselines; without these, it is impossible to assess whether the gains are statistically significant or merely artifacts of the specific track and parameter choices.
minor comments (2)
  1. Notation for the 3D single-track states and the exact list of omitted terms should be tabulated for clarity.
  2. Figure captions for the circuit model and trajectory plots should explicitly state the sampling rate and solver tolerances used in the high-fidelity simulator.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help us strengthen the rigor of the manuscript. We address each major comment below and indicate the revisions we will incorporate.

read point-by-point responses

  1. Referee: [§3] §3 (model derivation): The selective omission of terms with 'negligible dynamic influence' is stated without quantitative error bounds, sensitivity analysis with respect to road curvature/banking, or verification that residual dynamics remain inside the uncertainty set used for constraint tightening. On non-planar surfaces these omitted couplings (load transfer, slip-angle interactions) directly modulate the friction ellipse; their absence risks prediction bias precisely where the tightening margin is smallest, undermining the controllability claim.

    Authors: We acknowledge that the current derivation in §3 relies on order-of-magnitude arguments without explicit quantitative bounds or sensitivity plots. In the revised manuscript we will add a new subsection to §3 containing a sensitivity analysis that quantifies the effect of the omitted load-transfer and slip-angle coupling terms on the friction ellipse across representative ranges of road curvature and banking. We will also compare the reduced-order predictions against the full-order model on the same trajectories used in §5 and confirm that the observed discrepancies lie inside the uncertainty set employed for tightening. This directly addresses the concern that bias may occur where margins are smallest. revision: yes

  2. Referee: [§4.3] §4.3 (uncertainty-aware tightening): The scheme assumes the reduced-order 3D model supplies predictions accurate enough for the tightened constraints to keep the vehicle inside the feasible set, yet no a-priori guarantee, post-hoc prediction-error quantification, or comparison against the full-order model is provided. This is load-bearing for the robustness claim.

    Authors: The referee is correct that the robustness argument rests on the reduced-order model remaining sufficiently accurate relative to the tightening margins. While a formal a-priori guarantee for the nonlinear case is difficult to obtain, we will add to the revised §4.3 a post-hoc prediction-error study: we will roll out the reduced-order 3D model in open-loop against the high-fidelity double-track simulator over the closed-loop trajectories of §5, report the resulting lateral and longitudinal force prediction errors, and verify that these errors are bounded by the uncertainty set used for constraint tightening. This quantification will be presented alongside the existing closed-loop results. revision: partial

  3. Referee: [§5] §5 (simulation results): The reported accuracy improvements lack error bars, multiple randomized runs, or ablation studies isolating the effect of the 3D model versus planar baselines; without these, it is impossible to assess whether the gains are statistically significant or merely artifacts of the specific track and parameter choices.

    Authors: We agree that the simulation section would be strengthened by statistical measures and ablation studies. In the revised manuscript we will augment §5 with (i) ten additional simulation runs using randomized initial conditions and small parameter perturbations drawn from the uncertainty set, reporting mean and standard deviation of tracking errors; (ii) an explicit ablation comparing the proposed 3D dynamic single-track MPC against a planar single-track baseline with identical tightening; and (iii) a brief discussion of how the 3D effects contribute to the observed improvement on the non-planar circuit. revision: yes

Circularity Check

0 steps flagged

No circularity in first-principles derivation or MPC scheme

full rationale

The paper derives the 3D dynamic single-track model explicitly from first principles, then selectively omits terms judged to have negligible dynamic influence solely to retain real-time capability. The resulting MPC prediction model and uncertainty-aware constraint tightening are presented as direct consequences of this derivation plus an added robustness layer; validation occurs via independent high-fidelity double-track simulations on a real circuit. No equation, prediction, or performance claim reduces by construction to a fitted parameter, self-citation, or renamed input. The central controllability claim therefore rests on external simulation evidence rather than tautological re-use of its own modeling choices.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claim rests on first-principles vehicle dynamics with selective term omission and an uncertainty model for tire-road interactions; no explicit free parameters or invented entities are named in the abstract.

pith-pipeline@v0.9.0 · 5459 in / 1096 out tokens · 32781 ms · 2026-05-10T01:59:58.663561+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [1]

    Head-to-Head Autonomous Racing at the Limits of Handling in the A2RL Challenge

    Simon Hoffmann et al. “Head-to-Head Autonomous Racing at the Limits of Handling in the A2RL Challenge”. Submit- ted to Sci. Robot. 2025

    work page 2025

  2. [2]

    TUM autonomous motorsport: An autonomous racing software for the Indy Autonomous Chal- lenge

    Johannes Betz et al. “TUM autonomous motorsport: An autonomous racing software for the Indy Autonomous Chal- lenge”. In:J. Field Robot.40 (4 June 2023), pp. 783–809

    work page 2023

  3. [3]

    James B Rawlings et al.Model Predictive Control: Theory, Computation, and Design 2nd Edition. Tech. rep. 2020

    work page 2020

  4. [4]

    Model predictive path tracking control for automated road vehicles: A review

    P. Stano et al. “Model predictive path tracking control for automated road vehicles: A review”. In:Annu. Rev. Control 55 (January 2023 2023), pp. 194–236

    work page 2023

  5. [5]

    Online Time-Optimal Trajectory Planning on Three-Dimensional Race Tracks

    Matthias Rowold et al. “Online Time-Optimal Trajectory Planning on Three-Dimensional Race Tracks”. In:2023 IEEE Intell. V eh. Symp. (IV). IEEE, June 2023, pp. 1–8

    work page 2023

  6. [6]

    Three-Dimensional Vehicle Dynamics State Estimation for High-Speed Race Cars under varying Signal Quality

    Sven Goblirsch et al. “Three-Dimensional Vehicle Dynamics State Estimation for High-Speed Race Cars under varying Signal Quality”. In:2024 IEEE/RSJ Int. Conf. Intell. Robots Syst. (IROS). IEEE, Oct. 2024, pp. 3371–3378

    work page 2024

  7. [7]

    Autonomous Vehicles on the Edge: A Survey on Autonomous Vehicle Racing

    Johannes Betz et al. “Autonomous Vehicles on the Edge: A Survey on Autonomous Vehicle Racing”. In:IEEE Open J. Intell. Transp. Syst.3 (Feb. 2022), pp. 458–488

    work page 2022

  8. [8]

    Max Schwenzer et al.Review on model predictive control: an engineering perspective. Nov. 2021

    work page 2021

  9. [9]

    Tube model predictive control for an autonomous race car

    A. Wischnewski et al. “Tube model predictive control for an autonomous race car”. In:V ehicle Syst. Dyn.(2021)

    work page 2021

  10. [10]

    Autonomous Vehicle Trajectory Fol- lowing: A Flatness Model Predictive Control Approach With Hardware-in-the-Loop Verification

    Zejiang Wang et al. “Autonomous Vehicle Trajectory Fol- lowing: A Flatness Model Predictive Control Approach With Hardware-in-the-Loop Verification”. In:IEEE Trans. Intell. Transp. Syst.22.9 (Sept. 2021), pp. 5613–5623

    work page 2021

  11. [11]

    Autonomous racing using Linear Parameter Varying-Model Predictive Control (LPV-MPC)

    Eugenio Alcal ´a et al. “Autonomous racing using Linear Parameter Varying-Model Predictive Control (LPV-MPC)”. In:Control Eng. Pract.95 (Feb. 2020), p. 104270

    work page 2020

  12. [12]

    Motion Planning and Control for Multi Vehicle Autonomous Racing at High Speeds

    Ayoub Raji et al. “Motion Planning and Control for Multi Vehicle Autonomous Racing at High Speeds”. In:IEEE Int. Conf. Intell. Transp. Syst. (ITSC). IEEE, Oct. 2022, pp. 2775–2782

    work page 2022

  13. [13]

    Dynamic Constraint Tightening for Nonlinear MPC for Autonomous Racing via Contraction Analysis

    Joscha F. Bongard et al. “Dynamic Constraint Tightening for Nonlinear MPC for Autonomous Racing via Contraction Analysis”. In:IEEE Intell. V eh. Symp. (IV). IEEE, June 2025, pp. 2288–2295

    work page 2025

  14. [14]

    A three-dimensional free- trajectory quasi-steady-state optimal-control method for minimum-lap-time of race vehicles

    S. Lovato and M. Massaro. “A three-dimensional free- trajectory quasi-steady-state optimal-control method for minimum-lap-time of race vehicles”. In:V ehicle Syst. Dyn. 60 (5 May 2022), pp. 1512–1530

    work page 2022

  15. [15]

    Optimal Control of a Formula One Car on a Three-Dimensional Track - Part 2: Optimal Control

    D. J.N. Limebeer and G. Perantoni. “Optimal Control of a Formula One Car on a Three-Dimensional Track - Part 2: Optimal Control”. In:Journal of Dynamic Systems, Mea- surement and Control, Transactions of the ASME137 (5 May 2015)

    work page 2015

  16. [16]

    Model Predictive Stabilization Control of High-Speed Autonomous Ground Vehicles Considering the Effect of Road Topography

    Kai Liu et al. “Model Predictive Stabilization Control of High-Speed Autonomous Ground Vehicles Considering the Effect of Road Topography”. In:Applied Sciences8 (2018), p. 822

    work page 2018

  17. [17]

    Neural Network Autoregressive With Exogenous Input Assisted Multi-Constraint Nonlinear Pre- dictive Control of Autonomous Vehicles

    Hamid Taghavifar. “Neural Network Autoregressive With Exogenous Input Assisted Multi-Constraint Nonlinear Pre- dictive Control of Autonomous Vehicles”. In:IEEE Trans. V eh. Technol.68.7 (2019), pp. 6293–6304

    work page 2019

  18. [18]

    A Tube-MPC Approach to Autonomous Multi-Vehicle Racing on High-Speed Ovals

    Alexander Wischnewski et al. “A Tube-MPC Approach to Autonomous Multi-Vehicle Racing on High-Speed Ovals”. In:IEEE Trans. Intell. V eh.(2022)

    work page 2022

  19. [19]

    Pacejka.Tire and V ehicle Dynamics

    Hans B. Pacejka.Tire and V ehicle Dynamics. Elsevier, 2012, pp. 1–58

    work page 2012

  20. [20]

    Robust adaptive MPC using control contraction metrics

    Andr ´as Sasfi et al. “Robust adaptive MPC using control contraction metrics”. In:Automatica155 (September Sept. 2023), p. 111169

    work page 2023

  21. [21]

    acados—a modular open-source framework for fast embedded optimal control

    Robin Verschueren et al. “acados—a modular open-source framework for fast embedded optimal control”. In:Math. Program. Comput.14 (1 Mar. 2022), pp. 147–183

    work page 2022

  22. [22]

    A Software Architecture for an Au- tonomous Racecar

    Johannes Betz et al. “A Software Architecture for an Au- tonomous Racecar”. In:2019 IEEE 89th V ehicular Tech- nology Conference (VTC2019-Spring). IEEE, Apr. 2019, pp. 1–6

    work page 2019

  23. [23]

    Sampling-Based Motion Planning with Online Racing Line Generation for Autonomous Driv- ing on Three-Dimensional Race Tracks

    Levent ¨Ogretmen et al. “Sampling-Based Motion Planning with Online Racing Line Generation for Autonomous Driv- ing on Three-Dimensional Race Tracks”. In:IEEE Intell. V eh. Symp. (IV). IEEE, June 2024, pp. 811–818

    work page 2024

  24. [24]

    Real-time optimization and nonlin- ear model predictive control of processes governed by differential-algebraic equations

    Moritz Diehl et al. “Real-time optimization and nonlin- ear model predictive control of processes governed by differential-algebraic equations”. In:J. Process Contr .12 (4 June 2002), pp. 577–585

    work page 2002

  25. [25]

    HPIPM: a high- performance quadratic programming framework for model predictive control

    Gianluca Frison and Moritz Diehl. “HPIPM: a high- performance quadratic programming framework for model predictive control”. In:IF AC-Pap.53 (2 Mar. 2020), pp. 6563–6569

    work page 2020

  26. [26]

    Longitudinal Control for Autonomous Racing with Combustion Engine Vehicles

    Phillip Pitschi et al. “Longitudinal Control for Autonomous Racing with Combustion Engine Vehicles”. In:IEEE Intell. V eh. Symp. (IV). IEEE, June 2025, pp. 1070–1077

    work page 2025

  27. [27]

    Analyzing the Impact of Simu- lation Fidelity on the Evaluation of Autonomous Driving Motion Control

    Simon Sagmeister et al. “Analyzing the Impact of Simu- lation Fidelity on the Evaluation of Autonomous Driving Motion Control”. In:IEEE Intelligent V ehicles Symposium, Proceedings. Institute of Electrical and Electronics Engi- neers Inc., 2024, pp. 230–237

    work page 2024

  28. [28]

    Modeling aggressive maneuvers on loose surfaces: The cases of Trail-Braking and Pendulum- Turn

    Efstathios Velenis et al. “Modeling aggressive maneuvers on loose surfaces: The cases of Trail-Braking and Pendulum- Turn”. In:2007 European Control Conference (ECC). IEEE, July 2007, pp. 1233–1240

    work page 2007

  29. [29]

    Human-Inspired Autonomous Racing in Low Friction Environments

    Trey P. Weber et al. “Human-Inspired Autonomous Racing in Low Friction Environments”. In:IEEE Trans. Intell. V eh. (2024), pp. 1–14

    work page 2024