A Singularity Avoiding Track Model for Race Car Optimization

Adwait Verulkar; Behzad Danaei; Srikanth Sivaramakrishnan; Ryan Trinter

cs.ROOct 27, 2026

A Singularity Avoiding Track Model for Race Car Optimization

Authors: Adwait Verulkar, Behzad Danaei, Srikanth Sivaramakrishnan, Ryan Trinter

Organizations: General Motors, USA

Abstract

A track fitting pipeline is presented that converts a measured racetrack point cloud into a closed, $C^4$ differentiable, three-dimensional surface model suitable for high-fidelity Modelica race-car optimization. The surface is a Monge patch, parametrised by a curvilinear arc-length coordinate measured along the minimum-curvature line (MCL) and a signed lateral offset, with a height polynomial of user-selectable degree capturing the cross-sectional elevation profile. The choice of MCL as the reference line is deliberate since anchoring the same surface on the geometric centre-line can cause adjacent normals to cross inside the drivable region on tight corners. The resulting non-injective parametrisation assigns multiple curvilinear addresses to the same three-dimensional point and returns invalid road geometry. The MCL is obtained from a quadratic program that minimises integrated curvature subject to track-boundary, periodicity, and jerk-regularisation constraints. By reducing total curvature while remaining within the track bounds, the MCL moves the region of parametrisation breakdown outside the drivable strip on all circuits studied, keeps the normal map injective across the full track width, and yields a more linear time-to-arc-length transformation for minimum lap-time problems. The fitted surface and the associated track widths are exported as periodic piecewise-quintic $C^4$ B-splines, solved as a nonlinear program.

Figures & tables

Panel of Figure 1 — Figure 1: Three-dimensional schematic of the Monge-patch track model $\bm{r}_{b}(s,w)=\bm{r}(s)+w\,\bm{n}_{\!g}(s)+z(s,w)\,\bm{e}_{z}$ . The drivable corridor is delimited by the left/right track boundaries (grey); the geometric centre-line (dashed) runs midway between them, whereas the minimum-curvature reference line $\bm{r}(s)$ (black) is free to move laterally within the bounds. The reference-line frame $(\bm{n}_{\!g}(s),\bm{e}_{z})$ is drawn at a station along the line. At the marked station $s$ the vertical cross-section plane is spanned by the lateral normal $\bm{n}_{\!g}(s)$ (the $w$ axis) and $\bm{e}_{z}$ ; the road altitude along that section is the lateral polynomial $z(s,w)=\sum_{k=0}^{N}z_{k}(s)\,w^{k}$ (red) shows that the elevation of every surface point $(s,w)$ above the datum plane equals the value of that polynomial.

Figure 2: Geometric origin of the centre-line defect. Where the local radius of curvature

1/\kappa_{c}(s)

of the reference line falls below the half-width

b(s)/2

of the road, neighbouring lateral vectors

\bm{n}_{\!g}(s)

intersect inside the drivable region and the Frenet map

(s,w)\mapsto\bm{r}_{b}(s,w)

ceases to be injective.

Figure 3: Result of the proposed reference-line construction on the same hairpin region as Fig. 2 . The minimum-curvature line (MCL, red) stays strictly inside the track and the recovered laterals (grey) no longer intersect within the drivable region, restoring injectivity of the Frenet map.

Figure 4: Regular

(n_{\mathrm{long}}\times n_{\mathrm{lat}})

resampling of the surveyed point cloud on the minimum-curvature frame.

Figure 5: Lateral degree-

N

polynomial fit of the elevation

z(s_{i},w_{k})

at each longitudinal station.

Figure 6: IPOPT convergence trace of the minimum lap-time NLP on a circuit containing a hairpin tighter than the road is wide. The non-converging case uses the centre-line-anchored track joint of Limebeer et al. (2023) : the dual infeasibility jumps erratically across iterations and the problem fails to converge.

Figure 7: Optimal racing line of the minimum lap-time solution on the Singapore circuit, expressed as the lateral offset within the drivable strip.

Figure 8: Optimal speed profile of the minimum lap-time solution along the reference arc length

s

Curvature-Aware Optimization

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Oct 26, 2026

A 2-DOF MultiBody Track Joint for Race Car Optimization

TrackJoint is a Modelica MultiBody component that turns a three-dimensional road surface into a joint with two degrees of freedom. The two joint coordinates are the distance travelled along a reference line and the sideways offset from that line. Together they give the position, orientation and angular velocity of a single frame_b connector. All road geometry reaches the joint as a small set of periodic piecewise-quintic splines, so one component can represent any circuit whose splines have been fitted beforehand. The road constraint is closed with d'Alembert's principle. The two generalized forces that belong to the joint coordinates are set to zero, so no Lagrange multipliers are needed. The joint is connected to a multibody race car in Dymola and the pair is exported as one FMI3.0 FMU. The same FMU is then used for time-domain simulation and for CasADi/IPOPT minimum lap-time optimization, with no change to the model in between. This works because the joint keeps the arc-length coordinate as a real state and publishes its time derivative, which is the quantity that lets the problem be rewritten with distance We report results for minimum lap time simulations on the Singapore Formula1 circuit.

Srikanth Sivaramakrishnan, Behzad Danaei, Adwait Verulkar +1

General Motors, USA

Multibody Dynamics Vehicle Dynamics

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Feb 17, 2026

Spatially-Aware Adaptive Trajectory Optimization with Controller-Guided Feedback for Autonomous Racing

We present a closed-loop framework for autonomous raceline optimization that combines NURBS-based trajectory representation, CMA-ES global trajectory optimization, and controller-guided spatial feedback. Instead of treating tracking errors as transient disturbances, our method exploits them as informative signals of local track characteristics via a Kalman-inspired spatial update. This enables the construction of an adaptive, acceleration-based constraint map that iteratively refines trajectories toward near-optimal performance under spatially varying track and vehicle behavior. In simulation, our approach achieves a 17.38% lap time reduction compared to a controller parametrized with maximum static acceleration. On real hardware, tested with different tire compounds ranging from high to low friction, we obtain a 7.60% lap time improvement without explicitly parametrizing friction. This demonstrates robustness to changing grip conditions in real-world scenarios.

Alexander Wachter, Alexander Willert, Marc-Philip Ecker +1

Automation & Control Institute (ACIN), TU Wien, Vienna, Austria · AIT, Austrian Institute of Technology GmbH, Vienna, Austria

Trajectory Optimization Autonomous Racing Constrained Optimization Adaptive Control

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Jun 30, 2026

AD-MPCC: Adaptive Differentiable Model Predictive Contouring Control for Autonomous Racing

This paper presents Adaptive Differentiable Model Predictive Contouring Control (AD-MPCC), a framework for autonomous racing that integrates differentiable MPCC with online parameter estimation to handle varying road-surface conditions. For online parameter estimation, we leverage a parameterized Pacejka Magic Formula together with a regularized moving-horizon estimation scheme with exponentially decaying weights to capture road interactions and update parameters in real time. Furthermore, we propose a differentiable MPCC (Diff-MPCC) framework that enables optimal adjustment of objective weights based on predefined long-horizon performance costs. To implement Diff-MPCC for online objective weight adaptation, we propose a Pacejka-informed machine learning model that is trained in a supervised manner using data generated by Diff-MPCC to tune the objective weights. Simulation results demonstrate that AD-MPCC reliably ensures safety and achieves faster lap times compared to baseline controllers in both single-surface and multiple-surface scenarios.

Nam T. Nguyen, Binh Nguyen, Ahmad Amine +3

Department of Electrical and Computer Engineering, University of Central Florida, Orlando, FL 32816, USA. · Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA. · CTI Lab4EV, School of Electrical and Electronic Engineering, Hanoi University of Science and Technology, Hanoi 100000, Vietnam.

Model Predictive Control Nonlinear MPC Autonomous Racing Adaptive Control Vehicle Dynamics Moving Horizon Estimation+1

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Abstract

Curvature-Aware Optimization

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