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

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Abstract

A track fitting pipeline is presented that converts a measured racetrack point cloud into a closed, C4C^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 C4C^4 B-splines, solved as a nonlinear program.

Figures & tables

Figure 1: Three-dimensional schematic of the Monge-patch track model rb(s,w)=r(s)+w n ⁣g(s)+z(s,w) ez\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 r(s)\bm{r}(s) (black) is free to move laterally within the bounds. The reference-line frame (n ⁣g(s),ez)(\bm{n}_{\!g}(s),\bm{e}_{z}) is drawn at a station along the line. At the marked station ss the vertical cross-section plane is spanned by the lateral normal n ⁣g(s)\bm{n}_{\!g}(s) (the ww axis) and ez\bm{e}_{z} ; the road altitude along that section is the lateral polynomial z(s,w)=∑k=0Nzk(s) wkz(s,w)=\sum_{k=0}^{N}z_{k}(s)\,w^{k} (red) shows that the elevation of every surface point (s,w)(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/κc(s)1/\kappa_{c}(s) of the reference line falls below the half-width b(s)/2b(s)/2 of the road, neighbouring lateral vectors n ⁣g(s)\bm{n}_{\!g}(s) intersect inside the drivable region and the Frenet map (s,w)↦rb(s,w)(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 (nlong×nlat)(n_{\mathrm{long}}\times n_{\mathrm{lat}}) resampling of the surveyed point cloud on the minimum-curvature frame.
Figure 5: Lateral degree- NN polynomial fit of the elevation z(si,wk)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 ss .
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