Source-linked AI summary
A Singularity Avoiding Track Model for Race Car Optimization
Adwait Verulkar, Behzad Danaei, Srikanth Sivaramakrishnan, Ryan Trinter
TL;DR
Track-surface inaccuracies propagate into vehicle predictions, and centre-line coordinates can become non-injective inside tight-corner drivable regions. The paper fits a Monge-patch surface using a minimum-curvature reference line and exports periodic C4 quintic splines; with that reference, the lap-time NLP converges where the centre-line version does not.
Problem
Track-surface inaccuracies affect predicted lap time, tyre forces, suspension loads, and aerodynamic behaviour, while the centre-line Frenet map can become non-injective inside the drivable region on tight corners.
Source: §1 · p. 1, The Frenet singularity · p. 2Method
The pipeline fits a Monge patch anchored on a minimum-curvature line and exports a periodic, C4-continuous piecewise-quintic representation for Modelica vehicle simulation.
Source: §2.4 · p. 5Results
With the minimum-curvature joint, primal and dual residuals decrease monotonically to IPOPT tolerances and the NLP converges in a comparable number of iterations to well-conditioned track problems.
Source: §4 · p. 8Takeaways & Limitations
On processed circuits, the minimum-curvature line avoids the lateral centre-line distortion previously needed to push the Frenet fold outside the track width.
Source: §4 · p. 8The iterative curvature fit uses lateral bounds from the old reference line, which are approximate after the line moves and can leave the new line outside the polygon those bounds appeared to enclose.
Source: §2.2.2 · p. 5
Abstract
from arXiv · showhide
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.
1 Introduction
Track-surface errors propagate into vehicle loads, forces, and predicted lap times, while centre-line coordinates can become non-injective inside tight-corner drivable regions. This section motivates anchoring the Monge patch on a minimum-curvature reference line.
1 Introduction
- Track-surface inaccuracies propagate into predicted lap time, tyre forces, suspension loads, and aerodynamic behaviour because vehicle quantities depend on the road surface and its first two derivatives.Contact-patch normal load, gravitational projections, and wheel-on-road constraints are expressed using those road quantities.Source: §1 · p. 1
The Frenet singularity
- When the centre-line radius of curvature is smaller than the road’s half-width, lateral vectors can intersect inside the drivable region, making the Frenet map non-injective and road geometry multi-valued.The Jacobian vanishes at the fold and changes sign beyond it; the paper reports this failure on circuits with hairpins or tight slow-speed sequences.Source: The Frenet singularity · p. 2
- Near the fold, the time-to-arc-length transformation becomes ill-conditioned, producing a stiff spatial-domain NLP whose Hessian becomes singular at optimizer-relevant stations.On processed tracks, the minimum-curvature line keeps |1−κ(s)w| bounded away from zero across the track strip.Source: The Frenet singularity · pp. 2–3
- Centre-line offsets in cambered slow corners can force the lateral elevation polynomial to use larger high-order coefficients, producing spurious wiggles in its derivatives.The paper links these wiggles in ∂wz and ∂wwz to the tyre normal load.Source: The Frenet singularity · p. 3
Proposed solution
- The pipeline repurposes a minimum-curvature line as the Monge-patch coordinate reference, certifying Frenet-map injectivity across the full track strip rather than using it as a driving target.It fits lateral elevation polynomials and exports an arc-length-parameterised, periodic, C4-continuous piecewise-quintic representation for Modelica-based race-car optimisation.Source: Proposed solution · p. 3
2 Methodology
The pipeline converts a structured track point cloud into a periodic surface model by constructing a minimum-curvature reference frame, fitting elevation and track widths, and exporting smooth spline channels for Modelica.
2.1 Surface model
- The track surface is modeled as a Monge patch with user-selectable lateral polynomial degree; degree 3 captures camber and crown while keeping lateral derivatives well behaved on processed circuits.The surface derivatives and reference-frame tangents provide quantities needed for its first and second fundamental forms.Source: §2.1 · pp. 3–4
2.2 Minimum curvature reference line
- The minimum-curvature line minimizes integrated squared curvature within track bounds, typically hugs the inside through slow corners, and reduces lateral excursion relevant to fitting and time transformation.The recovered laterals are checked for intersection after recomputing the line and widths in the minimum-curvature frame.Source: §2.2 · p. 4
2.2.1 Quadratic program
- The quadratic program minimizes weighted squared discrete curvature subject to margin-shrunk lateral bounds, with length and jerk regularizers; the hairpin laterals no longer intersect inside the drivable region.The result is obtained from the proposed reference-line construction and restores Frenet-map injectivity in the illustrated region.Source: §2.2.1 · p. 4, Figure 3 · p. 4
2.2.2 Iteration, smoothing and containment
- A single QP solve is only a curvature-reducing step because the line-position problem is nonlinear under a frozen reference; the method refreshes arc-length sampling, normals, and ray-cast widths, with 3–5 iterations typically sufficient.The outer loop is a convex QP built on a linearisation of a nonlinear problem and iterated to a fixed point.Source: §2.2.2 · p. 5
2.2.3 Frenet frame and track widths
- A periodic quintic spline represents the closed minimum-curvature line, and ray-polyline intersections with both boundaries determine the left and right track widths.At each station, the smallest positive intersection parameter is used.Source: §2.2.3 · p. 5
2.3 Re-sampling and Monge-patch fit
- The surveyed point cloud is resampled on an orthonormal longitudinal–lateral grid in the minimum-curvature frame, then station elevations are fitted by least squares with a degree-N lateral polynomial.Elevation queries use Delaunay triangulation, with a thin-plate-spline radial-basis interpolant outside the triangulation's convex hull.Source: §2.3 · p. 5
2.4 Export: 𝐶4 piecewise quintic
- The export uses shared-knot periodic piecewise quintics with C4 continuity and periodic function values and first four derivatives, supporting continuous surface derivatives and real-time Modelica DAE evaluation.A zero-phase low-pass filter limits longitudinal elevation waves whose wavelength approaches the knot spacing.Source: §2.4 · pp. 5–6
3 Modelica Implementation
TrackJoint consumes the fitted track geometry to provide a differentiable road frame and constraints for Modelica vehicle optimization. Its spline and joint design supports smooth evaluation across the lap and a spatial-domain FMIOPT formulation.
3 Modelica Implementation
- The joint uses arc length s and lateral offset w as its coordinates, consumes geometry channels, and applies track widths separately as lateral path constraints.At N = 3, it consumes six geometry channels; the left and right widths bound w in the optimal-control problem.Source: §3 · p. 6, §3.1 · p. 6
3.1 Parameter record and spline storage
- A shared break vector lets the joint locate a segment once per step, while periodic values and derivatives through order four allow wraparound without a special case.Tunable coefficient matrices let one compiled FMU use a different circuit without recompilation.Source: §3.1 · p. 6
3.2 Segment lookup and Horner evaluation
- Uniform breaks enable constant-time segment lookup, and Horner-form channel derivatives keep evaluation branch-free in continuous variables for CasADi differentiation.The required fourth derivative makes piecewise quintics with C4 continuity the lowest-degree choice that keeps differentiated quantities continuous at knots; cubic splines can cause integrator chatter.Source: §3.2 · pp. 6–7
3.3 Surface geometry and the d’Alembert constraint
- The joint constructs surface position, tangents, and an analytic frame triad from the height polynomial, then uses d’Alembert relations to close the road constraints.The two wrench relations form an index-1 DAE and their Jacobian loses rank at the Frenet fold; the minimum-curvature reference line keeps the constraint solvable.Source: §3.3 · pp. 7–8
3.4 State selection and MultiBody assembly
- State selection keeps the track coordinates as continuous states, while heave and roll, pitch, and yaw joints express the chassis degrees of freedom relative to the track frame.The joint is configured as the orientation-graph root for its subtree.Source: §3.4 · p. 8
3.5 FMI 3.0 export and use in FMIOPT
- The FMI interface exports s-dot as vsTrack for FMIOPT’s time-to-space transformation and supports a feasible full-lap initial guess using a linear s ramp.The ramp sets other chassis coordinates and velocities to zero, placing the vehicle on the surface with its axes aligned to the track triad without preliminary simulation.Source: §3.5 · p. 8
4 Results
On the Singapore circuit, the minimum-curvature TrackJoint moves the Frenet fold outside the drivable strip, unlike the centre-line-anchored joint. The resulting lap-time problem converges, and the optimized solution includes racing-line and speed profiles.
4 Results
- On Singapore, the minimum-curvature reference line moves the Frenet fold outside the drivable strip without geometric distortion, even where corner radius is smaller than road width.The reported setup couples the TrackJoint to an F1 vehicle model in an FMI 3.0 FMU and solves a CasADi direct-collocation NLP with IPOPT.Source: §4 · p. 8
- The centre-line-anchored joint fails to converge on the tight-hairpin case, whereas the minimum-curvature joint reaches IPOPT tolerances with monotonically decreasing residuals in a comparable number of iterations.The centre-line case has erratic dual infeasibility; the minimum-curvature case makes previously intractable lap-time cases solvable without changing the vehicle model or solver settings.Source: §4 · p. 8, Figure 6 · p. 9
- The optimized solution reports the Singapore racing line as lateral offset within the drivable strip and the corresponding optimal velocity along arc length.Figure 7 shows the lateral offset; Figure 8 reports the speed profile.Source: §4 · pp. 8–9, Figure 7 · p. 9
5 Conclusions
The pipeline produces a closed, C4-continuous Monge-patch model from surveyed track data. Anchoring it on the minimum-curvature line removes the Frenet singularity and improves minimum-lap-time problem conditioning; the exported splines support Modelica vehicle simulation.
5 Conclusions
- Anchoring the Monge patch on the minimum-curvature line removes the Frenet singularity and improves minimum lap-time problem conditioning.The geometric centre-line is unsafe in corners tighter than the road is wide; the resulting curvature profile is close to what the vehicle experiences.Source: §5 · p. 9
- The track point cloud is converted into a closed, C4-continuous Monge-patch surface for high-fidelity Modelica vehicle simulation.The shared-knot, periodic piecewise-quintic splines are consumed by the TrackJoint component, which enforces the two-degree-of-freedom kinematic constraint.Source: §5 · p. 9
- The pipeline runs in roughly 5 minutes per circuit and reproduces surveyed elevations and bounds to an order of a few centimetres.Figure 8 presents the minimum-lap-time solution’s optimal speed profile along reference arc length.Source: §5 · p. 9, Figure 8 · p. 9