A 2-DOF MultiBody Track Joint for Race Car Optimization

Srikanth Sivaramakrishnan; Behzad Danaei; Adwait Verulkar; Ryan Trinter

cs.ROOct 26, 2026

A 2-DOF MultiBody Track Joint for Race Car Optimization

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

Organizations: General Motors, USA

Abstract

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.

Figures & tables

Figure 1: Schematic of the TrackJoint . The body frame_b is constrained to the Monge-patch surface $\mathcal{M}$ and is free to move along the reference line ( $s$ ) and laterally across it ( $w$ ). At the current $(s,w)$ the joint evaluates the six channel splines $\{r_{x},r_{y},r_{z},z_{1},z_{2},z_{3}\}$ to build the surface position $\bm{r}_{b}(s,w)$ and the right-handed surface triad $(\bm{n}_{x},\bm{n}_{y},\bm{n}_{z})$ , whose tangent $\bm{n}_{x}$ follows the reference line and whose normal $\bm{n}_{z}$ is the surface normal. The reaction wrench is resolved onto the two free directions through the d’Alembert projection.

Figure 2: Modelica validation experiment of a point mass driven around the track using TrackJoint .

Figure 3: Validation of the TrackJoint surface output against the measured track geometry of the Singapore Grand Prix Circuit from LiDAR point cloud.

Figure 4: Path of the point mass driven through the TrackJoint and the error in absolute height against the 3D mesh track surface of the Singapore Grand Prix Circuit constructed from LiDAR point cloud.

Figure 5: Optimized velocity profile of a multibody vehicle model driven around the Singapore Grand Prix Circuit using TrackJoint in a FMIOPT minimum lap-time problem.

Figure 6: Optimized racing line of a multibody vehicle model driven around the Singapore Grand Prix Circuit using TrackJoint in a FMIOPT minimum lap-time problem.

Circuit	$L$ [m]	$N_{\mathrm{seg}}$	$R_{\mathrm{min}}$ [m]	bank [ ∘ ]	margin
Monaco	3283	657	9.3	5.2	1.03
Zandvoort	4185	837	31.2	16.2	2.32
Singapore	4858	972	23.0	2.1	1.56
Spa	6934	1387	22.5	4.6	1.42
Silverstone	5798	1160	41.2	1.9	2.38
all 23	3283	657	9.3	1.5	1.03

Table 1: Geometry of the fitted circuits: track length

L

, number of quintic segments, minimum reference-line radius, maximum bank angle and injectivity margin

\min_{s}1/(|\kappa|b)

. Extremes of the 23-circuit set.

	Horner	expanded
IPOPT status	optimal	optimal
Iterations	264	249
IPOPT time [s]	464.4	442.5
Final NLP error	$8.0\times 10^{-4}$	$3.8\times 10^{-4}$
Lap time [s]	90.6462	90.6429

Table 2: Singapore minimum lap-time NLP with the two FMU builds, which differ only in the spline evaluation function. Identical track, setup, guess, mesh and solver options.

	Rail	TrackJoint
Free DOF	1 ( $s$ )	2 ( $s,w$ )
Geometry	space curve	Monge-patch surface
Orientation	rail tangent or	analytic triad from
	3 angle tables	surface partials
Input data	$xyz$ or $xyz\phi$	6 quintic splines
	sample table	$+$ shared knots

Table 3: Claytex Rail versus TrackJoint .

Multibody Dynamics

Multibody dynamics studies how connected rigid or flexible bodies move and interact under forces, constraints, impacts, and friction.

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A Singularity Avoiding Track Model for Race Car Optimization

Adwait Verulkar, Behzad Danaei, Srikanth Sivaramakrishnan, Ryan Trinter

General Motors, USA

Curvature-Aware Optimization Point Cloud Surface Reconstruction

Open paper AskPDF arXiv

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.

Abstract

Multibody Dynamics

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