Source-linked AI summary
A 2-DOF MultiBody Track Joint for Race Car Optimization
Srikanth Sivaramakrishnan, Behzad Danaei, Adwait Verulkar, Ryan Trinter
TL;DR
Race-car simulation and lap-time optimization need a road-surface joint that supplies geometry while constraining vehicle motion to the road. TrackJoint provides that joint from fitted splines and is used in a full-detail multibody vehicle optimization on Singapore, where a single solve converges. The reported validation and optimization studies remain limited in scope.
Problem
A reusable MultiBody component for a spline-defined racetrack surface is needed to support road-constrained vehicle simulation and optimization.
Source: §1 · p. 1Method
TrackJoint represents the road as a two-degree-of-freedom Monge-patch joint whose geometry is computed from fitted periodic piecewise-quintic splines.
Source: §1 · p. 1Results
A single converged FMIOPT direct-collocation solve produced the optimized racing line and velocity profile for a full-detail multibody vehicle on Singapore Grand Prix Circuit.
Source: §5.2 · p. 7Takeaways & Limitations
TrackJoint was used for full-circuit minimum lap-time optimization of a detailed multibody vehicle while also serving as a reusable component for time-domain simulation.
Source: §5.2 · p. 7Validation used one circuit and one point-mass run, while optimization used one vehicle, setup, and initial guess per circuit; no spline-resolution or multi-seed study was run.
Source: §6 · p. 8
Abstract
from arXiv · showhide
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 FMI~3.0 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 Formula~1 circuit.
1 Introduction
TrackJoint extends a one-dimensional rail joint into a reusable two-degree-of-freedom joint for a spline-defined racetrack surface. It consumes fitted road splines to provide geometry for multibody simulation and optimization.
1 Introduction
- Earlier work includes a symbolic path-following joint and the Claytex Rail, which constrains motion to a one-dimensional B-spline curve.The Rail exposes orientation through a heading frame or three independent angle splines, whereas TrackJoint represents a two-dimensional road surface.Source: §1 · p. 1
- TrackJoint extends the Claytex Rail concept from a one-degree-of-freedom curve to a two-degree-of-freedom Monge-patch racetrack surface.The body moves along a planar reference line and laterally across it; its orientation is constructed from surface partials, and two d’Alembert relations close the wrench.Source: §1 · p. 1
- The surface position is parameterized by distance along a planar reference line and lateral offset, with elevation represented by a cubic in the offset.The six periodic piecewise-quintic B-spline channels are fitted separately from measured racetrack point clouds and anchored on the circuit’s minimum-curvature line.Source: §1 · p. 1
- The contribution is a reusable Modelica MultiBody track joint intended for both time-domain simulation and optimization.Horner expansion, constant-time segment lookup, and a shared spline evaluator make repeated evaluation at CasADi/IPOPT collocation points inexpensive.Source: §1 · pp. 1–2
2 Surface model
The surface model uses along-track and lateral coordinates to locate a point on a smooth road patch. Its validity depends on avoiding a curvature fold within the drivable strip, and its spline continuity must support the joint’s high derivatives.
2 Surface model
- The joint represents a point on the road surface using along-track distance and lateral offset, with those coordinates and their time derivatives carried as states.The position and other quantities are evaluated from the surface splines at the current along-track coordinate.Source: §2 · p. 2
2.1 Regularity of the parametrization
- A reference-line fold inside the road strip makes the surface mapping non-one-to-one, so the joint’s input data must keep the fold outside that strip.Anchoring splines on the minimum-curvature line lowers curvature peaks and moves the fold outside the drivable strip.Source: §2.1 · p. 2
2.2 Continuity required of the splines
- Piecewise quintic splines with C4 continuity are required because the joint and DAE index reduction use derivatives through fourth order.Cubic splines leave a second derivative discontinuity at knots, which makes the integrator chatter at segment boundaries.Source: §2.2 · p. 2
3 Joint formulation
TrackJoint constrains a frame to a Monge-patch road surface while leaving distance along the reference line and lateral offset free. From those coordinates and the fitted splines, it reconstructs the frame pose and angular velocity, then closes the constraints through d’Alembert relations.
3.1 States and frame outputs
- The free coordinates (s,w) determine frame_b’s position, orientation and angular velocity on the road surface.The joint evaluates the road splines at the current coordinates to reconstruct the surface frame.Source: §3.1 · pp. 2–3, §3 · p. 2, Figure 1 · p. 3
3.2 Parametric rotation rates
- Closed-form parametric rotation rates are built from surface partials, combining in-surface rotation with the turn about the surface normal.The required spline derivatives extend to third order in s for the planar channels and second order for the height channels.Source: §3.2 · p. 3
3.3 Time-to-space transformation for minimum lap-time problems
- Using arc length s as the independent variable fixes the integration domain to [0,L], and the exported ˙s rescales vehicle-state derivatives for collocation.The time-to-space transformation uses track-triad velocity components, chassis yaw and reference-line curvature to obtain ˙s.Source: §3.3 · pp. 3–4
3.4 D’Alembert constraint
- Setting the generalized forces conjugate to (s,w) to zero yields two algebraic d’Alembert equations that close the DAE without multiplier variables.The equations are linear in the six wrench components.Source: §3.4 · p. 4
- The constraint system has rank two only when ∂srb and ∂wrb are independent; at the fold, the joint’s constraint Jacobian loses rank and the DAE cannot be solved.The cited spline condition keeps the constraint matrix full rank across the drivable strip.Source: §3.4 · p. 4
3.5 Compact form of the implementation
- The ground-plane left normal defines the lateral direction, so frame orientation follows from spline derivatives without a separate yaw spline or angle integration.The implementation uses the reference-line first and second s-derivatives to determine orientation.Source: §3.5 · p. 5
- The implementation evaluates the Monge cubic and its partials in Horner form for derivative-chain conditioning, although explicitly expanded polynomials compile faster.The equation block also assembles the surface triad, parametric rotation rates and d’Alembert relations.Source: §3.5 · pp. 4–5
- C4-continuous splines supply the second s- and first mixed sw-derivatives required when the solver differentiates the angular-velocity equation.The fitting pipeline enforces this continuity at interior knots and the periodic wrap.Source: §3.5 · p. 5
4 Implementation and FMI export
The implementation stores the road as shared-segment quintic spline channels, fixes the joint coordinates as continuous states, and integrates TrackJoint with a road-aligned chassis. The combined vehicle and joint are exported as a tunable FMI 3.0 model-exchange FMU.
4.1 Parameter record and spline storage
- Six road channels use degree-five coefficient matrices and a shared break vector; uniform breaks allow constant-time segment lookup, with binary search available for nonuniform breaks.Spline values and derivatives are evaluated through a custom Horner-form function.Source: §4.1 · p. 5
4.2 State selection and initialization
- The joint marks (s, ˙s, w, ˙w) as continuous states and exposes initial values as parameters, avoiding state-aliasing failures in the described chassis configuration.A boolean override lets users relax the forced state selection.Source: §4.2 · p. 5
4.3 MultiBody integration
- A surface-normal heave joint and track-relative roll, pitch and yaw joints follow frame_b, giving the chassis six degrees of freedom relative to the road.Heave is along the patch’s out-of-plane normal nz, while the three attitude angles are relative to the track triad.Source: §4.3 · pp. 5–6
4.4 FMI 3.0 export
- The vehicle and joint are exported together as one FMI 3.0 model-exchange FMU, with spline matrices exposed as tunable parameters for runtime circuit changes without recompilation.The exported state vector includes the joint coordinates and the stacked chassis position states and derivatives.Source: §4.4 · p. 6
5 Results
TrackJoint reproduces the measured Singapore road geometry in time-domain simulation and supports a converged full-circuit minimum lap-time optimization. Benchmarks report 23-circuit coverage and compare spline-evaluation speed, accuracy, and optimization outcomes.
5.1 Time-domain simulation
- The reconstructed Singapore surface closely tracks the LiDAR mesh in absolute height, grade, and bank angles across the lap.Source: §5.1 · p. 6, Figure 3 · p. 7, Figure 4 · p. 7
5.2 Minimum lap-time direct collocation
- On minimum-curvature-line circuits, the time-to-space denominator stays bounded away from zero and the NLP converges; geometric-centerline fits exhibit the documented convergence failure.Source: §5.2 · p. 6
- A zero chassis-coordinate guess with a linear arc-length ramp satisfies the road constraint at every collocation point and needs no preliminary simulation or circuit data beyond length.For Singapore, IPOPT reached “Optimal Solution Found” in 264 iterations.Source: §5.2 · pp. 6–7
- One converged FMIOPT solve produced the optimized racing line and velocity profile for a full-detail multibody vehicle on Singapore.The racing line and velocity profile are shown in Figures 6 and 5, respectively.Source: Figure 6 · p. 7, Figure 5 · p. 7, §5.2 · p. 7
5.3 Quantitative benchmarks
- 2.63 s versus 3.54 s: expanded polynomial evaluation cut joint-only integration time by 26%; FMU evaluations dominate the 464 s solve cost.The speed comparison used bit-identical trajectories; the solve-cost breakdown attributes 167 s to objective-gradient evaluations and 120 s to constraint evaluations.Source: §5.3 · pp. 7–8
- All 23 fitted circuits have injectivity margin above 1; Monaco is tightest at 1.03.The set includes very tight corners and Zandvoort banking up to 16.2°.Source: §5.3 · p. 7, Table 1 · p. 8
- Horner's maximum relative error was 2.6 × 10−12 versus 5.0 × 10−12 for expanded form; both FMUs converged, with lap times differing by 3.3 ms out of 90.6 s.Source: §5.3 · p. 8, Table 2 · p. 8
- Unlike the 1-DOF Claytex Rail, TrackJoint represents a surface and derives the orientation triad from surface partials using six scalar splines without extra tabulated data.Source: §5.3 · p. 8, Table 3 · p. 8
6 Limitations
TrackJoint’s scope is constrained by its dependence on externally fitted splines, limited validation, proprietary artifacts, and implementation assumptions.
Dependence on the fitting pipeline
- A faulty reference-line anchor can go undetected at runtime, while broken spline condition (10) causes the constraint Jacobian to lose rank.The proposed initialization guard on mins 1/(|κ|b) is not implemented.Source: §6 · p. 8
Validation scope
- Validation covers LiDAR terrain for one circuit and point-mass run, and optimization with one vehicle, setup, and initial guess per circuit.No spline-resolution or multi-seed convergence study was run, so robustness claims reflect experience rather than measurement.Source: §6 · p. 8
Availability
- The component, vehicle models, and fitted circuit data are proprietary and unavailable, although the formulation is provided for reimplementation.Source: §6 · p. 8
Implementation restrictions
- The implementation assumes uniform knots and a closed circuit, and four-wheel contact queries require four joint instances sharing one parameter record.TrackJoint exposes only one frame_b connector.Source: §6 · p. 8
7 Conclusions
TrackJoint packages a racetrack surface as a reusable MultiBody joint, with the spline regularity and track-relative chassis coordinates supporting well-posed optimization and initialization. The conclusions also identify computational findings and implementation extensions still to pursue.
7 Conclusions
- TrackJoint exposes one frame_b connector and two joint coordinates, packaging the road model as an ordinary MultiBody joint with six scalar splines and a shared knot vector.Source: §7 · pp. 8–9
- All 23 fitted circuits satisfy mins 1/(|κ|b) > 1, the condition for a well-posed DAE; Monaco is tightest at 1.03.The condition prevents the constraint Jacobian from losing rank on the Frenet fold.Source: §7 · p. 9
- Setting chassis coordinates to zero and ramping s initializes the vehicle on the road with aligned axes at every collocation point, without preliminary simulation.Source: §7 · p. 9
- About two thirds of a 464 s Singapore solve runs inside the FMU, and two mathematically identical polynomial forms showed a measured 26% gap.The authors note that generated code, rather than operation count on paper, determines the observed difference.Source: §7 · p. 9
- Next steps include asserting the injectivity margin at initialization, testing the lateral cubic, adopting adaptive knots, and sharing one parameter record across four contact-patch instances.Source: §7 · p. 9