Single-Track Model

This document defines the implemented single-track backend in src/apexsim/vehicle/single_track_model.py and src/apexsim/vehicle/_single_track_physics.py.

Terminology note:

• SingleTrack in this package corresponds to the "bicycle model" terminology frequently used in vehicle-dynamics literature.

The single-track backend is implemented as a physical extension of the point-mass physical core (PointMassPhysicalMixin), with single-track-specific lateral force and diagnostic equations layered on top.

1. Scope

The single-track model keeps the solver API contract and adds axle-level lateral tire force modeling and quasi-static load transfer, while transient runs expose dynamic yaw behavior through residual diagnostics.

State assumptions in the lap-time solver context:

• quasi-steady envelope evaluation (no transient tire relaxation in profile solve),
• lateral force from load-sensitive Pacejka at front and rear axle,
• friction-circle coupling between lateral demand and longitudinal capability,
• net along-track acceleration after drag and grade corrections.

2. Lateral Limit

At speed \(v\) and banking angle \(\beta\), the model solves lateral capacity by a fixed-point iteration because normal load and lateral force depend on the lateral-acceleration estimate itself [1, Ch. 10].

Given current iterate \(a_y^{(k)}\):

1. Estimate axle loads from quasi-static vertical balance and load transfer.
2. Evaluate axle lateral forces at representative peak slip angle \(\alpha_\text{peak}\).
3. Update the lateral limit with banking contribution. The dependence on the previous iterate is carried by the axle normal loads used in tire-force evaluation:

\[ a_y^{(k+1)} = \max\left( a_{y,\min}, \frac{2F_y\!\left(\alpha_\text{peak}, F_{z,f}\!\left(a_y^{(k)}\right)/2\right) + 2F_y\!\left(\alpha_\text{peak}, F_{z,r}\!\left(a_y^{(k)}\right)/2\right)}{m} + g\sin\beta \right). \]

Equivalently, writing axle-level lateral forces explicitly as \(F_{y,f}^{(k)}\) and \(F_{y,r}^{(k)}\):

\[ a_y^{(k+1)} = \max\left(a_{y,\min}, \frac{F_{y,f}^{(k)} + F_{y,r}^{(k)}}{m} + g\sin\beta\right), \]

with \(F_{y,f}^{(k)}, F_{y,r}^{(k)}\) computed from \(F_{z,f}(a_y^{(k)})\) and \(F_{z,r}(a_y^{(k)})\) [1, Ch. 10].

Stop criterion:

\[ \left|a_y^{(k+1)} - a_y^{(k)}\right| \le \text{tol}_y \]

or max iteration count.

3. Lateral Tire Force Model (Pacejka)

Each axle force is the sum of two equivalent tires:

\[ F_{y,f} = 2\,F_y(\alpha_f, F_{z,f}/2), \quad F_{y,r} = 2\,F_y(\alpha_r, F_{z,r}/2). \]

Per-tire Pacejka-style lateral force:

\[ F_y = D\left(\frac{F_z}{F_{z,\text{ref}}}\right)\mu_\text{scale}(F_z)\sin\left(C\,\arctan\left(\xi\right)\right), \]

\[ \xi = B\alpha - E\left(B\alpha - \arctan(B\alpha)\right), \]

with load sensitivity factor:

\[ \mu_\text{scale}(F_z) = \max\left(1 + s\,\frac{F_z - F_{z,\text{ref}}}{F_{z,\text{ref}}},\ \mu_{\min}\right). \]

Here, \(D\) is the peak lateral force at the reference load \(F_{z,\text{ref}}\) with unit newton [1], [2].

4. Quasi-Static Normal Loads

Total vertical load:

\[ F_{z,\text{tot}} = mg + F_\text{down}(v), \quad F_\text{down}(v)=\tfrac{1}{2}\rho C_L A v^2. \]

Front axle raw load with longitudinal transfer:

\[ F_{z,f}^\text{raw} = mg\,\phi_f + F_{\text{down},f} - \frac{m a_x h}{L}. \]

Rear axle load follows from equilibrium:

\[ F_{z,r} = F_{z,\text{tot}} - F_{z,f}. \]

Lateral transfer is distributed by effective front roll-stiffness share and split to left/right wheel loads while preserving axle totals [1, Ch. 9], [3].

5. Longitudinal Limits with Friction Circle

For required lateral acceleration magnitude \(|a_{y,\text{req}}|\):

\[ \lambda = \sqrt{\max\left(0,\ 1 - \left(\frac{|a_{y,\text{req}}|}{a_{y,\text{lim}}}\right)^2\right)}. \]

Drive and brake envelopes:

\[ a_{x,\text{drive}} = a_{x,\text{drive,max}}\,\lambda,\quad a_{x,\text{brake}} = a_{x,\text{brake,max}}\,\lambda. \]

Net along-track acceleration:

\[ a_{x,\text{net}} = a_{x,\text{drive}} - \frac{D(v)}{m} - g\gamma, \]

available deceleration magnitude:

\[ a_{x,\text{decel,avail}} = \max\left(a_{x,\text{brake}} + \frac{D(v)}{m} + g\gamma,\ 0\right), \]

with

\[ D(v)=\tfrac{1}{2}\rho C_D A v^2. \]

The coupling above follows the common friction-circle approximation used in race-vehicle performance analysis [3].

6. Diagnostics

The backend reports at each operating point:

• yaw moment according to solver-mode semantics:
• quasi-static mode: zero by steady-state model assumption,
• transient mode: dynamic residual [ M_z = I_z \dot r, ]
• front and rear axle normal loads,
• tractive power:

\[ P = \left(m a_x + D(v)\right)v. \]

The solver uses quasi-steady envelopes for speed profile generation; the 3-DOF single-track dynamics model is used primarily for physically meaningful analysis diagnostics (e.g., yaw-moment traces).

7. Equation-to-Code Mapping

• lateral limit fixed-point: SingleTrackModel.lateral_accel_limit(...)
• Pacejka lateral force: magic_formula_lateral(...)
• normal-load estimation: estimate_normal_loads(...)
• friction-circle scaling: EnvelopeVehicleModel._friction_circle_scale(...)
• longitudinal accel/decel: SingleTrackModel.max_longitudinal_accel(...), SingleTrackModel.max_longitudinal_decel(...)
• diagnostics: SingleTrackModel.diagnostics(...), SingleTrackDynamicsModel.force_balance(...)

8. Example

• Single-track standalone usage: examples/spa/spa_lap_single_track.py
• Side-by-side comparison against point-mass model: examples/spa/spa_model_comparison.py

References

[1] D. Schramm, M. Hiller, and R. Bardini, Vehicle Dynamics: Modeling and Simulation. Berlin, Heidelberg: Springer, 2014.

[2] H. B. Pacejka, Tyre and Vehicle Dynamics, 2nd ed. Oxford: Butterworth-Heinemann, 2006.

[3] W. F. Milliken and D. L. Milliken, Race Car Vehicle Dynamics. Warrendale, PA: Society of Automotive Engineers, 1995.