Skip to content

19. Tire — LuGre / brush-dynamic model

Learning objectives

After this chapter you can:

  1. Explain why quasi-static Pacejka becomes ill-conditioned as \(|v_x| \to 0\) (slip ratio / slip angle singularities) and what physical behaviour is missing.
  2. Derive the LuGre bristle state equation and identify the roles of \(z\), \(v_r\), \(\sigma_0\), \(\sigma_1\), \(\sigma_2\), and the breakaway envelope \(g\).
  3. Show how MF96 can serve as \(g(\cdot)\) so steady sliding recovers the same slip curve as Chapter 03.
  4. Contrast presliding (stick), micro-slip, and macro-slip (sliding) operating points without switching models.
  5. Describe VDSim's shipped adoption: decoupled long/lat LuGre, asymmetric breakaway floors, peak-axis friction ellipse, RK4 \(z\) sub-advances, ISO lateral sign, and opt-in via lugre.enabled.

Prerequisites

  • Chapter 01 — slip angle \(\alpha\), slip ratio \(\kappa\), ISO 8855 signs.
  • Chapter 03 — Pacejka MF96 pure slip, friction ellipse, relaxation length.
  • Chapter 11 — substep integration; why stiff states need implicit updates.
  • Design note — docs/design/LOW_SPEED_HANDLING.md (kinematic blend alternative in VDSim default config).

19.1 Motivation — one model from rest to slide

Chapter 03 treats tire force as an algebraic map \((\kappa, \alpha, F_z, \mu) \mapsto (F_x, F_y)\). That is correct in the steady sliding regime: the contact patch has reached a quasi-static deflection distribution and the Pacejka curve is an empirical fit to that limit.

Two gaps appear at low speed:

  1. Singularity — \(\kappa = (R\omega - v_x)/v_x\) and \(\alpha = \mathrm{atan2}(v_y, v_x)\) divide by \(v_x\). Any numerical floor on \(v_x\) injects noise into forces and yaw moment at parking speeds.
  2. Missing physics — before the patch slides, tread elements deflect elastically (brush / bristle behaviour). Static hold on a grade, stick-slip, and presliding hysteresis live in this regime; they are not in a pure slip curve.

The brush model is the physical origin of the slip curve: micro-elements in the adhesion zone deflect proportionally to slip; in the sliding zone they saturate at \(\mu F_z\). Integrating that picture over the patch yields a Pacejka-shaped steady state. Making bristle deflection a dynamic state (LuGre) unifies rest, presliding, and sliding in one ODE instead of blending to a kinematic bicycle at low speed.

  v_r ≈ 0, |z| small          v_r large, z → g/σ₀
  ─────────────────           ─────────────────────
  F ≈ σ₀ z  (elastic stick)   F ≈ g(v_r)  (MF96 envelope)
        presliding                    macro-slip

19.2 Assumptions

Assumption Meaning Breaks when
Point-contact bristle one \(z\) per force component per wheel patch length dynamics, carcass modes
\(g\) from steady MF96 breakaway force envelope from Ch.03 need temperature / wear in \(g\)
Decoupled long / lat (VDSim) separate \(z_x\), \(z_y\); ellipse rescale after combined brush (Deur 2004) preferred
Rigid wheel / flat road \(v_r\) from rigid kinematics large compliance, obstacle impacts
No camber-induced \(v_r\) coupling camber enters MF96 \(g\) only via Ch.03 inputs full 3D brush
Host integrates \(z\) ITireModel stays stateless —

19.3 Physical picture — brush and slip velocity

Imagine one representative tread element (bristle) in the contact plane.

  • Stick / presliding: the element tip deflects by \(z\) [m] while the base moves with the wheel; force is elastic \(F \approx \sigma_0 z\).
  • Sliding: deflection saturates; force approaches the friction limit \(g\).
  • Relative slip velocity \(v_r\) [m/s] drives how fast \(z\) tries to follow the kinematic mismatch between wheel and ground.

For one wheel in the wheel frame:

Direction Slip velocity \(v_r\) (VDSim)
Longitudinal \(v_{r,x} = R\omega - v_{x,\mathrm{wheel}}\)
Lateral \(v_{r,y} = v_{y,\mathrm{wheel}}\)

Sign convention follows Chapter 01 / 03: positive \(F_x\) drives the vehicle forward when \(\kappa > 0\); positive \(F_y\) is ISO 8855 lateral force (restoring — opposite to the direction of sideslip velocity at the contact).

19.4 LuGre state equations

Canudas de Wit et al. (1995) model friction through an internal deflection \(z\) with dynamics:

\[ \dot z = v_r - \frac{\sigma_0\,|v_r|}{g(v_r)}\,z \]
\[ F = \sigma_0\,z + \sigma_1\,\dot z + \sigma_2\,v_r \]
Symbol Role Typical unit
\(z\) bristle deflection (internal state) m
\(v_r\) relative slip velocity at contact m/s
\(\sigma_0\) bristle stiffness N/m
\(\sigma_1\) micro-damping (critical \(\approx 2\sqrt{\sigma_0 m_{\mathrm{eff}}}\)) N·s/m
\(\sigma_2\) viscous friction N·s/m
\(g(v_r)\) breakaway force magnitude (steady envelope) N

The nonlinear term \(\sigma_0 |v_r| z / g\) drives \(z \to g/\sigma_0\) when sliding persists, so \(F \to g\) — recovering a steady slip curve.

Operating limits

Rest (\(v_r \to 0\), constant \(z\)): \(\dot z = 0\) requires no motion of \(z\) from the ODE alone; force is \(F = \sigma_0 z\) (elastic stick). External load balance must set \(z\) — in practice initialization and load transients matter.

Steady sliding (\(v_r \neq 0\), constant \(z\)): \(z_{\mathrm{ss}} = g/\sigma_0\), hence \(F_{\mathrm{ss}} = g(v_r)\) when \(\sigma_2\) is small.

Low-speed singularity removed: forces use \(v_r\) directly, not \(\kappa = v_r/v_x\), so there is no \(1/v_x\) in the constitutive law.

19.5 Choosing \(g\) — Pacejka as steady-state envelope

VDSim sets the breakaway envelope from the existing MF96 evaluator (Chapter 03) at the current \((\kappa, \alpha, F_z, \mu_{\mathrm{long}}, \mu_{\mathrm{lat}})\), with axis-specific floors so presliding stiffness exists without phantom forces at rest.

Longitudinal

\[ g_x = \max\bigl(|F_x^{\mathrm{MF96}}|,\; 1\,\mathrm{N}\bigr) \]

There is no \(\mu_{\mathrm{long}} F_z\) floor on \(g_x\). At \(\kappa \approx 0\) the MF longitudinal force is near zero; a Coulomb floor would cap \(|z|\) as if the wheel were sliding while stationary, producing phantom \(F_x\) and wheel-spin chatter with zero throttle.

Lateral

\[ g_y = \max\Bigl(|F_y^{\mathrm{MF96}}|,\; w(\alpha)\,\mu_{\mathrm{lat}} F_z,\; 1\,\mathrm{N}\Bigr), \quad w(\alpha) = \exp\!\left(-\left(\frac{\alpha}{0.03\,\mathrm{rad}}\right)^2\right) \]

When \(\alpha \to 0\), MF96 sends \(|F_y^{\mathrm{MF}}| \to 0\) but cornering still needs presliding stiffness — the gated \(\mu_{\mathrm{lat}} F_z\) floor supplies it. When \(|\alpha|\) grows, \(w \to 0\) and \(|F_y^{\mathrm{MF}}|\) governs \(g_y\), so turn+throttle does not get an artificial lateral saturation from the floor.

LuGre changes how fast force approaches these envelopes (transient + low-speed), not the high-slip steady curve used in ISO validation.

ISO lateral sign

Raw bristle force from \(v_{r,y} = v_{y,\mathrm{wheel}}\) has the same sign as \(\alpha\) at moderate speed, while Pacejka uses \(F_y = -D\sin(\cdots)\). The host applies \(F_y = -F_{\mathrm{raw}}(z_{\mathrm{lat}}, v_{r,y}, g_y)\) so LuGre matches MF96 sign and magnitude bounds.

Combined slip

When combined_slip_enabled (catalog default), decoupled \((F_x, F_y)\) are rescaled onto the peak friction ellipse (Chapter 03 / task 15):

\[ F_{x,\max} = D_{\mathrm{long}}\,\mu_{\mathrm{long}}\,\mu_{\mathrm{eff}}\,F_z, \quad F_{y,\max} = D_{\mathrm{lat}}\,\mu_{\mathrm{lat}}\,\mu_{\mathrm{eff}}\,F_z \]

not onto instantaneous \(|F_x^{\mathrm{MF}}|\), \(|F_y^{\mathrm{MF}}|\). With LuGre on, the host also skips the extra circular \(\sqrt{F_x^2+F_y^2} \le \mu F_z\) clip so grip is not reduced three times in combined slip.

Aligning moment: \(M_z = -t_p(\alpha)\,F_y\) with pneumatic trail falloff (Chapter 03).

19.6 Time discretization — semi-implicit \(z\)

The term \(\sigma_0 |v_r| z / g\) makes \(\dot z\) stiff at large \(\sigma_0\). Explicit Euler on \(z\) can go unstable at \(\Delta t = 5\,\mathrm{ms}\) (race / real-time step).

VDSim uses the same semi-implicit bristle update as steering LuGre (Chapter 17):

\[ z_{k+1} = \frac{z_k + \Delta t\, v_r}{1 + \Delta t\,\sigma_0\,|v_r|/g} \]

then clamp \(|z_{k+1}| \le g/\sigma_0\). Force uses \(\dot z \approx (z_{k+1} - z_k)/\Delta t\) in the \(\sigma_1\) term.

Coupling to the vehicle integrator

Within one derivatives() call, \(z\) is frozen (all RK4 substates in that evaluation see the same bristle state).

For RK4, \(z\) is advanced after each of the four stages with \(\Delta t_{\mathrm{stage}} = h/4\), so the total bristle integration per substep is \(h\). There is no fifth advance at step end.

For Euler, one derivatives + one advance_lugre_states(h).

Wheel spin uses the same clipped longitudinal force as the body: \(d\omega = (T_{\mathrm{drive}} + T_{\mathrm{brake}} - F_x R) / I_{\mathrm{wheel}}\) with \(F_x\) from LuGre after the friction ellipse — no separate MF split on fx_kin.

19.7 Longitudinal / lateral coupling

Textbook target (Deur et al., 2004): a 2D bristle field in the contact plane so the friction ellipse emerges from geometry instead of a post-scale.

VDSim shipped model: independent LuGre channels per wheel for \(x\) and \(y\), then peak-axis friction-ellipse rescale when combined_slip_enabled. Simpler than Deur's 2D brush and tuned for RWD throttle+steer, grade hold, and coast without phantom longitudinal force — but does not capture shear coupling in the adhesion zone (e.g. simultaneous brake-in-turn transient path).

19.8 Comparison with VDSim default low-speed path

When lugre.enabled: false (catalog default), VDSim uses:

  • kinematic–dynamic blend on \((v_y, r)\) below \(V_{\mathrm{blend}} \approx 3\,\mathrm{m/s}\);
  • lambda fade on lateral (and display longitudinal) tire force;
  • viscous brake-hold damper on longitudinal creep.

When lugre.enabled: true, those paths are disabled; LuGre forces feed the full dynamic EOM at all speeds. Catalog default remains enabled: false so ISO and blend baselines are unchanged. Tuned preset: tire.lugre_on (\(\sigma_0 = 3\times 10^5\,\mathrm{N/m}\), \(\sigma_2 = 120\,\mathrm{N{\cdot}s/m}\)). Implementation detail: V0.2_TIRE_LUGRE.md.

Aspect MF96 + blend (default) LuGre + MF96 \(g\) (opt-in)
Low-speed lateral kinematic constraint bristle elasticity
Standstill on grade brake-hold damper (small creep) bristle \(z\) + presliding \(g_y\)
High-speed curve MF96 MF96 steady limit
State in host \(\alpha_{\mathrm{dyn}}\) optional \(z_x\), \(z_y\) per wheel
Singularity avoided by blend, not physics avoided in constitutive law

19.9 Validation and limits

Check Expectation (ctest LuGreTire/*)
Steady deflection \(z = g/\sigma_0 \Rightarrow F \approx g\)
High-speed cruise $
Grade creep less creep than blend (LessGradeCreepThanKinematicBlend)
Coast at rest no phantom \(F_x\) (CoastFlatNoPhantomLongitudinal)
Throttle + steer longitudinal not “iced” (LongitudinalWithSmallSteer)
RWD \(\kappa\) bounded under throttle (RearSlipBoundedUnderThrottle)
ISO \(F_y\) sign matches MF96 (LateralForceOpposesSlipIso8855)
Yaw from small \(r_0\) does not run away (LowSpeedYawDoesNotRunaway)

Not modeled: tire temperature, wear, pressure transient, belt dynamics, full MF2002, relaxation of carcass independent of LuGre \(z\), force-dependent camber feedback, reverse-driving \(\alpha\) wrap.

19.10 VDSim implementation notes

[VDSim impl] — state and parameters

  • TireParams.lugre: enabled, sigma0, sigma1, sigma2, m_eff.
  • YAML: lugre: under configs/parts/tire/*.yaml; default enabled: false.
  • Per wheel: z_long, z_lat in the dynamics host (not in ITireModel).

[VDSim impl] — force pipeline (lugre_tire.hpp)

  1. MF96 compute(in) → lugre_breakaway() → \(g_x\), \(g_y\).
  2. lugre_force(z, v_r, g, p) → \(F_x\); \(F_y = -F_{\mathrm{raw,lat}}\).
  3. lugre_friction_ellipse() if combined_slip_enabled.
  4. \(M_z = -t_p(\alpha)\,F_y\); fx_kin = Fx for wheel spin.

[VDSim impl] — hosts

  • L1 bicycle_dynamics.cpp, L2 seven_dof_dynamics.cpp, L3 → L2.
  • LuGre on: skip kinematic blend, \(\lambda\) fades, brake-hold; ignore relaxation_length_*.
  • LuGre + combined_slip_enabled: skip outer \(\mu F_z\) circular clip.

[VDSim impl] — time stepping

  • RK4: advance_lugre_states_(h/4) after k1, k2, k3, k4 (four times).
  • Euler: one advance after the step.
  • \(z\) constant inside each derivatives() evaluation.

[VDSim impl] — enable paths

  • Catalog tire.lugre_on, CLI --lugre / --no-lugre, GUI Tire panel (persisted in scene fleet_overrides), Python TireParams.lugre.enabled.

[VDSim impl] — tests & user guide

[VDSim impl] — steering LuGre (different module)

Chapter 17 §17.6: steering-column friction in N·m, Stribeck \(g(\omega)\). Do not mix with tire lugre (N/m, contact slip in m/s).

19.11 Self-check

  1. Why does \(\dot z = v_r - (\sigma_0 |v_r|/g) z\) imply \(F \to g\) in steady sliding?
  2. What is \(v_r\) for the rear wheel in a wheel-aligned frame when the vehicle sideslips but does not spin (\(\omega = 0\))?
  3. Why is semi-implicit integration preferred over explicit Euler for \(z\)?
  4. Why does VDSim omit the \(\mu F_z\) floor on \(g_x\) but gate it on \(g_y\) with \(w(\alpha)\)?
  5. Why is \(F_y\) negated after lugre_force for the lateral channel?
  6. What is lost by decoupled long/lat LuGre + peak ellipse vs Deur's 2D brush?

19.12 References

  1. Canudas de Wit, Olsson, Åström, Lischinsky, "A new model for control of systems with friction," IEEE TAC 45(3), 1995.
  2. Canudas-de-Wit, Tsiotras, Velenis, Basset, Gissinger, "Dynamic Friction Models for Road/Tire Longitudinal Interaction," Vehicle System Dynamics 39(3), 2003.
  3. Deur, Asgari, Hrovat, "A 3D brush-type dynamic tire friction model," Vehicle System Dynamics 41(2), 2004.
  4. Pacejka, Tire and Vehicle Dynamics, 3rd ed., Elsevier — Ch.3 brush model.
  5. VDSim design: docs/design/V0.2_TIRE_LUGRE.md.
  • Chapter 04–06 — LuGre forces enter the same Newton–Euler summation as MF96.
  • Chapter 15 — ISO step-steer transients: compare relaxation length (§3.9) vs LuGre when re-baselining validation.
  • Chapter 17 — actuator LuGre shares the semi-implicit \(z\) trick but not the tire kinematics.