Low-speed / standstill dynamics — research note + design

Status: implemented (2026-06) in L1/L2/L3. See "Implemented" below. Motivation: under manual low-speed steering the L2/L3 model lurches in yaw, drifts laterally, and oscillates left/right when coming to a stop. A series of band-aids (lateral-force fade, lateral-velocity damping) reduced but did not remove the artifacts — because this is a known structural problem, not a tuning bug.

The problem (root cause)

A slip-based tire model computes - slip ratio κ = (Ω·R − Vx) / Vx - slip angle α = arctan(Vy / Vx) (per axle: αf = (Vy + lf·r)/Vx − δ, αr = (Vy − lr·r)/Vx)

Both divide by the longitudinal speed Vx, so as Vx → 0 the slip terms blow up and the lateral/yaw states explode numerically. The current kSpeedEps floor on the denominator caps the magnitude but the resulting force is still ill-conditioned (snaps to the friction limit from velocity noise; over-large yaw moment with no speed-dependent yaw damping → the spin/wiggle).

Established solutions (literature)

1. Kinematic–dynamic blending (Kong et al. 2015; Polack et al. 2017). Below a speed threshold use the kinematic bicycle (all slip angles = 0; the car follows the steering geometry exactly: r = Vx·tan(δ)/L, β = arctan(lr·tan δ/L)), above it use the dynamic (force-based) model, blended on speed. Physically correct: at parking speed tires don't slip. Most widely used; removes the singularity by construction. Blend parameter is usually Vx (some use lateral acceleration).

2. Implicit / backward-Euler discretisation of the lateral states (arXiv 2011.09612, 2411.17334). Integrating Vy and r with backward Euler turns the update denominator from 1/Vx into m·Vx − Ts·(Cf+Cr) (and Iz·Vx − Ts·(lf²Cf+lr²Cr)), which stays finite as Vx → 0. Keeps the dynamic model at all speeds, no blend. Derived for a linear tire (constant cornering stiffness) — not directly applicable to VDSim's nonlinear Pacejka.

3. Low-speed damping (Pacejka MF-Tyre "VLOW"): add a force proportional to the slip velocity below a threshold. Numerical-stability band-aid only — does not give correct low-speed behaviour or standstill hold. (≈ what the current fade/damping patches do.)

4. Brush / LuGre stick (Canudas de Wit 1995; Deur; Velenis): a contact-patch bristle deflection gives true static friction / standstill hold (incl. holding on a slope). Most physical but needs the bristle state + careful params, and on its own (no yaw damping) it spun the vehicle at low speed in our tests.

Recommendation for VDSim

Adopt (1) kinematic–dynamic blending as the primary fix (works with the nonlinear Pacejka model; matches the physics of manual low-speed steering; is the textbook answer). Optionally layer (4) later for true slope standstill hold.

Concrete design (L2 7-DOF; L3 reuses the inner L2; L1 bicycle analogous):

• Kinematic targets from the front road-wheel angle δ and wheelbase L = a+b:
• r_kin = Vx · tan(δ) / L
• vy_kin = r_kin · b (lateral velocity at the CG, rear-ref bicycle)
• Blend factor λ = smoothstep(Vx / V_blend), V_blend ≈ 3 m/s (λ=0 at rest → 1 at V_blend).
• Apply as a constraint relaxation on the lateral state derivatives so it composes with RK4 (no post-hoc state jump):
• dVy += (1−λ) · (vy_kin − Vy) / τ
• dr += (1−λ) · (r_kin − r ) / τ, τ ≈ 0.05 s At rest (λ=0) this pulls (Vy, r) onto the kinematic curve → the car turns smoothly per the steering and cannot oscillate/spin; at speed (λ=1) the term vanishes and the validated dynamic model is unchanged.
• Remove the lateral-force fade and lateral-velocity damping band-aids.
• Keep the existing standstill force cleanliness (forces still computed; the kinematic constraint governs the motion).

Implemented (2026-06)

The shipped solution is the kinematic blend (1) plus a viscous brake-hold creep damper in place of the brush/LuGre stick (4). The stick was prototyped and dropped: an elastic bristle (sigma0*z) stores and releases energy and, layered on the blend, rang at the stop and re-accelerated the car. A purely viscous damper is unconditionally dissipative — it cannot ring — and gives a slope hold that is good enough (cm/s creep), so it is the better engineering choice here.

Applied identically in L1 (bicycle_dynamics.cpp), L2 (seven_dof_dynamics.cpp), and inherited by L3 (delegates the planar solve to L2):

• Blend factor lambda = smoothstep(|V| / kStickBlend), kStickBlend = 3 m/s.
• Lateral states are blended in the derivative, not added on top: dVy = lambda·dVy_dyn + (1-lambda)·(vy_kin - Vy)/tau, dr = lambda·dr_dyn + (1-lambda)·(r_kin - r )/tau, tau = 0.05 s, with r_kin = Vx·tan(delta)/L, vy_kin = r_kin·lr. At rest this is pure kinematic, so no force-induced yaw (the cause of the old spin/stop-oscillation); at speed (lambda=1) the validated dynamics are untouched. Longitudinal Vx stays fully dynamic.
• Brake-hold creep damper (longitudinal only): when on the brake and off the throttle, add -kStickC·v_x_wheel per wheel (kStickC = 6e4 N·s/m), gated by (1-lambda)·clamp(brake-throttle, 0, 1) and clamped to mu·Fz. A braked wheel does not fully lock at low speed, so its dynamic force fades to zero as Vx->0 and the car would creep down a grade; this damper opposes the residual creep, which settles at gravity/damping. It never fights drive torque (gated off when throttle > brake) and vanishes above the blend speed.
• kSpeedEps slip-denominator floor tightened 0.5 -> 0.15 (lateral noise is now handled by the blend, not the floor, so the floor can be smaller for a sharper low-speed longitudinal force).

LuGre opt-in (shipped 2026-06)

Default (2026-06): catalog tire.default_pacejka has lugre.enabled: true — LuGre at all speeds; MF96 shapes g() (asymmetric floors). Fallback: set lugre.enabled: false (tire.kinematic_fallback, --no-lugre) to use the blend + brake-hold path below.

Theory: docs/theory/19_lugre_dynamic_tire.md. Design: V0.2_TIRE_LUGRE.md, user guide docs/CATALOG_AND_PHYSICS.md.

• The lateral tire force is faded by lambda in the dynamics (not just the display): as Vx->0 the slip angle is ill-conditioned, so the raw lateral force chatters; left unfaded it leaks into ay and makes the left/right load transfer (Fz) tremble even though the car is at rest. Fading it keeps ay -> 0 at a standstill, so Fz is clean. The low-speed lateral motion is carried by the kinematic relaxation, which is the physically correct low-speed behaviour. At lambda=1 (above the blend speed) the validated force is unchanged.
• The reported longitudinal force additionally fades the raw slip part for the rendered arrows / logs (the slip ratio is ill-conditioned as Vx->0, so the per-wheel Fx chatters even though the net ax is smooth). This is display-only — the equations of motion use the full Fx so the car still self-arrests. The brake-hold damper keys off the body-longitudinal speed (not the steer-rotated wheel speed) so a steered front wheel cannot flip its sign.

Measured (sedan preset, mu=1.0): braked grade hold creep ~1-2 cm/s on 8%, ~2-3 cm/s on 12-20%; brake-to-stop settles with no left/right oscillation; no-brake car still rolls down a grade (no phantom hold); ISO 7401/4138/3888-2 and all 190 ctest checks unchanged (the blend is inactive above 3 m/s).

With LuGre off (default): a true zero-creep static hold still needs brush stick or brake static friction; the viscous damper leaves small grade-proportional creep. With LuGre on: presliding z + gated g_y give slope hold without the blend path — enable via tire.lugre_on or GUI toggle (see Ch. 19).

Sources

• Kong, Pfeiffer, Schildbach, Borrelli, "Kinematic and dynamic vehicle models for autonomous driving control design," IEEE IV 2015.
• Polack et al., "The kinematic bicycle model: a consistent model for planning feasible trajectories for autonomous vehicles?," IEEE IV 2017.
• "Numerically Stable Dynamic Bicycle Model for Discrete-time Control," arXiv:2011.09612.
• "An Explicit Discrete-Time Dynamic Vehicle Model with Assured Numerical Stability," arXiv:2411.17334.
• Pacejka, Tyre and Vehicle Dynamics (low-speed / MF-Tyre VLOW).
• Canudas de Wit et al., "A new model for control of systems with friction" (LuGre), IEEE TAC 1995.