Bicycle model and lateral dynamics

Why it matters

The bicycle (single-track) model is the simplest model that predicts how a car responds to the steering wheel: how much yaw rate and lateral acceleration a given steer input produces, how that changes with speed, and when the car becomes unstable. It is the reference model inside every ESC controller and the starting point for steering, chassis and ADAS lane-keeping design.

Key ideas

Simplifications. The two wheels of each axle are lumped into one wheel on the centre line, with the axle's total cornering stiffness (C_f, C_r). Forward speed u is constant; the road is flat; roll, pitch and load transfer are ignored; tyres are linear (F_y = C·α); steer and slip angles are small. What remains are two degrees of freedom in the road plane: lateral velocity v (or body sideslip angle β = v/u) and yaw rate r.

Geometry. The CG lies a distance a behind the front axle and b ahead of the rear axle, L = a + b. Because the body rotates at r, the front axle has lateral velocity v + a·r and the rear axle v − b·r. The tyre slip angles are therefore:

• front: α_f = δ − (v + a·r)/u
• rear: α_r = −(v − b·r)/u

Equations of motion. Newton's law laterally and Euler's law in yaw:

• m·(v̇ + u·r) = F_yf + F_yr — note the u·r term: lateral acceleration in a turn is the centripetal part u·r plus any change in v;
• I_z·ṙ = a·F_yf − b·F_yr. With F_y = C·α these are two linear first-order differential equations in v and r, driven by δ.

Steady-state response. Setting v̇ = ṙ = 0 gives the familiar result δ = L/R + K'·a_y, with a_y = u·r = u²/R and K' = (m/L)·(b/C_f − a/C_r) (rad per m/s²). Multiplying K' by g gives the understeer gradient K in rad/g from the previous topic (since W_f = m·g·b/L). The model therefore contains the whole steady-state understeer–oversteer story, plus the transient behaviour.

Gains.

• Yaw-rate gain r/δ = u/(L + K'u²). For neutral steer it rises linearly with speed; for understeer it peaks at the characteristic speed √(L/K') and then falls; for oversteer it becomes infinite at the critical speed √(−L/K').
• Lateral-acceleration gain a_y/δ = u²/(L + K'u²).

Body sideslip. In steady state β = b/R − α_r. At low speed β is positive (the CG velocity points inward of the body axis); as speed increases α_r grows and β turns negative, i.e. the nose points into the turn. Large sideslip angles (more than a few degrees) mean the car is sliding; ESC watches β as well as r.

Neutral steer point (NSP) and static margin. The NSP is the point on the body where a side force produces equal slip angles front and rear, so no yaw. It lies C_r·L/(C_f + C_r) behind the front axle. If the NSP is behind the CG (positive static margin) the car understeers, which is the same condition as K > 0. Typical static margins are about 0.05–0.10 of the wheelbase.

Transient behaviour. Immediately after a step steer, v and r are still zero, so only the front tyre has slip (α_f = δ). The car starts to yaw with ṙ = a·C_f·δ/I_z, then settles to the steady-state values with a response time and damping that depend on speed. At higher speed the yaw response becomes less damped.

Limits of the model. It ignores load transfer, tyre saturation, roll steer and longitudinal forces, so it is valid only up to roughly 0.4 g on dry roads. It is still accurate enough for controller design and for understanding stability.

Formulas

α_f = δ − (v + a·r)/u, α_r = −(v − b·r)/u — axle slip angles (rad); v lateral velocity (m/s), r yaw rate (rad/s), u forward speed (m/s), a, b CG-to-axle distances (m).

m·(v̇ + u·r) = C_f·α_f + C_r·α_r — lateral equation (N); m mass (kg).

I_z·ṙ = a·C_f·α_f − b·C_r·α_r — yaw equation (N·m); I_z yaw moment of inertia (kg·m²).

K' = (m/L)·(b/C_f − a/C_r) — understeer gradient (rad per m/s²); K = K'·g in rad/g.

r/δ = u / (L + K'·u²) — steady-state yaw-rate gain (1/s).

a_y/δ = u² / (L + K'·u²) — steady-state lateral-acceleration gain (m/s² per rad).

β = b/R − α_r — steady-state sideslip at the CG (rad); R = u/r.

x_NSP = C_r·L / (C_f + C_r); SM = (x_NSP − a)/L — neutral steer point measured behind the front axle (m) and static margin (dimensionless).

V_ch = √(L/K') (K' > 0), V_cr = √(−L/K') (K' < 0) — characteristic and critical speeds (m/s).

Worked examples

Example 1 (standard). A car has m = 1500 kg, a = 1.2 m, b = 1.4 m, axle cornering stiffnesses C_f = 100 000 N/rad and C_r = 110 000 N/rad. It travels at 25 m/s with a steady road-wheel steer angle of 0.02 rad. Find the yaw rate, lateral acceleration, turn radius and sideslip angle.

1. L = 2.6 m. K' = (m/L)(b/C_f − a/C_r) = (1500/2.6)(1.4/100 000 − 1.2/110 000) = 576.9 × (1.400 − 1.091)×10⁻⁵ = 1.783×10⁻³ rad per m/s² (K = 0.0175 rad/g → understeer).
2. r = u·δ/(L + K'u²) = 25 × 0.02/(2.6 + 1.783×10⁻³ × 625) = 0.5/3.7145 = 0.1346 rad/s.
3. a_y = u·r = 25 × 0.1346 = 3.37 m/s²; R = u/r = 185.7 m.
4. Slip angles: α_r = (m·a/L)·a_y/C_r = (1500 × 1.2/2.6) × 3.365/110 000 = 0.02118 rad; α_f = (m·b/L)·a_y/C_f = 0.02718 rad. Check: L/R + α_f − α_r = 0.01400 + 0.00600 = 0.0200 = δ ✓.
5. β = b/R − α_r = 1.4/185.7 − 0.02118 = −0.0136 rad (−0.78°) — the nose points slightly into the turn.

Example 2 (GATE level). For the same car with I_z = 2500 kg·m², find (a) the neutral steer point and static margin, (b) the characteristic speed, and (c) the initial yaw acceleration and lateral acceleration just after a sudden 0.02 rad steer at 25 m/s.

1. x_NSP = C_r·L/(C_f + C_r) = 110 000 × 2.6/210 000 = 1.362 m behind the front axle.
2. SM = (x_NSP − a)/L = (1.362 − 1.2)/2.6 = 0.062 — NSP behind CG, so understeer (consistent with K' > 0).
3. V_ch = √(L/K') = √(2.6/1.783×10⁻³) = 38.2 m/s (137 km/h).
4. At t = 0⁺, v = r = 0, so α_f = δ = 0.02 rad, α_r = 0. F_yf = 100 000 × 0.02 = 2000 N.
5. ṙ = a·F_yf/I_z = 1.2 × 2000/2500 = 0.96 rad/s²; v̇ = F_yf/m = 2000/1500 = 1.33 m/s².

Common mistakes

• Dropping the u·r term: lateral acceleration is v̇ + u·r, not v̇.
• Using per-tyre cornering stiffness in a model that lumps each axle into one wheel.
• Writing α_r = (v − b·r)/u with the wrong sign, which flips the stability conclusion.
• Applying the linear model near the grip limit, where tyres saturate.
• Confusing the steer angle at the road wheel with the steering-wheel angle (divide by the steering ratio, typically 14–18).

For GATE ME

Questions usually test steady-state results: yaw-rate or lateral-acceleration gain, understeer gradient from a, b, C_f, C_r, critical or characteristic speed, neutral steer point, and the sign of sideslip. Practise deriving slip angles from the velocity of each axle and checking δ = L/R + α_f − α_r.

Quick check

1. What are the two degrees of freedom of the linear bicycle model?
2. A neutral-steer car (L = 2.5 m) is steered 0.025 rad at 20 m/s. What is its yaw rate?
3. If the NSP lies ahead of the CG, does the car understeer or oversteer?
4. Why does the front tyre alone carry side force at the instant a step steer is applied?

Answers: 1. Lateral velocity (or sideslip) and yaw rate. 2. r = uδ/L = 0.2 rad/s. 3. Oversteer. 4. v and r are still zero, so only the front has slip angle (equal to δ).