Bicycle Model of Lateral Dynamics: Understeer and Steady-State Cornering

The problem

Moving heavy batteries and motors changes an EV's weight distribution and therefore its handling. Engineers use the bicycle model as the first tool to predict understeer, yaw-rate response and stability before any multibody model. It is also the reference model inside stability and torque-vectoring controllers, so understanding it is essential for later vehicle-dynamics work.

Objectives

• Derive and implement the two-degree-of-freedom linear bicycle model.
• Compute the understeer gradient and characteristic or critical speed.
• Simulate step-steer and sine-steer responses and steady-state cornering.
• Compare handling for different centre-of-gravity positions and tyre stiffnesses.
• Verify simulation results against analytical steady-state formulas.

What you need

Key concepts

Bicycle model

Left and right wheels are lumped into one front and one rear wheel, leaving two states: lateral velocity (or sideslip) and yaw rate. It is valid for small angles and moderate lateral acceleration.

Cornering stiffness

Lateral tyre force per unit slip angle in the linear region, Fy = Cα·α. It depends on tyre and load.

Understeer gradient

K = (m/L)·(lr/Cf − lf/Cr). Positive K means understeer: the vehicle needs more steering as speed rises. Negative K means oversteer and a critical speed above which it is unstable.

Yaw-rate gain

Steady yaw rate per steering angle, r/δ = v / (L + K·v²). It shows how responsive the vehicle is at each speed.

Step-by-step guide

1. 1

   Choose example parameters

   Define two example vehicles: a compact EV with a front motor and front-biased mass, and the same vehicle with the battery moved rearward. Set mass, yaw inertia, wheelbase, distances lf and lr from centre of gravity to axles and front and rear cornering stiffnesses, all labelled as example values. Compute static axle loads and check they add up to the vehicle weight.

   Checkpoint: A parameter script with two vehicle sets and axle loads that sum correctly.

2. 2

   Derive the equations of motion

   Write the lateral force and yaw moment balance: m·(dvy/dt + vx·r) = Fyf + Fyr and Iz·dr/dt = lf·Fyf − lr·Fyr, with slip angles αf = δ − (vy + lf·r)/vx and αr = −(vy − lr·r)/vx. Arrange them as a state-space model with states vy and r, input δ and constant forward speed. Write the A and B matrices symbolically in the report.

   Checkpoint: A and B matrices derived by hand and matching those built in MATLAB code at a test speed.

3. 3

   Implement in Simulink

   Build the model with Integrator blocks and gains, or with a State-Space block whose matrices are recomputed from speed. Add steering inputs: a step, a ramp for steady-state cornering and a sine. Log lateral acceleration ay = dvy/dt + vx·r, yaw rate and sideslip. Use a fixed-step solver such as ode4 with a 1 ms step or a variable-step solver.

   Checkpoint: Zero steering gives zero lateral motion, and a small step gives a smooth, bounded yaw-rate response at a moderate speed.

4. 4

   Verify steady-state results

   For each vehicle and several speeds, run a long constant-steer simulation and compare the final yaw rate with r = v·δ / (L + K·v²). Compute K from the parameters and the characteristic speed sqrt(L/K) for understeer, or the critical speed sqrt(−L/K) for oversteer. Tabulate simulated and analytic values. Use a small steering angle, for example 1 degree, so the linear model stays valid, and run long enough for transients to die out completely before reading the final value.

   Checkpoint: Simulated steady yaw rate matches the formula within one per cent at every speed tested.

5. 5

   Step-steer response

   Apply a step steer at 60 and 100 km/h and measure yaw-rate rise time, overshoot and settling time. Compare the two vehicles. Relate overshoot to the damping of the eigenvalues, which you compute from eig(A) at each speed. Explain how damping falls as speed rises. Use the same small steer amplitude for both vehicles and speeds. Present time histories on shared axes so differences are easy to see, and explain why higher speed gives a faster but less damped response.

   Checkpoint: A table of rise time, overshoot and eigenvalue damping for each vehicle and speed.

6. 6

   Eigenvalue analysis against speed

   Plot the eigenvalues of A as speed increases from 10 to 150 km/h. For the understeering vehicle, they stay in the left half plane; for an oversteering configuration, one eigenvalue crosses to the right at the critical speed. Create an oversteering set deliberately, for example by reducing rear cornering stiffness, to demonstrate the crossing. Mark the predicted critical speed on the plot.

   Checkpoint: A root-locus-style plot showing the crossing at the speed predicted by the critical-speed formula.

7. 7

   Steady-state cornering diagram

   Run a slowly increasing steering ramp at constant speed, or constant-radius cornering at increasing speed, and plot steering angle against lateral acceleration. The slope above the Ackermann angle L/R shows the understeer gradient. Mark where the linear model stops being credible, roughly above 0.4 g lateral acceleration for typical tyres. Compare the result with the analytical relation δ = L/R + K·ay and explain any deviation.

   Checkpoint: Steering-against-lateral-acceleration plots whose slope matches K for each vehicle.

8. 8

   Sensitivity to weight distribution

   Sweep the centre-of-gravity position between the axles and the rear-to-front cornering stiffness ratio. Plot understeer gradient and yaw-rate gain at 100 km/h. Discuss what battery placement and tyre selection do to handling, and why manufacturers often aim for mild understeer. Use a script that loops over CG positions and stiffness ratios, computes K analytically and confirms a few points by simulation. Mark the neutral-steer line where K is zero, and explain what crossing it would mean for the driver.

   Checkpoint: A contour or line plot of understeer gradient across CG position with the two example vehicles marked.

9. 9

   Report

   Present the derivation, implementation, verification against analytic results, transient and eigenvalue analysis, cornering diagrams and sensitivity study. State the limitations: linear tyres, constant speed, no load transfer and no roll. Suggest the next model step, such as adding a non-linear tyre model or roll dynamics. Include the derivation in an appendix and the scripts used to produce every figure, so another student can reproduce the work and extend it with the next model step.

   Checkpoint: A report with every simulated result checked against a formula where one exists.

Testing & validation

• Hand-derived matrices match code.
• Steady yaw rate matches the analytic formula.
• Critical speed from eigenvalues matches the formula for an oversteering set.
• Understeer gradient from the cornering diagram matches the parameter-based value.

Expected results

A verified bicycle model with understeer, transient and stability analysis for two EV layouts, showing quantitatively how battery placement and tyres change handling.

Take it further

• Add a non-linear Pacejka tyre model and compare at high lateral acceleration.
• Add lateral load transfer and roll.
• Use the model as a reference generator for a yaw-rate controller.
• Fit cornering stiffness to logged data from a real vehicle test.

Viva and review questions

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