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Vehicle dynamics for ADAS functions: tyres, bicycle model, understeer and longitudinal dynamics

Degrees of freedom according to ISO 8855, slip ratio and slip angle, cornering stiffness, bicycle model, understeer gradient and characteristic speed, resistances to motion, and how ACC, AEB, LKA and ESC use them.

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https://www.electronics-engineering.com/en/vehicle-dynamics-bicycle-model-tyres-adas-functions/

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Driver assistance functions (ADAS: adaptive cruise control, lane keeping, automatic emergency braking, stability control) are automatic control systems whose “process” is the vehicle. To design them, tune them and test them in simulation a model of the vehicle is needed: simple enough to be analysed and put into an ECU, faithful enough to predict real behaviour. This lesson presents the basic models: the forces exchanged by the tyres, the bicycle model for lateral dynamics, longitudinal dynamics and the way the main ADAS functions use them. The linked in-depth study develops the complete models, with proofs and code.

Reference frames and degrees of freedom

According to ISO 8855 the vehicle-fixed frame has the x axis forwards, y to the left and z upwards, with origin at the centre of gravity. A rigid vehicle has six degrees of freedom: longitudinal, lateral and vertical translations, and roll (about x), pitch (about y) and yaw (about z) rotations. For driving on a flat road three matter most: the longitudinal velocity v_x, the lateral velocity v_y and the yaw rate r = \dot\psi. Their ratio defines the vehicle sideslip angle \beta = \arctan(v_y/v_x): the angle between the direction the vehicle points and the direction it moves.

The tyre: all forces pass through it

The only horizontal forces that push, brake and turn a car (apart from the air) arise in the four contact patches, each the size of a postcard. A tyre produces force only if it slips a little relative to the road:

  • in the longitudinal direction the slip ratio \lambda = \frac{\omega R_e - v_x}{v_x} (in traction; in braking one divides by the wheel speed): the wheel turns a little faster or a little slower than it “should”;
  • in the lateral direction the slip angle \alpha: the angle between the direction the wheel points and the direction its centre moves.

For small slips the force is proportional: F_y = C_\alpha\,\alpha, with C_\alpha the cornering stiffness (typically 50-100 kN/rad per wheel of a car). For large slips the force reaches a maximum, about \mu F_z with \mu the friction coefficient (about 1 on dry asphalt, 0.5 on wet, 0.1-0.2 on snow and ice), and then drops slightly. Measured curves are described with Pacejka’s magic formula (in-depth study).

Lateral tyre force as a function of slip angle
For small angles the force grows linearly, dashed grey, with slope equal to the cornering stiffness; then it saturates at a value about equal to the friction coefficient times the vertical load. With a larger load, in green, the maximum force increases; with lower friction, dashed orange, it halves.

The bicycle model

To study cornering at constant speed the two wheels of each axle are lumped into one, at the centre: the vehicle becomes a “bicycle” with a steered front wheel (angle \delta) and a rear one. With a and b the distances of the centre of gravity from the axles, m the mass, I_z the yaw moment of inertia, C_f and C_r the axle cornering stiffnesses, and assuming small angles and constant speed v, the slip angles are

    \[ \alpha_f = \delta - \frac{v_y + a\,r}{v},\qquad \alpha_r = -\frac{v_y - b\,r}{v} \]

and Newton’s and the moment equations give

    \[ m(\dot v_y + v\,r) = C_f\alpha_f + C_r\alpha_r,\qquad I_z\dot r = a\,C_f\alpha_f - b\,C_r\alpha_r \]

A linear second-order system with input \delta and state (v_y, r): it is analysed with the tools of Control Systems (transfer function, poles, step response). The term m\,v\,r is the centripetal acceleration due to the rotation of the frame.

The bicycle model seen from above
The front wheel is steered by the angle delta; the centre of gravity is a from the front axle and b from the rear one. The vehicle moves with velocity v, which forms the sideslip angle beta with the longitudinal axis, and rotates with yaw rate r. The wheels move with velocities v f and v r, different from the direction they point: the angles between the two are the slip angles, which generate the lateral forces F yf and F yr.

Understeer and oversteer

In steady state, on a bend of radius R driven with lateral acceleration a_y = v^2/R, the model gives the required steering angle:

    \[ \delta = \frac{L}{R} + K_{us}\,a_y,\qquad K_{us} = \frac{m}{L}\left(\frac{b}{C_f} - \frac{a}{C_r}\right) \]

with L = a + b the wheelbase. L/R is the “geometric” (Ackermann) angle that would be needed at very low speed; K_{us} is the understeer gradient:

  • K_{us} \gt 0, understeering vehicle: the faster it goes, the more steering is needed for the same bend. This is the behaviour wanted in all production cars, because it is stable and the driver perceives it as “the car tends to run wide”;
  • K_{us} = 0, neutral;
  • K_{us} \lt 0, oversteering: above a critical speed v_{crit} = \sqrt{L/|K_{us}|} the vehicle becomes unstable and tends to spin.

For an understeering vehicle the steady-state yaw-rate gain \frac{r}{\delta} = \frac{v}{L + K_{us}v^2} has a maximum at the characteristic speed v_{ch} = \sqrt{L/K_{us}}: for an average car it is around 70-100 km/h.

Yaw-rate gain as a function of speed
For the understeering vehicle, in blue, the gain grows up to the characteristic speed and then decreases; for the neutral one, dashed, it grows linearly; for the oversteering one, in red, it tends to infinity at the critical speed.

Longitudinal dynamics

Along x, the traction (or braking) force F_x exchanged by the tyres with the road overcomes the resistances:

    \[ m\,\dot v = F_x - \underbrace{f\,m\,g}_{\text{rolling}} - \underbrace{\tfrac{1}{2}\rho\,C_d\,A\,v^2}_{\text{aerodynamic}} - \underbrace{m\,g\sin\theta}_{\text{gradient}} \]

with f \approx 0.01 the rolling resistance coefficient, \rho = 1.2 kg/m³ the air density, C_dA the drag area (about 0.6-0.7 m² for a car). Aerodynamic drag grows with the square of speed and the power needed with the cube: this is why motorway fuel consumption rises so quickly. In braking the maximum deceleration is about \mu g, limited by grip and by load transfer towards the front axle.

Resistances to motion as a function of speed
Rolling resistance, orange, is roughly constant; aerodynamic drag, blue, grows with the square of speed and becomes dominant above 20 metres per second; the red one is the sum.

How ADAS functions use them

  • ACC, adaptive cruise control: uses longitudinal dynamics. A radar measures the distance and speed of the vehicle ahead; the controller adjusts the speed keeping a distance that grows with speed, d = d_0 + h\,v, with a time gap h of 1-2 s.
  • AEB, automatic emergency braking: computes the time to collision TTC = d/v_{rel} and, below a threshold, warns and then brakes; it needs to know the maximum achievable deceleration.
  • LKA, lane keeping assist: uses the bicycle model written in error coordinates relative to the lane (lateral offset and relative heading angle) and applies a torque to the steering wheel.
  • ESC, electronic stability control: compares the measured yaw rate with the “reference” one given by the bicycle model, limited by grip (|r| \le \mu g/v); if the vehicle oversteers or understeers too much, it brakes a single wheel to create a corrective moment.
  • Trajectory planning and autonomous driving: at low speed the kinematic model (without slip) is used, with algorithms such as pure pursuit.

The linked in-depth study derives the bicycle model step by step, Pacejka’s tyre model, load transfers and ADAS controllers, with Python simulations; the exercises are on the linked page.

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ielettronica
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ielettronica
10 hours ago

Ciao, sono l'autore di «Dinamica del veicolo per le funzioni ADAS: pneumatici, modello a bicicletta, sottosterzo e dinamica longitudinale». Domande, dubbi, segnalazioni di errori o richieste di approfondimento sono benvenuti: scriveteli qui sotto e vi rispondo io. Potete anche votare la pagina con le stelle all'inizio dell'articolo.Hi, I am the author of “Vehicle dynamics for ADAS functions: tyres, bicycle model, understeer and longitudinal dynamics”. Questions, doubts, error reports and requests for more detail are all welcome: write them below and I will answer. You can also rate the page with the stars at the top of the article.

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