Section 48.7: Vehicle kinematics, dynamics, and control

Geometry decides where the path goes; the tires, the friction circle, and the actuator limits decide whether the car can actually follow it.

On vehicle dynamics and control

Kinematic Bicycle Model

The kinematic bicycle model is the first useful model for route following and local planning. With position $(x,y)$, heading $\psi$, speed $v$, wheelbase $L$, and steering angle $\delta$:

$$\dot x = v\cos\psi, \qquad \dot y = v\sin\psi, \qquad \dot\psi = \frac{v}{L}\tan\delta, \qquad \dot v = a.$$

This model exposes non-holonomic motion and curvature limits. It is not enough for high-speed handling, but it is the right bridge between geometric planning and controller design.

Dynamic Bicycle Model

The dynamic bicycle model adds lateral velocity, yaw rate, tire slip, and lateral tire forces. It matters when speed, friction, braking, and evasive maneuvers dominate. A planner that ignores these effects can choose trajectories that are geometrically collision-free but dynamically unsafe. In practice this usually appears first as an interaction between curvature and the friction circle: the demanded lateral acceleration $a_y = v^2 \kappa$ exceeds what the tire-road pair can deliver, so the "successful" geometric plan arrives at the controller already broken.

The model boundary should be explicit. At low speeds and modest curvature, the kinematic bicycle model is often enough. At higher speeds, low friction, emergency braking, or evasive steering, tire slip angle, lateral stiffness, yaw inertia, and friction limits become part of the planning problem.

Control Choices

Pure pursuit and Stanley control are useful geometric baselines. LQR and MPC expose the state-space and constrained-optimization view. MPC becomes especially important when the controller must trade route progress, comfort, lane keeping, collision margins, and actuator limits in one horizon.

# Kinematic bicycle rollout for a local planner sanity check.
# Roll out a constant-curvature lane-following segment.
# Use the result to sanity-check heading and path change before simulator runs.
import math

x, y, psi, v = 0.0, 0.0, 0.0, 8.0
L, dt = 2.8, 0.1
for _ in range(20):
    delta = math.radians(5.0)
    x += v * math.cos(psi) * dt
    y += v * math.sin(psi) * dt
    psi += (v / L) * math.tan(delta) * dt
print(round(x, 2), round(y, 2), round(math.degrees(psi), 2))

Expected output: the final heading should increase smoothly and the lateral displacement should stay consistent with a shallow arc rather than a sudden sideways jump. If the numbers imply an unrealistically sharp turn for the chosen steering angle and speed, the planner, units, or wheelbase model is inconsistent before closed-loop evaluation even starts.