What does the atan2(k_e·e, v) term do?

It converts cross-track error into a steering correction that shrinks as speed v rises, so the same lateral error commands less aggressive δ at high speed. k_e sets how strongly the controller hunts the path.

Why use a kinematic bicycle instead of a dynamic model?

At moderate speeds and teaching fidelity, geometry of L and δ already captures nonholonomic path constraints. Tire force dynamics, load transfer, and slip angles are omitted so the Stanley law stays transparent.

What if k_e is too large?

The vehicle overcorrects, weaving about the reference with large δ swings. Too small k_e leaves a persistent lag in e(t), especially on tight curves where heading error alone is not enough.

Bicycle Model & Stanley Controller

Kinematic bicycle (rear-axle): δ = θ_e + atan2(k_e·e, v) Stanley path-following law. Pick oval, race-track, lemniscate, sine-road or S-curve and tune k_e, v, L; live cross-track e(t) and steering δ(t).

About this model

The kinematic bicycle model places the state at the rear axle with wheelbase L and steering angle δ, approximating car-like nonholonomic motion without tire slip. Path following here uses the Stanley controller: δ = θ_e + atan2(k_e · e, v), combining heading error θ_e with a speed-scheduled correction of cross-track error e. Pick oval, race-track, lemniscate, sine-road, or S-curve references and tune gain k_e, speed v, and L while plotting e(t) and δ(t). Assumptions include pure rolling (no sideslip), instantaneous steering, flat ground, and no dynamics of inertia or actuators. Vary k_e and v to see overcut corners versus sluggish convergence to the path.

Who it's for: Vehicle dynamics, autonomous driving, mobile robotics, and advanced control courses.

Key terms

Kinematic bicycle model
Stanley controller
Cross-track error
Path following
Wheelbase
Nonholonomic vehicle

How it works

Kinematic bicycle model with the Stanley path-following controller used by Stanford's eponymous DARPA Grand Challenge winner. The control law combines two terms: align with the path heading (θ_e) and pull toward the path proportionally to cross-track error e measured at the front axle, scaled by 1/v: δ = θ_e + atan2(k_e · e, v). Increase k_e for tighter tracking on tight curves; decrease it to avoid oscillations. Try changing speed v with a fixed k_e to see why the atan2 normalization keeps tracking stable across speeds. Bottom panel plots the cross-track error e(t) in pink and the steering δ(t) in yellow.

Key equations

Bicycle: ẋ = v cosθ, ẏ = v sinθ, θ̇ = (v/L) tanδ

Stanley: δ = θ_e + atan2(k_e · e, v_x), δ ∈ [−δ_max, δ_max]

Frequently asked questions

What does the atan2(k_e·e, v) term do?: It converts cross-track error into a steering correction that shrinks as speed v rises, so the same lateral error commands less aggressive δ at high speed. k_e sets how strongly the controller hunts the path.
Why use a kinematic bicycle instead of a dynamic model?: At moderate speeds and teaching fidelity, geometry of L and δ already captures nonholonomic path constraints. Tire force dynamics, load transfer, and slip angles are omitted so the Stanley law stays transparent.
What if k_e is too large?: The vehicle overcorrects, weaving about the reference with large δ swings. Too small k_e leaves a persistent lag in e(t), especially on tight curves where heading error alone is not enough.

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