Bicycle Model & Stanley Controller

This interactive simulator explores Bicycle Model & Stanley Controller in Engineering. 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). Use the controls to change the scenario; watch the visualization and any graphs or readouts to connect the model with lectures, labs, and homework.

Who it's for: For learners comfortable with heavier math or second-level detail. Typical context: Engineering.

Key terms

bicycle
model
stanley
controller
bicycle stanley
engineering

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]

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