Marine robotics · Guidance, navigation and control

Line-of-Sight (LOS) Path Following

See why a lookahead point turns cross-track error into a desired course, how a heading controller drives the vehicle toward the path, and what changes when the lookahead distance, steering limits or water current change.

Offline · Single HTML file · Student laboratory

1 · Choose a scenario

Presets reset the experiment

Reference path

Guidance

Cross-current handling

LOS lookahead Δ 12.0 m

Through-water speed U 2.0 m/s

Heading gain kψ 1.8 s⁻¹

Max yaw rate 35 °/s

Cross-current (north) 0.0 m/s

Initial lateral offset 16 m

Initial heading 0 °

Playback speed 3×

Draw guidance arrows

2 · LOS geometry and vehicle motion

t = 0.0 s / 120 s

Desired pathVehicle trajectoryLOS point and rayCross-track errorDesired course

3 · Error, Lyapunov candidate and its derivative

Gray = saved reference run

Interpretation: This is a kinematic USV with heading-rate limits, not a hydrodynamic model. The tangent-line LOS preview is generally not exactly on a curved path. The Lyapunov quantity below measures geometric cross-track error only, not all vehicle dynamics.

Live geometry and dynamics

Ready

Signed cross-track error eᵧ0.0 mRelative to local path normal

Lookahead distance Δ12.0 mMeasured along path tangent

Desired course χ_d0.0°LOS guidance direction

Actual heading ψ0.0°Vehicle orientation

Ground course χ0.0°Direction of ground velocity

Heading command ψ_cmd0.0°Includes current correction if enabled

V = ½ eᵧ²0.0 m²Cross-track error energy

V̇ = eᵧėᵧ0.0 m²/sMonitors local error decrease

The desired LOS course aims toward a point Δ metres ahead of the orthogonal projection on the local path tangent.

Classical tangent-based LOS guidance

t = [cos χₚ, sin χₚ]ᵀ

n = [−sin χₚ, cos χₚ]ᵀ

eᵧ = nᵀ (p − pₚ)

pLOS = pₚ + Δ t

χd = χₚ − atan(eᵧ / Δ)

χₚ is the path-tangent angle, pₚ the nearest path point. The last formula is equivalent to the bearing from the vehicle to pLOS.

Cross-track Lyapunov candidate

V(eᵧ) = ½ eᵧ²

V̇ = eᵧ · nᵀv_ground

Ideal straight LOS, no current:

ėᵧ = − U eᵧ / √(Δ² + eᵧ²)

V̇ = −U eᵧ² / √(Δ² + eᵧ²) ≤ 0

Near-path ideal time constant: Δ / U ≈ 6.0 s.

Suggested student experiments

15–20 minutes

Lookahead: run straight-line convergence with Δ = 5 m and save the trace. Reset and compare with Δ = 30 m. Which has a larger near-path time constant?
Control lag: lower yaw-rate limit to 10°/s. Does a very short lookahead still give the smoothest response?
Why LOS? switch to “Path tangent” from an off-path initial position. Does following the path heading eliminate cross-track error?
Cross-current: apply +0.8 m/s north current. Compare uncompensated heading guidance against current-compensated LOS. Is χ equal to ψ?
Lyapunov: find a period where V̇ > 0 during heading transients. Why does ideal geometric LOS analysis not immediately prove stability of the full heading-controlled USV?

Instructor prompt

Reveal after prediction

Why doesn't a very large Δ converge quickly? The LOS guidance direction changes only weakly with eᵧ, so the vehicle steers gently and the lateral error usually decreases more slowly. With a very small Δ, correction is more aggressive but yaw-rate limits can produce overshoot or oscillation.

Tip: pause and click the map to reposition the USV, then press Start. Space = play/pause · R = reset.