LOS Guidance

t = 0.0 s

LOS, projected reference

LOS, propagated particle

| reference point on the path · → velocity v · ⇢ command v_d · ▲ travel direction

Task

U speed [m/s]

a_s ramp [m/s²]

Path

closed loop

R_min turn radius [m]

Guidance

Δ lookahead [m]

Δ_s particle sat [m]

k_s particle gain [1/s]

Vessel

τ velocity lag [s]

speed cap [m/s]

Simulation

time scale ×

v_d = U η_d, η_d = (Δ t − e)/√(Δ²+|e|²)
e = cross-track error to the reference point
projection: s_r = argmin |p − p_r(s)|
particle: ṡ_r = t·ṗ + Δ_s tanh(k_ss_e/Δ_s)
t = travel direction × path tangent

Cross-track error |e| [m] (last 120 s)

Speed |v| [m/s], dashed U

law	\|e\| m	s_e m	\|v_d\| m/s	\|v\| m/s	s_r m	U m/s
LOS, projected reference
LOS, propagated particle

How it works

Path. Straight legs between draggable waypoints, joined by the symmetric polynomial transition curves of Kobryń (2014). The curves have zero curvature at both ends and are sized by a minimum turn radius R_min. The path is a closed loop by default and can be opened; it can be followed in either direction. The dashed line joins the waypoints.
Guidance. Lookahead-based LOS guidance (Breivik and Fossen, 2005; Fossen, 2021) in vector form. The desired velocity is v_d = U η_d with the LOS direction η_d = (Δ t − e)/√(Δ² + |e|²), where t is the path tangent at the reference point, e the cross-track error to it and Δ the lookahead distance.
The reference point, two ways. Projected: the point on the path closest to the vessel. Simple, but it can jump when the vessel is far from the path or near a tight turn. Propagated particle: the reference moves along the path with its own dynamics, ṡ_r = t·ṗ + Δ_s tanh(k_ss_e/Δ_s), so it tracks the vessel's progress and pulls the along-track error s_e to zero without ever jumping.
Vessel. Drawn as a hull pointing along its velocity, but modelled as a point particle whose velocity follows v_d with a first-order lag τ and an optional speed cap. No heading dynamics, no environmental disturbances.

The plots show the cross-track error and the speed over the last 120 s; the table shows the live state of both vessels. Drag a square handle to move a waypoint, drag a vessel to move it; Space pauses. On a touch screen, touch a handle or a vessel to drag it, anywhere else to scroll.