≈Control & Guidance LabsInteractive course modules

English review draft. This brief imported guide is retained with corrections. The activity is ungraded practice. Read the expanded student notes.

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Module 02 · Marine guidance

Line-of-Sight Path Following

Student notes and an in-class experiment: how lookahead geometry turns lateral path error into a course command, and how motion constraints and cross-current affect the result.

Launch LOS simulator ↗Next: cooperative path following →

1. What does LOS solve?

A path specifies a geometric curve, not the time at which the robot must be at each point. LOS guidance seeks to reduce cross-track error while allowing the vehicle to move forward along the desired path. It is therefore a guidance method, not a complete vehicle-dynamics controller or formation-keeping method.

2. Lookahead geometry

Find the closest point on a sufficiently smooth reference path, construct the local tangent and its normal, and measure the signed cross-track error ey. Place a virtual lookahead point Δ metres ahead along the tangent. The angle from the current vehicle position to that point gives a commanded desired course.

χd = χp − atan(ey / Δ)

Here χp is the local path-tangent angle. With positive cross-track error on the left side of the path, the negative correction steers toward the path. Increasing Δ generally produces gentler steering and slower lateral convergence. Decreasing Δ produces more aggressive correction, which can be demanding when yaw rate is limited.

What the orange point means

In this demonstrator the LOS point lies on the local tangent line. For curved paths it will generally not lie exactly on the desired curve. Other LOS formulations use different lookahead constructions.

3. Vehicle and autopilot model

The simulator uses a constant through-water speed U and a first-order heading controller with a maximum yaw rate. A uniform cross-current vc can push the robot across the path.

ẋ = U cos ψ,   ẏ = U sin ψ + vc
ψ̇ = sat[kψ wrap(ψcmd − ψ), ±rmax]

The body heading ψ is not generally the ground course χ when water current is present. The simple current-compensated mode assumes that vc is known and that the current normal to the course is weaker than U and the resulting ground velocity has positive forward component along that course. This is known-current compensation, not integral or adaptive LOS.

4. Lyapunov connection

For a horizontal straight path with no current and ideal instantaneous heading tracking, ey = y. Choosing V(ey) = ½ey2 gives:

ėy = − U ey / √(Δ² + ey²)
V̇ = − U ey² / √(Δ² + ey²) ≤ 0

Thus the ideal reduced-order cross-track error decreases. Near ey = 0, the convergence rate is approximately U/Δ and the local time constant is Δ/U. These expressions are not a stability proof for the complete yaw-limited robot, especially during actuator saturation or with uncompensated current.

5. Ungraded practice (20 minutes)

  1. Start 16 m above a straight path. Run Δ = 5 m and save the reference trace. Repeat with Δ = 30 m. Which reaches |ey| < 1 m first?
  2. Reduce the maximum yaw rate to 10°/s. Compare the initial turn and overshoot. Does the earlier conclusion always hold?
  3. Choose Path tangent rather than LOS. Explain why a lateral offset can persist despite correct forward heading.
  4. Apply a northward current of +0.8 m/s. Compare the steady cross-track offset without and with current compensation.
  5. Examine V̇ after a disturbance. Can it be temporarily positive? What additional model states and assumptions are missing from the ideal Lyapunov argument?

Suggested record: one screenshot of the lookahead geometry, exported CSV, two comparative ey(t) traces, and a short explanation of the chosen Δ.

6. Recommended reading

The interactive simulator and these notes work offline. External university lecture resources require an internet connection.