LOS & coordinated virtual targets

Learning objectives

Explain the difference between geometric path following and progress coordination; compare LOS guidance and moving virtual targets; define leader–wingman slots along separate paths; and identify when coordination is limited by propulsion, communication and safety constraints.

1. What does cooperative path following mean?

For robot i, let the reference path be pᵢ(sᵢ), where sᵢ is a path-progress parameter. The local guidance task is to drive the vehicle toward its assigned path. The team-level task is to regulate how the progress variables of several vehicles evolve relative to one another.

Two USVs may both have negligible cross-track error yet be tens of metres apart longitudinally after one loses propulsion.

2. Line-of-sight guidance

A common tangent-based LOS rule steers toward a point ahead of the closest projection on the local tangent line. On a curved path that point generally lies off the curve. V4's separate LOS baseline instead uses the curve point at horizontal station x + 9.5 m; it is a different lookahead construction. With a declared positive-left cross-track error and counterclockwise tangent/course angles, the tangent-based command is:

The benefit is local simplicity: each vehicle can follow its path with little or no inter-vehicle communication. Standard LOS does not by itself regulate the progress of neighboring vehicles. A separate speed or synchronization controller is required for cooperative behavior.

3. Moving virtual targets

A virtual-target controller defines a point pᵢ(sᵢ) moving along each vehicle's path and designs guidance to reduce the vehicle-to-target error. The target's progress rate ṡᵢ can be modified by local tracking feedback and by fleet coordination signals.

A simple 2D tracker may combine target-velocity feedforward and positional feedback:

For a symmetric positive-definite gain matrix K with units s⁻¹, and under ideal realization of this Cartesian velocity, ėᵢ = −K eᵢ. Real robots have bounded speed and yaw rate, so perfect convergence cannot be assumed under all conditions. A lookahead variant intentionally maintains a forward gap and should not be confused with convergence to the target point.

4. Wingmen and progress coordination

Suppose the leader has desired lag d₁ = 0 and each wingman has desired lag dᵢ. Define aligned progress variables zᵢ = sᵢ + dᵢ. Synchronized formation progress corresponds to zᵢ ≈ zⱼ. A conceptual consensus term is:

Coordination acts on target progression and indirectly on physical speed. If a vehicle is speed-limited, reducing the other vehicles' progress may preserve formation at the expense of throughput. Communication loss can break the coupling while leaving local following operational.

5. Path shapes and phase offsets

One simulated sine path has the form:

The lane offset y₀,ᵢ moves the whole curve sideways; the x-shift aᵢ changes its phase along the horizontal station. Neither is the same as the desired formation lag dᵢ.

The global virtual-target speed is a nominal physical path speed: the code divides it by √(1 + yᵢ′(s)²) to obtain the horizontal-station rate before adding feedback and coordination corrections. Those corrections can change actual path speed. Per-vehicle pace multipliers, local tracking corrections and coordination can change the realized target speed. Speed saturation can prevent a vehicle from keeping up with its target.

6. Guided practice · 25 minutes

1. Baseline: Select three vehicles and mixed straight/sine/curved paths. Run independent LOS; record the vehicle x-station progress RMS.
2. Virtual target: Switch to independent virtual targets. Separate local vehicle-to-target convergence from team-level progress error.
3. Disturbance: Reduce the speed multiplier of one wingman. Predict which vehicle should slow down if formation is to be preserved.
4. Consensus: Enable coordinated virtual targets. Compare peer consensus against leader broadcast with the same disturbance.
5. Geometry: Set different sine x-shifts, then set nonzero desired lags. Explain why these are two different geometric choices.
6. Feasibility: Raise nominal target speed beyond what the actuators can achieve. Observe tracking error and state what would need to change in a real controller.

Suggested study record

For each controller, report approximate tangent cross-track error, vehicle-to-target error and x-station progress RMS. Include one screenshot or CSV plot and discuss communication, saturation and collision avoidance in 200–300 words.

Metric and fault definitions. V4 uses a local tangent approximation for cross-track error at the vehicle's x-station. Vehicle progress RMS and VT progress RMS are horizontal-station residuals, not rigid Euclidean formation errors. Minimum separation is between point centres; no hull shape is modelled. The fault is an instantaneous speed multiplier, and the communication switch removes coupling rather than simulating packet delay or a network. Peer/chain options are optional imported explorations outside the assigned MRS supplied-reference example.

7. Lecture slides & references

• Fossen — Guidance Systems, Ch. 12 (lecture PDF index)
• Fossen — Workshop Lecture on 2D/3D LOS Guidance (official site)
• Kaminer — Path Following and Time Coordination for Small UAVs (slides)

This is a simplified reduced-order classroom demonstration. It is not a verified marine dynamics, robust stability, collision-avoidance or communication-network simulator. An explicit safety layer is required before transferring this architecture to real vehicles.