Matrix Inversion & Least Squares in Thrust Allocation

Why It Matters

The technique enables precise, energy-efficient vessel control in harsh offshore environments, protecting costly infrastructure and boosting operational reliability.

Key Takeaways

• DP system uses real-time matrix inversion for thrust allocation.
• Least-squares method provides best-fit solution when perfect answer unavailable.
• Algorithm distributes thrust across actuators while minimizing energy consumption.
• Continuous calculations keep vessel position within a meter despite wind or currents.
• AHTS vessels use tension tow lines to fine-tune shuttle tanker positioning.

Summary

The video explains how dynamic positioning (DP) systems compute thrust allocation in real time, relying on matrix inversion to “undo” the set of force equations that govern a vessel’s motion.

When equations lack an exact solution—due to coupled thrusters or conflicting forces—the system switches to a least-squares approach, finding the solution that minimizes the error between commanded and achievable forces while also reducing energy use.

The presenter illustrates the method on a shuttle tanker being positioned for offshore loading. AHTS vessels employ tensioned tow lines from stern winches, nudging the tanker inch‑by‑inch to keep the loading hose intact even in rough seas.

By continuously recalculating position, heading and drift, the DP algorithm can lock a ship within a meter of its target, improving safety, reducing downtime, and lowering fuel costs for offshore operations.

Original Description

Once the DP system calculates the ship's absolute deviation, it must balance the vessel across six degrees of freedom: surge, sway, heave, roll, pitch, and yaw. The central computer resolves these complex multi-variable movement equations in real time by employing mathematical matrix inversion to parse exactly how to counter ocean forces.

Because real-world variables don't always align perfectly, the controller uses a least-squares optimization algorithm to find the absolute best-fit solution. This mathematical approach minimizes total positioning error and balances load distribution across the azimuth thrusters, tunnel thrusters, and main propellers to lock the hull in place with meter-level precision.