Ship Motion Compensation

What is modelled

A planar three-leg platform (three prismatic legs, 3-RPR) can move its deck in surge, heave and roll. The ship carries the leg bases in heave and roll. Inverse kinematics gives the leg lengths that keep the deck at its level reference pose; each leg is a second-order servo that tries to follow its target but cannot move faster than its speed limit. The actual deck pose follows from the real leg lengths by solving the forward kinematics with Newton iterations.

Equations

• Leg length li = |p + R(θ) ai − bi| for deck anchor a_i and base anchor b_i.
• Servo l″ = Kp(l* − l) − Kd l′, so H(s) = Kp / (s² + Kds + Kp) and the residual is |1 − H(jω)| times the leg motion.
• First-order leg motion for a heave A: stroke A · (rise / length), speed A ω · (rise / length).

Checks run in the page

• Inverse then forward kinematics round trip.
• Servo tracking error against |1 − H|.
• Leg stroke for a small heave against the geometric slope.
• Peak leg speed against A ω times the same slope.

Honest simplifications

Planar motion only, rigid links, ideal frictionless legs, perfect knowledge of the ship motion as a feed-forward command, no sensor delay and no payload dynamics. Real compensation systems use measured motion with latency, which limits performance at high frequency.

• Press Heavy swell, speed limited and watch the red legs: the deck then moves again.
• Press Soft servo and see how a low-stiffness servo lets heave through near its natural frequency.
• Press Stiff servo and see the residual fall toward zero at the wave frequency.