Considering a multi-body floating system consisting of N structures, let the initial estimate of the structure positions and orientations be represented by the vector ,

(11–9)

where are the coordinates of the j-th structure center of gravity with respect to the FRA, and are the finite angular rotations describing the orientation of this structure; the superscripts denote the iteration step. The displacement required in step 1 is given by

(11–10)

and the new position of the body is given by

(11–11)

The process is repeated until the m-th iterative step, when is smaller than the prescribed limit for convergence.

It is possible to have more than one equilibrium position: for example, a capsized ship can still float in equilibrium, if buoyancy is preserved. It is therefore important to start the iterative process with an initial estimate that is close to the required solution. Furthermore, because of the nonlinearities in the system, it is also possible to overshoot the intended equilibrium position. In practice, may therefore be scaled by a specified under-relaxation factor to ensure stability in the iterative scheme.

Before equilibrium is reached, a set of unbalanced residual forces and moments will act on the bodies. These include hydrostatic forces, weights of the structures, mooring tensions, wind and current drag, thruster forces, steady wave drift forces, and constraint reaction forces as described in External Static Forces and Articulations Between Structures.