Vessel theory: Vessel rotations

The orientation of a vessel is specified by 3 rotation angles, called Euler angles. There are various different ways of defining Euler angles and the conventions used by OrcaFlex are documented below. In all cases positive rotation angle means rotating clockwise about the relevant axis direction.

General rotation conventions

Order of application of rotations

When provided with a set of rotation angles, the orientation that they define is affected by the order in which the elemental rotations are applied.

Given rotations roll, pitch and yaw, they define elemental rotations which are about x, y and z axes respectively and which we denote Rx, Ry, Rz respectively. These rotations can be represented by simple rotation matrices $M\!x$, $M\!y$, $M\!z$, which are multiplied together to obtain a final orientation matrix $M$. Matrix multiplication is not, however, commutative, so the product $M\!x M\!y M\!z$ is not equal to $M\!y M\!x M\!z$. It is this property that makes defining the order of application (and thus the order of multiplication) for rotations so important.

Successive rotations' axes

When applying any rotation angle, an axis of rotation is defined. When applying multiple rotations, a starting frame of reference is available, and a final frame of reference is obtained once all the rotations have been applied. Moreover, there are also intermediate frames of reference.

Once the first rotation has been applied, the starting frame is still defined, and a rotated frame is also defined, but that rotated frame is not yet the final frame, because there are further rotations to make. Should the next rotation be applied about the appropriate axis of the intermediate frame, or of the starting frame?

In fact, both conventions are valid, but they will yield different final orientations. We label these conventions as using

• Rotated axes when each successive rotation is applied using the appropriate axis from the intermediate frame defined by the previous rotations.
• Original axes when each successive rotation is applied using the appropriate axis from the starting frame.

To fully specify a 3D rotation, as well as the given rotation angles the associated convention used (rotated or original) must also be known.

OrcaFlex conventions

Naming of OrcaFlex vessel rotation angles

To provide flexibility, a vessel in OrcaFlex can be given two types of motion, primary and superimposed. In many cases only one of the two types is needed. The total motion of the vessel is the combination of the two. To distinguish between the primary, superimposed and total motion, different names are used for the 3 rotation angles as follows:

• For primary motion the rotation angles are called primary rotation 1, primary rotation 2 and primary rotation 3. The initial orientation of the vessel specifies the initial values of the primary rotation angles, and these are called the initial heel, trim and heading.
• For displacement RAOs and harmonic motion they are called roll, pitch and yaw.
• The total motion results report the combination of the primary and superimposed motion. For this the angles are called rotation 1, rotation 2 and rotation 3.

Vessel rotations

For each set of vessel rotation angles listed above, the rotations are applied in the reverse order about rotated axes. That is, the rotation about the $z$-axis is done first, followed by the rotation about the new $y$-axis, and then finally the rotation about the new $x$-axis.

Here is more detail of how the vessel rotations are applied.

The rotations start with the vessel axes $V_\mathrm{x}$, $V_\mathrm{y}$, $V_\mathrm{z}$ aligned with the global axes directions $G_\mathrm{X}$, $G_\mathrm{Y}$, $\GZ$.

The primary motion rotation angles are applied first, in reverse order. That is, primary rotation 3 is applied about the initial $V_\mathrm{z}$-direction (= global $\GZ$ direction), then primary rotation 2 is applied about the resulting new $V_\mathrm{y}$ direction, and finally primary rotation 1 is applied about the resulting new $V_\mathrm{x}$ direction.

The superimposed motion is then applied. For superimposed displacement RAOs and harmonic motion the superimposed motion rotations are called roll, pitch and yaw. These rotations are applied starting from the primary low frequency heading frame, which is the vessel orientation resulting from only the primary low frequency yaw. The reverse order of application is used, so first the yaw rotation is applied as a rotation about the frame $z$-direction (= vertically upwards direction = global $\GZ$ direction), then the pitch rotation is applied as a rotation about the frame's new $y$-axis direction, and finally the roll is applied as a rotation about the frame's new $x$-axis direction.

For superimposed time history motion the rotations are applied about the primary frame axes. Once again, the reverse order of application is used.