**Axis System Conventions**

**Approach**

My first step in designing the hull is to define some axis systems I
can use throughout the design process, including defining the shape of
the hull, calculating its performance, and analyzing the dynamics of the
boat's motion.

Body axis systems are always fixed to the body - they move and rotate
with it. The reference point for the hull body axis system is the midpoint
of the hull, at the design waterline. The X, Y, and Z axes form a right-handed
coordinate system. The X axis is measured positive forward of midships,
and negative aft. The Y axis is measured positive to starboard, and negative
to port. The Z axis is measured positive down, and negative up. The position
of a hull is given by the position of the midpoint of the hull relative
to the reference point for the whole boat. For defining the geometry of
monohulls, trimarans, and proas, this is the reference point of the main
hull. For catamarans, this is the midships point, mid way between the two
hulls, at the waterline. The body axis system will be relocated to the
boat's center of gravity for dynamic analyses.

I've chosen the midpoint as the origin for the body axis system because
it is near the center of gravity and center of buoyancy, and it avoids
having all the X coordinates be negative numbers. An aft-starboard-up coordinate
system starting at the forward end of the design waterline would be just
as good for the geometry, but the boat's velocity would be negative, which
is also awkward.

Forces applied to the hull are also positive when acting forward, starboard,
and down, as are the velocity components, U (surge), V (sway), W (heave).
The moments about the X, Y, and Z axes, L, M, and N, and the angular rates,
P, Q, and R, are positive rolling to starboard, pitching bow up, and yawing
to starboard.

The earth axis system, used to orient the boat, is positive to North,
East, and down. The boat's location is measured from an earth axis system
that doesn't move, and a moving earth axis system that is attached to the
boat but doesn't rotate.

**Axes Rotations**

To compute the hydrostatics and motion for a complete multihull vessel,
it is necessary to assemble the hulls in relation to each other, as they
will be assembled in the complete boat. This will involve translation -
lateral, longitudinal, and vertical - to position the hulls, and rotation.
Rotation is necessary to consider hulls that are canted so as to be vertical
when the boat is heeled, but also for amas on a trimaran that may be pitched
up or hulls that are toed in or out, as with flip-tackers. Rotation of
the coordinates is also necessary to compute the hydrostatics when the
boat is in a heeled condition, or trimmed up or down by the bow.

Rotating the points is done by multiplying the X-Y-Z vector of coordinates
by a transformation matrix. There is a transformation matrix for rotations
about each axis, and these can be combined into a single transformation
matrix. These transformations are necessary to change from one axis system
to another, say from the body axis system (forward, starboard, keelward)
to the earth axis system (North, East, down). The order of the rotations
is important, since each successive rotation is done about an axis in an
intermediate axis system. These transformation matrices turn out to have
a special mathematical property - the inverse of the matrix is equal to
its transpose. This makes it easy to change from going from earth to body
axes to going from body to earth axes - just swap elements across the diagonal
and reverse the order of the matrix multiplications.

I have chosen to use the same convention as is used in aeronautical
practice in order to make it easy to use existing derivations for the equations
of motion when flying on hydrofoils. Going from the earth to the body axis
one rotates first in yaw about the Z axis- a change in the boat's heading.
Then one rotates in pitch about the new Y axis, and finally one rotates
in roll about the final X axis. There is some evidence that boats tend
to pitch in the heeled plane of symmetry, rather than in the vertical plane,
which would make a yaw-roll-pitch order more natural. However, I will be
sticking with the yaw-pitch-roll convention.

The individual rotations are: Earth to intermediate system 1

Intermediate system 1 to intermediate system 2

Intermediate system 2 to body axes

These can be combined together into one single rotation matrix (here's
where the order comes in):

Going from body to earth axes:

The procedure for assembling the hulls in an arbitrary arrangement is
to first rotate the hulls in toe (t, positive
bow to starboard), incidence (i, positive bow
up), and cant (k, positive rolled to starboard),
using the expression above. Then position the hull by adding to all the
coordinates the position of the reference center relative to the boat's
reference center.

Note that this can play havoc with the station definitions. If toe in/out
or incidence is used, the points for a given station will no longer be
all at the same longitudinal station for the whole boat (X coordinates
will not be the same for a given hull cross section). This doesn't cause
any fundmental difficulty, as long as one carries along the X coordinate
as well as Y and Z for each point.

I've found it useful to have a special case of the earth axis system,
which is aligned with the waterplane and the boat's X axis. Thus, the heading
angle, Y, is zero, but the pitch and roll angles
are the same as for the earth axis system. The origin of the waterplane
axis system is also at the mean water surface, so after rotating from body
to waterplane orientation, the coordinates are translated vertically to
account for the height of the boat's reference center relative to the waterplane.

A similar axis system is the stability axis system. The stability axis
system will be used to linearize the equations of motion to analyze the
boat's dynamics. The stability axis system is a body axis system, fixed
to the body at the boat's reference point, and moving and rotating with
the body. However, at some time, typically then the boat is in steady trimmed
motion, the orientation is frozen to be parallel to the waterplane axis
system: