The dynamics equations model rigid body rotation under the influence of applied moments from thrusters, water drag, and buoyancy moment. The equations do not include translational dynamics. The general expression describing the rotational motion of a rigid body [4] is

_{} 
(4.1) 
where _{ } is the total moment applied to the vehicle,_{ } represents its angular rate, and _{ } is the inertia matrix. The moments applied to the vehicle are summed to give a total moment.

_{} 
(4.2) 
_{}
is simply the moment commanded by the flight controller.
The drag moment _{ }
is calculated using

_{}. 
(4.3) 
The drag matrix D parameterizes the drag properties of the
vehicle. Each of the diagonal elements of the drag matrix indicates
the drag about an axis due to angular velocity about that same axis.
These values are always positive. Each of the off diagonal elements
indicates the drag about an axis due to angular velocity about one of
the other axes. An example of this would be roll moment caused by
drag on a stationary dexterous manipulator held extended during a
pitch maneuver. These off diagonal elements can be positive or
negative based on the direction of the moment induced. For example,
if the left dexterous manipulator were held extended, then a positive
pitch velocity (up) would cause a negative roll moment (left). This
drag would be parameterized by a negative value in _{ }.
In comparison, the parameter _{ }
represents a pitch moment caused by a roll velocity of the vehicle.
In this case, the same extended left dexterous manipulator would
produce a negative pitch moment when the vehicle had a positive roll
velocity. Therefore, this element would also be negative. The
magnitude of the linear velocity of various parts of the manipulator
are different when comparing a roll and pitch maneuver of the same
angular velocity because the manipulator links are not the same
distance from the two different axes of rotation. So while it makes
sense that symmetric off diagonal elements in the drag matrix should
have the same sign, it does not follow that they should have the same
magnitude.
The buoyancy moment _{ }
is calculated by taking the cross product of the buoyancy offset
vector, and the gravity vector..

_{} 
(4.4) 