4.1.1 Dynamics

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)