The time rate of change of the quaternion as a function of angular velocity is

_{}, 
(4.9) 
where _{ } is a 3x4 matrix given by

_{} 
(4.10) 
and

_{}. 
(4.11) 
In summary, equations (4.1)(4.4) describe the change in angular velocity due to moments applied to the vehicle, and equation (4.9) describes the change in the attitude due to the vehicle angular velocity. Together these equations completely describe the attitude motion of Ranger NBV.