4.7.2 Integration

The system dynamics are represented by a system of nonlinear time invariant equations, in the form of


,

(4.62)

where is the vehicle state vector from (4.12), and is a nonlinear function of the current state and the commanded moment that calculates the first derivative of each of the elements of . The change in angular velocity , , and are calculated by solving (4.63) for angular acceleration. First, expand (4.1).


.

(4.63)

Then, solve for angular acceleration.


(4.64)

where


(4.65)


(4.66)

and


(4.67)

To find the change in angular velocity, the algorithm first calculates the angular momentum using (4.66). This is then plugged into (4.65). This result, along with the result of (4.67) is then used in (4.64) to find the angular acceleration.
The change in attitude , , , and due to the current angular velocity are calculated using (4.9).
The Cartesian position and velocity are not modified in this simulation. The state is propagated forward through time by numerical integration of (4.62) using a fourth order RungeKutta algorithm.