Derivation of the Pseudo-Five-DoF Equations

Now we are ready to proceed with the derivation. First, let us develop the pseudo-five-DoF equations for the round rotating Earth and then simplify them for the flat Earth. Newton’s second law, Eq. (5.9), applied to a vehicle of mass mB, with external aerodynamic and propulsive forces / , and gravitational force /

yields

(9.3)

We shift to the velocity frame U using Euler’s transformation

and express the equation in inertial velocity coordinates

The rotational time derivative is simply

Because the aerodynamic and propulsive forces are usually modeled in body co­ordinates, they must be converted to velocity axes [fa, p]u = [T]BU[fa, p]B, as well as the gravity force, which is given in geographic coordinates [fg]u = [T]UG[fg]G. Before we can program the equations, we have to determine the coordinate trans­formation matrices [T]UI, [T]UG, and [T]BU.