The EOM come from applying the transport theorem to Newton's second law and its rotational equivalent, since neither force nor moment behaves simply when you're differentiating a vector in a rotating body frame instead of an inertial one. That's the whole reason this derivation matters practically, an IMU measures accelerations and rates in the body frame, but flight dynamics are naturally an inertial-frame problem. The transport theorem is what bridges the two, converting the body-frame rate of change into the inertial one by adding a rotation cross-product term.
Working from free body diagrams of the vehicle, I decomposed the gimbal thrust vector by two Euler-like deflection angles (pitch deflection δq, yaw deflection δr) to get thrust components in the body axes, combined those with gravity resolved through the vehicle's roll/pitch/yaw attitude, and left aerodynamic drag and moment symbolic pending better estimates of Cd and Cp. On the rotational side, I used the standard rigid body moment equations with a symmetric inertia tensor to get the roll, pitch, and yaw accelerations, including the coupling terms between axes.
The full state vector ends up being position (earth frame), velocity (body frame), Euler attitude (roll, pitch, yaw), and body rates (p, q, r), driven by gimbal deflection angles δq and δr as the control inputs.
Assumptions
Coming Soon!