Aerial Robotics (Formulation)

Now that we have looked at positions and velocities, it's time to study the dynamics of a quadrotor.
Once again, let's remind ourselves that we have two coordinate systems. One attached to the moving robot, and the other, the inertial coordinate system.
b1, b2, b3 constitute the set of unit vectors that describe the body fixed coordinate system, and likewise a1, a2, a3 describe a coordinate system that's fixed to the inertial frame. Recall that you have four rotors, each of which is independently actuated. r is the position vector of the center of mass, and we know expressions for the truss that the motors produce on the airframe.
We also know the expressions for the reaction moments, and both of these are proportional to the square of the angular speeds of the rotors.
We've seen how Euler Angles are used to represent rotations.
We will use the zxy-connection, the first rotation being about the z-axis to psi. The second one about the x-axis. And the third one about the y-axis. So, the first axis is yaw, the second one is roll, the third one is pitch.
Again just to remind ourselves, this is the Z-X-Y convention. The first rotation about the Z axis through psi, the second rotation about the X axis through phi, this is the roll angle, and finally, the pitch about the Y axis through theta.
Of course, there are singularities. Singularities occur when the roll angle Is equal to 0, which is phi equal to 0. And even when the angle phi is not equal to 0, you can have two sets of Euler angles for every rotation.
Let's look at the external forces and moments that act on the airframe.

(Formulation)