I am trying to simulate and implement the controller of the paper Geometric Tracking Control of a Quadrotor UAV on SE(3). To do this I need to first implement the dynamics of the quadrotor. According to the paper, the whole control structure looks as follows:

And the equations of motion are:

Now according to the setup given f and M as input I need to calculate the equations numbered 2 to 5. I see how we can calculate equation 2 and 3 and that is by calculating the acceleration and integrating over dt to get velocity and integrating again to get the position. But I don't understand how to calculate equations 4 and 5. In 5 I can calculate the angular acceleration from 5 by manipulating it but I don't know the angular velocity omega. Similarly in 4 I don't know the angular velocity omega to calculate R_dot. How do I calculate omega given just f and M? Is there a flaw in my understanding? Am I missing some piece of information? The link to the paper is here. Thanks in advance

Note that $\hat{.}$ is a skew symmetric operator, and retrieving the roll pitch yaw angle from $\mathbf{R}$ an be done easily based on the expression of some of its components. The tricky part is integrating the $\dot{\mathbf{R}}$
• The $\dot{R}$ is the time derivative of the rotation matrix, it's integration shall give you the possibility to retrieve the rotation angles used to generate the rotation matrix. – N. Staub Nov 21 '17 at 9:03
• you do not multiply it by $\Omega$ but by this skew symmetric matrix of the vector of your rotational velocity, This is the formula you get if you write down the derivation of the rotation matrix w.r.t. time. In general one prefer to use rotation matrix in order to avoid singularities (gimbal lock) and the dozen of conventions all called "Euler angles". Moreover angular velocity are given around 'fixed' axes of teh body frame, while Euler angles are successive rotation along intermediate frame axes. – N. Staub Nov 22 '17 at 10:43
I think what you're missing is initial conditions; you need to define $\Omega$ at $t=0$. Give that angular velocity, you can then easily solve (4) for $\dot{R}$ and (5) for $\dot{\Omega}$. From that point forward you always know what $\Omega$ is at the start of the timestep so can continue to solve in the same manner.