Quadrotor Dynamics: Calculating angular velocity of quadrotor

Question

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

Answer 1

If you want to compute the quadrotor dynamics you just need to plug M and f in (5) and (3) to get the accelerations, then you integrate the accelerations to get the velocities and you integrate the velocities to get the position/attitude. You will need to initialize the velocities and position/attitude to their initial values (typically zero velocity and initial position/attitude).

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 $\dot{\mathbf{R}}$

Answer 2

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.