Quadrotor SE3 Controller: Calculating Rd_dot

Question

I am trying to simulate and implement the controller in the paper Geometric Tracking Control of a Quadrotor UAV on SE(3). I have the dynamics implemented, however I am stuck at one part in the controller which is the calculation of the following equation:

$$e_\Omega=\Omega-R^TR_d\Omega_{d\cdot}$$

I have all the variables in equation (11) calculated except for $\Omega_d$ which is the desired angular velocity.

From equation (4) of the paper we know the relation:

$$\dot{R}_d=R_d\hat{\Omega}_d$$

However, I don't know how to calculate $\dot{R}_d$. Can someone give the exact equation for getting $\dot{R}_d$ so I can calculate $\Omega_d$ to get the error $e_\Omega$?

My code is available on github.

Answer 1

In this paper rotation matrices are used to describe the orientation of the quadrotor. This is chosen because each orientation as a single representation (contrary to quaternions) and gimbal lock is not possible (contrary to any Euler angle convention).

In the proposed controller, ones want to know how close the actual rotational velocities from the desired one, hence the equation (11) which provides this information as a vector from the matrix representation.

That being said, $\Omega_d$ is your desired angular velocity and should come from your planner as for the desired rotation transcribed in $R_d$. As you have the time evolution of $R_d$ you should be able to get its derivative $\dot{R}$.