I've posted a question regarding this matter that I couldn't solve. I'm reading this paper, the authors state
Linear $x$ and $y$ Motion Control: From the mathematical model one can see that the motion through the axes $x$ and $y$ depends on $U_{1}$. In fact $U_{1}$ is the total thrust vector oriented to obtain the desired linear motion. If we consider $U_{x}$ and $U_{y}$ the orientations of $U_{1}$ responsible for the motion through x and y axis respectively, we can then extract from formula (18) the roll and pitch angles necessary to compute the controls $U_{x}$ and $U_{y}$ ensuring the Lyapunov function to be negative semi-definite ( see Fig. 2).
The paper is very clear except in the linear motion control. They didn't explicitly state the equations for extracting the angles. The confusing part is when they say
we can then extract from formula (18) the roll and pitch angles necessary to compute the controls $U_{x}$ and $U_{y}$
where formula (18) is
$$ U_{x} = \frac{m}{U_{1}} (\cos\phi \sin\theta \cos\psi + \sin\phi \sin\psi) \\ U_{y} = \frac{m}{U_{1}} (\cos\phi \sin\theta \sin\psi - \cos\phi \cos\psi) \\ $$
It seems to me that the roll and pitch angles depend on $U_{x}$ and $U_{y}$, therefore we compute the roll and pitch angles based on the $U_{x}$ and $U_{y}$ to control the linear motion.