# A Comparative Study for Control of Quadrotor UAVs
## 2 Materials and Methods  
### System Model  
Quadrotor UAVs are mechatronic systems with two rigid arms arranged in a cross configuration and four rotors. Each pair of rotors is located at the arm's extremity and rotates in opposite directions. The speed of the rotors is controlled independently. A picture of a commercial quadrotor platform is shown in Figure 1. The movement of flying quadrotors derives from the total lift force produced by all rotors and from their capability of inclination, which is due to the resultant forces acting on the arms and to the resultant counter-torque caused by the pairs of rotors. The representation shown in Figure 2 is derived from the configuration of Figure 1, considered as a reference for the formulation of the quadrotor dynamical model. Figure 1: Picture of a commercial quadcopter platform [20]. Two reference frames are defined in order to characterize the position and the orientation of the UAV: a fixed inertial reference frame OXYZ and a body reference frame oxyz attached to the quadrotor. The relation between the two is expressed by the homogeneous transformation matrix \(\mathbf{T_{L}}\) derived with the Euler ZYX convention, also known as the RPY convention. The RPY convention involves three rotations. Firstly, a rotation of \(\psi\) (Euler yaw angle) about \(Z\) produces the frame OX\({}_{1}\)Y\({}_{1}\)Z. Secondly, a second rotation of \(\vartheta\) (Euler pitch angle) about Y\({}_{1}\) results in the frame OX\({}_{2}\)Y\({}_{1}\)Z\({}_{1}\). Lastly, a third rotation of \(\phi\) (Euler roll angle) about X\({}_{2}\) results in the frame OX\({}_{2}\)Y\({}_{2}\)Z\({}_{2}\) (coincident with oxyz). These rotations are performed according to a post multiplication rule,  
\[\mathbf{T_{L}}=\mathbf{T_{\psi}T_{\theta}T_{\phi}}= \begin{pmatrix}c_{\phi}&-s_ {\phi}&0\\ s_{\phi}&c_{\phi}&0\\ 0&0&1\\ \end{pmatrix}\begin{pmatrix}c_{\phi}&0&s_{\phi}\\ 0&1&0\\ -s_{\phi}&0&c_{\phi}\\ \end{pmatrix}\begin{pmatrix}1&0&0\\ 0&c_{\phi}&-s_{\phi}\\ 0&s_{\phi}&c_{\phi}\\ \end{pmatrix} = \begin{pmatrix}c_{\phi}c_{\phi}&c_{\phi}s_{\phi}s_{\phi}&-s_{ \phi}c_{\phi}&s_{\phi}s_{\phi}+c_{\phi}s_{\phi}c_{\phi}\\ s_{\phi}c_{\phi}&c_{\phi}+s_{\phi}s_{\phi}s_{\phi}&s_{\phi}s_{\phi}c_{\phi}-s_{ \phi}c_{\phi}\\ -s_{\phi}&c_{\phi}s_{\phi}&c_{\phi}\\ \end{pmatrix}  \tag{1}\]  
where c and s denote cosine and sine functions, respectively. \(\mathbf{T_{L}}\) transforms linear movement-related vectors from oxyz to OXYZ. Consequently, \(\mathbf{T_{R}}\) transforms rotational movement-related vectors from OXYZ to oxyz,  
\[\mathbf{\Delta}= \begin{pmatrix}\mathrm{P}\\ \mathrm{q}\\ \mathrm{r}\end{pmatrix} =\mathbf{T_{\Phi}^{T}} \begin{pmatrix}\phi^{\prime}\\ 0\\ 0\\ 0\end{pmatrix} +\mathbf{T_{\Phi}^{T}}\mathbf{T_{\theta}^{T}} \begin{pmatrix}0\\ 0\\ 0\\ \end{pmatrix} +\mathbf{T_{\Phi}^{T}}\mathbf{T_{\theta}^{T}} \begin{pmatrix}0\\ 0\\ 0\\ \end{pmatrix} = \begin{pmatrix}1&0&-s_{\phi}\\ 0&c_{\phi}&c_{\phi}s_{\phi}\\ 0&-s_{\phi}&c_{\phi}c_{\phi}\\ \end{pmatrix}\begin{pmatrix}\phi^{\prime}\\ \phi^{\prime}\\ \psi^{\prime}\end{pmatrix} =\mathbf{T_{R}} \begin{pmatrix}\phi^{\prime}\\ \phi^{\prime}\\ \psi^{\prime}\end{pmatrix}  \tag{2}\]  
where p, q and r denote the rotational velocities in the body frame, while \(\phi^{\prime}\), \(\theta^{\prime}\) and \(\psi^{\prime}\) denote the Euler rates. The Newton-Euler formalism is adopted to present the quadrotor dynamical model; for further details about modeling refer to [21; 22; 23; 24]. In the adopted model, the following assumptions are made:  
* The structure is perfectly symmetrical. * The structure is perfectly rigid. * The quadrotor's center of mass coincides with the origin of the body fixed frame. * The rotors' motors dynamics are neglected. * Complex phenomena of difficult modeling are neglected, such as the ground effect, blade flapping and any other minor aerodynamical or inertial contributions not explicitly taken into account.