# A Comparative Study for Control of Quadrotor UAVs
## 2 Materials and Methods
### System Model
It follows the translational dynamical model in the inertial fixed reference frame,  
\[\mathrm{m}\mathbf{R^{\prime}}=\mathbf{mg}-\mathrm{K_{d}^{L}R^{\prime}}-( \mathbf{F_{1}}+\mathbf{F_{2}}+\mathbf{F_{3}}+\mathbf{F_{4}})\mathbf{T_{L}} \tag{3}\]  
Figure 2: Schematic representation of the quadrotor configuration of reference. \[\mathbf{F_{i}}= \begin{pmatrix}0\\ 0\\ \mathrm{bw}_{1}^{2}\end{pmatrix} \,,\text{ with i }=1,2,3,4 \tag{4}\]  
where \(\mathbf{R}\) denotes the position vector in OXYZ, \(\mathbf{F_{i}}\) is the thrust force produced by rotor i, w\({}_{i}\) is the angular speed of rotor i, b is the thrust coefficient, m is the quadcopter's mass, \(\mathbf{g}\) is the gravity acceleration and K\({}_{\mathrm{d}}^{\mathrm{L}}\) denotes the aerodynamic thrust drag coefficient. It follows the rotational dynamical model in the body reference frame,  
\[\mathrm{I}\Delta^{\prime}=-(\mathbf{\Delta}\,\mathbf{x}\,\mathrm{I}\Delta)- \mathbf{M_{gy}}-\mathrm{K_{d}^{R}}\Delta+\mathbf{M} \tag{5}\]  
\[\mathbf{M_{gy}}=\mathrm{I_{R}}(\mathbf{w}_{1}+\mathbf{w}_{3}-\mathbf{w}_{2}- \mathbf{w}_{4})\ \mathbf{\Delta}\times \begin{pmatrix}0\\ 0\\ 1\end{pmatrix}  \tag{6}\]  
\[\mathbf{M}= \begin{pmatrix}\mathrm{L}(\mathrm{F_{2}}-\mathrm{F_{4}})\\ \mathrm{L}(\mathrm{F_{1}}-\mathrm{F_{3}})\\ \mathrm{d}\big{(}\mathbf{w}_{1}^{2}-\mathbf{w}_{2}^{2}+\mathbf{w}_{3}^{2}- \mathbf{w}_{4}^{2}\big{)}\end{pmatrix}  \tag{7}\]  
\[\mathrm{I}= \begin{pmatrix}\mathrm{I_{x}}&0&0\\ 0&\mathrm{I_{y}}&0\\ 0&0&\mathrm{I_{z}}\end{pmatrix}  \tag{8}\]  
with the quadrotor's arm length being L, the diagonal inertia matrix I, the inertia of the rotors I\({}_{\mathrm{R}}\), the drag coefficient d, the aerodynamic moment drag coefficient K\({}_{\mathrm{d}}^{\mathrm{R}}\), the rotors' induced moment vector \(\mathbf{M}\) and the gyroscopic moment vector \(\mathbf{M_{gy}}\); the latter is due to the fact that the axes of rotation of the propellers are attached to the quadrotor and rotate with it. The complete nonlinear dynamical model of the quadrotor is derived by evaluating Equations (3) and (5) and considering the transformations in Equations (1) and (2):  
\[\mathrm{X^{\prime\prime}}=-\frac{1}{\mathrm{m}}\mathrm{K_{d}^{L}}\mathrm{X^{ \prime}}-\frac{1}{\mathrm{m}}(\mathrm{s_{\phi}}\mathrm{s_{\phi}}+\mathrm{c_{ \psi}}\mathrm{s_{\theta}}\mathrm{c_{\phi}})\mathrm{U_{1}} \tag{9}\]  
\[\mathrm{Y^{\prime\prime}}=-\frac{1}{\mathrm{m}}\mathrm{K_{d}^{L}}\mathrm{Y^{ \prime}}-\frac{1}{\mathrm{m}}(\mathrm{s_{\theta}}\mathrm{s_{\psi}}\mathrm{c_{ \phi}}-\mathrm{s_{\phi}}\mathrm{c_{\psi}})\mathrm{U_{1}} \tag{10}\]  
\[\mathrm{Z^{\prime\prime}}=\mathrm{g}-\frac{1}{\mathrm{m}}\mathrm{K_{d}^{L}} \mathrm{Z^{\prime}}-\frac{1}{\mathrm{m}}\mathrm{c_{\phi}}\mathrm{c_{\phi}} \mathrm{U_{1}} \tag{11}\]  
\[\mathrm{p^{\prime}}=\frac{\mathrm{I_{y}}-\mathrm{I_{z}}}{\mathrm{I_{x}}} \mathrm{qr}-\frac{\mathrm{I_{R}}}{\mathrm{I_{x}}}\mathrm{q}(\mathrm{w_{1}}+ \mathrm{w_{3}}-\mathrm{w_{2}}-\mathrm{w_{4}})-\frac{1}{\mathrm{I_{x}}}\mathrm{ K_{d}^{R}}\mathrm{p}+\frac{1}{\mathrm{I_{x}}}\mathrm{LU_{2}} \tag{12}\]  
\[\mathrm{q^{\prime}}=\frac{\mathrm{I_{z}}-\mathrm{I_{x}}}{\mathrm{I_{y}}} \mathrm{pr}-\frac{\mathrm{I_{R}}}{\mathrm{I_{y}}}\mathrm{p}(\mathrm{w_{1}}+ \mathrm{w_{3}}-\mathrm{w_{2}}-\mathrm{w_{4}})-\frac{1}{\mathrm{I_{y}}}\mathrm{ K_{d}^{R}}\mathrm{q}+\frac{1}{\mathrm{I_{y}}}\mathrm{LU_{3}} \tag{13}\]  
\[\mathrm{r^{\prime}}=\frac{\mathrm{I_{x}}-\mathrm{I_{y}}}{\mathrm{I_{z}}} \mathrm{pq}-\frac{1}{\mathrm{I_{z}}}\mathrm{K_{d}^{R}}\mathrm{r}+\frac{1}{ \mathrm{I_{z}}}\mathrm{U_{4}} \tag{14}\]  
\[\mathrm{\phi^{\prime}}=\mathrm{p}+\mathrm{s_{\phi}}\mathrm{s_{\phi}}\mathrm{c _{\phi}}^{-1}\mathrm{q}+\mathrm{c_{\phi}}\mathrm{s_{\phi}}\mathrm{c_{\phi}}^{-1} \mathrm{r} \tag{15}\]  
\[\mathrm{\phi^{\prime}}=\mathrm{c_{\phi}}\mathrm{q}-\mathrm{s_{\phi}}\mathrm{r} \tag{16}\]  
\[\mathrm{\psi^{\prime}}=\mathrm{s_{\phi}}\mathrm{c_{\phi}}^{-1}\mathrm{q}+ \mathrm{c_{\phi}}^{-1}\mathrm{c_{\phi}}\mathrm{r} \tag{17}\]  
\[\mathrm{u}=