## 3 Application to the Trajectory Tracking of a Quadrotor
### Trajectory Tracking of a Quadrotor  
Let us first consider the \(z\) position dynamics in the quadrotor model (24)-(29), more precisely Equation (26). These dynamics have the form (2) with \(x=z\), \(u=u_{1}\) and  
\[f_{\chi}=f_{z}=-g,\quad g_{\chi}=g_{z}=\frac{cos(\phi)cos(\theta)}{m},\quad \Delta_{\chi}=\Delta_{z}. \tag{33}\]  
The tracking error \(e\) also takes the form \(e=e_{z}=z-z_{d}\) with \(\chi_{d}=z_{d}\) being the reference signal. In accordance with the methodology described in the previous section, the FOSMC that assures to have convergence to \(z\) towards \(z_{d}\) despite the presence of the perturbation term \(\Delta_{z}\) is given by  
\[u_{1}=\frac{m}{cos(\phi)cos(\theta)}[g+P_{z}(z,\dot{z},t)], \tag{34}\]  
where  
\[P_{z}(z,\dot{z},t)=P_{\chi}(\chi,\dot{\chi},t)=\ddot{z_{d}}-\lambda_{z}\dot{e_ {z}}-\sigma_{z}(D^{-\beta}sgn(s_{z}))-\mu_{z}s_{z}, \tag{35}\]  
with \(s_{z}=s_{\chi}=\dot{e}+\lambda_{\chi}e=\dot{e_{z}}+\lambda_{z}e_{z}\) as switching function (\(\lambda_{\chi}=\lambda_{z}\)) and \(\sigma_{z}=\sigma_{\chi}\), \(\mu_{z}=\mu_{\chi}\) in (35). \(\Delta_{z}=\Delta_{\chi}\) is assumed to satisfy (13), with a bound \(\kappa_{\Delta}=\kappa_{\Delta z}\). In order to design the control signals \(u_{2}\) and \(u_{3}\), a desired reference signal is first defined for the angles \(\phi\) and \(\theta\), this is \(\phi_{d}\) and \(\theta_{d}\), respectively. For doing this, we adopt the small angle approximation widely used in the literature of unmanned aerial vehicles (see for example [28; 30]). Under this assumption, the dynamics (24), (25), (27)-(29) take the form  
\[\ddot{x}\approx tan(\theta)(g+P_{z})+\Delta_{x}, \tag{36}\]  
\[\ddot{y}\approx-tan(\phi)(g+P_{z})+\Delta_{y}, \tag{37}\]  
\[\ddot{\phi}=\frac{u_{2}}{I_{xx}},\quad\ddot{\theta}=\frac{u_{3}}  
where \(\sigma_{x}\), \(\sigma_{y}\), \(\mu_{x}\) and \(\mu_{y}\) are positive real constants. The \((-\beta)\) order derivative of Equations (42) have the form (43), this is  
\[\dot{s_{x}}=-\sigma_{x}D^{-\beta}sgn(s_{x})-\mu_{x}s_{x},\quad\dot{s_{y}}=- \sigma_{y}D^{-\beta}sgn(s_{y})-\mu_{y}s_{y}. \tag{43}\]  
Also from the dynamics (36) and (37), with \(\Delta_{x}=\Delta_{y}=0\), one has that Equation (10) is given by  
\[\dot{s_{x}}=\ddot{e_{x}}+\lambda_{x}\dot{e_{x}}=tan(\theta)(g+P_{z})-\ddot{x_{ d}}+\lambda_{x}\dot{e_{x}}, \tag{44}\]  
\[\dot{s_{y}}=\ddot{e_{y}}+\lambda_{y}\dot{e_{y}}=-tan(\phi)(g+P_{z})-\ddot{y_{d} }+\lambda_{y}\dot{e_{y}}. \tag{45}\]  
The desired signals \(\theta_{d}\) and \(\phi_{d}\) can then be computed from (44) and (45) when the signals \(tan(\theta)\) and \(tan(\phi)\) are considered to be virtual input signals, as mentioned before. This consideration allows to have the following expressions for \(\theta_{d}\) and \(\phi_{d}\):  
\[\theta_{d}=arctan\left(\frac{P_{x}}{P_{z}+g}\right),\quad\phi_{d}=-arctan \left(\frac{P_{y}}{P_{z}+g}\right), \tag{46}\]  
where  
\[P_{x}=\ddot{x_{d}}-\lambda_{x}\dot{e_{x}}-\sigma_{x}D^{-\beta} sgn(s_{x})-\mu_{x}s_{x}, \tag{47}\] \[u_{1}=\frac{m}{cos(\phi)cos(\theta)}[g+P_{z}(z,\dot{z},t)], \tag{34}\]0  
Notice now that, when \(\theta\) and \(\phi\) are managed to approach \(\theta_{d}\) and \(\phi_{d}\) as soon as possible, the FOSMC strategy discussed above allows to reach the sliding surface \(s_{x}=0\) and \(s_{y}=0\) where the tracking errors \(e_{x}\) and \(e_{y}\) tend to zero exponentially. In this paper, a simple proportional and derivation action is implemented to allow that \(\theta\rightarrow\theta_{d}\), \(\phi\rightarrow\phi_{d}\) and \(\psi\rightarrow\psi_{d}\), this is  
\[u_{1}=\frac{m}{cos(\phi)cos(\theta)}[g+P_{z}(z,\dot{z},t)], \tag{34}\]1  
\[u_{1}=\frac{m}{cos(\phi)cos(\theta)}[g+P_{z}(z,\dot{z},t)], \tag{34}\]2  
\[u_{1}=\frac{m}{cos(\phi)cos(\theta)}[g+P_{z}(z,\dot{z},t)], \tag{34}\]3  
where \(e_{\phi}=\phi-\phi_{d}\), \(e_{\theta}=\theta-\theta_{d}\) and \(e_{\psi}=\psi-\psi_{d}\) are the tracking errors in the angles \(\phi\), \(\theta\) and \(\psi\) respectively. Notice in particular, that \(\phi_{d}\) is set to 0 in (51).