## 4 Simulation Results  
Some simulations were carried out to evaluate the performance of the control scheme proposed. The simulations were made in MATLAB by integrating the dynamic equations of the quadrotor (24)-(29) using a script that implements a Runge-Kutta method of fourth order. In order to calculate the fractional derivative used in Equations (35), (47) and (48) of the control scheme, the Grunwald-Letnikov numeric method is used, implementing the adaptive time step memory algorithm described in [31]. Different values of the fractional order \(\beta\) were tested, in order to show the performance of the control scheme.Also the signum function was implemented using the expression  
\[sgn(s_{z})=\frac{s_{z}}{|s_{z}|+\epsilon},\]  
where the small quantity \(\epsilon>0\) is a real positive constant. A piecewise trajectory was used as a tracking reference in the space as follows: the first 5 s are used for the take-off of the quadrotor until it reaches hover at 1 m, then a 2 m long lemiscata trajectory is followed once, with a period of 30 s per return, finally landing from 1 m to the floor is performed within the last 5 s.  
A perturbation term was introduced at the right hand side of the quadrotor dynamic model (24)-(29) at \(t=13\) s in order to simulate the effect of weather conditions, such as wind variations. This perturbation term used was given by  
\[\Delta_{z}=\Delta_{x}=\Delta_{y}=K_{a}+K_{b}sin(\frac{2\pi t}{T_{1}}),\]  
\[\Delta_{\phi}=\Delta_{\theta}=\Delta_{\phi}=K_{a}+K_{\beta}sin(\frac{2\pi t}{T _{2}}),\]  
with \(K_{a}=0.15\) [m/s\({}^{2}\)], \(K_{b}=0.01\) [m/s\({}^{2}\)], \(T_{1}=1.2\) [s], \(K_{\alpha}=0.087\) [rad/s\({}^{2}\)], \(K_{\beta}=0.015\) [rad/s\({}^{2}\)] and \(T_{2}=1.8\) [s]. The model parameters used in the simulation are shown in Table 1 while the control parameters are given in Table 2. The model parameters correspond to a real quadrotor that will be used in real experimentation. The quadrotor has 23 cm of arm longitud; this is the length from the center of the quadrotor to the place where each motor is mounted. The motors used are the Emax MT2213-935 KV; each motor can thrust up to 670 grams. An IMU um7 from RedShiftLabs is used and an Optitrack System will be used to locate the quadrotor in the space. The trajectory of the quadrotor in the space is shown in Figure 2. Nine fractional orders are being plotted (from \(\beta=0.1\) to \(0.9\)); the colors used in the figures for each value of \(\beta\) are listed in Table 3. Notice that the evolution of the variables with some values of \(\beta\) are indistinguishable, and the light green color, corresponding to the fractional order \(\beta=0.9\), is predominant. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Units** \\ \hline Mass m & 1.4 & kg \\ Gravity g & 9.81 & \(\frac{m}{s^{2}}\) \\ Ixx & 0.02 & kg\(\cdot\)m\({}^{2}\) \\ Iyy & 0.02 & kg\(\cdot\)m\({}^{2}\) \\ Izz & 0.04 & kg\(\cdot\)m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model Parameters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Value** \\ \hline \(\epsilon\) & 0.005 \\ \(\sigma_{z}\) & 0.055 \\ \(\lambda_{z}\) & 6.0 \\ \(\mu_{z}\) & 6.0 \\ \(\sigma_{y}\) & 0.055 \\ \(\lambda_{y}\) & 6.0 \\ \(\mu_{y}\) & 6.0 \\ \(\sigma_{x}\) & 0.055 \\ \(\lambda_{x}\) & 6.0 \\ \(\mu_{x}\) & 6.0 \\ \(Kd_{\phi}\) &  
The four control signals are depicted in Figures 12-15. It is important to mention that these control signals were limited to the values shown in Table 4[32].This values are similar to the ones that correspond to the quadrotor that will be used in the real experiments. Since in a real application the absolute value of those signals are limited due to the actuators (motors). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Signal** & **Maximal Value** & **Units** \\ \hline Thrust & 35 & N \\ Roll Torque & 4 & Nm \\ Pitch Torque & 4 & Nm \\ Yaw Torque & 2 & Nm \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Limitation of the control signals.