## 2 Materials and Methods
### Control Architecture
To test the step response of the Crazyflie 2.0 in indoor conditions, different additional weights (0, 4.7, 6.1, and 10.8 g) were considered. To calculate which parameters would be used for PID according to the Ziegler-Nichols, lambda, and AMIGO methods, the same program was used in Matlab. Four tests for each case were performed and they resulted in the mean values of the PID parameters according to Tables 4-7. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{K}\) & \(\mathbf{k_{i}}\) & \(\mathbf{k_{d}}\) \\ \hline \(1.2\cdot T_{100\%}\) & \(1.2\cdot T_{100\%}\) & \(1.2\cdot T_{100\%}\) \\ \hline \(K_{p}\cdot L\) & \(2\cdot K_{p}\cdot L^{2}\) & \(2\cdot K_{p}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of values for k, k\({}_{i}\), and k\({}_{d}\) according to the Ziegler-Nichols method. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{k}\) & \(\mathbf{k_{i}}\) & \(\mathbf{k_{d}}\) \\ \hline \(\dfrac{1}{K_{p}}\left(0.2+0.45\cdot\dfrac{T_{63\%}}{L}\right)\) & \(\dfrac{1}{\dfrac{K_{p}}{K_{p}}\left(0.2+0.45\cdot\dfrac{T_{63\%}}{L}\right)}\) & \(\dfrac{\left(0.2+0.45\cdot\dfrac{T_{63\%}}{L}\right)}{K_{p}}\cdot\dfrac{0.5 \cdot L\cdot T_{63\%}}{0.3\cdot L+T_{63\%}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of values for k, k\({}_{i}\), and k\({}_{d}\) according to the approximated \(M\)-constrained integral gain optimization (AMIGO) method. \begin{table}
\begin{tabular}{c c c} \hline \hline \(\mathbf{k}\) & \(\mathbf{k_{i}}\) & \(\mathbf{k_{d}}\) \\ \hline \(\dfrac{1}{K_{p}}\dfrac{\dfrac{L}{2}+T_{63\%}}{\dfrac{L}{2}+T_{cl}}\) & \(\dfrac{1}{K_{p}\cdot\left(\dfrac{L}{2}+T_{cl}\right)}\) & \(\dfrac{1}{K_{p}}\dfrac{\dfrac{L}{2}+T_{63\%}}{\dfrac{L}{2}+T_{cl}}\dfrac{As is shown in the experimental results, with an additional load of 0, 4.7, and 6.1 g in Tables 8-10, respectively, the AMIGO method in each case has a noticeably shorter settling time than the Ziegler-Nichols method (as is shown in Table 8, the lambda method is much slower than the others and is therefore excluded when placing an additional load of 4.7 and 6.1 g). The AMIGO method is considered the better of them even though in some cases it has a slower rise time and higher overshoot than the Ziegler-Nichols method, but these two parameters do not differ much from the different methods. As an example, we can observe the experimental results in Figures 3 and 4 for both a case without any additional weight and a case with a load of 4.7 g, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Rising Time (s)** & **Maximum Overshoot (cm)** & **Settling Time (s)** \\ \hline _AMIGO_ & 0.90 & 12.1 & 4.5 \\ _Ziegler-Nichols_ & 0.5 & 11.9 & 7.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The rising time, maximum overshoot, and settling time for the step response with an additional load of 6.1 g.  
Figure 3: Experimental results when following the predefined trajectory without any additional load, according to the AMIGO method. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Rising Time (s)** & **Maximum Overshoot (cm)** & **Settling Time (s)** \\ \hline _AMIGO_ & 0.90 & 19 & 4 \\ _Ziegler-Nichols_ & 0.55 & 14 & 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The rising time, maximum overshoot, and settling time for the step response with an additional load of 4.7 g.  
Therefore, based on the preliminary experimental results presented above, the AMIGO method was chosen as the primary tuning method as it has been proven to be a reliable method for tuning the PID controllers of the Crazyllie 2.0.