## 2 System Dynamic Model
### Variation in the System's Mass and Moments of Inertia Parameters  
To develop the model equations for time-varying mass and time-varying inertia, we have used SOLIDWORKS. The 3D CAD (three-dimensional computer-aided design) model of the quadrotor with payload developed in SOLIDWORKS is shown in Figure 1. The mass and MOI values are noted at different levels of the water in the liquid tank. The values recorded are listed in Table 2. The curve fitting toolbox available in MATLAB was used to find the model equations for mass and MOI. The root mean squared error (\(RMSE\)) was used to assess the goodness of fit. The variation in the values of the system's mass and MOI are shown in Figures 3 and 4. \begin{table}
\begin{tabular}{c c c c} \hline
**Quantity** & **Symbol** & **Value** & **Unit** \\ \hline Mass of quadrotor & \(\overline{\text{m}}\) & \(1.0497\times 10^{-1}\) & kg \\ Inertia about the x-axis & \(\overline{I_{x}}\) & \(1.1568\times 10^{-4}\) & kg m\({}^{2}\) \\ Inertia about the y-axis & \(\overline{I_{y}}\) & \(1.1378\times 10^{-4}\) & kg m\({}^{2}\) \\ Inertia about the z-axis & \(\overline{I_{z}}\) & \(1.2595\times 10^{-4}\) & kg m\({}^{2}\) \\ Length of an arm & \(1\) & \(6.24\times 10^{-2}\) & m \\ Rotor inertia & \(J_{r}\) & \(1.4338\times 10^{-4}\) & kg m\({}^{2}\) \\ Drag factor & \(k_{s}\) & \(2.8\times 10^{-2}\) & kg/m \\ Drag factor & \(k_{u}\) & \(5.4\times 10^{-2}\) & kg/m \\ Thrust coefficient & \(K_{f}\) & \(6.5330\times 10^{-4}\) & N/(rad\({}^{2}\)/s\({}^{2}\)) \\ Thrust coefficient & \(K_{m}\) & \(1.5769\times 10^{-6}\) & Nm/(rad\({}^{2}\)/s\({}^{2}\)) \\ Acceleration of gravity & g & \(9.81\) & m/s\({}^{2}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Quadrotor parameters with full payload. Figure 4: Variation in MOI with mass. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**S.No** & **Water Level (m)** & **Mass (kg)** & **I\({}_{\mathbf{x}}\) (kg m\({}^{2}\))** & **I\({}_{\mathbf{y}}\) (kg m\({}^{2}\))** & **I\({}_{\mathbf{z}}\) (kg m\({}^{2}\))** \\ \hline \(1\) & \(1.71\times 10^{-2}\) & \(1.0497\times 10^{-1}\) & \(1.1568\times 10^{-4}\) & \(1.1378\times 10^{-4}\) & \(1.2595\times 10^{-4}\) \\ \(2\) & \(1.52\times 10^{-2}\) & \(1.0141\times 10^{-1}\) & \(1.1311\times 10^{-4}\) & \(1.1254\times 10^{-4}\) & \(1.2396\times 10^{-4}\) \\ \(3\) & \(1.33\times 10^{-2}\) & \(9.7844\times 10^{-2}\) & \(1.1022\times 10^{-4}\) & \(1.1099\times 10^{-4}\) & \(1.2196\times 10^{-4}\) \\ \(4\) & \(1.14\times 10^{-2}\) & \(9.4281\times 10^{-2}\) & \(1.0697\times 10^{-4}\) & \(1.0906\times 10^{-4}\) & \(1.1994\times 10^{-4}\) \\ \(5\) & \(9.47\times 10^{-3}\) & \(9.0756\times 10^{-2}\) & \(1.0331\times 10^{-4}\) & \(1.067\times 10^{-4}\) & \(1.1794\times 10^{-4}\) \\ \(6\) & \(7.57\times 10^{-3}\) & \(8.7194\times 10^{-2}\) & \(9.910\times 10^{-5}\) & \(1.0380\times 10^{-4}\) & \(1.1590\times 10^{-4}\) \\ \(7\) & \(5.68\times 10^{-3}\) & \(8.3650\times 10^{-2}\) & \(9.431\times 10^{-5}\) & \(1.0028\times 10^{-4}\) & \(1.1386\times 10^{-4}\) \\ \(8\) & \(3.78\times 10^{-3}\) & \(8.0088\times 10^{-2}\) & \(8.876\times 10^{-5}\) & \(9.601\times 10^{-5}\) & \(1.1179\times 10^{-4}\) \\ \(9\) & \(1.86\times 10^{-3}\) & \(7.6488\times 10^{-2}\) & \(8.230\times 10^{-5}\) & \(9.080\times 10^{-5}\) & \(1.0967\times 10^{-4}\) \\ \(10\) & \(0\) & \(7.3002\times 10^{-2}\) & \(7.506\times 10^{-5}\) & \(8.477\times 10^{-5}\) & \(1.0760\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mass and inertia values at different water levels. Figure 3: Variation in mass with parameter \(\ell\). A parameter \(\ell\) has been introduced which is given as:  
\[\ell=\frac{\overline{h}-h}{\overline{h}} \tag{10}\]  
where \(h\) represents the level of the water in the tank and \(\overline{h}\) indicates the water level when the payload is filled to capacity. The equations of the mass and MOI obtained using curve fitting are:  
\[ \begin{cases}m=a_{1}\ell+a_{2}\\ I_{x}=b_{1}m^{2}+b_{2}m+b_{3}\\ I_{y}=c_{1}m^{2}+c_{2}m+c_{3}\\ I_{z}=d_{1}m+d_{2}\end{cases}  \tag{11}\]  
The derivatives are:  
\[