## 5 Measurement of Mass Characteristics of the Pico-Satellite
### Measurement of the Moment of Inertia of the Pico-Satellite  
The measurement system measured the moment of inertia of the pico-satellite 50 times in different positions and states, and the experimental values obtained from the angular acceleration results are shown in Figures 11-13. The curves in the figures represent the experimental values of the angular acceleration without the pico-satellite and with the pico-satellite, respectively. The dashed lines represent the averages of the angular acceleration.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**State** & **Mass Gap** & **1** & **2** & **3** & **4** & **5** & **Average Value (Qg)** & **Experimental Coordinate Value (mm)** & **Theoretical Coordinate Value (mm)** & **Deviation (mm)** \\ \hline \multirow{2}{*}{Vertical Position} & \(P_{A}\) & 7.008 & 7.008 & 7.009 & 7.045 & 7.045 & 7.069 & \multirow{2}{*}{\(x_{C}=-0.939\,y_{C}=9.959\)} & \(x_{C}=-0.173\,y_{C}=8.688\) & \(\Delta x=0.766\) \\  & \(P_{A}\) & 8.974 & 8.979 & 8.851 & 8.781 & 8.766 & 8.74 & \multirow{2}{*}{\(x_{C}=-0.939\,y_{C}=9.959\)} & \(x_{C}=-0.173\,y_{C}=8.688\) & \(\Delta x=0.766\) \\ \cline{2-9}  & \(P_{A}\) & 14.288 & 14.281 & 14.288 & 14.266 & 14.257 & 14.252 & \multirow{2}{*}{\(x_{C}=0.606\,z_{C}=580.988\)} & \(x_{C}=-0.173\,z_{C}=581.196\) & \(\Delta x=0.433\) \\ \cline{2-9}  & \(P_{A}\) & 5.266 & 5.314 & 5.299 & 5.213 & 5.204 & 5.223 & \\ \cline{2-9}  & \(P_{A}\) & 14.277 & 14.282 & 14.218 & 14.218 & 14.220 & 14.240 & 14.240 & \multirow{2}{*}{\(x_{C}=0.688\,z_{C}=581.196\)} & \(\Delta x=0.046\) \\ \cline{2-9}  & \(P_{A}\) & 5.241 & 5.208 & 5.235 & 5.226 & 5.226 & 5.227 & \multirow{2}{*}{\(x_{C}=0.689\,z_{C}=580.745\)} & \(x_{C}=0.688\,z_{C}=581.196\) & \(\Delta x=0.481\) \\ \cline{2-9}  & \(P_{C}\) & 5.342 & 5.168 & 5.372 & 5.376 & 5.372 & 5.326 & \\ \hline \hline \end{tabular}
\end{table}
Table 13: The centroid measurement results of the pico-satellite.  
The relationship between the angular velocity and the motor starting time can be obtained based on the average values of the angular acceleration, as shown in Figure 14. In Figure 14, \(V_{vu}\) is the relationship between the angular velocity and time when the pico-satellite is not loaded in the vertical position. \(V_{vl}\) is the relationship between the angular velocity and time when the pico-satellite is loaded in the vertical position. \(V_{au}\) is the relationship between the angular velocity and time when the pico-satellite is not loaded in the horizontal position-\(\alpha\). \(V_{al}\) is the relationship between the angular velocity and time when the pico-satellite is loaded in the horizontal position-\(\alpha\). \(V_{Bu}\) is the relationship between the angular velocity and time when the pico-satellite is not loaded in the horizontal position-\(\beta\). \(V_{Bl}\) is the relationship between the angular velocity and time when the pico-satellite is loaded in the horizontal position-\(\beta\).  
According to Equations (14)-(17), when \(t\) is 0.5 s, the corresponding angular velocity values of \(\omega_{2}\) and \(\omega_{3}\) can be obtained. The moment of inertia of the measurement system is \(l_{us}\) when the pico-satellite is not loaded, thus allowing us to determine the moment of inertia of the experimental value (\(J_{Gz}\)). According to Equations (18)-(20), the moment of inertia of the pico-satellite can be obtained when the coordinate origin is located at the center of mass. The comparison of the moment of inertia between the experimental and theoretical results of the pico-satellite is shown in Table 14.  
Figure 14: \(Cont.\)  
Figure 13: Angular acceleration in different states at horizontal position-\(\beta\).  
Table 14 also shows that the moment of inertia deviations of the pico-satellite are 1.7 \(\times\) 10\({}^{-3}\) kgm\({}^{2}\), 1.49 \(\times\) 10\({}^{-2}\) kgm\({}^{2}\), and 1.37 \(\times\) 10\({}^{-2}\) kgm\({}^{2}\), respectively. The measurement results can also satisfy the design requirements of the measurement system.