## 3 Stability Analysis Based on Lyapunov Exponential Method
### MATLAB Simulation Analysis
#### 3.3.1 Takeoff Stage  
When the tethered UAV is in the takeoff phase, the Liapunov exponential spectrum of the tethered UAV can be obtained by simply varying the density \(\rho\) of the tether cable, as shown in Figure 8. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Meaning** & **Parameter Value** \\ \hline \(m\) & UAV platform weight & 1.365 kg \\ \(g\) & Gravity acceleration & 9.8 m/s\({}^{2}\) \\ \(L\) & Boom length & 0.225 m \\ \(C_{T}\) & Lift coefficient & 1.0792 \(\times\) 10\({}^{-5}\) \\ \(C_{Q}\) & Reverse moment coefficient & 1.8992 \(\times\) 10\({}^{-5}\) \\ \(R\) & Blade radius & 0.127 m \\ \(Ix\) & Moment of inertia around the \(x\)-axis & 0.2868 kg/m\({}^{2}\) \\ \(ly\) & Moment of inertia around the \(y\)-axis & 0.3141 kg/m\({}^{2}\) \\ \(lz\) & Moment of inertia around the \(z\)-axis & 0.147 kg/m\({}^{2}\) \\ \(\rho_{a}\) & Air density & 1.293 kg/m\({}^{3}\) \\ \(\rho\) & Density of tethered cable & 1.29 kg/m\({}^{3}\) \\ \(s\) & Cross-sectional area of a tethered cable & 0.002 m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of tethered UAVs. Figure 7: Physical drawing of the tethered UAV prototype. Further, the results of Figure 8 are statistically analyzed. The results of the numerical quantitative analysis of the Lyapunov exponential spectrum are shown in Table 2, where the data in parentheses represent the percentage change in the Lyapunov exponent. (\(\dagger\): Indicates the percentage increase in the Lyapunov index; \(\downarrow\): Indicates percentage decrease in the Lyapunov index; \(\
eg\): Indicates that the Lyapunov Index remains unchanged). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & \(\rho\) **= 0.92** & \(\rho\) **= 1.38** & \(\rho\) **= 1.84** \\ \hline \(\lambda(x)\) & 0.42195 & 0.46652 (\(\uparrow\) 10.56\%) & 0.50592 (\(\uparrow\) 19.9\%) \\ \(\lambda(y)\) & 0.44627 & 0.49017 (\(\uparrow\) 9.84\%) & 0.52917 (\(\uparrow\) 18.58\%) \\ \(\lambda(z)\) & 0.36989 & 0.41851 (\(\uparrow\) 13.14\%) & 0.46032 (\(\uparrow\) 24.45\%) \\ \(\lambda(\varphi)\) & 0.02505 & 0.02505 (\(\
eg\)) & 0.02505 (\(\
eg\)) \\ \(\lambda(\theta)\) & \(1.39\times 10^{-8}\) & \(1.38\times 10^{-8}\) (\(\uparrow\) 0.53\%) & \(1.38\times 10^{-8}\) (\(\downarrow\) 0.92\%) \\ \(\lambda(\psi)\) & \(5.33\times 10^{-7}\) & \(5.30\times 10^{-7}\) (\(\uparrow\) 0.54\%) & \(5.28\times 10^{-7}\) (\(\downarrow\) 0.93\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Final values of Lyapunov exponent at different cable densities during the takeoff stage. Figure 8: Lyapunov exponential spectra at different cable densities during the takeoff stage. As can be seen from Table 2, \(\lambda(\varphi)\), \(\lambda(\theta)\) and \(\lambda(\psi)\) do not change much. When the density of the tethered cable is changed from \(\rho\) = 0.92 g/cm\({}^{3}\) to \(\rho\) = 1.38 g/cm\({}^{3}\), \(\lambda(\varphi)\), \(\lambda(\theta)\), and \(\lambda(\psi)\) increase by 10.56%, 9.84%, and 13.14%, respectively. When the density of the tethered cable changes from \(\rho\) = 0.92 g/cm\({}^{3}\) to \(\rho\) = 1.84 g/cm\({}^{3}\), \(\lambda(\chi)\), \(\lambda(\chi)\), and \(\lambda(\chi)\) increase by 19.9%, 18.58%, and 24.45%, respectively. It can be seen that as the density of the tethered cable increases by 50% and 100% from 0.92 g/cm\({}^{3}\) to 1.38 g/cm\({}^{3}\) and 1.84 g/cm3, the Lyapunov exponent of the tethered UAV increases in a near-equivalent scale. Based on the definition of the Lyapunov exponent, it can be seen that the stability of the tethered UAV decreases as the density of the tether cable increases. Assuming that the overall stability can be described using the sum of the percent change amplitude of all channel state quantities, the overall stability of the tethered UAV decreases by 32.47% when the tethered cable density changes from \(\rho\) = 0.92 g/cm\({}^{3}\) to \(\rho\) = 1.38 g/cm\({}^{3}\).