## 3 Stability Analysis Based on Lyapunov Exponential Method
### MATLAB Simulation Analysis
#### 3.3.3 Landing Stage  
When the tethered UAV is in the landing phase, the Liapunov exponential spectrum of the tethered UAV can be obtained by simply varying the tether cable density \(\rho\), as shown in Figure 10.  
Further, the results of Figure 10 are statistically analyzed, and the numerical quantification of the Lyapunov exponential spectrum is shown in Table 4.  
From Table 4, it can be seen that in the landing stage, when the tethered cable density changes from \(\rho\) = 0.92 g/cm\({}^{3}\) to \(\rho\) = 1.38 g/cm\({}^{3}\), \(\lambda(x)\) and \(\lambda(y)\) increase by 5.75% and 6.77%, respectively. When the tethered cable density is changed from \(\rho\) = 0.92 g/cm\({}^{3}\) to \(\rho\) = 1.84 g/cm\({}^{3}\), \(\lambda(x)\) and \(\lambda(y)\) rise by 10.96% and 12.76%, respectively. \(\lambda(z)\), \(\lambda(\varphi)\), \(\lambda(\theta)\), and \(\lambda(\psi)\) do not change much with the change in tethered cable density. Assuming that the overall stability can be described using the sum of the percent change amplitude of all channel state quantities, the overall strength of the tethered UAV decreases by 14.63% when the tether cable density is changed from \(\rho\) = 0.92 g/cm\({}^{3}\) to \(\rho\) = 1.38 g/cm\({}^{3}\). The overall stability of the tethered UAV decreased by 24.75% when the density of the tethered cable was changed from \(\rho\) = 0.92 g/cm\({}^{3}\) to \(\rho\) = 1.84 g/cm\({}^{3}\). The stability of the tethered UAV in the landing phase changed less with the tether cable density than in the takeoff phase.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & \(\rho\) **= 0.92** & \(\rho\) **= 1.38** & \(\rho\) **= 1.84** \\ \hline \(\lambda(x)\) & 0.41221 & 0.43593 (\(\uparrow\) 5.75\%) & 0.45740 (\(\uparrow\) 10.96\%) \\ \(\lambda(y)\) & 0.38481 & 0.41085 (\(\uparrow\) 6.77\%) & 0.43391 (\(\uparrow\) 12.76\%) \\ \(\lambda(z)\) & 1.22517 & 1.24377 (\(\uparrow\) 1.52\%) & 1.20319 (\(\downarrow\) 1.79\%) \\ \(\lambda(\varphi)\) & 0.06928 & 0.06928 (-) & 0.06928 (-) \\ \(\lambda(\theta)\) & \(2.78\times 10^{-8}\) & \(2.79\times 10^{-8}\) (\(\uparrow\) 0.28\%) & \(2.82\times 10^{-8}\) (\(\uparrow\) 1.37\%) \\ \(\lambda(\psi)\) & \(1.59\times 10^{-8}\) & \(1.60\times 10^{-8}\) (\(\uparrow\) 0.31\%) & \(1.61\times 10^{-8}\) (\(\uparrow\) 1.45\%) \\ \hline \end{tabular}
\end{table}
Table 4: Final values of Lyapunov exponent for different cable densities during the landing stage.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & \(\rho\) **= 0.92** & \(\rho\) **= 1.38** & \(\rho\) **= 1.84** \\ \hline \(\lambda(x)\) & 0.42187 & 0.46646 (\(\uparrow\) 10.57\%) & 0.50588 (\(\uparrow\) 19.91\%) \\ \(\lambda(y)\) & 0.44572 & 0.48955 (\(\uparrow\) 9.83\%) & 0.52851 (\(\uparrow\) 18.58\%) \\ \(\lambda(z)\) & 0.36967 & 0.41835 (\(\uparrow\) 13.17\%) & 0.46020 (\(\uparrow\) 24.49\%) \\ \(\lambda(\varphi)\) & 0.02244 & 0.02244 (–) & 0.02244 (–) \\ \(\lambda(\theta)\) & \(1.24\times 10^{-7}\) & \(1.24\times 10^{-7}\) (\(\uparrow\) 0.54\%) & \(1.23\times 10^{-7}\) (\(\downarrow\) 0.92\%) \\ \(\lambda(\psi)\) & \(1.25\times 10^{-7}\) & \(1.25\times 10^{-7}\) (\(\uparrow\) 0.54\%) & \(1.24\times 10^{-7}\) (\(\downarrow\) 0.94\%) \\ \hline \end{tabular}
\end{table}
Table 3: Final values of Lyapunov exponent for different cable densities during the hovering stage.