## 1 Introduction  
As a new type of rotary-wing unmanned aerial vehicle (UAV), the tethered UAV comprises three parts: A ground power supply platform, a tethered cable, and a UAV platform. Tethered UAVs are widely used in scenarios such as personnel search and rescue [1, 2], geological exploration [3, 4], and fire monitoring [5] because of the advantages of long endurance, which is a trend in the future development of UAVs. However, tethered UAVs also have the problem of stability and susceptibility to interference. During the flight of the tethered UAV, there are factors such as gusts of wind interference, ground effect [6], aerodynamic effect [7, 8], etc., that will cause the UAV to vibrate. At the same time, the uncontrolled instability caused by the above-influencing factors will continue to be amplified due to tethered cables. Tethered UAVs are expensive, and depending on the application, they can be equipped with expensive avionics such as searchlights, high-definition cameras, and high-performance sensors. To reduce the crash rate and improve the maneuverability of UAVs, it is necessary to study the influence of tethered cables on the motion stability of tethered UAVs. There are generally two methods for studying the motion stability of UAVs. (1) Solving dynamic differential equations. This method has particular difficulties dealing with rotorcraft systems with high coupling, many variables, nonlinearity, and underdrive, so its versatility and popularization could be better. (2) Lyapunov function analysis. This method generally does not require the solution of cumbersome aerodynamic equations and is also quite popular for aircraft stability analysis [9]. There are two types of Lyapunov function analysis methods: Direct and exponential. For the direct method, [PERSON] et al. [10] obtained the dynamic model of the unconventional dual-fan UAV by the Newton-Euler method, analyzed its stability by Lyapunov's direct method, and designed an integral backsteppingcontrol strategy to calm down the state quantities of the position and attitude loops on this basis. [PERSON] et al. [PERSON] et al. (2018) studied a quadrotor's nonlinear adaptive trajectory tracking problem under fault conditions based on the Lyapunov direct method. However, the Lyapunov direct method usually requires more work to construct a suitable function in the face of nonlinear and highly coupled systems. The ability to suppress external disturbances could be more vital, especially since giving quantitative system indicators is challenging. For the exponential method, the Lyapunov exponential method can quantitatively analyze the motion stability of the system and provide a precise evaluation index. [PERSON] et al. (2018) evaluated the performance of the vehicle control system for yaw stabilization and rollover prevention by the Lyapunov exponent method. [PERSON] et al. [PERSON] et al. (2018) established a quantitative relationship between structural parameters and the dynamic stability of a quadrotor UAV by employing the Lyapunov exponential method, which guided the optimization of the mechanical structure design of the vehicle and its stability control. [PERSON] et al. [PERSON] et al. (2018) investigated the quantitative relationship between the structural parameters of a quad-tilt wing-unmanned aerial vehicle and the dynamical stability of the system's transition mode using the Lyapunov exponential method. Through the above literature research, it is found that some scholars' research on aircraft stability mainly focuses on fixed-wing UAVs and quadrotor UAVs. There is no more in-depth analysis of tethered UAVs; stability studies do not explore the relationship with tethered cables. Based on the above analysis, this paper uses a method based on the Lyapunov exponent to quantify the motion stability of tethered UAVs. The Euler-Poincare equation and catenary theory are used to obtain the dynamic equation of the tethered UAV. Then, based on the Lyapunov exponential method, the motion stability of the tethered UAV is analyzed in the simulation environment.