## 1 Introduction  
Electricity serves as a central energy source for human civilization, and PTLs are a critical element of the electricity system [1, 2]. The operational integrity of PTLs is vulnerable to various potential failures, such as strand breakage, corrosion, and wear, which can be caused by environmental conditions, human error, and manufacturing defects [3, 4]. Therefore, regular inspections of PTLs are essential to maintain the long-term stability of the power system. Traditional manual inspections are inefficient, costly, and inaccurate [5], highlighting the need for robotic inspection. Currently, the main types of PTL inspection robots include aerial robot, climbing robots, and HIRs [6]. Aerial robots use drones as platforms, equipped with onboard inspection devices, to perform inspections while airborne [7, 8]. Climbing robots have two or more mechanical arms, oriented perpendicular to the PTL, with locomotion wheels mounted on these arms. These wheels interface with and move along the PTL, driven by motors [9]. HIRs integrate the functionalities of both climbing and flying inspection robots. Researchers typically use the flying function to suspend the robot on the PTL, subsequently utilizing the walking function to enable the robot to inspect along the line [10; 11; 12; 13; 14; 15; 16]. In summary, aerial robots operate autonomously but have limited endurance and cannot inspect along PTLs, while climbing robots have longer endurance but require human assistance. The HIR integrates the advantages of both, enabling inspections both on the PTL and in the air, thereby improving inspection efficiency. FPTLIR is an advanced HIR variant designed with a pendulum-like structure to increase stability. However, due to the flexibility and inclination of the PTL, the FPTLIR exhibits unpredictable dynamics during landing, leading to potential failure modes, such as collisions and falls, which may also compromise the insulation clearance between PTLs. Many previous studies have explored the HIR landing process to enhance its safety. [PERSON] et al. [12] developed a vision-based control system for semi-automatic landing. [PERSON] et al. [13] implemented a vision-based autonomous landing system, integrating multi-scale powerline detection and spatiotemporal tracking for stable landings. [PERSON] et al. [14] developed a discrete-time control algorithm for PTL tracking. [PERSON] et al. [15] also proposed a trajectory planning method for HIR. [PERSON] et al. [16] designed an over-actuated HIR with tilt rotors for stable catenary tracking without additional locomotion systems. Despite some studies enhancing the pre-landing safety of HIRs through visual localization and control [12; 13; 14], others pre-plan safe landing paths [15] or improve post-landing static stability through structural design [11; 16]. However, the impact of HIRs' dynamic behavior after contacting the PTL on safety remains a challenge. To study the dynamics of the HIR landing process, the PTL cannot be assumed to be a rigid body as in previous research [11; 12; 13; 14; 15]. Instead, a flexible body model incorporating its nonlinear characteristics is required [17]. [PERSON] et al. [18] developed a real-time interactive simulation system for wire harness assembly based on physical principles, using the mass-spring model to simulate the stretching, bending, and twisting behavior of wire harnesses. [PERSON] et al. [19] proposed a multi-body dynamics model based on the arbitrary Lagrangian-Eulerian method to simulate the dynamic behavior of aircraft-arresting cable systems. [PERSON] et al. [20] employed rigid-flexible coupling dynamic modelling and simulation methods, integrating the Lagrange equation, finite element method (FEM), and modal synthesis method to analyze the dynamic performance of an inspection robot on PTL. [PERSON] et al. [21] proposed a method to solve the problems of shape finding of a suspension bridge with spatial cables. [PERSON] et al.