## 4 Simulation and Experiment
### Span of 200 m PTL Case  
The span of PTLs typically ranges from 100 to 200 m. To further investigate the dynamic performance of the robot on long-span PTLs, a PTL with a span of 200 m was specified. The elastic modulus was set to 79.8 GPa, the diameter to 18 mm, and the linear density to 0.91 kg/m. The bending stiffness and axial stiffness could be derived from these parameters. The initial configuration of the PTL was set without any stretching, maintaining its original length. The simulation was run for 2 s with a convergence criterion of \(1\times 10^{-3}\). A total of 20 nodes were set up for the simulation. The initial condition for the FPTLIR's descent was set by establishing the steady state of the PTL. The FPTLIR was  
Figure 11: Dynamic behavior error of the FPTLIR: (**a**) displacement error; (**b**) roll error. Figure 10: ADAMS simulation workflow. designed to descend from point 9 of the PTL, with the centerline of the FPTLIR's travelling wheels aligned with the centerline of the PTL in the yz plane. To analyze the dynamic behaviors of the system, Figure 12 shows the Z-displacement of the FPTLIR and the variations in attitude angles (yaw, pitch and roll) during and after the descent. As the displacements in the X and Y directions during the descent were negligible, they were not included in the data collection. During the initial phase, the Z-displacement stabilized at \(-1.45\) m, indicating the steady-state condition where the PTL was only affected by gravity, with no interaction between the FPTLIR and the PTL. Then, within the time frame of 0-0.9 s, the FPTLIR rapidly descended to reach a peak value of \(-3.3\) m due to gravitational forces. Thereafter, the Z-displacement oscillated around \(-2.0\) m. Following 9.8 s, upon the interaction between the PTL and the FPTLIR beginning, the Z-displacement finally stabilized at \(-1.8\) m.  
The attitude angles exhibited varying amplitudes and oscillation frequencies. The roll angle started at \(0^{\circ}\), increased with amplified oscillations to a peak at 4.47 s, and transitioned to periodic fluctuations as the Z-displacement stabilized. This behavior can be attributed to the transformation of the system into a universal pendulum model after stabilization of the Z-displacement. Similarly, single-arm robotic structures also show an initial surge in roll angles at the start of the walking phase, followed by damped oscillations [33]. The yaw angle gradually rose from \(0^{\circ}\), reaching \(-1.2^{\circ}\) at 25 s. This change occurred slowly and without oscillation, showcasing the robust coupling between the yaw angle and the PTL. The PTL's integration into the FPTLIR's wheels ensured steady changes over time. The pitch angle exhibited oscillations between \(-0.1^{\circ}\) and \(0.1^{\circ}\), with minimal amplitude variations. These differences in amplitude levels arose from varying constraints and the FPTLIR's principal moments of inertia. The principal moment of inertia in the Z direction was the largest for the robot, and the constraints were also the strongest, resulting in minimal yaw angle fluctuations. In the X and Y directions, the moments of inertia were identical. During zero-velocity landing, the robot was mainly affected by gravity, resulting in no excitation in the Y direction. Consequently, compared to roll, the pitch fluctuation was much smaller, although its fluctuation frequency was similar to that of roll. Figure 12: Time history of the attitude and Z-displacement of the FPTLIR. The roll angle initially stabilized during oscillations, with the oscillation period gradually increasing. Once stabilization occurred, variations in the FPTLIR's attitude angles had a negligible impact on the Z-displacement. The FPTLIR's descent at a point 9 demonstrates the relationship between the Z-displacement, pitch angle, yaw angle, and roll angle as they change over time during the descent. Nevertheless, it should be noted that in some cases, descending at that point may not be a necessary action for the FPTLIR when operating on the PTL.