## 6 Conclusions  
This study has established a coupled dynamic model of the FPTLIR and PTL. By employing the ANCF, the nonlinear large deformation and large rotation behaviors of the PTL were accurately modeled. The Hunt-Crossley contact model was adopted to simulate the dynamic interaction between the FPTLIR and the flexible PTL during landing, effectively resolving the limitations of traditional approaches that oversimplify contact force dimensions and neglect groove angle effects. Experimental and simulation results validated the accuracy of the proposed dynamic model, establishing a robust foundation for future dynamic analyses and providing theoretical insights to ensure safe FPTLIR landing in complex environments. The main conclusions are given as follows:  
1. A coupled FPTLIR/PTL dynamic model was derived using the ANCF method. Shear and torsion were ignored using a Euler-Bernoulli beam to improve the efficiency. The coupled effect between the FPTLIR and PTL was formulated using the Hunt-Crossley contact model, considering the deflection angle and the groove of the of the travelling wheel to ensure the accuracy. 2. A modular simulation of the model was performed with different landing positions and FPTLIR masses. Compared with the FEM, the ANCF demonstrated an accuracy improvement exceeding 50%. The results show that all dynamic responses remained within acceptable ranges. As the landing position approached the tower, the amplitude of the \(Z\)-displacement wave decreased, with the peak \(Z\)-displacement at the closest point being 34.4% of that at the farthest point while the roll angle amplitude increased, with the maximum exceeding the minimum by 3.7%. The appropriate landing point should be selected by weighing the collision risk against the insulation risk. Additionally, increasing the FPTLIR mass amplified the \(Z\)-displacement and attitude angle waves, with the lightest robot achieving a \(Z\)-displacement peak 9.2% \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Hybrid Robot I [11]** & **LineDrone [12]** & **Hybrid Robot II [13]** & **FPTLIR** \\ \hline Hybrid layout & Bottom & Bottom & Bottom & Top \\ Landing & Yes & Yes & Yes & Yes \\ Wheel numbe & 3 & 4 & 2 & 3 \\ PTL model & Rigid & / & / & Flexible \\ Contact & Static & / & / & Dynamic \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison between hybrid robots. and a roll angle peak 12.8% of those of the heaviest. Therefore, both the collision risk and the insulation risk exhibited a positive correlation with the robot's mass. 3. An integrated landing test platform was constructed to observe the landing process of the FPTLIR and to validate the accuracy of the model. The time domain variations of the attitude, Z-displacements, and contact forces were measured from the test platform, with the contact forces recorded using six opposing FSRs. The results show that the relative errors for the roll angle and Z-displacement were found to be 0.004 and 0.044, respectively, validating the accuracy of the proposed method. The platform enabled accurate analysis and prediction of FPTLIR system responses during the landing process. This study provides a methodological framework applicable not only to HIRs, but also to other robotic systems in rigid-flexible coupling environments, such as space-deployable robots, surgical robots, soft continuum robots, and cable-driven inspection robots. Conceptualization, [PERSON], P.J. and X.Q. ; methodology, [PERSON], J.L., P.J. and X.Q. ; software, [PERSON], J.L. and P.J. ; validation, [PERSON], J.L., J.T. and P.J. ; formal analysis, [PERSON] and P.J. ; investigation, [PERSON], P.J., X.Q. and S.Z. ; resources, [PERSON], S.Z. and M.Z. ; data curation, [PERSON] and [PERSON]; writing--original draft preparation, [PERSON], [PERSON] and [PERSON]; writing--review and editing, [PERSON] and [PERSON]. All authors have read and agreed to the published version of the manuscript. This research was supported by the National Natural Science Foundation of China (grant nos.