## 6 Conclusions  
This study investigated the dynamic instability of a stair-climbing robot equipped with a curved-spoke wheel design and introduced a compliant spiral torsional suspension (C-STS) to address abrupt velocity drops--which lead to increased acceleration and inertial forces in the main body, thereby undermining stability--during the discontinuous contact state (DCS). Simulation analyses revealed that higher angular velocity \(\omega\) or taller stair height \(H\) tend to magnify the velocity loss (\(\Delta v\)) in the DCS phase, thereby reducing the overall stability. To alleviate this issue, we placed a spiral suspension between the motor shaft and the wheel shaft, with the aim of absorbing and releasing torsional energy during contact transitions so that the robot retains more of its body inertia, thereby reducing deceleration and ultimately enhancing the dynamic stability of both the chassis and its payload.  
Experimental findings confirm that C-STS significantly improves dynamic stability, but only above a certain angular velocity threshold. At medium- and high-speed conditions, suspensions with higher stiffness yielded meaningful reductions in deceleration (\(a\)). For instance, at high speeds, the high-k suspension reduced \(a\) by 23.7% relative to the basic setup, demonstrating notable performance gains. In addition, global metrics showed an 11.6% rise in mean velocity and a 17.5% drop in velocity standard deviation \(s\). In contrast, low-speed scenarios limited the kinetic energy and hindered effective energy release, sometimes causing lower wheel velocities or brief stalls--indicating that C-STS requires a minimum rotational speed (around 8.5 rad/s) to function properly.  
While the proposed spiral suspension thus enhances dynamic stability under mid- to high-speed conditions, it provides limited benefits at low speeds. Future work will aim to optimize the suspension's structure and stiffness to support a broader range of speeds and varying load conditions. Strategies may include multi-modal suspensions capable of deforming in multiple directions or actively tuning stiffness in real time, ensuring robust stair-climbing performances even when the robot faces changing loads or operates at lower speeds.  
Conceptualization: S.J. and Y.K.; Methodology: S.J.; Software: S.J.; Validation: S.J. and Y.K.; Formal Analysis: S.J.; Investigation: S.J.; Resources: Y.K.; Data Curation: S.J.; Writing--Original Draft Preparation: S.J.; Writing--Review & Editing: Y.K.; Visualization: S.J.; Supervision: Y.K.; Project Administration: Y.K.; Funding Acquisition: Y.K. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the National Research Foundation (NRF) of Korea grant funded by the Korea government (MSIT) (RS-2024-00359219).  
All data generated or analyzed during this study are available from the corresponding author upon reasonable request.  
The authors declare no conflict of interest.  
The following abbreviations are used in this manuscript:  
C-STS Compliant Spiral Torsional Suspension  
CCS Continuous Contact State  
DCS Discontinuous Contact State  
COR Center-of-Rotation