## 1 Introduction  
Service robots play an increasingly vital role in various applications, from domestic assistance to industrial logistics. For these robots to perform a wide range of tasks, they must be capable of negotiating everyday obstacles, among which stairs pose one of the most challenging barriers. Researchers have long sought ways to enable reliable stair climbing, leading to diverse approaches that balance stability, complexity, and versatility. A track-based system [1, 2, 3, 4, 5, 6] is a commonly adopted approach to negotiate rough or soft terrain. Its wide contact area allows the robot to move over loose or uneven surfaces with relative ease, distributing weight to reduce sinking and maintain traction. However, tracked platforms are often bulky and heavy, making them cumbersome to maneuver in confined indoor spaces. Furthermore, their reliance on skid steering can damage floor surfaces and is generally unsuitable for precise navigation, especially in narrow corridors or tight corners. In contrast, legged robots [7, 8, 9, 10, 11, 12, 13] can navigate a wide range of terrains by mimicking biological locomotion. Their multi-jointed limbs enable them to step over obstacles and maintain balance on uneven ground. Yet, this versatility comes at the cost of increased mechanical complexity and demanding control algorithms. As a result, the number of actuators and sensors required for smooth-legged motion can drive up both cost and system fragility. Another line of research, the wheel-linkage mechanism [14; 15; 16; 17; 18; 19], attaches linkages to circular wheels to effectively overcome certain obstacles by distributing loads. While promising, its performance on stairs can be heavily influenced by the shape of the stair's vertical face. Moreover, it depends strongly on friction, often causing slip during ascent or descent. Curved-spoke wheels [20; 21; 22; 23; 24; 25] have been proposed as a simpler alternative, where a specially contoured wheel can ascend stairs using only rotational motion. In previous studies [21], mechanical stoppers were introduced to address slip issues at the instant the contact point shifts from one step to the next. While these measures showed partial success, they did not fully resolve the underlying dynamic effects, such as abrupt velocity fluctuations due to the changing radius of curvature. Although curved wheels can generate a smooth center-of-rotation (COR) path, the linear velocity of the robot body tends to fluctuate. This fluctuation arises from the abrupt changes in the radius of the curvature, especially at the exact moment the contact point shifts from one part of the wheel to another. In practice, these discontinuities can lead to instability of any payload carried by the robot, making reliable stair climbing difficult. The situation worsens at higher speeds, where the dynamic impact of these abrupt changes is amplified, further compromising the robot's stability on the stairs. In addition to purely mechanical approaches, some researchers have attempted to reduce fluctuations in the wheel's COR linear velocity by calculating the required wheel rotation speed in reverse and then applying active velocity control [25]. Although this method can stabilize stair climbing to some degree, its effectiveness has been demonstrated only at extremely low speeds--around 0.01 m/s--due to control-cycle constraints and limited motor output. Such a slow speed is impractical for most real-world scenarios that demand more agile stair-climbing performance. Consequently, an alternative strategy that does not rely solely on low-speed control is needed to ensure dynamic stability and efficient climbing. Motivated by the need to improve dynamic stability during stair climbing, this work presents a Passive Compliant Spiral Torsional Suspension (C-STS) for curved-spoke wheels (Figure 1). The suspension stores energy during ground rolling and releases it at the critical moment of contact transition, assisting the wheel's ascent and mitigating abrupt deceleration.