## 5 Analysis and Discussion
### DCS-Related Dynamic Stability  
(1) Reduced Deceleration at Medium/High Speeds  
The experimental results in Figure 15b,c and Tables 7 and 8 show that, under medium- and high-speed conditions, a higher suspension stiffness consistently led to a noticeable reduction in deceleration during the DCS phase--for example, middle-k and high-k achieved average decreases of 9.17% and 22.12% at mid-speed and 23.74% and 21.8% at high speed, respectively, compared to the basic configuration. Notably, middle-k sometimes matched or surpassed high-k at higher speeds, suggesting that an overly stiff spring may release its energy at a slightly off-peak timing, whereas a moderate stiffness better aligns with the wheel's inertia and the contact transition moment. Consequently, the spiral spring can significantly counteract sudden velocity drops during the DCS interval, provided the spring stiffness, speed, and other conditions in a curved-spoke wheel are well matched. We hypothesize that this arises because the spring's deformation and release timing coincide more effectively with the required torque and inertial forces, though further studies would be needed to confirm the exact mechanism.  
(2) Limited Benefits of Low-k  
Under the same mid/high-speed conditions, the low-k suspension produced deceleration values comparable to basic (ranging from \(-\)1.94% to +0.43%). Our preliminary evaluation of \(\Delta t\) and \(\Delta v\) suggests that the velocity gap may either grow while the transition extends, or vice versa, effectively canceling out in terms of overall deceleration. This phenomenon likely arises because low stiffness cannot provide sufficient torque offset at the critical instant when the CoR velocity needs support, leading to only minimal net improvement in dynamic stability.  
(3) Negative Effects at Low Speeds  
Table 6 shows that low-k, middle-k, and high-k each produced deceleration values 30-37% higher than basic at low speeds, indicating that stiffer springs paradoxically worsened dynamic stability in this regime. A key factor appears to be premature spring \"unwinding\" when the wheel rotates slowly, briefly accelerating the CoR before the actual contact transition. During the rolling process (Figure 18a,b)), the required torque typically rises and then falls, and at low speeds, this fluctuation can trigger undesirable energy release just prior to contact transition--especially for stiffer springs that store more energy under the same angular displacement (Figure 14 shows that lower-speed runs with higher stiffness often reach a larger maximum velocity). As a result, the CoR velocity spikes at the wrong moment, and little or no energy remains to cushion the actual DCS. We occasionally observed short stops at the next stair edge (S2-S4, Figure 14), suggesting the wheel momentarily loses forward motion when it lacks sufficient \(\Delta\theta\) to deliver the needed torque for climbing.  
Figure 18: Comparison of required torque and spring deflection during the curved-spoke wheel’s climbing process. Here, \(l_{c1}\) and \(l_{c2}\) denote the distances from the contact point to the wheel’s CoR, while \(N_{1}\) and \(N_{2}\) represent the normal forces acting on the wheel at each contact point.  
By contrast, faster rotation shortens the window in which the spring can misfire its energy, making compliance more beneficial at higher speeds (as evidenced by Figure 15c, where low stiffness was not detrimental in mid/high-speed scenarios). Moreover, at higher speeds, inertia helps the wheel bypass such stalls, and the spring's compliance more effectively reduces deceleration (_a_).