## 3 Design of Passive Torsional Compliant Suspension
### Design of Compliant Spiral Torsional Suspension  
An exploded view of the C-STS mechanism developed in this study is shown in Figure 7. It primarily consists of the motor, the compliant spiral suspension module, the wheel, and a set of connecting components. As illustrated in Figure 8a, the compliant spiral suspension is composed of two main parts: an inner ring and an outer ring. The inner ring features a slot (Figure 8b) where the inner shaft holder is attached, which--together with a key--links the motor shaft to the suspension. Thus, the inner ring receives torque directly from the motor. The outer ring has six holes (Figure 8a), three of which are used to connect it to the outer hub, which in turn is fastened to the wheel hub via the outer shaft holder. In other words, the outer ring interfaces with the wheel, transmitting external loads to the suspension. To prevent deformations in any direction other than the torsional axis, two additional design features were implemented. First, to keep the inner ring from deforming radially, an inner-alignment ring was added to the rear of the inner ring (Figure 8c), and an inner-adapter hole (Figure 8f) was added to the rear of the outer hub. These two parts fit loosely together, allowing relative rotational motion between the compliant spiral suspension and the hub assembly. Second, to constrain deformation along the motor's rotational axis, an inner hub was attached to the suspension's face that couples with the motor, thereby blocking the inner ring from moving axially. As a result, the compliant spiral suspension is free to deform primarily in the torsional direction. In this study, we used PLA (polylactic acid) as the primary material, with an elastic modulus of \(E=3.5\times 10^{9}\)N/m\({}^{2}\). Our goal was to support a 6 kg curved-wheel robot, which requires an estimated 0.76 N\(\cdot\)m of static torque under worst-case stair-climbing conditions. To ensure safety, we set a factor of safety of five relative to the PLA's bending strength (\(89\times 10^{6}\) N/m\({}^{2}\)). This guided the initial geometry for our spiral spring, characterized by three key parameters: thickness t, length L, and width b. From practical trial-and-error, we identified a baseline that effectively moderated velocity fluctuations: t = 5 mm, L = 827.28 mm, and b = 15 mm. We used the classic flat spiral spring stiffness formula,  
\[k_{t}=\frac{Ebt^{2}}{12L} \tag{6}\]  
We calculated a theoretical stiffness of approximately 0.661 N\(\cdot\)m/rad. Two other variants were defined by modifying the thickness while keeping b = 15 mm and L = 1136.2 mm:  
Low-stiffness suspension: t = 4 mm, \(k_{t}\approx\) 0.246 N\(\cdot\)m/rad  
High-stiffness suspension: t = 6 mm, \(k_{t}\approx\) 1.157 N\(\cdot\)m/rad  
All three designs respected the safety margin criteria for a 6 kg load, with the intention of comparing different levels of torsional compliance. To confirm the actual stiffness, we measured the angular deflection by applying a 1-2 kg load at a radius of 0.11 m from the spiral spring's center. We then derived the experimental \(k_{t}\) from the resulting torque-angle relationship. For instance, the baseline  
Figure 8: Configurations of key components of the suspension system. (**a**) Front view of the compliant spiral suspension. (**b**) Front side view of the suspension. (**c**) Back side view of the suspension. (**d**) Inner hub. (**e**) Front side view of the outer hub (**f**) Back side view of the outer hub. Figure 7: Configuration of the compliant spiral suspension system. (**a**) Exploded top view. (**b**) Exploded side view. suspension (t = 5 mm, L = 827.28 mm) yielded an empirical stiffness of 0.547 N\(\cdot\)m/rad, notably less than the theoretical 0.661 N\(\cdot\)m/rad. Similarly, the low-stiffness and high-stiffness versions measured at 0.206 N\(\cdot\)m/rad and 0.909 N\(\cdot\)m/rad, respectively, again falling below their corresponding theoretical values. We attribute the lower-than-expected stiffness primarily to two factors.