## 3 Design of Passive Torsional Compliant Suspension
### Concept of Compliant Spiral Torsional Suspension (C-STS)  
The curved-spoke wheel undergoes sudden deceleration whenever its contact point transitions to a new step. As discussed earlier, this abrupt velocity change at the wheel's center of rotation (CoR) can compromise dynamic stability, particularly because the effective mass M concentrated at the CoR attempts to maintain its inertia. Figure 6 contrasts the conventional wheel (a1-a3)--which experiences abrupt deceleration--with the proposed wheel equipped with a torsional spring (b1-b3) that deforms to mitigate such instability. To reduce these nonlinear velocity fluctuations, we introduce torsional compliance between the motor shaft (\(\theta_{M}\)) and the wheel shaft (\(\theta_{W}\)). During normal rolling on the ground, the spiral spring (see Figure 6c) deflects as the wheel rotates relative to the motor, thereby storing energy. When the contact point shifts, the wheel would otherwise decelerate instantly, but here, it is free to rotate slightly further relative to the motor. This motion releases a portion of the stored spring energy (Figure 6d), helping to propel the wheel upward and maintain the CoR's inertia. For this study, we focus on the magnitude of the CoR's velocity, treating its trajectory as if it were nearly linear. In reality, the velocity's direction also changes abruptly at contact transitions. However, we assume these directional changes are relatively small or gradual  
Figure 6: Illustration of the proposed torsional suspension’s working principle and design space. (**a**) Conventional curved-spoke wheel (**a1–a3**): During obstacle climbing and contact transitions, the wheel’s center of rotation (CoR) experiences an abrupt deceleration, potentially causing instability. (**b**) Curved-spoke wheel with torsional spring (**b1–b3**): At the moment of contact transition, the torsional spring deforms to mitigate sudden deceleration and improve dynamic stability. (**c**) Simplified schematic of the spiral suspension mechanism: The difference in angular displacement between the motor (\(\theta_{M}\)) and the wheel (\(\theta_{W}\)) is absorbed by the spiral spring. (**d**) Energy storage and release through \(\Delta\theta\). In addition, the blue arrow at the CoR denotes the velocity’s magnitude and direction, while the green arrow in (**b,b1,b3**) indicates the torsional spring’s deflection direction. compared to the dominant effect of speed reduction. Since the CoR moves in a direction that increases the wheel's potential energy, the spiral spring's torque input is crucial for compensating this energy requirement. By design, the C-STS is deformed during rolling (increasing \(\Delta\theta\)) and then returns some of that energy when the wheel attempts to decelerate more than the motor shaft. In other words:  
1. Energy Storage: As the motor rotates further or faster than the wheel, \(\Delta\theta\) increases, and the spring deflects, storing energy. 2. Energy Release: When contact transitions occur, the wheel can briefly rotate relative to the motor shaft in the opposite direction, reducing \(\Delta\theta\) and releasing stored elastic energy. 3. Maintaining Inertia: By allowing the wheel to rotate somewhat independently of the motor shaft, the compliance helps the CoR preserve part of its original velocity, avoiding an extreme deceleration spike and thus improving dynamic stability. Hence, C-STS functions by absorbing sudden rotational differences between the motor and the wheel, and then releasing that energy at critical moments, softening the rapid velocity drop that would otherwise destabilize the system. A compliant spiral mechanism was selected as an alternative due to its distinct advantages. First, by forming the entire mechanism as a single compliant structure, it replaces multiple hinges, links, and bearings, thereby reducing component count and minimizing friction or backlash. In addition, the spiral geometry facilitates a gradual elastic deformation under external loads, providing enhanced energy absorption and dispersion.