## 2 Kinematic Analysis of Curved-Spoke Wheel
### MATLAB Simulation on Curved-Spoke Wheel  
Using MATLAB R2023a we conducted a kinematic analysis and simulation of the curved-spoke wheel climbing a representative stair. The stair dimensions were chosen as 300 mm \(\times\) 160 mm, which are commonly found in everyday environments. Consequently, under a constant angular velocity, the CoR's linear velocity \(v\) for the curved-spoke wheel follows the trend shown in Figure 3. The wheel's rotational speed and the CoR's linear velocity exhibit a nonlinear, discontinuous relationship. Over the course of stair climbing, changes in the CoR's linear velocity can be broadly divided into two phases. The first phase, continuous contact state (CCS, (2)\(\rightarrow\)(3)), occurs as \(r\) increases continuously from \(r_{min}\) to \(r_{max}\) while the wheel rolls forward. The second phase, discontinuous contact state (DCS, (3)\(\rightarrow\)(4)), takes place momentarily when the contact point shifts to the next step, causing \(r\) to drop abruptly from \(r_{max}\) to \(r_{min}\). Figure 2: Configuration and rolling motion of curved-spoke wheel while climbing the stair from step \(i\) to step \(i+1\). The red arrow labeled \(v\) indicates the CoR’s general velocity vector, while \(v_{max,i}\) denotes the peak velocity just before contact transition and \(v_{min,(i+1)}\) the minimum velocity after transition. During the continuous contact state (CCS), as \(r\) increases continuously, the CoR's linear velocity \(v\) likewise rises in a smooth manner. In contrast, during the discontinuous contact state (DCS), \(r\) abruptly drops from \(r_{max}\) to \(r_{min}\), causing the wheel's linear velocity to plunge from its pre-transition peak \(v_{max,i}=\omega r_{max}\) to \(v_{min,(i+1)}=\omega r_{min}\). The drop in velocity at this point is given by \(\Delta v_{i}=v_{max,i}-v_{min,(i+1)}\). Consequently, when the curved-spoke wheel climbs the stairs, a discontinuity in the curvature radius arises precisely at the instant the contact point transitions, and the CoR's linear velocity decreases discontinuously. In reality, however, due to inertia, the CoR cannot instantaneously slowdown, which can lead to payload instability or slip at the floor-wheel interface. To examine how the magnitude of this velocity discontinuity depends on the wheel's speed and stair dimensions, we compared \(\Delta v_{i}\)--the discontinuity occurring during the transition from one stair step to the next--under various rotation speeds and stair sizes. Table 1 summarizes the conditions tested: Condition 1 (C1) increases the wheel's angular velocity while keeping the stair dimensions fixed, whereas Condition 2 (C2) modifies the stair height while holding the wheel's angular velocity constant. The resulting trends in the CoR's linear velocity for each condition are shown in Figure 4 and Table 2.  
From C1, we observe that when the wheel's angular velocity increases at a fixed stair size, the velocity decrease \(\Delta v\) during the DCS phase also rises. Notably, as \(\omega\) increases, not only does \(\Delta v\) grow but also both the minimum and maximum velocities increase, as shown in Figure 4a. The C2 results, presented in Figure 4b, indicate that at a fixed angular velocity, \(\Delta v\) grows larger when the stair height is increased. These findings demonstrate that the nonlinear relationship between the wheel's rotation speed and the CoR's linear  
Figure 3: Simulation snapshots illustrating the stair-climbing dynamics. Each ascending phase ((1), (3), (5), and (7)) features a maximum rotational radius, while the subsequent frames ((2), (4), (6), and (8)) demonstrate the abrupt decrease in radius due to discontinuous radius of curvature of the spoke. The continuous contact state (CCS) refers to the wheel rolling smoothly from \(r_{min}\) to \(r_{max}\), whereas the discontinuous contact state (DCS) occurs momentarily at the contact transition, leading to a sudden velocity drop. velocity becomes more pronounced--and thus more destabilizing--when angular velocity or obstacle size increases.