## 2 Mechanical Modeling of the ISU System
### Kinematic Analysis  
#### 2.2.1 Crank-Housing Angle Calculation  
The key structural components of a typical ISU system, along with the coordinate system setup for mechanical design and dynamic analysis, are illustrated in Figure 2. Unlike the OSU system, the ISU housing itself functions as an arm. This implies that the housing undergoes rotational motion in response to the movement of the wheel. A local coordinate system was introduced to accurately describe the kinematic motion characteristics of the ISU, particularly to decouple the mechanical motion from the rotational movement of the arm. Figure 1: The assembled configuration of a typical ISU system. Figure 2: Structural components and global-local coordinate systems of a typical ISU system used in kinematic modeling. In Figure 3, the global coordinate system is represented using the X-Y coordinate system, where the origin is set at the rotation center of the ISU and the x-axis is parallel to the horizontal direction. The global coordinate system remains fixed and is independent of the movement of the robotic arm. The X-Y coordinate system is defined as the local coordinate system, which has the same origin as the global coordinate system. However, the x-axis is parallel to the installation direction of the ISU, and the local coordinate system rotates along with the movement of the robotic arm. By introducing the local coordinate system, the kinematic relationship between the crank and the main piston can be described independently of the rotation of the arm, as shown in Figure 3. The connecting rod aligned with the central axis of the main cylinder in both the static and the full jouce (F.J.) positions, as illustrated in the figure. This configuration was designed to ensure that the side force is reduced to zero both in the initial mounted state of the ISU system and when the nitrogen gas pressure reaches its maximum value at FJ. Therefore, the crank length \(L_{2}\) and the cylinder installation distance \(h\) can be calculated using the moment equilibrium equation about the rotational center of the arm in the static position. \[h=L_{2}\text{cos}\ \frac{1}{2}\theta_{FJ} \tag{1}\]  
The wheel position can be expressed as the distance \(x\) from the wheel to the horizontal reference line, as shown in Figure 4. The parameters \(x\) and \(w\) are defined as reference values indicating the position of the wheel. Specifically, \(x\) denotes the wheel position measured relative to the horizontal reference, while \(w\) represents the wheel position measured relative to the static position. The horizontal reference position was set as \(x_{\text{static}}=-0.1630\) m in the static state, whereas the static reference position was defined as \(w_{static}=0\). The relationship between \(x\) and \(w\) can be described by the following equation. \[x=w-x_{\text{static}} \tag{2}\]  
The angle \(\theta_{2}\) denotes the angle between the crank and the housing. It corresponds to the angle between the \(x\)-axis of the local coordinate system and \(L_{1}\) and the angle between \(L_{1}\) and \(L_{2}\) in Figures 3 and 4, respectively. Figure 3: Relationship between crank and main piston motions within the local coordinate system. The value of \(\theta_{2}\) varied with the wheel movement. Two new variables, \(\beta_{1}\) and \(\beta_{2}\), were introduced to facilitate the calculation of \(\theta_{2}\).