## 2 Mechanical Modeling of the ISU System
### Kinematic Analysis
#### 2.2.2 Piston Position Derivation  
The diagram in Figure 5 is a vectorized, simplified version of Figure 3, represented schematically for computational convenience. This diagram simplifies the geometric relationships between each component using vectors.  
The following expression can be obtained from Figure 5.  
\[r_{c}=r_{1}+r_{4}=r_{2}+r_{3} \tag{6}\]  
Figure 4: Analysis of the wheel position and motion.  
Figure 5: Kinematic relationship of the crank-connecting rod–piston system using vectors.  
Here, the vector representations are given as follows:  
\[\mathbf{r}_{1}=r_{1}\hat{\mathbf{i}}\]  
\[\mathbf{r}_{2}=L_{2}\big{(}\cos\ \theta_{2}\hat{\mathbf{i}}+\sin\ \theta_{2}\hat{\mathbf{j}} \big{)}\]  
\[\mathbf{r}_{3}=L_{3}\big{(}\cos\ \theta_{3}\hat{\mathbf{i}}+\sin\ \theta_{3}\hat{\mathbf{j}} \big{)}\]  
\[\mathbf{r}_{4}=h\hat{\mathbf{j}}\]  
where \(\hat{\mathbf{i}}\) and \(\hat{\mathbf{j}}\) are the respective unit vectors along the horizontal and vertical directions, while \(\theta_{3}\) denotes the deviation angle between the connecting rod and the main cylinder, and \(r_{1}\) corresponds to the piston position.  
The following equation was obtained by reorganizing these expressions.  
\[r_{1}\hat{\mathbf{i}}+h\hat{\mathbf{j}}=L_{2}\big{(}\cos\theta_{2}\hat{\mathbf{i}}+\sin \theta_{2}\hat{\mathbf{j}}\big{)}+L_{3}\big{(}\cos\theta_{3}\hat{\mathbf{i}}+\sin \theta_{3}\hat{\mathbf{j}}\big{)} \tag{7}\]  
By separately organizing the equations into their \(\hat{\mathbf{i}}\) and \(\hat{\mathbf{j}}\) components, the following relationships can be established.  
\(\hat{\mathbf{i}}\)-component:  
\[r_{1}=L_{2}\text{cos}\ \theta_{2}+L_{3}\text{cos}\ \theta_{3} \tag{8}\]  
\(\hat{\mathbf{j}}\)-component:  
\[h=L_{2}\text{sin}\ \theta_{2}+L_{3}\text{sin}\ \theta_{3} \tag{9}\]  
Since \(\theta_{3}\) is present in both equations, it can be eliminated to derive an equation for \(r_{1}\). The following expression was obtained from the \(\hat{\mathbf{j}}\)-component equation in the vertical direction:  
\[L_{3}\text{sin}\ \theta_{3}=h-L_{2}\text{sin}\ \theta_{2} \tag{10}\]  
The equation can be simplified down to the following expression by dividing both sides by \(L_{3}\):  
\[\text{sin}\ \theta_{3}=\frac{h-L_{2}\text{sin}\ \theta_{2}}{L_{3}} \tag{11}\]  
The following relationship can be derived by applying the trigonometric identity \(\text{sin}^{2}\ \theta+\text{cos}^{2}\ \theta=1\):  
\[\text{cos}\ \theta_{3}=\sqrt{1-\text{sin}^{2}\ \theta_{3}} \tag{12}\]  
The following result was obtained by substituting Equation (11) into Equation (12):  
\[\text{cos}\ \theta_{3}=\sqrt{1-\bigg{(}\frac{h-L_{2}\text{sin}\ \theta_{2}}{L_{3}}\bigg{)}^{2}} \tag{13}\]  
Substituting Equation (13) into Equation (8) results in a quadratic equation for the horizontal position \(r_{1}\) of the main piston. The following expression gives the solution to this equation:  
\[r_{1}=L_{2}\text{cos}\ \theta_{2}+\bigg{[}\Big{(}L_{2}\text{cos}\ \theta_{2} \big{)}^{2}-\Big{(}L_{2}^{2}+h^{2}-L_{3}^{2}-2L_{2}h\ \text{sin}\ \theta_{2}\Big{)}\ \bigg{]}^{1/2} \tag{14}\]  
The main piston displacement can be determined by substituting the values obtained from Equations (1) and (5) into Equation (14).  
The following result was derived from Equation (8):  
\[\text{cos}\ \theta_{3}=\frac{r_{1}-L_{2}\text{cos}\ \theta_{2}}{L_{3}} \tag{15}\]The following equation was obtained by combining Equations (11) and (15) and utilizing the trigonometric identity \(\tan\theta=\frac{\sin\ \theta}{\cos\ \theta}\):  
\[\tan\ \theta_{3}=\frac{h-L_{2}\text{sin}\ \theta_{2}}{r_{1}-L_{2}\text{cos}\ \theta_{2}} \tag{16}\]  
Finally, the value of \(\theta_{3}\) was determined by using the arctangent function.  
\[\theta_{3}=\tan^{-1}\left(\frac{h-L_{2}\text{sin}\ \theta_{2}}{r_{1}-L_{2}\text{ cos}\ \theta_{2}}\right) \tag{17}\]  
Meanwhile, the displacement of the hydraulic cylinder piston was determined using \(r_{1}\) and its static position \(r_{1\_\text{static}}\) based on Figure 3.  
\[\Delta\text{Y}=r_{1}-r_{1\_\text{static}} \tag{18}\]