## 2 Materials and Methods
### Inverse Kinematic Model  
Inverse kinematics focuses on determining the angles of the robot's joints that correspond to a desired position and orientation of the end-effector. This is essential for controlling the robot's motion along a predefined trajectory. Specifically, the objective is to calculate the input angles \(\theta_{1}\) and \(\theta_{4}\) for the actuators when the coordinates of point \(C\) (\(x_{c},y_{c}\)) are known. This analysis serves as the foundation for precise trajectory control of the robot's tool. Below, in Figure 3, a geometric scheme for the study of inverse kinematics is shown. Figure 3: Schematic of the parallel plane manipulator for the study of inverse kinematics. Points \(A\) and \(E\) are the fixed base joints of the manipulator; point \(C\) (\(x_{c},y_{c}\)) is the end-effector or terminal point. Links \(L_{1}\), \(L_{2}\), \(L_{3}\), and \(L_{4}\) connect the joints \(A\)–\(B\), \(B\)–\(C\), \(D\)–\(C\), and \(D\)–\(E\), respectively. The dashed lines \(L_{AC}\) and \(L_{EC}\) represent the distances from point \(C\) to base points \(A\) and \(E\). The angles \(\theta_{1}\) and \(\theta_{4}\) correspond to the input joint angles at the actuators located at \(A\) and \(E\). Angles \(\beta_{1}\) and \(\phi_{1}\) are intermediate angles used for computing \(\theta_{1}\), and \(\beta_{4}\) and \(\phi_{4}\) are used similarly for \(\theta_{4}\). Figure 2: Assembly modes: (**a**) upper configuration and (**b**) lower configuration. From the triangle \(\overline{AEC}\) and using the law of cosines,  
\[L_{EC}^{2}=L_{AC}^{2}+x_{E}^{2}-2L_{EC}x_{E}\text{cos}\,\phi_{1} \tag{24}\]  
Solving for \(\phi_{1}\),  
\[\phi_{1}=\text{cos}\left(\frac{L_{AC}^{2}+x_{E}^{2}-L_{EC}^{2}}{2L_{EC}x_{E}}\right) \tag{25}\]  
From the vector relationships in triangle \(\overline{AEC}\),  
\[\overline{AC}=\overline{AE}+\overline{EC} \tag{26}\]  
Decomposing this equation,  
\[L_{AC}\text{cos}\,\theta_{1}=x_{E}+L_{EC}\text{cos}\,\phi_{4} \tag{27}\]  
\[L_{AC}\text{sin}\,\theta_{1}=y_{E}+L_{EC}\text{cos}\,\phi_{4} \tag{28}\]  
From Equation (27), solving for \(\phi_{4}\),  
\[\phi_{4}=\text{cos}\bigg{(}\frac{L_{AC}\text{cos}\,\theta_{1}-x_{E}}{L_{EC}} \bigg{)} \tag{29}\]  
where  
\[L_{AC}=\sqrt{x_{c}^{2}+y_{c}^{2}} \tag{30}\]  
\[L_{EC}=\sqrt{\left(x_{C}-x_{E}\right)^{2}+\left(y_{C}-y_{E}\right)^{2}} \tag{31}\]  
Applying the law of cosines to triangle \(\overline{ABC}\),  
Figure 2:0  
Solving for \(\beta_{1}\),  
Figure 2:1  
Similarly, applying the law of cosines to triangle \(\overline{CDE}\),  
Figure 2:2  
Solving for \(\beta_{4}\),  
Figure 2:3  
Using the results from Equations (33) and (35), the input angles \(\theta_{1}\) and \(\theta_{4}\) are calculated as follows:  
Figure 2:4  
Figure 2:5  
From Equations (33)-(37), there are four possible solutions for the inverse kinematics problem, corresponding to different assembly modes of the manipulator (Figure 4). These configurations are as follows:  
* \"+ -\" model: \(\theta_{1}=\beta_{1}+\phi_{1}\), \(\theta_{4}=\phi_{4}-\beta_{4}\)
* \"--\" model: \(\theta_{1}=\beta_{1}-\phi_{1}\), \(\theta_{4}=\phi_{4}-\beta_{4}\)
* \"++\" model: \(\theta_{1}=\beta_{1}+\phi_{1}\), \(\theta_{4}=\phi_{4}+\beta_{4}\)These configurations are illustrated in Figure 4, where each model represents a different working mode of the manipulator. In this manner, the inverse kinematic model provides a framework for calculating the input angles \(\theta_{1}\) and \(\theta_{4}\) based on the desired position \((x_{c},y_{c})\) of the end-effector. By leveraging geometric constraints and trigonometric relationships, this study ensures precise control of the manipulator's motion. This inverse kinematic analysis is critical for determining the joint movements required for the end-effector to follow a desired trajectory.