## 2 Materials and Methods
### Workspace Analysis  
The workspace of the 2-RRR planar parallel manipulator is defined as the set of points that the end-effector can reach when the joint angles \(\theta_{1}\) and \(\theta_{4}\) vary between 0 and \(2\pi\). This analysis excluded any interference between the manipulator's links and ignores singularities. The workspace is bounded by surfaces that encompass all accessible points for the manipulator, whether it is fully extended or fully folded.  
It is important to note that not all points within the workspace have the same level of accessibility. Points on the boundary of the workspace, which can only be reached with specific joint configurations, represent areas of minimum accessibility.  
The workspace of the first kinematic chain \(\left(\overline{ABC}\right)\) was determined by two concentric circles:  
1. The outer circle, with a radius of \((L_{1}+L_{2})\), represents the maximum reach when the manipulator is fully extended: \[C_{1o}=x_{C}^{2}+y_{C}^{2}=\left(L_{1}+L_{2}\right)^{2}\] (38)
2. The inner circle, with a radius of \((L_{1}-L_{2})\), represents the minimum reach when the manipulator is fully folded: \[C_{1i}=x_{C}^{2}+y_{C}^{2}=\left(L_{1}-L_{2}\right)^{2}\] (39)  
These two circles define the workspace region of the first kinematic chain as the area between the circles \(C_{1o}\) and \(C_{1i}\).  
Similarly, the workspace of the second kinematic chain \(\left(\overline{EDC}\right)\) was defined by two concentric circles:  
3. The outer circle, centered at \(\left(x_{E},0\right)\), with a radius of \(\left(L_{4}+L_{3}\right)\): \[C_{2o}=\left(x_{C}-x_{E}\right)^{2}+y_{C}^{2}=\left(L_{4}+L_{3}\right)^{2}\] (40)
4. The inner circle, also centered at \(\left(x_{E},0\right)\), with a radius of \(\left(L_{4}-L_{3}\right)\): \[C_{2i}=\left(x_{C}-x_{E}\right)^{2}+y_{C}^{2}=\left(L_{4}-L_{3}\right)^{2}\] (41)  
The workspace region of the second kinematic chain is the area between the circles \(C_{2o}\) and \(C_{2i}\).  
When both kinematic chains are considered, the theoretical workspace of the 2-RRR manipulator is defined as the intersection of the workspace regions of the two chains. If the following condition holds, \(\frac{x_{E}}{2}>L_{1}+L_{2}\) and \(\frac{x_{E}}{2}>L_{4}+L_{3}\). No intersection exists, meaning the manipulator has no workspace. If \(\frac{x_{E}}{2}=L_{1}+L_{2}\) and \(\frac{x_{E}}{2}=L_{4}+L_{3}\), the workspace is reduced to a single point, located at the center of the line segment \(\overline{AE}\). If the condition \(L_{1}=L_{2}\), \(L_{4}=L_{1}\) and \(L_{3}=L_{2}\) is met, the theoretical workspace is a region that corresponds to the overlapping area of the two concentric circles, as illustrated in Figure 5.  
Figure 5: The shaded area represents the theoretical workspace of the 2-RRR planar parallel manipulator, determined by the intersection of the two workspace regions defined by Equations (38) and (40). Points A and E represent the fixed base joints of the manipulator. The curves \(C_{1o}\) and \(C_{1i}\) define the outer and inner boundaries, respectively, of the reachable region for the first kinematic chain (\(A\)–\(B\)–\(C\)). Similarly, curves \(C_{2o}\) and \(C_{2i}\) represent the outer and inner limits for the second kinematic chain (\(E\)–\(D\)–\(C\)). The intersection of these regions defines the total workspace where the end-effector (point \(C\)) can be located.  
In real-world applications, the actual workspace may be smaller than the theoretical workspace due to physical constraints, such as interference between the manipulator's links; singularities were induced by configurations where the manipulator loses degrees of freedom, and joint limits were caused by restrictions on the angles \(\theta_{1}\) and \(\theta_{4}\).