# Defining Feasible Joint and Geometric Workspaces Through Boundary Functions
## 4 Boundary Functions for Workspace Definition
Although multiple feasible regions exist, reaching them from the home pose of the planar robot would require crossing or making contact with boundary functions, which would impose constraints on the motion. For this reason, the selection of the feasible region shown in the figure is justified. As observed in Figure 4, the presence of multiple boundary functions complicates the identification of a unique solution. However, a feasible region can be constructed by identifying the intersection points of the selected boundary functions, forming a continuous boundary, as illustrated in Figure 5. This boundary is defined by six intersection points. Nevertheless, the nonlinearities of certain functions prevent their respective constraints from being fully satisfied. This issue can be addressed through two possible approaches. The first approach involves incorporating additional tangent points along the sections of the functions where the constraints are not met. The second approach, as depicted in Figure 6, consists of computing the centroid of the feasible region and scaling it with respect to this point. Figure 5: Construction of the feasible region by identifying the intersection points of the selected boundary functions. The continuous boundary is formed by six intersection points, ensuring a well-defined feasible area for the robot’s motion. Figure 6: Scaling of the feasible region with respect to its centroid. This approach ensures that the boundary functions do not directly imply collision while introducing a safety margin for the robot’s operation. The scaling factor acts as a safeguard to maintain a collision-free workspace. The latter approach also resolves an additional issue: while the boundary functions define the safe region, they inherently imply potential collisions. By applying a scaling factor, a safety margin is introduced between the robot and the boundary functions. This scaling coefficient acts as a safety factor, ensuring a collision-free operational space. Figure 6 not only presents the closed and continuous boundary of the joint limits but also proposes a sequence for traversing these limits. However, Figure 7 illustrates the trajectory of these six intersection points in the geometric space, along with a simplified diagram of the planar robot in its final pose after completing the trajectory. It is important to note that, although Figure 6 defines an ordered sequence for the boundary traversal, this does not inherently guarantee a structured or sequential trajectory in the geometric space. Figure 8 presents two plots. The first illustrates a nonclosed sequence of five points, while the second depicts the simplified diagram of the planar robot in its final pose after completing the trajectory. In this case, the trajectory is structured but remains open. This phenomenon can be explained as follows: the workspace with the greatest reach is directly linked to the robot's links when they become collinear. However, this configuration  
Figure 8: Nonclosed trajectory consisting of five points. The first plot represents the joint space sequence, while the second illustrates the planar robot in its final pose after completing the trajectory. Despite following an ordered sequence, the trajectory remains open. Figure 7: Trajectory of the six intersection points in the geometric space, including the simplified diagram of the planar robot in its final pose. results in a joint singularity. This issue was indirectly resolved by scaling the feasible joint region with respect to its centroid, ensuring a structured path. As a result, two distinct approaches exist for traversing the feasible geometric space along the right or the left side. This choice determines the final pose of the robot as well as its reach. Due to the limited workspace, the final link cannot transition to its mirrored pose within the same trajectory. This second alternative is illustrated in Figure 9, which contains two plots.