# Defining Feasible Joint and Geometric Workspaces Through Boundary Functions
## 4 Boundary Functions for Workspace Definition
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Right-Bounded Function**} & \multicolumn{2}{c}{**Left-Bounded Function**} \\ \hline \(\beta_{1,1,1}=\tan^{-1}\left(\frac{z_{1,2}}{x_{1,2}}\right)+\cos^{-1}\!\left( \frac{x_{0,6}}{\sqrt{x_{1,2}^{2}+z_{1,2}^{2}}}\right)\) & \(\geq q_{1}\geq\) & \(\beta_{1,1,2}=\cos^{-1}\!\left(\frac{x_{0,6}}{\sqrt{x_{1,2}^{2}+z_{1,2}^{2}}} \right)+\tan^{-1}\!\left(\frac{z_{1,2}}{x_{1,2}}\right)-2\pi\) \\ \(\beta_{1,3,1}=\pi+\tan^{-1}\!\left(\frac{-x_{1,2}}{x_{1,2}}\right)-\cos^{-1}\! \left(\frac{z_{01}-z_{0,6}}{\sqrt{x_{1,2}^{2}+z_{1,2}^{2}}}\right)\) & \(\geq q_{1}\geq\) & \(\beta_{1,3,2}=-\pi+\tan^{-1}\!\left(\frac{x_{1,2}}{x_{1,2}}\right)-\cos^{-1} \!\left(\frac{z_{01}-z_{0,6}}{\sqrt{x_{1,2}^{2}+z_{1,2}^{2}}}\right)\) \\ \(\beta_{2,1,1}=\tan^{-1}\!\left(\frac{z_{1,9}}{x_{1,9}}\right)+\cos^{-1}\! \left(\frac{x_{0,8}}{\sqrt{x_{1,9}^{2}+z_{1,9}^{2}}}\right)\) & \(\geq q_{1}\geq\) & \(\beta_{2,1,2}=\cos^{-1}\!\left(\frac{x_{0,8}}{\sqrt{x_{1,9}^{2}+z_{1,9}^{2}}} \right)+\tan^{-1}\!\left(\frac{z_{1,9}}{x_{1,9}}\right)-2\pi\) \\ \(\beta_{2,3,1}=\pi+\tan^{-1}\!\left(\frac{-x_{1,9}}{z_{1,9}}\right)-\cos^{-1} \!\left(\frac{z_{01}-z_{0,8}}{\sqrt{x_{1,9}^{2}+z_{1,9}^{2}}}\right)\) & \(\geq q_{1}\geq\) & \(\beta_{2,3,2}=-\pi+\tan^{-1}\!\left(\frac{-x_{1,9}}{z_{1,9}}\right)-\cos^{-1} \!\left(\frac{z_{01}-z_{0,8}}{\sqrt{x_{1,9}^{2}+z_{1,9}^{2}}}\right)\) \\ \(\beta_{3,1,1}=\pi+\tan^{-1}\!\left(\frac{z_{2,11}}{x_{2,11}}\right)-\cos^{-1} \!\left(\frac{x_{1,2}-x_{1,10}}{\sqrt{x_{2,11}^{2}+z_{2,11}^{2}}}\right)\) & \(\geq q_{2}\geq\) & \(\beta_{3,1,2}=-\pi+\tan^{-1}\!\left(\frac{z_{2,11}}{x_{2,11}}\right)-\cos^{-1} \!\left(\frac{x_{1,2}-x_{1,10}}{\sqrt{x_{2,11}^{2}+z_{2,11}^{2}}}\right)\) \\ \(\beta_{3,3,1}=\pi+\tan^{-1}\!\left(\frac{-x_{2,11}}{z_{2,11}}\right)-\cos^{-1} \!\left(\frac{z_{1,2}-z_{1,10}}{\sqrt{x_{2,11}^{2}+z_{2,11}^{2}}}\right)\) & \(\geq q_{2}\geq\) & \(\beta_{3,3,2}=-\pi+\tan^{-1}\!\left(\frac{-x_{2,11}}{z_{2,11}}\right)-\cos^{-1} \!\left(\frac{z_{1,2}-z_{1,10}}{\sqrt{x_{2,11}^{2}+z_{2,11}^{2}}}\right)\) \\ \(\beta_{4,1,1}=\pi+\tan^{-1}\!\left(\frac{z_{2,12}}{z_{2,13}}\right)-\cos^{-1} \!\left(\frac{x_{1,2}-x_{1,12}}{\sqrt{x_{2,13}^{2}+z_{2,13}^{2}}}\right)\) & \(\geq q_{2}\geq\) & \(\beta_{4,1,2}=-\pi+\tan^{-1}\!\left(\frac{z_{2,13}}{x_{2,13}}\right)-\cos^{-1} \!\left(\frac{x_{1,2}-x_{1,12}}{\sqrt{x_{2,13}^{2}+z_{2,13}^{2}}}\right)\) \\ \(\beta_{4,3,1}=\pi+\tan^{-1}\!\left(\frac{-x_{2,13}}{z_{2,13}}\right)-\cos^{-1} \!\left(\frac{z_{1,2}-z_{1,12}}{\sqrt{x_{2,13}^{2}+z_{2,13}^{2}}}\right)\) & \(\geq q_{2}\geq\) & \(\beta_{4,3,2}=-\pi+\tan^{-1}\!\left(\frac{-x_{2,13}}{z_{2,13}}\right)-\cos^{-1} \!\left(\frac{z_{1,2}-z_{1,22}}{\sqrt{x_{2,13}^{2}+z_{2,13}^{2}}}\right)\) \\ \(\beta_{5,1,1}=\pi-q_{1}-\cos^{-1}\!\left(\frac{x_{1,2}\cos(q_{1})-x_{\min,1}+z _{1,2}\sin(q_{1})}{x_{2,3}}\right)\) & \(\geq q_{2}\geq\) & \(\beta_{5,1,2}=\) \\ \(\beta_{6,1,1}=\pi-q_{1}-\cos^{-1}\!\left(\frac{x_{1,2}\cos(q_{1})-x_{\min,2}+z _{1,2}\sin(q_{1})}{x_{2,3}}\right)\) & \(\geq q_{2}\geq\) & \(-q_{1}-\pi-\cos^{-1}\!\left(\frac{x_{1,2}\cos(q_{1})-x_{\min,2}+z_{1,2}\sin(q_{1 })}{x_{2,3}}\right)\) \\ \(\beta_{7,3,1}=\sin^{-1}\!\left(\frac{z_{0,1}-z_{\min,1}+z_{1,2}\cos(q_{1})-x_{1,2}\sin(q_{1})}{x_{2,3}}\right)-q_{1}\) & \(\geq q_{2}\geq\) & \(-q_{1}-\pi-\sin^{-1}\!\left(\frac{z_{01}-z_{\min,1}+z_{1,2}\cos(q_{1})-x_{1,2} \sin(q_{1})}{x_{2,3}}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Boundary function bounds. Figure 4: Graphical representation of the boundary functions. Red lines indicate functions that are not considered in defining the feasible region, while cyan lines represent the functions used to construct it. The selected feasible region is determined by the boundary functions \(\beta_{1,3,1}\), \(\beta_{2,1,2}\), \(\beta_{3,3,2}\), \(\beta_{4,1,2}\), \(\beta_{7,3,1}\), and \(\beta_{7,3,2}\).