# Defining Feasible Joint and Geometric Workspaces Through Boundary Functions
## 4 Boundary Functions for Workspace Definition
Since these constraints refer to entire geometric boundaries rather than discrete points, the conditions imposed by each function may be satisfied independently. This distinction highlights the difference between collision detection at specific points and the enforcement of global geometric restrictions that ensure safe operation. The tables presenting the first four point boundary functions are labeled as Tables 3 and 4. The final table, labeled as Table 5, provides the geometric boundary functions that define broader workspace constraints. Together, these boundary functions offer a structured method for computing the complete feasible workspace, ensuring that the manipulator operates within a collision-free region while supporting trajectory planning and operational reliability. The boundary functions obtained from the previous analysis are presented as follows, encapsulating the constraints derived from the kinematic and geometric relationships of the planar robot. These functions define the conditions that must be satisfied to ensure a collision-free workspace while maintaining the feasible motion range of the manipulator, and they are presented below:  
\[\beta_{1,1} =x_{1,7}\cos(q_{1})-x_{0,6}+z_{1,7}\sin(q_{1}), \tag{29}\] \[\beta_{1,3} =z_{0,1}-z_{0,6}+z_{1,7}\cos(q_{1})-x_{1,7}\sin(q_{1}),\] \[\beta_{2,1} =x_{1,9}\cos(q_{1})-x_{0,8}+z_{1,9}\sin(q_{1}),\] \[\beta_{2,3} =z_{0,1}-z_{0,8}+z_{1,9}\cos(q_{1})-x_{1,9}\sin(q_{1}),\] \[\beta_{3,1} =x_{1,2}-x_{1,10}+x_{2,11}\cos(q_{2})+z_{2,11}\sin(q_{2}),\] \[\beta_{3,3} =z_{1,2}-z_{1,10}+z_{2,11}\cos(q_{2})-x_{2,11}\sin(q_{2}),\] \[\beta_{4,1} =x_{1,2}-x_{1,12}+x_{2,13}\cos(q_{2})+z_{2,13}\sin(q_{2}),\] \[\beta_{4,3} =z_{1,2}-z_{1,12}+z_{2,13}\cos(q_{2})-x_{2,13}\sin(q_{2}),\] Figure 3:0 Figure 3:1 Figure 3:2  
To further characterize the motion constraints imposed by these boundary functions, the next step involves defining upper and lower bounds for each function. These bounds, denoted as \(\beta_{i,j,k}\), represent the \(k\)-th constraint imposed on the \(j\)-th component of the boundary function associated with the \(i\)-th collision condition. By establishing these bounds, the feasible motion range of the planar robot is precisely quantified, ensuring a structured approach to workspace determination. Figure 3:3 Figure 3:4 \[\mathbf{r}_{0,6}= \begin{bmatrix}x_{0,6}\\ 0\\ z_{0,6}\end{bmatrix} ,\quad\mathbf{r}_{1,7}= \begin{bmatrix}x_{1,7}\\ 0\\ z_{1,7}\end{bmatrix} .\]  
\[B_{1}(\mathbf{q})=\mathbf{r}_{0,1}+R_{0,1}\mathbf{r}_{1,7}-\mathbf{r}_{0,6},\] \[ \begin{bmatrix}\beta_{1,1}\\ \beta_{1,2}\\ \beta_{1,3}\end{bmatrix} = \begin{bmatrix}x_{1,7}\cos(q_{1})-x_{0,6}+z_{1,7}\sin( q_{1})\\ 0\\ z_{0,1}-z_{0,6}+z_{1,7}\cos(q_{1})-x_{1,7}\sin(q_{1})\end{bmatrix} .\]  
\[x_{0,8}=-7.5\,\text{cm},\quad z_{0,8}=12.5\,\text{cm}.\] \[x_{1,9}=-1.5\,\text{cm},\quad z_{1,9}=12.41\,\text{cm}.\] \[\mathbf{r}_{0,8}= \begin{bmatrix}x_{0,8}\\ 0\\ z_{0,8}\end{bmatrix} ,\quad\mathbf{r}_{1,9}= \begin{bmatrix}x_{1,9}\\ 0\\ z_{1,9}\end{bmatrix} .\]  
\[B_{2}(\mathbf{q})=\mathbf{r}_{0,1}+R_{0,1}\mathbf{r}_{1,9}-\mathbf{r}_{0,8},\] \[ \begin{bmatrix}\beta_{2,1}\\ \beta_{2,2}\\ \beta_{2,3}\end{bmatrix} = \begin{bmatrix}x_{1,9}\cos(q_{1})-x_{0,8}+z_{1,9}\sin( q_{1})\\ 0\\ z_{0,1}-z_{0,8}+z_{1,9}\cos(q_{1})-x_{1,9}\sin(q_{1})\end{bmatrix} .\]  
\[x_{1,10}=17.82\,\text{cm},\quad z_{1,10}=15.87\,\text{cm}.\] \[x_{2,11}=9.09\,\text{cm},\quad z_{2,11}=-5.25\,\text{cm}.\] \[\mathbf{r}_{1,10}= \begin{bmatrix}x_{1,10}\\ 0\\ z_{1,10}\end{bmatrix} ,\quad\mathbf{r}_{2,11}= \begin{bmatrix}x_{2,11}\\ 0\\ z_{2,11}\end{bmatrix} .\]  
\[B_{3}(\mathbf{q})=\mathbf{r}_{1,2}+R_{1,2}\mathbf{r}_{2,11}-\mathbf{r}_{1,10},\] \[ \begin{bmatrix}\beta_{3,1}\\ \beta_{3,2}\\ \beta_{3,3}\end{bmatrix} = \begin{bmatrix}x_{1,2}-x_{1,10}+x_{2,11}\cos(q_{2})+ z_{2,11}\sin(q_{2})\\ 0\\ z_{1,2}-z_{1,10}+z_{2,11}\cos(q_{2})-x_{2,11}\sin(q_{2})\end{bmatrix}