# Defining Feasible Joint and Geometric Workspaces Through Boundary Functions
## 4 Boundary Functions for Workspace Definition
.\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Geometric Offsets and Boundary Functions** \\ \hline \hline \(x_{0,6}=7.5\,\text{cm},\quad z_{0,6}=12.5\,\text{cm}.\) \\ \(x_{1,7}=11.5\,\text{cm},\quad z_{1,7}=4.91\,\text{cm}.\) \\ \(\mathbf{r}_{0,6}= \begin{bmatrix}x_{0,6}\\ 0\\ z_{0,6}\end{bmatrix} ,\quad\mathbf{r}_{1,7}= \begin{bmatrix}x_{1,7}\\ 0\\ z_{1,7}\end{bmatrix} .\) \\ \(B_{1}(\mathbf{q})=\mathbf{r}_{0,1}+R_{0,1}\mathbf{r}_{1,7}-\mathbf{r}_{0,6},\) \\ \( \begin{bmatrix}\beta_{1,1}\\ \beta_{1,2}\\ \beta_{1,3}\end{bmatrix} = \begin{bmatrix}x_{1,7}\cos(q_{1})-x_{0,6}+z_{1,7}\sin( q_{1})\\ 0\\ z_{0,1}-z_{0,6}+z_{1,7}\cos(q_{1})-x_{1,7}\sin(q_{1})\end{bmatrix} .\) \\ \hline \(x_{0,8}=-7.5\,\text{cm},\quad z_{0,8}=12.5\,\text{cm}.\) \\ \(x_{1,9}=-1.5\,\text{cm},\quad z_{1,9}=12.41\,\text{cm}.\) \\ \(\mathbf{r}_{0,8}= \begin{bmatrix}x_{0,8}\\ 0\\ z_{0,8}\end{bmatrix} ,\quad\mathbf{r}_{1,9}= \begin{bmatrix}x_{1,9}\\ 0\\ z_{1,9}\end{bmatrix} .\) \\ \(B_{2}(\mathbf{q})=\mathbf{r}_{0,1}+R_{0,1}\mathbf{r}_{1,9}-\mathbf{r}_{0,8},\) \\ \( \begin{bmatrix}\beta_{2,1}\\ \beta_{2,2}\\ \beta_{2,3}\end{bmatrix} = \begin{bmatrix}x_{1,9}\cos(q_{1})-x_{0,8}+z_{1,9}\sin( q_{1})\\ 0\\ z_{0,1}-z_{0,8}+z_{1,9}\cos(q_{1})-x_{1,9}\sin(q_{1})\end{bmatrix} .\) \\ \hline \(x_{1,10}=17.82\,\text{cm},\quad z_{1,10}=15.87\,\text{cm}.\) \\ \(x_{2,11}=9.09\,\text{cm},\quad z_{2,11}=-5.25\,\text{cm}.\) \\ \(\mathbf{r}_{1,10}= \begin{bmatrix}x_{1,10}\\ 0\\ z_{1,10}\end{bmatrix} ,\quad\mathbf{r}_{2,11}= \begin{bmatrix}x_{2,11}\\ 0\\ z_{2,11}\end{bmatrix} .\) \\ \(B_{3}(\mathbf{q})=\mathbf{r}_{1,2}+R_{1,2}\mathbf{r}_{2,11}-\mathbf{r}_{1,10},\) \\ \( \begin{bmatrix}\beta_{3,1}\\ \beta_{3,2}\\ \beta_{3,3}\end{bmatrix} = \begin{bmatrix}x_{1,2}-x_{1,10}+x_{2,11}\cos(q_{2})+z_{2,11} \sin(q_{2})\\ 0\\ z_{1,2}-z_{1,10}+z_{2,11}\cos(q_{2})-x_{2,11}\sin(q_{2})\end{bmatrix} .\) \\ \hline \(x_{1,10}=17.82\,\text{cm},\quad z_{1,10}=15.87\,\text{cm}.\) \\ \(x_{2,11}=9.09\,\text{cm},\quad z_{2,11}=-5.25\,\text{cm}.\) \\ \(\mathbf{r}_{1,10}= \begin{bmatrix}x_{1,10}\\ 0\\ z_{1,10}\end{bmatrix} ,\quad\mathbf{r}_{2,11}= \begin{bmatrix}x_{2,11}\\ 0\\ z_{2,11}\end{bmatrix} .\) \\ \(B_{3}(\mathbf{q})=\mathbf{r}_{1,2}+R_{1,2}\mathbf{r}_{2,11}-\mathbf{r}_{1,10},\) \\ \( \begin{bmatrix}\beta_{3,1}\\ \beta_{3,2}\\ \beta_{3,3}\end{bmatrix} = \begin{bmatrix}x_{1,2}-x_{1,10}+x_{2,11}\cos(q_{2})+z_{2,11 }\sin(q_{2})\\ 0\\ z_{1,2}-z_{1,10}+z_{2,11}\cos(q_{2})-x_{2,11}\sin(q_{2})\end{bmatrix} .\) \\ \hline \(x_{1,10}=17.82\,\text{cm},\quad z_{1,10}=15.87\,\text{cm}.\) \\ \(x_{2,11}=9.09\,\text{cm},\quad z_{2,11}=\[x_{1,12}=4.83\,\text{mm},\quad z_{1,12}=23.37\,\text{mm}.\] \[x_{2,13}=9.00\,\text{mm},\quad z_{2,13}=5.25\,\text{mm}.\] \[\mathbf{r}_{1,12}= \begin{bmatrix}x_{1,12}\\ 0\\ z_{1,12}\end{bmatrix} ,\quad\mathbf{r}_{2,13}= \begin{bmatrix}x_{2,13}\\ 0\\ z_{2,13}\end{bmatrix} .\] \[B_{4}(\mathbf{q})=\mathbf{r}_{1,2}+R_{1,2}\mathbf{r}_{2,13}- \mathbf{r}_{1,12},\] \[ \begin{bmatrix}\beta_{4,1}\\ \beta_{4,2}\\ \beta_{4,3}\end{bmatrix} = \begin{bmatrix}x_{1,2}-x_{1,12}+x_{2,13}\cos(q_{2})+z _{2,13}\sin(q_{2})\\ 0\\ z_{1,2}-z_{1,12}+z_{2,13}\cos(q_{2})-x_{2,13}\sin(q_{2})\end{bmatrix} .\]  
Several of the boundary functions obtained in the analysis are expressed as linear combinations of cosine and sine terms. Therefore, it is necessary to establish a general method for determining the range of values that the variable \(q\) can assume while satisfying these constraints. The following theorem provides an explicit formulation for these bounds, enabling a systematic computation of the feasible joint configurations. Theorem 2.2: _Given the equation of the form_  
\[a\cos(q)+b\sin(q)=c, \tag{30}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Graphic** & **Geometric Offsets and Boundary Functions** \\ \hline \hline \(x_{1,12}=4.83\,\text{mm},\quad z_{1,12}=23.37\,\text{mm}.\) \\ \(x_{2,13}=9.00\,\text{mm},\quad z_{2,13}=5.25\,\text{mm}.\) \\ \(\mathbf{r}_{1,12}= \begin{bmatrix}x_{1,12}\\ 0\\ z_{1,12}\end{bmatrix} ,\quad\mathbf{r}_{2,13}=