## 3 Generic Model of the 3 RRR PPM
### Calculation of the Global CoM Location  
To calculate the global CoM location, Equation (21) can be used. It is necessary to calculate or assemble the global mass matrix first. Thus, it is important to note that point \(B_{1}\) is shared by bodies 1 and 2, point \(B_{2}\) is shared by bodies 3 and 4, point \(B_{3}\) is shared by bodies 5 and 6, point \(C_{1}\) is shared by bodies 2 and 7, point \(C_{2}\) is shared by bodies 4 and 7, and point \(C_{3}\) is shared by bodies 6 and 7. In this case, following the same order specified in the positions vector, Equation (29), the \((18\times 18)\) mass matrix is assembled as follows:  
\[\mathbf{M}= \begin{bmatrix}\mathbf{M}_{11}&\mathbf{M}_{21}&0&0&0&0&0&0&0\\ \mathbf{M}_{31}&\mathbf{M}_{41}+\mathbf{M}_{12}&\mathbf{M}_{22}&0&0&0&0&0&0\\ 0&\mathbf{M}_{32}&\mathbf{M}_{42}+\mathbf{M}_{17}&0&0&\mathbf{M}_{27}&0&0& \mathbf{M}_{37}\\ 0&0&0&\mathbf{M}_{13}&\mathbf{M}_{23}&0&0&0&0\\ 0&0&0&\mathbf{M}_{33}&\mathbf{M}_{43}+\mathbf{M}_{41}&\mathbf{M}_{24}&0&0&0\\ 0&0&\mathbf{M}_{27}&0&\mathbf{M}_{34}&\mathbf{M}_{44}+\mathbf{M}_{47}&0&0& \mathbf{M}_{57}\\ 0&0&0&0&0&0&\mathbf{M}_{15}&\mathbf{M}_{25}&0\\ 0&0&0&0&0&0&\mathbf{M}_{25}&\mathbf{M}_{45}+\mathbf{M}_{16}&\mathbf{M}_{26}\\ 0&0&\mathbf{M}_{37}&0&0&\mathbf{M}_{37}&0&\mathbf{M}_{36}&\mathbf{M}_{67}\\ \end{bmatrix}  \tag{35}\]  
In this case, matrix \(\mathbf{B}\) has nine \((2\times 2)\) identity matrices:  
\[\mathbf{B}=\left[\mathbf{I}_{2}\quad\mathbf{I}_{2}\quad\mathbf{I}_{2}\quad \mathbf{I}_{2}\quad\mathbf{I}_{2}\quad\mathbf{I}_{2}\quad\mathbf{I}_{2}\quad \mathbf{I}_{2}\quad\mathbf{I}_{2}\right] \tag{36}\]  
Multiplying the terms in Equation (21) and simplifying, the final expression to calculate the global CoM position is as follows:  
\[m\mathbf{g}=\mathbf{C}_{A1}\mathbf{q}_{A1}+\mathbf{C}_{B1}\mathbf{q}_{B1}+ \mathbf{C}_{C1}\mathbf{q}_{C1}+\mathbf{C}_{A2}\mathbf{q}_{A2}+\mathbf{C}_{B2} \mathbf{q}_{B2}+\mathbf{C}_{C2}\mathbf{q}_{C2}+\mathbf{C}_{A3}\mathbf{q}_{A3} +\mathbf{C}_{B3}\mathbf{q}_{B3}+\mathbf{C}_{C3}\mathbf{q}_{C3} \tag{37}\]where  
\[\mathbf{C}_{A1} = \begin{bmatrix}\left(1-\frac{x_{1}}{l_{1}}\right)m_{1}&\frac{y_{1}} {l_{1}}m_{1}\\ -\frac{y_{1}}{l_{1}}m_{1}&\left(1-\frac{x_{1}}{l_{1}}\right)m_{1}\end{bmatrix}  \tag{38}\] \[\mathbf{C}_{B1} = \begin{bmatrix}\frac{x_{1}}{l_{1}}m_{1}+\left(1-\frac{x_{2}}{l_{ 2}}\right)m_{2}&\left(\frac{y_{2}}{l_{2}}m_{2}-\frac{y_{1}}{l_{1}}m_{1}\right) \\ -\left(\frac{y_{2}}{l_{2}}m_{2}-\frac{y_{1}}{l_{1}}m_{1}\right)&\frac{x_{1}}{l _{1}}m_{1}+\left(1-\frac{x_{2}}{l_{2}}\right)m_{2}\end{bmatrix} \] (39) \[\mathbf{C}_{C1} = \begin{bmatrix}m_{7}+\left(\frac{k_{7}y_{7}}{k_{7}y_{7}}-\frac{x_ {7}}{l_{7}}-\frac{y_{7}}{l_{7}}\right)m_{7}+\frac{x_{2}}{l_{2}}m_{2}&-\frac{y_ {2}}{l_{2}}m_{2}\\ \frac{y_{2}}{l_{2}}m_{2}&m_{7}+\left(\frac{k_{7}y_{7}}{k_{7}l_{7}}-\frac{y_{7 }}{l_{7}}-\frac{y_{7}}{l_{7}}\right)m_{7}+\frac{x_{2}}{l_{2}}m_{2}\end{bmatrix} \] (40) \[\mathbf{C}_{A2} = \begin{bmatrix}\left(1-\frac{x_{3}}{l_{3}}\right)m_{3}&\frac{y_ {3}}{l_{3}}m_{3}\\ -\frac{y_{3}}{l_{3}}m_{3}&\left(1-\frac{x_{3}}{l_{3}}\right)m_{3}\end{bmatrix} \] (41) \[\mathbf{C}_{B2} = \begin{bmatrix}\left(1-\frac{x_{4}}{l_{4}}\right)m_{4}+\frac{x_ {3}}{l_{3}}m_{3}&\left(\frac{y_{4}}{l_{4}}m_{4}-\frac{y_{3}}{l_{3}}m_{3}\right) \\ -\left(\frac{y_{4}}{l_{4}}m_{4}-\frac{y_{3}}{l_{3}}m_{3}\right)&\left(1-\frac{ x_{4}}{l_{4}}\right)m_{4}+\frac{x_{3}}{l_{3}}m_{3}\end{bmatrix} \] (42) \[\mathbf{C}_{C2} = \begin{bmatrix}\left(\frac{x_{7}}{l_{7}}-\frac{y_{7}k_{7}}{l_{7 }l_{7}l_{7}}\right)m_{7}+\frac{x_{4}}{l_{4}}m_{4}&-\frac{y_{4}}{l_{4}}m_{4}\\ \frac{y_{4}}{l_{4}}m_{4}&\left(\frac{x_{7}}{l_{7}}-\frac{y_{7}k_{7}}{l_{7}l_{7 }}\right)m_{7}+\frac{x_{4}}{l_{4}}m_{4}\end{bmatrix} \] (43) \[\mathbf{C}_{A3} = \begin{bmatrix}\left(1-\frac{x_{5}}{l_{5}}\right)m_{5}&\frac{y_ {5}}{l_{5}}m_{5}\\ -\frac{y_{5}}{l_{5}}m_{5}&\left(1-\frac{x_{5}}{l_{5}}\right)m_{5}\end{bmatrix} \] (44) \[\mathbf{C}_{B3} =