## 2 Materials and Methods
### The SSM
#### 2.1.2 The Implementation of the SSM in the 3-RRR Planar Manipulator  
A general view of the 3-RRR planar manipulator and its dimensions and parameters are given in Figure 2.  
In Figure 2, \(\theta_{2}\), \(\theta_{4}\), and \(\theta_{6}\) are called active joint angles because they are coordinates defining the positions of driving links while \(\theta_{3}\), \(\theta_{5}\), and \(\theta_{7}\) are passive joint angles. Mathematical operations similar to what is done for the SP are also carried out for this manipulator. Since the SSM is a general method for this type of planar manipulator, for the completeness of the subject, it will be explained here in detail how to derive the kinematic constraint equations. From Figure 2, the coordinates of points C and F of the planar manipulator in Figure 2 are easily written as follows:  
\[ \begin{array}{l}\mathrm{X_{C}=r_{2}\cos\theta_{2}+r_{3}\cos\theta_{3}}\\ \mathrm{Y_{C}=r_{2}\sin\theta_{2}+r_{3}\sin\theta_{3}}\\ \mathrm{X_{F}=r_{4}\cos\theta_{4}+r_{5}\cos\theta_{5}+X_{D}}\\ \mathrm{Y_{F}=r_{4}\sin\theta_{4}+r_{5}\sin\theta_{5}}\\ \end{array}  \tag{19}\]  
The distance between C and F is:  
\[(\mathrm{X_{F}-X_{C}})^{2}+(\mathrm{Y_{F}-Y_{C}})^{2}=\overset{\rightarrow}{ \|\mathrm{CF}\|}^{2} \tag{20}\]  
Equation (20) can also be written as  
\[ \begin{array}{ll}\overset{\rightarrow}{\|\mathrm{CF}\|}^{2}-\mathrm{a}^{2 }=&\mathrm{A_{1}\cos\theta_{3}+B_{1}\sin\theta_{3}+C_{1}\cos\theta_{5}+D_{1} \sin\theta_{5}}\\ &+E_{1}\cos\theta_{3}\cos\theta_{5}+E_{1}\sin\theta_{3}\sin\theta_{5}+F_{1}=0 \end{array}  \tag{21}\]  
where the constant coefficients are:  
\[ \begin{array}{ll}\mathrm{A_{1}=}&(2r_{2}r_{3}\cos\theta_{2}-2r_{3}\mathrm{ X_{D}-2r_{3}r_{4}\cos\theta_{4}})\\ \mathrm{B_{1}=}&(2r_{2}r_{3}\sin\theta_{2}-2r_{3}r_{4}\sin\theta_{4})\\ \mathrm{C_{1}=}&(2r_{4}r_{5}\cos\theta_{4}+2r_{5}\mathrm{X_{D}-2r_{2}r_{5}\cos \theta_{2}})\\ \mathrm{D_{1}=}&(2r_{4}r_{5}\sin\theta_{4}-2r_{2}r_{5}\sin\theta_{2})\\ \mathrm{E_{1}=}&-2r_{3}r_{5}^{2}\\ \mathrm{F_{1}=}&r_{2}^{2}+r_{3}^{2}+r_{4}^{2}+r_{5}^{2}+X_{D}^{2}-\mathrm{a}^{ 2}-2r_{2}\mathrm{X_{D}\cos\theta_{2}}\\ &+2r_{4}\mathrm{X_{D}\cos\theta_{4}-2r_{2}r_{4}\cos(\theta_{4}-\theta_{2})} \end{array}  \tag{22}\]  
Similarly, the coordinates of J and C are:  
\[ \begin{array}{ll}\mathrm{X_{J}=r_{6}\cos\theta_{6}+r_{7}\cos\theta_{7}+X_{G }}\\ \mathrm{Y_{J}=r_{6}\sin\theta_{6}+r_{7}\sin\theta_{7}+Y_{G}}\\ \mathrm{X_{C}=r_{2}\cos\theta_{2}+r_{3}\cos\theta_{3}}\\ \mathrm{Y_{C}=r_{2}\sin\theta_{2}+r_{3}\sin\theta_{3}}\end{array}  \tag{23}\]  
The distance between J and C is:  
\[ \begin{array}{ll}(\mathrm{X_{C}-X_{J}})^{2}+(\mathrm{Y_{C}-Y_{J}})^{2}= \overset{\rightarrow}{\|\mathrm{JC}\|}^{2}\end{array}  \tag{24}\]  
Equation (24) can also be written as  
\[ \begin{array}{ll}\overset{\rightarrow}{\|\mathrm{JC}\|}^{2}-\mathrm{a}^{2 }=&\mathrm{A_{2}\cos\theta_{3}+B_{2}\sin\theta_{3}+C_{2}\cos\theta_{7}+D_{2} \sin\theta_{7}}\\ &+E_{2}\cos\theta_{3}\cos\theta_{7}+E_{2}\sin\theta_{3}\sin\theta_{7}+F_{2}=0 \end{array}  \tag{25}\]  
Figure 2: General views of the 3-RRR planar manipulator. \[ \begin{array}{l}\text{K}_{1}=-\text{A}_{2}-\text{C}_{2}+\text{E}_{2}+\text{F}_{ 2}\\ \text{K}_{2}=2\text{B}_{2}\\ \text{K}_{3}=\text{A}_{2}-\text{C}_{2}-\text{E}_{2}+\text{F}_{2}\\ \text{L}_{1}=2\text{D}_{2}\\ \text{L}_{2}=4\text{E}_{2}\\ \text{L}_{3}=2\text{D}_{2}\\ \text{M}_{1}=-\text{A}_{2}+\text{C}_{2}-\text{E}_{2}+\text{F}_{2}\\ \text{M}_{2}=2\text{B}_{2}\\ \text{M}_{3}=\text{A}_{2}+\text{C}_{2}+\text{E}_{2}+\text{F}_{2}\\ \\ U_{1}=-A_{3}-\text{C}_{3}+\text{E}_{3}+\text{F}_{3}\\ U_{2}=2\text{B}_{3}\\ U_{3}=A_{3}-\text{C}_{3}-\text{E}_{3}+\text{F}_{3}\\ V_{1}=2\text{D}_{3}\\ V_{2}=4\text{E}_{3}\\ V_{3}=2\text{D}_{3}\\ W_{1}=-A_{3}+\text{C}_{3}-\text{E}_{3}+\text{F}_{3}\\ W_{2}=2\text{B}_{3}\\ W_{3}=A_{3}+\text{C}_{3}+\text{E}_{3}+\text{F}_{3}\end{array}  \tag{36}\]  
As readily seen, the equation set of Equations (31) to (33) for the 3-RRR planar manipulator has the same form as in Equations (4)-(6).