## 3 Numeric Examples
### Numerical Examples for the SSM  
#### 3.1.1 Numerical Application of the SSM to the Stewart Platform  
The physical parameters of the Stewart platform manipulator are seen in Figure 1, and the coefficients in Equations (4)-(6) are given in Tables 1 and 2, respectively. From Equations (8) and (9), one finds:  
\[\mu=\frac{54\times 10^{5}\lambda+\sqrt{-9.7\times 10^{12}\lambda^{4}+1.17 \times 10^{13}\lambda^{2}-2.79\times 10^{12}}}{5.09\times 10^{6}\lambda^{2}+8.9 3\times 10^{5}}\]  
\[\
u=\frac{5.1\times 10^{6}\lambda+\sqrt{-1.73\times 10^{13}\lambda^{4}+1.26 \times 10^{13}\lambda^{2}-1.88\times 10^{12}}}{7.4\times 10^{6}\lambda^{2}+1.35 \times 10^{6}}\] \begin{table}
\begin{tabular}{c c c} \hline a = 600 mm & l\({}_{3}=792.13\) mm & h\({}_{1}=895.68\) mm \\ b = 200 mm & l\({}_{4}=836.39\) mm & h\({}_{2}=754.92\) mm \\ l\({}_{1}=1030.29\) mm & l\({}_{5}=713.19\) mm & h\({}_{3}=712.2\) mm \\ l\({}_{2}=900.27\) mm & l\({}_{6}=908.13\) mm & h \(=519.6\) mm \\ \hline \end{tabular}
\end{table} 
Table 1: The physical parameters of the SP. \begin{table}
\begin{tabular}{c c c} \hline G\({}_{1}=2,543,309.32\) & H\({}_{1}=2,211,400.81\) & I\({}_{1}=3,699,427.32\) \\ G\({}_{2}=446,541.81\) & H\({}_{2}=1,040,240.96\) & I\({}_{2}=1,167,240.32\) \\ G\({}_{3}=-5,409,318.49\) & H\({}_{3}=-4,301,537.69\) & I\({}_{3}=-5,103,577.62\) \\ G\({}_{4}=953,461.98\) & H\({}_{4}=194,392.77\) & I\({}_{4}=675,534.86\) \\ G\({}_{5}=1,561,353.72\) & H\({}_{5}=1,174,001.77\) & I\({}_{5}=695,136.68\) \\ \hline \end{tabular}
\end{table} 
Table 2: Coefficients of the Equations (4)–(6) for the SP. Figure 4: Variations of q and v versus \(\lambda\) for the planar manipulator. Substituting these expressions in Equation (5) and carefully rearranging yields the following polynomial:  
\[ \begin{array}{ll}\text{f}(\lambda)=&7.65\times 10^{9}\lambda^{32}-9.95\times 10^{9 }\lambda^{30}+1.06\times 10^{9}\lambda^{28}+2.77\times 10^{9}\lambda^{26}-5.63 \times 10^{8}\lambda^{24}\\ &-3.46\times 10^{8}\lambda^{22}+7.1\times 10^{7}\lambda^{20}+2.19\times 10^{7} \lambda^{18}-5.99\times 10^{6}\lambda^{16}-9.64\times 10^{5}\lambda^{14}\\ &+3.99\times 10^{5}\lambda^{12}+8.05\times 10^{4}\lambda^{10}-2240\lambda^{8}-10 20\lambda^{6}+67.4\lambda^{4}+20.7\lambda^{2}+1\end{array} \]  
This polynomial is the explicit form of Equation (16). The roots of this  
Using the expressions, the following 32 nd degree polynomial is found:  
\[ \begin{array}{ll}\text{f}(\lambda)=&306.8\lambda^{32}&-2985.2\lambda^{31}+13625.2\lambda^{30}-38493.7\lambda^{29}+73618.6\lambda^{28}-93686.7\lambda^{27}+5828 7.6\lambda^{26}\\ &+55911.05\lambda^{25}-216742.8\lambda^{24}+332487.6\lambda^{23}-304685.9 \lambda^{22}+107439.7\lambda^{21}\\ &+174373.6\lambda^{20}-390571.6\lambda^{19}+425937.7\lambda^{18}-276127.5 \lambda^{17}+40976.04\lambda^{16}\\ &+152053.9\lambda^{15}-228415.8\lambda^{14}+194143.3\lambda^{13}-106190.6 \lambda^{12}+23181.7\lambda^{11}\\ &+25090.4\lambda^{10}-38706.1\lambda^{9}+32255.2\lambda^{8}-20103.04\lambda^{ 7}+10088.5\lambda^{6}-4164.1\lambda^{5}\\ &+1412.02\lambda^{4}-386.5\lambda^{3}+82.1\lambda^{2}-12.35\lambda+1\end{array} \]  
The roots of the expression above are given in Table 7.  
As seen from Table 7, there are ten real number solutions of \(\lambda\). Complex ones are rejected. The polynomial has multiple roots. In short, there are four different real roots of the polynomial: \(\lambda_{1}=-0.97\), \(\lambda_{2}=0.97\), \(\lambda_{3}=0.498\), and \(\lambda_{4}=0.67\). For each \(\lambda\), two roots are obtained for each of \(\mu\) and \(\
u\), as seen in Table 8.  
Table 8 shows that eight roots (rows 1-4 and 13-16) are complex numbers. They are rejected. Eight of the roots are real numbers; eight possible solutions exist. However, it should be checked if the rigidity conditions satisfy each solution set of \(\lambda\), \(\mu\), \(\
u\). After this control, it is seen that only the \(\lambda\), \(\mu\), \(\
u\) triplets given in Table 9 correspond to feasible solutions.