# Automatic Generation of N-Bar Planar Linkages Containing Sliders
## 6 Results  
A general algorithm is presented to obtain an atlas of N-bar KCs containing sliders. The proposed algorithm has been applied successfully to generate an atlas of 6-bar and 8-bar chains with up to 3 sliders. Rules of adding prismatic joints introduced by [7] are considered. Therefore, some results are not coincident with [1]. The developed technique produces 37 and 1350 non-isomorphic KCs for 6-bar and 8-bar chains, respectively. The presented algorithm is also used to generate KCs with up to 3-sliders of a 10-bar chain for the first time. This produces 13, 73, and 222 KCs with 1, 2, and 3-sliders, respectively. Detailed results for 6-bar and 10-bar chains are presented in Table 1. Brief results of the proposed algorithm and other literature are shown in Table 2. Atlases of Stephenson chain containing 1, 2, and 3 sliders are shown in Figure 11. Atlases of Watt chain with 1, 2, and 3-sliders are shown in Figure 12. Some non-isomorphic 10-bar KCs with 1, 2, and 3-sliders are shown in Figure 13. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Stephenson**} & \multicolumn{2}{c}{**Watt**} & \multicolumn{3}{c}{**A 10-Bar Chain**} \\ \hline
**No. of Prismatic Pairs** & **1P** & **2P** & **3P** & **1P** & **2P** & **3P** & **1P** & **2P** & **3P** \\ \hline Enumeration Results & 7 & 21 & 35 & 7 & 21 & 35 & 13 & 78 & 286 \\ Rejected KCs & - & 4 & 19 & - & 4 & 20 & - & 5 & 64 \\ Available KCs & 3 & 8 & 8 & 4 & 10 & 9 & 13 & 73 & 222 \\ Non-isomorphic KCs & 4 & 9 & 8 & 3 & 7 & 6 & 13 & 73 & 222 \\ Total KCs & & 21 & & & 16 & & & 308 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of the generation of 6-bar and 10-bar KCs with 1,2 and 3 sliders. Figure 10: (a) [PERSON] chain with 2 sliders (b) [PERSON] chain with 3 sliders. Figure 11: (**a**). Atlas of Stephenson chain with 1-slider. **(b). ** Atlas of Stephenson chain with 2-sliders. **(c). Atlas of Stephenson chain with 3-sliders. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & \multicolumn{3}{c}{**Present Results**} & \multicolumn{3}{c}{**Other Results**} \\  & & **1P** & **2P** & **3P** & **1P** & **2P** & **3P** \\ \hline \multirow{3}{*}{6-bar chains} & [PERSON] & 4 & 9 & 8 & 3 [1], common [8] & 8 [1], common [8] & 12 [1], common [8] \\  & [PERSON] & 3 & 7 & 6 & 3 [1], common [8] & 8 [1], common [8] & 12 [1], common [8] \\ \hline \multirow{6}{*}{8-bar chains} & Chain-1 & 4 & 15 & 25 & 4 [1], common [8] & 16 [1], common [8] & 36 [1], common [8] \\  & Chain-2 & 6 & 22 & 43 & 6 [1], common [8] & 25 [1], common [8] & 64 [1], common [8] \\  & Chain-3 & 3 & 13 & 23 & 3 [1], common [8] & 10 [1], common [8] & 22 [1], common [8] \\  & Chain-4 & 6 & 23 & 43 & 10 [1], common [8] & 45 [1], common [8] & 120 [1], common [8] \\  & Chain-5 & 10 & 41 & 81 & 6 [1], common [8] & 25 [1], common [8] & 64 [1], common [8] \\  & Chain-6 & 6 & 22 & 43 & 2 [1], common [8] & 9 [1], common [8] & 18 [1], common [8] \\  & Chain-7 & 6 & 23 & 45 & 6 [1], common [8] & 25 [1], common [8] & 64 [1], common [8] \\  & Chain-8 & 7 & 24 & 48 & 5 [1], common [8] & 18 [1], common [8] & 42 [1], common [8] \\  & Chain-9 & 10 & 41 & 83 & 2 [1], common [8] & 8 [1], common [8] & 13 [1], common [8] \\  & Chain-10 & 5 & 22 & 39 & 5 [1], common [8] & 25 [1], common [8] & 60 [1], common [8] \\  & Chain-11 & 10 & 40 & 78 & 10 [1], common [8] & 45 [1], common [8] & 120 [1], common [8] \\  & Chain-12 & 10 & 40 & 78 & 8 [1], common [8] & 31 [1], common [8] & 76 [1], common [8] \\  & Chain-13 & 5 & 22 & 43 & 3 [1], common [8] & 12 [1], common [8] & 24 [1], common [8] \\  & Chain-14 & 10 & 40 & 79 & 8 [1], common [8] & 31 [1], common [8] & 76 [1], common [8] \\  & Chain-15 & 3 & 14 & 23 & 3 [1], common [8] & 8 [1], common [8] & 18 [1], common [8] \\  & Chain-16 & 7 & 24 & 42 & 3 [1], common [8] & 12 [1], common [8] & 23 [1], common [8] \\ \hline
10-bar chain & Chain-1 & 13 & 73 & 222 & New result & New result & New result \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Brief results of the proposed algorithm and other literature.