## 2 Kinematics Constraints and Robot Design Methodology
### Robot Design Methodology
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multicolumn{6}{c}{**Control Factors [mm]**} & \multicolumn{6}{c}{**Objective Function [mm]**} & **[dB]** \\ \hline Test & \(l_{1}\) & \(l_{2}\) & \(l_{4}\) & \(r_{1}\) & \(r_{2}\) & \(r_{3}\) & \(y_{1}\) & \(y_{2}\) & \(y_{3}\) & S/N \\  
\hline \end{tabular}
\end{table} 
Table 3: First iteration \(L_{27}(3^{6})\) array from Taguchi method. \begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{6}{c}{**Control Factors [mm]**} \\ \hline Level & \(l_{1}\) & \(l_{2}\) & \(l_{4}\) & \(r_{1}\) & \(r_{2}\) & \(r_{3}\) & \(y_{1}\) & \(y_{2}\) & \(y_{3}\) & S/N \\
1 & 75 & 65 & 120 & 38 & 40 & 50 \\
2 & & & & 42 & 42 \\
3 & & & & 45 & 45 & 45 \\ \hline \end{tabular}
\end{table} 
Table 5: Second iteration \(L_{9}(3^{2})\) array from the Taguchi method. The sensitive analysis from the second iteration is shown in Figure 7. To determine the optimal values for \(r_{1}\) and \(r_{2}\), we compare the S/N ratios from \(-77.6\) to \(-75.6\) (dB) in the first iteration, which reduces from \(-75.6\) to \(-74.6\) (dB) in the second iteration, these values denote better results than the previous ones. Then, we selected the optimal values in conjunction with the kinematic constraints. It is worth noting that the three levels in each parameter satisfy the kinematic constraints. Therefore, the optimal values are \(r_{1}=45\) and \(r_{2}=45\), even in the graph showing that \(r_{2}=40\) has a better S/N ratio; We selected 45 because when the robot is climbing up or down the high \((11\times 27)\) cm step, the chassis rubbed against the stair. Table 6 shows the comparison of the initial design and the optimized design, as well as the gain in (dB) being 0.84. Figure 8 shows the area between the path of the center of mass and the slope of each stair. The results show satisfactorily that the area decreased with the optimal design. Figure 8a,b shows the robot climbing up one step of (6 \(\times\) 17) cm, initial and optimal design, respectively; the area reduces from 1474 mm to 1227 mm. Figure 8c,d shows the robot climbing up one step of (7.5 \(\times\) 25) cm, initial and optimal design, respectively; the area reduces from 5603 mm to 4828 mm. Figure 8e,f shows the robot climbing up one step of (11 \(\times\) 27) cm, initial and optimal design, respectively; the area reduces from 8887 mm to 7624 mm. The next section details how the robot recognizes if it climbs up or down stairs. Additional emergency control prevents the robot from falling when the wheels lose contact with the ground. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Control Factors [mm]**} & \multicolumn{4}{c}{**Objective Function [mm]**} & \multicolumn{2}{c}{**[dB]**} \\ \hline Test & \(l_{1}\) & \(l_{2}\) & \(l_{4}\) & \(r_{1}\) & \(r_{2}\) & \(r_{3}\) & \(y_{1}\) & \(y_{2}\) & \(y_{3}\) & S/N \\ Initial & 70 & 70 & 125 & 45 & 45 & 45 & 1474 & 5603 & 8887 & \(-75.74\) \\ Optimal & 75 & 65 & 120 & 45 & 45 & 50 & 1296 & 5107 & 8074 & \(-74.91\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Initial and optimal design results of the wheeled robot after the second iteration. Figure 7: Second sensitive analysis between the control factors and S/N ratio.