## 2 Kinematics Constraints and Robot Design Methodology
### Robot Design Methodology
\(n\) is the number of experiments conducted at level \(i\), in our case, three sizes of stairs: \((6\times 17)\) cm, \((7.5\times 25)\) cm, and \((11\times 27)\) cm. \(y_{i}\) is the area between the slope of the stair and the center of mass trajectory. To find the optimal values, we expect to choose the most positive values of the S/N ratio in each parameter and check the kinematic constraints to ensure that the robot can satisfy the requirements to climb up and down stairs. The center of mass considered in the paper is the CM of the delivery package, including the load. Figure 5 shows the trajectory of the bogie, wheels, and center of the mass from the robot design while climbing up a step of \((11\times 27)\) cm, obtained by the kinematic relation between wheels and links. Furthermore, \(CM_{sim}\) denotes the trajectory of the center of mass obtained in the Gazebo simulator using ROS. Therefore, we can validate that the center of mass can be assumed to be fixed above Link 1 as long as the motor maintains the delivery package in a horizontal position during the robot's motion. From the orthogonal array \(L_{27}(3^{6})\) and the three levels of each parameter, we obtain \(y_{i}\). Therefore, we calculate S/N Equation (4). Table 3 shows the result of the Taguchi method. The result of the initial design is \(-75.74\) [dB]. This result is acceptable because prior to the construction of the robot and the stairs, we analyzed the kinematic constraints to know the maximum height the robot could climb. However, in real tests, the wheels lost contact with the ground, and inevitably the robot fell. Then, we decided to implement an emergency  
Figure 4: Robot design methodology based on the Taguchi method. controller to avoid falls, as detailed in the next section, and optimize the robot design in the current section. \[S/N=-10\log_{10}\left\{\sum_{i=1}^{n}\frac{y_{i}^{2}}{n}\right\} \tag{4}\]  
The sensitivity analysis of the Taguchi method is shown in Figure 6. The six control factors are evaluated based on the objective function. Higher values in the S/N ratio mean optimal values of the design parameters. However, suppose the slope between the levels in each parameter is high. In that case, it means having large sensitivity. Then, it is recommended to prepare a second iteration with new levels in the parameters with high sensitivity in conjunction with the optimal values in the parameters with low sensitivity effects. The parameters \(l_{1}\), \(l_{2}\), \(l_{4}\), and \(r_{3}\) have low sensitivity, and \(r_{1}\) and \(r_{2}\) have high sensitivity. From the first iteration results, we selected \(l_{1}=75\), \(l_{2}=65\), \(l_{4}=120\), and \(r_{3}=50\). For \(l_{4}\), even though 125 shows a higher value in (dB), we selected 120 based on kinematic constraints to reduce the length of the robot. For \(r_{3}\), the three values in (dB) are practically the same. Therefore, we decided to choose 50 because, during the descent of the 11 cm-high step, the rear wheel takes time to touch the ground, and the robot loses stability. Figure 5: The trajectory of the wheels and the CM of the delivery package while the robot climbs one step of \(11\times 27\) cm. Figure 6: Sensitive analysis between control factors and S/N ratio. The second iteration focuses on \(r_{1}\) and \(r_{2}\). The optimal values obtained from the first iteration, and the three new levels for \(r_{1}\) and \(r_{2}\) are shown in Table 4. The new orthogonal array is given by \(L_{9}(3^{2})\). We calculate \(y_{i}\) and the objective function S/N ratio, as show in Table 5. \begin{table}
\begin{tabular}{c c c c c c} \hline \multicolumn{6}{c}{**Control Factors [mm]**} \\ \hline Level & \(l_{1}\) & \(l_{2}\) & \(l_{4}\) & \(r_{1}\) & \(r_{2}\) & \(r_{3}\) \\
1 & 75 & 65 & 120 & 38 & 40 & 50 \\
2 & & & & 42 & 42 \\
3 & & & & 45 & 45 & \\ \hline \end{tabular}
\end{table} 
Table 4: Optimal values from the first iteration and three new levels for the second set of the experiment.