## 2 Kinematics Constraints and Robot Design Methodology
### Robot Design Methodology  
Designing a robot based on experience and trial and error does not ensure that the chosen parameters and the combination of these offer the best results for the robot's performance. The robot design implies multiple variables, objectives, restrictions, and evaluation criteria. The robot design methodology proposed in this paper is described in Figure 4. The flowchart shows how the designer obtained the optimal parameters for the wheeled robots. We identify the parameters of control, which are the length of the links and the wheel radius. These parameters are changeable and evaluated to improve the robustness of the robot. We selected three levels of each parameter based on the kinematic constraints. The noise factor is a variable that cannot be controlled. For example, we do not know the height and the minimum step that the robot can find on its way. Then, we chose three different stair sizes as noise factors to ensure the robot could climb up and down different sizes of steps. We selected the orthogonal array according to the number of control factors and the levels. Then, we calculated the signal-noise ratio according to the objective function and obtained the sensitivity analysis. If the results show higher slopes between the levels of each parameter, it means we can select new levels and evaluate them again. We can select the best values based on the objective function if the slopes are minimums. Finally, if the new design parameters meet the kinematic constraints, they can be considered optimal design parameters. The Korea Electronics Technology Institute (KETI) designed the six-wheeled mobile robot [24,25], and we adapted it to the robot shown in Figure 1. The optimization is carried out under the Taguchi method [26, 27, 28]. We find the optimal combination parameters and improve its performance based on minimizing the trajectory of the center of mass and the slope of 3 different stairs. The results obtained show the robustness of the robot through the analysis of the signal-to-noise (S/N) ratio. For the simplicity of the robot design, nine parameters are chosen, as shown in Table 1. For the methodology design, the six most relevant parameters shown in Table 2, these parameters represent the links and joints of the robot. Although the dynamic parameters affect the robot's behavior, we can safely create a proper robot control to climb up and down stairs with the kinematic analysis. The dimensions of the chassis and the delivery box remain fixed as \(l_{3}\), \(l_{5}\), and \(l_{6}\), since these values keep the original design. Therefore, the control factors are \(r_{1}\), \(r_{2}\), \(r_{3}\), \(l_{1}\), \(l_{2}\), and \(l_{4}\). In addition, three levels are determined for each parameter to optimize the design. These values were based on the kinematic analysis. Thus, it reduces the time of experimentation and analysis to find the optimal values. The levels are shown in Table 2. Since there are six parameters and three levels, we selected the orthogonal array \(L_{27}(3^{6})\). The evaluation of the three levels is given by the signal-to-noise ratio (S/N), where \"signal\" represents the desired value and \"noise\" represents the undesirable value. Lower signal noise means the design is robust. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{5}{c}{**Control Factors [mm]**} \\ \hline Level & \(l_{1}\) & \(l_{2}\) & \(l_{4}\) & \(r_{1}\) & \(r_{2}\) & \(r_{3}\) \\
1 & 70 & 70 & 125 & 45 & 45 & 45 \\
2 & 65 & 65 & 120 & 35 & 50 & 50 \\
3 & 75 & 75 & 115 & 30 & 35 & 40 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Wheeled robot parameters chosen as control factors at three different levels. The optimal design parameters are given by sensitive analysis based on the signal-noise ratio Equation (4). The goal is to reduce the trajectory of the center of mass as low as possible to the slope of the stair, maintaining the wheels' contact with the ground. Therefore, the robot is more robust and adapts to different sizes of stairs.