# Analysis and Design of a Minimalist Step Climbing Robot
## 4 Design Analysis  
In Section 3, the equations for describing the mathematical model of the proposed robot are discussed. The contribution of various parameters of the robot must be studied for successful operation of the robot on the stairs. In this section, the contribution of the parameters of a robot on the stairs is illustrated. The dimensions of the robot and the stairs are shown in Table 2. For the analysis, one of the parameters is varied while keeping the other parameters unchanged. When the robot climb stairs, the inclination angle of the robot changes significantly. With little consideration, it may be observed that the change in inclination angle depends on various parameters of the robot and the stair. One of the parameters that influence the variation in the inclination angle is the height of the step of a stair. Figure 9 illustrates the change in inclination angle of the robot at three initial steps of the stairs when the height of the step is varied from 70 mm to 160 mm. The three plots, step 1, step 2 and step 3, represent the variation of inclination angle when the front wheel is at the first, second and third steps of the stair and the height of the step is varied from 70 mm to 160 mm, respectively. It may be noticed that, in step 1, although the contact point of the step edge with the front will raise with an increase in step height, the robot inclination angle remains unaffected. It is observed that with the increase in step number, the slope of the plot, which represents the change in inclination angle of the robot, also increases. In Figure 9, as the height of a step increases, the torque required by the front wheels to overcome the step will also increase. Although higher torque required by the front wheels may be achieved by selecting higher torque motors, the maximum allowable torque on the front wheel is directly proportional to the product of the coefficient of static friction and the normal reaction of the wheel at the contact point between the edge and the wheel (Equation (12)). If the torque on the front wheels exceeds the maximum permissible limit, the front wheel will slip on the edge of the step, resulting in failure in climbing the stairs. The variations in the required torque and maximum allowable torque with respect to the change in step height at three initial steps are shown in Figure 10. In Figure 10a, the effect of step height on the torques of the front wheels is shown when the simulations are performed while controlling the inclination angle of the robot in climbing up the stairs, whereas Figure 10b illustrates the simulations without controlling the inclination angle. It \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Values** & **Parameters** & **Values** \\ \hline R & 300 mm & \(m_{1}\) & 10 kg \\ r & 200 mm & \(m_{4}\) & 5 + 45 (Pay load) kg \\ \(l_{b}\times h_{b}\) & 400 \(\times\) 300 mm & \(m_{b}\) & 10 kg \\ \(l_{L}\times h_{L}\) & 150 \(\times\) 200 mm & \(m_{L}\) & 150 kg \\ \(L_{3}\) & 800 mm & \(m_{3}\) & 5 kg \\ \(L_{x}\) & 325 mm & \(\mu_{s}\) & 0.7 \\ \(h_{s}\) & 160 mm & \(w_{s}\) & 240 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dimension of the robot for analysis. Figure 8: Free body diagram of wheel. may be observed that in both the controlled and uncontrolled simulations, the required torque increases along with the height of the steps, whereas the maximum allowable torque decreases. However, in the controlled case, the torque required to overcome the step is lower than the maximum allowable torque, whereas the required torque is higher than the maximum allowable torque in the uncontrolled case. This will result in slippage of the front wheel during uncontrolled operations. In Figure 10, the uncontrolled simulations are performed with passive rear wheels, whereas controlled simulations are performed with active rear wheels. It may be noticed that during uncontrolled simulations, the maximum allowable torque on the front wheels is less than the required torque, resulting in the slippage of front wheels.