# Mechanical Design and Gait Optimization of Hydraulic Hexapod Robot Based on Energy Conservation
## 4 Parameter Optimization
### Design Sensitivity Analysis  
The design sensitivity analysis provides the derivatives of certain output variables with respect to specified design parameters, which determine the most influential variables on the objective function. The mechanical structure and the gait parameter can be adjusted with the installation and manufacture requirements according to the influence of variables.  
It can be obtained as  
\[Z_{i}=\frac{\partial F(\mathrm{x})}{\partial(\mathrm{x}_{i})} \tag{44}\]  
The magnitude of the sensitivity coefficients indicates the relative importance of that variable on the variation of the objective function. Figure 12 shows the values of the sensitivity coefficients of the objective functions \(f_{1}(x),f_{2}(x),f_{3}(x)\), and \(F(x)\) on the optimal solution point. Combined with Equations (27) and (28), it shows that the objective function \(F(x)\) is most sensitive to the variation of design variables \(b_{1}\) and \(b_{2}\) which influence the arm of force of the actuators, and are limited by the constraints of pressure (Equation (37)), installation position (Equation (39)), and control accuracy (Equation (41)). The value of variables \(b_{1}\) and \(b_{2}\) cannot be decreased. The step length \(S\) is the second most important variable and has influence in all objective functions. As the step length increases and the forward speed remains unchanged, the energy consumption of the HHR decreases, the maximum pressure increases, and the stability margin decreases. In the rest variables, \(L_{1}\) and \(L_{2}\) affect the flow rate and maximum pressure of the hydraulic system; \(m\) and \(s_{i}\) (\(i\) = 1-6) affect the maximum pressure and motion stability, respectively. Variables \(a_{1}\), \(a_{2}\), \(e_{11}\), \(e_{12}\), \(e_{21}\), \(e_{22}\), and \(\varphi\) have less influence, which can be used for mechanical structure adjustment.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Design** & **Initial** & **GA** & **AMGA** \\
**Variables** & **Value** & **Value** & **Value** \\ \hline \(L_{1}\) (mm) & 400 & 415 & 420 \\ \(L_{2}\) (mm) & 400 & 408 & 448 \\ \(m\) (mm) & 0 & 0 & 0 \\ \(S\) (mm) & 200 & 306 & 318 \\ \(\varphi\) (rad) & 0.3491 & 0.4265 & 0.4096 \\ \(a_{1}\) (mm) & 300 & 322 & 311 \\ \(a_{2}\) (mm) & 300 & 322 & 311 \\ \(b_{1}\) (mm) & 80 & 61 & 60 \\ \(b_{2}\) (mm) & 80 & 61 & 60 \\ \(e_{11}\) (rad) & 1.3963 & 1.5707 & 1.5707 \\ \(e_{12}\) (rad) & \(-\)0.2618 & \(-\)0.2269 & \(-\)0.2269 \\ \(e_{21}\) (rad) & \(-\)0.2618 & \(-\)0.2269 & \(-\)0.2269 \\ \(e_{22}\) (rad) & 0.1745 & 0.3491 & 0.3491 \\ \(s_{1}\)/\(s_{2}\) (mm) & 0 & 106 & 120 \\ \(s_{3}\)/\(s_{4}\) (mm) & 0 & \(-\)8 & 18 \\ \(s_{5}\)/\(s_{6}\) (mm) & 0 & \(-\)197 & \(-\)203 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Optimization results.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Specifications** & **Initial Value** & **GA Value** & **AMGA Value** \\ \hline Flow rate (L/min) & 34.94 & 22.08 & 19.34 \\ Maximin pressure (MPa) & 14.9 & 14.6 & 15.6 \\ Stability margin (mm) & 115 & 102 & 113 \\ Fitness & \(2.30\times 10^{5}\) & \(1.50\times 10^{5}\) & \(1.32\times 10^{5}\) \\ Power (kW) & 9.32 & 5.89 & 5.16 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Specifications of different designs.