# Parameters Sensitivity Analysis of Position-Based Impedance Control for Bionic Legged Robots' [PERSON]
## 4 Methods of Sensitivity Analysis  
### Contrast between First-Order and Second-Order Sensitivity Analysis  
Desired stiffness \(K_{D}\) and desired damping C\({}_{D}\) are control parameters of the impedance control outer loop. Proportional gain \(K_{P}\) and integral gain \(K_{i}\) are control parameters of the position control inner loop. They affect the impedance control performance in different ways. So, in this paper, the system output position is mainly discussed, which is influenced by the variation of the four parameters. Because of space limitations, only one working condition (bias: 1500 N, amplitude: 1000 N, frequency: 2 Hz) is studied. The first-order and second-order trajectory sensitivity analysis methods researched in our previous works [15; 16] are adopted to analyze the position variation when the four parameters increase 10% to 20%. The contrast curves are shown in Figures 12-15. Figure 11: Experimental and simulation curves of sinusoidal response (frequency: 2 Hz, bias: 2500 N, amplitude: 2000 N). (**a**) Experimental curves; (**b**) Simulation curves. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Performance Index**} & \multicolumn{5}{c}{**No.**} \\ \cline{2-6}  & **1** & **2** & **3** & **4** \\ \hline \multirow{2}{*}{Amplitude reduction (mm)} & Experimental & \(-0.05\) & \(-0.09\) & \(-0.06\) & \(-0.11\) \\  & Simulation & \(-0.03\) & \(-0.06\) & \(-0.03\) & \(-0.06\) \\ \hline \multirow{2}{*}{Phase angle delay (\({}^{\circ}\))} & Experimental & 7.8 & 6.7 & 8.9 & 9.4 \\  & Simulation & 5.2 & 5.3 & 7.1 & 7.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mean values of performance indexes in different conditions. Figure 14: Position variation resulting from \(K_{P}\) variation. (**a**) Four parameters increase 10%; (**b**) Four parameters increase 20%. Figure 12: Position variation resulting from \(K_{D}\) variation. (**a**) Four parameters increase 10%; (**b**) Four parameters increase 20%. Figure 13: Position variation resulting from C\({}_{D}\) variation. (**a**) Four parameters increase 10%; (**b**) Four parameters increase 20%. Figure 15: Position variation resulting from \(K_{i}\) variation. (**a**) Four parameters increase 10%; (**b**) Four parameters increase 20%. The following conclusions can be reached from the four sets of curves. There is little difference between the first-order position variation and the second-order variation when the values of the four parameters increases 10%. Specifically, the curves of desired damping C\({}_{D}\) and integral gain \(K_{i}\) almost overlap. In the other two sets, the maximum error is no more than 10% of the amplitude. When the values of the four parameters increase 20%, the variations of first-order and second-order are still close, although there is some increase of their error. Specifically, for desired damping C\({}_{D}\) and integral gain \(K_{i}\), the variation curves of first-order and second-order are still close. As for the proportional gain \(K_{P}\), its maximum error is no more than 10% of the amplitude. When it comes to the desired stiffness \(K_{D}\), its maximum error is no more than 20% of the amplitude. The method of second-order trajectory sensitivity analysis has a very high accuracy, while its calculation is complicated and demands a lot of hard work. Particularly in this paper, when the four parameters increase less than 20%, the corresponding results of the first-order and second order sensitivity analysis method are close. So, in order to ensure the simplicity of calculation and application, the method of first-order sensitivity analysis is adopted to analyze the sensitivity of the four parameters under different working conditions. The method studied previously can precisely analyze the sensitivity of the parameters.