# Locomotion Efficiency Optimization of Biologically Inspired Snake Robots
## 4 Optimization Study
### Results for Underwater Snake Robots
We can also observe that the maximum forward velocity is achieved using the maximum values of \(\omega=210^{\circ}/\mathrm{s}\) for all the cases, while the obtained values of the amplitude, \(\alpha\), and the phase-shift, \(\delta\), are different for the investigated motion patterns. These observations are useful for both control design purposes and defining the range of the gait parameters for solving the optimization problem more precisely in the future. Figure 6: Optimal gait parameters for the lateral undulation motion pattern for the USR, considering the negative work effect: (**a**) 3D plots of optimal gait parameters for robot with \(n=5\) links; (**b**) 3D plots of optimal gait parameters for robot with \(n=10\) links; and (**c**) 3D plots of optimal gait parameters for robot with \(n=20\) links. Colors illustrate different weights of the objective functions. Figure 8: Optimal gait parameters for the lateral undulation motion pattern for the USR, without considering the negative work effect: (**a**) 3D plots of optimal gait parameters for robot with \(n=5\) links; (**b**) 3D plots of optimal gait parameters for robot with \(n=10\) links; and (**c**) 3D plots of optimal gait parameters for robot with \(n=20\) links. Colors illustrate different weights of the objective functions. Figure 7: Optimal gait parameters for the eel-like motion pattern for the USR, considering the negative work effect: (**a**) 3D plots of optimal gait parameters for robot with \(n=5\) links; (**b**) 3D plots of optimal gait parameters for robot with \(n=10\) links; and (**c**) 3D plots of optimal gait parameters for robot with \(n=20\) links. Colors illustrate different weights of the objective functions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(n\) & \(\alpha\) (deg) & \(\omega\) (deg/s) & \(\delta\) (deg) & \(\bar{\upsilon}\) (m/s) & \(P_{\text{avg}}\) (W) \\ \hline
5 & 79.21 & 210 & 34.23 & 0.38 & 3.26 \\
5 & 72.04 & 210 & 43.73 & 0.37 & 1.89 \\
5 & 69.69 & 210 & 50.09 & 0.36 & 1.39 \\
10 & 59.24 & 210 & 26.20 & 0.60 & 13.44 \\
10 & 52.61 & 210 & 33.47 & 0.58 & 7.88 \\
10 & 51.33 & 210 & 36.69 & 0.57 & 6.37 \\
20 & 55.13 & 210 & 23.97 & 0.77 & 36.08 \\
20 & 43.83 & 210 & 29.48 & 0.73 & 20.81 \\
20 & 42.03 & 210 & 33.86 & 0.68 & 14.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Multi-objective optimization results for the eel-like motion pattern for the USR with different numbers of links, considering the negative work effect. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(n\) & \(\alpha\) (deg) & \(\omega\) (deg/s) & \(\delta\) (deg) & \(\bar{\upsilon}\) (m/s) & \(P_{\text{avg}}\) (W) \\ \hline
5 & 57.68 & 210 & 26.82 & 0.46 & 4.83 \\
5 & 52.33 & 210 & 35.65 & 0.44 & 2.68 \\
5 & 51.68 & 210 & 39.33 & 0.43 & 2.20 \\
10 & 44.01 & 210 & 15.14 & 0.84 & 34.25 \\
10 & 34.92 & 210 & 20.32 & 0.81 & 18.92 \\
10 & 33.42 & 210 & 22.97 & 0.79 & 15.69 \\
20 & 41.21 & 210 & 20.00 & 1.06 & 74.01 \\
20 & 32.45 & 210 & 21.73 & 1.00 & 46.74 \\
20 & 29.16 & 210 & 24.01 & 0.95 & 36.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Multi-objective optimization results for the lateral undulation motion pattern for the USR with different numbers of links, considering the negative work effect. Figure 9: Optimal gait parameters for the eel-like motion pattern for the USR, without considering the negative work effect: (a) 3D plots of optimal gait parameters for robot with \(n=5\) links; (b) 3D plots of optimal gait parameters for robot with \(n=10\) links; and (c) 3D plots of optimal gait parameters for robot with \(n=20\) links. Colors illustrate different weights of the objective functions.