# Locomotion Efficiency Optimization of Biologically Inspired Snake Robots
## 4 Optimization Study
### Results for Land-Based Snake Robots  
In this section, optimization results for the land-based snake robots are discussed. The motion pattern is considered lateral undulation which is the most common motion pattern for land-based snake robots [1]. Figure 10 presents the Pareto fronts (trade off curves) for different numbers of links. Figure 11 illustrates the Pareto optimal average power consumption and forward velocity versus corresponding optimal gait parameters. Table 7 summarizes a selected number of Pareto optimal objective functions and related Pareto optimal gait parameters. For a land-based snake robot with 5, 10, and 20 links, the maximum forward velocity is equal to \(\bar{v}=0.46\) m/s, \(\bar{v}=0.94\) m/s, and \(\bar{v}=1.27\), respectively, and the corresponding average power consumption is equal to \(P_{\mathrm{avg}}=0.96\), \(P_{\mathrm{avg}}=6.86\), and \(P_{\mathrm{avg}}=14.62\), respectively. This means increasing the number of links from 5 to 10 leads to approximately a 104% increase in forward velocity, while the average power consumption is increased approximately seven times. We observe about 35% increment in the forward velocity by changing the number of links from 10 to 20, and as a result the average power consumption is increased approximately two times. From this table, we notice that the optimal frequency related to the maximum forward velocity is equal to \(\omega=210\)\({}^{\circ}\)/s for different numbers of links. The optimal maximum amplitude is equal to \(\alpha=44.32^{\circ}\), \(\alpha=25.27^{\circ}\), and \(\alpha=22.22^{\circ}\), respectively, for 5, 10 and 20 links. The optimal phase shift for 5, 10, and 20 links is defined equal to \(\delta=36.04^{\circ}\), \(\delta=18.02^{\circ}\), and \(\delta=20^{\circ}\), respectively. We observe from Figure 10 that one can obtain a significant decrease in the average power consumption as a trade-off for a negligible decrease in the achieved forward velocity. This trade-off is also stated in Table 7. For a snake robot with 5 links, the table gives a 40.79% decrease of the average power consumption (from \(P_{\mathrm{avg}}=0.96\) W to \(P_{\mathrm{avg}}=0.57\) W) results in just 4.35% decrease in the forward velocity (\(\bar{v}=0.46\) m/s to \(\bar{v}=0.44\) m/s). Another alternative is to decrease the average power \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(n\) & \(\alpha\) (deg) & \(\omega\) (deg/s) & \(\delta\) (deg) & \(\bar{v}\) (m/s) & \(P_{\mathrm{avg}}\) (W) \\ \hline
5 & 79.21 & 210 & 34.23 & 0.38 & 3.45 \\
5 & 69.37 & 210 & 45.38 & 0.37 & 1.89 \\
5 & 66.13 & 210 & 50.18 & 0.35 & 1.52 \\
10 & 59.24 & 209.98 & 26.20 & 0.60 & 14.96 \\
10 & 51.83 & 210 & 34.41 & 0.58 & 8.53 \\
10 & 50.17 & 209.98 & 37.48 & 0.56 & 7.14 \\
20 & 55.13 & 210 & 23.97 & 0.77 & 36.78 \\
20 & 43.49 & 210 & 29.28 & 0.73 & 22.21 \\
20 & 40.39 & 209.95 & 31.23 & 0.70 & 18.62 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Multi-objective optimization results for the eel-like motion pattern for the USR with different numbers of links, without 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) & \(\bar{v}\) (m/s) & \(P_{\mathrm{avg}}\) (W) \\ \hline
5 & 58.05 & 210 & 26.68 & 0.46 & 5.33 \\
5 & 51.16 & 210 & 35.17 & 0.44 & 3.18 \\
5 & 48.14 & 210 & 39.34 & 0.42 & 2.54 \\
10 & 44.02 & 210 & 15.17 & 0.84 & 34.92 \\
10 & 34.76 & 210 & 20.98 & 0.81 & 19.31 \\
10 & 32.26 & 210 & 24.68 & 0.77 & 14.92 \\
20 & 40.96 & 210 & 20.00 & 1.06 & 83.45 \\
20 & 34.03 & 210 & 21.07 & 1.02 & 57.09 \\
20 & 29.81 & 210 & 22.64 & 0.97 & 43.92 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Multi-objective optimization results for the lateral undulation motion pattern for the USR with different numbers of links, without considering the negative work effect. consumption from \(P_{\text{avg}}=0.96\) W to \(P_{\text{avg}}=0.44\) W (about 54% increment) by decreasing the forward velocity from \(\tilde{v}=0.46\) m/s to \(\tilde{v}=0.42\) m/s (8.69% decrease).