# Locomotion Efficiency Optimization of Biologically Inspired Snake Robots
## 4 Optimization Study
### Results for Land-Based Snake Robots
Similar results are also stated for a robot with 10 and 20 links. For a snake robot with 10 links, we see that the robot is able to consume 45.63% and 56.26% less power by reducing the forward velocity by 4.26% and 7.45%, respectively. Similarly, for a snake robot with 20 links, we can reduce the power consumption by 34.82% and 46.64% by reducing the achieved forward velocity by 4.72% and 10.23%, respectively. Consequently, based on these results, the operators can select the gait parameters in a way that would result in a significant decrease of the consumed power, while giving only a slight decrease of the forward velocity. As we can see in Figure 11, the forward velocity and the power consumption decrease when the value of the gait parameter \(\delta\) increases. In addition, we observe that the optimal value of the gait parameter \(\alpha\) is greater than \(14^{\circ}\), \(17^{\circ}\) and \(13^{\circ}\) in all weighting sets for the robot with \(n=5\), \(n=10\) and \(n=20\) links, respectively. In particular, the gait parameter \(\alpha\) takes values in the range of \([14^{\circ},45^{\circ}]\), \([17^{\circ},30^{\circ}]\) and \([13^{\circ},21^{\circ}]\) for all optimization trials for the land-based snake robot studied in this paper with \(n=5\), \(n=10\) and \(n=20\) links, respectively. The values of the phase shift parameter \(\delta\) are in the range of \([36^{\circ},72^{\circ}]\), \([18^{\circ},78^{\circ}]\) and \([20^{\circ},60^{\circ}]\) for the robot with \(n=5\), \(n=10\) and \(n=20\) links, respectively. The maximum forward velocity is achieved for the maximum values of \(\omega=210\)\({}^{\circ}\)/s. Note that these insights are essential for control design purposes for bio-inspired snake robots. Furthermore, they can be used to customize the necessary constraints for future optimization investigations. Note that the Pareto fronts illustrated in Figures 4, 5 and 10 constitute an informative visualization tool, which helps control system designers to get full information about objective values (power consumption and velocity) and objective tradeoffs. Consequently, depending on the control objectives and the available power of the robot, the operators or designers of bio-inspired snake robots can choose the optimal operational point using the Pareto fronts. Afterward, the corresponding gait parameters can be obtained. We observe in Tables 3-7 that the frequency parameter \(\omega\) tends to stay at the maximum value. Therefore, there is a possibility to reduce the dimension of the search space, \(n_{c}\), to 2 by eliminating \(\omega\) from the decision variables and setting this parameter equal to the maximum possible value. This observation can also be embedded in the design of the bio-inspired snake robots. This means one should consider the maximum possible value for the frequency during actuation mechanism selection. As a result, it motivates designers to investigate possible high frequency actuation solutions in control and design of underwater and land-based snake robot. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(n\) & \(\alpha\)**(deg)** & \(\omega\)**(deg/s)** & \(\delta\)**(deg)** & \(\delta\)**(m/s)** & \(P_{\text{avg}}\)**(W)** \\ \hline
5 & 44.32 & 210 & 36.04 & 0.46 & 0.96 \\
5 & 37.81 & 210 & 43.78 & 0.44 & 0.57 \\
5 & 33.92 & 210 & 46.78 & 0.42 & 0.44 \\
10 & 25.27 & 210 & 18.02 & 0.94 & 6.86 \\
10 & 20.87 & 210 & 23.29 & 0.90 & 3.73 \\
10 & 19.61 & 210 & 25.57 & 0.87 & 3.00 \\
20 & 22.22 & 210 & 20.00 & 1.27 & 14.62 \\
20 & 15.87 & 210 & 20.02 & 1.21 & 9.53 \\
20 & 14.73 & 210 & 21.38 & 1.14 & 7.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Multi-objective optimization results for the land-based snake robot with different numbers of links. Figure 11: _Cont._  
Figure 10: Pareto front for the land-based snake robot using the lateral undulation motion pattern: **(a)** Pareto front for robot with \(n=5\) links; **(b)** Pareto front for robot with \(n=10\) links; and **(c)** Pareto front for robot with \(n=20\) links.