# Locomotion Efficiency Optimization of Biologically Inspired Snake Robots
## 4 Optimization Study
### Results for Underwater Snake Robots
Similar results are also obtained for an underwater snake robot with 10 and 20 links. For the robot with 10 links, we can observe that the  
Figure 4: Pareto front for the USR 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. Colors illustrate different weights of the objective functions. robot can consume 44.76% (or 54.19%) and 41.40% (or 52.60%) less power by reducing the forward velocity by 3.57% (or 5.95%) and 3.33% (or 5%), for lateral undulation and eel-like motion, respectively. If we neglect negative work effect, the robot consumes 44.70% (or 57.27%) less power for lateral undulation and 42.98% (or 52.27%) less power for eel-like motion pattern. For the robot with 20 links with lateral undulation pattern, we can reduce the power consumption by 36.85% and 50.34% by reducing the achieved forward velocity by 5.66% and 10.38%, respectively. For the eel-like motion pattern, Figure 5c shows that the average power consumption can be reduced by 42.32% and 59.23% while the corresponding forward velocity is decreased only by 5.19% and 11.69%. If we calculate the power consumption without considering negative work effect, for the mentioned decrease of the forward velocity we can save 31.59% and 47.40% power for lateral undulation, and 39.61% and 49.37% power for eel-like motion pattern. In conclusion, this observation may attract the attention of designers to study Pareto fronts in order to reduce power consumption significantly while decreasing the corresponding forward velocity slightly. Figure 5: Pareto front for the USR using the eel-like 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. Colors illustrate different weights of the objective functions. Figures 6-9 present the inverse correlation of the parameter \(\delta\) and the forward velocity or the power consumption. Furthermore, we can also observe in Figure 6 that, for the case of considering the negative work effect, the gait parameter \(\alpha\) takes values in the range of \([50^{\circ},66^{\circ}]\), \([31^{\circ},69^{\circ}]\) and \([25^{\circ},59^{\circ}]\) for all optimization trials for the underwater snake robot studied in this paper with \(n=5\), \(n=10\) and \(n=20\) links, respectively, for lateral undulation. Figure 7 shows that the range of change of the gait parameter \(\alpha\) is equal to \([31^{\circ},80^{\circ}]\), \([51^{\circ},90^{\circ}]\) and \([42^{\circ},70^{\circ}]\) for a robot with \(n=5\), \(n=10\) and \(n=20\) links, for an eel-like motion pattern. The values of the phase shift parameter \(\delta\) are in the range of \([26^{\circ},90^{\circ}]\), \([15^{\circ},90^{\circ}]\), \([20^{\circ},90^{\circ}]\) and \([34^{\circ},90^{\circ}]\), \([25^{\circ},90^{\circ}]\), \([23^{\circ},90^{\circ}]\) for the robot with \(n=5\), \(n=10\), \(n=20\) links for lateral undulation and an eel-like motion pattern, respectively. In the case that the negative work effect is not considered, the maximum amplitude \(\alpha\) for lateral undulation pattern is in the range of \([22^{\circ},49^{\circ}]\), \([19^{\circ},45^{\circ}]\), and \([14^{\circ},41^{\circ}]\) when the number of links is, respectively, equal to 5, 10, and 20. For the eel-like motion pattern, the range of \(\alpha\) is equal to \([23^{\circ},80^{\circ}]\), \([36^{\circ},60^{\circ}]\), and \([28^{\circ},56^{\circ}]\), respectively, for \(n=5,10\), and 20. For lateral undulation pattern, the phase shift \(\delta\) changes in the range of \([26^{\circ},80^{\circ}]\), \([15^{\circ},62^{\circ}]\), and \([20^{\circ},64^{\circ}]\) for \(n=5,10\), and 20, respectively. For the eel-like motion pattern, the range of phase shift is equal to \([34^{\circ},81^{\circ}]\), \([26^{\circ},78^{\circ}]\), and \([23^{\circ},64^{\circ}]\) for \(n=5,10\), and 20, respectively.