# Locomotion Efficiency Optimization of Biologically Inspired Snake Robots
## 4 Optimization Study
### Results for Underwater Snake Robots  
In this section, the results of motion efficiency optimization of underwater snake robots are presented for both lateral undulation and eel-like motion patterns. The parameters of the robot are specified in Section 4.1. Pareto fronts are presented in Figures 4 and 5 for a different number of links for lateral undulation and eel-like motion pattern, respectively. In these figures, we also investigate the negative work effect in the calculation of average power consumption (see Equations (16) and (17)). Figures 6-9 show Pareto optimal gait parameters of the sinusoidal motion pattern vs. Pareto optimal average power consumption and achieved forward velocity. As it was expected, the robot consumed the maximum power when moving with the highest forward velocity, while zero power is consumed when the robot is not moving (i.e., for zero forward velocity). To discuss some of the observations, selected Pareto optimal points are compared in Tables 3-6. The maximum achieved forward velocities and corresponding optimal parameters for different numbers of links are presented in these tables. The maximum forward velocity for the underwater snake robot with lateral undulation motion pattern is equal to \(\bar{v}=0.46\) m/s, \(\bar{v}=0.84\) m/s, and \(\bar{v}=1.06\) m/s, respectively, for a number of links equal to 5, 10, and 20. The maximum average power consumption considering the negative work effect is equal to \(P_{\text{avg}}=4.83\) W, \(P_{\text{avg}}=34.25\) W, and \(P_{\text{avg}}=74.01\) W for a number of links equal to \(n=5,10\), and 20, respectively. If we neglect negative work effect the maximum average power consumption is increased to \(P_{\text{avg}}=5.33\) W, \(P_{\text{avg}}=34.92\) W, and \(P_{\text{avg}}=83.45\) W, respectively, for a number of links equal to 5, 10, and 20. The increment of average power consumption as a result of disregarding negative work effect is equal to 10.35%, 1.97% and 12.76%. As expected, the maximum forward velocity for eel-like motion pattern is lower than that of snake robot with a lateral undulation pattern: \(\bar{v}=0.38\) m/s for \(n=5\), \(\bar{v}=0.60\) m/s for \(n=10\), \(\bar{v}=0.77\) m/s for \(n=20\). The maximum average power consumption for eel-like motion pattern considering negative work effect is equal to \(P_{\text{avg}}=3.26\) W, \(P_{\text{avg}}=13.44\) W, \(P_{\text{avg}}=36.08\) W for \(n=5,10\) and 20, respectively. In the case of disregarding negative work effect, the maximum power consumption is changed to \(P_{\text{avg}}=3.45\) W, \(P_{\text{avg}}=14.96\) W and \(P_{\text{avg}}=36.78\) W, for a number of links equal to 5, 10, and 20, respectively. This means neglecting negative work effect results in 5.83%, 11.31% and 1.94% increase of power consumption for \(n=5,10\) and 20, respectively. One can observe from all Pareto optimal points of both motion patterns, the robot consumes more energy when the negative work effect is not considered. It is important to consider this observation in the development of underwater snake robots. problem. The optimum value of the number of links gives useful inputs in the design stage and the implementation of the swimming snake robots. Figures 4 and 5 show that it is possible to obtain a significant decrease in the consumed power by a slight decrease in the achieved forward velocity. This observation is also clearly presented in Tables 3-6. For the lateral undulation motion pattern of the snake robot with five links, 44.51% (or 54.45%) decrease in the average power consumption leads to a slight decrease of 4.35% (or 6.52%) in the forward velocity. Similarly for the eel-like motion pattern, by decreasing 42.02% (or 57.36%) in the power consumption, we have 2.63% (or 5.26%) decrease in the forward velocity. Tables 5 and 6 give similar results for the case that the negative work effect is not considered. In this case, we see that it is possible to decrease the consumed power about 40.34% (or 52.35%) and 45.22% (or 55.94%) for lateral undulation and eel-like motion pattern, respectively.