Snake robots constitute bio-inspired solutions that have been studied due to their ability to move in challenging environments where other types of robots, such as wheeled or legged robots, usually fail. In this paper, we consider both land-based and swimming snake robots. One of the principal concerns of the bio-inspired snake robots is to increase the motion efficiency in terms of the forward speed by improving the locomotion methods. Furthermore, energy efficiency becomes a crucial challenge for this type of robots due to the importance of long-term autonomy of these systems. In this paper, we take into account both the minimization of the power consumption and the maximization of the achieved forward velocity in order to investigate the optimal gait parameters for bio-inspired snake robots using lateral undulation and eel-like motion patterns. We furthermore consider possible negative work effects in the calculation of average power consumption of underwater snake robots. To solve the multi-objective optimization problem, we propose transforming the two objective functions into a single one using a weighted-sum method. For different set of weight factors, Particle Swarm Optimization is applied and a set of optimal points is consequently obtained. Pareto fronts or trade-off curves are illustrated for both land-based and swimming snake robots with different numbers of links. Pareto fronts represent trade-offs between the objective functions. For example, how increasing the forward velocity results in increasing power consumption. Therefore, these curves are a very useful tool for the control and design of snake robots. The trade-off curve thus constitutes a very useful tool for both the control and design of bio-inspired snake robots. In particular, the operators or designers of bio-inspired snake robots can choose a Pareto optimal point based on the trade-off curve, given the preferred number of links on the robot. The optimal gait parameters for the robot control system design are then directly given both for land-based and underwater snake robots. Moreover, we are able to obtain some observations about the optimal values of the gait parameters, which provide very important insights for future control design of bio-inspired snake robots.

Underwater snake robots constitute a bio-inspired solution within underwater robotics. Increasing the motion efficiency in terms of the forward speed by improving the locomotion methods is a key issue for underwater robots. Furthermore, the energy efficiency is one of the main challenges for long-term autonomy of these systems. In this study, we will consider both these two aspects of efficiency, which in some cases can be conflicting. To this end, we formulate a multi-objective optimization problem to minimize power consumption and maximize forward velocity. In particular, the optimal values of the gait parameters for different motion patterns are calculated in the presence of trade-offs between power consumption and velocity. As is the case with all multi-objective optimization problems, the solution is not a single point but rather a set of points. We present a weightedsum method to combine power consumption and forward velocity optimization problems. Particle Swarm Optimization (PSO) is applied to obtain optimal gait parameters for different weighting factors. Trade-off curves or Pareto fronts are illustrated in a power consumption-forward velocity plane for both lateral and eel-like motion pattern. They give information on objective trade-offs and can show how improving power consumption is related to deteriorating the for-

discrete, continuous, and adaptive) using several numerical experiments, followed by a discussion of the properties of the proposed reduced random sampling approach and comparison with global optimization techniques. The method is applied to several numerical experiments, including case studies involving vertical, horizontal, and lateral wells, to evaluate its performance. The results from these experiments indicate that the reduced random sampling approach can provide significant computational gain with minimal impact on the attained optimization performance.

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