# Locomotion Efficiency Optimization of Biologically Inspired Snake Robots
###### Abstract
Furthermore, we here take into account the negative work effect of underwater snake robots, which was not considered in [16]. The optimization framework proposed in this paper is thus an extension of the method presented in [16] and may be used for bio-inspired snake robots locomotion both on land and in water. In this paper, we solve a motion optimization problem to obtain the optimal gait parameters based on a dynamic model of USR. The optimization problem is challenging as the USR model includes highly nonlinear hydrodynamic effects. Several models have been proposed for USRs in previous studies[7; 8; 17; 18; 19; 20]. A closed form model for USRs was presented in [6; 21; 22], where both the resistive and reactive fluid forces were considered. In particular, the proposed model in [6; 21; 22], considers several hydrodynamic effects such as the linear and the nonlinear drag forces, the added mass effect, the fluid moments and current effects. This model is general in the sense that it can be used to simulate the behavior of snake robots moving on land or swimming in the water by considering either ground friction [1] or fluid friction model [6]. In this paper, the optimization results are obtained based on this model. To optimize both the power consumption and the forward velocity, a weighted sum of both objective functions is proposed as a single goal. Optimization algorithms can be divided into two main classes: gradient-based and derivative free algorithms. Gradient-based algorithms are generally faster than the derivative free ones. However, the optimization solution in the gradient-based methods often depends on the initial points, since these algorithms are more prone to get trapped in local optima [23]. Therefore, derivative-free and more specifically stochastic derivative-free methods are reported to be used more often in problems with highly non-smooth objective functions containing multiple optima, because these types of algorithms can avoid local solutions by their stochastic nature. Consequently, as also mentioned in [8], we find that stochastic derivative-free methods constitute a proper optimization algorithm for the motion optimization problem. In [24], a genetic algorithm (GA) is applied to a three-link swimmer to solve the swimming gait optimization problem. Moreover, pattern optimization of anguilliform swimming is studied in [25]. In particular, a combination of an evolutionary optimization algorithm and a three-dimensional numerical solution are implemented. Two objective functions of efficiency and velocity are considered in [25], and the optimization problem is addressed by solving Navier-Stokes equations. Because of the computational cost of solving the Navier-Stokes equations, only one optimization run is considered for each fitness goal. In [8], optimization is used to generate efficient hyper-redundant mechanism swimming gaits. In particular, the gait parameters were optimized in order to minimize the total required energy over a given distance for a desired swimming velocity. Therefore, a penalty term was added to the objective function to set the velocity to the desired value. Since the penalty factor is chosen relatively large, the velocity tends to stay at the desired value, and thus it is difficult to observe the trade-off between the velocity and the required energy. However, we will show in this paper that a slight reduction of velocity may result in a significant saving in power consumption. In particular, we propose to optimize both the energy and the velocity, simultaneously. Therefore, there is no need to tune the penalty factor or define the desired velocity. We apply the particle swarm optimization (PSO) algorithm to optimize the power consumption and the forward velocity. PSO iterates on the gait parameters in order to improve a candidate solution. [PERSON] and [PERSON] [26] developed the PSO algorithm inspired by the social behavior of animals.