# High-Efficiency Closed-Loop Control of a Robotic Fish via Virtual Musculoskeletal Methodology
## 1 Introduction  
Robotic fish, which is usually driven by motors in the joint space, is a kind of bionic device that imitates the body and the control mechanism of fish in nature. Robotic fish are widely applied in many fields, such as marine exploration, underwater rescue, fish breeding, and biotechnology research [1]. As robotic fish has a limited power supply, long working time in these applications requires it to swim efficiently. Prior works on the propulsion efficiency of robotic fish mainly focus on two aspects: (1) generating fish-like swimming patterns by learning the swimming model from the fish in nature and (2) designing control algorithms according to the characteristics of the robots and the effects of the flow [2, 3]. The swimming model, including the kinematic and the dynamic model, is the basic challenge for the efficient propulsion of robotic fish. In [4], the multiple-joint kinematic model is presented to imitate the carangoid. Then, the control of the joints in this model is learned from the swimming data of the carangoid by the central pattern generator (CPG). Afterward, the work in [5] discussed the relationship between the CPG-based locomotion control and the energy consumption of the miniature self-propelled robotic fish, which provided a reference for improving the energy efficiency and locomotion performance of the versatile swimming gaits. The work in [6] proposes a function of the relationship between the head and the whole body to simulate the swimming pattern of the carangoid. Moreover, the angles of the joints in this function are discretized to form a lookup table, which achieves cruise and turning patterns in the experiment. Meanwhile, the work in [7] designs the caudal fin by combining the skeleton and the tail fin. The skeleton moves relatively while the fin swings to affect the efficiency. Although the experiment shows how the stiffness, the waveform, and the frequency of the skeleton significantly impact thepropulsion efficiency, the mathematical model of the parameters is still unclear. Similarly, [PERSON] uses three-dimensional particle image velocimetry (PIV) to record the flow generated by different fins. The results indicate that the shape nonlinearly affects the hydrodynamics, and the length influences the forward speed [8]. The literature emphasizes the hydrodynamic performance of the wake vortex when robotic fish swims by analogizing the movement of the caudal fin [9]. In [10], an elaborate model with actuator dynamics was introduced to demonstrate the swimming behavior of the carangiform locomotion-type fish robots. However, there are still some uncertainties. Additionally, researchers have done many simulations and experiments to find out the relationship between typical wake vortex and propulsive efficiency [11; 12]. The control algorithms of high propulsive efficiency are also widely studied recently. The work in [13] introduced the mainstream robotic fish propulsion mechanism and control method, and guided the robotic wave fish's kinematic optimization and motion control. The work in [14] designed an extremum-seeking method to control the robotic fish to swim along a predetermined trajectory independently on the model. Moreover, a robotic fish consisting of a body, pectoral fins, and a caudal fin is controlled by a CPG model to swim point-to-point with the feedback from the visual system and the gyroscope [15]. In [16], researchers applied two control methods named computed torque method and feed-forward method on a four-joint, six-degrees-of-freedom robotic fish with a horizontal caudal fin. The work in [3] addresses an iterative learning control (ILC) method to a two-link carangiform robotic fish. The P-type ILC algorithm works well for the highly nonlinear model of the robotic fish and the nonaffine in the input. [PERSON] et al. [17] define the natural oscillation of a class of flat-body fishes as a free response to the flow field under the damping compensation.