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performance gap between robot fishes and fish.

Currently, there are three main research methods to study the swimming mechanism of robot fishes. The
strengths and weaknesses of the three research methods are summarized in Table 3. The first method is
theoretical analysis. In this method, the swimming equations of robot fishes are established by mathematical
and physical models. The method is very adaptable, but it is mathematically challenging. Further, the
difficulty lies in the need to establish equations that can be solved and correctly describe the complex
swimming of robot fishes. The second method is experimental observation. This method uses particle image
velocimetry (PIV) or other special equipment to observe robot fishes or fish. The conclusions of the
research are highly accurate due to real-world observations, but they have poor universality due to the
experimental setting’s restrictions. The third method is numerical simulation, which uses computers to
numerically solve existing models to predict the swimming characteristics of robot fishes. The method is
low cost and accurate, but it cannot solve some complex swimming problems that lack a perfect
mathematical model. We can see that each of the three research methods has strengths and weaknesses, and
combining these methods can yield complementary benefits.

3.1. Theoretical analysis
The swimming of robot fishes mimics that of fish. A better understanding of the fish motion mechanism
aids in the design of robot fishes. There are numerous theories about fish swimming, but only a few widely
accepted ones are discussed here. In 1970, Lighthill proposed the “elongated-body theory” . This theory
[33]
only investigates the role of the fish’s caudal cross-section in swimming, ignoring the effect of the caudal
vortex. As a result, the swimming performance obtained by this theory is only related to the flow parameters
in the cross-section of the fish’s caudal. Furthermore, the theory is only applicable to analyzing the
swimming of fish with small amplitude. One year later, the “large-amplitude elongated-body theory” was
further proposed by Lighthill . In 1991, Tong et al. developed the “three-dimensional waving plate theory”
[34]
based on the “two-dimensional waving plate theory” of Wu [35,36] . This theory simplifies the swimming of a
fish to a flexible deformed plate oscillating in a wave-like motion. It is worth noting that the tail vortex effect
is considered, which makes the calculation results closer to the real swimming of the fish. This theory is
applicable to fish swimming with small amplitude. It can be extended to the accelerated swimming of fish
and large-amplitude non-linear swimming.

In recent years, there have been new developments in the theory of robot fishes’ swimming. They are
mainly a supplement to the previous theories and thus solve some practical problems. Wang et al.
incorporated the robot fish’s head oscillation equation into the kinematic model based on the elongated-
body theory [33,37] . The improved kinematic model was established successfully. The results show that the
maximum swing angle of the head was reduced to 86% of its original value, while the swimming speed was
increased by 17%. Kirchhoff’s equations of motion were utilized by Kopman et al. to show the dynamics of
[38]
frontal link . Caudal fin oscillation was modeled by Euler-Bernoulli beam theory. The influence of the
fluid around the robot fish was described by the Morison equation. Finally, the dynamic equation of the
robot fish propelled by soft fin was established.

3.2. Experimental observation
With the emergence of new experimental equipment, experimental observation has become more popular.
PIV is the most effective experimental method. It is a method of measuring flow velocity that involves
recording the position of particles in the flow field with multiple cameras and analyzing the images
captured. The basic idea is to spread tracer particles in the flow field and then inject a pulsed laser into the
measured flow field area. The images of the particles are recorded by two or more consecutive exposures.
Zhu et al. visualized the flow field by PIV and obtained the flow field image of Tunabot during the caudal