A Novel Untethered Robotic Fish with an Actively Deformable Caudal Fin

Based on the observation that live fish caudal fins exhibit softness, flexibility, and active deformation, which are essential for generating thrust, stability, and maneuverability, herein, a comprehensive study on a novel type of bionic robotic fish with an actively deformable caudal fin is presented. The article first presents the design of the robot. Next, the quasisteady model theory is used to establish the hydrodynamic force for modeling the deformable caudal fin. Furthermore, three deformation modes are studied and compared: the conventional nondeformable mode, the sine‐based mode, and the instantaneous mode. Finally, a series of extensive experiments are conducted to evaluate various performance metrics of this innovative untethered biomimetic robotic fish, including thrust, swimming speed, yaw stability, turning radius, and turning rate. The results demonstrate that the introduction of active deformation of the caudal fin significantly enhances the swimming performance in the aforementioned indices when compared to the conventional nondeformable mode. Notably, the instantaneous mode exhibits best performance in terms of thrust, swimming speed, turning radius, and turning rate, while the sine‐based mode demonstrates the best yaw stability. Consequently, this research contributes to the advancement of robotic fish design and the development of underwater biomimetic robots.

and the fan fin for maneuvering. [13]Another parameter characterizing the caudal fin shape is the aspect ratio, calculated as the length (chord) divided by the width (span). [14]Lower aspect ratios were found to contribute to higher swimming speeds and increased efficiency due to reduced drag. [15]Boute et al., compared a boxfish with a folded caudal fin to one with an unfolded caudal fin and discovered that the caudal fin served not only as a course stabilizer but also as a rudder during yaw motion. [16]uring fish swimming, the active or passive deformation of various flexible fins leads to enhanced performance compared to rigid nondeformable fins. [17]While research on active deformation primarily focuses on auxiliary fins such as pectoral fins [18,19] and dorsal fins, [5,20,21] the study of actively deformable caudal fins remains relatively limited.Liu et al., developed a novel variable area robotic fin to investigate the effect of surface area changes on hydrodynamic forces.The study revealed that the fin with a larger surface during in-strokes achieved a favorable balance between thrust and efficiency. [22]However, the surface area modulation in this study was achieved by creating holes in the caudal fin, which lacked biomimetic characteristics.Another noteworthy deformable robotic fish was developed by Yang et al., capable of transitioning from a crescent shape to a fan shape to modulate surface area and aspect ratio.It was discovered that the oscillating kinematic parameters were intricately coupled with the controlling modes of the fin shape, and the selection of an appropriate combination could improve thrust generation. [23]Nonetheless, this robotic fin did not deform in a manner consistent with live fish.Similar to the caudal fin deformation in robotic fish, deformation of webbed feet has also been explored in the design of robotic frogs. [24,25]Both the deformable caudal fin of the biomimetic robotic fish and the webbed feet of the robotic frog share a similar structure, consisting of fin bones (referred to as phalanges in the case of the frog) covered by a membrane (referred to as web in the case of the frog).The deformation in both cases is achieved by rotating the fin bones or phalanges.This mechanism allows for active changes in the area of the caudal fin or the webbed feet during a single flapping cycle.
The existing literature on the caudal fin has predominantly focused on its shape rather than its deformation, although some insights into the working principles have been gained.In our previous work, we proposed an actively deformable caudal fin and investigated three deformation modes in terms of thrust. [26]uilding upon this research, the present study introduces a novel untethered robotic fish driven by the actively deformable caudal fin.This work makes two key contributions.First, a more comprehensive investigation is conducted, encompassing additional performance metrics such as swimming speed, yaw stability, turning radius, and turning rate.The findings reveal that the instantaneous mode exhibits superior performance in thrust, swimming speed, turning radius, and turning rate, whereas the sine-based mode achieves optimal yaw stability.Second, we present the dynamic model of this actively deformable biomimetic robotic caudal fin, which is validated through experiments.
The structure of this article is organized as follows.Section 2 outlines the design of the untethered robotic fish.Section 3 presents the dynamic modeling of the deformable caudal fin.Section 4 describes the experimental setup and presents results related to thrust, swimming speed, yaw stability, turning radius, and turning rate.Section 5 concludes the article by summarizing the key findings and discussing future directions for research.

The Design of the Untethered Robotic Fish
The detailed design of the novel robotic fish with an actively deformable caudal fin is presented in Figure 1.As shown in Figure 1a, the robotic fish measures 785 in length, 380 in width, and 200 in height.It comprises three main components: a rigid head, a flapping body, and the actively deformable caudal fin.The deformable caudal fin is connected to the rigid head through a connecting plate, a flange, a servo motor, and a soft cover.The design of the deformable caudal fin is inspired by seal swimming.During one flapping cycle, the caudal fin of a robotic fish generates both thrust and drag.To maximize the mean driving force, the caudal fin is designed to increase its area during the time slice that generates thrust, while reducing the area during the time slice that produces drag.
As shown in Figure 1b, the rigid head serves as the base of the robotic fish and features a sealed cabin to provide a dry environment for electronic components, including control circuits, batteries, a communication module, and an inertial measurement module (IMU).Two servo motors drive the pectoral fins, enabling vertical swimming movements such as rising, cruising, or diving.To control the buoyancy, a buoyancy block is incorporated, carefully designed to match the density of the surrounding aquatic environment.
The flapping body consists of a main servo motor (type: DYNAMIXEL XW540-T140-R), a flange, and a soft cover with accordion folds.This configuration facilitates the generation of the flapping motion, contributing to the locomotion of the robotic fish.
The design and functionality of the deformable caudal fin are described in detail in another study. [26]It deforms in the vertical plane of the robotic fish.Notably, this implementation represents the first instance of an actively deformable caudal fin being applied to an untethered biomimetic robotic fish.The deformation of the caudal fin is characterized by three distinct modes: the conventional nondeformable mode, the sine-based mode, and the instantaneous mode.By formulating these three deformation modes, the caudal fin exhibits versatile and adaptable behavior.An auxiliary motor works in the range from 0°to 50°, controlling the caudal fin to change from folded state to unfolded state.In addition, the caudal fin has a lunar shape, 205 in length and 350 mm in width.The design assumes that when the caudal fin is unfolded, the membrane is completely stretched without any wrinkles.However, as the caudal fin starts to fold, the membrane takes on an accordion-like shape.It is crucial to ensure that the height of this accordion shape remains within acceptable limits.As previously mentioned, an excessive height could potentially impact the behavior of the robotic fish.The determination of the accordion shape's height relies on factors such as the number of fin bones and the overall shape of the caudal fin. Figure 1c shows the prototype of the novel biomimetic robotic fish.

Modeling of the Actively Deformable Caudal Fin
In the realm of caudal fin dynamics, two distinct types of motion are observed: body flapping and caudal fin deformation.The former involves the rhythmic oscillation of the fish's body, while the latter refers to the shape deformation of the caudal fin itself.To establish a reference point for analysis, we introduced a fixed coordinate system centered at the main servo motor's axis, depicted in Figure 2a.According to the traveling wave model, [27,28] flapping of the fish follows a sinusoidal-like pattern.Thus, flapping of the deformable caudal fin is given as follows.
where θ is the flapping angle (rotational angle of the main servo motor), ϑ b is the flapping amplitude, f is the flapping frequency, and t represents time.
In the deformation of the caudal fin, rotational angle of the auxiliary servo motor, α, is given as follows.
where ϑ c is the deformation amplitude, f deform is the deformation frequency, and C θ is the phase difference between flapping and deformation in the sine-based mode.Please take note that the deformation of the caudal fin exhibits symmetry with respect to the main axis of the fish's body.Consequently, the frequency of deformation is twice that of the flapping motion, denoted as A schematic is shown in Figure 2b, where l b is the length of the flapping body, l c is the length of the cauda fin, F b is the force exerted on the fixed bracket by the flapping body, r : ⊥ χ ð Þ is the velocity perpendicular to the deformable caudal fin, f b χ ð Þ is hydrodynamic forces exerted on the flapping body, and f c χ ð Þ is hydrodynamic forces exerted on the caudal fin.Please note that the viscous force is neglected in this article.
u and v are the unit vectors parallel and perpendicular to the actively deformable caudal fin, respectively, which can be expressed as The position of a specific point on the caudal fin is where χ is the distance from one specific point on the device to the origin (o).Thus, the velocity and the acceleration are It is found that the direction of r : χ ð Þ is perpendicular to the caudal fin.As a result, r : χ ð Þ is also noted as r According to the quasisteady model theory [29] , hydrodynamic forces exerted on the flapping body (f b χ ð Þ) and the caudal fin where C h is the force coefficient, which can be obtained from another study. [30]It is intricately linked to the shape of the flapping part.Typically, this parameter is determined empirically through experimental investigations.ρ is the water density, S b is the side projection area of the flapping body, which keeps constant, and S c is the area of the caudal fin, which changed in distinct deformation mode.It has been observed that the hydrodynamic force is linearly correlated to the square of the velocity perpendicular to the caudal fin, with their directions being opposite to each other.During the outward flapping phase (beat phase) of the caudal fin, the device generates resistance.In contrast, during the inward flapping phase (restore phase), it produces thrust.To optimize the performance, the area of the deformable caudal fin is controlled to decrease during the beat phase, reducing resistance.In contrast, it is controlled to increase during the restore phase, enhancing thrust generation.The configuration of the caudal fin, as depicted in Figure 2c, exhibits a shape that can be divided into three sections: S 1 , S 2 , and S 3 .It is noteworthy that S 2 remains constant throughout, while S 1 and S 3 can be dynamically altered synchronously using the auxiliary servo motor.a, b, d, and γ are the geometric parameters of the deformable caudal fin.The respective areas of these three sections are as follows Assuming that the gear ratio between auxiliary servo motor and outer fin bones is g, the relation between the rotational angle of auxiliary servo motor (α) and geometric parameter (γ) can be depicted as Therefore, the area of the caudal fin is The force exerted on the fixed bracket by the whole actively deformable caudal fin is where m b is the mass of the flapping body, δ b is the distance between the center of mass of the flapping body to the origin, m c is the mass of the caudal fin, and δ c is the distance between the center of mass of the caudal fin to the origin.The thrust and the lateral force can be expressed as where x and ŷ are unit vectors of x-axis and y-axis in the coordinate system (xoy) in Figure 2b. Figure 3 illustrates a representative simulation example showcasing the thrust behavior.The simulation parameters employed in this study are detailed in Table 1.Notably, the flapping amplitude (ϑ b ) remains constant at 45°, while the flapping frequency ( f ) is varied at three specific values: 0.20, 0.25, and 0.30 Hz.An observation emerges from the results: the conventional nondeformable mode exhibits the most substantial resistance, followed by the sine-based mode and the instantaneous mode, in descending order.Furthermore, the positive thrusts generated by these three modes are comparable.Therefore, a preliminary qualitative analysis suggests that the instantaneous mode may achieve the highest thrust output.

Experiments
In Figure 4a, the experimental setup employed to measure thrust is presented.The actively deformable caudal fin was subjected to testing within a water tank measuring 2 m (Length) Â 1 m (Width) Â 0.6 m (Height).The fin was affixed to a load cell (DYLY-102, DAYSENSOR Co., Ltd.) using custom-made connectors.The load cell captured the thrust generated, which was subsequently processed by a signal conditioner and transmitted to a data acquisition board (PXI-6289).The data acquisition board was integrated into a PXI system, operating at a sample rate of 1000 points per second (pts s À1 ).Additionally, a real-time controller (PXI-8106) was employed to generate two channels of signals: one for controlling the main servo motor and the other for controlling the auxiliary servo motor.The user interface (UI) for the experiment was programmed using LabVIEW.
In Figure 4b, the experimental platform devised to assess swimming speed, yaw stability, turning radius, and turning speed is showcased.The biomimetic robotic fish was allowed to freely swim within a larger water tank measuring 5.5 m (length) Â 2 m (width) Â 1.5 m (height).A high-speed camera was positioned above the tank to record the fish's motion and trajectory.
Due to the constraints imposed by the size of the water tank, higher frequencies resulted in increased noise in the thrust signal.To mitigate the influence of this disturbance, flapping frequencies were chosen at relatively low levels, specifically 0.2, 0.25, and 0.3 Hz.In contrast to thrust-related experiments, other investigations involving swimming speed, yaw stability, and turning motion were conducted with the biomimetic robotic fish freely swimming, thereby suffering less impact of water waves.Consequently, for these experiments, higher flapping frequencies were selected (0.25, 0.50, and 0.75 Hz) to facilitate differentiation and analysis of performance differences.Please note that the pectoral fins were fixed throughout the whole following experiments.experimental and simulation results, particularly for lower flapping frequencies.However, as the flapping frequency increases, a deviation between the experimental and simulation data becomes apparent.Additionally, the experimental thrust measurements exhibit more noise as the flapping frequency increases.This behavior can be attributed to the heightened water disturbance experienced by the actively deformable caudal fin at higher locomotion speeds.Furthermore, it is worth noting that the resistance generated by the instantaneous mode is found to be the smallest, whereas the positive thrust of the three deformation modes is comparable.This trend aligns with the observations made in the simulation example (Figure 3).In conclusion, the analysis indicates that the instantaneous mode yields the highest thrust output, followed by the sine-based mode, while the conventional nondeformable mode exhibits the lowest thrust performance.

Test of Swimming Speed
The swimming speed experimental results are depicted in Figure 6, where flapping amplitudes ranging from 10°to 40°a nd flapping frequencies of 0.25, 0.50, and 0.75 Hz are tested.The swimming speed is determined based on the trajectory captured by an overhead camera positioned above the experimental water tank.A distance of 2 m is recorded during stable locomotion.To ensure reliability, each test is repeated five times.The findings clearly demonstrate that the robotic fish equipped with an actively deformable caudal fin exhibits superior swimming speed compared to its nondeformable counterpart.Furthermore, the impact of caudal fin deformation becomes increasingly significant as both the flapping amplitude and flapping frequency increase.
To further evaluate the swimming speed performance, Table 2 presents a comparison between the sine-based mode and the conventional nondeformable mode, while Table 3 focuses on the comparison between the instantaneous mode and the conventional nondeformable mode.Table 2 showcases instances where the swimming speed is either increased or decreased with caudal fin deformation, illustrating the potential of caudal fin deformation to enhance or hinder swimming speed depending on the specific parameters.In contrast, Table 3 exclusively displays positive values, indicating that the instantaneous mode consistently enhances swimming speed across all parameter combinations.Notably, the instantaneous mode outperforms the sine-based mode for every specific parameter set examined.Overall, the analysis reveals that the instantaneous mode yields the highest swimming speed, followed by the sine-based mode, while the conventional nondeformable mode exhibits the slowest swimming speed.This trend aligns with the thrust performance of these three modes, as previously discussed.

Test of Yaw Stability
Figure 7 presents the yaw stability for different flapping amplitudes (10°, 20°, 30°, 40°) and flapping frequencies (0.25, 0.5, 0.75 Hz).In this study, yaw stability refers to the head-shaking of the robotic fish during cruising, which is quantified as the yaw angle.A smaller yaw angle indicates reduced head shaking and better yaw stability.The yaw angle of the robotic fish is obtained using an onboard IMU.Only the data recorded during stable locomotion is considered for analysis.To ensure accuracy and reliability, each condition is repeated five times.As depicted in Figure 7, at a flapping frequency of 0.25 Hz, no significant distinctions are observed among the three flapping modes in terms of yaw angle.However, when the flapping frequency increases to 0.75 Hz, the sine-based mode generally exhibits the smallest yaw angle, suggesting that the sine-based mode possesses the highest yaw stability.

Test of Turning Motion
Maneuverability plays a crucial role when fish navigate through confined spaces or evade predators.Among the key metrics that measure maneuverability, turning radius and turning rate stand out as significant indices.This particular set of tests aims to assess the impact of different deformation modes on the turning radius and turning rate.The rotational angles of the servo motors, specifically the main servo motor (represented by the red dashed line) and the auxiliary servo motor (represented by the blue solid line), are illustrated in Figure 8.It should be noted that a 0°rotational angle for the main servo motor corresponds to the flapping body resting at the midline of the fish's body.Similarly, a 0°rotational angle for the auxiliary servo motor indicates the fully folded configuration of the actively deformable caudal fin, whereas a 50°rotational angle signifies the unfolded state of the caudal fin.
The turning radius and turning rate are determined using the trajectory captured by the overhead camera positioned above the experimental water tank, similar to the method used for calculating swimming speed.By fitting the turning trajectory using a circle, the turning radius can be obtained.The turning rate is computed as the average rate within one complete revolution.
Figure 9 showcases the swimming trajectories of the untethered robotic fish, with a flapping frequency of 0.75 Hz and a flapping amplitude of 40°.Each red point represents the geometric center point of the robotic fish at a specific time stamp.The time interval between two adjacent red dots is 1 s.In Figure 10, a comparison of the turning radii is presented.The conventional nondeformable mode exhibits the largest turning radius, measuring 632 mm.This is followed by the sine-based mode with a turning radius of 405 mm and the instantaneous mode with a turning radius 358 mm.In other words, the sine-based mode demonstrates a 36% reduction in radius compared to the conventional nondeformable mode, while the instantaneous a 43.4% reduction in radius.The average turning rate of one circle can be determined by analyzing the data from the onboard IMU.The results reveal that the average turning rates for the conventional nondeformable mode, the sine-based mode, and the instantaneous mode are 18°s À1 , 20.5°s À1 , and 23.2°s À1 , respectively.
Table 4 presents a comprehensive comparison of various performance metrics, including thrust, swimming speed, yaw stability, turning radius, and turning rate, among the conventional nondeformable mode, the sine-based mode, and the instantaneous mode.It is evident that introducing caudal fin deformation significantly enhances performance across all listed aspects.Specifically, the instantaneous mode outperforms the other modes in terms of thrust, swimming speed, turning radius, and turning rate, while the sine-based mode exhibits the best yaw stability.

Conclusion
This article presents a comprehensive investigation of a novel untethered biomimetic robotic fish equipped with an actively deformable caudal fin.The design and modeling of both the robotic fish and the actively deformable caudal fin are elaborated upon.Extensive experimental analyses are conducted to evaluate key performance parameters such as thrust, swimming speed, yaw stability, turning radius, and turning rate.The findings demonstrate a significant enhancement in swimming performance when compared to the conventional nondeformable mode, thanks to the incorporation of active deformation in the caudal fin.Specifically, the instantaneous mode exhibits superior performance in terms of thrust, swimming speed, turning radius, and turning rate, while the sine-based mode displays the best yaw stability.
In future research, advanced control technologies will be applied to further enhance the performance of this untethered biomimetic robotic fish and its actively deformable caudal fin.Additionally, computational fluid dynamics (CFD) simulations will be employed to investigate the underlying solid-liquid coupling mechanisms in greater detail.

Figure 1 .
Figure 1.Design of the robotic fish.a) Computer aided design (CAD) model illustrating the detailed structure of the robotic fish.b) Internal configuration showcasing the arrangement of components within the robotic fish.c) Physical prototype.

Figure 2 .
Figure 2. Schematic representation illustrating the motion of the actively deformable caudal fin.a) Depiction of the deformable caudal fin motions.b) Planar motion diagram.c) Caudal fin shape.

Figure 5 Figure 3 .
Figure 5 showcases the thrust performance for various flapping frequencies ( f ) at a fixed flapping amplitude (ϑ b ) of 45°.The experimental results are represented by the blue solid line, while the simulation results are depicted by the red solid line.Generally, a good agreement is observed between the

Figure 4 .
Figure 4.The experimental platforms.a) Thrust measurement platform.b) Platforms for swimming speed, yaw stability, turning radius, and turning speed testing.

Table 1 .
Parameters for simulation.

Table 2 .
Comparison of swimming speed between the sine-based mode and the conventional nondeformable mode.

Table 3 .
Comparison of swimming speed between the instantaneous mode and the conventional nondeformable mode.

Table 4 .
Summary of the thrust, swimming speed, yaw stability, turning radius, and turning rate under three deformation modes.