Optimum Curvature Characteristics of Body/Caudal Fin Locomotion

Abstract

Fish propelled by body and/or caudal fin (BCF) locomotion can achieve high-efficiency and high-speed swimming performance, by changing their body motion to interact with external fluids. This flexural body motion can be prescribed through its curvature profile. This work indicates that when the fish swims with high efficiency, the curvature amplitude reaches a maximum at the caudal peduncle. In the case of high-speed swimming, the curvature amplitude shows three maxima on the entire body length. It is also demonstrated that, when the Reynolds number is in the range of 10⁴–10⁶, the swimming speed, stride length, and Cost of Transport (COT) are all positively correlated with the tail-beat frequency. A sensitivity analysis of curvature amplitude explains which locations change the most when the fish switches from the high-efficiency swimming mode to the high-speed swimming mode. The comparison among three kinds of BCF fish shows that the optimal swimming performance of thunniform fish is almost the same as that of carangiform fish, while it is better not to neglect the reaction force acting on an anguilliform fish. This study provides a reference for curvature control of bionic fish in a future time.

1. Introduction

After hundreds of millions of years of evolution, fish show superiority in high-efficiency and high-speed motion, which has attracted the attention of many researchers. The study on bionic fish can lead to a new design of Automatic Underwater Vehicles (AUV). In the nature, most fishes use the body and/or caudal fin (BCF) propulsion to generate thrust. These fishes are distinguished into three categories: anguilliform mode, carangiform mode, and thunniform mode (Figure 1). The paramount differences between the three modes are the amplitude envelope of the propulsive wave and the wavelength [1,2].

A fish uses both the active and passive mechanisms while swimming [3]. Passive swimming reacts to external flow with no muscle activation [4]. In contrast, active swimming implies swimming generated through muscular activation [5]. BCF fish can autonomously control their curvature profile to achieve the compliant body motion and realize active/passive swimming [6,7,8,9,10,11]. Hence, the research on curvature profile is important to understand the fish locomotion mechanism. The curvature distribution is closely related to the materials and shape of the fish body, including soft tissue (tendons, muscle, and ligaments) and hard parts (fin rays and vertebral column). From the biological perspective, the curvature is controlled mostly by internal muscular activation. Specifically, fish can receive the stretch receptors that depend on body curvature to the central nervous system and then drive the calcium dynamics to affect the muscle activity, which can further modulate the swimming behavior [12,13]. Many in vivo experiments and electromyogram analyses have been conducted to discover the neuro–musculo–mechanical model, and much progress on the model has been made [14,15,16,17,18]. Williams and Bowtell developed a simple dynamic model to describe the interaction between body curvature and muscle activation [19,20]. McMillen et al. suggested that fish neuromuscular systems produce an intrinsic preferred curvature that can be derived from in vivo experiments [21,22]. Considering the complexity and the difficulty of the comprehensive model, some researchers turn to study the curvature profile directly instead of the connection to muscular activation. Van Rees et al. and Eloy combined different hydrodynamic models and bi-objective optimization algorithms to obtain the optimal curvature profiles and body shapes for undulatory swimming [23,24]. Such numerical optimization can help overcome some limiting constraints in reality. Nevertheless, these optimized body shapes show a slightly unsmooth or strange geometry when compared to real fish. One of the possible reasons is that the fish in nature faces many constraints in the process of evolution. In this paper, empirical body shapes obtained by biological data of BCF fish are used to avoid this problem and the corresponding analysis of optimal curvature profiles can provide a more reasonable value of information.

Certain techniques have been used to achieve variable curvature structures [3,25,26,27,28,29]. A common method is to tune the pressure of the soft pneumatic actuator. White et al. changed the local curvature by developing a Tunabot with different degrees of freedom [28,29]. In addition, some accurate curvature control methods have been proposed and applied in the field of soft robotics [30,31,32]. The developments in structure fabrication and the control methods, in turn, demand the further investigation of optimal curvature distribution of BCF fish. Therefore, swimming performance optimization through curvature control will become the focus of future related research on bionic robotic fish. The premise of curvature control is to understand the impact of varying the curvature distribution on swimming performance, which is discussed in this study.

This paper is organized as follows: Section 2 describes the optimization problem of swimming performance. Section 3 presents the optimization results including the swimming motions, curvature profiles, the frequency effect, the change mechanism of curvature amplitude, and comparison among BCF fishes. Section 4 discusses the similarities and differences between our optimization and other optimizations. Finally, conclusions are summarized in Section 5.

2. Materials and Methods

2.1. Optimization Problem Definition

Here we take the carangiform fish as an example for optimization. The optimization of the other two kinds of BCF fish will be conducted in Section 3.4. The shape of carangiform fish is determined by the empirical equations in [33] (see Figure 2). The views are set out in the global coordinate system OXYZ where Y-axis is horizontal and Z-axis is vertical. The X-axis is determined by the right-handed rule. The swimming direction is along with the negative Y-axis.

The midline kinematics can be described by a curvature profile κ(s,t), defined as an amplitude function K(s) multiplied by a traveling wave [23], as:

$$\kappa (s,t)=K(s)\cos [2\pi (\frac{t}{T}-\tau (s)\frac{s}{L})]$$

(1)

where t is the time, T = 1 s is the tail-beat cycle, and τ(s) determines the wavelength. Both K(s) and τ(s) are functions fitted at n = 11 interpolation points that are evenly distributed on the midline. Considering the boundary conditions, K(0) = K(L) = τ(0) = 0, the kinematic optimization problem includes 19 degrees of freedom in total. According to the biological data gathered by Zuo Cui, the range of τ(s) is set as 0.5–2 [34].

If a specific curvature profile is given, the swimming kinematics can be determined according to the reduced-order dynamic model developed by Eloy [24]. The general introduction of this model is as follows (shown in Figure 3). Once the initial curvature profile is set, the angle θ between the tangent to the midline and horizontal axis is obtained based on the mathematical definition of curvature. The position of any point r = (x, y) on the midline is then derived by employing the integral to a trigonometric function of θ. As a result, the velocity components on the tangential and normal direction at any point of the midline, namely u and v, can be calculated through the differential to the position function and the projection analysis.

The forces acting on the fish body that include reaction force F_m, skin-friction drag F_//, form drag F_form, and resistive force F_⊥, can be expressed as the functions of the velocity components. Their empirical formulas are shown in Equations (2)–(5), respectively. The reaction forces are derived by the large-amplitude elongated-body theory [35]. The description of skin-friction drag and resistive force is obtained by the hydrodynamics analysis of Taylor [36]. The empirical formula of form drag caused by the non-streamlined body is concluded by Hoerner [37].

$$\mathrm{d}{F}_{\mathrm{m}}=(\frac{\partial (muvn-0.5m{v}^{2}t)}{\partial s}-\frac{\partial mvn}{\partial t})\mathrm{d}s$$

(2)

$$\mathrm{d}{F}_{//}=-2.9\rho \sqrt{\left|\nu av\right|}u\mathrm{d}s t$$

(3)

$$F_{\mathrm{form}}=0.33\rho U_0{}^2\times \frac{\max (b)}{L}S e_Y$$

(4)

$$\mathrm{d}{F}_{\perp }=-C_{D}a\rho v\left|v\right|\mathrm{d}s n$$

(5)

where t and n are the unit vectors of tangential and normal directions of midlines, respectively. $\rho$ = 1000 Kg/m³ is the density of both fish and fluid. m = $\rho$ a² is the added mass of each section. ν = 10⁻⁶ m²/s is the kinematic viscosity of the fluid. C_D = 2 − b/a is the approximation of the resistive force coefficient. U₀ is the mean tangential velocity at the fish head. S = 0.0043 is a number that describes the streamlining of the fish body.

The constants of integration that emerged during the process, such as θ₀, x₀, and y₀, can be solved by the conservation of momentum and angular momentum (Equations (6) and (7)). Thus, the undulatory behavior is fully understood.

$$F_{\mathrm{form}}+{\displaystyle {\int }_0^L\mathrm{d}{F}_{//}+\mathrm{d}{F}_{\mathrm{m}}+\mathrm{d}{F}_{\perp }}-M\frac{{\mathrm{d}}^{2}r}{\mathrm{d}{t}^2}\mathrm{d}s=0$$

(6)

$${\displaystyle {\int }_0^{L}I\frac{{\mathrm{d}}^2\theta }{\mathrm{d}{t}^2}{e}_{\mathrm{z}}\mathrm{d}s+(\mathrm{d}{F}_{//}+\mathrm{d}{F}_{\mathrm{m}}+\mathrm{d}{F}_{\perp }}-M\frac{{\mathrm{d}}^{2}r}{\mathrm{d}{t}^2})\times r\mathrm{d}s=0$$

(7)

where M = $\rho$πab is the fish mass of each section. I = 0.25$\rho$πab³ is the moment of inertia of each section.

The indices used in the simulation to measure the swimming performance are shown in Equations (8) and (9). Consequently, each curvature profile has its corresponding swimming performance. The bi-objective optimization was adopted for obtaining the maximum relative swimming speed U* and the minimum Cost of Transport (COT), deriving the optimal curvature profile.

$$U^{*}=U/L$$

(8)

$$\mathrm{COT}=\frac{P_{\mathrm{tm}}}{M_{\mathrm{tot}}gU}=\frac{P_{\mathrm{s}}+P_{\mathrm{musc}}}{M_{\mathrm{tot}}gU}=\frac{0.1327M_{\mathrm{tot}}{}^{0.8}+5P_{\mathrm{tot}}}{M_{\mathrm{tot}}gU}$$

(9)

where g is the acceleration of gravity and M_tot is the total fish mass. The total metabolic power P_tm is the sum of standard metabolic rate P_s and the metabolic power P_musc consumed by the swimming muscles [38,39]. The unit of U* is Body Length/s, namely, BL/s. COT quantifies the total energy consumption per unit mass and distance [40]. The higher the value of COT, the lower the efficiency. P_tot serves as the total power required for swimming, which is described in Equation (10).

$$P_{\mathrm{tot}}=P_{\mathrm{m}}+P_{\perp }+P_{//}+P_{\mathrm{form}}+P_{\mathrm{i}}$$

(10)

where P_m, P_∥, P_form, and P_⊥ are the powers consumed by reaction force, skin-friction drag, form drag, and resistive force, respectively. These powers are easy to calculate by multiplying the forces and the corresponding velocities. P_i is the internal dissipation power (see Equation (11)).

$$P_{\mathrm{i}}=\langle {\displaystyle {\int }_0^L\mu I{(\frac{{\mathrm{d}}^2\theta }{\mathrm{d}s\mathrm{d}t})}^2\mathrm{d}s}\rangle$$

(11)

where μ = 1000 Pa s is the internal viscosity of fish and the brackets mean time averaging.

The optimization problem is concluded here. Given the shape of carangiform fish, (K₁, K₂, …, K_n−1, τ₁, τ₂, …, τ_n) were chosen as variables and the two objective functions were the maximum U* and the minimum COT.

In the process of parameter settings, if the K_max value is too high, the search space will become wider and the simulation time will be longer, whereas, if the K_max value is too low, true optimization results may not be obtained. Therefore, it is necessary to set the appropriate value of K_max. Gazzola et al., Van Rees et al. and Eloy set this value to 2π/L, 3π/L, 10/L, respectively [23,24,41]. In this paper, the K_max value was initially set to 10π/L (a number high enough to produce simulation results), while the maximum values of K corresponding to different lengths could be thus obtained once optimization was completed. The simulation showed that K_max was approximately inversely proportional to L. The inverse proportional function was fitted as K_max = 8.4/L. This restriction can be relaxed in some cases, which are later discussed in Section 3.3.

2.2. Optimization Algorithm

The NSGA-II algorithm for bi-objective optimization was used in this paper. NSGA-II is nowadays one of the most popular and efficient multi-objective algorithms [42,43]. Before introducing NSGA-II, there are two important concepts to understand: dominance and Pareto front. For two individuals G1 and G2, if U*(G1) > U*(G2), COT(G1) ≤ COT(G2) or U*(G1) ≥ U*(G2), COT(G1) < COT(G2), G1 is said to dominate G2. Therefore, the set of all non-dominated points is called the Pareto front. Like genetic algorithms, the NSGA-II procedure is roughly as follows (Figure 4). First, an offspring population was generated from the parent population through crossover and mutation. The probability for crossover was 0.8; next, offspring and parent population were combined into a mixed population and sorted according to non-dominated sorting (the conception of crowding distance was not used in our simulation); following, the set of all non-dominated points was selected for the next parent generation; finally, the above steps were repeated until the termination condition was met. The initial size and the maximum size of the population were set as 20 and 2000, respectively. The detailed algorithm implementation was coded using the open-source software Python.

3. Results

3.1. Reference Case

The case of the tail-beat frequency f = 1 Hz was taken as a reference case. Setting φ(s) = −2π·τ(s)·s/L, φ(s) was the phase of the curvature profile. For the swimming forms of minimum COT and maximum swimming speed in the reference case, the distributions of τ and φ along the body length are both depicted in Figure 5. To validate the effectiveness of the simulation results, a single-objective optimization algorithm, covariance matrix adaptation evolution strategy (CMA-ES), was used to obtain the distributions of τ for the above-mentioned forms. It can be seen from Figure 5 that the numerical solutions based on CMA-ES were in good agreement with these based on NSGA-II. Furthermore, the distributions of τ and φ were similar for the two cases. τ increased sharply at first, followed by a decrease, but then again increased in the last fifth of body length. The phase distribution presented a step shape in general. As for the disagreement between algorithms around three-tenths of the body length for the minimum COT case (Figure 5a), it can be explained as follows. The curvature amplitude in this location was near zero for the minimum COT case (shown in Figure 6). Therefore, according to Equation (1), the change in τ value in this location had little effect on swimming performance. In spite of the disagreement of τ value in this location, the swimming performance obtained by the two algorithms was almost equal.

Six typical swimming forms A-F were extracted from the reference case (the speed range was divided into fifths). Figure 6 presents the amplitude envelopes of these curvature profiles. When U* was small (Case A in Figure 6), the curvature amplitude was near zero along with the first 2/3 of the body length, while its peak value appeared at the caudal peduncle. When U* reached the maximum (Case F in Figure 6), there were three local peaks of curvature amplitude in the whole body length, which were at the head, at the 2/3 of the body length, and at the caudal peduncle, respectively. It indicates that the undulatory motion mainly concentrates in the last third of the body length to achieve minimum COT, and the large undulation also exists in the head part to reach maximum speed. In nature, the fish head needs to keep rigid to protect the brain tissue, so the curvature of the head part cannot reach the simulation value of the swimming form F. It can therefore be assumed that the evolution of natural fish inclines towards the direction of maximizing swimming efficiency. However, this does not mean that the simulation result of F is meaningless. If different kinds of soft materials are applied in the fabrication of biomimetic fish to realize the curvature distribution of F, the simulated maximum speed can also be achieved in reality.

3.2. Frequency Effect

Yue and Tokić showed that when the fish mass is about 0.5 kg (the fish mass in this paper was 0.48 kg), the range of tail-beat frequency is 1.25–4.5 Hz [44]. Therefore, optimization results of U* and COT at various tail-beat frequencies within the range of 1 Hz–4 Hz were obtained (Figure 7). Figure 7 and Figure 8 demonstrate that the maximum U* and the maximum COT increased linearly with frequency while the minimum U* and the minimum COT remained constant. Real fish swimming data observed by Videler and Hess [7] indicate that the swimming speed is proportional to frequency when the frequency is in the range of 1 Hz–14 Hz, also supporting the above result. Another evaluation index often used in other references is introduced here: the stride length U’ = U*/f [7,24]. The stride length is a dimensionless number that is the number of fish lengths travelled during one tail-beat period. In our simulation, the maximum U’ remained stable at around 1.42, and the minimum U’ decreased from 0.84 to 0.245, with the increasing frequency.

Swimming kinematics with different frequencies are shown in Figure 9. In the case of minimum COT, the amplitude envelopes decreased with frequency. Especially, at 3 Hz–4 Hz, fish relied solely on the tail to generate thrust. However, at the maximum U*, the amplitude envelopes remained the same.

There are other ways to study the frequency effect on swimming performance. For example, in the robotic fish swimming experiment presented in [45], the fishtail is driven by the steering gear. Liu et al. kept the swing angle amplitude of steering gear constant, meaning swimming kinematics do not change. Only the frequency is changed to investigate the swimming performance. Similar to this method, this paper studied the relationships of U*, U’, A*, and COT with frequency while keeping the swing angle of the fishtail (θ_tail) constant. θ_tail can be derived based on the different undulatory locomotion of the Pareto fronts. The premise is that these parameters should have a one-to-one mapping with θ_tail. The relationships of U*, U’, A*, and COT with θ_tail were checked, and the one-to-one mapping was generally satisfied.

Figure 10 displays the trends of parameters U*, U’, A*, and COT vs. frequency when θ_tail was 25° and 35°, respectively. U* and COT values showed a sharp increase along with the rise in frequency while U’ and A* values exhibited a lighter increase. This shows that the swimming speed, stride length, and COT are all positively correlated with frequency, which agrees well with the trend of these metrics demonstrated in the experiments of White et al. [29].

Nevertheless, the above simulation result does not mean that the swimming speed increases across the frequency spectrum indefinitely. It is worth noting that the proposed optimization model is suitable for Re within the range of 10⁴ to 10⁶. If Re lies beyond this range, the calculation formula of forces will change, the model will no longer be valid and the relationship between swimming speed and frequency will also change.

3.3. Curvature Amplitude

Figure 11 indicates that as the fish length increased from 0.3 m to 0.7 m, the minimum COT decreased, so the swimming efficiency increased, which is consistent with the optimization results presented in [44]. Two aspects give reason to this occurrence. First, as K_max decreased with length, the lateral amplitude at the fish head decreased while that at the tail showed a small increase. Moreover, the slope of the amplitude envelope of swimming kinematics became smoother when the length increased (Figure 12). These all led to a reduction in both the recoil motion and the internal diffusion. Furthermore, though U* decreased with length, the actual swimming speed increased with length, leading to an increase in Re. The drag coefficient C_d~Re^−0.5 also decreased with length [44].

Next, the sensitivity analysis of K(s) was conducted, namely, the impact of varying a single feature of K(s) on swimming performance was studied. As displayed in Figure 6, from the swimming form of high-efficiency A to the swimming form of high-speed F, the values of K at the second, eighth, and ninth interpolation point (represented as K₂, K₈, K₉, respectively) increased at to different extent. On the other hand, the values of K at five interpolation points, from the third one to the seventh one, increased synchronously roughly. Thus, the joint sensitivity analysis of these K values (represented as K_3–7) was conducted. To study the influence of K_max on swimming performance, the K value at the tenth interpolation point (represented as K₁₀) was also included in the discussion. The variations of the aforementioned K values on the swimming performance are shown in Figure 13. For convenience, the curvature amplitudes in the case of minimum COT (Case 0) were a reference for other cases.

K₁₀ was set to 12, 16, 21, 24, 28, and 32 from case −3 to case 2, respectively. K₁₀ appeared to have little effect on the swimming performance, as illustrated in Figure 13a. As K₁₀ increased, both speed and efficiency slightly increased at first and then decreased a little, implying that there must exist some K₁₀ value that maximizes the speed and efficiency. Therefore, although the curvature profile changed from A to F in Figure 6, the value of K₁₀ kept constant. Since the value of K₁₀ affects the swimming performance a little, the restriction of K_max can be relaxed appropriately.

K₉ was set to 5, 8, 10.7, 13, and 16 from case −2 to case 2, respectively (Figure 13b). K₉ had a great influence on swimming speed. Higher K₉ caused a dramatic improvement (up to 50%) in speed and a small change (up to 10%) in efficiency. In the bi-objective optimization process, case 0, case 1, and case 2 were all non-inferior solutions while both case -2 and case -1 were inferior solutions. Consequently, the minimum K₉ illustrated in Figure 6 was the K₉ value in case 0.

K₈ was set to 2, 4.1, 10, 15, and 20 from case −1 to case 3, respectively (Figure 13c). K₈ had a great influence on both speed and efficiency. Higher K₈ value resulted in a significant increase in speed and COT, by 62% and 68%, respectively. This change in COT meant that the swimming efficiency was significantly reduced. Therefore, the value of K₈ constantly increased from A to F in Figure 6. The second-highest K value, K₈, thus became the most important factor in the selection of K_max. The minimum K_max value must be higher than that of K₈. Considering that the maximum K₈ value in Figure 6 was less than 20, K_max could be relaxed to 6/L.

K_3–7 (the average value of the sequence from K₃ to K₇) was set to 0, 0.6, 1.4, 2.2, and 3 from case −1 to case 3, respectively (Figure 13d). The increase in K_3–7 had a similar effect to the increase in K₈, leading to a marked rise in speed and COT, by 30% and 26%, respectively. Since the increase in K_3–7 was smaller than that of K₈, the swimming performance was more sensitive to K_3–7.

K₂ was set to 0, 0.85, 5, 10, 15, and 20 from case −1 to case 4, respectively (Figure 13e). K₂ had little effect on speed. It meant that although K₂ increased greatly from A to F in Figure 6, it had little effect on improving the swimming speed. COT increased by 15% along with K₂, indicating that the swimming efficiency decreased moderately.

3.4. Comparison among BCF Fish

Two other kinds of fishes, anguilliform fish, and thunniform fish, are discussed in this part. The shapes of anguilliform fish and thunniform fish are described in [46] and [33], respectively. Based on the above two body shapes, optimization results of U* and COT are plotted in Figure 14. Interestingly, for the same U*, carangiform fish was more efficient, although thunniform fish is more streamlined in a general sense. This reveals that swimming performance optimization is a comprehensive balance of the body shape, kinematics, dynamics, not just some aspect among them. For the same COT, carangiform fish and thunniform fish were faster than anguilliform fish. The reason is that the caudal fin propulsion adopted by the former two can provide a higher reaction force than the body propulsion used by anguilliform fish.

Portions of different powers included in the total power are plotted in Figure 15, where P_D = P_// + P_form is the power consumed by the total drag force. Not surprisingly, the change in power ratios of carangiform fish with the swimming speed was in agreement with that of thunniform fish. The greatest difference between anguilliform fish and the other two fishes was that it had a larger P_⊥ ratio and a small P_m ratio. This indicates that anguilliform fish swimming is mainly a body propulsion mode while carangiform fish and thunniform fish combine body propulsion and caudal fin propulsion. As U* increased, all the elements of P_D decreased, while all the elements of P_⊥ increased. The main cause was that higher values of U* enhanced the lateral motion amplitude of the fish body, bringing about the increase in P_⊥. Moreover, with the increase in U*, P_i ratios in thunniform fish and carangiform fish increase rapidly, reaching 66% and 50%, respectively. The P_m ratio in anguilliform fish was about 8%, half that of the ratio in the other two fish types. The reaction force F_m at the tail is often neglected in some related studies of anguilliform fish swimming [17,18,47,48]. Boyer et al. used Poincaré–Cosserat equations to solve the problem of anguilliform fish swimming [48]. Compared to CFD simulation results, the force in the swimming direction is quite different. The omittance of the reaction force may be a factor attributing to this discrepancy.

4. Discussion

Triantafyllou et al. discovered that the St (St = fA/U, f is the tail-beat frequency, A is the peak-to-peak amplitude of the tail) range of swimming animals with foil-like tails is 0.2–0.4 [49]. In our simulation, the St range of carangiform fish was 0.19–0.30 and that of thunniform fish was 0.255–0.360, both of which can be identified for efficient propulsion. However, the St range of anguilliform fish was 0.42–0.5 and was beyond the St range proposed by Triantafyllou et al. On the other hand, Long et al. determined that St equals 0.56 based on the observation data of hagfish swimming and the range calculated by Kern et al. through computational fluid dynamics is 0.59–0.67 [9,50]. Thus, it is considered that the optimal St range, proposed by Triantafyllou et al., may not apply to anguilliform fish swimming.

Moreover, Wiens and Hosoi found that optimally efficient swimming kinematics could be characterized by a non-dimensional variable Ψ (Equations (12) and (13)) [51]. When Ψ is between 0.3 and 1.0, the swimming efficiency is near-optimal. The Ψ range of carangiform fish in this simulation was 0.44–0.94. These analyses reflect the rationality of simulation.

$$\Psi =1-\frac{\sin (\beta )-\pi St\cos (\beta )}{\beta -\pi St}$$

(12)

$$\beta =\pi \frac{U}{V}St$$

(13)

where V is the wave propagation speed. U/V is the slip ratio that has been widely used and detailed calculation can be found in previous studies [51,52].

The sensitivity analysis of curvature amplitude showed that K₉, K₈, and K_3–7 were the most significant variables, which can instruct the stiffness adjustment strategy for robotic fish. On the other hand, it can be seen that K₂ should always be near zero if the gentle effect on speed is ignored. In other words, there is no need to improve the speed by only one percent at the expense of keeping K₂ so large. Following this way, the large curvature amplitudes mainly existed in the last third of the fish body, which conforms to the biological observation in reality. Furthermore, this discovery also reduced the difficulty of the stiffness distribution implementation.

The three differences between our model and Eloy’s model are as follows. First, the frequency was set constant in our model, which can help investigate the frequency effect; second, the fish body shape was determined by the empirical equations. Strange shapes (like Figure 5e,f in [24]) can be avoided through this way; third, COT, the commonly used measure in other references, was chosen as the efficiency measure instead of a novel measure E* first adopted by Eloy. The discussion on the differences between the two efficiency measures is not the scope of this paper. The phenomenon of bifurcation in the Pareto front that ever emerged in Eloy’s model did not occur in our model. The main cause may be the biological fish shape adopted in our model.

In the case of the highest efficiency, the swimming speed and the midline kinematics obtained by our model are similar to those obtained by Eloy’s model (the relative swimming speed in Eloy’s model is 0.727 and the swimming kinematics are illustrated in Figure 16). The similarity shows that the fish body shape used in our model is near the desired body shape for the most efficient swimming. The midline kinematics of Saithe was also used to make a comparison with the results of these two models (the relevant data is extracted from [53]). The corresponding amplitude envelopes are depicted in Figure 17. There was no big difference between the three cases. The differences in the head and tail amplitudes may result from the fish body shape. It can be concluded that for the most efficient swimming mode, the results obtained from models and the data observed in reality reached an agreement, and the consistency was not only reflected in body shape but also in swimming kinematics.

5. Conclusions

Optimal curvature profiles were obtained by combining bi-objective optimization and the reduced dynamic model proposed by Eloy [24]. As the swimming mode of the carangiform fish transitioned from high-efficiency mode to high-speed mode, the number of local maxima of the curvature amplitude changed from one to three along with the body length.

The frequency effect on swimming performance was examined. The maximum U* and the maximum COT increased linearly with the tail-beat frequency, while the minimum U* and the minimum COT remained constant. On the other hand, when swimming kinematics remained unchanged, swimming speed, stride length, and COT were all positively correlated with the tail-beat frequency.

Based on the sensitivity analysis, the changing mechanism of curvature amplitude regarding swimming speed was revealed. The curvature amplitude at nine-tenths of body length (K₁₀) had little effect on both efficiency and speed. The curvature amplitude at four-fifths of body length (K₉) had a great influence on speed but had little effect on efficiency. The curvature amplitude at seven-tenths of body length (K₈) and the curvature amplitude between fifths and three-fifths of body length (K_3–7) both had a great impact on speed and efficiency. The curvature amplitude at tenths of body length (K₂) had little effect on speed, although it increased from high-efficiency mode to high-speed mode. In addition, it was found that the swimming efficiency increased as the fish body was longer.

Optimization in the cases of the other two kinds of BCF fishes (anguilliform fish and thunniform fish) indicated that anguilliform fish mainly adopts body propulsion, while carangiform fish and thunniform fish combine body propulsion and caudal fin propulsion. Portions of different powers included in the total power were analyzed. It was indicated that the lateral motion of anguilliform fish is large and the reaction force of anguilliform fish is low. However, considering that the proportion of the power caused by the reaction force (P_m) of anguilliform fish accounted for about 8%, it is better not to neglect it in the accurate swimming analysis of anguilliform fish.

The above discussion shows that if real-time and accurate curvature control can be achieved, bionic robot fish has the potential to surpass the swimming performance of real fish in terms of swimming efficiency and speed. This study lays a theoretical foundation for the design and control of high-speed and efficient AUVs.