Speed and Energy Efficiency of a Fish Robot Featuring Exponential Patterns of Control

Abstract

Fish in nature have evolved more efficient swimming capabilities compared to that of propeller-driven autonomous underwater vehicles. Motivated by such knowledge, we discuss a bionic (bio-memetic) autonomous underwater vehicle—a fish robot—that mimics the swimming of rainbow trout (Oncorhynchus mykiss) in nature. The robot consists of three (anterior, posterior, and tail) segments, connected via two (anterior and posterior) actuated hinge joints. We divided the half-period of undulation of the robot into two phases—thrusting and braking. In addition, we hypothesized that an asymmetric duration—a short period of thrusting and a long period of braking—implemented as an exponential (rather than “canonical”, sinusoidal) control would favorably affect the net propulsion of these two phases. The experimental results verified that, compared to sinusoidal undulation, the proposed exponential control results in increased speed of the robot between 1.1 to 4 times in the range of frequencies of undulation between 0.4 Hz and 2 Hz, and improved energy efficiency from 1.1 to 3.6 times in the same frequency range.

1. Introduction

Fish in nature are known to have evolved an undulating swimming propulsion mechanism that is about two times more energy-efficient than that of autonomous underwater vehicles (AUVs) driven by screw propellers [1,2,3]. This inspired our research on AUVs that mimic the natural undulating propulsion of fish. Such bionic (biomimetic) AUVs (BAUVs) can potentially reach the same level of performance as their natural biological counterparts (e.g., fish) [4,5,6,7,8,9]. Moreover, BAUVs offer additional, significant advantages—that can favorably impact their applicability in eco-friendly monitoring, marine exploration, and the management of underwater living systems—such as weak wake, low noise, and lack of cavitation. However, optimization—at the level of control [4,10] or morphology [11,12,13] (or both)—of BAUVs might be needed before they reach the same level of performance and efficiency—tuned over millions of years of evolution—of their natural counterparts.

As a case study of BAUVs, in our research, we adopted an in-house-built fish robot (Fishbot) that models the natural rainbow trout (Oncorhynchus mykiss) [4]. Its swimming locomotion belongs to the group of the subcarangiform, in which the propulsion is mainly generated by the caudal fin and the posterior part of the body [14]. The bot consists of three (anterior, posterior, and tail) segments that are connected via two (anterior and posterior) actuated hinge joints. In our previous work, we heuristically optimized its main control parameters—the amplitude and frequency of undulation—via genetic algorithms (GAs) [4,15]. Also, we proposed an approach to solving the most significant challenge of genetic algorithms (and evolutionary computing in general)—the significant runtime of the evolution of real, physical (rather than simulated) robots—by minimizing the runtime of the most time-consuming stage of evolution—the fitness trials [16]. In addition, we proposed a novel morphology for the robot—a limitedly underactuated posterior joint—and verified that, compared to its fully actuated implementation, such a morphology yields both a higher speed and better energy efficiency of the bot [13].

However, all of our previous efforts were focused on how to optimize the morphology of the robot. Conversely, in our current research, we intend to explore the possibility of improving the performance (e.g., speed and energy efficiency) of the bot by enhancing its control and specifically—by adopting an exponential pattern of control (i.e., the pattern of the target turning angles of servo motors and, consequently, the joints of the bot) actuators as an alternative to “canonical” sinusoidal control. Sinusoidal (harmonic) patterns have been commonly considered for the control of BAUVs because of their natural plausibility. Indeed, these patterns are a good approximation of the patterns observed in natural fish [17]. Moreover, such harmonic patterns result in low acceleration and low jerk of the movements of the joints, which, in turn, minimizes the wear-and-tear of the electro-mechanical components of the robot.

The inspiration for our current work is fueled by our assumption that during each half of the undulation cycle, the thrust is mainly generated by the movement of the posterior segment—the caudal (tail) fin—of the bot during the initial phase of undulation. In the remaining undulation, the fin is subjected to a hydrodynamic drag as it moves against the flow of the surrounding water. Based on such a decomposition of the undulation into thrusting- and braking phases, we hypothesize that an increased angular speed (and the corresponding linear speed of the tip of the posterior fin) would enhance the effect of the thrusting phase, while, at the same time, a reduced speed during the braking phase would diminish the negative effect of the latter.

An additional interest of this research stems from our observations that the downstroke (i.e., the thrusting phase of undulation) of some water creatures, such as the red-eared slider (Trachemys scripta elegans, Figure 1a) and the moon jellyfish (Aurelia aurita, Figure 1b), is quicker (i.e., shorter in time) than the upstroke—the braking phase of undulation. Despite the fact that we could not verify any asymmetry in the duration of the undulatory phases of natural fish, we were curious about the feasibility of adopting an analogy of such asymmetry, induced by the natural evolution, in the control of the Fishbot.

Our objective is to explore whether the exponential control—as a case of control that offers such a difference in the speed (and, consequently, asymmetry in duration) of thrusting and braking phases of undulation—would facilitate a higher net effect of the two phases of undulation that could be manifested by an increased swimming speed or (and) better energy efficiency of the robot. To the best of our knowledge, we are not aware of any implementations of such exponential control of fish robots. As possible alternatives to sinusoidal undulations (Figure 2a), “square-like” (Figure 2b) and “triangular-like” (Figure 2c) motions have been previously considered [18]. None of these alternative, non-sinusoidal patterns of undulation, however, feature any time asymmetry of the main phases of undulation. Indeed, in both “square-like” and “triangular-like” undulations, the thrusting and braking phases of undulation (at least theoretically) have identical durations. Moreover, both the theoretical and experimental analysis of the advantages of these non-sinusoidal undulations do not necessarily require a decomposition of the undulation into distinct phases.

The remainder of the paper is organized as follows. In the second section, we elaborate on the proposed concept of exponential control and why we believe that it should have a positive net effect on the speed and energy efficiency of the robot. In the same section, we introduce the experimental setup. In Section 3, we present the experimental results that verify the beneficial effect of the proposed exponential control of the speed and energy efficiency of the robot. In Section 4, we discuss some limitations of the proposed approach. Finally, in Section 5, we draw conclusions.

2. Materials and Methods

2.1. The Design and Control of the Fishbot

We adopted an in-house built two-joint Fishbot that models the natural rainbow trout fish (Oncorhynchus mykiss) [4]. The bot comprises three (anterior, posterior, and tail) segments, connected via two (anterior and posterior) actuated hinge joints, as illustrated in Figure 3.

The two joints are undulated by two servo motors. Each motor is originally governed by a (“canonical”) sinusoidal control signal (generated by the controller—the central pattern generator) that sets the instant values of the angles α₁ and α₂ of the joints with a sampling interval of 20 ms, as expressed in the following Equations (1) and (2):

α₁(t) = A₁ × sin(2 × π × f × t)

(1)

α₂(t) = A₂ × sin(2 × π × f × t + β)

(2)

where f is the frequency of undulation, A₁ is the angular amplitude of the anterior joint, A₂ is the angular amplitude of the posterior joint, and β is the phase shift between the two signals, α₁ and α₂. Details on the design and control of the robot are elaborated in [4].

In our previous research on the bot, we used genetic algorithms to evolve the optimal values of the four main parameters of the sinusoidal control, A₁, A₂, f, and β, that yield the fastest speed of swimming [4,15,16]. In the current work, we used the evolved optimal values of these parameters, as well as the resulting speed of locomotion (and associated energy efficiency) of the bot featuring such sinusoidal control as a benchmark for comparison with the experimental results of the newly proposed (exponential) control.

2.2. Analysis of Undulation

The propulsion of the swimming Fishbot is generated by the undulations of the two actuated hinge joints. As illustrated in Figure 4, a single undulation can be split into the following two phases [19]: beginning (thrusting, Figure 4a), which could be seen as an analog to the downstroke of the red-eared slider and the moon jellyfish (Figure 1), and ending (braking, Figure 4c), similar to the upstroke as illustrated in Figure 1. The smooth transition between these two phases is manifested by a short intermediate phase, as shown in Figure 4b.

According to the dynamic lift theory of fish locomotion [20], at the beginning of undulation (Figure 4a), the moving caudal fin—due to the certain angle of attack—produces a lift (thrust) L by pushing the water in the backward direction:

L = 0.5 × C_L × ρ × U²_T × A

(3)

where C_L is the coefficient of lift (thrust) of the fin, ρ is the density of the fluid (water), U_T is the average velocity of the fin, and A is the planform area of the fin. Thus, the change in thrust impulse ΔP_T during the thrusting phase can be expressed as follows:

ΔP_T = L × T_T = 0.5 × C_L × ρ × U²_T × A × T_T

(4)

where T_T is the duration of the thrusting phase of undulation. Considering that U_T = l_T/T_T, where l_T is the distance the caudal fin travels during the thrusting phase, Equation (4) can be rewritten as follows:

ΔP_T = 0.5 × C_L × ρ × l²_T × A/T_T

(5)

For symmetrical hydrofoils, such as the caudal fin of fish, the coefficient of lift C_L in the above Equation (5) can be approximated as follows:

C_L ≈ 2 × π × α_a

(6)

where α_a is the angle of attack of the fluid. However, the above Equation (6) holds only for small values (up to the critical value of about 15°) of α_a, i.e., only during the first phase (beginning) of undulation (Figure 4a).

For values of α_a that are higher than the critical one, a stall of the fin occurs, which is manifested by the separation of the (turbulent) flow behind the fin. The stall results in an abrupt reduction in the lift (thrust) and a significant increase in the hydrodynamic drag. Starting from the intermittent phase of undulation (Figure 4b,c), in which the fin moves at a blunt angle relative to the water flow, the caudal fin stalls and is not able to produce any thrust. Moreover, as the caudal fin moves in the opposite direction to the water flow (Figure 4c), it is subjected to a high-pressure build-up at the front part of the fin (i.e., the “lower” part of the hydrofoil) and a low-pressure area in the rear part of the fin (the “upper” part of the hydrofoil). Such a pressure difference will result in turbulent fluid flow between the front and rear parts of the fin, resulting in flow separation and, ultimately, hydrodynamic pressure (form) drag D that can be expressed as follows [21]:

D = 0.5 × C_D × ρ × U²_B × A

(7)

where C_D is the coefficient of drag, ρ is the density of the fluid (water), U_B is the average velocity of the fin, and A is the planform area of the fin. The hydrodynamic drag D would yield a detrimental braking effect and a corresponding change in the braking impulse ΔP_B:

ΔP_B = D × T_B = 0.5 × C_D × ρ × U²_B × A × T_B

(8)

where T_B is the duration of the thrusting phase of undulation. Because U_B = l_B/T_B, where l_B is the distance the fin travels during the braking phase, Equation (8) can be rewritten as follows:

ΔP_B = 0.5 × C_D × ρ × l²_B × A/T_B

(9)

The overall propulsion force depends on the ratio of changes in the thrust and braking impulses—as expressed in Equations (5) and (9) in the above-mentioned two phases of undulation [21,22]. Consequently, enhancing the thrust or (and) diminishing the braking associated with these phases of undulation might result in a favorable net increase in propulsion. We shall elaborate in the next subsection on how the proposed exponential control could achieve such an overall increase in thrust.

For simplicity, we assume that the water flow in the front part of the caudal fin is both (i) laminar and (ii) in a direction nearly parallel to the longitudinal axis of the anterior segment of the robot. In reality (e.g., due to the vortices, Coanda effect, etc.) this might not be exactly the case. Also, in Equation (7), we consider the pressure drag only, ignoring the second component of the overall drag—the skin friction drag. We assume that the latter is negligible due to the streamlined body of the Fishbot and its smooth fairings. Nevertheless, we believe that these assumptions do not compromise the fidelity of the proposed approach.

2.3. The Concept of the Exponential Control of the Fishbot

The proposed approach assumes that, as mentioned above, the net propulsion of the Fishbot would depend on the ratio of the change in thrust and braking impulses (as expressed by Equations (5) and (9), respectively) during the corresponding two phases of undulation. Thus, we argue that maximizing this thrust-to-brake ratio of changes in impulses would result in an increase in the net propulsion of the robot. From Equations (5) and (9), we obtain this thrust-to-brake ratio R_TTB as follows:

R_TTB = ΔP_T/ΔP_B = (C_L/C_D) × (l²_T/l²_B) × (T_B/T_T)

(10)

Assuming that l_T ≈ l_B, the above Equation (10) could be rewritten as follows:

R_TTB = (C_L/C_D) × (T_B/T_T)

(11)

As follows from the above Equation (11), a hypothetical control signal that maximizes the ratio T_B/T_T would, in theory, result in a higher thrust-to-brake ratio R_TTB, that, in turn, would yield a higher speed and, presumably, better energy efficiency of the robot. Notice that, for the commonly adopted sinusoidal undulation, as expressed in Equations (1) and (2) and illustrated in Figure 2a, as well as the alternative “square-like” (Figure 2b) and “triangular-like” (Figure 2c) undulations, there is no such asymmetry in the duration of the two phases of undulation, T_B and T_T. Consequently, the ratio T_B/T_T is equal to 1 in these undulations.

In our quest for control of both actuators of the Fishbot, which would result in an increased ratio T_B/T_T, we shall consider a mathematical function that fulfills the following three intuitive criteria:

• While an increased ratio T_B/T_T is sought, the sum T_B + T_T should be equal to half of the period of undulation;
• An increased ratio T_B/T_T implies that the average derivative (i.e., gradient) of the function of the target turning angle of servo motors during the thrusting phase should be higher than that of the braking phase;
• The transition between the two (thrusting and braking) phases should be smooth, with the higher instant derivative of the function of the target turning angle of servo motors during the thrusting phase gradually approaching the lower values pertinent to the braking phase of undulation;
• The target turning angle of servo motors of both phases of undulation should (preferably) be expressed by the same (monotonous) function.

We propose an exponential function of control, as shown in Figure 5, because it fulfills the above-mentioned criteria. The proposed exponential control signal can be analytically expressed as follows:

$$\alpha =-A_{\mathit{SM}}+(A_{\mathit{SM}}+A_{\mathit{PS}}) \times {(1-e}^{-t/\mathsf{\tau }})$$

(12)

where α is the instant value of the turning angle of (either anterior or posterior) servo motors, A_SM is the amplitude of undulation (i.e., the maximum value of the turning angle of the servo motor), A_PS is the amplitude of the imaginary pulse signal that defines the asymptotic value of α, t is the time, and τ is a time constant (Figure 6).

In order to investigate the effect of the ratio T_B/T_T on the speed and energy efficiency of the bot, for a given frequency (and corresponding duration of the half-period T_HP) and amplitude A_SM of undulation, we would need to define the two main parameters—A_PS and τ, respectively—of the exponential function, as expressed in Equation (12). We can define these two unknown parameters from the following system of two Equations (13) and (14). These two equations express the value of the exponential function at the points of time A and B, respectively, as shown in Figure 6:

$$\alpha =-A_{\mathit{SM}}+(A_{\mathit{SM}}+A_{\mathit{PS}}) \times {(1-e}^{-T_{\mathit{HP}}/\mathsf{\tau }})=A_{\mathit{SM}}$$

(13)

$$\alpha =-A_{\mathit{SM}}+(A_{\mathit{SM}}+A_{\mathit{PS}}) \mathbf{\times } {(1\mathbf{-}e}^{-T_{\mathit{HP}}/\left(\right(R+1)\times \mathsf{\tau })})=0$$

(14)

where R = T_B/T_T. The analytically obtained solution of the above system can be expressed in a general form as follows:

A_PS = k_ps × A_SM

(15)

τ = k_τ × T_HP

(16)

Consequently, Equation (12) can be rewritten as follows:

$$\alpha =-A_{\mathit{SM}}+(A_{\mathit{SM}}+k_{\mathit{ps}}\times A_{\mathit{SM}}) \times (1-e^{-t/(k_{\mathsf{\tau }}\times T_{\mathit{HP}})})$$

(17)

The concrete analytical solutions—the obtained values of the coefficients k_ps and k_τ— that define the values of parameters A_PS and τ, respectively, of the exponential function are shown in

Table 1. Values of coefficients k_ps and k_τ defining the main parameters A_PS and τ of the exponential control of the Fishbot for various ratios of R = T_B/T_T.

R = TB/TT	kps(APS = kps × ASM)	kτ(τ = kτ × THP)
2	1.618	0.693
3	1.192	0.410
4	1.078	0.305
6	1.017	0.209
8	1.004	0.161
10	1.001	0.131

for different values of the ratio T_B/T_T.

Consequently, for example, for R = T_B/T_T = 2, the control of the servo motors for a given amplitude (A_SM) and half period (T_HP) of undulation would be defined by the following Equation (18):

$$\alpha =-A_{\mathit{SM}}+(A_{\mathit{SM}}+1.618\times A_{\mathit{SM}})\times (1-e^{-t/(0.693\times T_{\mathit{HP}})})$$

(18)

Analogically, for example, for the ultimate value of R = T_B/T_T = 10, the control of the servo motors would be expressed as follows:

$$\alpha =-A_{\mathit{SM}}+(A_{\mathit{SM}}+1.001\times A_{\mathit{SM}})\times (1-e^{-t/(0.132\times T_{\mathit{HP}})})$$

(19)

The forms of the control signals, defined by the obtained values of the main parameters of the exponential control of undulation A_PS and τ for A_SM = 45°, T_HP = 0.25 s (f = 2 Hz), and different values of the ratio R = T_B/T_T are shown in Figure 7.

3. Experimental Results

3.1. Experimental Setup

To investigate the effect of the proposed exponential control—in which the instant values of turning angles of both servo motors of the Fishbot are set according to Equation (17)—on both the speed of locomotion and energy efficiency of the bot, we conducted experiments with six different values of the ratio R = T_B/T_T, equal to 2, 3, 4, 6, 8, and 10. For each value of this ratio, we ran four trials (to account for the effect of environmental noise) of the bot with a given frequency of undulation f (i.e., a given half-period of undulation—the parameter T_HP in Equation (17)), with the maximum amplitude of undulation of both servo motors (i.e., parameter A_SM in Equation (17) set to 45°), and no phase shift between the exponential signals of the two servo motors. Our decision to run just four trials is based on the preliminary investigation of the consistency of the results obtained from each trial. The maximum value of the standard deviation of the speed and energy efficiency is just about 2% of the experimentally obtained values—much lower compared to the differences in speed and energy efficiency due to the proposed exponential control of the robot.

We verified in our previous research that the maximum amplitude of undulation does indeed result in maximum speed and energy efficiency of the bot [4,15,16]. We tested five different frequencies f ranging from 0.4 Hz (T_HP = 1.25 s) to 2.0 Hz (T_HP = 0.25 s) with a discretization step of 0.4 Hz. The results of the four runs are aggregated based on the calculated mean of the achieved performance (e.g., speed and energy efficiency).

The experimental system consists of the Fishbot, its controller, and an optical tracker, as depicted in Figure 8. The controller consists of an abstract layer and a physical layer. The former generates the exponential control signals—the instant values of the turning angles of servo motors, as of Equation (17)—with a sampling interval of 20 ms. The physical layer converts the values of the turning angles of servo motors, generated by the abstract layer into pulse-width modulated signals that physically govern the instant angular positions of the servos. In addition, in order to measure the electrical power consumed by the robot during the trials, we connected a current sensor serially to the power supply (Ucc = 5 V) line of the robot. We adopted the Micro Maestro 6-Channel USB Servo Controller and ACS724—both produced by Pololu Robotics and Electronics [23,24]—as a hardware implementation of the physical layer and the current sensor, respectively.

In addition to the Fishbot and its controller, the experimental system also includes an optical tracker, which uses a video feed from an overhead video camera to track (via a bright red LED mounted on the bot, as shown in Figure 3) the timestamped 2D positions of the swimming Fishbot. From the timestamped 2D coordinates of the bot, obtained from the optical tracker, the controller estimates the instant speed of the bot during the 5 s swimming trial. The maximum value of the instant speed is assumed to be the actual steady-state swimming speed of the Fishbot. Also, based on the readings from the current sensor (denoted as CS in Figure 8), and the speed of the bot, the controller calculates the energy efficiency of the latter, as elaborated below in Section 3.3. The third version of the software of the abstract layer of the controller, and the optical tracker, both developed by the author, was used. The details of the experimental setup are elaborated in [3,13,16].

3.2. Swimming Speed

Figure 9 illustrates the experimentally obtained speed of locomotion as a function of the frequency of oscillations f and the value of the ratio R = T_B/T_T. As shown in Figure 9a, for frequencies up to f = 1.6 Hz, the increase in the ratio R = T_B/T_T results in an increase in the swimming speed of the robot. The maximum speed-up of four times, compared to the canonical sinusoidal control, is achieved for R = 10 and f = 0.4 Hz. With the increase in the frequency of undulation, however, the favorable effect of the exponential control gradually decreases, and, moreover, at the maximum frequency of 2 Hz, the fastest speed is achieved for R = 8 (Figure 9a), with a corresponding drop in the speed-up (compared to the sinusoidal control featuring the same main parameters of oscillation, such as amplitude, frequency, and no phase shift between the signals of the two servo motors) to about 1.15 times. Nevertheless, virtually for all frequencies of undulation, and all values of the ratio R (with the only exception of R = 2 for frequencies f = 1.6 Hz and f = 2 Hz), the exponential control is superior to that of the sinusoidal one in that it yields a faster robot swimming speed.

3.3. Energy Efficiency

The energy efficiency E_E is estimated as the distance D the Fishbot can swim autonomously with a given energy (e.g., stored in its batteries) E_B:

E_E = D/E_B

(20)

The above Equation (20) could be rewritten as follows:

E_E = (V × T)/(I_AVR × U_CC × T)

(21)

and

E_E = V/(I_AVR × U_CC)

(22)

where T is the time the bot would eventually travel before it exhausts its available energy, V is the swimming speed, I_AVR is the average electrical current consumed by the bot during the trial, and Ucc is the constant power supply voltage (5 V), respectively. We estimated the energy efficiency according to the above Equation (22).

As shown in Figure 10, compared to the sinusoidal control, the proposed exponential one results in improved (experimentally obtained) energy efficiency.

Virtually for all frequencies f, the increase in the ratio R = T_B/T_T yields better energy efficiency, with the only exception observed for the maximum frequency f = 2 Hz, and the maximum value of R = 10. Compared to the canonical sinusoidal control, the maximum increase in energy efficiency of about 3.7 times is achieved for R = 10 and f = 0.4 Hz (Figure 9a,b). The maximum absolute value of the energy efficiency of about 177 mm/s/W is achieved for R = 10 and a frequency between 1.2 Hz and 1.6 Hz (Figure 9a). With the increase in the frequency of undulation, however, the energy efficiency of the exponential control gradually decreases; moreover, the exponential control with lower values of R is even less energy efficient than that of the sinusoidal one.

4. Discussion

Compared to the sinusoidal control, the proposed exponential control yields improved performance (e.g., swimming speed and energy efficiency) of the Fishbot. However, the favorable effect of the control somehow diminishes at higher frequencies of undulation. We argue that there are two possible causes for this: (i) the flexibility of the caudal fin and (ii) exceeding the technical specifications of the servo motors. Regarding the former cause, being subjected to excessive forces during the thrusting phase of the high-frequency undulations, the fin flexes too much, which, in turn, reduces the angle of attack α_a which—according to Equation (6)—results in a corresponding reduction in the coefficient of lift (thrust) of the fin. Therefore, in the future, it would be interesting to repeat our experiments with a stiffer robot caudal fin.

As for the latter cause—exceeding the technical specifications of the servo motors as a result of the more demanding exponential control—it is important to stress that the controller of the bot—according to Equation (17)—sets the instant values of the desired (target) turning angles of servo motors rather than the actual one. Typically, the servo motors lack the capability to provide any feedback to the controller about the instant values of their actual turning angles. However, as a result of the interplay between (i) the dynamics of the desired turning angle, (ii) the extent of the external forces exerted on the moving segments of the bot, and (iii) the capabilities of the motor (according to its technical specifications) and, especially the maximum torque exerted by the servo and the maximum angular speed, the instant value of the actual turning angle of the motor may intermittently differ from the value of the desired one. This, in turn, would distort the actual control signal, as its rate of change would be lower than that of the desired one. The maximum angular speed (i.e., the rate limit) of the servo motors, used to build the Fishbot, is 600°/s [25], which, as depicted in Figure 11, for the frequency of undulation f = 2 Hz, is exceeded by the desired angular speed. The latter is calculated as a derivative of the desired turning angle of the servo motors (as illustrated in Figure 7). As shown by the shaded areas in Figure 10, the peaks of the desired angular speed at the very onset of the thrusting phase of undulation intermittently exceed the speed limit of the servo motors (i.e., 600°/s) for the ratios R = T_B/T_T equal to 4, 6, 8, and 10. Moreover, the increase in the value of R results in an increase in both (i) the difference between the desired and actual (presumably, 600°/s) angular speeds and (ii) the duration during which the desired speed exceeds the actual one. Therefore, the increase in the ratio R, especially in high-frequency undulations, would result in increased severity of distortion of the intended exponential control signals. In our future research, it will be important to conduct additional experiments with an increased rate limit of the servo motors. One possible approach to achieving this would be simply raising the DC voltage of the power supply of the servo from the currently set Ucc = 5 V.

The potential real-world applications of the proposed exponential control could be in the energy-efficient monitoring and management of underwater living systems. Faster speed of locomotion might also favorably impact the range of the bot in upstream underwater currents. There are remaining challenges—such as navigation, communication, collecting and transmitting sensory information, and limited payload—that have to be addressed before the widespread adoption of BAUVs.

5. Conclusions

We decomposed the undulation of the Fishbot into two phases—thrusting and braking—and proposed an alternative—exponential (rather than sinusoidal)—pattern of control. The proposed control features an asymmetric duration of these two phases of undulation—shorter (faster) thrusting and longer (slower) braking. The experimental results suggest that, compared to bots featuring a sinusoidal undulation, the proposed exponential control favorably impacts the net effect of these two phases. Compared to the sinusoidal control (featuring identical values of the main parameters of oscillation, such as amplitude, frequency, and no phase shift between the signals of the two servo motors), the proposed exponential control yields between 1.1 and 4-times faster speeds in the range of frequencies of undulation between 0.4 Hz and 2 Hz and between 1.1 and 3.6 times higher energy efficiency in the same frequency range.

In our future work, we plan to address the challenge of the diminishing effect of the proposed exponential control at higher frequencies of undulation. In particular, we are interested in whether a stiffer caudal fin would be more responsive at the onset of the thrusting phase when the external forces exerted on the fin are highest. Also, we would like to consider increasing the angular speed limits of the servo motors in order to minimize the possible distortion of the intended exponential control signals caused by intermittently exceeding these limits during the high-speed undulations.