A Controllable Nonlinear Bistable “Fishtail” Boosting Robotic Swimmer with Excellent Maneuverability and High Energy Efficiency

Abstract

High maneuverability and energy efficiency are crucial for underwater robots to perform tasks in engineering practice. Natural evolution empowers aquatic species with skills of agile and efficient swimming, which can be deliberately employed for better robotic swimmers. A critical issue for efficient robotic swimmers is the design and control of an appropriate propulsion system. This study, therefore, presents a completely different realization of a highly flexible and controllable bistable nonlinear mechanism as a “fishtail.” The mechanism combines an elastic spine and a lightweight parallel linkage mechanism. Through active control of the endpoint of the elastic spine, the compliant tail can be empowered with exceptional controllability and tunable bistability for a much more efficient and also the first-ever accurately controlled bistable elastic propulsion system. Experimental results demonstrate that the new bistable fishtail can achieve a faster speed of its size (up to an average speed of 0.8 m·s⁻¹) with an associated higher energy efficiency (corresponding cost of transport as low as 9 J·m⁻¹·kg⁻¹), and greater maneuverability (with an average turning speed of up to 107°/s at a much smaller turning radius of 0.31 body length). This study will definitely provide an efficient controllable and feasible approach to the design of nonlinear compliant propulsion systems for underwater vehicles by exploring nonlinear dynamics.

Introduction

Body-and-caudal-fin propulsion is a popular biomimetic approach for driving bioinspired underwater robots due to its simple implementation, reasonable speed, and high energy efficiency.^1–3 This propulsion method produces very little noise and interacts harmoniously with aquatic environments, rendering it more suitable for the close-up observation of marine animals compared to traditional propeller-based systems.^4,5 Notably, fish possess an innate capacity for agile swimming maneuvers, facilitated by their sophisticated tail locomotion control.⁶ In complex and unstructured underwater environments, exceptional maneuverability is essential for robots to execute diverse tasks.⁷ Therefore, it is imperative to endow robotic swimmers with analogous swimming capabilities through ingenious tail design and control.

Many robotic swimmers have been designed in the past two decades with various fish morphologies, which can be classified into discrete and continuous bodies. A discrete body design, such as the multi-joint-link mechanism,^8–12 may result in some salient drawbacks such as discrete kinematics and large friction loss, leading to low propulsion efficiency and performance limitations. Compared to rigid structures, soft materials offer benefits including continuous morphology and safe, adaptive interaction with harsh environments. Researchers thus attempt to use soft materials to achieve continuous compliant fish bodies, such as hydraulic or pneumatic soft actuators,^13–16 smart soft actuation materials like ionic polymer–metal composites,^17–19 macro fiber composites,^20,21 and dielectric elastomer actuators (DEAs).^22,23 Although these materials have been successfully employed in devising diverse swimming robots, a noticeable issue with these soft mechanisms is that they tend to produce relatively weak thrust and low speed. Additionally, it is difficult to accurately control their body motion for high agility and maneuverability, especially in dynamic water environments. For instance, the hydraulically actuated soft robotic fish, SoFi, achieves a swimming speed of 0.235 m/s (0.5 BL/s), an angular speed of 15°/s, and a turning radius of 78.2 cm.²⁴ The eel-inspired soft robot, actuated by soft pneumatic elastomer actuators, reaches a maximum velocity of 0.19 m/s (0.36 BL/s) with a corresponding cost of transport (CoT) of 10.72.¹³ The soft robotic fish with DEAs achieves a top speed of 76 mm/s (0.76 BL/s), a peak thrust of 14 mN, and a turning radius of 1.25 BL.²³ The reported results indicate that the performance of most of these robots is much worse than that of their biological counterparts.

To address the shortcomings of inadequate force and slow response in existing soft robots, various studies have been conducted. The elastic instability of bistable or multistable structures can induce an interesting snap-through phenomenon, quickly storing and releasing strain energy.²⁵ During this process, remarkable force amplification and rapid morphing can be achieved, which are two characteristics of elastic instability that can be further utilized to enhance the performance of soft robots effectively.²⁶ For example, a robotic insect designed by Koh et al. can achieve jumping on water by leveraging the bistable impulsive response of the catapult mechanism consisting of composite materials and planar shape memory alloy actuators.²⁷ Tang et al. proposed a pyramid-like folding structure driven by the magnetic field, achieving an ultrafast bistable soft jumper, which is advanced in both jumping capability and response time.²⁸

Bistable mechanisms have also been widely adopted in the design of swimming robots in recent years. Chen et al. designed a soft and untethered robot using shape memory polymers as the muscle connected to a small buckling beam, exploiting bistable actuation to amplify its directional propulsion.²⁹ Tang et al. utilized rigid linkages and a linear spring to create bistable linkages as the spine of soft terrestrial and aquatic robots to improve their speed and force of locomotion.³⁰ Soft swimming robots built with hair-clip mechanisms take advantage of bistable actuation to increase speed and efficiency.^31,32 Wang et al. devised a soft robotic jellyfish driven by pneumatic bistable actuators, demonstrating improved efficiency and maneuverability compared to the robot without a bistable configuration.³³ Bambrick et al. developed a swimming mechanism using four magnets to realize the bistability and experimentally proved that the well-tuned bistable transmission can increase thrust, swimming speed, and efficiency.³⁴ Although all these mentioned soft swimming robots can leverage elastic instability to enhance their performance to some extent, they all lack dexterous and accurate controllability, as well as online tunability of the underlying bistable nonlinear dynamics. As a result, they can only exhibit unstable locomotion mode without offline structural adjustment, and their performance, in general, is still far from that of natural fish, posing challenges for potential practical deployment. Thus, the objective of this work is to explore an online controllable and tunable bistable mechanism for achieving agile and efficient swimming robots.

In this study, we propose a highly controllable elastic tail for an untethered robotic swimmer, which is composed of a parallel linkage mechanism and an elastic spine, as shown in Figure 1. By accurately controlling endpoint trajectory of the spine via the parallel mechanism, the tail possesses tunable bistability, consequently enabling flexible switching between motion modes (referring to Fig. 2). The robot can perform smooth swing motion of the monostable mode and rapid impulsive motion of the bistable mode (referring to Fig. 2C and F). With the controllable bistability, the propulsion system demonstrates significantly improved forward and turning speeds (referring to Figs. 4D and 5B), a smaller turning radius (referring to Fig. 5C), and lower energy consumption across a wide range of speeds through motion mode switching (referring to Fig. 4F). It is also noted that the high controllability and continuous morphology of the elastic spine allow the robot to steer with the most effective tail beating trajectory (referring to Fig. 6F), thus leading to good maneuverability.

FIG. 1.

System design of the robot: it consists of a rigid head, an active-controlled flexible tail, and a passive compliant caudal fin. (A) Mechanical structure. (B) Electronic components. (C) Structure of passive joint. a: robot head, b: active joints, c: parallel mechanism, d: caudal fin, e: elastic spine, f: passive rotational joint, g: antenna of the receiver, h: Arduino nano, i: TTL/USB converter, j: servo motors, k: battery for microcontroller, l: current sensors, m: battery for motors, n: microcontroller, o: bearing, p: shaft.

FIG. 2.

Working modes of the tail: the tunable bistability of spine can be controlled by the predefined moving trajectory of the tail. (A) Defined parameters for the kinematics of parallel mechanism. (B–D) Monostable motion mode: the tail beat trajectory is defined by three parameters (γ, r, and α_c) (B); the deformed shape (C) and strain energy (D) of the spine during one cycle of monostable motion. (E–H) Bistable motion mode: compressing the spine to form a buckling beam and the tail beat trajectory can be derived by the parameters in (E); the deformed shape (F) and strain energy (G) of the spine during one cycle of bistable motion; and (H) different compression and thickness of the spine result in different energy barrier ΔE for snapping.

Extensive experimental results (referring to Figs. 4 and 5) show that the maximum average swimming speed of the robot can reach 0.8 m·s⁻¹, the energy consumption to maintain the maximum swimming speed is as low as 9 J·m⁻¹·kg⁻¹, and the maximum average turning rate is 107°/s with a minimum turning radius of 0.31 body length (BL), demonstrating its higher energy efficiency and better maneuverability than most existing swimming robots (referring to the comparisons in Fig. 7).

Results

Design and working principles

The designed robotic fish is shown in Figure 1. The robot mainly consists of three parts: a rigid robotic head, an elastic tail, and a compliant caudal fin. The rigid head is a waterproof container to house the electronic components. For the tail, a steel strip connected to the head is adopted as the elastic spine and is actively actuated by a parallel linkage mechanism which is a double-layer X structure composed of eight bars. One side of the parallel mechanism is connected to the head by two active rotational joints, and another side is connected to the spine by a passive rotational joint. The caudal fin is fixed on the passive rotational joint opposite to the spine. The dimensions of the entire robot are 550(L) × 106(W) × 120(H) mm³ and it weighs 1.8 kg. The detailed hardware implementation of the robot is described in Supplementary Data S1. The working modes of the tail are shown in Figure 2. Through the accurate control of the parallel mechanism, the elastic spine connected to the passive rotational joint can follow any motion trajectories to swing. This is a significantly different and important feature compared to all existing nonlinear elastic propulsion mechanisms in the literature. Due to the high controllability of the tail, the robot is capable of swimming in two different locomotion modes: the monostable mode and the bistable mode (referring to Fig. 2). Based on the predefined trajectory, the inverse kinematics (see Supplementary Data S1) can be used to obtain the input of two motors in the active rotational joints to control the parallel mechanism, and the defined parameters of the parallel mechanism for the derivation of inverse kinematics are shown in Figure 2A.

Figure 2B demonstrates the trajectory generation for the monostable mode, which is part of a circle and controlled by three parameters as follows. Here, γ is the distance between the circle center and the starting point of the elastic spine. The distance from the circle center to the controlled rotational joint (point E) is the radius of the trajectory, denoted by r. α _c is the angle between the midline of the robot and the radius vector that connects to the endpoint of the trajectory, and it is used to control the tail beat amplitude. The deformed shape of the elastic spine during the motion of the monostable mode is presented in Figure 2C. For each motion cycle of monostable mode, point E passes the point where the elastic spine is in the undeformed state. Hence, the minimum strain energy of the elastic spine is 0 and is located at only one point of the trajectory of the monostable mode as seen in Figure 2D. The theoretical model of the monostable mode is provided in Supplementary Data S1.

Figure 2E illustrates the realization of the bistable locomotion mode. In the initial stage of the bistable locomotion mode, there is a prestored strain energy of the elastic spine due to compression (resulting in a pinned–pinned buckled beam). It is known that there are two stable positions of this mechanism, which are symmetrical about the horizontal line. When the input pivot ( $ϕ_{1}$ or $ϕ_{2}$ ) of the elastic spine is actuated by rotation, a phenomenon of instantaneous and rapid change in position and speed, namely snap-through, will be triggered between the switching of two stable states. Figure 2f and g show the deformed shape of the elastic spine and its strain energy variation during this process. When the instantaneous and rapid release of the strain energy occurs, the elastic spine snaps from shape II to shape III or from shape IV to shape I. The gap between the maximum and minimum energy is defined as the energy barrier that needs to be overcome to achieve snapping. Figure 2H shows that different precompressions and thicknesses of the elastic spine can lead to different energy barriers. The theory model of the bistable mode is described in Supplementary Data S1.

Stiffness coordination between the spine and the caudal fin

The stiffness plays a crucial role in achieving efficient swimming of fish.^35,36 In the proposed fishtail, two compliant elements need to be considered, including the elastic spine and the compliant caudal fin. The stiffness of the spine is not only one of the dominant factors affecting the energy barrier of the bistable locomotion mode, but also together with the compliance of the caudal fin, determines the kinematics of the tail during swimming. Therefore, the stiffness coordination between the spine and the caudal fin is important for the swimming performance of the robot, and its influence is first explored for both the monostable and bistable locomotion modes. The stiffness is adjusted by varying the thickness, and other dimensions of the elastic spine and the caudal fin remain unchanged as shown in Figure 3A. The thickness sets of the spine and the caudal fin are {0.25 mm, 0.30 mm, 0.35 mm} and {0.3 mm, 0.4 mm, 0.5 mm}, respectively. So, nine cases in total of thickness combinations for the spine and the caudal fin are tested as presented in Figure 3B. Here, only three cases (spine–caudal–fin combination [SCFC] 1, SCFC5, and SCFC8) are selected to compare, which represent the best swimming performance for the robot with the elastic spine of three different thicknesses. More comparisons are provided in Supplementary Figures S1 and S2 (see Supplementary Data S1).

FIG. 3.

Comparison of the swimming performance between the fishtail with different thickness combinations of the elastic spine and the caudal fin, revealing the stiff spine and the medium stiff caudal fin enable the robot to swim faster with higher energy efficiency. (A) Dimensions of the elastic spine and caudal fin. The unit is the millimeter. (B) Different thickness combinations of the elastic spine and caudal fin. SCFC, spine–caudal–fin combination. (C) Swimming speed under different frequencies. The control parameters of the monostable mode are given as γ = 2 cm, α_c = 40°. And the precompression length of the elastic spine of the bistable mode is set as ΔL = 10 mm. (D) Cost of transport (CoT) under different frequencies. CoT is defined as the consumed power for the robot moving over a distance of 1 m.

Figure 3C (dashed line) and D (dashed line) demonstrate the results of the average swimming speed and the cost of transport (CoT) for the monostable mode. The robot with SCFC1 swims the slowest under different tail beat frequencies. Before the frequency of 2.5 Hz, the robot with SCFC5 is the fastest, however, the speed of the robot with SCFC8 surpasses that of the robot with SCFC5 when the frequency rises above 2.5 Hz, and the SCFC8 achieves the highest speed among all cases for the monostable mode at the frequency of 3 Hz. For the results of CoT, no one case is always the most energy efficient. When the swimming velocity is under 0.25 m/s, the CoT of the robot with SCFC1 is the lowest. When the speed is in the range between 0.25 and 0.44 m/s, the most energy-efficient one turns into SCFC5. And as the speed continues to increase, the best one is SCFC8.

Figure 3C (solid line) and D (solid line) shows the average swimming speed and the CoT for the bistable mode. Similar conclusions about the effect of different cases of stiffness combinations on the monostable swimming mode can be found with the bistable mode, but the differences between different cases are more obvious. When the tail beat frequency is below 2 Hz, the swimming speeds of the robot with SCFC5 and SCFC8 are close to each other, and the robot with SCFC5 is slightly faster than the case of SCFC8, while SCFC8 performs much better on the velocity than other two cases when the frequency is over 2 Hz. And the maximum speed is achieved with SCFC8 as well, which is about 0.60 m/s at the frequency of 3 Hz. When it comes to the CoT, SCFC1 achieves the minimum CoT at speeds below 0.37 m/s. From the speed of 0.37 m/s to 0.53 m/s, the CoT of the robot with SCFC5 is minimal. When the speed exceeds 0.53 m/s, SCFC8 is most energy efficient.

According to the analysis above, no case can produce the best performance both in swimming speed and CoT over the studied range of tail beat frequencies. However, SCFC8 enables the robot to obtain the maximum speed and the lowest CoT at high speeds. Thus, for the subsequent tests, SCFC8 is selected for the robot.

Effects of different trajectories on forward swimming performance

We explored the forward swimming performance of the robot when the tail swings with different trajectories. Due to the structure limitations ( $θ_{1} < 77 °$ , $θ_{2} > - 77 °$ ), the reachable parameter space of each trajectory needs to be investigated. Figure 4A shows the results of reachable spaces for trajectories of both monostable and bistable modes. The reachable space presents the maximum value of $α_{c}$ for each trajectory of the monostable mode and the allowable precompression length $Δ L$ of the elastic spine for the bistable mode. Four different demanded trajectories were selected for the forward swimming tests as shown in Figure 4B. Two trajectories belong to the monostable mode, and the trajectory with larger $γ$ is not included because the amplitude is too small to get good performance on the speed. The other two trajectories pertaining to the bistable mode have different precompressions of the elastic spine. The resultant trajectories were measured and compared with the demanded trajectories as shown in Supplementary Figure S3(see Supplementary Data S1), which indicates the consistency between the inputs and outputs of control.

FIG. 4.

Forward swimming performance of the robot with different control trajectories, revealing the bistable mode enables the robot to swim faster, and the switchable motion modes allow the robot to maintain higher energy efficiency over a wide range of speeds. (A) The reachable space of control parameters for the monostable mode (top) and the bistable mode (bottom). (B) Different control trajectories for forward swimming. (C) Comparison of swimming speed between the bistable and monostable modes at 1 Hz in the time domain. (D) Average forward swimming speed comparisons of the robot with different control trajectories. (E) Comparison of consumed current between the bistable and monostable modes at 1 Hz in the time domain. (F) Cost of transport (CoT) comparisons of the robot with different control trajectories. (G) Comparison of the swimming between the bistable mode (top) and the monostable mode (bottom). The tail beat frequency is 1 Hz. For bistable mode, the precompression length ΔL is 2 cm, and for monostable mode, the γ of the trajectory is 2 cm.

Figure 4C demonstrates the swimming speed comparison between bistable ( $Δ L = 2$ cm) and monostable ( $γ = 2$ cm, $α_{c} = 60 °$ ) modes in the time domain at the frequency of 1 Hz. In the steady part of the speed, the minimum instantaneous speeds of the two modes are almost same, while the maximum instantaneous speed of bistable mode is much higher than that of the monostable mode. As a result, the average swimming speed in bistable mode is higher. The results of the swimming speed under various tail beat trajectories and frequencies are shown in Figure 4D. For the monostable locomotion mode, the speed increases with the decrease of the value of $γ$ , this is because with the same control angle $α_{c}$ , the amplitude is larger for the trajectory with a smaller $γ$ . The maximum average swimming speed of the monostable mode is around 0.65 m/s when the robot swims with the trajectory of $γ = 2 cm$ and the frequency of 3.5 and 4 Hz. More speed comparisons of each trajectory with different amplitudes for the monostable mode are provided in Supplementary Figure S4 (see Supplementary Data S1). As for the bistable swimming mode, the average speed is dramatically increased compared to the monostable mode. With a higher compression rate of the elastic spine, more improvement of the speed can be achieved. The reason is that a larger compression rate induces more prestored elastic strain energy of the spine, which will boost more the thrust. The highest average swimming speed that can be achieved by the bistable mode is about 0.80 m/s at the frequency of 3.5 Hz and the precompression length $Δ L$ equal to 2.0 cm. To explore the impact of the precompressed length of the spine on the passive rotation joint’s stiffness, the relation of moment and rotational angle under different precompression lengths was calculated as shown in Supplementary Figure S5 (see Supplementary Data S1), whose slope represents stiffness. During one cycle of bistable motion, the stiffness is alternately positive and negative, and there are positive and negative stiffness mutations, which results in the snap-through motion. Comparing the curves of two precompression lengths, the stiffnesses are almost the same at each motion stage, while more compression rate leads to larger snapping angle and more strain energy. Thereby, the interaction between the caudal fin and the surrounding water is stronger and generates a larger fluid force.

The consumed currents of two servos for bistable and monostable modes at the frequency of 1 Hz are illustrated in Figure 4E. There is one peak current for each motor in one bistable motion cycle. During one half of the motion cycle, the consumed current of one motor reaches the maximum value, while the other motor gets the minimum current value. It is opposite in the other half of the motion cycle. The results indicate that generating the needed torque to overcome the energy barrier for achieving snapping motion on one side mainly relies on one motor. The curve shape of the consumed current in one monostable motion cycle is significantly different from that of bistable motion, which exhibits two peak values for each motor. In a half of the motion cycle, two motors both reach the maximum current and contribute equally. The average current value of bistable mode is higher. Because the working load on the motors increases due to the precompression of the spine. It is worth noting that a higher current cannot be always considered a drawback. According to the torque-speed-efficiency characteristic of a servo motor as shown in Supplementary Figure S6 (see Supplementary Data S1), the working efficiency of the motor shows the first rising and then dropping trend with the current increasing. Thus, the proposed fishtail mechanism could allow the motor to operate with high efficiency when the tail swing trajectory is properly selected. Figure 4F compares the results of CoT when the robot swims with different tail-flapping trajectories. When the robot swims in the monostable mode, the trajectory with a large $γ$ exhibits a higher energy efficiency than that of the trajectory with a small $γ$ . The trajectory of $γ$ = 6 cm is more energy efficient than the other trajectory. Regarding the bistable locomotion mode, a large compression rate of the elastic spine leads to higher energy consumption. Among the two trajectories of the bistable mode, the one with the compression length $Δ L$ of 1 cm achieves the lowest CoT. Through further comparisons between the CoT of the monostable and bistable modes, we find that the robot can swim energy efficiently with the monostable mode at low speeds and with the bistable mode at high speeds. For example, it is better for the robot to use the monostable mode with the trajectory of $γ$ = 6 cm when the swimming speed is below 0.40 m/s and to use the bistable mode with the trajectory of $Δ L$ = 1 cm when the swimming speed is higher than 0.40 m/s.

According to the computation method described in equations (28) and (29) of Supplementary Data S1, the specific swimming performance improvement of the bistable mode relative to the monostable mode under diverse frequencies was calculated and summarized in Supplementary Table S1 (see Supplementary Data S1). There is an improvement of about 10–25% in swimming speed across the entire range of testing frequencies, benefiting from the bistable mode. While the monostable mode is more efficient at a frequency lower than 1.5 Hz. When the frequency is over 2 Hz, the swimming efficiency of bistable mode gets an improvement about 5–10% compared to that of the monostable mode. Figure 4G shows the snapshots of swimming comparison between the bistable and monostable modes under the frequency of 1 Hz (Supplementary Videos SV1 and Video SV2). And the swimming comparison under the frequency of 4 Hz is provided in Supplementary Video SV3. Other than the forward swimming speed and swimming efficiency, the thrust of both bistable ( $Δ L = 2$ cm) and monostable ( $γ = 2$ cm, $α_{c} = 60 °$ ) modes was also tested as shown in Supplementary Figure S7 (see Supplementary Data S1). Supplementary Figure S7 illustrates the comparison of thrust in one motion cycle between the bistable and monostable modes. There are three local thrust peaks for both modes during a half of the motion cycle. The first two local thrust peaks of the bistable mode are much higher than those of the monostable mode, which are the main sources of the higher average thrust. Particularly, the second local thrust peak of the monostable mode is almost negligible. According to one study,³⁷ the second local thrust peak occurs at the transition stage of the caudal fin’s attack angle. During this transition stage, the snap-through motion happens. Thus, the thrust results prove that the energy storage and release offered by bistability is also the source of thrust gains. Supplementary Figure S7 demonstrates that the average thrusts of the bistable mode are higher than those of the monostable mode under various frequencies. Supplementary Figure S7illustrates the thrust improvement of the bistable mode compared to the monostable mode. At 1 Hz, the bistable mode gains the largest thrust improvement. And the average value of the thrust improvement of all tested frequencies is about 30.6%. The calculation method is depicted in equations (30) and (31) of the Supplementary Data S1.

Effects of different trajectories on the turning performance

Next, we compared the turning performance of the robot with different tail beat trajectories. Four trajectories are selected with $γ$ varying from 2 to 12 cm for the turning performance exploration as shown in Figure 5A. Only the results of the trajectories with their maximum control angle $α_{c}$ are presented here, and the comparisons for several trajectories with different $α_{c}$ are described in Supplementary Figure S8 (see Supplementary Data S1).

FIG. 5.

Turning performance of the robot with different trajectories, revealing the robot can achieve the fastest turning rate and smallest turning radius simultaneously with the tail beat trajectory of a small radius and a large bending angle (red line in A). (A) Different control trajectories for turning. (B) Average turning speed comparisons of the robot with different control trajectories. (C) Turning radius comparisons of the robot with different control trajectories. (D and E) Comparisons of the turning angle in the time domain between the trajectory of 2 cm γ and 60° α_c and the trajectory of 12 cm γ and 150° α_c when tail beat frequencies are 1 and 4 Hz, respectively. (F) Comparison of the turning between the trajectory of a large radius (left) and the trajectory of a small radius (right). The tail beat frequency is 4 Hz. For the trajectory of a large radius, γ = 2 cm, and for the trajectory of a small radius, γ = 12 cm.

Figure 5B shows the results of the average turning speed for the robot under various control trajectories. We find that the variation trend of the average turning rate drops initially and then goes up significantly with the decrease in the radius of the tail beat trajectory. The maximum turning speed achieved by the trajectory of $γ$ = 6 cm is smaller than that of the other trajectories. And among all trajectories, the highest average turning speed is around 107°/s when the robot turns with the trajectory of $γ$ = 2 cm and the frequency of 4 Hz. Similar to the effect of the tail trajectory on the turning rate, the turning radius presents earlier increase and later decrease trend (Fig. 5C). The robot turns in the path with the largest radius when the $γ$ of the trajectory is 6 cm, while the robot driven by the trajectory of $γ$ = 2 cm shows the minimum turning radius at the frequency of 4 Hz. The minimum turning radius is 17 cm, equivalent to 0.31 BL per second.

Figure 5D further compares the turning angle in the time domain between the robot driven by the trajectories of large ( $γ$ = 2 cm) and small ( $γ$ = 12 cm) radii. At the frequency of 1 Hz, one cycle of the tail flapping can generate 49° of the turning angles during the loading stroke with a recovery angle of 26° during the return stroke for the robot with the control trajectory of $γ$ = 2 cm. While for the trajectory of $γ$ = 12 cm, the robot can turn 58° with one stroke of the tail flapping, and the recovery angle is just a half compared to the robot turns under the trajectory of $γ$ = 2 cm. When the frequency is 4 Hz, one stroke of the tail flapping generates almost the same turning angle for the robot with the control trajectories of $γ$ = 2 cm and $γ$ = 12 cm (28° and 29°, respectively). However, during the return stroke, the recovery angle of the trajectory of $γ$ = 12 cm is much less than that of the trajectory of $γ$ = 2 cm, which are 2° and 13°, respectively.

Figure 5F demonstrates the snapshots of the turning performance comparison between the robot turns with the tail beat trajectories of large and small radii under the frequency of 4 Hz (Supplementary Video SV5). And the turning comparison under the frequency of 1 Hz is provided in Supplementary Video SV4.

Furthermore, the deformed shapes of the elastic spine under different control trajectories of point E for the turning motion are shown in Figure 6. According to the results, as the values of both $γ$ and $α_{c}$ increase, the elastic spine exhibits increased curvature and decreased swing amplitude. Different deformed shapes of the elastic spine result in reorienting forces generated by the caudal fin. Larger curvature and smaller tail beat amplitude can reorient forces to achieve smaller forward thrust and generate a larger bending angle, which is beneficial for enhancing angular velocity and decreasing the turning radius, thus improving maneuverability. This advantage is due to the high controllability, continuous morphology, and elasticity of the spine, which is challenging to achieve using either discrete rigid structures or passive compliant mechanism.

FIG. 6.

Deformed shapes of the elastic spine under different control trajectories of point E for turning, demonstrating that increasing γ and α_c results in a larger bending angle and a smaller swing amplitude of the spine.

Performance comparisons with the reported robotic swimmers

The performance of the robot in this article is compared with several similar reported robotic swimmers.^{24,31,37–47} The performance indicators include swimming speed, energy efficiency, turning speed, and turning radius. To make the evaluation benchmark more reasonable, the differences in the achievable maximum tail beat frequency and the mass of robots need to be taken into consideration. We use the stride length ( $SL = V / f$ ) to reflect the swimming speed of the robotic swimmers, which denotes the traveled distance of the robot during one complete tail beat period.⁴⁸ Energy efficiency is characterized by the consumed energy for moving a unit mass over a unit distance (CoT/mass).⁴⁹

Figure 7A shows the comparison of the stride length. The average SL of fish varies from 0.35 to 0.93 BL/cycle,⁵⁰ the part highlighted by the gray color in the figure. Our robotic swimmer achieves the fastest average swimming velocity in the bistable locomotion mode when the precompression length of the spine is 2 cm. Thus, the maximum SL under each tested tail beat frequency ranges from 0.35 to 0.55 BL/cycle, which shows comparable swimming speed with the real fish in nature. And the maximum stride length (0.55 BL/cycle) achieved at the frequency of 1 Hz is higher than most compared platforms and only lower than one platform, active compliant propulsion mechanism (ACPM) robotic fish.⁴¹ But the length of the ACPM robotic fish is much smaller than that of our robot. Figure 7B compares the energy efficiency when the robots swim at the maximum velocity. It is clear to see that the robotic swimmer in this work consumes the minimal energy at its maximum average swimming speed when compared to other platforms, which means our designed robot can achieve fast speed with high energy efficiency. Our robot also exhibits excellent maneuverability when compared with other robotic swimmers (Fig. 7C). The fastest turning rate with the minimal turning radius demonstrates that its maneuverability is strongly superior to other platforms.

FIG. 7.

Performance comparisons between the robot in this study and similar robotic swimmers in the literature, demonstrating superior maneuverability and higher energy efficiency of the proposed “fishtail.” (A) Comparison of stride length. The part in gray color (0.35–0.93) is the average stride length of fish in nature. (B) Comparison of energy consumption for moving a unit mass over per unit distance (CoT/mass) at the maximal average swimming speed of the robots. (C) Comparison of maneuverability in terms of turning speed and turning radius.

Discussion and Conclusion

In this study, we developed a compliant robotic fish tail based on the bistable nonlinear mechanism, which combines a parallel linkage mechanism and an elastic spine. The active-controlled elastic spine endows the robot tail with high controllability and flexibility, which allows the robot to swim and turn with different predefined tail beat trajectories. The tunable and controllable bistability of the robot tail can perform two different motion modes freely and accurately (monostable and bistable modes).

First, the study on different stiffness combinations of the elastic spine and the caudal fin reveals that the spine thickness and caudal fin stiffness are two fundamental factors for improving swimming performance (see Supplementary Figs. S1 and Figs. S2), and it is noted that a soft spine combined with a soft caudal fin performs better in both speed and CoT when compared to a soft spine combined with other two stiffer caudal fins (see Supplementary Figs. S1 and Figs. S2). It is also noted that a stiff spine with a caudal fin of medium stiffness enables the robot to swim faster (see Fig. 3C), but no combination can exhibit the best energy efficiency over a wide range of swimming speeds (see Fig. 3D). At a lower speed, the robot with a soft spine and a soft caudal fin (compared to stiff or medium ones in Fig. 3B) is more energy efficient, while at a higher speed, the energy efficiency of the robot with a stiff spine and a moderately stiff caudal fin is better.

Second, tunable tail beating modes, benefiting from the accurately controlled spine, not only enable the robotic fishtail to take advantage of the bistability for effectively amplifying the thrust force (see Supplementary Fig. S7) by altering the caudal fin’s kinematics, such as varying the angle of the passive joint ( $ϕ_{2}$ ) through different compressed lengths of the spine and producing snapping motion, but also allow the robot to maintain higher energy efficiency at different swimming speeds through switching modes (see Fig. 4F). Compared to the traditional soft swimmers either using passive compliant tail⁵¹ or actuated by hydraulic soft tail,²⁴ our robot shows much better propulsive performance owing to its controllability of the tail trajectory and nonlinear stiffness. Compared with other bistable swimmers,^31,32 the tunable and controllable bistability makes the robot tail more flexible and powerful, which is beneficial to real applications.

Third, the maneuverability of the robotic swimmer can be largely enhanced by selecting the most suitable tail beat trajectory for turning. The actively controlled spine allows the robot to simulate the smooth and continuous motion morphology of fish, which is difficult to be realized by other robotic fish built by discrete rigid linkages.¹⁰ This property enables the robot to turn with the most efficient tail motion morphology like a real fish (the red tail beat trajectory in Fig. 5A). Based on the results of turning tests, the tail beat trajectory of a small radius and a large bending angle empower the robotic swimmer to have a faster turning rate (see Fig. 5B) and a smaller turning radius (see Fig. 5C).

The robotic swimmer successfully demonstrates its superior performance in terms of swimming speed, maneuverability, and energy efficiency through extensive comparisons with other similar robots in the literature (see Fig. 7). In the comparisons, several platforms have similar tail actuation mechanisms but can only realize single motor-actuated reciprocating motion with high tail beat frequency.^37,43–46 Thus, the swimming speed of these robots can be faster. However, the robots of this kind are difficult to realize turning motion, as a result, their maneuverability is relatively worse and some even can only swim forward. Moreover, most electronic components of a platform, Tunabot,³⁷ are placed outside the robot. All these mentioned drawbacks show a high restriction on practical applications. It is noted that the robot of this study can achieve a reasonable forward speed (although not the fastest because of the tail beat frequency), and it is more applicable to marine environments due to these features as discussed before including self-contained compact system, high energy efficiency, and excellent maneuverability.

Therefore, the proposed tail mechanism provides a more reliable way for reproduction of real fish swimming skills and the results definitely unveil a benchmark solution for the development of soft actuation of underwater robots. For example, using two such tails to imitate frog swimming, or using three or four tails to simulate the motion of octopus, etc., which will be investigated in further studies. In this article, we preliminarily and partially investigated the optimal configuration and control approach, including the preferable stiffness combination of the elastic spine and the caudal fin, switching motion mode to maintain efficient swimming across a wide range of swimming speed, and the most effective swing trajectory and deformed shape of the spine for achieving excellent steering capability. A more general structural and control optimization using model-based method or learning method will be conducted in future work. Additionally, to further improve the performance by reducing drag force, we plan to develop a more compact prototype using components with small size and cover the tail structure with an appropriate soft material in the future as well.

Footnotes

Acknowledgments

The authors appreciate and acknowledge the constructive comments and suggestions from anonymous reviewers.

Authors’ Contributions

X.C. developed the concept, conducted the simulations, designed the experiments, analyzed the data, and wrote the original draft. I.M. helped with the experiments. D.N.A. supervised and reviewed the draft. X.J. conceptualized and supervised the research; prepared, reviewed, and revised the article; and funded all materials and facilities needed.

Author Disclosure Statement

No competing financial interests exist.

Funding Information

This study was supported by a Shenzhen-HongKong-Macau Scheme-C fund (9240115), an Innovation and Technology Fund of Hong Kong Innovation and Technology Commission (ITP/003/24LP), a booster fund of City University of Hong Kong (7030015), and a startup fund from City University of Hong Kong (Ref. 9380140).

Supplementary Material

References

Sfakiotakis

, Lane

, Davies

JBC

. Review of fish swimming modes for aquatic locomotion. IEEE J Oceanic Eng, 1999; 24(2):237–252.

Wang

, Wang

, et al. Development and motion control of biomimetic underwater robots: A survey. IEEE Trans Syst Man Cybern, Syst, 2022; 52(2):833–844.

Di Santo

, Goerig

, Wainwright

, et al. Convergence of undulatory swimming kinematics across a diversity of fishes. Proc Natl Acad Sci USA, 2021; 118(49):e2113206118.

Scaradozzi

, Palmieri

, Costa

, et al. BCF swimming locomotion for autonomous underwater robots: A review and a novel solution to improve control and efficiency. Ocean Engineering, 2017; 130:437–453.

Marras

, Porfiri

. Fish and robots swimming together: Attraction towards the robot demands biomimetic locomotion. J R Soc Interface, 2012; 9(73):1856–1868.

Lauder

, Anderson

, Tangorra

, et al. Fish biorobotics: Kinematics and hydrodynamics of self-propulsion. J Exp Biol, 2007; 210(Pt 16):2767–2780.

, Chao

, Hameed

, et al. Biomimetic omnidirectional multi-tail underwater robot. Mech Syst Signal Process, 2022; 173:109056.

, Liu

, Wang

, et al. Bottom-level motion control for robotic fish to swim in groups: Modeling and experiments. Bioinspir Biomim, 2019; 14(4):046001.

Huang

, Ma

, Lin

, et al. Impact of caudal fin geometry on the swimming performance of a snake-like robot. Ocean Eng, 2022; 245:110372.

10.

Crespi

, Ijspeert

. Online optimization of swimming and crawling in an amphibious snake robot. IEEE Trans Robot, 2008; 24(1):75–87.

11.

Yan

, Wu

, Wang

, et al. Efficient cooperative structured control for a multijoint biomimetic robotic fish. IEEE/ASME Trans Mechatron, 2021; 26(5):2506–2516.

12.

, Nagy

, Graving

, et al. Vortex phase matching as a strategy for schooling in robots and in fish. Nat Commun, 2020; 11(1):5408.

13.

Nguyen

, Ho

. Anguilliform swimming performance of an eel-inspired soft robot. Soft Robot, 2022; 9(3):425–439.

14.

Marchese

, Onal

, Rus

. Autonomous soft robotic fish capable of escape maneuvers using fluidic elastomer actuators. Soft Robot, 2014; 1(1):75–87.

15.

Zhou

, Li

. Modeling and analysis of soft pneumatic actuator with symmetrical chambers used for bionic robotic fish. Soft Robot, 2020; 7(2):168–178.

16.

Yuk

, Lin

, Ma

, et al. Hydraulic hydrogel actuators and robots optically and sonically camouflaged in water. Nat Commun, 2017; 8(1):14230.

17.

Guo

, Fukuda

, Asaka

. A new type of fish-like underwater microrobot. IEEE/ASME Trans Mechatron, 2003; 8(1):136–141.

18.

Aureli

, Kopman

, Porfiri

. Free-locomotion of underwater vehicles actuated by ionic polymer metal composites. IEEE/ASME Trans Mechatron, 2010; 15(4):603–614.

19.

Chen

, Shatara

, Tan

. Modeling of biomimetic robotic fish propelled by an ionic polymer–metal composite caudal fin. IEEE/ASME Trans Mechatron, 2010; 15(3):448–459.

20.

Lou

, Gu

, Chen

, et al. Effects of actuator-substrate ratio on hydrodynamic and propulsion performances of underwater oscillating flexible structure actuated by macro fiber composites. Mechanical Systems and Signal Processing, 2022; 170:108824.

21.

Barbosa

, da Silva

. Macro fiber composite-actuated soft robotic fish: A gray box model-predictive motion planning strategy under limited actuation. Soft Robot, 2023; 10(5):948–958.

22.

, Chen

, Zhou

, et al. Self-powered soft robot in the Mariana Trench. Nature, 2021; 591(7848):66–71.

23.

Wang

, Zhang

, et al. Fast-swimming soft robotic fish actuated by bionic muscle. Soft Robot, 2024; 11(5):845–856.

24.

Katzschmann

, DelPreto

, MacCurdy

, et al. Exploration of underwater life with an acoustically controlled soft robotic fish. Sci Robot, 2018; 3(16):eaar3449.

25.

Tissot-Daguette

, Schneegans

, Thalmann

, et al. Analytical modeling and experimental validation of rotationally actuated pinned–pinned and fixed–pinned buckled beam bistable mechanisms. Mech Mach Theory, 2022; 174:104874.

26.

Currier

, Lheron

, Modarres-Sadeghi

. A bio-inspired robotic fish utilizes the snap-through buckling of its spine to generate accelerations of more than 20g. Bioinspir Biomim, 2020; 15(5):055006.

27.

Koh

J-S

, Yang

, Jung

G-P

, et al. Jumping on water: Surface tension–dominated jumping of water striders and robotic insects. Science, 2015; 349(6247):517–521.

28.

Tang

, Zhang

, Pan

, et al. Bistable soft jumper capable of fast response and high takeoff velocity. Sci Robot, 2024; 9(93):eadm8484.

29.

Chen

, Bilal

, Shea

, et al. Harnessing bistability for directional propulsion of soft, untethered robots. Proc Natl Acad Sci U S A, 2018; 115(22):5698–5702.

30.

Tang

, Chi

, Sun

, et al. Leveraging elastic instabilities for amplified performance: Spine-inspired high-speed and high-force soft robots. Sci Adv, 2020; 6(19):eaaz6912.

31.

Xiong

, Su

, Lipson

. Fast Untethered Soft Robotic Crawler with Elastic Instability. IEEE: 2023.

32.

Chi

, Hong

, Zhao

, et al. Snapping for high-speed and high-efficient butterfly stroke–like soft swimmer. Sci Adv, 2022; 8(46):eadd3788.

33.

Wang

, Qiao

, Li

, et al. Jellyfish-inspired soft robot driven by pneumatic bistable actuators. Soft Robot, 2024.

34.

Bambrick

, Viquerat

, Siddall

. Does bistability improve swimming performance in robotic fish? Advanced Intelligent Systems, 2024; 6(6).

35.

Zhong

, Zhu

, Fish

, et al. Tunable stiffness enables fast and efficient swimming in fish-like robots. Sci Robot, 2021; 6(57):eabe4088.

36.

Chen

, Wu

, Dong

, et al. Exploration of swimming performance for a biomimetic multi-joint robotic fish with a compliant passive joint. Bioinspir Biomim, 2020; 16(2):026007.

37.

Zhu

, White

, Wainwright

, et al. Tuna robotics: A high-frequency experimental platform exploring the performance space of swimming fishes. Sci Robot, 2019; 4(34):eaax4615.

38.

Chen

, Jiang

. Swimming performance of a tensegrity robotic fish. Soft Robot, 2019; 6(4):520–531.

39.

Liu

, Liu

, Liang

, et al. Tunable stiffness caudal peduncle leads to higher swimming speed without extra energy. IEEE Robot Autom Lett, 2023; 8(9):5886–5893.

40.

Chen

, Jiang

. Body stiffness variation of a tensegrity robotic fish using antagonistic stiffness in a kinematically singular configuration. IEEE Trans Robot, 2021; 37(5):1712–1727.

41.

Zhong

, Li

, Du

. A novel robot fish with wire-driven active body and compliant tail. IEEE/ASME Trans Mechatron, 2017; 22(4):1633–1643.

42.

Liao

, Zhou

, Zou

, et al. Dynamic modeling and performance analysis for a wire-driven elastic robotic fish. IEEE Robot Autom Lett, 2022; 7(4):11174–11181.

43.

, Chen

, Zhang

. Snapp: An agile robotic fish with 3-D maneuverability for open water swim. IEEE Robot Autom Lett, 2023; 8(10):6499–6506.

44.

Chen

, Wu

, Meng

, et al. Development of a high-speed swimming robot with the capability of fish-like leaping. IEEE/ASME Trans Mechatron, 2022; 27(5):3579–3589.

45.

Wang

, Zhou

, Wang

, et al. Toward propulsive performance evaluation of a robotic tuna based on the damping-elastic composite mechanism. IEEE/ASME Trans Mechatron, 2024; 29(3):1809–1819.

46.

Liao

, Zhou

, Wang

, et al. A wire-driven dual elastic fishtail with energy storing and passive flexibility. IEEE/ASME Trans Mechatron, 2024; 29(3):1914–1925.

47.

, Wu

, Wang

, et al. Design and control of a two-motor-actuated tuna-inspired robot system. IEEE Trans Syst Man Cybern, Syst, 2021; 51(8):4670–4680.

48.

White

, Lauder

, Bart-Smith

. Tunabot flex: A tuna-inspired robot with body flexibility improves high-performance swimming. Bioinspir Biomim, 2021; 16(2):026019.

49.

Bujard

, Giorgio-Serchi

, Weymouth

. A resonant squid-inspired robot unlocks biological propulsive efficiency. Sci Robot, 2021; 6(50):eabd2971.

50.

Videler

, Wardle

. Fish swimming stride by stride: Speed limits and endurance. Rev Fish Biol Fisheries, 1991; 1(1):23–40.

51.

Kopman

, Porfiri

. Design, modeling, and characterization of a miniature robotic fish for research and education in biomimetics and bioinspiration. IEEE/ASME Trans Mechatron, 2013; 18(2):471–483.

Supplementary Material

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