Does Bistability Improve Swimming Performance in Robotic Fish?

A bistable mechanism has two stable states with energy input required to move from one stable state to another. This energy barrier allows for energy storage and release which can be used to improve systems characteristics. Bistability has been used to increase the frequency range over which a kinetic energy harvester is effective, and it has been proposed that bistability can increase the efficiency of biomimetic swimming robots. However, experiments involving bistable swimming robots have typically used bistability as a means of overcoming limitations inherent to soft actuators, rather than to increase overall performance. This article implements bistability into a swimming robotic and compares performance with and without bistable action. The static thrust generation and power consumption for bistable and nonbistable configurations for five different tail morphologies are compared. Bistability is generally found to increase the system efficiency, particularly at lower frequencies where increases are observed up to 250%. The untethered swimming speed of the robot in open water is also found to increase by approximately 30%. The results show that bistability can offer direct performance benefits for biomimetic swimming, but that the bistable transmission must be well tuned to the dynamics of the rest of the system.


Introduction
Unmanned underwater vehicles (UUVs) are a class of autonomous vehicles that have become increasingly important for use in marine observation, [1] surveillance, [2] reconnaissance, [3] and research. [4]UUVs can adopt a biomimetic propulsion system, swimming through oscillatory fanning.This swimming type is ubiquitously found across most fish species.Several recent studies have investigated and compared marine thruster propulsion and oscillatory fanning for use in marine observation UUVs.The advantages of oscillatory propulsion systems include reduced vibrations due to lower operating frequencies, which cause less disturbance to marine behavior and improved sensor data quality; [5] removed risk of damage to environment/ marine life by propellers; and higher operating efficiencies, allowing for longer mission durations or reduced energy consumption. [6]he utility of a biomimetic fish robot will depend greatly on its efficiency and endurance.One potential method to increase the efficiency of an oscillatory fanning swimmer is the implementation of bistability into the swimming mechanism.Bistability refers to a mechanism with two distinct stable points, or energy wells.Between these two points, there will exist a local energy maximum, or metastable point.At the metastable point, any perturbation will cause the mechanism to snap into one of its stable positions, releasing stored energy.By using this energy release to increase the maximum velocity of the tail through the water, bistability has the potential to improve the swimming efficiency and swimming performance of oscillatory swimmers, as well as allowing the use of novel actuators using soft and smart materials with otherwise unsuitable force-displacement characteristics.
This article aims to investigate the incorporation of bistability into the oscillating tail of a biomimetic robotic swimmer and its impact on the efficiency and swimming characteristics of the oscillatory propulsion system.

Swimming Mechanisms
Fish locomotion is typically either ostraciiform, involving only the tail and caudal fin, or carangiform, with movement across the body (with the greatest amplitude at the caudal fin). [7,8]his article focuses on ostraciiform locomotion for its mechanical simplicity and potential in robotic swimmers, specifically examining the caudal fin as it generates the majority of thrust (anywhere from 45% to 84% of thrust depending on the species and testing methodology [9] ).
While bistability is not a feature in natural fish locomotion (which favors more sinusoidal motion patterns), bistability has improved energy harvesters, [10] and its application in robotic swimming may enhance thrust by maximizing tail velocity. [11]tudies show that bistability could broaden the operational frequency range of energy harvesters and increase power output, suggesting similar available benefits for robotic swimming by converting oscillatory motion into propulsive thrust more efficiently. [12]

Bistability in Swimming Robots
Biomimetic fish-like swimming is an active topic of research across several fields and is a point of focus for soft robotics research, [13] owing to the importance of tuned compliance to effective thrust production. [14]Several studies have investigated the implementation of bistability in swimming robots.Tang et al. [15] investigated the impact of implementing bistability into a crawler, a swimmer, and a gripper.Within this study, bistability was implemented using a preloaded spring and actuated via pneumatic channels.This study found that the implementation of bistability in the robotic swimmer increased the swimming speed from 0.59 BL s À1 (body lengths per second) to 0.78 BL s À1 for an equal driving frequency and actuation pressure (a 32.2% increase at 1.3 Hz).However, there is no discussion regarding any tuning or impedance matching of the system and bistable mechanism, with only one configuration tested.In another study, [16] the effect of bistability in a rajiform soft swimmer was investigated.In this study, bistability was induced using precurved polyester ribbons connecting to two pneumatic actuators.This study generated the fastest robotic swimmer published at the time of writing, with a maximum speed of 3.67 BL s À1 achieved.Both studies on the implementation of bistability into a robotic swimmer aimed to use bistability to address one of the shortcomings of soft robotic mechanisms: the inability to generate high-speed high-force action.As such they were not trying to use bistability to provide maximal efficiency in an absolute sense, but rather circumvent limitations in a particular type of actuator.
For an object propelled by an undulating foil, the force produced is proportional to the kinetic energy of the fluid moved downstream by the foil's beating, and thrust scales with the amplitude of the oscillation multiplied by the angular velocity squared. [17]The addition of bistability with a fixed maximum amplitude increases the peak velocity of the foil, essentially changing the shape of the motion from a sinusoidal waveform to a clipped triangular wave, and in the limiting case, a square wave.A given increase in velocity at a fixed frequency will result in a proportionate increase in the time at which the foil is stationary at the limits of its travel (and not producing thrust), but a squared increase in thrust produced while the foil is in motion, so to a first approximation, bistability should increase thrust overall.However, if a passive, flexible foil is used, the increased velocity may also result in inefficient kinematics and lost thrust due to fluid being accelerated in the wrong direction.A fixed amplitude actuator reaching the limits of its travel at high speed and dissipating energy as friction will further decrease efficiency.
This study aims to examine whether bistability is a fundamentally useful feature of swimming mechanisms or rather a tool to overcome other mechanical or actuation limitations.It investigates and quantifies the effects of introducing bistability into the swimming mechanism of a simple bioinspired propulsion system driven by a single electric motor.It includes the modular design of a swimming mechanism featuring a bistable transmission that can be removed/replaced; thrust and power data from the swimming performance of this mechanism in its bistable and nonbistable drive configurations, showing improvements in both thrust and efficiency; and validation of the performance of this propulsion system in a robot freely swimming outdoors, showing a speed increase of 30% for the same input motion.

Experimental Section
A test platform that could be switched from a bistable to nonbistable configuration was constructed to isolate the effect of adding bistability to various swimming robot morphologies.

Bistable Swim Mechanism Design
When designing the experimental model, various options were considered, with a primary focus on selecting a suitable bistability implementation method.Bistability in a joint can be achieved through several techniques, commonly involving materials and mechanisms that enable energy storage and release.These techniques include adding elastically deformable structures such as coil springs and permanent magnets.In this case, magnets were chosen for their simplicity and lack of moving parts, although a mechanical solution would also suffice.
Previous bistable robotic projects have typically employed either rotary actuation, such as motors and servos, or pneumatic actuation using airlines and pumps.The primary concern in the selection of power delivery was the ability to untether the system for free-floating experiments.Pneumatic/hydraulic actuation necessitates pumps, tubing, batteries, and circuitry, whereas motor actuation simplifies the system.A single servo was chosen over multiple antagonistic actuators [18] for simplicity.Figure 1 shows the resulting propulsor design.It was decided that for ease of manufacturing the motor itself would be waterproofed with the control circuitry sitting above the water level during testing.
The nonbistable (hereafter referred to as "direct drive") configuration of the swimming robot was based on a pin slot mechanism, selected for its simplicity.The nonbistable mechanism is shown in Figure 1D.The pin in the direct-drive mechanism, as shown above, was manufactured using an M3 hex bolt with the threads filed and polished to reduce friction within the slot.
The bistable model features four 10 Â 2 mm cylindrical neodymium magnets, with a 12.5 N peak holding force (see Appendix 1).Two of these magnets are fixed into the tail on either side and two are fixed into the body on either side of the fishtail hinge (Figure 1C).The magnets are oriented to attract each other, making the tail unstable in its center position.The two walls of the experimental model that limit the rotation of the model were designed to be 60°apart.Attached to the servo motor is a 3D-printed fork that pushes the tail between its two stable states.The fork is designed such that the tail could be free to snap through between its two stable states unimpeded (Figure 1E).
To determine how the tail geometry could impact the implementation of bistability, several different tail types were designed, manufactured, and evaluated to determine their thrust-generating characteristics.Of these tails, three were fully rigid with lengths of 78, 98, and 118 mm, and two were created with varying stiffness to include the compliant peduncle found in many fish species.The compliant sections are formed from 0.25 mm carbon fiber sheet, while rigid sections are manufactured using a FDM 3D printer and PLA filament.All other custom components are also printed in PLA, with M3 screws used for fastening.
The motor used is an IP67-rated brushless servo (DFRobot SER0062), with an unloaded speed of 83 rpm, a stall torque of 4.4 Nm, and a peak mechanical power output of 9.6 W. A brushless model was chosen so that there was no potential short circuit from a brushed commutator, and the servo was waterproofed [19] to survive continuous immersion by coating the internal PCB in acrylic lacquer and filling the rest of the servo cavity with mineral oil.The servo is supplied with 5 V by either a mains adaptor or a 3.7 V lithium battery and boost converter.The servo's motion is controlled by a Cortex-M0 32-bit SAMD21 microcontroller (Arduino Nano IoT), with a current sensor monitoring motor power consumption and a magnetic angle sensor (AMS AS5048B) recording the motion of the tail output.The additional magnets used to create bistability did not have an observable effect on the magnetic encoder readings.
Within the experimental model, the oscillation frequency and amplitude could be controlled to assess the ranges where potential advantages may be observed.The code for the swimming robot was written in Cþþ to control the movement of the tail and to process the data from the current and position sensors.The motion of the driving servo was programmed to be sinusoidal.Sensor data were recorded in MATLAB over a serial connection.

Tail Position Tracking
The magnetic angle sensor allowed for analysis of the tail's position, velocity, and acceleration while swimming.The sensor magnet was connected to the tail over the axis of rotation and the sensor was mounted on the body of the swimming robot using two printed brackets (Figure 2A).
Using the position sensor, displacement responses for the experimental model were collected with the fish oscillating in the air.In Figure 2B,C, the displacement response of the swimming robot in both the direct-drive and bistable configuration moving at 1.5 Hz can be seen.It is observed that the tail experiences a rapid deceleration as it collides with the body of the fish.This rapid deceleration is audible in the air and from the displacement curve induces some vibration into the tail.It is also seen that due to the implementation of bistability, the amplitude of oscillation is increased to the full bounds of AE30 ∘ .
The average angular velocity of the displacement curve was then calculated and compared for the direct-drive and bistable configuration with the robot immersed in water (Figure 1B), across 10 repeated cycles of 50 000 data points, or approximately 66 oscillations.Table 1 gives the root mean square (RMS) angular velocities of the 10 repetitions.From the angular displacement tables, the angular velocity of the tail is higher in the bistable configuration than in the direct-drive configuration.The bistable tail displacement resembles a sawtooth wave, due to the additional acceleration provided by the magnets.

Thrust and Current Measurements
To determine the potential advantages of bistability in the robotic swimmer the robot was tested in a series of underwater swimming experiments.An experimental apparatus consisting of a water tank and a 10 kg load cell was designed and built to facilitate these experiments (Figure 3).The arm used to mount the swimming robot to the axle also acted as a moment arm, increasing the measured force and reducing noise.Using this testing apparatus, the thrust force generated and the associated current draw could be gathered and analyzed.Testing was controlled by a MATLAB script which automatically cycled through the drive frequencies while collecting current and force data.Within this experiment, the fish is constrained from rotating and moving in all but the transverse direction to the tail's oscillation.While this is very useful to isolate the thrust as a parameter in the system, it does prevent the more complex dynamics of the system such as any yawing or spinning generated due to the oscillation of the tail.To measure the effect of these dynamics a further, free-floating experiment was conducted (see Section 4).

Results
An automated script ran the thrust experiments, cycling through 16 frequency values and amassing 8000 data points for each tail and bistability configuration, gathering 80 000 data points in total.Each 300-point collection spanned roughly 10 s, depending on the computer's processing speed, with a 30-second pause added between frequency changes for system equilibriation.This pause, when extended, did not significantly affect the results' standard deviation.Mean and standard deviation were calculated for the current and force readings per frequency, averaged over five cycles.Initial tests without the tail attached (Figure 3C) set a baseline current draw of 21 mA due to internal motor friction, but no net on the load cell confirming the rig's accuracy.
The results show the thrust generated by the short rigid tail.The results (Figure 4A) indicated a steady increase in thrust and current in the direct-drive mode, contrasting with the variable thrust in the bistable mode.For example, at 1 Hz, the direct drive generated 0.35 N of thrust against 0.15 N in the bistable configuration; at 1.1 Hz, the bistable configuration unexpectedly jumped to 0.55 N compared to the direct drive's 0.45 N.Both configurations showed a similar current draw pattern, increasing from about 150 to 900 mA.The medium-length rigid tail (Figure 4B) in the direct-drive configuration shows a smooth increase in the thrust generated as the frequency increases.In the bistable configuration, the thrust generated increases at a faster rate.Furthermore, while in the direct-drive configuration, the thrust plateaus at around 0.85 N at 1.4 Hz, in the bistable configuration, the thrust generated increases almost linearly with the increase of the driving frequency.At 1.5 Hz, the bistable configuration generates just above 1 N of thrust compared to 0.8 N for the direct-drive configuration.This is also achieved at a slightly lower average current draw at just below 900 mA compared to around 950 mA for the direct-drive configuration.
For the long-length rigid tail (Figure 4C) in the direct-drive configuration, the thrust generated rises sharply initially before tapering out as the driving frequency increases.In the bistable configuration, the thrust increases more linearly with the increase of frequency before reaching a maximum thrust value of 1.15 N at 1.2 Hz and decreasing sharply above 1.3 Hz.For 1.2 Hz, the thrust in the bistable configuration is 15% higher than that of the direct-drive configuration for a current draw that is 5% lower.The standard deviation for the two thrust curves is slightly larger in the bistable configuration for the middle to lower frequency values however for the higher frequencies the direct-drive configuration has a higher deviation between cycles.
Different tails showed distinct trends: the medium-length rigid tail in bistable mode generated thrust more rapidly than in direct drive, and with a slightly lower current draw.A partially flexible peduncle attached to a medium-length tail (Figure 4D) resulted in the smoothest thrust increase in direct drive, while the bistable mode showed erratic behavior with a higher thrust peak.The long-length rigid tail demonstrated a sharp initial thrust increase in direct drive, which plateaued, whereas, in the bistable mode, it increased linearly before dropping after 1.3 Hz.
Finally, with the long-length fully flexible tail in the directdrive configuration (Figure 4E), the thrust generated quickly rises to a maximum thrust value of around 1.1 N at a frequency of 1.2 Hz.In the bistable configuration, the maximum thrust generated is slightly lower at a maximum thrust of just under 0.9 N generated at 0.9 Hz.The current draw for the two configurations is very similar, rising almost linearly with the bistable current being slightly lower than the direct drive current across all frequency values.

Comparison of Direct-Drive and Bistable Transmissions
Figure 5 and 6 show a comparison between the direct-drive and bistable mechanism for the different tail configurations.These plots use the same data as the previous set of plots, however, a "thrust efficiency" coefficient has been calculated here, taking the thrust value divided by the input power, giving a parameter in terms of Newtons per Watt with a high value indicating better efficiency of the mechanism in generating thrust.Bistability provided thrust efficiency improvements at certain frequencies for all tail types, with significant improvements at lower speeds.In all cases other than the fully flexible tail (Figure 5C), a bistable drive produced the highest achievable thrust efficiency.The partially flexible medium-length tail showed increased efficiency at low frequencies (Figure 6B), while the fully flexible long-length tail reached the maximum efficiency found across all tests (for both bistable and direct drive), but was noticeably less efficient at higher speeds, highlighting the importance of stiffness tuning to undulatory propulsion.

Untethered Swimming Experiment
To test the nonconstrained swimming performance of the swimming robot, a free-floating swimming test was conducted.To facilitate this test, the swimming body of the fish was attached to a model boat to provide buoyancy and stability.The model was then allowed to swim in open water to analyze its swimming performance.The robot setup can be seen in Figure 7A.
The tests were conducted over two days at the University of Surrey Lake.This was used as it was the nearest body of water without any significant water current.In Figure 7B, the swimming location can be seen from a satellite view with the swimming path highlighted in red.
To remotely control the swimming robot, the robot driving code was altered to facilitate wireless control via a Bluetooth controller (taken from an Xbox One).Using this controller, the device parameters could be altered such as the oscillation center point, the driving frequency, and the driving amplitude without requiring a connection to a computer.
The swimming tests were captured using a video camera on a tripod and the footage was analyzed using the open-source software "Tracker" by Open-Source Physics (example tracking is shown in Figure 7D).From the videos, an average speed can be calculated for the swimming robot.Within the below swimming runs, the swimming robot was run at 1.5 Hz with the medium-length partially flexible tail affixed.
The swimming tests were conducted six times for each configuration over two days.Three of the tests were conducted east to west, the other three west to east.This was done to remove the   possibility of environmental factors such as wind and water currents affecting the swimming speed of the fish.
The results of the outdoor tests (Figure 7C) indicate that through the implementation of bistability, the average swimming speed of the swimming robot was increased.The significance of the results gathered was verified with a two-sample t-test, showing a statistically significant change in swim speed (p = 0.001).It is important to note that this swimming test was only conducted for one frequency and one tail type.In the thrust-current experiment, this configuration did not show significant thrust improvements for this tail configuration and frequency (1.5 Hz, medium-length partially flexible tail, see Figure 6B).Despite this, the average swimming velocity is significantly increased for this free-floating case, showing that the static thrust tests do not capture all of the potential performance changes from a bistable drive (Table 2).

Discussion
The primary objective of this article is to evaluate any enhancement of the swimming efficiency of a biomimetic swimming robot through the implementation of bistability.This enhancement would be manifested in a UUV as an increase in running time or swim speed.
From the current-thrust experiment results shown in Section 2.3, a significant increase in the thrust generated by the swimming robot in the bistable configuration can be seen.Furthermore, there is generally a slight reduction in the current draw for the bistable configuration.These two factors combine to enhance the efficiency coefficient for four out of the five tail configurations within a certain frequency range.
The thrust-current tests indicate that the partially flexible tail configuration is the most effective in maximizing the thrust output of the swimming robot.With this tail, bistable configuration shows increased efficiency within the lower operating frequency range.However, in the higher frequency range (0.9-1.5 Hz), the efficiency coefficient becomes similar for both bistable and nonbistable configurations.
Another potential benefit of incorporating bistability into the system is the ability to increase the oscillating amplitude without requiring a larger range of movement from the motor.The bistable snap-through action allows for a lower range of servo motor movement while still achieving a full-range oscillation.This not only offers potential power savings but also enables higher flapping frequencies due to the reduced travel requirement of the motor.
The implementation of bistability also requires a tuning of the bistable forces to ensure that kinetic energy built up through energy release is not wasted through a collision with the retaining wall of the mechanism.The addition of an elastic energy return at the boundary walls of the bistable model could facilitate an increase in efficiency by reducing the energy lost to the walls of the model at frequencies where the tail reaches the limits of its motion and the paired magnets collide.This could be achieved using a spring mounted at either wall or an elastic material damper fixed to the wall.
The implementation of bistability into the swimming mechanism of a robotic swimmer has drawbacks for the swimming dynamics.One issue is the need to match the bistable behavior to the system's dynamics.The fully flexible tail configuration (Figure 4E) shows a loss of thrust in the bistable configuration at higher driving frequencies.This occurs because a flexible tail cannot transmit enough force to the caudal fin from the bistable hinge, and when the mechanism snaps through, the tail moves too slowly and occasionally remains on the wrong side resulting in chaotic motion.This effectively creates a maximum drive frequency and ultimately will limit robot speed.
Another disadvantage of implementing bistability is the mechanical disconnection required between the actuating motor and the tail to allow unrestrained snap-through.This reduces control over the tail amplitude and limits the ability to deflect the tail statically to one side (e.g., for steering).Avoiding this would be possible, but may be mechanically complex.Ideally, a bistable mechanism would provide for adjustable oscillation amplitude, coupled with the ability to enable or disable the bistable action to some extent.
The most effective bistability stiffness would be a function of the damping force, the bistable stiffness, the driving frequency, and the driving amplitude.The driving frequency of the tail is the parameter most likely to be variable throughout the operation of the mechanism, as this would be the main means of changing swimming speed in a robot.Because of this, the bistable force would be required to change as a function of the frequency for the efficiency advantages to be maintained when adjusting frequency.This implements bistability to improve swimming efficiency difficult as the common methods of creating bistability are not adjustable, including the system presented here.

Conclusion and Future Work
From the experiments conducted, it is apparent that the implementation of bistability can increase the thrust generated in a swimming mechanism, via an increased maximum speed and an ability to accelerate the output at high frequencies despite limited input torque.While the overall gains demonstrated in this article are modest, the system presented is primarily intended to offer a side-by-side comparison, and despite not being tuned for optimal performance, at certain speeds bistability offers efficiency gains of up to 250%.Given that the implementation of bistability is a low-cost design addition compared to a motor upgrade, it seems likely that there would be circumstances where bistability is a logical implementation.While using a more capable motor could increase the system's maximum velocity, acceleration, and operating efficiency, it would also raise the overall system cost.Therefore, including bistability presents a cost-effective option to enhance the operating performance of a system limited by the characteristics of its driving motor.
For the bistable implementation to be most effective at increasing the efficiency of the swimming system, the bistability must be tuned to match the system.Otherwise, the additional energy input by the motor will only be lost through a kinetic collision with the system geometry, or output motion that is poorly matched to the physics of the output motion.
To further develop this area of research, several improvements could be made to the experimental model.The first improvement is adding the ability to adjust the bistability strength.This would enable tuning of the system through the adjustment of the bistable force.This could potentially be achieved using a spring with adjustable pretension or an electromagnet with adjustable strength, although the latter would be energetically unsuitable for a mobile robot.A second improvement to the experimental model could be the ability to adjust the angle of the tails' sweep between the bistable limits.Using this, the bistable amplitude could be adjusted to match that of the most efficient swimming parameters.
Additionally, the servo used throughout the project to power the experimental model featured an internal position controller.It is possible that this controller created changes in the actuation force profile that were not captured by the experiment.The setup could potentially be improved with a custom drive train where all underlying parameters were known and adjustable.
Further, in-water testing would be required to fully understand the effects of bistability in the swimming mechanism of a swimming robot.Conducting further tests of the swimming robot, potentially in a large indoor tank free of external factors like wind and current would better allow for the detection of subtle differences in the swimming dynamics of the bistable and nonbistable system.
Although this work primarily focuses on oscillatory swimmers, the concept of bistability and its influence on oscillating motion extends to various mechanisms, robots, and machines.Gaining a better understanding of bistability and its effective utilization may lead to significant advancements in diverse fields, including energy-efficient transportation systems, [20] adaptive structures, [21] and innovative medical devices. [22]

Figure 1 .
Figure 1.Schematic of propulsion system used for testing the effects of bistability on swimming.A) Computer aided design (CAD) model of fish setup, showing servomotor, tail, and transmission.B) Image of propulsion system in a water tank.C) Detailed view of the bistable transmission, showing paired magnets used to create bistability, with one fixed magnet attracting another magnet attached to the tail hinge.D) Undulation cycle in direct-drive mode, with the tail connected directly to servomotor via a slotted link mechanism.E) Undulation cycle in bistable drive mode, with the tail hinge pushed by the servo output, but free to accelerate toward the fixed magnet once it is pushed past the center point.

Figure 3 .Figure 2 .
Figure 3. Experimental setup for thrust testing fish propulsor.A) Schematic of the setup, with the fish attached to a rotating rod, pushing against a load cell.B) Photograph of the test rig.C) Test with no tail attached to the motor, showing no net thrust measured and the current used by the motor overcoming its own internal friction at increasing drive frequencies.

Figure 6 .Figure 7 .Figure 5 .
Figure 6.The effect of bistability on swimming performance at different tail stiffnesses (note that the data in Panel A is repeated from Figure 5C): A) Rigid tail.B) Partially flexible tail.C) Fully flexible tail.

Table 1 .
RMS angular velocity measurements for direct-drive and bistable transmissions.

Table 2 .
Testing the direct-drive and bistable tails in a free-swimming test shows a 30% mean increase in speed using bistable actuation (p = 0.001, Student's t-test).