The propulsors of organisms from paramecia to dolphins have ball-and-socket jointed bases that allow large-amplitude, low-friction swings. Their olivo-cerebellar control also remains unchanged. Yet, the propulsive surfaces of small animals vary widely from flagellar filaments (0 < Re < 5) to flapping fins (Re > 20) with an intermediate range of Reynolds number (5 < Re < 20) where both types are present in the same swimming animal. Analysis suggests that these unsteady surfaces are mechanical oscillators coupled to their nonlinear wakes. A low-friction-driven oscillator that can interact with the oscillators of models or live swimming and flying animals could help us understand the hydro-structural events prompting the evolution of such surfaces at specific Re values. A gearless underdamped (in air) hemispherical motor oscillator is described where energetic efficiency increases by a factor of eight as the forces drop by a factor of ten from 10 N. The electrical efficiencies at 0.8 N are comparable to the total thermal efficiencies of flies, and the quality factor is comparable. The continuously varying fin oscillation of penguin fins and abruptly varying fin oscillations of Clione antarctica and flies are reproduced. When flapping at 0.3 Hz, the oscillator would respond to all wake nonlinearities. Abrupt fin turning is modeled by switching the roll and pitch phase difference between $\u2212\pi /2$ and $\pi /2$ in successive quadrants. Defining the fish-wake lock-in error as the difference between Triantafyllou's fish Strouhal number and the tangent of the vortex-shedding angle, an experiment is discussed for measuring the minimum drag of live fish.

## Introduction

In the context of swimming, hydrodynamics helps us to go to the right place, at the right time, at the right cost, and in the right manner. Hydrodynamic laws give us the forces and moments produced on the platform that carry us, and the oscillatory forces and acoustic radiation they produce in the surrounding, particularly in the wake. During the motion, the body interacts with the medium usually in unsteady manner. However, in modeling or design, fluid-structure interaction is frequently ignored, or insufficiently treated, for lack of knowledge of unsteady hydrodynamics, of time-responsive but fatigue-resistant materials, and of control that is highly responsive. These interacting complexities have been addressed optimally in swimming animals. Invariably, the motion of these animals is unsteady. Engineering may benefit by understanding the underlying principles of such unsteady motion. For example, if we design oscillatory surfaces to interact with the fluid *freely*, then the hydrodynamic efficiency is likely to go up than our current abilities.

Conventional underwater fluids engineering propulsors have time-independent propulsive surfaces. This contrasts with swimming animals which have oscillatory propulsive surfaces. This is because the motion controller in animals is oscillatory, and the propulsive surface and its wake have similar self-regulating mechanisms [1–4]. Engineers avoid loading oscillatory surfaces for fear of fatigue failure. However, oscillatory surfaces allow quick response at small spatial scales. In our laboratory, it has been possible to determine the force and moment control laws of rigid cylindrical and delta shaped bodies of 1 m in length to which multiple large flapping fins are attached [4–6]. Animal-like spatio-temporal responses have been routinely demonstrated both in laboratory and in littoral environments [4]. However, this progress is limited to flapping fins whose chord (*c*) Reynolds number (Re* _{c}*), which is the ratio of destabilizing inertial forces to stabilizing viscous forces, is greater than 40 × 10

^{3}[7]. It was observed that for Re

*< 40 × 10*

_{c}^{3}, the propulsion mechanism is chaotic and not well understood. This range is “gray” in the sense that there is competition between the destabilizing and stabilizing forces, with sensitivity to initial conditions. In other words, the individual thrust oscillations do not “lock-in” because of the excess energy in the system per cycle of oscillation [7]. In a recent study in our laboratory on a conventional propulsor but with vibrating blades at transitional values of 3.75 × 10

^{3}≤ Re

_{c}≤ 3.75 × 10

^{4}, it has also been found that the emergence of patterns of disorganized limit cycles of thrust is episodic and not continuous [8]. This suggested that smaller animals have more initial condition-dependent mechanisms. It is then justified to suggest that

*new types of flapping fins evolved intermittently*. This extraordinary sensitivity to initial conditions may have engineering potential in terms of platform response to transients. The present work offers an oscillator for the exploration of these chaotic behaviors of small swimming and flying animals.

For a propulsive surface to be truly chaotic with large excursions like a double pendulum in air, its internal frictional damping must be low. Because of high internal friction, the geared orthogonal cylindrical motor assembly that we use at Re* _{c}* > 40 × 10

^{3}to produce roll ($\varphi $) and pitch ($\theta $) oscillations in flapping fins has very low quality factors (a ratio of total to dissipative energy per cycle). For this reason, for Re

*< 40 × 10*

_{c}^{3}, we seek to explore a gearless pendulum-like configuration that will permit abandoning the common cylindrical motors.

Moreover, in our work so far at Re* _{c}* > 40 × 10

^{3}using single flapping fins, we generate planar (two-dimensional) yawing but not three-dimensional turning as in barrel roll maneuvers [5]. Hence, it would be useful to make our proposed oscillator with higher degrees-of-freedom (DOFs) so that it would be amenable to three-dimensional oscillations. All of these requirements lead to a hemispherical motor design much like the joint many animals possess to actuate their pectoral limbs. A ball-and-socket joint is critical to such a design (Figs. 1(a)–1(b)). The ball-and-socket joint occurs widely in swimming, flying, and other animals. This suggests a common evolutionary origin.

### This Work.

Here, we develop an underdamped hemispherical motor [9] for biorobotic experiments and show that the typical motion trajectories of propulsive actuators in swimming animals over a wide range of laminar and transitional inertial forces can be reproduced in principle with comparable efficiency, and friction as compared with current alternatives plus several advantages over those alternatives. We model the hydrodynamics of ball-and-socket-jointed mechanical oscillators and verify the damped oscillation properties.

In fish swimming and unsteady wake flows, we offer a new definition of the Strouhal number (St) based on the wake vortex shedding angle as opposed to that based on the properties set by the platform/fish as is commonly done. Using the difference between the St values based on the fish and its wake, it is discussed how the low-friction-driven pendulum oscillator may allow feedback fish-wake lock-in experiments with models of propulsive surfaces or tethered live animals. The oscillator is particularly suitable for flying insects or small fish at very low Reynolds numbers where there is no such general oscillator readily available.

### Past Work on Spherical Motors.

Spherical motors are in the research stage of development for automation purposes, but, although their compactness is appealing, they are not very efficient or reliable. Because of material issues, low electrical efficiencies (0.01) have been reported [10] for torques of <10^{–2} N·m. Elimination of the harmonic components of the detent force produces smoother motions [11]. Analytical models of torque estimation have been given for producing pitch, roll, and torque in a single wrist-like joint, which helps to plan and control trajectories, as compared to commonly used three-joint systems [12].

## Results

### Ball-and-Socket Joint Design is Common in the Pectoral Actuators of Swimming and Flying Animals.

Figure 1 shows that the ball-and-socket design of the propulsive actuator is present in both the flagellar filaments (cilia) of paramecia and the pectoral fins of dolphins over a factor of 10^{5}–10^{6} for their length-based Reynolds numbers irrespective of their size, hardness, and kinematics, which means that they have a common evolutionary ancestor. The ball-and-socket structure of the pectoral fin joint in the dolphin is pivoted at a virtual origin. The presence of a real or virtual origin of oscillation and the ball-and-socket construction suggest that the pectoral fins are mechanical resonant oscillators with low damping and with disturbance rejection properties [4,6].

### Ball-and-Socket Joints Have a Three-Dimensional Force Profile.

We compare the design of a spherical system for a propulsive oscillatory actuator vis-a-vis a cylindrical system as currently used [5,7]. The two mechanical oscillators are shown in Fig. 2. The number of fundamental frequencies of oscillation ($fo$) is given by the number of degrees-of-freedom. In the orthogonal cylindrical system, two values of $fo$ and its harmonics are possible. To minimize the secondary vortex losses, in practice, we keep $fo$ the same in both oscillators. On the other hand, in the spherical system, many values of $fo$ are possible, although the number can be reduced to a manageable level by powering off most of the electromagnets in a hemispherical motor drive (discussed later). Assuming the oscillator in Fig. 1(b) to be a network of both series and parallel spring-mass-damper systems, notice that the motion is preferentially free in the spring and damping constants.

Due to the similarities, we use a pendulum framework for understanding the dynamics of the oscillatory propulsive actuator. In the present work, we have also used the hemispherical motor in the vertical pendulum-like configuration. A pendulum is a widely occurring gravity-damped oscillatory system that becomes nonlinear if the commonly used string or wire holding the ball mass is replaced by one or more articulated solid arms restrictively simulating the digits of fins or hands. Chaotic motions of high amplitude can be produced if the pendulum is disturbed from its equilibrium position by a large amount. Large deflections can be produced with very little internal friction.

Notice that we have arbitrarily set the phase difference with roll motion to be $\u2212\psi $. Nutation increases the degree-of-freedom and would allow yawing in an orthogonal plane. We model the effects of the ball-and-socket arrangement on the fin forces and motion by applying quasi-steady assumptions [13].

*f*= 0.5 Hz, $\theta bias=20deg$, $\epsilon bias=15deg$, and forward velocity

*U*= 1 unit, the average radius is $Ravg$ where the force is 1 unit, and the spanwise location where the angle of attack is being calculated is $s\u2032$ = 1 unit. These fin kinematics are similar to our flapping fin propulsion work at $Rec>40\xd7103$. The angle of attack is given by

*ω*is the angular frequency,

*2πf*. Further, force coefficients are modeled as follows from the measured relationships of the flapping force lift and the drag versus the angle of attack [13]:

*m*is the slope of the measured relationship between $CL$, and $\alpha $ for a flapping fin [13] is given as follows:

Forces are normalized by dynamic force ($12\rho U\xafwing2Aplanform$), where $\rho $ is the density of the medium, $U\xafwing$ is the average wing velocity, and $Aplanform$ is the wing planform area.

Figure 2(c) shows the modeled effects of $\theta Bias=30deg$. The normal force $FN$ is no longer symmetric about the zero angle of attack but shifts to positive values. As a result, yawing will be produced as the measurements also show [5]. Similarly for $\epsilon Bias$, yawing in the vertical plane would be produced. With positive and negative bias angles, a motion toward any of the four quadrants ahead can be generated. In particular, with simultaneous $\theta Bias$ and $\u03f5Bias$, the forward or reverse barrel roll trajectory of motion can be generated in the ball-and-socket jointed fins.

Figure 2(d) shows the relationships of the oscillation angles in the ball-and-socket joint case in Fig. 2(b). A point on the fin, or a fluid particle close by, experiences a spiraling motion both in ($\varphi $, $\theta $) and ($\varphi $, $\epsilon $). However, in ($\theta $, $\epsilon $), it is not a spiral but a zigzag that is oscillatory (side to side). The vector diagram shows the torques for the right-hand rule for cross products. When the kinematic relationship is a spiral, thrust is generated. This happens only by taking the cross product of pitch or nutation with the rolling motion.

Applying Taylor's assumption of frozen flow over short distances, write $U=x/t$, where *U* is the forward speed of the flapping fin, *x* is the forward distance, and the primary motion is forward, for constant *U* (cruising), $x\u221dt$ in Fig. 2(d). Figure 2(e) is produced by resolving the motions in Fig. 2(d). A meandering induced jet interspersed between reverse Kármán vortex trains is shown. The spiraling core of the jet interconnects the shed elliptic vortices. The synthesis of the simple relationship between the angular kinematics in Fig. 2(d) gives an early indication of how thrust is produced; this is shown in Fig. 2(e).

Longitudinal vortices last for long distances, whereas cross-stream vortices dissipate in short distances. Conventional propulsors produce long-lasting longitudinal vortices. In the flapping case in Fig. 2 where the span is finite, Kármán and Lighthill [2] have shown the vortices to be elliptically closed (Fig. 2(e) shows a horizontal cut revealing the lower half of the ellipse). So, the zigzag jet in Fig. 2(e) is expected to have a slow spiraling motion on receiving its energy from the vortices, which then wither [14]. Reported fish wake measurements have time-averaged out this spiraling motion. In analogy, in a single-engine propeller aircraft, the effects of such spirals are nulled by setting the tail rudder at a small angle of attack. Overall, Fig. 2 may be seen as a heuristic method of flow visualization or synthesis where there is an underlying limit cycle organization [8].

The spirals in Fig. 2(d) are similar to what the tips of steady conventional engineering underwater propulsors produce when seeded with cavitation air. The difference is, in the present case, these spiraling motions can be generated temporally in three-dimensional directions, including a sign change that allows significant advances in spatio-temporal maneuverability. In our laboratory, we have demonstrated unprecedented planar maneuverability in a 1 m-long cylinder [5]. The hemispherical oscillator theoretically extends that range of motion to three dimensions.

### Flapping Oscillators are Resonant Which Means They Have Low Friction.

It has been shown that vortex shedding in oscillatory wakes affects the upstream angle of attack on the hydrofoil [2]. The jetting reverse Kármán vortex train is a nonlinear oscillator [2,7,15] and, ideally, the flapping fin system needs to couple to that oscillator. However, due to the constrained nature of the gearing and the high levels of damping, the cylindrical actuator (Fig. 3(a)) is not free to allow feedback to couple to the wake. On the other hand, due to the absence of these shortcomings, the presently proposed hemispherical motor oscillator system (Fig. 3(b)) might allow real-time coupling with the wake. Such feedback coupling would make the engineered system more animal-like, with rapid response capability produced with less energy consumption. This is considered in Sec. 3.

The damping of an oscillator is commonly measured by the quality factor *Q* defined as the ratio of resonant frequency to bandwidth or as the ratio of the energy of the total system to that lost per radian due to damping. Below, we discuss the damping in swimming and flying animals and the two engineering systems shown in Fig. 3. Define the nondimensional Strouhal number $St=fA/U$, where *f* is the fin oscillation frequency, *A* is the excursion, and *U* is the velocity.

Swimming animals mostly have a constant Strouhal number St of 0.30 [16,17], meaning that they are similar resonant systems [2]. However, straight-flying (cruising) birds have St of 0.20 and insects have St of 0.30. Swimming and flying animals have their own typical cruising speeds and flapping amplitudes, meaning that St is proportional to their cruising flapping frequency. In swimming, then, the bandwidth ($\Delta f$) of the resonant frequencies is also of similar proportion. If *f* drops, $\Delta f$ will also drop, meaning that the tuning is narrower, with excellent disturbance rejection properties. If we leave aside the case of flies, this discussion of slightly larger animals suggests that the bandwidth of tuning is a function of the material properties (density and viscosity) of the medium.

At the resonant frequency, the amplitude of oscillation is the highest. Therefore, for the same forward speed, a high amplitude of oscillation can be achieved at a lower frequency by keeping St constant. Since larger flying animals flap their wings at lower frequencies, we expect larger flying animals to be more narrowly tuned. This is, in fact, supported by the following data on flying animals. For flying insects, Ellington [18] gives these values of *Q*: 6.5 (fruit fly), 10 (hawkmoth), and 19 (bumblebee).

### Force and Efficiency of the Hemispherical Motor Oscillator.

The orthogonal configuration of the current oscillator for producing oscillatory motion in fins and the hemispherical motor are shown in Figs. 3(a) and 3(b), respectively. The latter is more compact.

*F*is given by

where *n* is the number of turns in the coil in each electromagnet, *i* is the current through the coil, $\mu o$ is the magnetic permeability in a vacuum, *A* is the area, and $\u2113gap$ is the air gap between the magnet and the steel in the direction of the magnetic flux.

The relationship between the force and the gap is shown in Fig. 4(c). Compared to $\mu o$, the magnetic permeability of cold rolled steel is 2000 times larger. For $\u2113gap$ of 3 to 4 mm at *i* = 1.5 A, both Eq. (9) and the measurement of maximum force lead to a force of 0.74 N. The conclusion is that the design would produce the required motions of small swimming animals.

The sequence of turning the electromagnets on and off is calculated for the required mechanical oscillator, such as penguin pectoral fins [2,7] or the pectoral fins of *Clione antarctica* [19–21] or fly wings [22] or the cilia of a paramecium [23,24]. Then, the power supply to the electromagnets is sequenced by using relays (Fig. 4(d)).

where $V$ is the system voltage, *I* is the system current, and the subscript *o* denotes conditions where the fin/wing is stationary. The hemispherical motor was used to generate an oscillatory electro-magneto-motive force, and *U* is the velocity of the oscillator beam. Figure 5 shows the measurements of total electrical efficiency of the hemispherical motor for producing oscillatory forces. As the force produced drops from 10 N to 1 N, the efficiency increases from 0.01 to 0.08 which is in the range of the total efficiency of flying animals as well as robotic flapping fin platforms [25].

### Characterization of the Damped Oscillations of the Hemispherical Motor Oscillator.

Figures 3(a) and 3(b) show the cylindrical and hemispherical systems, respectively, of fin actuation. Compare their damped oscillations. The cylindrical system in Fig. 3(a) is overdamped (*Q* < 0.5), and it does not freely oscillate at all. When the fin in the hemispherical system in Fig. 3(b) is released from its position of maximum excursion ($\xb170deg$, 15 cm radius) in air where potential energy is maximum and kinetic energy is zero, it swings six times with decreasing amplitudes before coming to a complete stop. This is shown in Fig. 6(a). The times in seconds are stamped at the bottom of the fin. The normalized excursion ($X/Xo$) versus time is shown in Fig. 6(b) (symbols).

*x*is the horizontal excursion in Fig. 6(a),

*γ*is the damping coefficient,

*t*is the time, $xo$ is a normalizing initial maximum excursion, $a$ is a phase constant, and the frequency is given by $\omega 1=\omega o2\u2212\gamma 2$. The peak positive normalized amplitudes in each damped oscillation are given by the envelope

Figure 6(b) shows that, for $\gamma =0.5$, $\omega o=10.8$, and $a=1.1$, the hemispherical motor in Fig. 3(b) has a damped oscillation property in air that is in excellent agreement with the theory up through the first five oscillations. The period of the damped motion is 0.60 s. The period of the undamped motion should be marginally lower. The conclusion is that the hemispherical motor in Fig. 3(b) is an underdamped oscillator, whereas the cylindrical system in Fig. 3(a) is an overdamped oscillator. The hemispherical system is, therefore, a significant improvement in the reduction of friction. The pectoral fins of animals are likely critically damped because that would require very little torque to impart oscillation and also would not have undue oscillation when stopped or switched to a different gait.

Defining *Q* as the ratio of total energy *E* in the mechanical oscillator in air, with the 15 cm-long fin attached, to the energy lost per cycle, estimated *Q* = 5. (Sometimes, this definition of *Q* is multiplied by $2\pi $.) This value of 5 is close to 6.5 for fruit flies [18]. Later, we compare the total efficiency with that of flies.

The time-averaged thrust of flapping fins does not have the nonlinearities of the propulsion mechanism. Experiments have shown that up to five Fourier coefficients are needed to capture the dynamics for control purposes [26]. The five subharmonics are the feedback from the wake. Although the oscillator in Fig. 3(a) would produce the subharmonics, it would not respond to them, meaning that they would not lock-in to the wake. This is an important shortcoming in all swimming or flying model experiments to date. The power spectral density matches up to the peaks of frequency/$fo$ = 5 [26], which gives a harmonic time period of 1/1.5 s = 0.66 s: this is near the first peak in the damped oscillation in Fig. 6(b). Hence, the damped oscillation response in Fig. 6(b), on the other hand, shows that the hemispherical oscillator would allow lock-in with the first five harmonics of the wake if the oscillation frequency were 0.3 Hz ($=fo$), which is commonly used in 1 m scale platforms [4].

*m*is the mass, $h$ is the radius of gyration, and the external input force is given by

where $b\varphi \u02d9t$ is the damping torque from mechanical friction with air and water and muscle active damping, and $q(t)$ is the control torque input (note their opposite signs). In the cylindrical system in Figs. 2(a) and 3(a), frictional damping is very high due to the presence of gears for speed reduction from the usual high levels of efficient motors (10^{2}–10^{4} Hz) to the common low (<1 Hz) values of $fo$ in water. On the other hand, in the spherical system, there is no gearing for speed reduction. For this reason, the external input torque $\tau o$ in the cylindrical systems can be expected to be much greater than that in the spherical system. Equations such as Eq. (13) can also be written for $\theta $ and $\epsilon $. However, for oscillatory propulsion in water, the energy consumption due to roll motion is 90% of the total, whereas it is only 10% for pitching and twisting oscillation [7]. For nutation ($\u03f5$), we expect the energy consumption to be about similar to that for pitching. $Qext$ may be used to relate to the magneto-electromotive forces in the hemispherical motor.

### Mapping of the Fin Motion on the Hemispherical Motor Oscillator.

Figures 7 and 8 are views of the stators and rotors, respectively. Figures 7(a) and 8(a) show the underside of the diametric cover plate. Figure 7(a) shows a gimbal support and a single ball mass. Figure 8(a) shows a three-ball-mass assembly with two sets of orthogonal bushings to support roll and pitch oscillations. The central ball is used to traverse a rolling motion, while the other two balls, placed diametrically oppositely each other, produce an orthogonal pitching oscillation. Figures 7(b) and 8(b) show the 35-electromagnet assembly exposed. Figure 7(b) shows a side view of the stator. The single ball in Fig. 7(a) and the three balls in Fig. 8(b) skirt the hemispherical surface, as a result of the assembly shown in Fig. 4(b), with a small clearance.

The electromagnet assembly of the hemispherical motor with a single ball shown in Fig. 7 reproduces the motion of the base of a cilium (that is, of the pivot shown in Fig. 1(a)) of a paramecium whose axial temporal posture is shown in Fig. 7(c) [23]. Next, the single ball was replaced by a three-ball mass. The bearing apparatus supporting the balls was rebuilt as shown in Fig. 8(a) to generate the rolling and pitching motions of flapping fins.

Here, *r* is the radius of the sphere, and (*x, y, z*) are the Cartesian coordinates denoting the position of the axes of the flapping fin mapped on the hemisphere.

Figure 9(a) shows the direction of oscillation of the central roll ball ($\varphi $) and the two oppositely placed pitch balls ($\theta $). Figure 9(b) shows how the spherical ball paths become eccentric with increasing pitch bias. Figure 9(c) shows the “bent bow” effects of phase difference ($\psi $) on the tracks of the balls. The effect of $\psi $ is to change the slant of the zigzag in Fig. 2(d). For cruising, both bow bending and eccentricity are zero, and the loading is balanced, thereby minimizing muscle work and fluid momentum. This is the arrangement for maximum efficiency. Both kinds of variations in the tracks of fins can be accommodated in the hemispherical enclosure.

The calculated tracks of the balls are laid on the numbered electromagnet grid shown in Fig. 4(a). To power the electromagnets, the computer controlled relays in Fig. 4(d) are turned on in the sequence generated in Fig. 9.

One hemispherical motor can provide both orthogonal roll and pitch motions to a fin (Fig. 3(b)), whereas two orthogonally situated, conventional cylindrical motors are needed to produce similar motions (Fig. 3(a)) in current biorobotics [4,5]. Further, the gears and high rotational speeds of motors are sources of friction and mechanical vibration in conventional cylindrical motors such as the Maxon motors that we have used to produce roll and pitch motions for biorobotic purposes. The Maxon EC-max-30 272763 24V motor, with a maximum continuous torque of 61.3 mN/m and a planetary gearhead GP32C with a 111:1 gear ratio, has been used at 486–6075 rpm to produce roll oscillations; the Maxon EC-max-16 283835 24V motor, with a maximum continuous torque of 8.2 mN/m and a planetary gearhead GP16A with a 104:1 gear ratio, has been used at 521–5212 rpm to produce pitch oscillations. With the hemispherical motors, no gearing is needed (Fig. 3(b)). In conventional motors (Fig. 3(a)), the motor rotational speed is geared down to match the low flapping frequencies (<1 Hz) of swimming animals by factors of 10^{3}–10^{4}. In insects, however, the flapping frequencies are in the hundreds of Hz. The hemispherical motor drive is more direct, and the layout is more animal-like.

In the biorobotics of underwater flapping fin propulsion, the laws of forces and moments are given in Ref. [3]. For swimming, we keep the flapping frequency below 1 Hz when the forces are low (1 N). However, since force is proportional to the square of frequency, the flapping frequency should be below 0.5 Hz at higher loads (10 N). Generally, we maintain a constant value of the Strouhal number (generally between 0.20 and 0.40, the ideal target being 0.30), and also a fixed value of the amplitude of flapping, and vary only the frequency of flapping to vary forward speed.

### Modeling of the Abrupt Turning of Pectoral Fins of *C. Antarctica*.

Both *C. antarctica* and flies swim and fly in the range 5 < Re < 20, and they exhibit an abrupt turning of their fins at the extremities of their excursions. These abrupt motions are modeled before they are reproduced in the hemispherical motor oscillator.

It has been possible to accurately model the position and velocity of the cilia of the paramecium, which remains self-similar along its length [23]. This uses the slow ionic oscillator (Ca_{2}-dependent) part of the FitzHugh–Nagumo (FN) model of the inferior-olive neurons of motion control of animals. The model also correctly predicts that the rigidity of the cilia switches between the power and return strokes when the acceleration becomes very high at one instant of the time period.

Note that, in Ref. [23] as summarized earlier, the actual deflection of the propulsive surface is not being directly modeled, but it is indirectly calculated by modeling the input ionic olivo-cerebellar control signal at the base (Fig. 1(a)). The implication is that the structural properties of the cilium are such that the fluid structure interaction and the control input are balanced in a manner that remains self-similar along the length of the cilium. The present hemispherical motor oscillator is an important apparatus for exploring the various self-regulating aspects of such fluid structure and control interactions.

Since olivo-cerebellar dynamics remain invariant in animals, we use the FN equations [23] to model the kinematics of the propulsive surfaces of *C. antarctica*. The orthogonal states of the *Ca _{2}* oscillator are ($z,z\u02d9$). Now, ($z,z\u02d9$) may be written as ($z,w$), that is, as roll and pitch oscillations. Write $z=sin(\theta )$ for an amplitude of 1.0 and $w=sin(\theta +\psi )$, also for an amplitude of 1.0 where $\psi $ is the phase difference. In Fig. 10(a), if we set $\psi =90deg$ for $270deg$ < $\theta $ < $90deg$, and $\psi $ = $\u221290deg$ for $90deg$ < $\theta $ < $270deg$ (arrows through symbols in Fig. 10(a)), we get the pattern of flapping pectoral fins of

*C. antarctica*shown in Fig. 10(a). The model agrees well with the measurements in Fig. 10(b) [21]. At the extremities of the excursions, the sign of the phase difference $\psi $ is simply toggling.

The value of $\psi =90deg$ gives maximum efficiency. The phase change in $\psi $ may define a feature of the evolutionary change across the hydrodynamic bifurcation from laminar jetting to reverse Kármán transitional vortex dynamics jetting in the range 5 < Re < 20 [19].

The value of $\psi $ abruptly shifts at the end of the excursions. Such a change is also present in models of fruit flies but not in our modeling of penguin wings where $\psi $ is held at $90o$ and rotations are continuous [7,13]. Therefore, this abrupt phase shift may be a feature of small swimming and flying animals whose length-based Reynolds number is straddling the hydrodynamic bifurcation (5 < Re < 20). The sum of the phase shift is zero per cycle in *C. antarctica* but not in penguins. There is some simultaneous simplification in the control algorithm and in the time-dependence of torsional rigidity of the propulsive surfaces as loading increases with Re.

### Reproduction of the Wing Kinematics of *C. Antarctica*, Flies, and Penguins.

We reproduce the penguin pectoral fin tracks given by the sinusoidal relationships (Eqs. (1) and (2)) [2,7]. In contrast to the continuous variation of the pitch angle in penguins, the pitch angle in *C. antarctica* [7] and in flies changes sign near the extremities of the wing travel [21,22]. The differences are shown in the time traces in the upper diagrams in Fig. 11. The calculated tracks of the roll and the paired pitch balls (Fig. 8(a)) are overlaid on the layout of the electromagnets of the hemispherical motor (Fig. 4(a)). They are shown in the lower diagrams in Fig. 11. The temporal description of the penguin and fly wing track simulations are shown in Videos 1 and 2, respectively (see Appendix for descriptions of the videos). The video files have been uploaded at Google Drive address.^{2}

Figure 12 shows a sequence of frames of the positions of the simulated dolphin fin in air. Figure 13 shows a sequence of frames of the positions of the simulated *C. antarctica* and fly wing in air. Video 3 shows the simulated penguin wing motion in air. Video 4 shows the simulated penguin wing motion in water. Video 5 shows the motion of the simulated fly wing in air. The response of the medium (air or water) would be continuous although the fin motion is discrete.

### Comparison of the Efficiency of the Hemispherical Motor Oscillator With Flies.

Figure 14 compares the total efficiencies of the hemispherical motor with the total efficiency of the flies [18,25]. For the hemispherical motor, the total efficiency is the electrical power efficiency. For the flies, the total efficiency is the thermal power efficiency after the flies have been exercised in a box. The efficiencies compare well.

## Discussion of a Wake-Based Strouhal Number and Fish-Wake Lock-in Error

The measurement of the drag and efficiency of live animals is a difficult task. Estimates of drag using coefficient-based modeling have been erroneous because they do not account for unsteady or quasi-steady effects [13]. As a result, claims about the energetic superiority of swimming animals over steady engineering means, strictly speaking, remain unverified. That swimming and flying animals flap their tails or wings at preferred values of St to maximize their efficiency remains a plausible hypothesis but not a hard fact because of the lack of measurements of efficiency. Moreover, how new propulsive surfaces evolved at specific Reynolds numbers remains a mystery. There is laboratory evidence that hydrodynamics below Re* _{c}* of 40 × 10

^{3}is initial-condition dependent. Our understanding of these basic aspects will improve if a closed-loop oscillator apparatus becomes readily available.

where $\theta $ is the included angle of the wake vortex train (Fig. 15(a)). The property of this definition is that it requires a snapshot of the vortex train but does not require any platform-based (fish-based) knowledge that the fish or the operator will set. $StWake$ is the response of the flow to $StFish$. They will be equal when there is perfect lock-in. Ideally, that will give the optimal St for cruising, where efficiency is maximum and drag is minimum. The wake-flow-based definition of St is allowing us to clarify this fundamental aspect of what optimal or efficient swimming is.

In the light of how a fish locks-in to its wake, in Fig. 15(b), we give a closed-loop control scheme for experiments with small biorobotic models or live swimming and flying animals that use the proposed oscillator, which follows the control scheme used in our past work for guidance [5].

The right box in Fig. 15(b) shows a tethered biorobotic unsteady propulsive device, or a live animal swimming or flying in a stream. Sensors detect the states such as position, velocity, and acceleration of the animal. Define error as the difference between the desired and the actual state variables (Eq. 16). Values of voltage and current to the electromagnets are calculated to bring the error to zero. The presently used relay solver gives the sequence and rate for powering the electromagnets. Calibration with known forces may be used for live drag measurements.

An emerging understanding of unsteady hydrodynamics is that the unsteady laminar [23] and transitional flapping fin propulsion of animals is self-regulating in nature [2–4,7,15]. Unsteady low-Reynolds-number turbulent boundary layers also have similar deterministic properties [28]. This brings the subject of unsteady (oscillatory) hydrodynamics in direct correlation with olivo-cerebellar control [6,29–31]. Since self-regulation implies closed-loop control with upstream effects that we have shown theoretically [2], the present oscillator could prove useful in this emerging basic area of hydrodynamics research.

The biorobotic approach to the actuation of propulsive surfaces using a hemispherical motor is different from the contemporary engineering approaches that use cylindrical rotating or oscillating geared-shaft motors and linear proportional-integral-and-derivative controllers. The present approach is animal-like and may be scalable.

Future work may include enhancing the efficiency of the motor at higher force levels, miniaturizing the hemispherical motor, and increasing the density of the electromagnets for smoother fin trajectories. Research on cooled magnets for higher forces, embedded sensors, and a resident controller for a portfolio of tracks will broaden the biorobotic scope.

## Conclusions

A novel hemispherical motor oscillator for experiments on the swimming and flying of small animals where forces are small (<1 N) has been built. The motor is gearless and has a pendulum-like configuration. The following conclusions have been drawn:

The total electrical efficiency at 0.8 N is comparable to the thermal efficiencies of flies [18].

The damped oscillations in air are in excellent agreement with spring-mass-damper theory indicating that the oscillator is underdamped. In other words, it has very low internal friction. In air, the estimated quality factor (*Q*) is 5. This is comparable to that for flies. In contrast, current geared cylindrical motor oscillators are overdamped (*Q* < ½).

The oscillator will respond to five harmonics at a flapping frequency of 0.3 Hz while capturing all nonlinear effects important to control.

The continuously rolling and pitching fin kinematics of penguins have been reproduced. The fin motion abruptly changes at the extremities of its excursions in *C. antarctica* and in flies, and this aspect also has been reproduced. This abrupt change in *C. antarctica* is modeled by switching the roll-pitch phase difference alternately by $\pi /2$ and $\u2212\pi /2$ in each quadrant, and the agreement with measurements [21] is good. The olivo-cerebellar link of this modeling is notable.

Ball-and-socket joints are present in organisms from tiny cilia to dolphin pectoral fins. Current orthogonal motor oscillators produce only planar motions that generate cruising and yawing in the horizontal plane. However, with ball-and-socket joints, the present hemispherical motor oscillator would produce yawing motions in both the horizontal and vertical planes that may lead to the barrel-roll type of motion that is found in terminal escape maneuvers. The present oscillator increases the degrees-of-freedom in fin motion.

An alternative to Triantafyllou and Triantafyllou's [16] definition of the fish Strouhal number is given that is based on the wake and not directly on the fish kinematics. It is shown that the tangent of the vortex shedding angle can be defined as a wake Strouhal number. This definition does not require any knowledge of the flow velocity or flapping frequency. The difference between the two definitions represents the error in lock-in between the fish and the wake.

Assuming perfect lock-in between an animal and its wake, we expect the mean vortex shedding angle in swimming animals and flies to be $16.7deg$ and $11.3deg$ in large flying animals.

In the past, coefficient-based modeling of the drag of live oscillatory swimming animals has proved to be unreliable. A closed-loop block diagram is discussed in which a robotic or live swimming or flying animal tethered to the hemispherical motor forms part of the loop. When calibrated, the oscillator could be used to measure drag directly. One could evaluate whether the drag is at the minimum when the error between the flow and fish Strouhal numbers is zero. The present oscillator gives a control-based approach to the examination of questions regarding mechanisms of swimming and flying in a laboratory environment.

## Acknowledgment

William Nedderman carried out the industrial design and fabrication of the hemispherical motor and the fin. William Wilkinson designed the independent rolling and pitching bearings with computer-aided design software and fabricated them. Aren M. Hellum carried out the MATLAB simulations of the roll and pitch ball motions required to emulate the penguin and fly fin/*C. antarctica* pectoral fin trajectories. Albert R. Fredette wrote the LabVIEW programs for stepping the relays and data acquisition.

## Funding Data

The U.S. Office of Naval Research, Biology-Inspired Autonomous Systems Program (ONR 341) (Grant No. N0001414AF00002).

## Nomenclature

*a*=constant in damped oscillation equation

*A*=fin flapping excursion length, or magnet flux cross-sectional area

- $Aplanform$ =
fin planform area

*b*=damping parameter

*c*=fin chord

*C*=coefficient of force

*F*=magneto-electromotive force

*f*=frequency of oscillation

- $fo$ =
fundamental frequency of oscillation

*g*=acceleration due to gravity

*h*=radius of gyration

*i*=current through the electromagnetic coil

*I*=mass moment of inertia

*I*=total hemisphere current; subscript o is for condition when fin is stationary

- $\u2113gap$ =
air gap between stator and rotor magnets

*m*=mass

*n*=number of turns of copper wire electromagnet

*q*=control torque input

*Q*=quality factor

- $Qext$ =
external input force to driven oscillator

*r*=radius of hemisphere

- $Rec$ =
Reynolds number based on chord, $Uc/\nu $

- $s\u2032$ =
fin span

- St =
Strouhal number $fA/U$

*t*=time

*U*=forward velocity

- $U\xafwing$ =
time-averaged wing velocity

*V*=total hemisphere voltage; subscript o is for condition when fin is stationary

*x*=forward distance

*X*=fin excursion in damped oscillation

*x*=_{o}normalizing initial maximum excursion

*x, y, z*=Cartesian coordinate system

- $\u02d9$ =
first time derivative

- $\xa8$ =
second time derivative

- $\alpha $ =
angle of attack

- $\Delta f$ =
bandwidth of tuning

*γ*=damping coefficient in damped oscillation equation

- $\epsilon $ =
nutation angle (orthogonal to pitch and roll angles)

- $\eta e$ =
total electrical efficiency

- $\theta $ =
pitch angle

- $\mu o$ =
magnetic permeability in vacuum

- $\nu $ =
kinematic viscosity

- $\rho $ =
density of the medium

- $\varphi $ =
roll angle

- $\tau o$ =
external input torque

- $\psi $ =
phase difference with roll oscillation

- $\omega $ =
angular frequency, $2\pi f$

### Appendix

The readers may obtain copies of the below-described videos from the Google Drive address.^{2}

Description of Videos:

**Video 1: Penguin relay activation sequence**

Description: This video shows the relay jumping sequence required to generate the continuously varying penguin pectoral wing track. The video shows the temporal sequences of the positions of travel of the single roll ball and the paired pitch balls. The numbering of the electromagnets is as shown in Fig. 4. The radial and circumferential matrix of the electromagnets is overlaid on the hemispherical surface. The roll and pitch time traces shown in the upper part of the views are being reproduced by the ball positions shown below.

**Video 2: C. antarctica and fly relay activation sequence**

Description: This video shows spatio-temporal mapping of roll and pitch oscillations of the hemispherical motor oscillator required for simulating the abruptly varying fin motions of *C. antarctica* and a fly. The fin pitch changes abruptly at the ends of the excursions, unlike in the penguin case where the change is continuous. The roll and pitch time histories are shown in the strip chart on top. The mapping of the sequence of displacement of the roll ball and of the pair of pitch balls on the electromagnet array is shown in the lower part of the frames. The numbering of the electromagnets is as they appear in Fig. 4. This map of the ball displacement is used to turn the relays of electromagnets on and off.

**Video 3: Simulation of penguin wing flapping in air**

Description: This video shows the reproduction of the roll and pitch motions of the pectoral fins of a penguin in air in the hemispherical motor oscillator. The relays in Fig. 4 are being jumped to power the electromagnets in a sequence by implementing the tracks of the roll and pitch balls shown in Video 1.

**Video 4: Simulation of penguin wing flapping in water**

Description: This video shows the simulation of the penguin wing track in water. The relays in Fig. 4 are being jumped to power the electromagnets in a sequence implementing the tracks of the roll and pitch balls shown in Video 1. This simulation is similar to Video 4 except that the fin is in water.

**Video 5: Simulation of fly wing flapping in air**

Description: This video shows the reproduction of the roll and pitch motions of the pectoral fins of *C. antarctica* and the wing of a fly in air by the hemispherical motor oscillator. The relays in Fig. 4 are being jumped to power the electromagnets in a sequence the tracks of the roll and pitch balls shown in Video 2.