Metachronal coordination enables omnidirectional swimming 3 via spatially distributed propulsion

Abstract


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Metachronal coordination of appendages is seen in many aquatic organisms spanning a wide range 54 of sizes and body plans, including shrimp, krill, polychaetes, and even aquatic insects [1,2]. 55 Organisms sequentially actuate a row of appendages (pleopods, ctenes, legs, parapodia, cilia, et 56 al) to generate fluid flow via drag-based paddling; hydrodynamic interactions between the paddles 57 can improve overall efficiency and flow speed [3]. The generated flow can be used for swimming 58 or for pumping to aid in feeding, clearance of wastes, and other functions [4,5]. This technique is 59 highly scalable, with metachronally coordinated appendages ranging from microns to centimeters 60 in length. Studies of metachronal locomotion have thus far focused primarily on overall swimming 61 ability [6-9], but some metachronal swimmers are also capable of surprising agility. Here, we 62 examine a highly maneuverable and agile metachronal swimmer: ctenophores, or comb jellies. 63 Ctenophores swim at Reynolds numbers on the order of 1-1000 [6]; both inertia and viscosity 64 impact their movements considerably. Locomotion is driven via eight metachronally coordinated 65 rows of paddles (ctenes), which are made up of bundled millimeter-scale cilia [10]. Fig 1 shows  66 the eight ctene rows circumscribing a lobate ctenophore's approximately spheroidal body and its 67 general morphology. The coordination between ctene rows allows ctenophores to turn tightly 68 around many axes, but not their axis of symmetry-that is, ctenophores can yaw and pitch, but 69 they cannot roll. This is not the case for all swimmers: animals that rely on paired appendages or 70 a single row of appendages tend to display maximum turning performance around a single axis, 71 depending on the appendages' positions along the body [11][12][13]. Some swimmers exploit the 72 flexibility of their bodies to turn, but these usually have anisotropic bending characteristics, and 73 thus have a preferential turning direction [14,15]. Only a small number of animals have completely 74 axisymmetric bending characteristics, which allow them to turn across the entire range of pitching 75 and yawing motions; jellyfish are one example [16], and even jellyfish cannot effectively rotate 76 about their axis of symmetry. However, the single-jet propulsion used by jellyfish medusae has a 77 notable disadvantage: with this strategy, an animal cannot easily reverse its swimming direction. 78 Ctenophores, by contrast, can quickly reverse their swimming direction by changing the power 79 stroke direction of their ctenes [17].

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Though ctenophores are primarily planktonic, they also swim actively and are capable of agile 83 maneuvering as described above. However, their turning behavior has only been described 84 qualitatively [18]. Existing quantitative information on ctenophore swimming trajectories comes 85 from single-camera (2D) experiments, and has focused on straight swimming [6,[19][20][21]. There 86 are no explicit quantitative data on ctenophores' turning, nor any direct measurements of ctene 87 beating frequencies in the context of turning. Full three-dimensional turning information is not 88 available for any metachronal swimmers, though several organisms have been noted for their 89 agility [9]. We therefore know little of the control strategies-that is, the coordination of 90 frequencies and other parameters between rows-used by ctenophores (or other metachronal 91 swimmers) while performing turning maneuvers. 92 Two important variables describe turning performance: maneuverability and agility. 93 Maneuverability refers to the ability to turn sharply within a short distance and is typically 94 quantified by the swimming trajectory's radius of curvature (usually normalized by body length) 95 [22]. Agility, however, is not clearly or consistently defined in the animal locomotion literature. A 96 widely used definition is the ability to rapidly reorient the body [23], quantified by the maximum 97 observed angular velocity. However, the angular velocity on its own does not speak to whether the 98 animal needs to stop or slow to perform a turn, which is another colloquial definition of agility. 99 An animal's translational speed while performing a turn can give insight into its agility [13, 24, 100 25]. Here, we will use the average speed during the turn ( � ) as a measure of agility, and the average 101 normalized radius of curvature ( ⁄ ������ , where is the radius of curvature and is the body length) 102 during the turn as a measure of maneuverability. We can examine a large number of discrete turns 103 to build a Maneuverability-Agility Plot (MAP), plotting / ����� vs. � for a given organism. 104 In this study, we explore the three-dimensional maneuverability and agility of freely swimming 105 ctenophores, and the control strategies used to produce the observed trajectories. We use 106 multicamera high-speed videography and three-dimensional kinematic tracking to correlate 107 overall trajectories with the beating frequencies of the ctene rows, and identify three distinct 108 turning modes. We also use a 3D reduced-order analytical model to explore the kinematics 109 resulting from the range of physically possible beat frequencies for each turning mode. We use the 110 MAP to explore the observed and hypothetical turning performance of ctenophores, showing how 111 they can sharply turn at high speeds relative to their top speed. In addition, by reconstructing B. 112 vitrea's "reachable space," also known as the Motor Volume (MV) [26], we show that ctenophores 113 have the potential to reorient in almost any direction within a small space over a short timeframe 114 (omnidirectionality). Our experimental and analytical results also provide a basis for comparison 115 to other animals known to have high agility and maneuverability, and suggest that ctenophores are 116 worthy of further study as a model for the development of small-scale bioinspired underwater 117 vehicles. 118

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Ctenophore morphometric and kinematic parameters 120 To describe the overall ctenophore propulsion system, we define nine morphometric and five 121 kinematic parameters. These parameters are listed in Table 1, along with a brief description, while 122 Fig 2 shows a graphical description of some parameters. Finally, Table 2 shows the average of the 123 morphometric parameters as measured from eight studied individuals. 124   From 27 recorded sequences, we observed four different appendage control strategies. These 138 strategies differ categorically in the total number and the geometrical arrangement of the rows 139 actively beating. The first three strategies are used to turn, with rows on the outside of the turn 140 beating at a higher frequency than the rows on the inside of the turn ( > ). In the first 141 strategy (mode 1), two adjacent rows beat at some frequency and the two opposite rows beat 142 at a lower frequency while the remaining four rows are inactive. In the second strategy (mode 143 2), the four outer rows beat at approximately the same frequency, which exceeds the frequency 144 used by the four rows on the opposite side. each of the body quadrants formed by the sagittal and tentacular planes (see Fig 2A), but the two 151 rows in each quadrant beat at approximately the same frequency. Table 3 shows a summary of the 152 control strategies and the number of times each was observed. The recorded beat frequencies range 153 from 0 to 34.5 . Examples of the four strategies can be seen in S2-S5 Videos. 154 156 To explore the turning performance of B. vitrea, we use the observed 3D swimming trajectories 157 and the mathematical model to build a maneuverability-agility plot (MAP). In Fig 3,  considerable speeds-that is, they have both high maneuverability and high agility. 170 We use the mathematical model to expand our analysis of B. vitrea's turning performance by 171 simulating all possible configurations of modes 1, 2, and 3. We ran a total of 612 simulations 172 covering the range and resolution of the beat frequencies reported in Table 4 Our model predicts that B. vitrea's locomotor system can reach ⁄ ������ = 0.08 at a speed of � = 177 0.58 / (lower-left corner of the MAP, maximizing maneuverability). However, the system is 178 also capable of significant maneuverability at high speeds: in the lower-right corner of the MAP 179 (highly maneuverable and agile), the system can reach a speed of � = 2.33 / for ⁄ ������ = 0.98. 180 These two data points range from 24% to 93% of the simulated top speed ( = 2.49 / , 181 with eight rows beating at 34 ), while still maintaining a turning radius of less than one body 182 length. The model results confirm that ctenophores' metachronal rowing platform is highly 183 maneuverable and agile, with performance limits that may extend beyond our experimental 184 observations. 185 Table 4. Range and resolution of the frequencies used in the analytical simulations. which illustrates the maneuvering capabilities of the ctenophore locomotor system (Fig 4). 199 Conceptually, the MV represents the reachable space of a swimming ctenophore over a given time 200 horizon. To build the MV, we translated and rotated the observed swimming trajectories so that 201 (at the start of the trajectory) the tentacular plane is aligned with the x-y plane, the midpoint 202 between the tentacular bulbs is at the origin, and the aboral-oral axis of symmetry is aligned with 203 the x-axis with the oral end facing the positive x-direction. From this starting position, the positive 204 x-direction is forward swimming (lobes in front) and the negative x-direction is backward 205 swimming (apical organ in front). Fig 4 shows the rearranged swimming trajectories (black lines) 206 and the volume swept by the animals' bodies (gray cloud). Each animal body was estimated as a 207 prolate spheroid based on its unique body length and diameter ( , ). In our observations, 208 animals swam freely (without external stimuli), and the trajectories were recorded through the time 209 period that the animal was in the field of view. Therefore, each observation has a different initial 210 speed and total swimming time (see Table 5). This is therefore not a direct comparison of different 211 appendage control strategies, since observed maneuvers have different initial speeds and durations. 212 We also note that because we only observed animals who freely swam through the field of view, 213 the dataset is biased towards animals who had a nontrivial initial swimming speed, leading to a 214 stretching of the MV along the x-axis. Nonetheless, the observed MV shown in Fig 4 provides  215 some visualization of the 3D maneuvering capabilities of B. vitrea's locomotor system. 216 which we define as the ability to move in any direction from a given initial position within a 220 relatively small space and short time. As expected by the number of active rows, mode 1 is the most maneuverable of the three (shortest 238 trajectories, Fig 5A); while mode 2 and mode 3 reach higher speeds while turning (longer 239 trajectories, Fig 5A). This suggests that activating only two ctene rows (mode 1) could be best 240 suited for fine orientation control (for example, when maintaining a vertical orientation when 241 resting/feeding) [18]. The higher number of active appendages used in modes 2 or 3 could be used 242 for escaping, where both high speed and rapid reorientation are needed [20]. A front view of all 243 modes (that is, the y-z plane) displays the range of swimming directions which are accessible from 244 a given initial position (Fig 5B. This MV-which captures only a fraction the full capability of the 245 swimming platform-shows the omnidirectionality of the ctenophore metachronal locomotor 246 system, achieved only by constant pitching and yawing. In an actual swimming trajectory, a 247 ctenophore can change the active rows, the frequency, or the turning mode over time, resulting in 248 much more complex maneuvers (as in Fig 7B).

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To fully explore the maneuvering capabilities of the ctenophore body plan, we will explore the 256 hypothetical case in which there is independent control of each ctene row. Fig 6 shows  For the body plan studied here, which is typical of lobate ctenophores, we found that the 274 asymmetric placement of ctenes within each row (i.e., ctenes distributed closer to the aboral than 275 the oral end) enabled sharper turns during backward swimming when compared to forward 276 swimming (Fig 5A). Ctene row asymmetries between the sagittal and tentacular rows of B. vitrea 277 are due to the presence of the lobes (see Fig 1), which are used to create highly efficient feeding 278 currents [33]. However, cydippid ctenophores such as Pleurobrachia sp. feed by capturing prey 279 with their tentacles, then bringing the prey to their mouth by rotating their bodies [18]. In 280 Pleurobrachia and other cydippids, ctenes are approximately symmetrically arranged from the 281 oral to aboral end, which may eliminate the trajectory asymmetries observed in lobate ctenophores. 282 It is likely that cydippid ctenophore swimming may be even more omnidirectional. To accomplish 283 their stereotypical rotating behavior, cydippid ctenophores also reverse the direction of the power 284 stroke on the inner ctene rows, potentially leading to even tighter turns that are not captured in our 285 model. Another lobate ctenophore genus, Ocyropsis, contracts its lobes (like the bell of a jellyfish 286 medusa) to increase its escape velocity, while still using ctene rows for orientation [21]; this 287 indicates that ctene rows can be coupled with other propulsive strategies to achieve goals beyond 288 that of maximizing maneuverability (e.g. to increase overall swimming speed). Extinct 289 ctenophores had as many as 80 ctene rows, increasing the number of reachable turning planes. Our results illustrate that the ctenophore body plan is highly agile and maneuverable, with the 295 ability to turn sharply without slowing down, reverse directions easily, and turn about many planes, 296 which enables them to access a nearly-unconstrained region of space from a given initial position 297 over relatively short time horizons. This body plan could be used as inspiration for millimeter-298 scale robotic platforms, with the potential to rapidly reorient into any direction from an initial 299 position. The reduced-order model presented here can be used in the design phase to estimate the 300 general swimming dynamics and inform future robots' control requirements. However, further 301 work should include the fluid-structure interactions between the appendages/body and the 302 surrounding flows. Soft robotic models and CFD simulations would provide tools for a more 303 controllable exploration of metachronal swimming strategies and their concomitant maneuvering 304 capability. However, it is clear that metachronal locomotion -with its scalability, efficiency, and 305 (as we have shown here) high degree of maneuverability and agility-represents a promising new 306 direction for bioinspired technology. 307 tracking approach using the apical organ and the two tentacular bulbs of the ctenophores (Fig 7B). 326

Materials and methods
We recorded 27 free-swimming sequences from eight individuals (B. vitrea). We note that the 327 camera system is also described in ]. However, a simplified modeling approach is still attractive due to the large and multivariate 345 parameter space we seek to explore. We therefore develop a reduced-order analytical model based 346 on known empirical expressions for fluid drag-an approach that has been previously used to study 347 metachronal rowing in 1D for low and intermediate Reynolds numbers [1,28,43,44]. This class 348 of analytical model is limited because it does not consider hydrodynamic interactions between the 349 propulsors, and therefore cannot fully reproduce key features (such as enhanced swimming 350 efficiency) of metachronal swimming. However, it can still reasonably predict swimming 351 kinematics, and (most importantly) it provides a useful tool for comparing the relative effects of 352 the many morphometric and kinematic parameters involved in metachronal swimming without 353 prohibitive computational cost. 354 In this section, we expand the 1D formulation found in [28] to three dimensions, and use it to study 355 ctenophore maneuverability. Unlike several similar models, here we fully incorporate the 356 combination of viscous and inertial effects which arises at intermediate Reynolds numbers by 357 ensuring that relevant drag and torque coefficients are a function of the instantaneous speed and 358 geometry of both the body and the ctenes. Based on the average body and appendage length (Table  359 2), the maximum swimming speed (2.7 / ) and maximum beat frequency (34 ), we 360 calculate body and appendage-based Reynolds numbers of 157 and 57 ( = ⁄ , = 361 We model the ctenophore as a self-propelled spheroidal body suspended in a quasi-static flow, 363 whose motion is governed by the balance between the propulsive and opposing forces and torques. 364 Table 6 lists all the model parameters. To describe the motion of the spheroidal body, we require 365 two coordinate systems: a global (fixed) coordinate system, in which a vector is expressed as ⃗ = 366 11 + 22 + 33 , and a body-based coordinate system in which ⃗ ′ = 1 ′̂1 ′ + 2 ′̂2 ′ + 3 ′̂3 ′ (see 367   � � �⃗̇′�. We will define each one of these terms in the following subsections. However, we direct the 387 reader to the supplementary material for details of the solution procedure, the numerical 388 implementation, the formulations for various coefficients, and the validation of the model against 389 experimental data. 390 (2) 391 Expressions for propulsive forces and torques 393 As seen in Fig 9B, the ctene tip follows a roughly elliptical trajectory during the power-recovery 394 cycle. During the power stroke, the paddle is extended and moving quickly; during the recovery 395 stroke, the paddle is bent and moving slowly. Such a cycle is both spatially asymmetric (higher 396 flow-normal area on the power stroke vs. the recovery stroke) and temporally asymmetric (power 397 stroke duration shorter than recovery stroke duration). To model this, we consider each ctene as 398 an oscillating flat plate with a time-varying length, whose proximal end oscillates along a plane 399 tangent to the body surface and whose distal end traces an ellipse (Fig 9D). The coordinates of the 400 distal end � ( ), ( )� are prescribed parametrically, and are constrained by five parameters: the 401 maximum length of the ctene ( ), the stroke amplitude (Φ), the beat frequency ( ), the temporal 402 asymmetry ( ), and the spatial asymmetry ( ). Further details are found in the supplementary 403 material (S1 Text). 404 We "place" a modeled ctene in each of the ctene positions (determined by , , and , coupled 405 with the body geometry) around the spheroidal body (Figs 9A and 9C). Each ctene oscillates 406 around its initial position (Fig 9D), creating a force tangential to the body surface (S1 Video). The 407 total propulsive force of the ℎ ctene row is modeled as the negative of the drag force summed 408 over each of oscillating plates: 409 where is the fluid density and is the number of ctenes in a given row ( is the index of the 411 ctene). The flow-normal area is given by the plate width (assumed to be 0. where is the angle defining the tangent to the body surface at the th plate (see Fig 9D). 420 Metachronal coordination is incorporated by dephasing the plate kinematic variables ̇ and 421 by an amount ( − 1) , where = • . Considering all ctene rows, the net propulsive 422 force is 423 We note that a one-dimensional model of a single ctene row is described in [28] and the sensitivity 425 of the net force to various model inputs is explored therein. 426 Propulsive torque is calculated as the cross product of the ctene's position relative to the centroid 427 of the body and the force generated by the ctene: 428 where ⃗ ′ is the position vector of the ℎ ctene in the ℎ row (relative to the body centroid), and 430 the bracketed term is the ctene propulsion force in the global frame of reference. To calculate the 431 propulsive torque, the propulsive force must be expressed in the body frame of reference; hence, 432 we multiply it by the transformation matrix . 433

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Expressions for resistive forces and torques 443 The drag force on the 3D spheroidal body is: 444 Because the body is spheroidal, we must consider two drag coefficients: ∥ is the drag coefficient 446 for the longitudinal movements (roll axis, ̂1 ′ ), and ⊥ is the drag coefficient for the lateral 447 movements (pitch and yaw axes, ̂2 ′ & ̂3 ′ ). Because we are in the viscous-inertial (intermediate 448 Reynolds number) regime, ∥ and ⊥ are each a function of both speed and geometry [47]. These 449 coefficients are multiplied by the respective velocity squared components (transformed to the body 450 frame of reference by the transformation matrix ), the corresponding flow normal area ( 2 , for 451 || , and , for ⊥ ), the fluid density, and a factor of 1/2. Finally, to transform the components 452 of the drag force back to the global frame of reference, we multiply by the transpose of the 453 transformation matrix . The drag force on the ctenes has already been incorporated as part of ⃗ , 454 which opposes the direction of motion during the ctene's recovery stroke. 455 The acceleration reaction (added mass) force is calculated as , where is the fluid density, 456 is the body volume, and is the added mass coefficient, which depends on the body shape and 457 the direction of motion [48]. We need two added mass coefficients for our spheroidal body: ∥ , 458 for motion along to the roll axis, and ⊥ , for motion along to the pitch/yaw axes [49] . Similar to 459 the derivation of the drag force (Equation (8) where is the equivalent sphere diameter (i.e., the diameter of a sphere with the same volume as 468 the spheroid), ∥ is the torque coefficient for rolling, and ⊥ for pitch and yaw. Both coefficients 469 are a function of angular speed and geometry (see S1 Text). The sign function is introduced so that 470 the resistive torque always opposes the body motion. 471 We use our reduced-order model to create simulated trajectories of freely swimming ctenophores 472 to explore the omnidirectional capability (MV) and general turning performance (MAP) of all the 473 possible turning strategies of the ctenophore locomotor system across the available parameter 474 space. The morphometric parameters for the model are based on the mean values of our 475 experimental observations (Table 2). The kinematic parameters (stroke amplitude Φ, phase-lag , 476 temporal asymmetry , and spatial asymmetry ) cannot be directly measured from the video 477 recordings of the swimming trajectories. However, kinematic parameters for the same set of 478 animals, filmed at a higher spatial resolution, are reported in [28]. Based on these measurements, 479 we use the following representative values: Φ = 112°, = 13.2%, and (based on the model 480 validation, see S1 Text) = 0.3, and = 0.3. 481