Kinematics of swimming of the manta ray: three-dimensional analysis of open-water maneuverability

Unsteady swimming is a vital aspect of the locomotor repertoire of aquatic animals (Webb, 1997, 2006; Fish and Domenici, 2015). Unsteady motions are related to changes in speed, orientation and direction (e.g. turning, acceleration). By utilizing self-induced instabilities (i.e. muscular effort), animals can produce changes in the state of the center of mass (COM) and effect maneuvers. Maneuvers are important for catching prey, escape from predators, negotiating obstacles in complex spatial environments, avoiding environmental disturbances, and ritualistic displays and mating (Howland, 1974; Webb, 1976; Weihs and Webb, 1984; Domenici and Blake, 1997; Maresh et al., 2004; Walker, 2004; Ware et al., 2014; Fish and Hoffman, 2015). Turning has been the focus of the majority of research on maneuvers, particularly with respect to lateral or yawing turns (i.e. rotation around the vertical axis normal to the axis of motion; Walker, 2000; Webb, 1994, 2006).

Turning performance is assessed by measurements of maneuverability and agility. Maneuverability is the capability to turn in a confined space and is measured as the length-specific radius of the turn trajectory (R/L, where R is the radius of the turn and L is total body length); agility is the rate of turn measured as the angular velocity (ω) (Norberg and Rayner, 1987; Webb, 1994; Walker, 2000). Norberg and Rayner (1987) considered an inverse relationship between maneuverability and agility, whereby an animal can turn tightly at a low speed or make a wide turn at higher speeds. However, compared with large-bodied swimmers, swimmers of small size can be highly maneuverable and agile, because turning radius is proportional to body length and agility decreases with increasing size (Webb, 1994; Fish and Nicastro, 2003; Parson et al., 2011). In addition, swimmers with greater flexibility of the body are more agile and have higher maneuverability than rigid-bodied animals of equal size (Fish, 2002; Fish and Nicastro, 2003; Parson et al., 2011).

The majority of studies of turning performance by marine vertebrates (e.g. fish, turtles, penguins, sea lions, dolphins) have been confined to animals that were examined in laboratory or aquarium settings (Webb and Keyes, 1981; Webb, 1983; Hui, 1985; Blake et al., 1995; Webb et al., 1996; Domenici and Blake, 1997; Gerstner, 1999; Schrank et al., 1999; Walker, 2000; Webb and Fairchild, 2001; Fish et al., 2003, 2012; Domenici et al., 2004; Rivera et al., 2006; Cheneval et al., 2007; Danos and Lauder, 2007; Parson et al., 2011). In certain cases, these studies were able to determine maximal performance in terms of maneuverability and agility by manipulation of the test environment or through training. Unrestrained swimming by vertebrates in nature is difficult to examine because of factors such as infrequent observations, size of the animal, and difficulty setting up reliable recording equipment. Recently, the maneuvering performance has been recorded for large cetaceans by multi-sensor tags and underwater cameras that were mounted on the whales (Hazen et al., 2009; Ware et al., 2011, 2014; Wiley et al., 2011; Goldbogen et al., 2013; Williams et al., 2015). However, this new methodology has limitations owing to the inability to accurately determine position and velocity, and to place the animal in a fixed frame of reference. As a result, measures of maneuverability and agility may not have the accuracy of direct observation. Furthermore, the association of kinematics of the mobile control surfaces (e.g. flukes, flippers) with the turning radius and rate are limited.

The use of three-dimensional videography provides the opportunity for enhanced accuracy and precision to examine maneuvering performance of aquatic animals (Boisclair, 1992; Hughes and Kelly, 1996). The swimming kinematics of the body and fins of fishes have been examined with three-dimensional videography for steady swimming and maneuvering (Boisclair, 1992; Tytell et al., 2008; Blevins and Lauder, 2012). However, these studies were performed in the laboratory in confined spaces. The only field studies using three-dimensional video tracking were for trout (Salmo sp.) that were foraging in a river (Boisclair, 1992; Hughes and Kelly, 1996; Hughes et al., 2003). The movements of relatively small (range 0.55–0.63 m) trout were examined to determine the swimming speeds and reaction volumes used to detect and acquire prey.

Mantas (Mobula birostris and Mobula alfredi; family Myliobatidae) are the largest of the over 500 elasmobranch species of batoid fishes (skates and stingrays), which have relatively rigid dorsoventrally compressed bodies and expanded pectoral fins (Rosenberger, 2001; Douady et al., 2003; Marshall et al., 2009; Kitchen-Wheeler, 2010). The manta can weigh over 1580 kg and is reported to have high-aspect-ratio (3.5, ratio of span to chord) pectoral fins that can reach a span of over 9 m (Perlmutter, 1961; Deacon et al., 1997; Compagno, 1999; Homma et al., 1997; Last and Stevens, 2009; Fontanella et al., 2013). Mantas use oscillatory locomotion (mobuliform mode) where there is a small undulatory component (the wavelength of the undulation is greater than the chord length of the pectoral fin), and swim primarily by flapping the pectoral fins dorsoventrally, analogous to the flight of birds (Klausewitz, 1964; Webb, 1994; Rosenberger, 2001).

Mantas are able to execute turning maneuvers (Clark, 1969; Duffy and Abbott, 2003; Fish et al., 2012). The pectoral fins exhibit a range of both symmetrical and asymmetrical motions (Homma et al., 1997), which can assist in turning. Although capable of leaping behavior, the most notable swimming maneuver by manta is a looping or backward somersault behavior used in feeding and mating (Bigelow and Schroeder, 1953; Homma et al., 1997; Duffy and Abbott, 2003; Fish et al., 2012).

Although swimming and maneuvering capabilities by the manta indicate a high level of performance (Homma et al., 1997; Duffy and Abbott, 2003; Fish et al., 2012), previous studies were limited by the lack of quantitative measurements on maneuverability and agility. Furthermore, swimming performance has not been analyzed in the field. Previous work by Parson et al. (2011) on maneuvering rays was only performed on small species in an aquarium setting using a single camera. Data on maneuvering performance are limited if video recordings are taken with a single camera and the animal swims outside the focal plane of the camera. The size of the manta and its operation within a large three-dimensional space present a challenge to measure turning capabilities in open water. In this study, we quantitatively analyzed the trajectory of the manta with stereovideography underwater to detail three-dimensional maneuvering performance in its natural environment. This is the first time that the maneuvering capabilities of fish of large size in the field have been described with a stereovideography system. The methodology of analysis of the three-dimensional data involved complex computation and statistical techniques beyond what had previously been performed for a two-dimensional study by Parson et al. (2011). The turning performance of the manta was compared with that of other swimmers to evaluate the effects of large body size as well as a stiff body. As the body of the manta is rigid along its longitudinal axis, it was predicted that the manta would have a turning performance similar to that of other rigid-bodied swimmers.

Video recordings of mantas (Mobula birostris Walbaum 1792) in the field were undertaken in 2008 and 2009. An evaluation of the molecular phylogeny of the genus previously called Manta considers that Mobula should be used as it has taxonomic priority (Poortvliet et al., 2015; White et al., 2017). However, the common name manta will be used throughout the article as the batoid classification is currently in transition. Data collection of the maneuvering behavior of mantas was performed in an open water environment around the island of Yap, Micronesia. The location has at least 65 known mantas, which frequent the lagoon inside the reef that surrounds the island (Homma et al., 1997). Mantas come inside the reef for mating activities and to visit cleaning stations (Homma et al., 1997). Video recordings of mantas were confined to a location on the north side of the island called Manta Ridge, which has access to the ocean through Mill Channel. A cleaning station was located at Manta Ridge at a depth of 24 m. Individuals were identified by the pigmentation pattern on the venter of the body (Homma et al., 1997).

To detail the kinematics of maneuvering by mantas, scaled stereo video recordings (30 frames s⁻¹) of mantas were made. With divers on SCUBA, two video cameras (Sony Handycam, model HDR-SR11, 10.2 megapixel resolution, Sony, Tokyo, Japan) in underwater housings (Dive Buddy, model SR11/12, Amphibico, Montreal, Quebec, Canada) were used for video recording. Each camera was mounted on a tripod. The cameras were positioned on the bottom of the lagoon around a cleaning station approximately 5 m apart with overlapping fields of view. The two video recordings were synchronized by periodic discharges of an underwater strobe (Sea&Sea YA-40A, Tokyo, Japan), which was triggered by an underwater still camera (Sea&Sea Motor Marine 35, model MX-10). A three-dimensional calibration device was positioned within the center of the field of view for both cameras at a distance of approximately 6 m. The calibration device consisted of six aluminum rods (9.7 mm diameter, 0.61 m long) that were screwed into each face of a 50 mm nylon cube, providing an orthogonal arrangement of the rods. Two 38 mm diameter nylon balls were attached to each rod with one ball at the end of the rod and another positioned equidistantly on the rod at 0.3 m between the centers of the terminal ball and the central cube.

The research was approved by the West Chester University Institutional Animal Care and Use Committee (Fish 08-01).

Video recordings were analyzed using Proanalyst Professional (Xcitex, Cambridge, MA, USA) to manually track eight distinct points on the body in three dimensions. The eight points included right and left cephalic lobes, right and left eyes, right and left pectoral fin tips, dorsal fin tip and base of the tail. The base of the tail coincided with the posterior margin of the dorsal fin. The body length (L) was measured as the distance from the eye to the base of the tail. Manta body length was estimated as 1.25±0.12 m.

The digitized points through a maneuvering sequence were assigned X, Y and Z coordinates. The pectoral fin closest to the center of a turn was referred to as the inboard fin and the fin furthest from the turn center was the outboard fin in accordance with the nomenclature of Fish and Nicastro (2003). The frequency of pectoral fin strokes was determined as the inverse of the period of at least one complete stroke cycle, where the tip of one fin could be observed. The difference in vertical displacements between the inboard and outboard fins when turning was measured when the fin tips of both fins could be observed through at least one half cycle. Maximum bank angles of the mantas executing a turn were measured from video with a protractor when the ray was oriented with the tips of the pectoral fins perpendicular to the camera view.

Each trajectory was tracked for intervals of 0.33 s in three dimensions. However, one dimension was solely due to the angle of the camera and did not contribute to the turning motion of the manta ray. Because this dimension was not consistent between trajectories, the points were projected from three dimensions into the two-dimensional turn using principal component analysis (PCA). PCA is a popular approach for deriving a lower-dimensional set of features from a larger-dimensional space. This method systematically rotates the points onto new axes, each explaining less of the overall variation in the data set than the last. This process continues until all of the variance has been explained by the orthogonal components. Fig. 1 shows the three-dimensional turning trajectories of four manta rays projected onto their respective first two principal components. Over 99% of the total variation in the three dimensions was explained by the first two principal components for all 30 trajectories.

The trajectory of the manta was defined in rectangular principal coordinates by the plane curve c(t)=[x(t), y(t)], where c is the position, t is time, and x and y are coordinates. Each of the 30 turning maneuvers was visualized as a trajectory in the plane. To fully capture a turn, the turning rate (ω, deg s⁻¹) and turning radius (R, m) were estimated instantaneously every 0.33 s throughout the entirety of the turn. ω(t) was defined as the derivative of the unit tangent vector to the curve, T(t). R(t) was defined as the radius of the osculating circle, which was the best-fit circle to the curve at a given point with the same unit tangent vector and curvature. Curvature, κ(t), was defined as the rate that the unit tangent vector changed with respect to arc length. Swimming speed (m s⁻¹) was calculated as 2πRω/360.

The observables were time averaged to provide a single estimate of the turning rate, radius, swimming speed and centripetal acceleration for each of the 30 turning sequences. To examine the maximal turning performance by the mantas, data were expressed as maximum and minimum values, means±1 s.d., median values and the means of the extreme 20% of values (i.e. minimum R, maximum ω). Choice of the extreme 20% of values was considered arbitrary but was used previously for comparisons of turning performance (Webb, 1983; Gerstner, 1999; Fish et al., 2003; Fish and Nicastro, 2003). Regression equations and correlation coefficients, r, were computed using KaleidaGraph Version 4.5 (Synergy Software, Reading, PA, USA). Statistical significance was determined at the P≤0.5 level.

There were two sources of variability in the data set. The first was due to the movement of the unsteady motion of the mantas. The second was because points on the manta to be tracked on the two synchronized videos were manually tracked in the three-dimensional space. In order to obtain smooth trajectories of the rays from the data, smoothing splines were used in both principal component dimensions separately. Splines are a general class of piecewise polynomial functions. As is most commonly the case, cubic splines were used. This manipulation produced a continuous curve with existing first and second derivatives. Smoothing splines also have the advantage of a regularization term that shrinks many of the estimated coefficients to zero. Instead of determining the smoothing parameter directly, the effective degrees of freedom were chosen based on the number of observed points for each ray. Based on the empirical data, the square root of the number of observations resulted in a trade-off between smooth curves and capturing local turns. The trajectories projected on to the first two principal components and the smoothed curves are displayed in Fig. 1. It can be seen that this method removes fluctuations that are most likely due to the movement of the mantas, while also preserving the overall smooth turn of the ray.

As two points on the body were measured for each ray every 0.33 s throughout the entire turn, the cross-correlation between the trajectories was analyzed. The cross-correlation measures the linear relationship between two time series for the shift in time between the two trajectories. The goal is to find the value of the shift that maximizes the correlation coefficient. In order to obtain a more stable estimate of the cross-correlation between two series, any trend and auto-correlation must be removed within the individual series. This removal is necessary because any inherent trend or auto-correlation within the individual series can artificially inflate the cross-correlation between any two series. We removed artificial inflation by using autoregressive moving average (ARMA) models. These models have two parts: regressing the outcome on time-shifted values (AR) and then modeling the error terms as a moving average (MA) of other error terms of time-shifted values. The order (i.e. number of shifted values) for each part can be found by examining the auto-correlation function (ACF) and partial auto-correlation function (PACF), respectively. Orders of each shifted value were chosen such that they were the largest time shift with values of ACF and PACF still significantly different from zero at the 95% level. After estimating the parameters via maximum likelihood estimation, residual values were examined between the two time series. Finally, the cross-correlation between the turning rates at both points of the body was examined.

Fig. 1 illustrates an example of the data processing (i.e. PCA, smoothing splines and cross-correlation analysis) from beginning to end of a representative trajectory. Two trajectories of points on the body of a single manta were analyzed.

Mantas were recorded singly or in pairs as they maneuvered about the cleaning station. Mantas swam 1–2 m above the substratum. A total of seven individual mantas were recorded for 30 turning sequences. The presence of the divers and calibration device did not appear to negatively affect behavior, as mantas are inquisitive (Last and Stevens, 2009). Each manta swam with an oscillatory, wing-like motion (Movie 1) as has previously been described by Klausewitz (1964) and Fish et al. (2016).

Yawing turns were accompanied with banking (rolling) of the body by 3–80 deg from the vertical (Fig. 2). A combination of powered (active fin strokes) and unpowered (gliding) movements was used to produce turning. Powered portions of turns were associated with asymmetrical motions of the pectoral fins. The outboard fin always moved through a greater vertical excursion than the inboard fin by 1.7±0.6 times. The frequency of the pectoral stroke was 0.32±0.11 Hz.

The mantas swam slowly at 1.42±0.50 m s⁻¹ (range: 0.46–2.51 m s⁻¹). Speed changed through the maneuver, which was associated with active fin movements interspersed with glides. Glides were executed at the end of the up-stroke. During periods of gliding, the pectoral fins were held with a dihedral orientation (Fig. 2).

Frequency histograms for the estimated ω and R are shown in Fig. 3. The data show a very skewed right distribution for both ω and R. Neither mean nor median values for ω and R were significantly correlated with swimming speed. The mean ω over 30 turning sequences was 18.26±5.90 deg s⁻¹ with a median value of 10.93 deg s⁻¹. The highest measured ω was 67.32 deg s⁻¹ and the mean ω for the highest 20% of values was 42.65±16.66 deg s⁻¹. The mean R over 30 turning sequences was 5.82±2.16 m with a median of 1.75 m. The smallest measured R was 0.48 m and the mean R for the smallest 20% of values was 2.05±1.26 m. The minimum length-specific radius (R/L) was 0.38 and the mean R/L for the smallest 20% of values was 1.64.

The mean and median ω decreased curvilinearly with increasing R (Fig. 4). Mean and median ω were significantly correlated with R (mean: r=0.81, P<0.05, d.f.=5; median: r=0.87, P<0.05, d.f.=5). The regression equations were: mean ω=51.58R^–0.74 and median ω=17.1R^–0.62.

The centripetal acceleration (a_c) was low, ranging from 0.01 to 0.18 g for mean values and 0.00 to 0.03 g for median values. a_c increased linearly with ω (Fig. 5), where the regression of the mean values was significantly correlated (r=0.76, P<0.025, d.f.=5), but the median values were not significantly correlated (r=0.61, n.s., d.f.=5). The regression equations were: mean a_c=0.009+0.002ω and median a_c=0.004+0.0003ω.

Of the 30 turns recorded, 17 sequences had enough overlap between two points on the body to perform a cross-correlation analysis. Of those 17, 10 had a cross-correlation value that was only significantly different from zero at a time shift of zero (i.e. the two series had a strong linear relationship at the exact same time). The remaining seven sequences only included one that had a cross-correlation that was significantly different from zero at a time shift within a reasonable amount of time (2 s). Indeed, one sequence resulted in a significant cross-correlation when the base of the tail lagged the left cephalic lobe by 0.66 s. The other significant time shifts occurred after more than 2 s and were very infrequent. These types of time shifts suggest that they were spurious associations between two points on a large body. The fact that there were mainly significant cross-correlations between ω at a time shift of zero implies that the two points on the body were effectively turning at the same rate. This was expected for a rigid-bodied animal.

The maneuvering performance of the manta was assessed by the use of stereovideography in the field. This was the first such analysis of a large aquatic animal in its natural surroundings. As opposed to laboratory studies and the use of tags in the field, stereovideography has three distinct advantages (Boisclair, 1992). First, analysis employing stereovideography allows a detailed qualitative and quantitative assessment of movement. The body orientation and movement of fins can be directly observed and thus used to determine their influence on maneuverability and agility. When appropriately scaled, speed can be measured and changes in speed can be associated with distinct fin movements. Second, this procedure allows measurement of the animal's movement in three dimensions. As fish are capable of swimming and maneuvering in three dimensions, the method can capture movements in their natural habitats. This was advantageous for examination of the manta owing to its size and operation in a large volume space. Third, in contrast to the attachment of tags on large aquatic animals, stereovideography can be used without influencing behavior, increasing drag and thus swimming performance, and injuring the animal (Read and Westgate, 1997; Watson and Granger, 1998; Chilvers et al., 2001; Pavlov et al., 2007; Dewar et al., 2008; Balmer et al., 2010; Jones et al., 2011).

Despite the advantages, there are also limitations to the use of stereovideography in the field. Boisclair (1992) considered that visibility was the most serious limitation in video recording underwater, which would be associated with low light intensity owing to turbidity, time of day and depth. Regarding the swimming by the mantas in the present study, visibility affected the distance from the cameras that movements of the mantas could be recorded. In addition, the low light levels made it difficult to manually track the distinct points on the animals. This reduction in accuracy potentially added to the error in digitizing the trajectory of a maneuver by the mantas. Therefore, mathematical analysis was required to statistically determine the swimming path of the manta. Another limitation was that video recording was confined to spatially limited locations where mantas were known to frequent, but may not have displayed maximal maneuvering performance. As a result, the data were highly skewed. Mantas swimming and foraging in the open ocean may have different degrees of maneuvering performance compared with movement around reef locations.

Another logistical limitation to stereovideography in the field was related to maintaining a fixed frame of reference. Cameras were positioned on tripods that were placed on the bottom. The rigid positioning of the cameras constrained what videos could be used and was dependent on the animal performing predictably in the chosen location. Stereovideography could not be performed in the mid-water column in the field owing to independent movements by the divers and independent movements by the mantas against a background with no fixed markers.

As displayed from videos of mantas in areas around the cleaning stations, these pelagic fish demonstrated maneuverability and agility that would have been considered uncharacteristic of such large animals. Mantas generally swim slow around cleaning stations and will hover above the bottom to be cleaned by wrasse (Labroides dimidiatus) or small shrimp (Homma et al., 1997). The speeds that were used to perform the turns (1.42±0.50 m s⁻¹) were higher than swimming speeds reported for mantas while foraging (range 0.25–0.47 m s⁻¹) and migrating (0.97 m s⁻¹) when measured from satellite tags (Graham et al., 2012; Jaine et al., 2014). The low speeds from satellite tags may be due to the intermittent uploading of positional data that does not take into account maneuvers and changing depth. However, Yano et al. (1999) reported that mantas can swim at 2.78–4.17 m s⁻¹ when males are chasing females during mating. In addition, mantas perform aerial leaps clearing the water (Coles, 1916; Rayner, 1986; Homma et al., 1997; Duffy and Abbott, 2003; De Boer et al., 2015). The physics of such leaps requires high-speed swimming (Rohr et al., 2002). By conservation of energy, a 2722 kg manta leaping 1 m into the air would require an initial speed of 4.43 m s⁻¹, which is 3.1 times the mean speed for mantas performing turning maneuvers in the present study.

The turns performed by mantas were produced by a combination of translational and rotational movements. As mantas have a rigid body, the turning performance measured from anterior and posterior points on the body were the same and thus reflect the turning performance of the COM. The COM would be on a line between the two points of measurement. For mantas as well as other mobuliform rays, the COM is located at 47% of body length (Fontanella et al., 2013).

The mantas were able to execute turns by unpowered glides and powered flapping motions. Unpowered glides have been observed in myliobatid rays that swim with an oscillatory, lift-based propulsion (Rosenberger, 2001; Parson et al., 2011). Gliding turns have also been noted for other oscillatory, lift-based swimmers including penguins, cetaceans and sea lions (Hui, 1985; Fish, 2002; Fish et al., 2003). In powered turns by the manta and other myliobatid rays, propulsive motions of the pectoral fins were evident throughout the entire turn (Parson et al., 2011; present study). Turning was accomplished by differential movements of the pectoral fins, with the outboard fin flapping at a higher frequency than the inboard fin.

Unpowered turns would rely on banking. Banking was required to generate the centripetal force to produce a curved trajectory. Banking is a rolling maneuver that provides a greater projected area facing the axis of the turn (Fish et al., 2003, 2012; Parson et al., 2011). Lift is generated primarily from the pectoral fins and additionally from the cambered body (Parson et al., 2011; Fish et al., 2012, 2016). The lift vector is perpendicular to the frontal plane of the body and fins of the ray (Fig. 6). The lift opposes a hydrodynamic force generated from the combined forces of the drag on the body and fins in the transverse plane and weight of the negatively buoyant animal. When banking through the turn, the lift vector is canted at an angle, which is directed toward the inside of the turn (Fish et al., 2012). The horizontal component of the lift vector generated by the pectoral fins produces a centripetal force, which turns the manta. Banking is a more economical means of moving in a circular path than generating an asymmetric thrust (Weihs, 1981). Another myliobatid ray, the cownose ray (Rhinoptera bonasus), banks by 65 and 70 deg in yawing turns (Parson et al., 2011). High bank angles are characteristic of aquatic animals that lack median fins and turn using elongate paired fins (Fish and Battle, 1995; Fish, 2002; Fish et al., 2003, 2012).

The wide variation in R was due to the movements of the mantas within a large volume. The turns may not represent the minimum R that could potentially be accomplished. Small R appeared to be associated with slow swimming speeds. The mean R for the smallest 20% of values was 2.05±1.26 m performed at a swimming speed of 1.06±0.39 m s⁻¹, whereas the mean R for the largest 20% of values was 9.31±1.73 m with a mean swimming speed of 1.46±0.57 m s⁻¹. Indeed, the minimum R of 0.48 m for a manta occurred at the slowest recorded swimming speed of 0.45 m s⁻¹. Fish and Nicastro (2003) reported that turning radius was directly but weakly related to swimming speed in whirligig beetles (Dineutes horni).

The highest turning rates (ω) for mantas were similar to the performance of cownose rays. Parson et al. (2011) found that cownose rays performed yawing turns at a rate of 44.4 deg s⁻¹ (Parson et al., 2011). This value of ω was only 1.04 times greater than the value for the mantas, despite the extreme size difference.

Aside from yawing turns, mantas are able to turn by pitching to execute somersaults in a looping circular trajectory (Coles, 1916; Clark, 1969; Duffy and Abbott, 2003; Fish et al., 2012). This somersault maneuver is used by mantas for feeding on concentrations of krill (Bigelow and Schroeder, 1953; Homma et al., 1997; Duffy and Abbott, 2003). Fish et al. (2012) video recorded somersaulting of a manta in an aquarium and measured the ray's turning performance for a single maneuver. The R/L of the loop was approximately 0.7, and the ω of the maneuver was 41.5 deg s⁻¹ (Fish et al., 2012). This turning performance was greater than yawing turns in terms of R/L by 1.7 times, but approximated the mean ω for the highest 20% of yawing turns for mantas measured in the present study.

The difference in performance for R/L represents limitations owing to flexibility of the pectoral fins versus the body that differentially affect movement about the rotation axes (pitch, yaw, roll). In yawing turns, torques can be generated from both passive and active synchronization and movements of the fins in addition to a centripetal force from banking. For pitching turns, the body is rigid in the sagittal plane resisting pitch (Fontanella et al., 2013). However, by simultaneous movements and chordwise bending of the flexible pectoral fins, mantas can execute small radius turns in the sagittal plane. This action was demonstrated in a YouTube video entitled, ‘Book'em Danno: Klepto Manta Mugs Cameraman’ (https://www.youtube.com/watch?v=I5wY38dhFPA), at 0:24 s displayed at normal speed and at 0:32 s in slow motion. In this case, the manta was estimated to make a 180 deg turn at 104 deg s⁻¹.

The ability to maneuver is a fundamental locomotory behavior. Maneuvering has been studied for a number of aquatic animals in both the laboratory and the field. In general, the size of turning radii is largely dependent on body size. Small animals can perform smaller radius turns than large animals (Howland, 1974; Fish and Nicastro, 2003). However, when the turn radius is scaled to body length, this generality does not always hold. Circular turns by small fish typically have radii of less than 0.2L (Webb, 1983; Domenici and Blake, 1997; Gerstner, 1999). The 12.4 mm whirligig beetle (D. horni) displayed a minimum turn radius of 0.24L (Fish and Nicastro, 2003), whereas the minimum turn radius was 0.09L for a 1.89 m sea lion (Zalophus californianus) (Fish et al., 2003) and 0.11L for 5.1 m killer whale (Orcinus orca) (Fish, 2002). In comparison, mantas displayed a  relatively large length-specific turn radius of 0.37L.

The limited maneuverability of the manta may have been due in part to motivation by the animal. As the mantas were freely swimming in an unconstrained environment, there was no incentive to make the tightest turn possible. Experiments performed on fish stimulated to yield fast-starts demonstrated that fish could turn with radii down to 0.06L (Domenici and Blake, 1997). In addition, the mantas are filter feeders and do not have to elicit small radius turns to acquire their planktonic food. The radius of the pitching somersault maneuver used by feeding mantas was 70% greater than the minimum yawing turn radius for the mantas observed around Yap (Fish et al., 2012).

Another constraint on turning performance was the relatively inflexible body of the manta. The large proportion of the body devoted to the branchial apparatus and the extended base of the thick pectoral fins limit flexibility in both the sagittal and frontal planes. The turn rate, ω, for mantas was lower than for similarly sized flexible-bodied swimmers (Fig. 7). Similarly, a_c for the manta was lower than for flexible-bodied animals (Fish and Nicastro, 2003). This assertion was largely a consequence of the wide turn radius and slow swimming speed for the manta compared with flexible-bodied animals.

Fish and Nicastro (2003) compared the ω of flexible- and rigid-bodied animals of different body sizes. When a line was drawn on a logarithmic scale between the maximum ω for the smallest (whirligig beetle: 4437.5 deg s⁻¹) and largest (the submarine USS Albacore: 3 deg s⁻¹) rigid bodies, flexible-bodied animals (e.g. fish, dolphins, sea lions) were positioned above the line and rigid-bodied animals (i.e. boxfish, squid, turtle, rays) were positioned below the line (Fish and Nicastro, 2003; Rivera et al., 2006; Parson et al., 2011). It was shown that for equivalent body sizes, flexible-bodied swimmers had higher turning rates than rigid-bodied animals. The humpback whale (Megaptera novaeangliae) is a flexible-bodied filter feeder with highly mobile flippers that has high agility for its size (Edel and Winn, 1978). Using tags with a three-axis magnetometer and accelerometers on humpback whales, Wiley et al. (2011) determined that a whale feeding on schools of small fish with an upward-spiral bubble-net feeding behavior had a maximum turn rate of 13.8 deg s⁻¹. Assuming a mature humpback whale of 12 m in length (Winn and Reichley, 1985), the turn rate of the whale was 11% higher than for an equivalent sized rigid-bodied swimmer.

Mantas have a low ω compared with that of various flexible-bodied animals (Fig. 7). The relatively rigid body, particularly with regard to lateral bending, would limit yawing turns. In this regard, the position of mantas below the line for rigid-bodied swimmers was consistent with previous studies (Fig. 7) (Fish and Nicastro, 2003; Rivera et al., 2006; Parson et al., 2011). However, mantas can compensate for their inflexibility by using the flexible distal portions of the pectoral fins or asymmetrical fin movements, and banking maneuvers (Fish et al., 2012, 2016; Russo et al., 2015). As a result, ω for mantas is only 22.7% lower than the predicted ω for an equivalent rigid-bodied swimmer. In addition to the ω of the manta being located closer to the line separating rigid-bodied from flexible-bodied swimmers than the ω of other batoids, the maximum ω of the manta was 40.3% and 110.4% greater than that of oscillatory- and undulatory-swimming rays, respectively (Parson et al., 2011). Walker (2000) insisted that ω for rigid-bodied swimmers was at least six to eight times lower than that predicted for flexible-bodied animals.

The analysis of the three-dimensional turning performance of animals in the field presents a number of challenges. The use of stereovideography has advantages that permit the ability to determine movements in three-dimensional space as well as the associated body movements. Using this technique, maneuverability and agility of manta rays were analyzed in the natural habitat of the species. Mantas were shown to bank and make small radius turns with turn rates as high as 67.32 deg s⁻¹. The data support the general conclusion that the rigid body of manta rays limits turning performance compared with flexible-bodied animals. In addition, the flexibility in the pectoral fins displayed by the manta allowed these large rays to perform turning maneuvers at a higher rate than other batoid rays. Such limitations are important as attempts are made to produce aquatic bio-robots inspired by the morphology and swimming kinematics of rays (Moored et al., 2011; Fish et al., 2016; Park et al., 2016).

We wish to express our appreciation for the assistance of the Manta Ray Bay Resort and Yap Divers and their staff, Bill Acker, John Chomed, Henry Erecheliug, Julie Morgan, R.J., Jan Siedsens and William Seiwemai. We also are grateful to Elizabeth Barchi, Janet Fontanella, Alex Meade and Rachel Nichols for data collection and analysis. We also wish to express our appreciation to Robert Bizzolara (program manager).