Does a Rigid Body Limit Maneuverability?

Most literature on the locomotion of fishes has focused on the maximum performance of either rectilinear swimming at uniform velocities or the fast-start escape response. In contrast, routine swimming performance is poorly understood, despite the greater frequency of routine swimming relative to the higher-performance behaviors (Webb, 1994a,b). Routine swimming in many fish species consists of frequent maneuvers, including turning laterally (yaw) and dorsoventrally (pitch), rolling, starting, braking and hovering (Breder, 1926). Most studies of animal maneuvering performance have focused on lateral turning. In fact, Norberg and Rayner (1987) formally defined maneuverability as the ability to turn in a confined space and have used the length-specific minimum radius of the turning path (r_path/L, where L is total body length) as a measure of this performance. Webb (1994a) has advocated this limited definition of maneuverability for fishes.

Morphologies that facilitate or limit turning have been widely discussed, but the effects of many design features on turning performance remain poorly resolved (Gerstner, 1999; Schrank and Webb, 1998; Schrank et al., 1999; Webb et al., 1996). It has been suggested that a stiff body limits turning performance (Aleev, 1969; Fish, 1997, 1999). The results of several studies are consistent with this hypothesis. The r_path/L of a fish with a relatively stiff body, the yellowfin tuna Thunnus albacores (0.47), is an order of magnitude greater than the relative turning radii of two fish with more flexible bodies, the knifefish Xenomystus nigri (0.065) and the angelfish Pterophyllum eimekei (0.055) (Blake et al., 1995; Domenici and Blake, 1997). Fish (1997) showed that the more flexible-bodied sea lion Zalophus californianus had a smaller r_path/L than seven more stiff-bodied species of cetaceans. Finally, Fish (1999) showed that the rigid-bodied whirligig beetle Dineutus horni has a higher r_path/L than more flexible fishes and aquatic mammals.

Nevertheless, Blake (1977) illustrated turning maneuvers of apparently small radii in two fishes with rigid bodies, the boxfishes (Ostraciidae) Lactoria cornuta and Tetrasomus gibbosus. Boxfishes are shallow-water, temperate and tropical fishes that are encased by a carapace of thickened, sutured scales modified into bony plates (Randall, 1972; Tyler, 1980). Because of this rigid armor, boxfishes cannot undulate their body anterior to the caudal peduncle. Consequently, boxfishes power both slow rectilinear swimming and turning maneuvers by oscillating or undulating the fan-shaped pectoral, dorsal and anal fins (Blake, 1977).

If boxfish can indeed turn with a small r_path, this would suggest that r_path is not a good measure of the actual radius of the space required to turn. A turn requires both the rotation of the animal and a translation of the center of rotation along a turning path. Letting r_e equal the effective radius of the turning animal (half the maximum projected diameter of the animal normal to the turning path), the minimum radius of the space required to turn, r_space, is r_e+r_path. For a rigid-bodied animal turning with r_path=0 (spinning with no translation), the minimum r_space is L/2. At the least, a flexible fish with the same L and r_path as a rigid fish should be able to turn in a smaller space than the rigid fish simply by having a smaller r_e.

Both measures of maneuverability, r_path and r_space, are important for the biology of fishes (see Discussion). To test whether and how body stiffness limits these two aspects of maneuverability, I measured both r_path and r_space in routine turns of a spotted boxfish Ostracion meleagris (Randall, 1972).

A single, female Ostracion meleagris (Fig. 1) was purchased from a wholesaler in Chicago (total length, L=11.6 cm). The fish was allowed 2 days to acclimate to the test tank, a 37.8 l aquarium fitted with a mirror oriented at 45 ° to the tank bottom. Water temperature was 25 °C. Dorsal views of turning sequences were filmed with a Red Lake high-speed digital camera at 125 Hz and saved as AVI files. There were no explicit attempts to elicit turning behavior by startling the fish.

AVI files were converted to Apple QuickTime format and digitized in a version of the public-domain NIH Image software (developed at the US National Institutes of Health and available on the Internet at http://rsb.info.nih.gov/nih-image/) modified to facilitate the digitization of landmarks (available from the author upon request). Twelve turning sequences in which there was little dorsoventral translation were analyzed. Both sharp and gradual turns were analyzed. All analyzed turns were preceded by forward swimming and were through a cumulative turning angle of >150 °. Four landmarks were digitized every fifth frame (Fig. 1): the tip of the snout, a spot on the dorsal midline immediately anterior to the dorsal fin, the base of the caudal fin and the posterior tip of the caudal fin.

The snout and caudal fin tip paths were used to estimate the boundaries of the turning space. These boundaries were computed as the maximum perpendicular distance to either side of the midline axis in the initial frame (Fig. 2). These distances were found by rotating the snout and caudal fin tip landmarks by the angle between the midline axis of the first frame and a positive x vector. r_space was then calculated as half the difference between the maximum and minimum y coordinates.

In addition to these two turning radii, several other variables were measured. Instantaneous turn angle, θ, was estimated as the angle between the midline segment connecting the snout and dorsal fin at time zero and the segment at time t. θ was differentiated with respect to time using a quintic spline algorithm (Walker, 1998a) to determine the turning velocity, ω. A normal acceleration coefficient, C_g, was computed as v² /r_pathg (Bandyopadhay, 1997), where v is the tangential velocity at κ_max and g is the acceleration due to gravity. Finally, the instantaneous angle, α, of the caudal fin, relative to the midline axis of the body was measured.

Turn magnitudes ranged from 153.2 to 250.4 ° (Table 1). The distance traveled by the center of rotation ranged from 5.2 to 15.8 cm (0.44–1.36L). The time for these turns ranged from 1.04 to 2.76 s. Mean turn rate ranged from 76.1 to 147 ° s⁻¹, while mean tangential velocity ranged from 2.7 to 8.1 cm s⁻¹ (0.23–0.7 L s⁻¹ ).

Although O. meleagris can perform hovering turns (rotation of the body without translation of the center of rotation), all analyzed turns included both translational and rotational components. Turning paths were not along the arc of a circle, which would require constant κ throughout the turn, but were characterized by an initial section with a low κ, a middle loop with a high κ, and a final section with a low κ (Figs 3, 4). While most of the sequences had single peaks of maximum curvature, three of the 12 sequences had two peaks of high curvature (Table 1; Fig. 4). In these double-peak turns, the loop section was characterized by an initial sharp 90 ° turn, followed by a short, straight path and then by a second sharp 90 ° turn. The fish rotated during the intervals with low curvature, indicating that the body was spinning while sliding, or slipping, through the water (Fig. 3). κ was not constant during the loop of high curvature but instead peaked approximately half-way through the loop. r_path was taken as the inverse of maximum curvature, κ_max (Table 1). Minimum r_path/L of the loop section ranged from 0.0005 to 0.11. Minimum effective length, L_e, during a turn was 8.3 cm. Minimum r_space/L ranged from 0.38 to 0.57, while r_space/L_e, which has a theoretical minimum of 0.5, ranged from 0.53 to 0.8.

Angular velocities changed throughout the turn. Peak ω, or ω_max, ranged from 107 to 218 ° s⁻¹. ω_max always occurred at or close to t_max, the time at which the turning curvature reaches its maximum. Tangential velocities changed throughout the turn, generally reaching a minimum near t_max. C_g values, which ranged from 0.01 to 0.16, were computed using the tangential velocity at t_max.

Howland (1974) suggested that r_path should be independent of the tangential velocity entering the turn. Indeed, neither r_path nor r_space was correlated with the tangential velocity entering the turn (r_path, r=—0.24, P=0.43; r_space, r=—0.17, P=0.59). In contrast, both r_path and r_space were correlated with the tangential velocity at t_max (r_path, r=0.96, P<0.0001; r_space, r=0.87, P=0.0003). This correlation is expected; at the limit, when κ_max is infinite, the tangential velocity must fall to zero as the body stops and reverses direction. ω_max was correlated with the tangential velocity entering the turn (r=0.69, P=0.01) but not with the tangential velocity at t_max (r=0.49, P=0.11) nor with either r_path or r_space (r_path, r=0.36, P=0.25; r_space, r=0.41, P=0.18).

The angle, α, of the caudal fin relative to the midline axis of the body changed throughout the stroke, reaching a minimum, α_min, at or near the time of ω_max. In some turns, the minimum formed a distinct inverted peak, while in others the minimum was within a broad inverted plateau. α_min ranged from 46.9 to 68 ° for turns with single high-curvature peaks (Table 1). α_min was 58.4 °, 81.2 ° and 89.9 ° for the turns with two curvature peaks. Within turns, correlations between α and ω ranged between —0.8 and —0.94. Among turns, α_min was significantly correlated with r_path and r_space (r_path, r=0.74, P=0.006; r_space, r=0.75, P=0.005). Excluding the double high-curvature turns, α_min was significantly correlated with ω_max (r=—0.77, P=0.015). These correlations suggest that the O. meleagris is partially controlling its turning kinematics by using the caudal fin as a rudder.

The single O. meleagris individual sufficiently demonstrated that a rigid-bodied fish can perform lateral turning maneuvers of approximately 180 ° during forward locomotion with minimum turning radii, r_path, approaching zero. This observation should not surprise anyone who has watched boxfish swim. The O. meleagris turns question the use of r_path as a measure of turning space. r_path will only approximate the radius of the actual turning space if the midline axis of the fish follows the turning path throughout the turn. For tight turns, the midline axis of the O. meleagris cannot follow the turning curve because of its rigid body. Instead, the snout and caudal fin tip paths sweep out much larger spaces than the center-of-rotation path (Fig. 3).

The actual radius of the space required to turn is captured by r_space. The theoretical minimum r_space for a turn is half the effective length of the fish, where the effective length is the major axis of the bent body. This minimum assumes that the fish is bent with constant curvature along its length and can maintain this curvature throughout the turn. A fish could turn with still smaller r_space if it effectively folded its body by passing a single, short wave of very high curvature down its body, as seems to occur in turning maneuvers of elongated, highly flexible fishes.

In the O. meleagris examined here, observed r_space for the turns with single high-curvature peaks ranged from 6 to 60 % above the theoretical minimum. The observed values are higher than the minimum value, even for the turns with low r_path, because the fish failed to maintain a low r_path throughout the turn. Casual observations suggest that O. meleagris can maintain a low r_path throughout a turn of 180 ° (and much higher), but these behaviors were not captured on digital video.

How do the minimum r_path/L (0.0005) and r_space/L (0.38) measured for this individual compare with those of other animals? For the stiff-bodied whirligig beetle Dineutus horni, minimum r_path/L was measured as 0.24 (Fish, 1999). For more flexible animals, r_path/L can be much lower. Mean r_path/L for the angelfish Pterophylum eimekei performing fast-start escape maneuvers was 0.065, but a minimum value has not been reported (Domenici and Blake, 1991). Domenici and Blake (1997) report an r_path/L of 0.055 for the knifefish Xenoystus nigri. Gerstner (1999) found minimum values of r_path/L between 0.02 and 0.05 for the median- and paired-fin (MPF)-powered turns and between approximately 0 and 0.05 for body- and caudal-fin (BCF)-powered turns of four species of coral reef fishes (foureye butterflyfish, Chaetodon capistratus; beaugregory damselfish, Stegastes leucostictus; ocean surgeonfish, Acanthurus bahanieusis; bluehead wrasse, Thalassoma bifasciatus). Minimum r_path/L was 0.09 for the California sea lion Zalophus californianus (Fish, 1997).

Turning curvatures along a turning path in the O. meleagris were not constant but were characterized by a sharp maximum approximately half-way through the turn. Among different animals, there is a continuum of turning styles, from quasi-uniform curvature turns along an arc of a circle to the highly non-uniform curvature turns of the boxfish. Unfortunately, there are not enough quantitative data from the literature to explore the relationship between maximum curvature within a path and the variability of curvatures along the path.

The only r_space values that I have found reported in the literature are from Schrank et al. (1999), who cleverly measured the minimum space required to turn by coercing fish to turn within tubes of different diameter. For turns of 180 °, mean r_space/L was approximately 0.11 for goldfish (Carassius auratus), 0.24 for silver dollar (Metynnis hypsauchen) and 0.26 for angelfish (Pterophylum scalare). The highest value, that for P. scalare, is more than three standard errors below 0.36, which is the minimum theoretical value for the individual O. meleagris analyzed here. Clearly, the flexible fishes can turn in smaller spaces than the rigid boxfish.

Webb (1994a) defined agility as the ability to quickly reorient the body and suggested turning rate, ω, as a measure of agility. The O. meleagris performed slow turns; the maximum instantaneous turning rate was 218 ° s⁻¹, while the maximum turning rate averaged over the entire turn, ω_avg, was 147 ° s⁻¹. By comparison, ω_avg for the whirligig beetle ranged from 400 to 4428 ° s⁻¹. ω_avg for the four coral reef fishes in the study of Gerstner (1999) ranged from approximately 100 to 1200 ° s⁻¹ for MPF turns and from approximately 650 to 9200 ° s⁻¹ for BCF turns. The sea lion Z. californianus turned at rates ranging from 150 to 690 ° s⁻¹ (Fish, 1997). Five of the seven cetaceans (with generally stiffer bodies than the sea lion) analyzed by Fish (1997) had low ω_avg, ranging between 25 and 200 ° s⁻¹, although one of the species (the white-sided dolphin Lagenorhynchus obliquidens) had a maximum ω_avg of 453 ° s⁻¹.

In a comparison of ω_avg across a broad size range of flexible, aquatic animals, Fish (1999) showed that ω_avg decreases with L. The ω_avg of two stiff-bodied vehicles, the whirligig beetle and the submarine USS Albacore, were six and eight times slower than the expected ω_avg of flexible animals (based on a deviation from the least-squares regression). In comparison, the maximum ω_avg for the O. meleagris was 26 times slower than the expected ω_avg. The measurement of maximum ω_avg for boxfish, in general, was not the goal of this study and there was no attempt to stimulate faster turns. The turning rate data are comparable, then, with the values for the reef fishes (Gerstner, 1999), cetaceans (Fish, 1997) and the whirligig beetle (Fish, 1999), but not with the fast-start escape values (Domenici and Blake, 1997), for which maximal performance is expected. Variation in ω_avg among multiple individuals or species of Ostraciidae was also not measured. Nevertheless, the available O. meleagris data, in combination with those of Fish (1999), are consistent with the hypothesis that a stiff body limits agility (Fish, 1997, 1999).

There are at least three reasons why the rigid body of the boxfish should limit agility. First, the bony exoskeleton precludes the myotomal muscles from bending the cranial end of the body into the turn and the caudal end of the body away from the turn. Second, the rigid body results in a relatively high pressure drag resisting rotation because the angle of attack between the body and the local flow is close to 90 ° along the entire length of the body. Finally, the inability to bend the body does not allow the fish to reduce the body’s second moment of area about the dorsoventral rotational axis, resulting in high inertial resistance to rotation.

Using r_path as a measure of maneuverability, the O. meleagris was maximally maneuverable despite its stiff body. Using r_space, the O. meleagris was a poor maneuverer relative to more flexible fish. Importantly, both performance variables are important measures of maneuverability in its broader sense. r_space should be associated with the ability effectively to exploit confined spaces created by complex structure, such as the vegetated margins of lakes and rock and coral reefs. The ability to rotate with a low r_path, in contrast, allows an individual precise control of its orientation, which could be important for sampling individual prey, especially in environments with rapidly changing flow velocities.

Boxfish power turns by the combined oscillation and undulation of the pectoral, dorsal and anal fins (Blake, 1977). The high correlations between α and ω within turns and between α_min and r_path/L among turns suggest that the caudal fin has a large influence on turning control by effectively acting as a rudder (Blake, 1977). The multiple control devices employed by maneuvering boxfish allow precise control of position and orientation, despite the rigid body.

Many reef fishes escape predation by rapidly seeking refuge within narrow spaces created by corals (Hixon and Beets, 1993), a behavior that is facilitated by extremely flexible (Hixon and Beets, 1993) or narrow (Webb et al., 1996) bodies. While the bony exoskeleton results in a rigid body that appears to limit the ability of boxfish to turn rapidly and in tight spaces, two behaviors necessary for effective refuging, the combination of the exoskeleton and the stress-induced secretion of ostracitoxin (Sancho, 1998; Thompson, 1964), provide an effective (but not perfect, see Sudekum et al., 1991), non-locomotor, anti-predator mechanism. By circumventing the need to refuge, the boxfish avoids the potential costs of refuging behavior (Godin, 1997).

R. Blob and two anonymous reviewers greatly improved the manuscript. This research was supported by Berkeley Research Associates and Office of Naval Research award N00014-99-1-0184 to M. W. Westneat and J.A.W. The initial ideas for this project were conceived while J.A.W. was supported by a National Science Foundation Postdoctoral Research Fellowship in the Biosciences Related to the Environment.