Three-dimensional kinematics of hummingbird flight

Hummingbirds (Trochilidae) are best known for their sustained hovering abilities, but they also regularly use forward flight. Both types of flight are central to the ecology of hummingbirds. While they routinely hover during foraging and displaying (Greenewalt,1960a; Wells,1993; Stiles,1982; Altshuler and Dudley,2002), they must also fly between foraging locations, at which time their velocities range from 5 to 11 m s^-1(Gill, 1985). Laboratory tests have shown that they are capable of even greater forward velocities, exceeding 13 m s^-1 (Greenewalt,1960a; Chai and Dudley,1999; Chai et al.,1999). Some species migrate long distances; the total migration distance for the rufous hummingbird (Selasphorus rufus) may exceed 6000 km, and the ruby-throated hummingbird Archilocus colubrismigrates, non-stop, over the Gulf of Mexico(Calder, 1993; Robinson et al., 1996).

Many aspects of hovering have been studied in hummingbirds, including kinematics (Stolpe and Zimmer,1939; Greenewalt,1960a; Greenewalt,1960b; Weis-Fogh,1973), physiology (Weis-Fogh,1972; Epting,1980; Bartholomew and Lighton,1986; Wells, 1993; Chai and Dudley, 1996) and aerodynamics (Warrick et al.,2005). Indeed, hummingbirds have emerged as a model assemblage for investigating maximal power capacity during hovering and climbing flight through the use of hypoxic, hypobaric and load-lifting protocols(Chai and Dudley, 1995; Chai and Dudley, 1999; Chai and Millard, 1997; Chai et al., 1999; Altshuler and Dudley, 2002; Altshuler et al., 2004). A prominent feature of wing motion during hovering in hummingbirds is pronounced supination about the long axis of the wing during upstroke, which is associated with a `figure-8'path of the wingtip in lateral projection(Stolpe and Zimmer, 1939; Greenewalt, 1960a). It was formerly thought that hovering downstroke and upstroke were kinematically and aerodynamically symmetrical, with each supporting body weight equally over each half of the wingbeat cycle (Stolpe and Zimmer, 1939; Greenewalt,1960a; Greenewalt,1960b; Weis-Fogh,1972; Weis-Fogh,1973).

This assumption of symmetry between half strokes needs to be revisited with new data, particularly because the aerodynamics of hovering are likely to be unsteady and, therefore, challenging to predict from kinematics(Spedding, 1993, Dickinson et al., 1999). Kinematics of similarly sized hovering hawkmoths Manduca sexta(Willmott and Ellington,1997b) lead to an estimate that the upstroke produces more lift than downstroke, while flow visualization about a dynamically scaled model indicates the downstroke produces more lift due, in part, to leading-edge vorticity (Van den Berg and Ellington,1997; Willmott et al.,1997). For the hovering hummingbird, the camber of the wing is different during the two halves of the wingbeat, and wake measurements indicate the majority (75%) of the weight support is provided by downstroke,while upstroke supports 25% (Warrick et al., 2005).

In contrast with the wealth of data available on hovering performance in hummingbirds, relatively little is known about how hummingbirds accomplish forward flight (Altshuler and Dudley,2002). Greenewalt (Greenewalt,1960a) presents lateral-view illustrations of wing paths and body postures of ruby-throated hummingbirds flying at 0, 4 and 13 m s^-1along with brief descriptions of wing kinematics at these velocities that include wingbeat frequency, downstroke fraction of the wingbeat cycle, and advance ratio. Data on long-axis rotation of the wing (pronation or supination) are lacking. Recent studies report maximum forward-flight velocities but not the associated wing kinematics(Chai et al., 1999; Chai and Dudley, 1999). From Greenwalt's description (Greenwalt, 1960a), it would appear that hummingbirds vary flight velocity primarily by changing the angle of their body relative to horizontal (β), thereby effecting a change in stroke-plane angle, as well as potentially varying their wingbeat amplitude (ϕ), which is known to vary according to power demands during hovering(Altshuler and Dudley, 2002; Altshuler and Dudley, 2003). Lateral-view figures representing the path of the wingtip hint (assuming the birds are drawn to the same scale) that ϕ may be less during forward flight at intermediate velocity compared with during hovering or fast flight(Greenewalt, 1960a). Paths of the wing in lateral projection also change from a figure-8 shape to an ellipse as hummingbirds change from hovering to forward flight(Greenewalt, 1960a).

No significant variation is reported for wingbeat frequency (f),the relative duration of the downstroke, or wing length (l) within the wingbeat cycle (Greenewalt,1960a; Greenewalt,1960b). Constant l would mean that upstroke:downstroke span ratio would be 100%, at all flight velocities, and this has important implications for the aerodynamics of flapping flight. Most birds in forward flight decrease their span ratio using wing flexion during upstroke to avoid producing negative thrust that is of equal magnitude to the positive thrust that they produced during downstroke(Rayner, 1986; Rayner, 1988; Tobalske, 2000). Assuming hummingbirds do not flex their wings and reduce their span during upstroke,they must change other parameters such as angular velocity or angle of attack of the wing to reduce upstroke circulation and lift so that they can sustain forward flight. Variation in circulation about the wings would lead to the shedding of a ladder-like wake structure with `rungs' of cross-stream starting and stopping vortices into the wake and inflation of induced drag relative to a constant-circulation wake (Rayner,1986). Recent kinematic evidence suggests that angle of attack and circulation varies throughout the wingbeat cycle in cockatiels Nymphicus hollandicus (Hedrick et al.,2002), and studies of wake structure reveal that regular shedding of cross-stream vortices is a characteristic of fast flight in passerines(Spedding et al., 2003; Hedenström et al.,2006).

Advances in technology since Greenewalt's pioneering research(Greenewalt, 1960a; Greenewalt, 1960b) allow us to explore more fully how kinematics of the wings and body of hummingbirds might vary with flight velocity. Given that, even during hovering, hummingbird flight is more consistent with the flight of other birds than previously believed (Warrick et al.,2005), we predicted that hummingbirds would vary aspects of their wing kinematics in a manner similar to other bird species(Brown, 1953; Brown, 1963; Tobalske and Dial, 1996; Tobalske, 2000; Park et al., 2001; Hedrick et al., 2002; Tobalske et al., 2003a; Tobalske et al., 2003b; Rosén et al.,2004).

Five female rufous hummingbirds Selasphorus rufus Gmelen 1788(body mass 3.4 g, Table 1) were captured from the wild under permits from the US Fish and Wildlife Service and Oregon Department of Fish and Wildlife. All housing and experimental protocols were approved by the University of Portland Institutional Animal Care and Use Committee. During captivity, birds were housed in 1 m×1 m×1 m flight cages with ad libitum access to food and water in the form of Nektar-Plus (NEKTON®; Günter Enderle, Pforzheim,Baden-Württemberg, Germany) or a 20% sucrose solution.

We measured the morphology of the birds with their wings spread as in mid-downstroke using standard techniques(Tobalske et al., 1999)(Table 1). Linear measurements(mm) were obtained using digital calipers and metric rulers. Areas(mm²) were measured from digital images using a known pixel-to-metric conversion. Average wing chord (mm) was calculated as wing area divided by wingspan. Aspect ratio (dimensionless) was calculated as wingspan divided by wing chord. Disc loading (N m^-2) was computed as body weight divided by disc area (S_d). For this instance, we assumed S_d=π(b/2)²,with b=wing span. Wing loading (N m^-2) was computed using body weight (N) divided by combined wing area, including the projected surface area of the body between the wings.

Our wind tunnel was designed for studies of avian flight at the University of Portland (Tobalske et al.,2005a). The tunnel is an open circuit with a closed jet, featuring a 6:1 contraction ratio and a total length of 6.1 m. The working section in which the bird flies is square in cross-section, 60 cm×60 cm×85 cm inner diameter at the inlet, with clear lexan walls, 6-mm thick, used to provide views inside the working section. The flight chamber increases to a 61.5 cm×61.5 cm outlet to accommodate boundary-layer thickening. Air is drawn through the tunnel using 7.5 kW (10 horsepower) direct current motor and a 0.75 m diameter fan assembly (AFS-75 Series, SMJ Incorporated, Grand Junction, CO, USA). Velocity (V) is selected as equivalent air velocity rather than true air velocity(Pennycuick et al., 1997; Hedrick et al., 2002). With a protective screen of vertical wires in place at the inlet of the working section, maximum deviations in velocity within a cross section are <10% of the mean, the boundary layer is <1 cm thick, and turbulence is 1.2%(Tobalske et al., 2005a).

Birds were acclimated to the flight chamber of the wind tunnel using an interval of 2-3 h, during which the birds were in still air with free access to a feeder and a perch. After this acclimation, the birds would sustain flight for 1 h or longer during hovering (0 m s^-1) and 2 m s^-1 and 10 min or longer during forward flight at faster velocities(4-12 m s^-1).

We allowed the birds to rest on the perch between trials, and we controlled access to the feeder to motivate flight. As the feeder was used in all trials,it undoubtedly affected inflow on the bird. Therefore, caution is warranted when extrapolating from our results to estimate performance in free flight. The feeder was a modified 1 ml Tuberculin syringe, 6.6 mm in diameter, with tabs removed and tip opening enlarged and painted red using fingernail polish. It was aligned parallel with inflow and suspended from the ceiling of the flight chamber using a steel wire (1.5 mm in diameter). The tip of the feeder was located in the midline of the flight chamber, approximately 10 cm back from the inlet and 20 cm down from the ceiling. Between trials, the feeder was manually refilled as necessary using Nektar-Plus or 20% sucrose. The perch was constructed of a 15 cm length of steel wire, 1.5 mm in diameter, and supported 20 cm up from the floor of the flight chamber between steel supports (12 mm×12 mm in cross section). The perch was oriented perpendicular to inflow, and at a minimum distance of 50 cm behind the tip of the feeder.

The combined frontal area of the feeder and perch (∼0.008 m²) was only 2% of the 0.36 m² area of the inlet of the flight chamber, so it was not necessary to correct tunnel velocity for blocking effects (Barlow et al.,1999).

We measured wing and body movement using digital video and three-dimensional (3D) reconstruction(Warrick and Dial, 1998; Hedrick et al., 2002). Digital video recordings, 2-4 s in duration, were obtained during longer intervals of sustained flight. We used two synchronized Redlake cameras, a PCI-2000 and PCI-500 (Redlake MASD LLC, San Diego, CA, USA) sampling at 500 frames s^-1 and with a shutter speed of 1/2500 s. Images were stored to computer using PCI-R v.2.18 software. Flights were illuminated using four 650-W halogen lights (Lowel Tota-light, Lowel-Light Manufacturing, Inc.,Brooklyn, NY, USA) distributed around the outside of the flight chamber.

We digitized anatomical landmarks and accomplished 3D reconstruction using custom M-files (available: http://www.unc.edu/~thedrick/)in MATLAB v.6.5 and v.7.0 (The Mathworks, Inc., Natick, MA, USA). To identify anatomical landmarks for digitizing, we marked the birds prior to the experiments using 1.5-mm dots of non-toxic white paint on the feathers over the spine (approximately over 1st thoracic vertebrae), dorsal and right-lateral base of tail, and, on the right wing: shoulder, wrist, distal tip of 1st secondary and distal tip of 9th primary.

For 3D reconstruction, we merged two-dimensional (2D) coordinates from each camera plane into a single 3D space using the Direct Linear Transform coefficients derived from a sixteen-point calibration frame(Hatze, 1988). For digitized points on all birds, median RMS error was <1 mm. Occasionally a point was not in the view of both cameras, resulting in a gap in the reconstructed point sequence. Point interpolation and filtering were accomplished using a quintic spline fit to known RMS error using the Generalized Cross Validatory/Spline program (Woltring, 1986).

Subsequent kinematic analysis used 3D coordinates of anatomical landmarks and software including MATLAB, IGOR Pro. (v.4.061, Wavemetrics, Inc.,Beaverton, OR, USA) and Excel (v.2003, Microsoft Corp, Renton, WA, USA). We utilized a bird-centered coordinate system (x, y and z axes centered at shoulder) to measure kinematics relative to the body and a global-coordinate system to measure aerodynamically relevant kinematics(Gatesy and Baier, 2005; Rosén et al., 2004)that require an estimate of translation of the body due to flight velocity. We accomplished transitions between coordinate systems using translation along x, y and z axes and rotation about Euler angles that described body pitch about the y axis and yaw about the zaxis. The anatomical landmarks we used did not permit measurement of roll, so we assumed that no roll was present and visually inspected video samples to eliminate from analysis any intervals of flight in which the birds appeared to maneuver. Thus, the observed lateral midline of the body(Fig. 1A), between the shoulder and the base of the tail, represented the mid-frontal plane, and the dorsal mid-line of the body, between the spine and the base of the tail, represented the mid-sagittal plane.

Movement at the wrist was used to identify characteristic portions of wingbeats including the transitions between upstroke and downstroke and the middle of upstroke and downstroke. At the start of each half-stroke, wrist span began increasing (the wrists were abducted from the mid-sagittal plane of the body). Downstroke started with wrist depression and upstroke began with wrist elevation. Mid-half-strokes occurred as the wristspan was at a local maximum, and with wrist elevation at the mid-frontal plane of the body. Body angle (β, deg.) was the acute angle between horizontal and the mid-frontal plane of the body (Fig. 1A). The stroke plane of the wing connected the shoulder, wingtip at start of downstroke and wingtip at the end of downstroke. We measured`anatomical' stroke-plane angle (Gatesy and Baier, 2005) relative to the mid-frontal plane of the body(γ<sub>b</sub>, deg.) and a `tracking' stroke plane angle relative to the horizontal plane (γ_h, deg.). Wing chord was a line connecting the wrist and the distal tip of the 1st secondary, and chord angle(α_c, deg.) was the cranially oriented acute angle formed between this lead line and the mid-frontal plane of the bird. This angle was<0° for pronation and >0° for supination.

We calculated wing and wrist span (mm) as double the perpendicular distance between the line connecting the spine and dorsal base of tail (hereafter`dorsal midline') and the distal tip of the 9th primary or the wrist,respectively. Span ratios for the wingtips and wrists (dimensionless) were spans at mid-upstroke divided by spans at mid-downstroke.

The inverse of the duration of the wingbeat gave wingbeat frequency(f, Hz). Wingbeat amplitude (ϕ, deg.) was measured in the bird-centered coordinate system as the acute angle formed between the line connecting the shoulder and wingtip at the start of downstroke and the line connecting the shoulder and the distal wingtip at the end of downstroke. Duration (s) of downstroke and upstroke were used to compute the proportion of time spent in downstroke (%). Average angular velocity of the wing (rad s^-1) was ϕ divided by duration of each downstroke or upstroke.

For each kinematic variable, we computed the mean value within each bird(N=5-10 wingbeats) for each velocity (N=7). We then used these means to test for a significant effect of flight velocity upon each variable using a univariate repeated-measures ANOVA (d.f. 4, 6) and StatView v.5.0.1 (SAS Institute, Inc., Cary, NC, USA). We also computed an overall mean± s.d. among the 7 flight velocities.

To describe variation in α_c and α within wingbeats,we converted time within a wingbeat cycle into a percentage of the full cycle that began with the start of downstroke and ended with the start of the subsequent downstroke. We then interpolated observed α_c andα using cubic-spline fitting with 100 points per curve. We computed an average and s.d. among birds for each of the 100 points.

Values are presented as means ± s.d.

Hummingbirds accomplished flight at different velocities by significantly varying the majority (75%) of the bird-centered kinematics variables that we measured (Table 2, Fig. 2).

Dorsal and lateral projections of the paths of the wingtips and wrists revealed gradual changes across the range of velocities(Fig. 2). During upstroke of slow flight (0 and 2 m s^-1), the tips and wrists traced in reverse nearly the same paths that were exhibited during downstroke. Lateral views of the paths at these slow flight velocities reveal an upwardly concave path of the wrists and wingtips, with the path of the wingtips also describing a horizontal figure-8 pattern in which the tip was higher during early downstroke, dipped down during late downstroke, raised at the start of upstroke, and dipped low during mid- and late upstroke. Wing movement at flight velocities of 4-12 m s^-1created two patterns. In dorsal view, the path of the wrists, but especially the wingtips, described a figure-8 loop with maximal wing span always exhibited during the middle of downstroke, and with the figure-8 loops becoming progressively more obvious as velocity increased. In lateral view, the paths of the wrists and wingtips were ellipses, with the wrists and wingtips positioned more cranially during downstroke and more caudally during upstroke.

As flight velocity increased from 0 to 12 m s^-1, body angle(β) decreased from 50±2° to 13±5° and tracking stroke-plane angle (γ_h) increased from 15±4° to 68±5° (Fig. 3A). These changes in β and γ_h produced a minimum anatomical stroke-plane angle (γ_b) of 61±5° during flight at 6 m s^-1 and a maximum γ_b of 82±3°during flight at 12 m s^-1.

Chord angle (α_c) varied significantly among flight velocities, with P=0.0168 for α_c at mid-downstroke and P<0.0001 for α_c at mid-upstroke(Table 2, Fig. 3B). Pronation of the wing, relative to the mid-frontal plane, of the body, occurred through most of downstroke. Consequently, α_c was negative during the majority of this phase of the wingbeat. Regardless of flight velocity, at the start of downstroke, α_c was near 0°, reached a minimum of between-13±17° and -36±5° at mid-downstroke, and returned to 0° at approximately 40% of the wingbeat cycle, 10% of the cycle duration before the onset of upstroke (Fig. 4). In contrast with downstroke, the wing was supinated during most of upstroke, resulting in positive values of α_c. Peak positive values of α_c were exhibited at mid-upstroke, at a time 65-70% into the full wingbeat cycle. Mid-upstroke α_cdecreased with increasing flight velocity from a maximum of 93±4°during hovering (0 m s^-1) to 26±10° and 26±8°during flight at 10 and 12 m s^-1.

Interactions among frequency (f), amplitude (ϕ), and the percentage of the wingbeat cycle spent in downstroke produced significant variation in the angular velocity of the wing among flight velocities(Fig. 5). With an average of 41±1 Hz, f was among the minority of bird-centered kinematics that did not vary significantly with flight velocity (P=0.1668). Wingbeat amplitude (ϕ) varied according to a U-shaped curve, with ϕbetween 100±6 and 103±19° at velocities of 2-8 m s^-1 and ϕ from 109±12 to 126±12° at 0, 10 and 12 m s^-1. The percentage of the wingbeat spent in downstroke increased with increasing flight velocity from 48% at 0 and 2 m s^-1to 53±1% at 12 m s^-1. For both downstroke and upstroke, the angular velocity of the wing varied with velocity according to U-shaped curves. Average angular velocity among all flight velocities was similar between downstroke (155+16 rad s^-1) and upstroke (156+19 rad s^-1). Local minima and maxima for each half of the wingbeat cycle occurred at different velocities. Specifically, angular velocity during downstroke varied from 140±6 rad s^-1 at 8 m s^-1to 178±21 rad s^-1 at 0 m s^-1, while minimum angular velocity during upstroke was 134±24 rad s^-1 at 4 m s^-1 and maximum was 195±23 rad s^-1 at 12 m s^-1.

All variables related to wing and wrist spans exhibited relatively low variability among flight velocities (Table 2); nevertheless, velocity had a significant effect upon wingspan at mid-downstroke (P=0.0014) and the span ratios of the wingtips(P=0.0072) and wrists (P=0.0332). At mid-downstroke,wingspan was slightly less during flight at 0 and 2 m s^-1(93±3 mm and 95±5 mm, respectively) compared with flight at velocities of 4-12 m s^-1, where mid-downstroke spans were between 100±3 mm and 102±4 mm. In general, span ratio for the wingtips and the wrists decreased as flight velocity increased. A minor deviation from this overall pattern was an observed increase in average span ratio of the wrist from 98±2 to 99±1% between 4 and 6 m s^-1. Among flight velocities, span ratio for the wrists was greater (98±1%) than for the wingtips (93±3%).

All of the measured global-coordinate wing kinematics varied significantly with flight velocity (Table 3;Figs 7, 8, 9). Dorsal and lateral projections of the paths of the wingtips and wrists revealed that overlap in wrist and tip excursion occurred between subsequent downstrokes at 0 and 2 m s^-1, and overlap in the paths of the wingtips was also apparent at 4 m s^-1. At each flight velocity, downstroke covered a larger area than upstroke. However, the difference between the means was less than 1 s.d. during hovering and was accounted for solely by slight flexion of the wings during upstroke, with span ratio of 98±4%(Fig. 6B). Downstroke plane area increased uniformly with velocity from 3059±411 mm² at 0 m s^-1 to 15296±1370 mm² at 12 m s^-1. Upstroke-plane area was 2830±380 mm² during hovering,approximately 4700 mm² during flight at 2-4 m s^-1, and increased with increasing velocity to reach a maximum of 10011±278 mm² during flight at 12 m s^-1.

Along with changes in stroke plane area(Fig. 7 and Fig. 8A), the angle of the aerodynamic stroke plane changed with velocity(Fig. 7 and Fig. 8B). A slight increase inΨ _d occurred as velocity increased, with a mimimum of 14±4° during flight at 2 m s^-1 increasing to 23°during flight from 8 to 12 m s^-1. In dramatic contrast,Ψ _u decreased with each incremental increase in velocity, from 162±4° during hovering to 35±3° during flight at 12 m s^-1.

Angle of attack (α) varied among velocities and within wingbeats(Table 3; Fig. 8C and Fig. 9). Mid-downstroke αwas greatest during hovering (23±5°), decreased with increasing velocity, and varied from 11-13° during faster flight (6-12 m s^-1). During mid-upstroke of flight at 0 and 2 m s^-1,α was negative, indicating an inverted airfoil (i.e. with the curved,anatomical underside of the wing facing in the same direction as presumed low pressure above the wing), whereas α was positive at mid-upstroke during flight from 4 to 12 m s^-1. Negative values for mid-upstroke αproduced an overall mean among flight velocities (9±22°) that was atypical of any given velocity. Ignoring which surface of the wing was uppermost, |α| was more useful as an average descriptor of the relative direction of incurrent air; mid-upstroke |α|averaged 22±4°. Average α at mid-downstroke 14±1°.

Within wingbeats, α varied more during slow flight (0 and 2 m s^-1) compared with flight at 4-12 m s^-1(Fig. 9). For example, αranged from -32±14° to 56±7° during flight at 0 m s^-1, and from 11±4° to 33±11° during flight at 12 m s^-1. During hovering, α varied between downstroke and upstroke approximately in a symmetrical manner about 0°. At the middle of each half stroke, |α| was 22-23°. Negative α was exhibited from slightly before the end of downstroke (44% of wingbeat cycle)to slightly before the end of upstroke (93% of cycle). The interval of negative α was more contracted in duration at 2 m s^-1 and occurred during the middle of upstroke, between times at 62% and 87% of the full cycle. Peak positive α was exhibited during early downstroke of flight at 0 m s^-1, 9% into the wingbeat cycle. In contrast, for all other flight velocities, peak positive α was exhibited at 51-58% of the wingbeat cycle, concurrent with, or slightly after, the transition between downstroke and upstroke.

Our experiments revealed that hummingbirds change a variety of bird-centered kinematic parameters among flight velocities and within wingbeats (Table 2, Figs 2, 3, 4, 5, 6), which causes changes in global-coordinate variables that are likely relevant to their aerodynamic performance (Table 3; Figs 7, 8, 9). Thus, our experiments provide new details about wing kinematics that should help improve understanding of hummingbird flight.

Three of the patterns that we observed were consistent with the kinematics previously reported for ruby-throated hummingbirds(Greenewalt, 1960a; Greenewalt, 1960b): fdid not vary significantly among flight velocities(Table 2, Fig. 5A), while β andγ _h exhibited significant variation. The lack of variation in f among flight speeds is also consistent with hummingbird behavior during hovering. When adjusting to varying power demands of load lifting or hovering in hypodense air, hummingbirds modulate ϕ to a much greater degree than f (Chai and Dudley,1995; Chai et al.,1997; Chai and Millard,1997; Altshuler and Dudley,2002; Altshuler and Dudley,2003).

A mechanical oscillator hypothesis has been proposed for modeling hummingbird wing motion (Greenewalt,1960a; Greenewalt,1960b). This describes the wing as a damped, driven oscillator,and one prediction from the model is that fl^5/4is constant. Among flight velocities studied here, this product did not vary significantly using f and mid-downstroke l(F=1.779, P=0.1461, d.f. 4, 6), or using f and mid-upstroke l (F=0.888, P=0.5188, d.f. 4, 6). Thus, our results provide support for the hypothesis, and demonstrate how hummingbirds modulate the orientation and trajectory of their mechanical oscillator to accomplish a change in velocity. Additional tests of the assumptions of the hypothesis are necessary, however, including the contribution of profile drag to damping and whether muscle force is proportional to strain (Greenewalt,1960b).

Insight into muscle force production is also necessary to evaluate whether the U-shaped curve exhibited for angular velocity of the wing(Fig. 5C) is representative of the shape of the mechanical power curve for hummingbird flight. If muscle force is proportional to strain, as assumed by the mechanical oscillator hypothesis (Greenwalt, 1960b), and strain in the primary flight muscles is proportional to ϕ (Fig. 5A), our kinematic data indicate that the mechanical-power curve for hummingbird flight has a pronounced U-shape, similar to curves exhibited by other bird species (Tobalske et al.,2003a).

The general pattern of bird-centered wingtip paths in lateral projection(Fig. 2B) was similar to that reported for ruby-throated hummingbirds(Greenewalt, 1960a), but our data furnish new insight into lateral-projections of wrist motion as well as the path followed by the tips and wrists in dorsal projection(Fig. 2A). Changing velocity of flight in the wind tunnel in 2 m s^-1 increments resulted in gradual but consistent shifts in wingtip and wrist paths. The figure-8 path of the tips that was apparent in lateral view at 0 and 2 m s^-1, and, to a lesser extent, during flight at 4 m s^-1, is similar to that exhibited during slow flight by other bird species that possess pointed wings or wings of relatively high aspect ratio (AR=5.5-7), including the rock dove(Columba livia, hereafter `pigeon') and cockatiel(Brown, 1953; Brown, 1963; Scholey, 1983; Tobalske and Dial, 1996; Tobalske, 2000; Tobalske et al., 2003a). Hummingbirds also share with pigeons characteristics of wing motion apparent in faster flight, specifically with the tips and wrists following an elliptical trajectory in lateral view and a figure-8 trajectory in dorsal view (Tobalske and Dial, 1996)(Fig. 2B).

The wing paths that hummingbirds share with other, distantly related species highlights that the evolution and functional significance of the wingtip-reversal upstroke used by other species merits additional study. Flow visualization shows that the upstroke in hovering hummingbirds, featuring negative angle of attack and chord angle (α and α_c;Figs 3, 4, 5), provides 25% of the weight support required from a full wingbeat cycle(Warrick et al., 2005). In comparison, some studies suggest that the tip-reversal upstroke of other bird species is an aerodynamically inactive recovery stroke(Spedding et al., 1984; Hedrick et al., 2004), while others indicate that the proximal wing may continue to support body weight while the distal wing provides thrust(Corning and Biewener, 1998; Usherwood et al., 2003; Usherwood et al., 2005). An extended, tip-reversed wing may also facilitate maneuvering at low speeds(Warrick and Dial, 1998). At present, the relationship between tip reversal and the phylogeny of birds is unknown, but it is possible that a tip-reversal upstroke may represent a precursor style of wing movement that an ancestral form exhibited early during the evolution of hovering within the Trochilidae.

Our measurements of α_c indicated continuous variation within (Fig. 4) and among(Table 2, Fig. 3B) flight velocities. Contrary to an expectation from earlier reports(Stolpe and Zimmer, 1939; Greenewalt, 1960a; Greenewalt, 1960b) that the two halves of the hovering wingbeat would be symmetrical, mirror images of one another, |α| during upstroke and downstroke exhibited greater disparity at slow velocity (0-4 m s^-1) than during faster flight (6-12 m s^-1). The only comparable values ofα _c are available for zebra finch at mid-downstroke(Tobalske et al., 1999), and the coefficient of variance (CV=s.d./mean) for α_c at mid-downstroke is greater in hummingbirds (29%) than in zebra finch (17%).

Zebra finch use an extremely different flight style compared with rufous hummingbirds (Tobalske et al.,1999; Tobalske et al.,2005a), and further comparison illustrates that the relative magnitude of variation for a given bird-centered kinematic variable differs between these two species depending upon the variable under consideration. Zebra finch have low-aspect ratio, rounded wings (AR=4.5), regularly flex their wings to adduct them almost against their body on upstroke (average span ratio=19%) and regularly use intermittent bounds (40-45% of total time during fast flight). Compared with zebra finch(Tobalske et al., 1999),hummingbirds exhibited greater variation in the percentage of their wingbeat spent in downstroke (CV=8% vs 3%), γ_h (CV=48%vs 24%) and γ_b (CV=12% vs 4%). In contrast,variation is greater in zebra finch for f (CV=5% vs 2%),ϕ (CV=15% vs 8%), β (CV=45% vs 12%), and wingspan at mid-upstroke (CV=13% vs 2%). Angular velocity of the wing during downstroke exhibits a CV of 11% in both species.

A comprehensive comparative analysis for all of these kinematic parameters awaits further study; however, one pattern that readily emerges from the data available for these and other bird species is that hummingbirds exhibit the highest span ratio, and the lowest variation in span ratio, of any species studied to date (Fig. 10). Their wingtip span ratio averaged 93% among velocities, and span ratios in other species vary from 17% to 80%(Tobalske and Dial, 1996; Park et al., 2001; Tobalske et al., 2003b; Rosén et al., 2004). Although this supports the view that hummingbirds have a kinematically `rigid'wing, broadly consistent with the mechanical oscillator hypothesis(Greenewalt, 1960a; Greenewalt, 1960b), a span ratio of less than 100% is evidence that they flex their wings on upstroke(Fig. 2A, Fig. 6 and Fig. 7).

Given that average wingtip span ratio in rufous hummingbirds was less than wrist span ratio (Table 2; Fig. 6), it appears that the majority of wing flexion during upstroke in hummingbirds occurs at the wrist,rather than at the shoulder or elbow. Further, the decrease in tip span ratio with flight velocity was, in roughly equal measure, a result of increased wrist flexion during upstroke, and increased extension during downstroke. The increase in downstroke span with increasing forward flight velocity may have allowed for the greater translational velocity required for a useful α,as the vector of incident air becomes increasingly dominated by forward flight velocity (V). The change in span ratio may also provide new insight into the functional morphology of the muscular and skeletal elements of the hummingbird wing, which feature a proportionally short humerus and long distal(hand wing) bones (Stolpe and Zimmer,1939; Greenewalt,1960a). Hummingbirds also have exaggerated curvature of their radius and ulna that is hypothesized to accommodate proportionally large muscles to control extension, flexion and long-axis rotation of the hand wing(Dial, 1992b).

Species that are closely related to each other, hypothesized to be within the same avian order (Sibley and Ahlquist,1990), appear to group similarly according to their characteristic span ratios (Fig. 10). Fully testing phylogenetic influences(Felsenstein, 1985; Garland et al., 1992) on this pattern relative to wing design is beyond the scope of our present analysis. Regardless of aspect ratio or wing shape, there is a general trend to decrease span ratio with increasing velocity among the rufous hummingbirds studied here, as well as in previously studied Columbiformes (pigeons and ringed turtle doves Streptopelia risoria), Psittaciiformes (budgerigars Melopsittacus undulatus and cockatiels) and two species of Passeriformes [barn swallows Hirundo rustica and black-billed magpies Pica hudsonica (Tobalske and Dial, 1996; Park et al.,2001; Tobalske et al.,2003b)]. This trend may be explained by the need to limit negative thrust during a lifting upstroke at fast velocities(Rayner, 1986; Rayner, 1988; Tobalske, 2000; Hedrick et al., 2002). Two passerine species, the zebra finch and thrush nightingale (Luscinia luscinia), exhibit an increase in span ratio with increasing velocity(Tobalske et al., 1999; Rosén et al., 2004). At least for the zebra finch, we predict the increase in span ratio is due to an effort on the part of the bird to enhance weight support by moving toward a lifting upstroke because the bird can potentially use intermittent bounds to offset an increase in average thrust requirements as velocity increases(Tobalske et al., 1999).

Other than a report of `undulating' during flight(Murphy, 1913), there have been no published accounts of hummingbirds using intermittent flight. Given the widespread use of this flight style in other species of small birds(Rayner, 1985, Tobalske, 2001), it is therefore noteworthy that two of the five birds in our study regularly used intermittent glides and bounds when flying at velocities between 8 and 12 m s^-1 (94±9% of their total number of flights at these velocities).

Continuous variation in angle of attack (α) within wingbeats(Fig. 9) indicated that bound circulation on the wing is likely to vary throughout the wingbeat cycle of hummingbirds, and this variation should be evident in the wake as cross-stream starting and stopping vortices that parallel the long axis of the wings(Rayner, 1986; Rayner, 1988; Tobalske, 2000). We interpret that this will give rise to some form of `ladder' appearance in the wake similar to the structure described for the wake of the thrush nightingale during fast forward flight (Spedding et al., 2003). Our ongoing research into the wake of flying hummingbirds using digital particle image velocimetry (DPIV) appears to support this expectation (Tobalske et al.,2005b; Warrick et al.,2005).

Relative to incurrent air, the wing functioned as an inverted airfoil during the upstroke of flight at 0 and 2 m s^-1. The opposite sign,nearly symmetrical pattern of α between the two halves of the wingbeat cycle during hovering would appear to support a hypothesis that the two halves of the wingbeat are functionally identical(Stolpe and Zimmer, 1939; Greenewalt, 1960a; Weis-Fogh, 1972; Wells, 1993), but a variety of small differences in kinematics help to explain why the hovering upstroke produces less lift than the downstroke(Warrick et al., 2005). These include slightly greater values of |α| during early downstroke versus upstroke (Fig. 9), a different trajectory of the wingtips and wrists apparent in lateral projection (Fig. 2A and Fig. 7), and marginally greater angular velocity of the wing during downstroke(Fig. 5C).

Because α and α_c were measured at the middle of the wing as a line between the wrist and the tip of the first secondary, we are unable to address any potential effects of long-axis twisting of the wings(Bilo, 1971; Bilo, 1972; Willmott and Ellington,1997a). Particularly during upstroke, the hummingbird wing twists so that the proximal and distal sections of the wing exhibit different angles relative to the body and, potentially, the incurrent air(Warrick et al., 2005). Data from other species (Bilo, 1971; Bilo, 1972; Hedrick et al., 2002) suggest that twist about the long-axis may be present during the entire wingbeat. For example, changes in α differ in magnitude and relative timing between the proximal and distal wing of the cockatiel(Hedrick et al., 2002).

New research on the patterns of wing motion in birds would, therefore,benefit from adopting a more complete protocol for measuringα _c and α along the entire wing. The strip method recommended for use in studies of insect flight may be appropriate(Willmott and Ellington,1997a). A challenge for applying this method to birds is that cross-sectional profile and camber vary greatly along the avian wing, but laser scanning shows promise for creating useful 3D models(Liu et al., 2004). Profile,camber, α_c and α may vary throughout the wingbeat of birds (Bilo, 1971; Bilo, 1972) (Figs 4, 5 and 9). This variation is probably due aeroelasticity (Bisplinghoff et al.,1955), neuromuscular control(Dial, 1992a) and interactions between the two (Natarajan,2002). Consequently, an effort to measure additional details of wing kinematics will remain pivotal to advancing understanding of bird flight.

With similar consequences for the predicted likelihood of unsteady flow patterns, k in hummingbirds varied from a maximum of 0.66±0.08 during flight at 2 m s^-1 to 0.11 during flight at 12 m s^-1 (k is undefined for hovering, Eqn 5). Spedding(Spedding, 1993) suggests that unsteady effects are likely to be significant to the flow about the wings when k>1 and that they may be ignored when k<0.1. Hummingbirds did not achieve k<0.1 even at their fastest sustained velocity. Thrush nightingales also do not achieve this value, with a minimum k of 0.2 during flight at 10 m s^-1(Rosén et al., 2004). Given the complexity of the avian wingbeat, k is only a rough predictor for the likelihood of unsteady flow(Rosén et al., 2004). Nonetheless, our results reveal that the potential contribution of unsteady flow should be taken into consideration in future experiments on flight aerodynamics in hummingbirds.

List of symbols

 
• α

  angle of attack of wing

 
• αc

  chord angle relative to mid-frontal plane of body

 
• β

  body angle relative to horizontal

 
• γb

  anatomical stroke-plane angle relative to mid-frontal plane of body

 
• γh

  tracking stroke-plane angle relative to horizontal

 
• ρ

  air density

 
• ϕ

  wingbeat amplitude

 
• Ψd

  global downstroke-plane angle relative to horizontal

 
• Ψu

  global upstroke-plane angle relative to horizontal

 
• AR

  aspect ratio

 
• b

  wing span

 
• c

  wing chord

 
• CV

  coefficient of variation

 
• f

  wingbeat frequency

 
• g

  gravitational acceleration

 
• k

  reduced frequency

 
• l

  wing length

 
• M_b

  body mass

 
• J

  advance ratio

 
• S_d

  disc area

 
• T

  duration of wingbeat

 
• V

  free-stream velocity

 
• V_i

  induced velocity

We thank Jim Usherwood and an anonymous referee for their helpful comments on a previous draft of the manuscript. Supported by Murdock Grants 99153 and 2001208 to B.W.T. and NSF Grants IBN-0327380 and IOB-0615648 to B.W.T. and D.R.W.