Morphology and motion: hindlimb proportions and swing phase kinematics in terrestrially locomoting charadriiform birds

Since the early 1900s, functional anatomists have proposed that limb morphology reflects the locomotor capacity of tetrapods. Taxa with more lightweight and distally light legs should possess lower locomotor costs and may attain higher speeds (Lull, 1904; Gregory, 1912; Howell, 1944; Smith and Savage, 1956; Gray, 1968; Gambaryan, 1974; Hildebrand, 1984, 1985, 1988; Hildebrand and Hurley, 1985; Steudel, 1990; Wickler et al., 2004; Raichlen, 2005, 2006; Browning et al., 2007). In contrast, taxa with more massive and distally heavy legs should incur higher locomotor costs or might be restricted to slower speeds of terrestrial locomotion (e.g. Kilbourne and Hoffman, 2013). Limbs with greater and more distal concentrations of mass should have higher moments of inertia (MOI) – an increased resistance to angular acceleration or, more simply, to swinging the limb back and forth (Steudel, 1990; Wickler et al., 2004; Kilbourne and Hoffman, 2013, 2015). This constitutes a link between morphology and locomotor capacity. Thus, limb MOI reflects morphological characteristics (overall limb mass and its distribution) that should directly influence the energetic costs of the limb's swing phase during locomotion.

How exactly differences in limb morphology relate to limb kinematics – particularly swing phase kinematics – has been little explored. Dissimilar proportions of segment lengths curtail the range of kinematics that species can employ during terrestrial locomotion, resulting in dissimilar excursion angles of individual segments as a result of geometric constraints (Pike and Alexander, 2002; Gatesy and Pollard, 2011). Given the same angular excursion in the proximal joint, a longer segment will displace the distal end of the segment further. Differences in the proportions of segment masses and the overall mass distribution of the limb may affect the motions of the limb during swing phase (Hildebrand and Hurley, 1985; Wickler et al., 2004; Nelson and Roberts, 2008). Distal limb segments with a high mass result in a longer radius of gyration and, consequently, a greater limb MOI (e.g. Kilbourne, 2014). However, reducing the radius of gyration via kinematic adjustments (e.g. more pronounced flexion) during swing phase is a means to reduce limb MOI.

Birds are one of the largest tetrapod lineages and represent an impressive range of hindlimb functions and specializations, making them an ideal group in which to study how limb morphology influences locomotor capability, particularly in terms of kinematics (Abourachid and Höfling, 2012; Stoessel et al., 2013). Within Aves, the Charadriiformes – or the shorebird clade – comprise one of the major avian clades (Hackett et al., 2008; Jetz et al., 2012), encompassing a diversity of hindlimb proportions (Barbosa and Moreno, 1999). The combination of close phylogenetic affinity and high limb morphological diversity makes Charadriiformes a group well suited for studying how length and mass proportions are reflected in hindlimb swing phase kinematics as predicted by the above considerations of limb MOI.

In the present study, we tested for differences in hindlimb swing phase kinematics among three species of charadriiform birds differing in their hindlimb morphology: northern lapwings [Vanellus vanellus (Linnaeus 1758)], Eurasian oystercatchers (Haematopus ostralegus Linnaeus 1758) and pied avocets (Recurvirostra avosetta Linnaeus 1758) (Fig. 1). Segmental mass and length data were available for one specimen per species: lapwings have relatively short limbs (152.6 mm anatomical hindlimb length) with a low mass (hindlimb mass 10.75 g), whereas oystercatchers have only moderately longer limbs (172.8 mm), but with a much higher mass (31.38 g). In contrast, avocets possess relatively lightweight limbs (16.94 g) that are, however, elongate and slender (225.7 mm). Moreover, avocets have a distal concentration of limb mass (Kilbourne, 2013; Fig. 1).

By studying these three closely related species of differing limb segment proportions and limb mass distribution, we tested whether differences in limb morphology are associated with differences in hindlimb kinematics reflecting adjustments to reduce MOI. Additionally, we derived the hindlimb's MOI of representative swing phases using in vivo kinematic data and compared them with the hindlimb MOI of the previously published ‘neutral pose’ (this pose is determined by manually extending the limbs to their maximum length and then releasing the limb to passively flex of its own accord; data from Kilbourne, 2013).

Our sample allowed the comparison and discussion of two specific test cases: (1) similar limb segment proportions, but different neutral pose MOI; and (2) similar neutral pose MOI, but different limb segment proportions. In the first case, lapwings and oystercatchers have similar limb segment proportions, but a stark difference in neutral pose MOI as a result of the overall higher limb mass in the oystercatcher. For similar limb flexion, animals with overall more massive but otherwise similar limbs (e.g. in terms of limb segment proportions) should exhibit greater swing durations, as increasing mass itself also increases limb MOI (Kilbourne, 2013; Kilbourne and Hoffman, 2013). In the second case, oystercatchers and avocets display similar neutral pose MOI because of the distal concentration of limb mass in the avocet; however, these species possess highly different limb segment proportions. Species with distally heavy limbs are expected to display relatively longer swing phase durations (Nelson and Roberts, 2008). Flexing their limbs is a means for birds with distally heavy limbs, such as the avocets of our sample, to effectively achieve a reduction of the limb's MOI.

Four individuals each of avocet, oystercatcher and lapwing were obtained from a local breeder approved by Friedrich-Schiller-Universität Jena (Table 1). All animals were legally bred and ringed in 2010 and 2011. When not performing in experiments, the animals were kept in the animal house of the Institute for Systematic Zoology and Evolutionary Biology with access to food and water ad libitum. During experiments, the birds were given ample opportunity for rest and had free access to water when not on the treadmill. Animal husbandry and all experimentation with the birds followed animal welfare regulations of the state of Thuringia, Germany, and were approved by the responsible authorities (registry number: 02-47/10).

Cineradiographic videos were recorded for each bird locomoting on a treadmill using a fully digital Neurostar fluoroscope (Siemens AG, Erlangen, Germany). To prevent the birds from flying away from the treadmill, a Plexiglas enclosure was placed atop the treadmill. At the rear end of the treadmill, before the belt passes under the carriage, a softly padded surface was placed so as to avoid injury to the birds.

Videos were recorded at 1000 Hz for each bird in lateral view. The digital image for each frame possessed a resolution of 1536×1024 pixels, and the X-ray source for the fluoroscope operated at 40 kV and 95 mA. Cineradiographic recordings suffer from typical fluoroscope distortion (e.g. Brainerd et al., 2010). Therefore, prior to digitization of skeletal landmarks, cineradiographic videos were undistorted using a MATLAB (The MathWorks Inc., Natick, MA, USA) routine (www.xromm.org), one of the freely available tools of the X-ray reconstruction of moving morphology (XROMM) technique developed at Brown University, RI, USA (for details, see Brainerd et al., 2010). We selected for analysis only trials in which a bird's torso was in full view in the front of the image intensifier.

From each video, the following skeletal landmarks were digitized: the cranial extreme of the pelvis and the proximal and distal ends of the femora, tibiotarsi and tarsometatarsi (Fig. 2). The distal extreme of digit III was also digitized; however, this landmark often went in and out of the camera image during the stride cycle, preventing the inclusion of this landmark during data analysis. From the skeletal landmarks, the angles of the hip, knee and intertarsal (‘ankle’) joints were calculated throughout the entire swing phase. For each joint, minimum and maximum joint angles and ranges of motion (RoM) were determined for each stride's swing phase using custom-written code in R version 2.15.1 (R Development Core Team, 2012). For the thigh, shank and pes, we also calculated the maximum excursion of these segments during swing phase by obtaining the angle the segment swings through about its proximal pivot. With this, we calculated the angular velocity of the thigh, shank and pes by means of the first derivative of joint angles using the R package pspline (Ramsey, 2015). Stride frequency, duty factor and stance and swing durations were also determined.

For each of the three species, maximum and minimum joint angles and RoM, stride frequency, duty factor and stance and swing durations were plotted against both absolute and dimensionless speed. The relationship between each parameter and speed was then fitted with linear and power functions. These two functions were chosen given their prevalence in describing relationships between biological traits, and for their ease of facilitating comparisons across species. To determine the function that best fitted our data, we calculated corrected Akaike information criterion (AICc) scores (Akaike, 1973, 1974; Hurvich and Tsai, 1989) using the R package qpcR (Speiss, 2014). When a given trait was best fitted by the same curve type (e.g. a power function) across more than one species, 95% confidence intervals for regression parameters (e.g. intercept, slope, etc.) were compared to determine whether those species differed in their relationships between the above locomotor parameters and speed.

To further determine whether the three species differed in their kinematics, we compared the maximum and minimum joint angles and RoM within specific speed ranges to investigate how limb flexion varies between the differing species. Minimum joint angles represent maximum flexion, whereas maximum joint angles represent maximum extension. To parse out any effect of speed, we only examined joint angles over ranges of absolute (0–1.70 m s⁻¹) or dimensionless speeds (0–1.32) shared by all species. We then divided these common speed ranges into three categories of low, medium and high speed. For absolute speed, low, medium and high categories, respectively, ranged from 0 to 0.56 m s⁻¹, 0.57 to 1.13 m s⁻¹ and 1.14 to 1.70 m s⁻¹. For dimensionless speed, low, medium and high categories, respectively, ranged from 0 to 0.44, 0.45 to 0.88 and 0.89 to 1.32. Using a Kruskal–Wallis test, we tested for significant differences in joint maximum and minimum angles and RoM between species for each speed interval. For the Kruskal–Wallis test, the independent factor corresponded to the differing species. Our level of significance was P<0.05; however, after applying a Bonferroni correction to account for our comparison of three species, P_significant<0.0167. Post hoc tests were used to compare individual species. Kruskal–Wallis tests and their associated post hoc tests were performed in R, using the packages pgirmess (Giraudoux, 2015) and coin (Hothorn et al., 2006, 2008).

It should be noted that we used both absolute and dimensionless speed for our analysis. Dimensionless speed, which is the square root of the Froude number, standardizes absolute speed (u) to hip height (h) and the gravitational constant (g=9.81 m s⁻²) using the equation: û=u[gh]^−0.5. Standardizing to hip height minimizes differences in hip height as a confounding factor in the different ranges of locomotor speed exhibited by locomoting tetrapods (Gatesy and Biewener, 1991). Also, as the Froude number represents the ratio of inertial to gravitational forces (Raichlen et al., 2013), dimensionless speed may also be an indicator of this ratio of forces. However, an implicit assumption in using dimensionless speed is that the effect of size is ostensibly consistent across all morphological and locomotor traits – the ‘larger’ species will have a greater body mass and hip height and use a higher range of speeds (for an example, see the species compared in Gatesy and Biewener, 1991). Moreover, this assumption is explicit with regards to Froude number as proposed by Alexander and Jayes (1983), though the authors argue for the metric's utility even if this assumption is not strictly followed. In light of this, we also analyzed our data using absolute speed. Absolute speed arguably is the more important metric of speed with regards to how animals function in their environment, especially as it relates to prey capture and predator avoidance or the movement of herds composed of different ontogenetic stages of one species and/or species of differing sizes. While the focus of our study is not ecology, including absolute speed as a variable here may allow future studies to address how morphology and locomotor performance relate to animals functioning in their natural environment and allow for inferences as to why species of differing morphologies and locomotor abilities may or may not co-exist in particular habitats.

One individual of each species was killed and dissected (Table 1). Hindlimb elements were detached at the joints and frozen. We then weighed each element and used a pendulum method to determine each element's CoM position (Table 2; Nyakatura et al., 2012). We also measured segment length using calipers. We used these data to model each hindlimb segment as a three-dimensional geometric solid using the CAD-software Inventor^®. The femur and tibiotarsus were modeled as truncated elliptic cones, while the tarsometatarsus and digits were modeled as cylinders. From these solids we estimated the MOI of each segment (J_i) about its CoM (Table 2). The moment of inertia of each segment with respect to the hip was then computed frame by frame based on the parallel-axis formula: ⁠, where m_i is the mass of the segment and s_i the distance between each segment's CoM and the hip. Finally, the sum of these segmental MOI yields the MOI of the limb with respect to the hip.

Raw kinematic data for avocets, oystercatchers and lapwings are available in spreadsheet format in Tables S1, S2 and S3, respectively. To determine significant differences among species in how a given temporal parameter varies with speed, we used 95% confidence intervals to compare parameters of fit curves (e.g. the exponent b of a power function). For Kruskal–Wallis tests comparing among-species differences in joint kinematics and segment excursions and angular velocity, our level of significance was P<0.05; however, after applying a Bonferroni correction to account for our comparison of three species, our level of significance became P_significant<0.0167.

All three species largely overlap in plots of temporal kinematic traits versus speed (Fig. 3; Fig. S1), with the variation of stride frequency, duty factor, swing duration and stance duration with increasing speed being best described by a power function (Table S4). While analyses involving absolute and dimensionless speed show differences in the values of the parameters (Table 3), the overall results do not differ between the uses of these two measures of speed. It is worth noting that the difference in AICc scores is seemingly trivial between linear and power fits for some temporal parameters (e.g. swing duration in avocets; Table S1). Significant differences among all three charadriiform species exist with regard to how stride frequency varies with speed (Table 3). Lapwings tend to utilize higher stride frequencies at lower speeds than oystercatchers, though the two species converge upon similar stride frequencies with increasing speed. Compared with lapwings and oystercatchers, avocets shallowly increase stride frequency with increasing speed.

Duty factor decreases with increasing speed in a largely parallel manner among all three species (Fig. 3B; Fig. S1B), though significant differences exist among the species (Table 3). However, note that lapwings tend to use overall lower duty factors than the remaining two species (Fig. 3B; Fig. S1B).

In spite of being largely invariant with speed in all three species (Fig. 3C; Fig. S1C), swing duration minimally (but nonetheless significantly) varies with speed in lapwings and oystercatchers (Table 3), while it does not in avocets. At higher speeds, avocets tend to employ slightly longer swing durations than the other two charadriiforms, owing to their higher constant a (Table 3).

Avocets tend to use significantly higher stance durations than the remaining two charadriiforms (Fig. 3D; Fig. S1D; Table 3). Lapwings and avocets do not significantly differ in the exponents of their power functions (parameter b; Table 3); however, these two taxa do differ in their constant (parameter a). These results indicate that although lapwings and avocets vary their stance duration with speed in indistinguishable ways, the absolute values of their stance durations differ. Oystercatchers differ from these two species in both coefficient a and exponent b.

The overall kinematic pattern during swing phase is very similar in the three analyzed species (Fig. S2). Across all absolute speed categories, oystercatchers and avocets flex their hips more than do lapwings, whereas lapwings extend their hips more than the other species (Fig. 4A,B). Post hoc tests indicate that maximal hip flexion significantly differs among all three species (P<0.0167). Likewise, maximal hip extension also significantly differs among all three species (P<0.0167) across the three speed categories. Hip RoM significantly differs among the species (P<0.0001; Fig. 4C), though the difference is not significant between avocets and oystercatchers at low speeds (P>0.0167). These results do not change when using dimensionless speed to define our three speed categories (Fig. S3).

Across all absolute speed categories, maximum knee flexion significantly differs amongst all three charadriiforms (P<0.0167). Post hoc tests reveal that lapwings flex their knees less than the other two charadriiforms (P<0.0167; Fig. 4D), whereas the difference between avocets and oystercatchers is not significant at low and medium speeds (P>0.0167). Differences in knee maximal extension are significant (P<0.0167; Fig. 4E). Avocets have a lower maximal knee extension compared with the remaining two species across all speed categories (P<0.0167). In contrast, lapwings and oystercatchers do not differ in maximal knee extension within any speed category (P>0.0167). Knee RoM differs among the species for all speed categories (P<0.0167), though post hoc tests reveal that only oystercatchers and lapwings consistently differ across all speed categories (P<0.0167), with lapwings having a smaller knee RoM (Fig. 4F). When using speed categories defined by dimensionless speed, the only change in the results is that knee RoM does not differ between lapwings and avocets for low and medium speed categories (Fig. S3; P>0.0167).

Across all absolute speeds, maximal intertarsal joint flexion significantly differs among the three species (P<0.0167); however, the only consistent difference across speed categories is significantly greater flexion in oystercatchers compared with the other two species (P<0.0167; Fig. 4G). Across all speeds, maximal intertarsal joint extension significantly differs between lapwings and the other two charadriiforms (P<0.0167; Fig. 4H), while avocets and oystercatchers do not differ in maximal intertarsal joint extension for any speed category (P>0.0167). Intertarsal joint RoM significantly differs between oystercatchers and the other two charadriiforms across all speed ranges (P<0.0167; Fig. 4I), with the exception that oystercatchers and lapwing do not significantly differ within the high speed range (P>0.0167). Differences in intertarsal joint RoM do not differ between avocets and lapwings for any speed range (P>0.0167). At low and medium dimensionless speeds, maximum flexion significantly differs among all three species, though the differences between avocets and lapwings are not significant at high dimensionless speed (Fig. S3; P>0.0167). Across all dimensionless speed categories, maximum intertarsal extension is significantly greater in lapwings than in avocets and oystercatchers. Differences in RoM among species when using dimensionless speed mirror those found when using absolute speed.

At low absolute speeds, thigh maximum angular velocity does not significantly differ among the three species (P>0.0167; Fig. 5A). Significant differences do occur at both medium and high absolute speeds (P<0.0167). This is solely due to avocets having a significantly lower velocity than the other two species (P<0.0167), as the difference between lapwings and oystercatchers is not significant at medium and high speeds (P>0.0167). Across all dimensionless speed categories, significant differences are present among species, with oystercatchers having higher thigh angular velocities at low speeds than the remaining two species (P<0.0167; Fig. S4). However, as when using absolute speed, avocets exhibit significantly lower thigh angular velocities than oystercatchers and lapwings (P<0.0167).

Thigh angular excursion tends to increase more gradually with speed in avocets than in lapwings or oystercatchers (Fig. 5B). Thigh angular excursion does not significantly differ among the three species at low absolute speeds (P>0.0167; Fig. 5B), although this trait significantly differs among all species at medium and high speeds (P<0.0167). Across all dimensionless speed categories, significant differences exist among species (Fig. S4; P<0.0167), with avocets having significantly lower excursion angles than lapwings (all speeds) and oystercatchers (only medium and high speeds). At high speeds, lapwings and oystercatchers also significantly differ in thigh angular excursion (P<0.0167).

Oystercatchers tend to employ greater shank angular velocities than either avocets or lapwings (Fig. 5C), and shank angular velocity differs among the three charadriiforms across all three absolute speed categories (P<0.0167). Post hoc tests indeed reveal that oystercatchers have greater shank angular velocities than the remaining two species across all speed categories (P<0.0167). At low and medium speeds, angular velocity does not significantly differ between lapwings and avocets (P>0.0167). Across all dimensionless speed categories, oystercatchers have a significantly higher shank angular velocity than the remaining two species (Fig. S4; P<0.0167). However, significant differences among all three species were only found at low dimensionless speeds (P<0.0167).

Shank angular excursion significantly differs among individual species, with the exception of avocets and lapwings at high speeds (P<0.0167; Fig. 5D; Fig. S4D), regardless of whether absolute or dimensionless speed is used to define speed categories.

Pes angular velocity differs among the three species for all absolute speed categories (P<0.0167; Fig. 5E). However, at low speeds, the difference between avocets and lapwings is not significant (P>0.0167); within the remaining speed categories, the differences among all three species are significant (P<0.0167). Notably, oystercatchers have a higher pes angular velocity across the entire analyzed speed range. Across all dimensionless speeds, oystercatchers have significantly higher pes angular velocity than avocets and lapwings (Fig. S4E; P<0.0167); however, the difference between avocets and lapwings is only significant at high speeds (P<0.0167).

Pes angular excursion significantly differs among the three species for all absolute speed categories (P<0.0167; Fig. 5F). However, the only consistent difference across speed categories is that the pes excursion of oystercatchers is higher than that of avocets. We found differences in pes angular excursion only at low and high dimensionless speeds. At low dimensionless speeds, oystercatchers have a significantly higher pes angular excursion than avocets and lapwings (P<0.0167; Fig. S4F), whereas at high dimensionless speeds, avocets have a significantly lower pes angular excursion than lapwings and oystercatchers (P<0.0167).

Among the three species, lapwings and oystercatchers tend to use higher stride frequencies than avocets, particularly at moderate and high speeds (Fig. 3). An important factor influencing stride frequency is limb length (Gatesy and Biewener, 1991). It therefore is not surprising that the species with the overall shortest limbs uses the highest stride frequency, whereas the species with the longest limbs displays the lowest stride frequency. With lapwings and oystercatchers, two species can be compared that have similar limb segment proportions and overall length, but highly different limb MOI (test case 1). In contrast to our expectations, the two species displayed highly similar swing phase durations. The short swing phase displayed by lapwings can probably be explained by the low overall rotational inertia of their hindlimbs. However, the similarly short swing phase durations of oystercatchers are probably due to more massive joint flexors and extensors, which may afford them the ability to rapidly accelerate their limbs (see discussion below). In particular, as the knee is the primary pivot of the hindlimb (Gatesy, 1990, 1999a,b; Gatesy and Biewener, 1991; Carrano, 1998; Reilly, 2000; Stoessel and Fischer, 2012), the greater shank segmental velocities (Fig. 5) of oystercatchers may enable their short swing phase durations (though note that the hip's contribution to hindlimb kinematics increases with increasing speed; Gatesy, 1999a; Nyakatura et al., 2012; Stoessel and Fischer, 2012; Andrada et al., 2013). Moreover, oystercatchers displayed overall larger segment excursions than the lapwings, which resulted in higher segment velocities given the similar swing duration in the two species. Our expectations were met in regard to test case 2 (similar MOI, but different limb segment proportions). Avocets, with their relatively more slender and elongate but distally heavy segments (Fig. 1), use longer swing durations, as they have a high hindlimb rotational inertia (Kilbourne, 2013) and probably a relatively low amount of muscle mass to accelerate and decelerate the hindlimb.

Across absolute and dimensionless speed, the largely invariant swing phase durations of the three charadriiforms agree with previous studies which have found this trait to be nearly constant with increasing speed during terrestrial locomotion of birds (Gatesy, 1999a; Reilly, 2000; Verstappen et al., 2000; Abourachid, 2001; van Coppenolle and Aerts, 2004; Rubenson et al., 2004; White et al., 2008; Nudds et al., 2010; Nyakatura et al., 2012; Stoessel and Fischer, 2012). The cost of swinging the limbs has been directly measured as being 26% of the total metabolic energy expended in running guinea fowl (Marsh et al., 2004); the invariant swing duration with increasing speed suggests that birds are limited in being able to more rapidly swing their limbs and thus rotational inertia of swinging limbs may be an important limiting factor in terrestrial locomotion (see also Rocha-Barbosa et al., 2005; Kilbourne, 2013; Kilbourne and Hoffman, 2013, 2015).

Larger bodied birds tend to increase their velocity through increases in stride length as opposed to increases in stride frequency, whereas the converse is true in small-bodied birds (Gatesy and Biewener, 1991). Muscle fiber area (which serves as a proxy for muscle force) scales isometrically with body mass (Maloiy et al., 1979; Bennett, 1996), resulting in large-bodied birds having muscles that produce less force relative to their body mass than small-bodied birds, as, under isometric scaling, body mass is proportional to length³; in contrast, muscle fiber area is proportional to length² under isometric scaling. As limb MOI is known to scale either with isometry (Maloiy et al., 1979) or positive allometry (Kilbourne, 2013), and limb and segment masses are known to scale with positive allometry (Kilbourne, 2013, 2014), large-bodied birds in general may be severely hampered in swinging their limbs more rapidly, as these traits all scale with exponents much higher than 1.0. As an allometric exponent of 2/3 characterizes how muscle fiber area scales with increasing body size under isometric scaling, the discrepancies between exponent values of 2/3 and those greater 1.0 indicate that limb MOI and segment masses increase disproportionately in comparison to the force of muscles as birds increase in size.

Recent studies have found (Marsh et al., 2004; Doke et al., 2005; Arellano and Kram, 2014) or estimated (Pontzer, 2007; Umberger, 2010) the cost of swinging the limbs to be non-trivial. However, this still leaves the majority of locomotor costs to be due to stance phase. It remains poorly understood whether tetrapods employ mechanisms or actions to lower their limb MOI and consequently reduce the amount of torque required, which would result in lower swing phase costs. Our findings indicate that increased limb flexion is probably a means to lower swing phase locomotor costs in taxa with limbs that have greater overall mass or are distally heavier. Increased flexion of the limb brings the mass of the distal limb segments closer to the limb's pivot, thereby reducing the limb's MOI (as a result of a reduction in the radius of gyration) and thus the torque and, consequently, the metabolic energy required to swing it. More pronounced hindlimb flexion in the avocets and oystercatchers (Fig. 4) results in a larger reduction of rotational inertia in comparison to the more extended limbs of lapwings. Thus, with regard to test case 1, oystercatchers showed more pronounced flexion than lapwings even though limb proportions and relative distribution of mass are similar in the two species. With regard to test case 2, our expectation was met: avocets with relatively heavier distal limbs segments flexed the hip and knee (though not the intertarsal joint) the most during swing. For species with distally heavy limbs, the benefits of flexion should be largest. Pronounced limb flexion is used regardless of whether the greater MOI is due to greater mass (e.g. oystercatchers) or a longer radius of gyration (e.g. avocets) (Fig. 1).

Displaying MOI over the duration of swing phase in representative trials reveals the close link to the swing phase joint kinematics (Fig. 6). Flexion of the limb joints shortens the radius of gyration and thus the MOI. When the limb is extended in preparation of the subsequent stance phase, the limb's MOI is increased almost to or even above the level of the neutral pose MOI (Kilbourne, 2013). This increase of MOI may also help to slow down protraction at the end of swing.

In spite of our findings regarding joint flexion, we found no consistent response among the three charadriiforms regarding limb segment angular velocity. Oystercatchers tend to exert higher angular velocities on their shank and pedal segments than the remaining two species, though these differences are not always significant (Fig. 5; Fig. S4). Thus, it appears that greater limb segment masses do not limit the ability to swing the limb segments at higher speeds. This result contradicts the implications of scaling and suggests that oystercatchers deviate from the expected trend (see discussion above; Kilbourne, 2013). A possible explanation for the high angular velocities observed in this species is that the greater mass of the hindlimb segments in oystercatchers (Fig. 1) is due at least in part to greater muscle mass. This apparently enables them to partly offset detrimental scaling effects. Our finding that oystercatchers tend to have higher segment angular velocities when using dimensionless speed to partition speed categories further underscores this possibility. Given that dimensionless speed is meant to mitigate the influence of size upon locomotor velocity observed for different species, the higher angular velocity of oystercatchers is unlikely to be purely a consequence of differences in body size.

A greater muscle mass distributed across joint flexors and extensors is likely to be linked to greater force production and postural/joint stability and would allow oystercatchers to better accelerate their hindlimb segments and attain greater segmental velocities. Estimates of segment soft tissue mass – which should largely reflect differences in muscle mass – in our three sampled species give credence to the possibility that oystercatchers have a greater hindlimb muscle mass than avocets and lapwings (Table 4). Given that oystercatchers locomote over a similar range of absolute velocities to lapwings in spite of their greater body mass (Figs 1 and 3), it seems likely that they may need greater hindlimb extensor mass to accelerate their larger bodies to a similar speed.

Greater segment mass being due to greater muscle mass underscores that overall limb morphology may be intimately tied to both swing and stance phase kinematics, and, as such, there may be a trade-off in morphological design between swing and stance phase performance. Such a trade-off has been noted several times in tetrapods, in particular with regard to minimizing the mass of the distal limb segments through digit reduction and/or a concentration of limb mass proximally along the limb (Howell, 1944; Gray, 1968; Coombs, 1978; Hildebrand, 1988; Alexander, 1997; Pasi and Carrier, 2003; Kemp et al., 2005). A concentration of mass towards the limb's proximal end offers greater out-lever velocities (via more proximal muscle insertions) and lower limb rotational inertia and is therefore advantageous for swing phase. Conversely, proximal mass concentration and proximal muscle insertions offer less leverage to muscles for torque production and postural stability (i.e. mechanical advantage) during stance (Gray, 1968; Biewener, 1989). It should also be noted that morphology advantageous for greater swing phase economy might compromise limb function not only for stance phase but also for other functions, such as digging and swimming. Digging and swimming, as well as prey subdual, rely upon more distal muscle insertions along the limb for increased mechanical advantage and, consequently, a more distal concentration of muscle mass (Smith and Savage, 1956; Pasi and Carrier, 2003; Moore et al., 2013; Kilbourne and Hoffman, 2015).

Does increased limb inertia necessarily entail greater flexion? Unfortunately, the majority of studies conducting multi-species comparisons of kinematics have focused on stance phase and/or have not considered limb inertia. As an exception to this, Ren et al. (2008) measured the limb kinematics of African and Asian elephants and compared them with previously published data on other mammals. They found that in spite of being graviportal and their reputation for ‘columnar’ limbs, elephants use RoMs during swing phase similar to other, smaller mammals. While similarity in RoM does not necessarily entail similarity in maximum limb flexion, it does suggest such a similarity is plausible. Thus, differences in limb segment proportions and limb mass distribution may not necessarily entail differences in limb flexion.

On the one hand, flexing the limbs more strongly, so as to draw the mass of distal limb segments closer to the limb's pivot, would decrease the radius of gyration of the limb and reduce its MOI, resulting in the limb requiring less energy to swing forward (Wickler et al., 2004). On the other hand, limb flexion during swing phase in birds seems to be achieved in large part by active muscle shortening (especially the m. iliotibialis cranialis and m. iliotibialis lateralis pars preacetabularis; see Gatesy, 1999b) with an as yet unresolved role of passive mechanisms. Thus, excessively flexing the limbs during swing phase is also likely to be connected to additional muscle work and costs, and thus an optimum probably exists for the degree of limb flexion during swing. Intriguingly, swing phase duration in many mammals and birds seems to be highly invariant intraspecifically and often hardly changes with speed (e.g. Gatesy and Biewener, 1991; Abourachid, 2001; Fischer et al., 2002; Stoessel and Fischer, 2012). Potentially, this reflects that animals operate close to an optimum in the trade-off between saving energy by reducing a limb's MOI and spending extra energy to achieve this. Interspecifically, however, morphological differences can be expected to be reflected in different degrees of limb flexion and temporal swing phase characteristics. Distally heavy limbs are most expensive to flex, but flexing distally heavy limbs also offers the largest benefit in terms of reducing MOI. The work to flex the limb is proportional to the product of force (i.e. mass) and distance, whereas the MOI is proportional to the product of mass and the square of distance [i.e. (radius of gyration)^2.0]. Clearly, this trade-off is likely to be skewed towards reducing the radius of gyration, given that this distance is squared.

To gain insight into how differences in limb mass and mass distribution influence the metabolic energy consumed during locomotion, Taylor et al. (1974) monitored oxygen consumption in cheetahs, gazelles and goats – three taxa similar in body mass yet different in limb mass and its distribution – locomoting over a similar range of speeds. Surprisingly, they found that differences in oxygen consumption (a direct measure of energy expenditure) were negligible among the three species. One possible reason for minimal differences in locomotor costs among these taxa might be differences in flexion of the limbs (not measured by Taylor et al., 1974), with the more distally heavy limbs of the cheetah being flexed more in comparison to species with more distally light limbs (i.e. kinematic adjustments that influence the MOI during swing).

In the current study, the increased flexion of the hindlimbs during swing phase as a means to reduce MOI and consequently the cost of swing phase is explored. But do other mechanisms exist? One possible mechanism explored in previous studies – though unexplored in birds – is protraction of the limb through elastic energy storage (Taylor et al., 1980). Elastic energy storage has been proposed for the mammalian forelimb in the tendon of m. biceps brachii (Wilson et al., 2003; Lichtwark et al., 2009) and also for the epaxial muscles (Alexander et al., 1985). However, these mechanisms obviously benefit quadrupeds not bipeds. Elastic energy storage has also been proposed for the tensor fascia latae of the mammalian hindlimb (Bennett, 1989). In these mechanisms, muscles and their associated fascia and tendons undergo substantial stretching during stance phase. In addition to limb protraction, the stretching of such elastic tissues during late stance may also aid the flexion of more distal limb joints during the start of swing phase. Therefore, other means of elastic energy storage should be investigated in birds, ideally alongside measurements of locomotor costs. For example, in the ostrich – the largest living biped capable of endurance runs at high velocity – a passive engage–disengage mechanism has been identified and has been shown to contribute to rapid intertarsal flexion and extension during swing phase (Schaller et al., 2009).

To conclude, when compared with a neutral pose, expected and observed flexing of the limb during swing phase in three species of morphologically diverse shorebirds results in a reduced MOI throughout most of swing phase. This mechanism presents a possible way to lower the metabolic costs of swinging the limb, as a trade-off between investing additional muscle work to flex the limb and the potential reward via a reduction of the limb's MOI should be skewed towards reducing MOI because of MOI being a function of the square of the radius of gyration.

We would like to thank Ingrid Weiß and Rommy Petersohn for their expertise and technical assistance in collecting cineradiographic footage. We would also like to thank Irina Mischewski, Patricia Henze, Itziar Candeal and Marc Thomsen for their diligence in digitizing cineradiographic videos, and Yefta Sutedja, Ben Derwel and Anvar Jakupov for their help in estimating the segmental MOIs. We would also like to thank Sylke Frahnert and Pascal Eckhoff for additional access to skeletal specimens. Alexander Stößel helped to design this study. Anonymous reviews improved the clarity of previous versions of this manuscript.