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Apples, oranges, and angles: Comparative kinematic analysis of disparate limbs
Journal of Theoretical Biology 282 (2011) 7–13
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Apples, oranges, and angles: Comparative kinematic analysis of disparate limbs Stephen M. Gatesy a,n, Nancy S. Pollard b a b
Department of Ecology and Evolutionary Biology, Box G-B209, Brown University, Providence, RI 02912, USA Robotics Institute and Computer Science Department, EDSH 227, Carnegie-Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA
a r t i c l e i n f o
abstract
Article history: Received 2 March 2011 Received in revised form 3 May 2011 Accepted 6 May 2011 Available online 14 May 2011
Tetrapod limbs exhibit diverse postures and movements during terrestrial locomotion. As with morphological traits, the history of kinematic evolution should be accessible to reconstruction through analysis of limb motion patterns in a phylogenetic framework. However, the angular data comprising most kinematic descriptions appear to suffer from limitations that preclude meaningful comparison among disparate species. Using simple planar models, we discuss how geometric constraints render joint and elevation angles independent of neither morphology, degree of crouch, nor one another during the stance phase of locomotion. The implicit null hypothesis of potential similarity is invalidated because angular data are not viably transferable among limbs of dissimilar proportion and/or degree of crouch. Overlooking or dismissing the effect of constraints on angular parameterization hampers efforts to quantitatively elucidate the evolution of locomotion. We advocate a search for alternative methods of measuring limb movement that can decouple intersegmental coordination from morphology and posture. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Locomotion Motion analysis Evolution Angle Comparative methods
1. Introduction Terrestrial locomotion is a complex, coordinated activity involving the dynamic interaction of numerous parts (e.g., Bernstein, 1967). During walking and running, a tetrapod’s multi-segmented limbs continuously change configuration as they oscillate with each cycle. Motion can be quite diagnostic, such that a person’s stride reveals clues about age (Elble et al., 1991), sex (Troje, 2002), health (Manor and Li, 2009), emotion (Michalak et al., 2009), and even individual identity (Richardson and Johnston, 2005). Yet despite such rich intraspecific variation, human limb movements would never be confused with those of a chimp, cat, or quail, which have their own characteristic motion profiles. How, why, and when limb kinematic patterns have changed along various lineages are major questions in functional morphology, comparative biomechanics, and paleontology (Biewener, 1989; Irschick and Jayne, 1999; Hutchinson and Gatesy, 2000; Blob, 2001; Gasc, 2001; Larson et al., 2001; Russell and Bels, 2001; Fischer et al., 2002; Schmidt, 2008; Hutchinson and Allen, 2009). The evolution of locomotor movements should be traceable
n
Corresponding author. Tel.: þ1 401 863 3770; fax: þ1 401 863 7554. E-mail addresses: [email protected] (S.M. Gatesy), [email protected] (N.S. Pollard). 0022-5193/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2011.05.009
throughout the tree of life using comparative methods (e.g., Harvey and Pagel, 1991). Just as for morphological, genetic, and behavioral traits, combining data from extant species with an explicit phylogenetic hypothesis should reveal evidence of movement patterns likely present in hypothetical common ancestors. Such inferences should help clarify trends along specific lineages and inform reconstruction of fossil taxa. In order for limb motion to be compared and mapped out through time, movement patterns must be quantified. Most kinematic descriptions rely on external and/or internal angular data to characterize a limb’s changing pose throughout the stride cycle. ‘‘Elevation’’ or ‘‘segment’’ angles are external angles that quantify the orientation of a limb segment (femur, tibia, metatarsus, etc.) relative to vertical or horizontal. ‘‘Joint’’ angles (hip, knee, ankle, etc.) are internal angles that measure the relative orientation of adjacent segments. Both types of angles are typically compared at equivalent times within the stride cycle. The mid-stance pose is frequently emphasized because the ground reaction force is often largest at this time (e.g., Biewener, 1989). Excursion angles between touch-down and lift-off are one way to summarize the amount of rotation a segment or joint undergoes during the stance phase. Angles are routinely compared among species to answer a range of questions. For example, Fischer et al. (2002) measured limb motion in eight species of small mammals to see if all share a basic kinematic pattern. Ashley-Ross (1994) compared data from
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walking salamanders to hind limb angles of a broad sampling of amniotes to highlight common features. More statistical treatments include a study of sprinting in five lizard species (Irschick and Jayne, 1999), an assessment of joint angles among cercopithecine primates (Polk, 2002), and a kinematic analysis of scaling in nine felids (Day and Jayne, 2007). As these and countless other publications attest, angles are ubiquitous. Herein, we attempt to shed light on previously overlooked limitations of such angular parameterization for evolutionary analyses of terrestrial locomotion.
2. Transferability—a prerequisite for comparison Comparative kinematic studies are founded on a concept of similarity. Evaluating whether a parameter such as hip angle, for example, differs among two or more species presumes that their hip angles could be the same. If two species are incapable of angular similarity, testing for significant differences is misguided (or, at best, unguided). Potential similarity is implied even when angles from two species are plotted on the same graph for more qualitative comparison. Yet this implicit premise – that the disparate limbs of different taxa can utilize the same angles – remains unexplored by zoologists studying locomotor evolution. Is such a null hypothesis realistic? We suggest a simple test of whether two limbs can operate with the same angular kinematic pattern. If the locomotor angles of species A are being compared to those of species B, can species A reasonably function using the angles of species B and vice versa. Following researchers in computer animation and robotics, we refer to such angle swapping as ‘‘motion transfer’’ (e.g., Bodenheimer et al., 1997; Sturman, 1998; Riley et al., 2003). Successful motion transfer (hereafter, ‘‘transferability’’) requires that each limb move viably when driven by the other’s angles.
In the following section we use minimal 2-D models of moving limbs to explore transferability of angular data among several bipeds.
3. Terrestrial limbs as constrained kinematic chains We represent the main components of the hind limb (femur, tibia, metatarsus) as three articulated line segments (Fig. 1A) that vary only in relative length. In the plane formed by the forward velocity and gravity vectors, each segment’s orientation is measured by its elevation angle (yf, yt, ym) with respect to vertical. The 2-D position of the most proximal or ‘‘root’’ joint (hip) determines the limb’s translation. Note that the location of the distal ‘‘end effector’’ (metatarso-phalangeal or MP joint) is not explicitly specified, being entirely dependent on the root position, segment lengths, and elevation angles of the chain. Theoretically, species with different limb proportions are free to employ identical angles. More realistically, however, two disproportioned limbs can only use the same root position and angles if the location of the end effector is unrestricted. A general tenet in robotics and animation (e.g., Shin et al., 2001) is that two articulated chains with different segment lengths can either have identical angles or identical root and end effector positions, but not both. Therefore, any constraints on the positions of the root and end effector are potential impediments to successful motion transfer using angles. 3.1. Ground constraints During a stance phase on relatively firm substrates, the limb normally remains fixed on the ground without significant lifting, sinking, or sliding. Thus, distal stability acts as a basic geometric constraint on the position of the limb’s end effector. How do such
Hip = "root" femur Knee tibia Ankle
metatarsus
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Guineafowl run
Human walk
Flamingo: human hip and angles
Flamingo: guineafowl hip and angles
Flamingo: human MP and angles
Flamingo: guineafowl toe and angles
Fig. 1. Artifacts from interspecific motion transfer using angles: (A) three-segment limb with four joints acts as a kinematic chain. (B) Stance phase poses of a human walking stride. (C) Transfer of human angles and hip translations to a flamingo limb causes unrealistic footskate and penetration of the MP. (D) Rerooting to insure MP stability adversely affects hip height. (E) Stance phase poses of a guineafowl running stride. (F) Transfer of guineafowl angles and hip translations also fail for the flamingo’s ground interaction. (G) Maintaining a static toe tip creates an arcing hip height.
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ground-based limits impact interspecific transfer of angular kinematic data? We first illustrate this issue by transferring the motion of a human limb to that of a Greater Flamingo (Phoenicopterus ruber). Lower limb movement was collected from a freely walking woman by optical motion capture at 60 Hz using 12 Vicon cameras and Vicon IQ software. For every other frame, coordinates of the right hip, knee, ankle, and MP joints were plotted in lateral view as a three-segment stick diagram. A 2-D articulated model was created in Maya 2010 (Autodesk, Inc.), superimposed on the pose sequence, and animated to match the position and orientation of the femur, tibia, and metatarsus through time (Fig. 1B). When human elevation angles and hip translations are applied to a limb model with flamingo proportions, the resulting motion sequence fails miserably (Fig. 1C). The MP joint deviates wildly from its formerly stable pattern. In early and mid-stance the MP joint slides forward with respect to the substrate, an artifact known to computer animators as ‘‘footskate’’ (Kovar et al., 2002). In late stance the MP joint penetrates deeply below the ground plane. Changes in overall limb scaling can reduce, but not eliminate, such flaws. The extreme proportional differences between a human (long femur, long tibia, and short metatarsus) and a flamingo (short femur, long tibia, and long metatarsus) magnify motion transfer artifacts. Yet flaws also occur when motion is swapped among species of birds. A Helmeted Guineafowl (Numida meleagris) running on a treadmill at 1.5 m/s was recorded in lateral perspective by video fluoroscopy at 60 Hz and converted into a sequence of JPEG images, which served as reference for Maya animation of a four-segment model (femur, tibia, metatarsus, and third toe). Guineafowl stance phase motion (Fig. 1E) creates smaller, yet obvious, errors when applied to the flamingo model (Fig. 1F). Rather than remain stable, the tip of the third toe describes an arcing path with footskate, penetration, and lifting violating the ground constraint. 3.2. Center of mass and hip constraints A basic goal of steady, non-turning locomotion is to move the body forward at a relatively constant speed and a relatively constant height. Although the body’s center of mass (CoM) undergoes important vertical oscillations and changes in forward speed with each stride, the magnitude of these deviations compared to forward progress during the stance phase of normal gaits is relatively small. Animals are not geometrically required to use such restricted CoM trajectories, but the energetic benefits of doing so are thought to explain the retention and convergent evolution of the inverted pendulum and mass-spring strategies (Cavagna et al., 1977; Blickhan and Full, 1987; Srinivasan and Ruina, 2005; Geyer et al., 2006). Given that a limb’s root joint is relatively fixed with respect to the CoM, the path of the hip is likewise constrained. Obviously, restrictions on CoM and hip motion are not as physically concrete as the ground constraint, but their consistent influence in extant animals makes them no less significant. One solution to footskate and ground penetration caused by motion transfer is to make the flamingo’s end effector retain the original, stable positions used by the human or guineafowl. Such ‘‘re-rooting’’ of the kinematic chain makes the hip’s location entirely dependent on distal joint (MP or toe tip) position, segment lengths, and angles in the chain. Here we focus on relative hip height (a dimensionless degree of crouch), which equals absolute hip height divided by the sum of the femoral, tibial, and metatarsal segment lengths (Alexander, 1977). The hip’s trajectory is negatively impacted by MP stability during motion transfer from human to flamingo. In the original motion (Fig. 1B), relative hip height only varies from 0.863 to
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0.886 ( þ3%) during stance. By contrast, the flamingo’s relative hip height increases from 0.62 to 0.87 (þ 40%) when driven by human elevation angles (Fig. 1D). Such marked ascent makes it impossible for the two hind limbs to achieve double support in early and late stance. Given these deformations of hip motion, it would be extremely difficult for the flamingo’s CoM to follow a trajectory conducive to recovering energy by an inverted pendulum mechanism. Using running guineafowl angles and toe tip positions (Fig. 1E), the flamingo’s hip path is less distorted, but is distinctly arced rather than level (Fig. 1G). Compared to a 4% drop followed by recovery in the original, relative hip height rises from 0.837 to 0.892 ( þ7%) before falling to 0.718 ( 24%) in late stance. Although such a compass-like path could be viable for acting as an inverted pendulum, we are unaware of any species of bird using such large hip height fluctuations during walking, much less running. Without exaggerated pitching of the body or other radical compensations, the CoM will follow a similarly unrealistic path. Once again, overall scaling of limb size does not correct these artifacts. 3.3. Ground and CoM/hip constraints Both human and guinefowl stance phases (Fig. 1B and E) are characterized by a relatively stable end effector (MP or toe tip) and relatively constant hip height. To move beyond stick figure representations, we measured the elevation angles of the three main limb elements (yf, yt, ym) for each pose (Fig. 2A). Graphs of elevation angle versus time show overlap in the ranges of yt and ym (Fig. 2B and C), but no poses are shared. Not surprisingly, 3-D angle–angle– angle plots of yf, yt, and ym for the two species form distinct, wellseparated trajectories (Fig. 2D). But are human and guineafowl limbs restricted to discrete regions of configuration space? Could these and other disparate species potentially move with the same angles? If both ground and CoM/hip constraints are operating during stance, then the articulated hind limb chain is severely restricted at each end. To move naturally, the toes must remain relatively stable while the hip moves forward at a more or less even height. Such root and end effector constraints interact with segment lengths to limit the angular combinations a limb can achieve. This geometric relationship is most easily demonstrated using the three main limb segments. We calculated all possible configurations of elevation angles that each limb could assume while maintaining a static MP and a fixed relative hip height. As described by three elevation angles, the complete solution sets form surfaces (ranging from ellipsoidal to spindle-shaped) that closely approximate the actual sequence of stance phase poses (Fig. 3A and B). Additional restrictions imposed by the ground, articular geometry, and soft tissues will further limit limbs to a subset of these combinations, but complete surfaces are shown. Possible stance phase poses for the human limb (0.865 relative hip height) form a vertical ellipsoid that is longest in the metatarsal elevation angle dimension (Fig. 3A). In this constrained chain, the relatively long femur and tibia are confined to a more limited range of elevation angles than the relatively shorter metatarsus. The guineafowl surface (0.780 relative hip height) is more spherical, owing to more equally proportioned limb segments (Fig. 4B). Based on images of walking flamingos, we estimated a relative hip height of 0.9. Viable combinations of flamingo elevation angles form an ellipsoid that is longest along the femoral elevation angle axis (Fig. 3C). The relatively long tibial and metatarsal segments cannot deviate very far from vertical and still satisfy the hip height constraint. Like the short human foot, the flamingo’s small thigh segment is free to use a wider range of orientations. Realized human motion involves a sequence of poses near the bottom of the human ellipsoid (Fig. 3A), far below the flamingo
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Fig. 2. Human and guineafowl limbs use distinct combinations of elevation angles: (A) the orientation of each segment is measured as an elevation angle (yf, yt, and ym) with respect to vertical. Graphs of human (B) and guineafowl (C) elevation angles throughout the stance phase have some overlapping ranges, but clearly differ. (D) A 3-D configuration space of yf, yt, and ym elevation angles reveals very different pose trajectories (arrows) for these species. Ticks mark 101 increments from 901 to 901.
θm
θm
θt
θf
θt
θm
θf
θt
θf
Guineafowl
θt
θf
guineafowl
human Human
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Flamingo
Multi-species
Fig. 3. Interspecific comparison among actual and potential poses using a simple 2-D model of the stance phase: (A) assuming a fixed MP and constant hip height, permissible combinations of human elevation angles form a vertical ellipsoid in which the shortest segment (metatarsus) can rotate the most. The actual human sequence (filled circles) falls very close to this surface. (B) Actual guineafowl poses (filled cones) lie along a more spherical potential surface. (C) In contrast, the flamingo’s potential surface is longest in the femoral dimension. Neither human nor guineafowl poses come close, resulting in motion transfer artifacts. (D) With minimal intersections and no shared poses among these three taxa, potential configuration surfaces reveal why the null hypothesis of transferability is unsupported. Ticks mark 101 increments from 901 to 901.
potential surface (Fig. 3C). A flamingo limb cannot reach such flatfooted poses with its relatively long metatarsus without unnaturally reducing relative hip height. Likewise, poses used by the running guineafowl (Fig. 3C) fall well outside the flamingo ellipsoid. Although based on simplified models, these plots highlight how end point constraints interact with proportions to confine human, flamingo, and guineafowl limbs to very limited regions of configuration space during the stance phase. Poses are shared along limited surface intersections, but most options are species-specific (Fig. 3D). None is common to all three. Such mismatched surfaces signal failure of transferability, and thus a violation of the null hypothesis of similarity implicit in direct interspecific comparison of angles. Differing degree of crouch also precludes use of the same poses. Behavioral variation in cats provides a dramatic example (Trank et al., 1996). The limb exhibits smaller elevation angles during normal walking (Fig. 4A; ca. 0.81 relative hip height), than during crouched walking (Fig. 4B; ca. 0.48 relative hip height). Poses change similarly through elevation angle space during both
movements (Fig. 4C), but normal and crouched trajectories remain well separated. Plotting the potential poses each limb could achieve (Fig. 4D) reveals the geometric restrictions on angular combinations imposed by end point constraints alone. Large relative hip heights can only be produced by small elevation angles, thus confining the normal walking ellipsoid nearer the origin. A more crouched posture expands the configuration surface outward because segments can, and often must, achieve larger angles. These surfaces do not intersect. Proportionally identical limbs with dissimilar degrees of crouch can have one or even two angles in common, but are geometrically precluded from sharing all three angles simultaneously.
4. Discussion In this study we carried out simple, 2-D examples of interspecific motion transfer. When hind limb angles from a walking human and running guineafowl were applied to a flamingo limb,
S.M. Gatesy, N.S. Pollard / Journal of Theoretical Biology 282 (2011) 7–13
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θm
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Fig. 4. Actual and potential poses in cats with different degree of crouch: (A) stick diagrams of normal (A) and crouched (B) walking (modified from Tranke et al. (1996). Elevation angle plots versus time (A and B) and versus one another (C) show similar patterns but distinct combinations. (D) The spindle-shaped potential configuration surface for crouching completely engulfs the ellipsoidal surface for normal walking, precluding shared angles. Ticks mark 101 increments from 1801 to 1801.
undesirable motion artifacts were introduced. Footskate, ground penetration, lifting, and extreme deviations in hip height all indicate failure of transferability. Thus, although it may always be theoretically possible for a flamingo to use the same angles as a human or guineafowl and vice versa, motion transfer exposes the kinematic consequences of accepting the null hypothesis. Nontransferability highlights several geometric limitations that undermine the direct comparison of angles among disparate taxa in an evolutionary context. 4.1. Angles are not independent of morphology, degree of crouch, or one another Human angles produce correct hip trajectories for humans, but are inappropriate for a limb with non-human proportions. The same holds true among species of birds. Following the research of Alexander and others (Alexander and Jayes, 1983; Hodgins and Pollard, 1997), animals with disparate proportions are not expected to walk or run in a dynamically similar fashion using identical angles. Our examples show that due to geometric constraints, it may not be energetically plausible for them to do so. Such results may appear so obvious as to be dismissed as trivial, but we contend that their implications for comparative motion analysis are significant. Angles may be dimensionless, but within a constrained system they are not independent of segment length. We consider the conservative motion of the CoM and hip in diverse species to be evidence of a proximal constraint. Likewise, stance phase stability of the end effector represents a distal constraint. When restricted at both ends, simple geometry dictates that angles must be matched to a limb’s proportions in order to satisfy basic spatial requirements. CoM, hip, and ground constraints pertain equally in 3-D. Angles will likewise be non-transferable among disparate ‘‘sprawling’’ tetrapods that exhibit more spatially complex limb motion. Even with fixed proportions, limbs with different relative hip height require different angular combinations (Trank et al., 1996; Grasso et al., 2000). Although it is reasonable to regard degree of crouch as a consequence of adopting specific sets of joint or
elevation angles, we find it fruitful to treat relative hip height as an independent variable that is both behaviorally and evolutionarily labile. Just as a crouching cat is prohibited from using normal walking poses, angles from a crouched ancestor are not directly transferable to a more erect descendant. Increases or decreases in degree of crouch along a lineage can, therefore, violate the null hypothesis and make angular comparisons among species biased toward difference. Our model of a planar, three-segment limb constrained at each end (Figs. 3 and 4D) accounts for the majority of angular change during the stance phase in our human, guineafowl, and cat case studies. In this simplified representation, a limb is restricted to a surface in elevation angle configuration space with only two degrees of freedom (DoF). The first DoF permits the hip joint to move forward and backward. The second DoF allows the limb to internally reconfigure among a family of poses for each hip position. A three-segment limb with fixed ends has redundancy, but angles are not free to change independently. Segments must reorient in a coordinated fashion. 4.2. Significance for comparative kinematic analysis How does non-transferability impact evolutionary functional morphology and comparative biomechanics? Simply put, the direct comparison of joint and elevation angles among species is not as straightforward as has been assumed. Kinematic data are often treated as independent from morphological data, but motion transfer artifacts highlight an intrinsic link between shape, posture, and movement. We contend that ground and CoM constraints during the stance phase are strong enough to impose significant limits on MP and hip position. Such limits compel the articulated limb chain to act more as a coordinated whole with reduced DoF than as three autonomous segments with independent rotational freedom. If interspecific differences in proportions and/or degree of crouch necessitate distinct limb configurations, any joint by joint or segment by segment assessment of angular data becomes an apples to oranges comparison that confounds elucidation of locomotor evolution. We are not aware of any zoologists attempting motion transfer to test transferability prior to comparison. Normally, motion
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Guineafowl run (source)
Bustard
Guan
Jacana
Macaw
Allosaurus
Fig. 5. Interspecific angular motion transfer artifacts vary with disparity. Stance phase angles of a running guineafowl (A) are successfully transferred to a similarly proportioned guan (B). Hip paths become distorted for limbs of other birds (C–E) and the extinct theropod, Allosaurus (F).
analysis of multiple species yields angular variables, which are then compared statistically or more qualitatively. Yet the null hypothesis of potential kinematic similarity remains unsubstantiated. Functional morphologists do not try to drive the limb of one species with angles from another as done here, and so remain unaware of any inherent conflict. Researchers in computer animation and robotics are sensitive to this predicament because reusing motion can avoid having to animate a new character entirely from scratch, a traditional technique that is slow and costly. Gleicher (1998) called this the ‘‘motion retargeting problem’’ because the original motion must be edited or ‘‘retargeted’’ before it can be successfully transferred to a second character of dissimilar shape. Viewers are incredibly adept at perceiving very minor aberrations of an animated character’s stance foot. Even the seemingly simple transfer from one adult human to another typically requires motion retargeting to correct for individual differences and small errors. As in the human to flamingo transfer, reuse of an actor’s motion for an animal, alien, or cartoon character with dissimilar proportions is especially fraught. Clearly, motion artifacts will vary depending on the level of disparity. Minor morphological or postural differences will not yield dramatic motion transfer artifacts and in such cases the null hypothesis can be accepted. For example, guineafowl angles and toe tip translations (Fig. 5A) appear to be transferable to the limb of a Crested Guan (Penelope purpurascens; Fig. 5B), a species in the same order (Galliformes) with comparable proportions. If actual guans move using a similar relative hip height, then the ellipsoids for these two species would come close enough (within measurement error) to share poses and directly compare angles. But for birds such as a Madagascar Jacana (Actophilornis albinucha; Fig. 5C), Houbara Bustard (Chlamydotis undulata; Fig. 5D), or Scarlet Macaw (Ara macao; Fig. 5E), proportional differences cause transfer failure. If the limb motion of these species is quantified one day, we are compelled to reject the null hypothesis of angular similarity a priori and predict distinct patterns. The same holds for inferences about extinct taxa, such as the theropod dinosaur, Allosaurus (Fig. 5F; Gatesy et al., 2009).
Yet for many comparisons, the judgment of transferability will be less clear-cut. How much hip oscillation or toe slip can motion transfer inflict and still remain viable? Our goal here is not to establish strict boundaries to unambiguously define transferability. We leave such judgment to individual researchers, whom we encourage to transfer motion among their study species. Rather, we wish to raise awareness about angles and their geometric inter-relationship with proportions, degree of crouch, end point constraints, and segment interdependence. Before raw angular data are compared, it seems reasonable to pause and consider, given the constraints on the limb, if two or more species could use the same poses. Kinematic distinctions are most meaningful when limbs could use the same motion, yet do not. 4.3. Intersegmental coordination and a call for alternative parameterization Despite quantitative angular differences, many species likely share qualitatively similar limb movement. What underlying patterns might proportional and postural disparity be masking? We believe these deeper patterns fall under the term ‘‘coordination’’ (e.g., Bernstein, 1967), specifically intersegmental, interjoint, or intralimb coordination. A quantitative characterization of coordination would be of value for evolutionary analysis because such patterns need not be tied to specific morphologies. Theoretically, intersegmental coordination can pass unchanged from ancestor to descendant or acquire novelty like any other character, even as the limbs evolve different segment lengths, degree of crouch, size, or other features. Unfortunately, existing methods for visualizing and evaluating coordination are largely based on angles. Bivariate and trivariate plots of joint angles (so-called cyclograms, Grieve, 1968; Charteris et al., 1979) and elevation angles (e.g., Borghese et al., 1996) are not conducive to unbiased comparison. Constrained limbs may not have equal access to angular configuration space (Figs. 3 and 4), thereby precluding similarity. We hope that highlighting some shortcomings of angular comparison will motivate others to seek creative new methods to quantify limb movement. Specifically, we advocate for a search for novel parameterizations of intersegmental coordination that
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are either derived from angular data or are non-angular. An ideal format for comparative studies would allow ‘‘coordination transfer’’ among both closely and distantly related species. If successful, having the same coordination pattern (due to retention of the primitive condition or to homoplasy) could serve as a productive null hypothesis in ways that angular equality cannot.
5. Conclusion In this paper, we have tried to show that despite being well established and convenient, angles are not directly equivalent among disparate, constrained limbs. Yet from discussions with our colleagues (functional morphologists, biomechanists, animators, roboticists, paleontologists), it is clear that angles hold a central place in motion analysis. Indeed, the conceptual link between rotary motion and angular measurement can sometimes be so strong that the terms become one and the same—rotations are angles. But must angles be the best and only way to measure limb motion for comparative analysis? Our warning is not meant to discourage comparison of limb kinematics in an evolutionary context. On the contrary, this paper is motivated by a desire to see more such studies (albeit carried out with additional regard for geometric constraints). Although we have focused on terrestrial locomotion where ground and CoM restrictions are well-established, limbs acting as wings and flippers likely operate under endpoint constraints as well. A search for alternative motion parameterizations may lead to the discovery of viable null hypotheses of intersegmental coordination. Such methods may pave the way for discovering similarity amidst disparity and novelty within uniformity, thus lending a new clarity to tetrapod locomotor evolution.
Acknowledgments We thank Kevin Middleton, Jessica Hodgins, Young-Hui Chang, and members of the Brown Evolutionary Vertebrate Morphology Group for helpful discussion and critical advice, as well as the Ornithology Department at the Museum of Comparative Zoology, Harvard University for access to avian skeletons. Three anonymous reviewers improved the manuscript. Supported by National Science Foundation grants DBI-9974424 and IOS-0925077, the Bushnell Faculty Research Fund, and Autodesk, Inc. (to S.M.G.) as well as National Science Foundation grants IIS-0205224 and CCF-0702443 (to N.S.P.). References Alexander, R.Mc.N., 1977. Mechanics and scaling of terrestrial locomotion. In: Pedley, T.J. (Ed.), Scale Effects in Animal Locomotion. Academic Press, London, pp. 93–110. Alexander, R.Mc.N., Jayes, A.S, 1983. A dynamic similarity hypothesis for the gaits of quadrupedal mammals. J. Zool. 201, 135–152. Ashley-Ross, M.A., 1994. Hindlimb kinematics during terrestrial locomotion in a salamander (Dicamptodon tenebrosus). J. Exp. Biol. 193, 255–283. Bernstein, N.A., 1967. The Coordination and Regulation of Movements. Pergamon Press, London. Biewener, A.A., 1989. Scaling body support in mammals—limb posture and muscle mechanics. Science 245, 45–48. Blickhan, R., Full, R.J., 1987. Locomotion energetics of the ghost crab II. Mechanics of the centre of mass during walking and running. J. Exp. Biol. 130, 155–174.
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Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Apples, oranges, and angles: Comparative kinematic analysis of disparate limbs Stephen M. Gatesy a,n, Nancy S. Pollard b a b
Department of Ecology and Evolutionary Biology, Box G-B209, Brown University, Providence, RI 02912, USA Robotics Institute and Computer Science Department, EDSH 227, Carnegie-Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA
a r t i c l e i n f o
abstract
Article history: Received 2 March 2011 Received in revised form 3 May 2011 Accepted 6 May 2011 Available online 14 May 2011
Tetrapod limbs exhibit diverse postures and movements during terrestrial locomotion. As with morphological traits, the history of kinematic evolution should be accessible to reconstruction through analysis of limb motion patterns in a phylogenetic framework. However, the angular data comprising most kinematic descriptions appear to suffer from limitations that preclude meaningful comparison among disparate species. Using simple planar models, we discuss how geometric constraints render joint and elevation angles independent of neither morphology, degree of crouch, nor one another during the stance phase of locomotion. The implicit null hypothesis of potential similarity is invalidated because angular data are not viably transferable among limbs of dissimilar proportion and/or degree of crouch. Overlooking or dismissing the effect of constraints on angular parameterization hampers efforts to quantitatively elucidate the evolution of locomotion. We advocate a search for alternative methods of measuring limb movement that can decouple intersegmental coordination from morphology and posture. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Locomotion Motion analysis Evolution Angle Comparative methods
1. Introduction Terrestrial locomotion is a complex, coordinated activity involving the dynamic interaction of numerous parts (e.g., Bernstein, 1967). During walking and running, a tetrapod’s multi-segmented limbs continuously change configuration as they oscillate with each cycle. Motion can be quite diagnostic, such that a person’s stride reveals clues about age (Elble et al., 1991), sex (Troje, 2002), health (Manor and Li, 2009), emotion (Michalak et al., 2009), and even individual identity (Richardson and Johnston, 2005). Yet despite such rich intraspecific variation, human limb movements would never be confused with those of a chimp, cat, or quail, which have their own characteristic motion profiles. How, why, and when limb kinematic patterns have changed along various lineages are major questions in functional morphology, comparative biomechanics, and paleontology (Biewener, 1989; Irschick and Jayne, 1999; Hutchinson and Gatesy, 2000; Blob, 2001; Gasc, 2001; Larson et al., 2001; Russell and Bels, 2001; Fischer et al., 2002; Schmidt, 2008; Hutchinson and Allen, 2009). The evolution of locomotor movements should be traceable
n
Corresponding author. Tel.: þ1 401 863 3770; fax: þ1 401 863 7554. E-mail addresses: [email protected] (S.M. Gatesy), [email protected] (N.S. Pollard). 0022-5193/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2011.05.009
throughout the tree of life using comparative methods (e.g., Harvey and Pagel, 1991). Just as for morphological, genetic, and behavioral traits, combining data from extant species with an explicit phylogenetic hypothesis should reveal evidence of movement patterns likely present in hypothetical common ancestors. Such inferences should help clarify trends along specific lineages and inform reconstruction of fossil taxa. In order for limb motion to be compared and mapped out through time, movement patterns must be quantified. Most kinematic descriptions rely on external and/or internal angular data to characterize a limb’s changing pose throughout the stride cycle. ‘‘Elevation’’ or ‘‘segment’’ angles are external angles that quantify the orientation of a limb segment (femur, tibia, metatarsus, etc.) relative to vertical or horizontal. ‘‘Joint’’ angles (hip, knee, ankle, etc.) are internal angles that measure the relative orientation of adjacent segments. Both types of angles are typically compared at equivalent times within the stride cycle. The mid-stance pose is frequently emphasized because the ground reaction force is often largest at this time (e.g., Biewener, 1989). Excursion angles between touch-down and lift-off are one way to summarize the amount of rotation a segment or joint undergoes during the stance phase. Angles are routinely compared among species to answer a range of questions. For example, Fischer et al. (2002) measured limb motion in eight species of small mammals to see if all share a basic kinematic pattern. Ashley-Ross (1994) compared data from
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walking salamanders to hind limb angles of a broad sampling of amniotes to highlight common features. More statistical treatments include a study of sprinting in five lizard species (Irschick and Jayne, 1999), an assessment of joint angles among cercopithecine primates (Polk, 2002), and a kinematic analysis of scaling in nine felids (Day and Jayne, 2007). As these and countless other publications attest, angles are ubiquitous. Herein, we attempt to shed light on previously overlooked limitations of such angular parameterization for evolutionary analyses of terrestrial locomotion.
2. Transferability—a prerequisite for comparison Comparative kinematic studies are founded on a concept of similarity. Evaluating whether a parameter such as hip angle, for example, differs among two or more species presumes that their hip angles could be the same. If two species are incapable of angular similarity, testing for significant differences is misguided (or, at best, unguided). Potential similarity is implied even when angles from two species are plotted on the same graph for more qualitative comparison. Yet this implicit premise – that the disparate limbs of different taxa can utilize the same angles – remains unexplored by zoologists studying locomotor evolution. Is such a null hypothesis realistic? We suggest a simple test of whether two limbs can operate with the same angular kinematic pattern. If the locomotor angles of species A are being compared to those of species B, can species A reasonably function using the angles of species B and vice versa. Following researchers in computer animation and robotics, we refer to such angle swapping as ‘‘motion transfer’’ (e.g., Bodenheimer et al., 1997; Sturman, 1998; Riley et al., 2003). Successful motion transfer (hereafter, ‘‘transferability’’) requires that each limb move viably when driven by the other’s angles.
In the following section we use minimal 2-D models of moving limbs to explore transferability of angular data among several bipeds.
3. Terrestrial limbs as constrained kinematic chains We represent the main components of the hind limb (femur, tibia, metatarsus) as three articulated line segments (Fig. 1A) that vary only in relative length. In the plane formed by the forward velocity and gravity vectors, each segment’s orientation is measured by its elevation angle (yf, yt, ym) with respect to vertical. The 2-D position of the most proximal or ‘‘root’’ joint (hip) determines the limb’s translation. Note that the location of the distal ‘‘end effector’’ (metatarso-phalangeal or MP joint) is not explicitly specified, being entirely dependent on the root position, segment lengths, and elevation angles of the chain. Theoretically, species with different limb proportions are free to employ identical angles. More realistically, however, two disproportioned limbs can only use the same root position and angles if the location of the end effector is unrestricted. A general tenet in robotics and animation (e.g., Shin et al., 2001) is that two articulated chains with different segment lengths can either have identical angles or identical root and end effector positions, but not both. Therefore, any constraints on the positions of the root and end effector are potential impediments to successful motion transfer using angles. 3.1. Ground constraints During a stance phase on relatively firm substrates, the limb normally remains fixed on the ground without significant lifting, sinking, or sliding. Thus, distal stability acts as a basic geometric constraint on the position of the limb’s end effector. How do such
Hip = "root" femur Knee tibia Ankle
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Flamingo: guineafowl hip and angles
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Flamingo: guineafowl toe and angles
Fig. 1. Artifacts from interspecific motion transfer using angles: (A) three-segment limb with four joints acts as a kinematic chain. (B) Stance phase poses of a human walking stride. (C) Transfer of human angles and hip translations to a flamingo limb causes unrealistic footskate and penetration of the MP. (D) Rerooting to insure MP stability adversely affects hip height. (E) Stance phase poses of a guineafowl running stride. (F) Transfer of guineafowl angles and hip translations also fail for the flamingo’s ground interaction. (G) Maintaining a static toe tip creates an arcing hip height.
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ground-based limits impact interspecific transfer of angular kinematic data? We first illustrate this issue by transferring the motion of a human limb to that of a Greater Flamingo (Phoenicopterus ruber). Lower limb movement was collected from a freely walking woman by optical motion capture at 60 Hz using 12 Vicon cameras and Vicon IQ software. For every other frame, coordinates of the right hip, knee, ankle, and MP joints were plotted in lateral view as a three-segment stick diagram. A 2-D articulated model was created in Maya 2010 (Autodesk, Inc.), superimposed on the pose sequence, and animated to match the position and orientation of the femur, tibia, and metatarsus through time (Fig. 1B). When human elevation angles and hip translations are applied to a limb model with flamingo proportions, the resulting motion sequence fails miserably (Fig. 1C). The MP joint deviates wildly from its formerly stable pattern. In early and mid-stance the MP joint slides forward with respect to the substrate, an artifact known to computer animators as ‘‘footskate’’ (Kovar et al., 2002). In late stance the MP joint penetrates deeply below the ground plane. Changes in overall limb scaling can reduce, but not eliminate, such flaws. The extreme proportional differences between a human (long femur, long tibia, and short metatarsus) and a flamingo (short femur, long tibia, and long metatarsus) magnify motion transfer artifacts. Yet flaws also occur when motion is swapped among species of birds. A Helmeted Guineafowl (Numida meleagris) running on a treadmill at 1.5 m/s was recorded in lateral perspective by video fluoroscopy at 60 Hz and converted into a sequence of JPEG images, which served as reference for Maya animation of a four-segment model (femur, tibia, metatarsus, and third toe). Guineafowl stance phase motion (Fig. 1E) creates smaller, yet obvious, errors when applied to the flamingo model (Fig. 1F). Rather than remain stable, the tip of the third toe describes an arcing path with footskate, penetration, and lifting violating the ground constraint. 3.2. Center of mass and hip constraints A basic goal of steady, non-turning locomotion is to move the body forward at a relatively constant speed and a relatively constant height. Although the body’s center of mass (CoM) undergoes important vertical oscillations and changes in forward speed with each stride, the magnitude of these deviations compared to forward progress during the stance phase of normal gaits is relatively small. Animals are not geometrically required to use such restricted CoM trajectories, but the energetic benefits of doing so are thought to explain the retention and convergent evolution of the inverted pendulum and mass-spring strategies (Cavagna et al., 1977; Blickhan and Full, 1987; Srinivasan and Ruina, 2005; Geyer et al., 2006). Given that a limb’s root joint is relatively fixed with respect to the CoM, the path of the hip is likewise constrained. Obviously, restrictions on CoM and hip motion are not as physically concrete as the ground constraint, but their consistent influence in extant animals makes them no less significant. One solution to footskate and ground penetration caused by motion transfer is to make the flamingo’s end effector retain the original, stable positions used by the human or guineafowl. Such ‘‘re-rooting’’ of the kinematic chain makes the hip’s location entirely dependent on distal joint (MP or toe tip) position, segment lengths, and angles in the chain. Here we focus on relative hip height (a dimensionless degree of crouch), which equals absolute hip height divided by the sum of the femoral, tibial, and metatarsal segment lengths (Alexander, 1977). The hip’s trajectory is negatively impacted by MP stability during motion transfer from human to flamingo. In the original motion (Fig. 1B), relative hip height only varies from 0.863 to
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0.886 ( þ3%) during stance. By contrast, the flamingo’s relative hip height increases from 0.62 to 0.87 (þ 40%) when driven by human elevation angles (Fig. 1D). Such marked ascent makes it impossible for the two hind limbs to achieve double support in early and late stance. Given these deformations of hip motion, it would be extremely difficult for the flamingo’s CoM to follow a trajectory conducive to recovering energy by an inverted pendulum mechanism. Using running guineafowl angles and toe tip positions (Fig. 1E), the flamingo’s hip path is less distorted, but is distinctly arced rather than level (Fig. 1G). Compared to a 4% drop followed by recovery in the original, relative hip height rises from 0.837 to 0.892 ( þ7%) before falling to 0.718 ( 24%) in late stance. Although such a compass-like path could be viable for acting as an inverted pendulum, we are unaware of any species of bird using such large hip height fluctuations during walking, much less running. Without exaggerated pitching of the body or other radical compensations, the CoM will follow a similarly unrealistic path. Once again, overall scaling of limb size does not correct these artifacts. 3.3. Ground and CoM/hip constraints Both human and guinefowl stance phases (Fig. 1B and E) are characterized by a relatively stable end effector (MP or toe tip) and relatively constant hip height. To move beyond stick figure representations, we measured the elevation angles of the three main limb elements (yf, yt, ym) for each pose (Fig. 2A). Graphs of elevation angle versus time show overlap in the ranges of yt and ym (Fig. 2B and C), but no poses are shared. Not surprisingly, 3-D angle–angle– angle plots of yf, yt, and ym for the two species form distinct, wellseparated trajectories (Fig. 2D). But are human and guineafowl limbs restricted to discrete regions of configuration space? Could these and other disparate species potentially move with the same angles? If both ground and CoM/hip constraints are operating during stance, then the articulated hind limb chain is severely restricted at each end. To move naturally, the toes must remain relatively stable while the hip moves forward at a more or less even height. Such root and end effector constraints interact with segment lengths to limit the angular combinations a limb can achieve. This geometric relationship is most easily demonstrated using the three main limb segments. We calculated all possible configurations of elevation angles that each limb could assume while maintaining a static MP and a fixed relative hip height. As described by three elevation angles, the complete solution sets form surfaces (ranging from ellipsoidal to spindle-shaped) that closely approximate the actual sequence of stance phase poses (Fig. 3A and B). Additional restrictions imposed by the ground, articular geometry, and soft tissues will further limit limbs to a subset of these combinations, but complete surfaces are shown. Possible stance phase poses for the human limb (0.865 relative hip height) form a vertical ellipsoid that is longest in the metatarsal elevation angle dimension (Fig. 3A). In this constrained chain, the relatively long femur and tibia are confined to a more limited range of elevation angles than the relatively shorter metatarsus. The guineafowl surface (0.780 relative hip height) is more spherical, owing to more equally proportioned limb segments (Fig. 4B). Based on images of walking flamingos, we estimated a relative hip height of 0.9. Viable combinations of flamingo elevation angles form an ellipsoid that is longest along the femoral elevation angle axis (Fig. 3C). The relatively long tibial and metatarsal segments cannot deviate very far from vertical and still satisfy the hip height constraint. Like the short human foot, the flamingo’s small thigh segment is free to use a wider range of orientations. Realized human motion involves a sequence of poses near the bottom of the human ellipsoid (Fig. 3A), far below the flamingo
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Fig. 2. Human and guineafowl limbs use distinct combinations of elevation angles: (A) the orientation of each segment is measured as an elevation angle (yf, yt, and ym) with respect to vertical. Graphs of human (B) and guineafowl (C) elevation angles throughout the stance phase have some overlapping ranges, but clearly differ. (D) A 3-D configuration space of yf, yt, and ym elevation angles reveals very different pose trajectories (arrows) for these species. Ticks mark 101 increments from 901 to 901.
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Fig. 3. Interspecific comparison among actual and potential poses using a simple 2-D model of the stance phase: (A) assuming a fixed MP and constant hip height, permissible combinations of human elevation angles form a vertical ellipsoid in which the shortest segment (metatarsus) can rotate the most. The actual human sequence (filled circles) falls very close to this surface. (B) Actual guineafowl poses (filled cones) lie along a more spherical potential surface. (C) In contrast, the flamingo’s potential surface is longest in the femoral dimension. Neither human nor guineafowl poses come close, resulting in motion transfer artifacts. (D) With minimal intersections and no shared poses among these three taxa, potential configuration surfaces reveal why the null hypothesis of transferability is unsupported. Ticks mark 101 increments from 901 to 901.
potential surface (Fig. 3C). A flamingo limb cannot reach such flatfooted poses with its relatively long metatarsus without unnaturally reducing relative hip height. Likewise, poses used by the running guineafowl (Fig. 3C) fall well outside the flamingo ellipsoid. Although based on simplified models, these plots highlight how end point constraints interact with proportions to confine human, flamingo, and guineafowl limbs to very limited regions of configuration space during the stance phase. Poses are shared along limited surface intersections, but most options are species-specific (Fig. 3D). None is common to all three. Such mismatched surfaces signal failure of transferability, and thus a violation of the null hypothesis of similarity implicit in direct interspecific comparison of angles. Differing degree of crouch also precludes use of the same poses. Behavioral variation in cats provides a dramatic example (Trank et al., 1996). The limb exhibits smaller elevation angles during normal walking (Fig. 4A; ca. 0.81 relative hip height), than during crouched walking (Fig. 4B; ca. 0.48 relative hip height). Poses change similarly through elevation angle space during both
movements (Fig. 4C), but normal and crouched trajectories remain well separated. Plotting the potential poses each limb could achieve (Fig. 4D) reveals the geometric restrictions on angular combinations imposed by end point constraints alone. Large relative hip heights can only be produced by small elevation angles, thus confining the normal walking ellipsoid nearer the origin. A more crouched posture expands the configuration surface outward because segments can, and often must, achieve larger angles. These surfaces do not intersect. Proportionally identical limbs with dissimilar degrees of crouch can have one or even two angles in common, but are geometrically precluded from sharing all three angles simultaneously.
4. Discussion In this study we carried out simple, 2-D examples of interspecific motion transfer. When hind limb angles from a walking human and running guineafowl were applied to a flamingo limb,
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Fig. 4. Actual and potential poses in cats with different degree of crouch: (A) stick diagrams of normal (A) and crouched (B) walking (modified from Tranke et al. (1996). Elevation angle plots versus time (A and B) and versus one another (C) show similar patterns but distinct combinations. (D) The spindle-shaped potential configuration surface for crouching completely engulfs the ellipsoidal surface for normal walking, precluding shared angles. Ticks mark 101 increments from 1801 to 1801.
undesirable motion artifacts were introduced. Footskate, ground penetration, lifting, and extreme deviations in hip height all indicate failure of transferability. Thus, although it may always be theoretically possible for a flamingo to use the same angles as a human or guineafowl and vice versa, motion transfer exposes the kinematic consequences of accepting the null hypothesis. Nontransferability highlights several geometric limitations that undermine the direct comparison of angles among disparate taxa in an evolutionary context. 4.1. Angles are not independent of morphology, degree of crouch, or one another Human angles produce correct hip trajectories for humans, but are inappropriate for a limb with non-human proportions. The same holds true among species of birds. Following the research of Alexander and others (Alexander and Jayes, 1983; Hodgins and Pollard, 1997), animals with disparate proportions are not expected to walk or run in a dynamically similar fashion using identical angles. Our examples show that due to geometric constraints, it may not be energetically plausible for them to do so. Such results may appear so obvious as to be dismissed as trivial, but we contend that their implications for comparative motion analysis are significant. Angles may be dimensionless, but within a constrained system they are not independent of segment length. We consider the conservative motion of the CoM and hip in diverse species to be evidence of a proximal constraint. Likewise, stance phase stability of the end effector represents a distal constraint. When restricted at both ends, simple geometry dictates that angles must be matched to a limb’s proportions in order to satisfy basic spatial requirements. CoM, hip, and ground constraints pertain equally in 3-D. Angles will likewise be non-transferable among disparate ‘‘sprawling’’ tetrapods that exhibit more spatially complex limb motion. Even with fixed proportions, limbs with different relative hip height require different angular combinations (Trank et al., 1996; Grasso et al., 2000). Although it is reasonable to regard degree of crouch as a consequence of adopting specific sets of joint or
elevation angles, we find it fruitful to treat relative hip height as an independent variable that is both behaviorally and evolutionarily labile. Just as a crouching cat is prohibited from using normal walking poses, angles from a crouched ancestor are not directly transferable to a more erect descendant. Increases or decreases in degree of crouch along a lineage can, therefore, violate the null hypothesis and make angular comparisons among species biased toward difference. Our model of a planar, three-segment limb constrained at each end (Figs. 3 and 4D) accounts for the majority of angular change during the stance phase in our human, guineafowl, and cat case studies. In this simplified representation, a limb is restricted to a surface in elevation angle configuration space with only two degrees of freedom (DoF). The first DoF permits the hip joint to move forward and backward. The second DoF allows the limb to internally reconfigure among a family of poses for each hip position. A three-segment limb with fixed ends has redundancy, but angles are not free to change independently. Segments must reorient in a coordinated fashion. 4.2. Significance for comparative kinematic analysis How does non-transferability impact evolutionary functional morphology and comparative biomechanics? Simply put, the direct comparison of joint and elevation angles among species is not as straightforward as has been assumed. Kinematic data are often treated as independent from morphological data, but motion transfer artifacts highlight an intrinsic link between shape, posture, and movement. We contend that ground and CoM constraints during the stance phase are strong enough to impose significant limits on MP and hip position. Such limits compel the articulated limb chain to act more as a coordinated whole with reduced DoF than as three autonomous segments with independent rotational freedom. If interspecific differences in proportions and/or degree of crouch necessitate distinct limb configurations, any joint by joint or segment by segment assessment of angular data becomes an apples to oranges comparison that confounds elucidation of locomotor evolution. We are not aware of any zoologists attempting motion transfer to test transferability prior to comparison. Normally, motion
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Guineafowl run (source)
Bustard
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Jacana
Macaw
Allosaurus
Fig. 5. Interspecific angular motion transfer artifacts vary with disparity. Stance phase angles of a running guineafowl (A) are successfully transferred to a similarly proportioned guan (B). Hip paths become distorted for limbs of other birds (C–E) and the extinct theropod, Allosaurus (F).
analysis of multiple species yields angular variables, which are then compared statistically or more qualitatively. Yet the null hypothesis of potential kinematic similarity remains unsubstantiated. Functional morphologists do not try to drive the limb of one species with angles from another as done here, and so remain unaware of any inherent conflict. Researchers in computer animation and robotics are sensitive to this predicament because reusing motion can avoid having to animate a new character entirely from scratch, a traditional technique that is slow and costly. Gleicher (1998) called this the ‘‘motion retargeting problem’’ because the original motion must be edited or ‘‘retargeted’’ before it can be successfully transferred to a second character of dissimilar shape. Viewers are incredibly adept at perceiving very minor aberrations of an animated character’s stance foot. Even the seemingly simple transfer from one adult human to another typically requires motion retargeting to correct for individual differences and small errors. As in the human to flamingo transfer, reuse of an actor’s motion for an animal, alien, or cartoon character with dissimilar proportions is especially fraught. Clearly, motion artifacts will vary depending on the level of disparity. Minor morphological or postural differences will not yield dramatic motion transfer artifacts and in such cases the null hypothesis can be accepted. For example, guineafowl angles and toe tip translations (Fig. 5A) appear to be transferable to the limb of a Crested Guan (Penelope purpurascens; Fig. 5B), a species in the same order (Galliformes) with comparable proportions. If actual guans move using a similar relative hip height, then the ellipsoids for these two species would come close enough (within measurement error) to share poses and directly compare angles. But for birds such as a Madagascar Jacana (Actophilornis albinucha; Fig. 5C), Houbara Bustard (Chlamydotis undulata; Fig. 5D), or Scarlet Macaw (Ara macao; Fig. 5E), proportional differences cause transfer failure. If the limb motion of these species is quantified one day, we are compelled to reject the null hypothesis of angular similarity a priori and predict distinct patterns. The same holds for inferences about extinct taxa, such as the theropod dinosaur, Allosaurus (Fig. 5F; Gatesy et al., 2009).
Yet for many comparisons, the judgment of transferability will be less clear-cut. How much hip oscillation or toe slip can motion transfer inflict and still remain viable? Our goal here is not to establish strict boundaries to unambiguously define transferability. We leave such judgment to individual researchers, whom we encourage to transfer motion among their study species. Rather, we wish to raise awareness about angles and their geometric inter-relationship with proportions, degree of crouch, end point constraints, and segment interdependence. Before raw angular data are compared, it seems reasonable to pause and consider, given the constraints on the limb, if two or more species could use the same poses. Kinematic distinctions are most meaningful when limbs could use the same motion, yet do not. 4.3. Intersegmental coordination and a call for alternative parameterization Despite quantitative angular differences, many species likely share qualitatively similar limb movement. What underlying patterns might proportional and postural disparity be masking? We believe these deeper patterns fall under the term ‘‘coordination’’ (e.g., Bernstein, 1967), specifically intersegmental, interjoint, or intralimb coordination. A quantitative characterization of coordination would be of value for evolutionary analysis because such patterns need not be tied to specific morphologies. Theoretically, intersegmental coordination can pass unchanged from ancestor to descendant or acquire novelty like any other character, even as the limbs evolve different segment lengths, degree of crouch, size, or other features. Unfortunately, existing methods for visualizing and evaluating coordination are largely based on angles. Bivariate and trivariate plots of joint angles (so-called cyclograms, Grieve, 1968; Charteris et al., 1979) and elevation angles (e.g., Borghese et al., 1996) are not conducive to unbiased comparison. Constrained limbs may not have equal access to angular configuration space (Figs. 3 and 4), thereby precluding similarity. We hope that highlighting some shortcomings of angular comparison will motivate others to seek creative new methods to quantify limb movement. Specifically, we advocate for a search for novel parameterizations of intersegmental coordination that
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are either derived from angular data or are non-angular. An ideal format for comparative studies would allow ‘‘coordination transfer’’ among both closely and distantly related species. If successful, having the same coordination pattern (due to retention of the primitive condition or to homoplasy) could serve as a productive null hypothesis in ways that angular equality cannot.
5. Conclusion In this paper, we have tried to show that despite being well established and convenient, angles are not directly equivalent among disparate, constrained limbs. Yet from discussions with our colleagues (functional morphologists, biomechanists, animators, roboticists, paleontologists), it is clear that angles hold a central place in motion analysis. Indeed, the conceptual link between rotary motion and angular measurement can sometimes be so strong that the terms become one and the same—rotations are angles. But must angles be the best and only way to measure limb motion for comparative analysis? Our warning is not meant to discourage comparison of limb kinematics in an evolutionary context. On the contrary, this paper is motivated by a desire to see more such studies (albeit carried out with additional regard for geometric constraints). Although we have focused on terrestrial locomotion where ground and CoM restrictions are well-established, limbs acting as wings and flippers likely operate under endpoint constraints as well. A search for alternative motion parameterizations may lead to the discovery of viable null hypotheses of intersegmental coordination. Such methods may pave the way for discovering similarity amidst disparity and novelty within uniformity, thus lending a new clarity to tetrapod locomotor evolution.
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