Coupling length: a generalized gleno-acetabular distance measurement for interpreting the size and gait of quadrupedal trackmakers

Abstract

The gleno-acetabular distance D_GA, a conventional proxy for the size of a quadrupedal trackmaker, is often estimated as the distance GA between the midpoint between a left and right pair of pes tracks and the midpoint between a selected pair of left and right manus tracks. While frequently used to estimate trackmaker size from fossil trackways, the relationship between GA and D_GA depends upon the gait (which is unknown for extinct trackmakers), and is subject to multiple additional sources of uncertainty including which specific pair of manus tracks to associate with a given pair of pes tracks. Here a generalization is introduced, termed coupling length, which does not require any presumption about trackmaker gait of the degree of overstepping. On the contrary, a systematic analysis of a trackway in terms of coupling length can permit estimation of both the size and the gait with which the trackmaker progressed. Coupling length can be computed at successive points along a trackway, allowing exploration of a range of hypothetical gaits and body sizes for the trackmaker responsible. A fitness function quantifying persistent variation in coupling length along a trackway is used to indicate whether a given trackway could have been created by a fairly consistent gait, and if so, a range of high-fitness solution gaits and their associated D_GA. The method was applied to selected quasi-regular sauropod trackways and a solution found for a narrow range of gaits with limb phase of about 0.3 and D_GA = 1.6 ± 0.2 m. This is the first estimation of sauropod trackmaker gait, and introduces a novel method by which irregularity along a trackway is used as a source of information to constrain inferences of trackmaker behavior. The computed D_GA for this sauropod suggests significantly smaller trackmakers than conventional estimations based on track dimensions and hip height estimates. Size estimation by this approach offers greatly reduced uncertainty compared to conventional estimates.

Introduction

Summary

An extensive collection of sauropod trackways uncovered near Porrentruy in the Canton Jura (NW Switzerland) provided a rich dataset of thousands of well-preserved sauropod tracks, comprising about 260 trackways, many of which extended over hundreds of meters (Marty, 2008; Marty et al., 2003, 2004, 2010). A thorough statistical analysis was performed, which led to a novel probabilistic method whereby variations in nearly-regular trackway yielded inferences about the size and gait of the now-extinct trackmakers. The method is offered as a refinement of the conventional approach to gleno-acetabular distance estimation, but it is restricted to gaits with duty factors greater than 0.5, and presumes the trackmaker was engaged in a symmetrical gait. This approach was documented in the final report on these ichnoassemblages (Paratte et al., 2018; Stevens & Ernst, 2017); see also (Stevens et al., 2017), wherein we reported upon this statistically-independent method for estimating trackmaker size (gleno-acetabular length) that is in close agreement with the estimate provided by conventional heuristics, but with finer numerical precision. The method simultaneously solved for the sauropod’s gait and suggested a narrow range of limb phases spanning about 0.25–0.3 (i.e., from a lateral sequence singlefoot to a lateral sequence diagonal couplet) (Stevens et al., 2017). The method presumes the use of explicit estimates of track measurement uncertainty (Paratte et al., 2018; Stevens & Ernst, 2017). The uncertainty in localizing tracks along a trackway imposes practical limits on the precision with which gait can be inferred.

Conventional methods for estimating trackmaker size

As a quadruped traverses a substrate that preserves its passage, the two forelimbs form a succession of manus tracks which is soon followed by the pes tracks created by the two hindlimbs. A regular, constant gait results in a repeating pattern of interleaved pes and manus tracks, the spacing and arrangement of which depending upon the separation between the forelimb pair and the hindlimb pair, the length of the steps it takes, and the relative timing of the placement of the four limbs (i.e., the gait). An important goal of trackway interpretation, especially for fossil trackways, is the inference of those properties of the trackmaker from measurements of the trackway. In the absence of the (now extinct) trackmaker, the inferences are grounded variously by analogy with the proportions, kinematics, dynamics, and behaviors of extant trackmakers. A few of the most common inferences attempt to rather directly relate trackway measurements to corresponding trackmaker measurements, relying on assumed constants of proportionality, heuristics about the gait that creates a given track pattern, and other rules of thumb. The more assumptions required of a given method, the more uncertain are the predictions it provides.

There are two conventional approaches towards trackmaker estimation from fossil trackways: track-based and trackway-based. The track-based approach estimates trackmaker hip height H_A from measured pes track length PL (or pes width PW), based on a presumed linear relationship ratio H_A/PL (or H_A/PW). Inferred hip height is then used to estimate trackmaker size (e.g., gleno-acetabular distance D_GA) based on another presumed linear proportionality (D_GA/H_A) and estimated hip height is also used to estimate trackmaker speed based on other heuristics (e.g., Alexander, 1976; Lockley, 1991; Thulborn, 1990)—see Eq. 1, below. Estimated hip height, size, and speed are obviously subject to the uncertainty associated with those unknown proportionality constants, the validity of assuming linear proportionality in body dimensions (which disregards allometry and relies on assumptions about the ichnotaxon), and the measurement uncertainty. We consider the degree of uncertainty associated with such track-based assumptions in the discussion.

Alternatively, the conventional trackway-based approach estimates the separation between the pectoral and pelvic girdles by measuring their so-called “manus-pes” distance (e.g., Leonardi, 1987). In Fig. 1, a pair of manus tracks {LM2, RM2} is assumed to correspond to the locations of the forelimbs when the hindlimbs correspond to the pes tracks {LP1, RP1}. Each pair of tracks is connected by a line segment, the midpoint of which would indicate the center of the corresponding girdle. Those midpoints are then connected by a line segment GA. The length of GA (the “manus-pes distance”) would presumably correspond to D_GA for a trackmaker standing stationary in those four tracks, or in location provided the gait has an interval in which all four limbs are simultaneously in support phase. The gait, however, is unknown, and while some gaits such as a trot do have such an interval during the step cycle, many quadrupedal gaits have a maximum of only three limbs in support (Hildebrand, 1965, 1976). Alternative formulae have thus been proposed that add a heuristic correction factor such as “half the stride length plus the manus-pes distance”, or “about three-quarters the stride length plus the manus-pes distance”, “plus one or two stride lengths to the body lengths depending on the degree of overlap” (Demathieu, 1970; Farlow et al., 1989; Halfpenny, 1987; Leonardi, 1987; Padian & Olsen, 1984). Clearly a heuristic that adds as much as three-quarters of a stride length to account for the unknown gait, plus perhaps one or two integral stride lengths to account for the hypothesized degree of overlap, adds considerable uncertainty to the estimate.

Fig. 1 [Images not available. See PDF.]

In the conventional practice of quadrupedal trackmaker size estimation, a line segment is constructed between two pes tracks (LP1 and RP1, above) and another between two manus tracks (LM2 and RM2). The separation GA between the centers of these two line segments is often used as the basis for estimating the trackmaker’s gleno-acetabular distance D_GA, subject to a correction to account for the presumed gait (see text), and having selected the correct pair of manus tracks to associate with the given pair of pes tracks. Perhaps contrary to intuition, an exquisitely preserved and regular trackway such as BEB-500-S4 (lower), one of the trackways used in this study, could have been created by a quadruped of many possible sizes and many possible gaits

A source of greater uncertainty is the choice of manus tracks to associate with a given pair of pes tracks (Leonardi, 1987; Peabody, 1959). In Fig. 1, instead of choosing the manus pair {LM2, RM2} as diagrammed, the manus pair {LM1, RM1} were selected, GA would be implausibly short, but another choice possible pairs of manus tracks such as {LM3, RM3} cannot be ruled out a priori, and would suggest a body length a full stride length longer. Other sources of uncertainty can be identified, such the unknown degree of lateral flexion of the vertebral column during a step cycle, and differences in the amount of limb protraction versus retraction, as occurs in crouched or sprawling animals (Bonnan et al., 2016; Fischer et al., 2002; Irschick & Jayne, 1999). Finally, there is uncertainty in an estimate that arises from basic measurement error, such as locating track centers. While measurement error in this case is minor compared to other sources of uncertainty just mentioned, the compounding of uncertainty in nontrivial mathematical computations, so-called error propagation (Taylor, 1997), will be demonstrated to be a limiting factor in the current study, and has perhaps an under-appreciated importance to quantitative ichnology in general.

The current study provides an alternative to the conventional track- and trackway-based approaches. Our analysis, by design, makes only minimal assumptions about the trackmaker. It does not presume any heuristic coefficients to scale hip height or body length; it also does not presume the trackmaker was engaged in any particular gait, other than it being a symmetrical and without an aerial phase, as will be discussed further). It relies on selecting a sufficiently regular and complete segment of a trackway, and while we make no assumption about gait or body size, we presume that the trackmaker maintained an approximately constant gait and body length D_GA. We then search through a space of possible solutions for a schematic trackmaker to the given track position data and select that solution (or set of similar solutions) that minimizes the apparent variation in body length. As will be shown, however, even with regular trackways and excellent track preservation, error propagation that starts with an inevitable degree of trackway measurement uncertainty becomes one of the most significant factors that limits the estimation of body size and the inference of the gait that best accounts for the given trackway pattern.

While this study was devised specifically for sauropod dinosaur trackway interpretation, it is not narrowly restricted to sauropods, and while the methods we present have yet to be applied to extant quadrupeds for empirical validation, we are able to show that this novel method provides an independent means for the sort of trackmaker estimates that are often made based on a few static measurements of track dimensions and the separations between successive tracks.

Limb phase and duty factor

In a regular quadrupedal gait, the four limbs maintain a fixed relative timing of footfalls. We will refer to the left and right manus as M_L and M_R and the left and right pes as P_L and P_R, and we adopt the conventional formulation where a step cycle begins when the left pes just contacts the ground (Hildebrand, 1965, 1976, 1989). The start of the support for the other three limbs is thus measured relative to P_L = 0. The fraction of the step cycle that each limb remains in contact (or support) is the duty factor DF (or duty cycle). This study presumes DF > 0.5, hence at least three limbs are simultaneously in support at any time throughout the step cycle. Additionally, we presume the duty factor for the forelimbs and hindlimbs are identical. Moreover, in this study we are concerned with symmetrical regular gaits, i.e., gaits in which the contralateral limbs are out of phase by one half cycle, therefore the relative timing of the footfalls for all four limbs for a regular gait can be characterized by a single parameter, limb phase LP, the relative fraction (0 ≤ LP ≤ 1.0) of a complete step cycle of the footfall for M_L. The four gaits pace, walk, trot, and amble are separated by 0.25 in limb phase (see Fig. 2). The pace (LP = 0.0) corresponds to the left forelimb and hindlimb moving together in synchrony, and in counterphase relative to the right forelimb and hindlimb. In a trot (LP = 0.5) the left forelimb and the right hindlimb in synchrony and in counterphase with the right forelimb and left hindlimb. A walk (LP = 0.25), which is intermediate between a pace and a trot, is a ‘lateral sequence singlefoot’, so named as the next footfall after the left hind limb is on the same side, and only one foot touches down at a time (Hildebrand, 1965).

Fig. 2 [Images not available. See PDF.]

The step cycles for gaits G0–G7 are plotted for a duty factor DF = 0.5 (on left) and DF = 0.6 (right). Blue indicates the limb is in support, red indicates transition. For DF = 0.6 only G0 and G4 provide intervals with four limbs simultaneously in support

Between the pace and the walk is an intermediate gait (LP = 0.125), a ‘lateral-sequence lateral-couplet’ gait favored by dogs engaged in walking at a steady speed (Griffin et al., 2004). Likewise, between the walk and the trot is an intermediate gait (LP = 0.375), a ‘lateral-sequence diagonal-couplet’ that is sometimes called the ‘4-beat trot’, or ‘trot-like walk’ or even ‘dirty’ trot (Lammers & Biknevicius, 2004; Reilly & Biknevicius, 2003; Zips et al., 2001). To include these four intermediate gaits for a total of eight gaits, we use a simple numbering convention G0–G7 (see Table 1).

Table 1. Gaits are conventionally defined by the timing of the footfall pattern

Limb phase	Gait	Abbreviation	Sequence	Couplets
0.0	Pace	G0	Lateral	Lateral
0.125	‘Pacing Walk’	G1	Lateral	Lateral
0.25	Walk	G2	Lateral	Singlefoot
0.375	‘Walking Trot’	G3	Lateral	Diagonal
0.5	Trot	G4	Diagonal	Diagonal
0.625	‘Trotting Amble’	G5	Diagonal	Diagonal
0.75	Amble	G6	Diagonal	Singlefoot
0.875	‘Ambling Pace’	G7	Diagonal	Lateral

A step cycle can be regarded as starting with the left pes beginning the support phase. For symmetrical gaits, the left and right hindlimbs P_L and P_R are out of phase by 0.5, as are M_L relative to M_R. Limb phase refers to the fraction of the cycle at which the left manus M_L begins its support for the left pes P_L. Four additional intermediate gaits are included to provide a denser sampling of this continuum for our analysis

While there is universal understanding of ‘pace’, ‘walk’, and ‘trot’ (sensu Hildebrand, 1965), a ‘walking’ gait can also refer to any gait that uses inverted-pendulum (vaulting) mechanics as opposed to a ‘running’ gait that uses bounding (bouncing) mechanics (Biknevicius & Reilly, 2006; Cavagna et al., 1977). A naming quagmire has thus resulted, wherein “… some gaits are theoretically possible as both walks and runs (trots and singlefoots) whereas other gaits occur primarily as runs (pace) or as walks (all other symmetrical gaits)” (Biknevicius & Reilly, 2006). Since a trot (as traditionally defined by the footfall pattern) might also be either a walk or a run based on its whole-body dynamics, we will follow the restricted usage of these gait terms to only describe footfall patterns as suggested by Biknevicius and Reilly (2006). There is still a matter of how to refer to the four intermediate gaits (G1, G3, G5, and G7) in a way that reflects their footfall patterns. The G1 gait is referred to here as a ‘pacing walk’ as it is intermediate between a pace and a walk, and similarly G3 is a ‘walking trot’ as it is intermediate between a walk and a trot, and essentially a diagonal-couplet.

The literature also has some potential confusion in the use of the term ‘amble’. An amble can be either a diagonal sequence gait G6 (LP = 0.75) or a lateral sequence gait (LP = 0.25), the latter having some of the dynamics of a run, with as few as one foot in support for short duty factors (Gambaryan, 1974; Hutchinson et al., 2003, 2006; Schmitt et al., 2006). Unfortunately, the term amble has also been used as a postulated sauropod gait (Casanovas et al., 1997; González Riga & Tomaselli, 2019; Vila et al, 2013), but their “amble pace gait” and “amble walking” are apparently synonymous with the G0 pace.

We note that while given discrete names, behaving quadrupeds vary their limb phases by roughly ± 0.06 from stride to stride (Hildebrand, 1965, 1976, 1989). To explore possible sauropod trackmaker gaits, it is assumed that limb phase was at least 0.5, and therefore at least three limbs are in support during the step cycle. Hence we examine all combinations of eight discrete limb phases, three choices for overstepping (0, 1, 2), and three choices of duty factor (DF = 0.5, 0.6, 0.75). A given choice is reflected by the trial name, such as G2-1-60.

Track measurement uncertainty

The trackway data for the present study is from the massive, extraordinarily well-documented collection of Late Jurassic (Kimmeridgian) dinosaur ichnoassemblages near Porrentruy in the Canton Jura (NW Switzerland) (Marty, 2008; Marty et al., 2003, 2004, 2010; Paratte et al., 2018). During the excavation of Highway A16, over 14,000 tracks of both sauropod and tridactyl dinosaurs were discovered, distributed across 50 ichnoassemblages, and comprising about 230 tridactyl trackways and 260 sauropod trackways. Some paleosurfaces covered more than 4000 m² in area, the largest of which containing nearly uninterrupted trackways that extended for hundreds of meters.

Figure 3 shows an orthophotograph of the Courtedoux-Béchat Bovais tracksite (Early/Late Kimmeridgian Reuchenette Formation) of the Canton Jura, NW Switzerland (Marty et al., 2007). Note that some of the sauropod trackways show both manus and pes tracks while others are pes-only. Given that a manus track may either be missing or overprinted, this study selected only trackways containing both manus and pes tracks, to avoid that source of spatial uncertainty. Three trackway segments of the 500-level of this tracksite were selected, each approximately straight, with a regular pattern, and devoid of missing or overprinted tracks, and of sufficient length to analyze multiple complete step cycles: BEB-500-S1, BEB-500-S3, and BEB-500-S4, referred to here as S1, S3, and S4.

Fig. 3 [Images not available. See PDF.]

An orthophotograph of a small portion of the BEB-515 level of the Courtedoux-Béchat Bovais tracksite. Note that multiple trackways were identified with differing trackway patterns (some have distinct manus and pes tracks, and others have only pes tracks). This image covers an area of 20 m by 10 m

For the current study, a trackway is represented by the locations of its tracks. A physical track is a depression of the substrate that is often—but not invariably—surrounded by a raised displacement rim, and within which are sometimes preserved impressions of digital and metapodial pads and unguals Since the outline of a track is often difficult to distinguish from the surrounding substrate, and frequently is partly obliterated by another track, the preferred reference point marking the location of a track is the track ‘center’ (Leonardi, 1987; Lockley, 1991; Thulborn, 1990). A robust means to define the track center is provided by the following geometric construction (Marty, 2008). The sauropod tracks (both manus and pes) in this assemblage have displacement rims that are generally elliptical and approximately bilaterally symmetrical: a line segment can be constructed to represent the axis of symmetry which roughly bisects the track. A second line, perpendicular to the first, spans the track at its greatest width, and their intersection marks the track center. This geometric construction can be applied to both sauropod manus and pes tracks, especially those that are closed and isolated. Measurement uncertainty is greater for shallow tracks with an incompletely-closed displacement rim. The location of the center of each track was assigned an uncertainty from 1 cm (for tracks with the most symmetrical and sharply-defined displacement rims) to 5 cm (or more, in the case of the poorly-preserved tracks) to provide an estimation of the precision of each center; see measurement methods and track statistics in (Paratte et al., 2018; Stevens & Ernst, 2017).

While the dimensions of the displacement rim are greater than those of the trackmaker’s foot, and its roughly elliptical shape is only a rough indicator of the foot morphology, the ‘center’ of the displacement rim permits a reasonably repeatable measurement of track location (Fig. 4). Pooling the 4207 sauropod pes tracks in the assemblages studied, the uncertainty-weighted median ratio PL/PW = 1.3 ± 0.13, matching the ratio 1.31 measured for a well-preserved natural cast of a sauropod pes (Platt & Hasiotis, 2006), where the length included the claw marks. The median measurement uncertainty in PL and PW was about 2 cm. A track is therefore represented as a mathematical point with associated uncertainties (x ± dx, y ± dy) that reflect confidence in that location. Even a modest spatial uncertainty of a few percent, however, can become a significant limiting factor in the reliability of a computation due to error propagation.

Fig. 4 [Images not available. See PDF.]

Orthophotographs of well-preserved sauropod tracks from the BEB-500-S1 trackway of the Courtedoux-Béchat Bovais tracksite (Marty et al., 2007). The displacement rims (solid line) are roughly elliptical and the manus tracks are partly overprinted. Within the roughly elliptical displacement rim, the actual outline of the foot is often, but not invariably, apparent (outlined by dashed lines). The displacement rim is more reliably present and therefore used as the basis for locating track ‘centers’. The red lines are separated by 50 cm

Method

In a precursor to the current study (Stevens et al., 2016), a synthetic trackway generator showed that estimating gait (LP and DF) from a regular trackway pattern depends upon trackmaker size D_GA and yet trackmaker size cannot be sufficiently-precisely estimated from conventional track-based heuristics to disambiguate gait (see the Discussion section regarding the estimation of the uncertainty associated with those heuristics). It would appear a stalemate, for even if pes length were measured precisely, that precision would be overshadowed by the uncertainty in the proportionality constants used to estimate trackmaker size. Perhaps unintuitively, a source of trackway irregularity, which usually just adds statistical uncertainty to a trackway measurement (median stride length) might be used as a source of gait information to break this stalemate (Stevens et al., 2017).

Rather than consider a perfectly regular trackway, we examine a segment of an actual, hence slightly-irregular, trackway. An individual track (such as LP1 in Fig. 1) is represented by a mathematical point indicating its center, with an uncertainty associated with each coordinate. A trackway is represented by four ordered sequences of tracks (e.g., LP1, LP2, etc., for the left pes tracks in Fig. 1). The corresponding trackmaker is represented by the manus locators M_L and M_R and the pes locators P_L and P_R. Each locator is a mathematical point that represents the location of limb distal point of contact and which alternates between two states: support while that limb contributes to supporting the body, and transition as the limb takes the next step, according to a specified gait. When a locator is in support, its coordinates (and spatial uncertainty) correspond to that of the specific track it stands upon. When in transition, its coordinates are interpolated towards the next track, and assigned greater uncertainty during protraction. The trackmaker is simply a set of four independently moving locators, each given a specified sequence of tracks to step upon, and a prescribed timing and duration with which to do so.

A trial starts for a specified gait (LP, DF) at t = 0.0 with P_L having just stepped onto LP1; it remains stationary until t = DF, whereupon it proceeds towards LP2, arriving at t = 1.0 (one step cycle later). The left manus M_L can similarly be modeled according to those gait parameters, stepping from one manus track to the next according to its assigned track sequence. M_L begins support at t = DF in the first cycle, and, depending upon the (unknown) degree of overstepping, it might initially step upon LM1, LM2, or even LM3 (Fig. 1). The trial G2-1-60 would correspond to M_L upon LM2. These alternatives are examined in different trials rather than make any assumption about the degree of overstepping. For symmetrical gaits, the two hindlimbs are in counterphase, as are the two forelimbs, therefore P_R and M_R can be assigned their initial tracks and timing for t = 0.0.

This pairing between the two hindlimbs is represented by a pes coupler γ_pes (a line segment connecting P_L and P_R, the midpoint of which approximates the location of the acetabulum), and likewise a manus coupler γ_manus represents the coupling between M_L and M_R, and locates the pectoral girdle). The separation between locators γ_pes and γ_manus is termed the coupling length CL (Fig. 5) and is a generalization of the conventional computation of GA (Fig. 1). The derivation of CL is provided in Appendix 1 (see Eq. 4).

Fig. 5 [Images not available. See PDF.]

A diagrammatic representation of a segment of a quadrupedal trackway as it is being traversed by a schematic trackmaker governed by a specific choice of gait parameters for this trial. The track locations are from the BEB-500 S1 trackway; the direction of travel was from left to right (closed shapes indicate the tracks, with pes tracks in blue and orange, and manus tracks in green and mauve). In this trial the gait is a walk (LP = 0.25, DF = 0.5). The trackmaker is indicated by the four locators (with open symbols, a blue ring for left pes, etc.). At the moment depicted, P_R, P_L, and M_R are in support and standing on tracks, and M_L is in transition. Line segments represent the manus and pes couplers, and the line segment CL that connected their midpoints represents the distance between the two couplers. The gray boxes represent the spatial uncertainty of each coupler

The position of each locator can be sampled at two moments within the step cycle: when it just reaches its next track, and when it just lifts from that track. During its interval of support, that locator has the spatial certainty of the track on which it stands. When that locator is in transition towards its next track its location is interpolated between its previous and next track with an uncertainty corresponding to the stride length (it could be anywhere between those tracks). Each coupler therefore alternates between having low spatial uncertainty when both locators are in support, and higher uncertainty otherwise. Coupling length, being the instantaneous distance between the two couplers, thus has variable uncertainty within each step cycle along a given trackway.

Even the most regular of trackways show subtle shifts in the placement of manus tracks relative to pes tracks—irregularities that reflect some unknown behavioral event such as a slight hesitation, change of speed, or reaction to the movements of other trackmakers. While the instantaneous separation between the girdles naturally varies during a step cycle (due to axial skeleton flexion, pectoral girdle mobility, etc.), it can be regarded as some (unknown) constant for the duration of the trackmaker’s progression down a trackway. If a regular trackway shows a perturbation (e.g., shorter strides for a few cycles) that irregularity is attributed to a momentary change in gait, not a change in gleno-acetabular length. If we then monitor the instantaneous coupling length for a trackmaker for a given hypothesized gait (e.g., G2-1-60), the trackway perturbation will be reflected in a corresponding perturbation in CL. Depending on the choice of gait, that variation might be “taken in stride” (with little variation in computed coupling length CL) or, for another choice of gait would require (in effect) a sudden acceleration or deceleration of one or more limbs to adjust the stride to accommodate the track positions in the vicinity of that irregularity, which is reflected in a sudden change in CL. By quantifying the magnitude of the perturbation in CL, we can compare the fitness of various hypothesized gaits for the given trackway.

The range of computed CL across the space of possible gaits is considerable, providing a basis for comparing alternatives despite the limitations posed by measurement uncertainty. Figure 6 illustrates alternative CL computations for an idealized trackway given different combinations of two duty factors (DF = 0.50, left column, and DF = 0.75, right column) and the eight gaits defined in Table 1.

Fig. 6 [Images not available. See PDF.]

Coupling length computations for the hypothetical regular trackway, for two choices of duty factor (DF = 0.5 on the left and 0.75 on the right) and the eight gaits (G0–G7). Each trial depicts the same point in a step cycle at which both hind limbs and the right forelimb are in support and the left pes is about to start transition. For the high duty factor trials (right column), note that G0, G1, and G2 are consistent with the same CL (and likewise G4, G5, and G6 would be consistent with a constant, but shorter, CL)

The range of possible CL interpretations increases considerably when one then considers alternative pairs of manus tracks to associate with a given pair of pes tracks. For the lateral sequence gaits (G0–G3) in Fig. 6 the M_L is placed upon the second manus track ahead of the pes track on which P_L is standing. This is called ‘tertiary overstep’ (Peabody, 1959). We use a simple ‘track separation’ (of 0, 1, or 2) to indicate this choice of manus track, i.e., the number of manus tracks between P_L and M_L. For example, in the top row of Fig. 6, the gait is a G0 pace, and the track separation is 1, so the gait is termed G0-1, to indicate one intervening manus track, as distinct from the (improbably) short-coupled G0-0, or the longer G0-2. To explore the space of possible computations of CL for various hypothetical combinations of track separation, limb phase and duty factor, these variables were combined into a single experimental variable, α-phase. Coupling length CL could then be computed along a given real trackway, with two computations per step cycle, for any specific choice of α-phase.

A graph of CL along the trackway results in a curve that fluctuates about a median value. The median value corresponds to the gleno-acetabular distance for a trackmaker proceeding with that combination of limb phase and duty factor on that trackway. To compare CL across different hypothetical gaits for the same trackway, a fitness function was designed to quantify the local deviations of CL from the median value. The derivation of the fitness function is provided in Appendix 2 (see Eq. 20).

Results

Figure 7 shows the trackway segment S1 from track LP11 to track LP22. Given that coupling length can be measured twice per step cycle, we could gather 20 values of CL along this trackway segment. While at first glance S1 appears quite regular, note that stride length gradually decreases towards LP18 then increases from that point on.

Fig. 7 [Images not available. See PDF.]

A segment of trackway BEB-500-S1 of the Courtedoux-Béchat Bovais tracksite (Marty et al., 2007), selected for coupling length computations for various combinations of gait and duty factor. While substantially straight and regular, the trackway shows gradual variations in stride length. All tracks are very well preserved, with location uncertainties of only a few cm. The top row shows the S1 trackway segment for this study

The present method was first developed during the analysis of the S1 trackway, then it was applied to the S3 and S4 trackways. Coupling length CL was computed along the length of trackway S1 for each combination of the eight gaits (G0–G7), the three track separations (0, 1, or 2), and the three duty factors (DF = 0.5, 0.6, and 0.75). For example, for a G4 trot, we examined all combinations of three track separations and three duty factors (G4-0-50, G4-0-60, G4-0-75, G4-1-50, etc.). We then excluded those combinations that would have placed the pes ahead of the manus at some point in the step cycle, resulting in a total of 40 trials.

The S1 trackway consists of 10 complete step cycles, sampled twice per cycle, resulting in 20 coupling length measurements per trial. We accumulated 40 such sets of CL measurements, and needed a way to compare them. We focussed upon the amount CL varied from cycle to cycle along the trackway in response to the slight irregularity in the trackway pattern. A fitness function was created to ‘reward’ or provide a better fitness score to trials in which CL showed less extreme relative fluctuation along the trackway (see Eq. 20, Appendix 2).

Figure 8 shows the resulting fitness functions values (ordinate) sorted by coupling length (abscissa). Note that the solution fitness is best in the vicinity of a coupling length of about 1.5 m, corresponding to a short-coupled G3-0 walking trot (for three duty factors 0.5, 0.6, and 0.75) and a G2-1 walk (for duty factors 0.5, and 0.6). These five trials were selected to consider in greater detail.

Fig. 8 [Images not available. See PDF.]

Solution fitness

The five selected solutions pertain to only two close-related gaits: the G2-1 walk and the short-coupled G3-0 walking trot, which vary by only 0.125 in limb phase LP (from 0.25 to 0.375). The five solutions also represent a range of hypothesized duty factors from 0.5 to 0.75. As shown in Table 2, these combinations result in overlapping solutions that span only a narrow range of median CL from 1.37 to 1.67 ± 0.07 m, with a best fit length of 1.6 ± 0.2.

Table 2. Resultant coupling length values (with uncertainties) for the highest-fitness solutions for BEB-500 S1

Trial	Minimum CL (m)	Median CL (m)	Maximum CL (m)
G3-0-50	1.22 ± 0.08	1.37 ± 0.07	1.49 ± 0.08
G3-0-60	1.26 ± 0.08	1.42 ± 0.07	1.53 ± 0.08
G2-1-50	1.4 ± 0.08	1.57 ± 0.07	1.68 ± 0.08
G3-0-75	1.4 ± 0.08	1.57 ± 0.07	1.68 ± 0.08
G2-1-60	1.49 ± 0.08	1.67 ± 0.07	1.78 ± 0.08

The two closely-related gaits, the G2-1 walk and the G3-0 ‘walking trot’, have similar predictions for coupling length CL for a range of duty factors. The minimum and maximum values refer to the extremes of CL for each trial, straddling the median

To appreciate the subtle differences in these solutions, Fig. 9 shows the placement of the four locators in the case of the G3-0-50 walking trot versus the G2-1-50 walk for the same duty factor (DF = 0.5). The solutions are diagrammed for the midpoint of the trackway segment, see Fig. 7), where a minimum in CL was observed (see Appendix 2, Fig. 17). The computation of CL along the trackway is shown in Fig. 10 for the five highest-fitness trials.

Fig. 9 [Images not available. See PDF.]

The G3-0–50 ‘walking trot’ (upper) and the G2-1-50 walk (lower). Note the subtle difference in the position of the forelimbs and the roughly 20 cm difference in inferred trackmaker CL

Fig. 10 [Images not available. See PDF.]

Coupling length computations for the five trials showing the best solution fitness (Table 2). These correspond to a small range of gaits (G2 walk and G3 walking trot) and duty factors

In addition to trackway S1, two additional trackways were analyzed: BEB-500-S3 and BEB-500-S4 of the same 500-level Courtedoux-Béchat Bovais tracksite. They are referred to here as the S3 and S4 trackways. The three trackways, S1, S3, and S4, had very similar track morphology and dimensions, but were following separate paths. S1 and S3 were slightly convergent straight paths about 20 m apart; S1 and S4 were straight, parallel, and separated by about 70 m.

The trackway pattern in S3 (Fig. 11), appears very similar to that of the S1 trackway (Fig. 7). The variation along the trackway is difficult to discern visually, but nonetheless, leads to a very distinctive signature in the fitness distribution (Fig. 11). As in the analysis of the S1 trackway, S3 revealed a cluster of high-fitness solutions around a coupling length of 1.6 ± 0.2. In fact, the distributions in Fig. 8 and Fig. 11 are statistically indistinguishable. The trackmakers appeared to have been of very similar size and engaged in basically the same gait.

Fig. 11 [Images not available. See PDF.]

The S3 sauropod trackway, a contiguous segment of BEB-500-S3 of the Courtedoux-Béchat Bovais tracksite (Marty et al., 2007). The results of the best-fit solutions are statistically indistinguishable from those of the S1 trackway (Fig. 8), suggesting the two trackmakers had very similar size and gait

Next, we analyzed the extraordinarily regular S4 segment (of BEB-500-S4 of the same 500-level Courtedoux-Béchat Bovais tracksite). Figure 12 shows the trackway pattern and the fitness results. Note the absence of a cluster of high-fitness solutions. The S4 segment was apparently too regular, and any ‘signal’ to indicate the trackmaker’s gait was masked by measurement uncertainty. The gradual increase in fitness with CL reflects a residual scaling effect (see Appendix 2).

Fig. 12 [Images not available. See PDF.]

The S4 trackway, a contiguous segment of BEB-500-S4, presented insufficient irregularity to provide information about the gait or size of the trackmaker based on coupling length

Discussion

High-fitness solutions

Evaluating the fitness function across the combinations of gait and duty factor applied to BEB-500 S1 and S3, it was found that those combinations corresponding to a median gleno-acetabular distance of about 1.5 m show markedly less persistent deviation from the median (hence they represent better solutions) than those that produced by either shorter or longer-coupled trackmakers. In other words, if the trackmaker had a D_GA of about 1.5 m and proceeded with a gait with limb phase between 0.25 and 0.375 (i.e., a G2 or G3 gait) with relatively relaxed duty factors of between 0.5 and 0.75, that it could have created the variations along the trackway in S1 or S3 with the least variation in its gait. The fitness function showed which combinations of gait parameters resulted in the least variation in apparent axial length from hip to shoulder. Stated another way, presuming that the trackmaker maintained a constant body length, if it were to have walked with any other gait, it would have had to shift those gait parameters much more while only producing the subtle variation we observe along the S1 or S3 trackway. The failure to detect a high fitness solution for S4 demonstrates that the method is not robust. This method, without further constraints applied, only admits a resolution of CL under ‘Goldilocks’ conditions (i.e., trackways that are regular, but not too regular, relative to the spatial uncertainty created by measurement error).

For both S1 and S3 the best fit solutions were closely clustered, with three G3-0 trials (with DF = 0.5, 0.6, and 0.75), and two G2-1 trials (for DF = 0.5 and 0.6). Bearing in mind that the approach is probabilistic, these alternative solutions correspond to nearly the same footfall pattern and imply a trackmaker of nearly the same length for each best fit solution (Fig. 9). This analysis cannot resolve limb phase more precisely than to bracket it between a lateral-sequence singlefoot (walk) and a lateral-sequence diagonal-sequence gait (a ‘walking trot’ as it has characteristics of both a walk and a trot). The high-fitness S1 and S3 solutions have an honorable mention: the G1-1-50 (‘walking pace’), which corresponded to a slightly longer D_GA of about 1.8 m. It is noteworthy that all the high-fitness solutions are in the range of G1–G3. Given the limited precision by which limb phase can be resolved using this method, it is helpful to consider what the analysis rules out, namely the G4 trot and any other diagonal-sequence gait (G4–G7). It also indirectly rules out any solutions for trackmaker length other than those within a narrow band of D_GA. There was also no support for the suggestion that sauropods engaged in a G0 pace, contra (Casanovas et al., 1997; González Riga & Tomaselli, 2019; Vila et al., 2013).

Diagonal couplet gaits (G3 through G5) are regarded as primitive for tetrapods, while lateral couplet gaits (G1 through G2) may be a “true mammalian innovation” (Wimberly et al., 2021). Sauropods, on that basis, might have been expected to have used a G4 trot, but the elephant, being the largest extant terrestrial vertebrate, has been a more frequent model for sauropod locomotion, and so sauropods are usually depicted with a lumbering walk. Elephants always use lateral sequence gaits that vary continuously from a lateral couplet (G1 with LP about 0.2) at slower speeds to a singlefoot walk (G2 with LP = 0.25) (Genin et al., 2010; Hutchinson et al., 2006). The result of our analysis suggests the sauropod trackmakers for the S1 and S3 trackways were engaged in lateral sequence gaits. Specifically, the set of highest likelihood solutions consisted of one G1, two G2, and three G3 gaits. Duty factor in some solutions was a brisk 0.5, and in one solution, a sauntering 0.75.

Gait naming conventions tend to suggest sharp categorical distinctions (e.g., between lateral couplet and singlefoot, or between singlefoot and diagonal couplet), slight variations in limb phase in the vicinity of a walk produce very similar trackway patterns, and very similar footfall patterns in the trackmaker. Whether a sauropod was engaged in a “true” G2 singlefoot or a G3 diagonal couplet cannot be resolved by our analysis, unfortunately. What can be offered, however, is that the high-fitness solution set includes several potential gaits that are regarded as more mammalian than archosaurian.

Estimation of trackmaker size

Track-based estimation of body size proceeds in two heuristic steps. First, hip height H_A is estimated by multiplying measured pes track length PL by an assumed proportionality constant H_A/PL. Secondly, body length D_GA is estimated by multiplying the hip height estimate by another presumed linear proportionality D_GA/H_A:

1

$$D_{\mathit{GA}}=\frac{H_A}{\mathit{PL}}\times \frac{D_{\mathit{GA}}}{H_A}\times PL$$

Each term has associated uncertainty. Proposals for the first constant, H_A/PL, range from as low as 2.8 (Tschopp et al., 2015) to as high as 5.9 (Thulborn, 1990), with most estimates around 4 (Alexander, 1976; González Riga, 2011; Ishigaki, 1988; Vila et al., 2013). This large range reflects uncertainty in the contribution of soft tissues, the degree of digidigrady in the pes, and taxonomic and ontogenetic variations (e.g., Gallup, 1989; Bonnan, 2005; Wilhite, 2005; Schwarz, Wings, et al., 2007a; Schwarz, Ikejiri, et al., 2007b; Bonnan et al., 2010; Holliday et al., 2010; González Riga, 2011). Since neither the taxonomy nor the maturity of the trackmakers responsible for the trackways in this study is known, the heuristic H_A/PL is roughly 3.3 ± 0.5 (i.e., an uncertainty of about 15%).

The second proportionality constant, D_GA/H_A, also varies considerably across sauropod taxa, ontogeny, and the skeletal reconstruction. Proposals for this constant based on earlier skeletal reconstructions vary from 0.92 to 1.2 (González Riga, 2011; González Riga & Tomaselli, 2019; Mazzetta & Blanco, 2001; Vila et al., 2013), but those reconstructions have evolved over the last century, with more recent reconstructions corresponding to D_GA/H_A of 1.0 or less, especially in subadults and juveniles due to their relatively longer limbs (Lovelace et al., 2007; Schwarz, Ikejiri, et al., 2007b; Stevens, 2013; Stevens et al., 2016; Woodruff & Foster, 2017; Woodruff et al., 2018). Here we assume D_GA/H_A = 1.0 ± 0.1. Given that the median pes track length PL along the S1 trackway was 0.48 ± 0.08 m (i.e., an uncertainty of about 17%), the estimate for trackmaker size, incorporating error propagation in Eq. 1 is:

2

$$D_{\mathit{GA}}=\frac{H_A}{\mathit{PL}}\times \frac{D_{\mathit{GA}}}{H_A}\times PL=3.3\times 1.0\times 0.48=1.6\pm 0.4\text{m}(\text{i}.\text{e}.,\sim 25\%\text{uncertainty})$$

Of course, the uncertainty would be less for a known trackmaker of known maturity, but for the sauropod trackways such as those in the assemblages studied here both those factors are unknown. The best fit trackmaker length D_GA based on coupling length was 1.6 ± 0.2 m. Given the very different sources of information on which the two measurements are based, it is noteworthy that the estimates are so similar. The estimate of D_GA based on coupling length is thus about twice as precise.

Conclusions

It is computationally difficult to rigorously infer the size and gait pattern of a trackmaker from a trackway pattern. The extraordinary preservation of the tracks on the BEB-500 level provided numerous trackway segments that were promising but for the absence of a manus print or two. Perhaps those tracks were overprinted, or perhaps they were simply not preserved. Given the positional uncertainty with which even the best-preserved tracks could be located, we were limited to only those trackway segments that were complete in providing every manus and pes track along a contiguous length of the trackway. Of those, we were further limited to only those segments where the trackway indicated by its regularity a constant forward progression of steps, i.e., a constant gait. Two of these, trackways S1 and S3, were indeed useful and yielded results, but the especially regular trackway S4 reminded us that a regular trackway is fundamentally ambiguous (Stevens et al., 2016), i.e., any repeating pattern of tracks could have been produced by some corresponding combination of trackmaker size and gait parameters.

Our probabilistic approach incorporated the conventional numerical method of explicit error propagation, which allowed quantifying the confidence with which a signal could be resolved distinctly from the noise. It struck us as noteworthy that trackway irregularity, usually a source of obfuscation when attempting to derive insights by statistics, actually carries a useful signal to be interpreted. The strategy described here was the first use of track irregularities to resolve gait ambiguity and permit trackmaker size estimation. It was originally presented in 2017 (Stevens et al., 2017), and the coupling length method was described in the final report summarizing our work with the work on the sauropod trackway assemblages in the Canton Jura (Paratte et al., 2018; Stevens & Ernst, 2017). We concluded at the time that the sauropod trackmakers were engaged in a lateral sequence gait with limb phase corresponding to somewhere between a singlefoot (G2 walk) and a diagonal couplet (G3), and as the method was founded upon a careful use of explicit error estimation, the estimate of D_GA provided by our method is consistent with, but offers greater precision than, the track-based methods which rely on heuristic coefficients of proportionality. The small absolute size of these sauropods, with a D_GA of roughly 1.6 m, was somewhat contrary to the usual expectation for gigantism in sauropods.

The present method was developed to provide some traction to the estimation of trackmaker size and gait in the case of extinct, graviportal, quadrupedal dinosaurs. Being graviportal, we assume these trackmakers were not engaging in running gaits, but that they used a symmetrical gait. Instead of assuming a specific gait (as a requisite for estimating trackmaker size from GA), the coupling length computation represents a trackway-based measurement that generalizes over gait. The variation in CL along a trackway provides a means to explore the goodness of fit of various hypothetical choices of trackmaker gait. Our method also provides independent support for recent suggestions that H_A/PL is less than often proposed, and perhaps about 3.3 ± 0.7.

Clearly, there is much more to be done to study the robustness of this method. It is fragile, and it has not yet been tested against the trackways created by extant quadrupeds, for which both gait and size are known. On the other hand, this method attempts to infer the most that can be derived with the least assumptions one may make, namely that trackmaker had a relatively fixed body length and was engaged in a specific gait that could take in stride the slight variations observed in those trackways.

We emphasize again that the method does not model any kinematic chain (e.g., from manus locator to manus coupler to pes coupler to pes locator). This was intentional, for one can always create an increasingly specific model provided one wishes to introduce increasingly specific assumptions (for example, that the trackmaker was of one or another specific sauropod taxon).

Acknowledgements

We gratefully acknowledge the funding by the Swiss Federal Roads Office and Canton Jura, and the Office de la culture—Paléontologie A16. Excavations and scientific documentation of Highway A16 dinosaur tracksites and related research by the Paléontologie A16 (Section d’Archéologie et Paléontologie, Office de la culture) were funded by the Swiss Federal Roads Office (FEDRO, 95%) and the Canton Jura (5%), and this important funding is acknowledged very much indeed. We thank all technicians, photographers, geometers, drawers, collection managers, and preparators involved during the excavation and documentation of the tracksites. We greatly appreciate the suggestions of the two anonymous reviewers.

Author contributions

DM and the Office de la culture documented the extensive trackway assemblages of the A16 Highway project by site maps, spreadsheets, and photographs. SE developed the framework that led from those documents to a database from which selected trackway segments could be analyzed in this study. SE developed the theoretical and probabilistic analyses of the Coupling Length model. KAS developed 3D graphics software to interactively populate this database with track measurements (plus quantified uncertainty estimates), mapping over 6000 sauropod tracks from the site maps and photographs. The three authors collaborated equally to refine the process of track and trackway measurement uncertainty, the selection of trackway segments for analysis, and confirmation of measurements and uncertainties with reference to the original material. KAS and SE shared in the writing and illustrating the paper. All authors read and approved the final manuscript.

Funding

This research was supported by the Swiss Federal Roads Office and Canton Jura, and the Office de la culture—Paléontologie A16. Excavations and scientific documentation of Highway A16 dinosaur tracksites and related research by the Paléontologie A16 (Section d’Archéologie et Paléontologie, Office de la culture) were funded by the Swiss Federal Roads Office (FEDRO, 95%) and the Canton Jura (5%).

Availability of data and materials

The trackway data is available in its entirety in the Final Report “Statistical analyses of Highway A16 dinosaur trackways” by Kent A. Stevens and Scott Ernst, Office de la culture, Porrentruy, Canton Jura, CH. The datasets used for the current study are available from the corresponding author.

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare that they have competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

Alexander, RM. Estimates of speeds of dinosaurs. Nature; 1976; 261, pp. 129-130. [DOI: https://dx.doi.org/10.1038/261129a0]

Belvedere, M. Ichnological researches on the Upper Jurassic dinosaur tracks in the Iouaridène area (Demnat, Central High-Atlas, Morocco) Ph.D. Thesis; 2008; Università degli Studi di Padova: 121.

Biewener, AA. Allometry of quadrupedal locomotion: The scaling of duty factor, limb curvature and limb orientation to body size. Journal of Experimental Biology; 1983; 105, pp. 147-171. [DOI: https://dx.doi.org/10.1242/jeb.105.1.147]

Biknevicius, AR; Reilly, SM. Correlation of symmetrical gaits and whole body mechanics: Debunking myths in locomotor biodynamics. Journal of Experimental Zoology; 2006; 305A, pp. 923-934. [DOI: https://dx.doi.org/10.1002/jez.a.332]

Bonnan, MF. Carpenter, K; Tidwell, V. Pes anatomy in sauropod dinosaurs: Implications for functional morphology, evolution, and phylogeny. Thunder-lizards: The sauropodomorph dinosaurs; 2005; Indiana University Press: pp. 346-380.

Bonnan, MF; Sandrik, JL; Nishiwaki, T; Wilhite, DR; Elsey, RM; Vittore, C. Calcified cartilage shape in archosaur long bones reflects overlying joint shape in stress-bearing elements: Implications for nonavian dinosaur locomotion. The Anatomical Record: Advances in Integrative Anatomy and Evolutionary Biology; 2010; 293, pp. 2044-2055. [DOI: https://dx.doi.org/10.1002/ar.21266]

Bonnan, MF; Shulman, J; Varadharajan, R; Gilbert, C; Wilkes, M; Horner, A; Brainerd, E. Forelimb kinematics of rats using XROMM, with implications for small eutherians and their fossil relatives. PLoS ONE; 2016; 11, 3 e0149377. [DOI: https://dx.doi.org/10.1371/journal.pone.0149377]

Casanovas, M; Fernández, A; Pérez-Lorente, F; Santafé, JV. Sauropod trackways from site El Sobaquillo (Munilla, La Rioja, Spain) indicate amble walking. Ichnos: an International Journal of Plant & Animal; 1997; 5, 2 pp. 101-107. [DOI: https://dx.doi.org/10.1080/10420949709386409]

Cavagna, GA; Heglund, NC; Taylor, CR. Mechanical work in terrestrial locomotion: Two basic mechanisms for minimizing energy expenditure. American Journal of Physiology; 1977; 233, pp. 243-261.

Demathieu, GR. Les empreintes de pas de vertébrés du Trias de la bordure nord-est du Massif Central; 1970; Cahiers de Paléontologie, Éditions du Centre National de la Recherche Scientifique: 211.

Farlow, JO; Pittman, JG; Hawthorne, JM. Gillette, DD; Lockley, MG. Brontopodus birdi, lower cretaceous sauropod footprints from the U.S. Gulf Coastal Plain. Dinosaur tracks and traces; 1989; Cambridge University Press: pp. 371-394.

Fischer, MS; Schilling, N; Schmidt, M; Haarhaus, D; Witte, H. Basic limb kinematics of small therian mammals. Journal of Experimental Biology; 2002; 205, 9 pp. 1315-1338. [DOI: https://dx.doi.org/10.1242/jeb.205.9.1315]

Gallup, MA. Functional morphology of the hindfoot of the Texas sauropod Pleurocoelus sp. indet. Paleobiology of the Dinosaurs. Geological Society of America Special Paper; 1989; 238, pp. 71-74. [DOI: https://dx.doi.org/10.1130/SPE238-p71]

Gambaryan, PP. How animals run; 1974; Wiley: 365.

Genin, JJ; Willems, PA; Cavagna, GA; Lair, R; Heglund, NC. Biomechanics of locomotion in Asian elephants. Journal of Experimental Biology; 2010; 213, 5 pp. 694-706. [DOI: https://dx.doi.org/10.1242/jeb.035436]

González Riga, BJ. Speeds and stance of titanosaur sauropods: Analysis of Titanopodus tracks from the Late Cretaceous of Mendoza, Argentina. Annals of the Brazilian Academy of Sciences; 2011; 83, pp. 1-12. [DOI: https://dx.doi.org/10.1590/S0001-37652011005000002]

González Riga, BJ; Tomaselli, MB. Different trackway patterns in titanosaur sauropods: Analysis of new Titanopodus tracks from the Upper Cretaceous of Mendoza, Neuquén Basin, Argentina. Cretaceous Research; 2019; 93, pp. 49-59. [DOI: https://dx.doi.org/10.1016/j.cretres.2018.08.021]

Griffin, TM; Main, RP; Farley, CT. Biomechanics of quadrupedal walking: How do four-legged animals achieve inverted pendulum-like movements?. Journal of Experimental Biology; 2004; 207, pp. 3545-3558. [DOI: https://dx.doi.org/10.1242/jeb.01177]

Halfpenny, JC. A field guide to mammal tracking in North America; 1987; 2 Johnson Books: 176.

Hildebrand, M. Symmetrical gaits of horses. Science; 1965; 150, pp. 701-708. [DOI: https://dx.doi.org/10.1126/science.150.3697.701]

Hildebrand, M. Herman, RM; Grillner, S; Stein, PSG; Stuart, DG. Analysis of tetrapod gaits: General considerations and symmetrical gaits. Neural control of locomotion; 1976; Plenum: pp. 203-236. [DOI: https://dx.doi.org/10.1007/978-1-4757-0964-3_9]

Hildebrand, M. The quadrupedal gaits of vertebrates. BioScience; 1989; 39, 11 pp. 766-775. [DOI: https://dx.doi.org/10.2307/1311182]

Holliday, CM; Ridgely, RC; Sedlmayr, JC; Witmer, LM. Cartilaginous epiphyses in extant archosaurs and their implications for reconstructing limb function in dinosaurs. PLoS ONE; 2010; 5, e13120. [DOI: https://dx.doi.org/10.1371/journal.pone.0013120]

Hutchinson, JR; Famini, D; Lair, R; Kram, R. Biomechanics: Are fast-moving elephants really running?. Nature; 2003; 422, pp. 493-494. [DOI: https://dx.doi.org/10.1038/422493a]

Hutchinson, JR; Schwerda, D; Famini, D; Dale, RHI; Fischer, M; Kram, R. The locomotor kinematics of African and Asian elephants: Changes with speed and size. Journal of Experimental Biology; 2006; 209, pp. 3812-3827. [DOI: https://dx.doi.org/10.1242/jeb.02443]

Irschick, DJ; Jayne, BC. Comparative three-dimensional kinematics of the hindlimb for high-speed bipedal and quadrupedal locomotion of lizards. Journal of Experimental Biology; 1999; 202, 9 pp. 1047-1065. [DOI: https://dx.doi.org/10.1242/jeb.202.9.1047]

Ishigaki, S. Les empreintes de dinosaures du Jurassique inférieur du Haut Atlas Central marrocain. Notes Service Geologuique Du Maroc; 1988; 44, pp. 79-96.

Lammers, AR; Biknevicius, AR. The biodynamics of arboreal locomotion: The effects of substrate diameter on locomotor kinetics in the gray short-tailed opossum (Monodelphis domestica). Journal of Experimental Biology; 2004; 207, pp. 4325-4336. [DOI: https://dx.doi.org/10.1242/jeb.01231]

Leonardi, G. Glossary and manual of tetrapod footprint palaeoichnology; 1987; Publicação do Departemento Nacional da Produção Mineral Brasil: 117.

Lockley, MG. Tracking Dinosaurs: A new look at an ancient world; 1991; Cambridge University Press: 238.

Lovelace, DM; Hartman, SA; Wahl, WR. Morphology of a specimen of Supersaurus (Dinosauria, Sauropoda) from the Morrison Formation of Wyoming, and a re-evaluation of diplodocid phylogeny. Arquivos Do Museu Nacional, Rio De Janeiro; 2007; 65, 4 pp. 527-544.

Marty, D. Sedimentology, taphonomy, and ichnology of Late Jurassic dinosaur tracks from the Jura carbonate platform (Chevenez-Combe Ronde tracksite, NW Switzerland): Insights into the tidal-flat palaeoenvironment and dinosaur diversity, locomotion, and palaeoecology. GeoFocus; 2008; 21, pp. 1-278.

Marty, D; Ayer, J; Becker, D; Berger, J-P; Billon-Bruyat, J-P; Braillard, L; Hug, WA; Meyer, CA. Late Jurassic dinosaur tracksites of the Transjurane highway (Canton Jura, NW Switzerland): Overview and measures for their protection and valorization. Bulletin for Applied Geology; 2007; 12, pp. 75-89.

Marty, D; Cavin, L; Hug, WA; Jordan, P; Lockley, MG; Meyer, CA. The protection, conservation and sustainable use of the Courtedoux dinosaur tracksite, Canton Jura, Switzerland. Revue De Paléobiologie; 2004; 9, pp. 39-49.

Marty, D; Hug, WA; Iberg, A; Cavin, L; Meyer, CA et al. Preliminary report on the Courtedoux dinosaur tracksite from the Kimmeridgian of Switzerland. Ichnos; 2003; 10, pp. 209-219. [DOI: https://dx.doi.org/10.1080/10420940390256212]

Marty, D; Paratte, G; Lovis, C; Jacquemet, M; Meyer, CA. Extraordinary sauropod trackways from the Late Jurassic Béchat Bovais tracksite (Canton Jura, NW Switzerland): Implications for sauropod locomotor styles; 2010; 8th Annual Meeting of the European Association of Vertebrate Palaeontologists: 56.

Mazzetta, GV; Blanco, RE. Speeds of dinosaurs from the Albian-Cenomanian of Patagonia and sauropod stance and gait. Acta Palaeontologica Polonica; 2001; 46, pp. 235-246.

Padian, K; Olsen, PE. Footprints of the Komodo monitor and the trackways of fossil reptiles. Copeia; 1984; 1984, pp. 662-671. [DOI: https://dx.doi.org/10.2307/1445147]

Paratte, G; Lapaire, M; Lovis, C; Marty, D. Traces de dinosaures jurassiques Contexte et méthode (Catalogues du patrimoine paléontologique jurassien—A16); 2018; Office de la culture—Paléontologie A16:

Peabody, FE. Trackways of living and fossil salamanders. Univ Calif Publ Zool; 1959; 63, pp. 1-72.

Platt, BF; Hasiotis, ST. Newly discovered sauropod dinosaur tracks with skin and foot-pad impressions from the Upper Jurassic Morrison Formation, Bighorn Basin, Wyoming, USA. Palaios; 2006; 21, 3 pp. 249-261. [DOI: https://dx.doi.org/10.2110/palo.2004.p04-69]

Reilly, SM; Biknevicius, AR. Bels, VL; Gasc, JP; Casinos, A. Integrating kinetic and kinematic approaches to the analysis of terrestrial locomotion. Vertebrate biomechanics and evolution; 2003; Bios Scientific Publishers Ltd.: pp. 243-265.

Schmitt, D; Cartmill, M; Griffin, TM; Hanna, JB; Lemelin, P. Adaptive value of ambling gaits in primates and other mammals. Journal of Experimental Biology; 2006; 209, 11 pp. 2042-2049. [DOI: https://dx.doi.org/10.1242/jeb.02235]

Schwarz, D; Ikejiri, T; Breithaupt, BH; Sander, PM; Klein, N. A nearly complete skeleton of an early juvenile diplodocid (Dinosauria: Sauropoda) from the Lower Morrison Formation (Late Jurassic) of North Central Wyoming and its implications for early ontogeny and pneumaticity in sauropods. Historical Biology; 2007; 19, pp. 225-253. [DOI: https://dx.doi.org/10.1080/08912960601118651]

Schwarz, D; Wings, O; Meyer, CA. Super sizing the giants: First cartilage preservation at a sauropod dinosaur limb joint. Journal of the Geological Society, London; 2007; 164, pp. 61-65. [DOI: https://dx.doi.org/10.1144/0016-76492006-019]

Stevens, KA. The articulation of sauropod necks: Methodology and mythology. PLoS ONE; 2013; 8, 10 e78572. [DOI: https://dx.doi.org/10.1371/journal.pone.0078572]

Stevens, KA; Ernst, S. Statistical analyses of highway A16 dinosaur trackways: Final report; 2017; Paléontologie A16 Étude Intermédiaire 81: 5895.

Stevens, KA; Ernst, S; Marty, D. Falkingham, PL; Marty, D; Richter, A. Uncertainty and ambiguity in the interpretation of sauropod trackways. Dinosaur tracks. The next steps; 2016; Indiana University Press: pp. 227-243.

Stevens, KA; Ernst, S; Marty, D. The importance of being uncertain: Probabilistic computation of trackmaker size, gait and gauge. Meeting program and abstracts; 2017; Society of Vertebrate Paleontology 77th annual Meeting: 198.

Taylor, J. Introduction to error analysis, the study of uncertainties in physical measurements; 1997; 2 University Science Books: 327.

Thulborn, RA. Dinosaur tracks; 1990; Chapman & Hall: 410. [DOI: https://dx.doi.org/10.1007/978-94-009-0409-5]

Tschopp, E; Wings, O; Frauenfelder, T; Brinkmann, W. Articulated bone sets of manus and pedes of Camarasaurus (Sauropoda, Dinosauria). Palaeontologia Electronica; 2015; [DOI: https://dx.doi.org/10.26879/559]

Vila, B; Oms, O; Galobart, À; Bates, KT; Egerton, VM et al. Dynamic similarity in titanosaur sauropods: Ichnological evidence from the fumanya dinosaur tracksite (Southern Pyrenees). PLoS ONE; 2013; 8, 2 [DOI: https://dx.doi.org/10.1371/journal.pone.0057408]

Wilhite, RD. Carpenter, K; Tidwell, V. Variation in the appendicular skeleton of North American sauropod dinosaurs: Taxonomic implications. Thunder-lizards: The sauropodomorph dinosaurs; 2005; Indiana University Press: pp. 268-301.

Wimberly, AN; Slater, GJ; Granatosky, MC. Evolutionary history of quadrupedal walking gaits shows mammalian release from locomotor constraint. Proceedings of the Royal Society B; 2021; 288, 20210937. [DOI: https://dx.doi.org/10.1098/rspb.2021.0937]

Woodruff, DC; Carr, TD; Storrs, GW; Waskow, K; Scannella, JB; Nordén, KK; Wilson, JP. The smallest diplodocid skull reveals cranial ontogeny and growth-related dietary changes in the largest dinosaurs. Scientific Reports; 2018; 8, 1 pp. 1-12. [DOI: https://dx.doi.org/10.1038/s41598-018-32620-x]

Woodruff, DC; Foster, JR. The first specimen of Camarasaurus (Dinosauria: Sauropoda) from Montana: The northernmost occurrence of the genus. PLoS ONE; 2017; 12, 5 e0177423. [DOI: https://dx.doi.org/10.1371/journal.pone.0177423]

Zips, S; Peham, C; Scheidl, M; Licka, T; Girtler, D. Motion pattern of the toelt of Icelandic horses at different speeds. Equine Veterinary Journal; 2001; 33, pp. 109-111. [DOI: https://dx.doi.org/10.1111/j.2042-3306.2001.tb05371.x]