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+ + + https://doi.org/10.1098/rsos.201441 + + + + + Natural Frequency Method: estimating the preferred walking + speed of Tyrannosaurus rex based on tail natural + frequency + + + 2021 + + + + +

royalsocietypublishing.org/journal/rsos

+

Research

+

Cite this article: van Bijlert PA, van Soest AJK, Schulp AS. 2021 Natural Frequency + Method: estimating the preferred walking speed of Tyrannosaurus + rex based on tail natural frequency . R. Soc. Open Sci.8 : + 201441 . https://doi.org/10.1098/rsos.201441

+

Received: 12 August 2020

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Accepted: 15 March 2021

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Subject Category:

+

Locomotor energetics are an important determinant of an animal’ s ecological niche. + It is commonly assumed that animals minimize locomotor energy expenditure by + selecting gait kinematics tuned to the natural frequencies of relevant body parts. + We demonstrate that this allows estimation of the preferred step frequency and + walking speed of Tyrannosaurus rex , using an + approach we introduce as the Natural Frequency Method. Although the tail of + bipedal dinosaurs was actively involved in walking, it was suspended passively by + the caudal interspinous ligaments. These allowed for elastic energy storage, + thereby reducing the metabolic cost of transport. In order for elastic energy + storage to be high, step and natural frequencies would have to be matched. Using a + 3D morphological reconstruction and a spring-suspended biomechanical model, we + determined the tail natural frequency of T. rex + (0.66 s −1 , range 0.41–0.84), and the corresponding walking speed (1.28 m s −1 , + range 0.80–1.64), which we argue to be a good indicator of preferred walking speed + (PWS). The walking speeds found here are lower than earlier estimations for large + theropods, but agree quite closely with PWS of a diverse group of extant animals. + The results are most sensitive to uncertainties regarding ligament moment arms, + vertebral kinematics and ligament composition. However, our model formulation and + method for estimation of walking speed are unaffected by assumptions regarding + muscularity, and therefore offer an independent line of evidence within the field + of dinosaur locomotion.

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Organismal and evolutionary biology

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Subject Areas:

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biomechanics/palaeontology/biomechanics

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Keywords:

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locomotion, cost of transport, theropoda, tetanurae, optimal walking speed

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Author for correspondence:

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Pasha A. van Bijlert e-mail: pasha.vanbijlert@naturalis.nl

+

Electronic supplementary material is available online at + https://doi.org/10.6084/m9.figshare.c.

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5369033.

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Natural

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Frequency

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Method:

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estimating

+

the

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preferredwalking

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speed

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of

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+ Tyrannosaurus rex +

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based

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on tail

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natural

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frequency

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Pasha A. van Bijlert1,3, A. J.‘Knoek’van Soest1,4

+

and Anne S. Schulp2,3,5

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1Department of Human Movement Sciences, Faculty of Behavioural and Movement Sciences, + and2Department of Earth Sciences, Faculty of Science, Vrije Universiteit + Amsterdam, Amsterdam, The Netherlands

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3Naturalis Biodiversity Center, Leiden, The Netherlands

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4Research Institute Amsterdam Movement Sciences, Amsterdam, The Netherlands

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5Faculty of Geosciences, Utrecht University, Utrecht, The Netherlands

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PAvB, 0000-0002-5567-7022 ; AJKvS, 0000-0002-1959-1061 ;

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AS, 0000-0001-9389-1540

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© 2021 The Authors. Published by the Royal Society under the terms of the Creative + Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which + permits unrestricted use, provided the original author and source are + credited.

+

1. Introduction

+

Animals display a variety of gaits and walking speeds. Two locomotor extremes are of + particular interest: the maximal speed, and the energetically optimal walking + speed, at which the metabolic cost of transport (MCOT, in J m −1 kg −1 ) is + minimal for walking gait. In the absence of external task constraints, optimal + walking speed is very close to preferred walking speed (PWS) in humans [1,2], + ratites [ 3 ], horses [ 4 ] and elephants [ 5 ]. It is likely that they are + related in all terrestrial animals, and that animals would tend to forage at this + speed. During locomotion at preferred speed, animals tend to make use of resonance + by matching locomotor frequencies to the (undamped) natural frequencies of their + relevant body parts, sometimes tuned through muscular contractions [ 6 – 13 ]. + Indeed, it has been demonstrated that mammalian quadrupedal taxa have convergent + natural frequencies between fore- and hindlimbs despite differing limb + morphologies [14,15]. An important advantage of moving a body part close to its + natural frequency is that this reduces mechanical work [16,17]. Based on the + commonly adopted idea that PWS is chosen to minimize MCOT [ 1 – 5 ], it has been + argued that preferred step frequencies are close to the natural frequencies of + relevant body parts [7,11,12]. Determination of the natural frequency of a body + part relevant to locomotion can, therefore, constrain preferred (and presumably, + optimal) step frequencies and walking speeds of extinct taxa [ 10 ].

+

Theropod dinosaurs had tails that were actively involved in locomotion [18,19], but + were passively supported through the caudal interspinous ligaments [ 20 – 22 ], + thus forming a mass-spring system. Tyrannosaurus + rex has been the focus of many locomotor studies [ 23 – 32 + ]. Its largest muscle, M. caudofemoralis longus (CFL), retracted the femur to + produce forward propulsion for locomotion [18,25,33–35] ( ). The tail was subject + to flexion torque due to gravity and CFL contractions, which was counteracted at + zero metabolic cost by the caudal interspinous ligaments, leading to metaplastic + adaptations [ 20 – 22 ] ( ). Elastic energy storage in these ligaments improved + locomotor efficiency, and like a mass on a spring, the tail would have oscillated + at the step frequency. This is conceptually similar to elastic storage in ungulate + neck movement [ 36 – 38 ]. The tail would resonate if step frequency was matched + to the tail natural frequency, and this would maximize strain and thus elastic + storage [ 39 ]. The principle of elastic energy storage has previously been used + to simulate more realistic running gaits of the smaller theropod + Allosaurus fragilis [ 40 ]. Elastic storage + of the caudal interspinous ligaments likely played a smaller role in quadrupedal + dinosaurs, due to the less pronounced centre of mass excursions. However, because + extant quadrupeds display similar energysaving mechanisms [ 36 – 38 ], our method + should also be applicable to the locomotion of quadrupedal dinosaurs, provided + their necks or tails were passively suspended by ligamentous structures.

+

In recent biomechanical models of non-avian dinosaurs, the focus has been on + detailed hindlimb muscular reconstructions, while the tail was simplified to a + single rigid structure (for instance: [30,41]). It is our view, however, that tail + flexibility is an essential aspect of the locomotion of nonavian dinosaurs. We, + therefore, chose to focus on the tail, because investigating the dynamic effects + of a compliant tail may provide interesting insights into locomotor capabilities. + The primary goal of this study was to develop a reductionist method to estimate + PWS for non-avian dinosaurs. To this end, we performed a detailed morphological + reconstruction of the tail, including the caudal interspinous ligaments. By + incorporating their spring-like properties in a biomechanical model, we + subsequently estimated the natural frequency of the vertical swaying of the tail. + Using this as an analogue for step frequency, we then combined the tail natural + frequency with trackway data to estimate the PWS. We will refer to this approach + as the Natural Frequency Method (NFM), and demonstrate its application by + determining the PWS for T. rex . We have also + investigated which morphological features of the tail have the largest impact on + tail natural frequency. In doing so, we hope to encourage researchers to + incorporate a non-rigid tail into their locomotor simulations, while also + demonstrating that the relatively simple NFM can be a valuable expansion of the + toolkit of palaeo-biomechanists.

+

2. Material and methods

+

We estimated tail natural frequency of T. rex , by + numerically determining the lowest eigenfrequency of a biomechanical model. + Essentially, we constructed this model by estimating inertial parameters of the + tail, dividing it into five segments, and then fitting joint spring parameters + based on inverse-dynamic relationships of different postures of the skeleton.

+

Inertial parameters of this model were based on a 3D volumetric musculoskeletal + reconstruction. This reconstruction was performed on adult T. + rex specimen RGM.792000 (nicknamed + ‘Trix’, ), in the collection of Naturalis Biodiversity Center in Leiden, The + Netherlands . This is a large, adult specimen with + exceptional surface preservation, making it possible to accurately distinguish the + attachment sites of the interspinous ligaments. We articulated 3D scans of the + caudal skeleton ( , electronic supplementary material, ), after which we + reconstructed the major caudal musculature to estimate inertial parameters ( , + electronic supplementary material, 4). Using osteological landmarks combined with + the physical articulation of 3D prints, we determined the axes of rotation between + the vertebrae (electronic supplementary material, figures S5 and S6). + Subsequently, we reconstructed the caudal interspinous ligaments ( ), which + enabled the determination of moment arms and cross-sectional area ( , electronic + supplementary material, figure S7). The morphological reconstruction of the + ligaments made it possible to quantify the kinematic relation between vertebral + flexion/extension and length of the individual ligaments (i.e. strain), which was + used to construct the biomechanical model. Parameters of the reconstruction are + provided in electronic supplementary material, table S1 .

+

Taking the morphological reconstruction as starting point, we then defined a + simplified biomechanical model consisting of five rigid bodies, connected in hinge + joints, with a nonlinear rotational spring at each joint ( and for angle + definitions). During walking, the most prominent movements and forces occur in the + sagittal plane, and the moment arm of the CFL is also the largest in this plane. + Furthermore, the exchange between gravitational and elastic potential energy is + most meaningfully studied in this plane. Therefore, similar to previous + researchers [27,30], we elected to perform a sagittal plane analysis. Inertial + properties of the rigid bodies were acquired from the corresponding segments of + the morphological reconstruction. Each of the rotational springs was assumed to + generate a torque that increased quadratically with joint angle when stretched [ + 42 ]. As a result, each joint spring has two parameters: joint angle at which the + spring torque is zero, and a stiffness parameter. To assign values to these + parameters, the biomechanical model was aligned with the morphological + reconstruction in two different postures in which the spring torques were known ( + ). We acquired the first posture by defining passive horizontal equilibrium, which + implies that the ligaments are strained in horizontal posture to counteract + gravity ( ). It has been shown that the interspinous ligaments could generate + ample force to maintain this pose [ 20 ], which our models confirm (see the + electronic supplementary material). In the second posture, all interspinous + ligaments were at resting length, and thus, all spring torques equalled zero. We + determined this posture on the basis of an assumption regarding the ligament + strain in horizontal equilibrium. Tendons and ligaments can be roughly divided + into low-stress and high-stress varieties, with the high-stress varieties + providing more energy savings at the cost of lower safety factors [ 39 ]. We + imposed an intermediate 4% strain in horizontal equilibrium ( εhor = 0.04), and + used this to find a pose where the ligaments are not strained ( ). This is + analogous to passive equilibrium in the absence of gravity, or a T. + rex lying on its side. After aligning the biomechanical + model with the morphological reconstruction in both postures, the mechanical + equilibrium conditions were used to calculate the parameter values of each + rotational spring.

+

In the absence of damping, the resonance of the tail would occur at the undamped + natural frequency ( fn ) of the system [ 43 ]. In reality, damping is omnipresent. + In that case, resonance occurs at a forcing frequency of fn · (1 – 2 β2 ) 0.5 , + with β being the damping ratio [ 43 ]. Ligaments (like tendons) display relatively + little energy loss during work-loop experiments [39,44], so they have low damping + ratios. However, damping of dorsoventral oscillations would be dependent on more + than just the ligaments, and a structure-based estimate of damping would be + difficult, if not impossible. Therefore, we will use the undamped natural + frequency of the biomechanical model as a proxy for the resonance frequency. To + calculate the natural frequency of the biomechanical model, we first linearized + the equations of motion in horizontal equilibrium using standard numerical + methods. Next, the five pairs of conjugated purely imaginary eigenvalues of the + system matrix were determined using the ‘eig’ function in MATLAB. Subsequently, + the natural frequency was calculated as the oscillation frequency pertaining to + the fundamental resonance mode, i.e. the mode with the lowest eigenfrequency. + Finally, walking speed was estimated by multiplying the natural frequency of each + model by step length, defined as half the distance between two consecutive + footfalls of the same foot. We determined a step length of 1.94 m by scaling a + large tyrannosaurid trackway [ 45 ] based on footprint length of RGM.792000 (see + the electronic supplementary material for details).

+

As was done in previous studies on dinosaur dimensions [27,33,46,47], we subjected + the major inputs to a sensitivity analysis. Since we hope to see future studies of + dinosaur locomotion incorporate our tail model, we intentionally chose wide bounds + for the sensitivity analysis. This serves to inform researchers which individual + steps in the modelling process induce the largest variation in the result. These + steps could then be the focus of future research in an attempt to reduce the + uncertainty.

+

To account for uncertainty in inertial estimates, we individually varied length (and + thus moment of inertia) and mass. Because we assumed 4% ligament strain in + horizontal equilibrium, we investigated the effects of low- and high-stress + ligaments on the result, by imposing 3% and 5% strain in horizontal equilibrium, + respectively [ 39 ]. Although each ligament is well-bounded by the skeleton, the + effective point of force application of each ligament was unknown. In the baseline + model, we based the moment arms on the area centroids of the ligaments. We used + the ligament reconstruction to determine bounds for the moment arms (electronic + supplementary material, figures S5 and S7). Lastly, whereas the proximal tail + shows pronounced articulations between the zygapophyses, after approximately the + 13th caudal this is no longer the case (electronic supplementary material, figure + S5). This shifts the rotational axes in the distal tail ventrally, towards the + vertebral centra, with the vertical position dependent on whether the tail is in + flexion or extension (electronic supplementary material, figure S6 shows the + baseline and two extreme possibilities). We, therefore, bounded the rotational + axes in tail segments 3 and 4, which affected joint springs 3–5. The dorsal axes + are unrealistically high, ensuring a wide bound for the sensitivity analysis. The + ventral axes are located at the vertebral articulation with the chevrons, and + represent their mechanical effect when the tail is flexed. In total, the + aforementioned bounds led to 11 different models.

+

To encourage the adoption of our compliant tail model for dinosaur locomotion, we + have provided in-depth descriptions of reconstruction and biomechanical modelling + steps, mathematical derivations, scaling relationships and supplementary results + in a combined document titled ‘Supplementary texts’ as electronic supplementary + material. Reconstructions, 3D modelling and measurements were all done in + Rhinoceros 6 (McNeel and Associates, Seattle, WA, USA). The biomechanical model + was constructed using custom-code written in MATLAB 2019b (MathWorks, Natick, MA, + USA). A 3D model of the iliosacrum and caudal skeleton of RGM.792000, and all + custom MATLAB scripts are provided as electronic supplementary material.

+

3. Results

+

Natural frequencies and walking speeds determined in this study, as well as key + outcomes of the sensitivity analysis, are reported in table 1 . The natural + frequency of our baseline model of the tail was 0.66 s −1 , which we interpreted + as the preferred step frequency for T. rex . This + corresponds to a PWS of 1.28 m s −1 . To provide the reader with an impression of + the dynamic behaviour that results, we have provided a simulation of a lightly + damped version of the baseline model (electronic supplementary material, video + S1). In this simulation, the only input to the tail was a sinusoidal motion at the + base of the tail with an amplitude of 0.08 m. Assuming pendular walking, this is + the vertical motion that results from taking 1.94 m long steps at a hip height of + 3.1 m. No muscle forces were included, so the phase relationship between the + vertical oscillations of the hip and tail may not be representative of a + muscle-actuated tail. In this simulation, a small amount of damping was introduced + in order to obtain a finite tail amplitude. The damping ratio of the fundamental + eigenmode was 0.16, which has a negligible effect on the resonant frequency [ 43 + ].

+

The effects of changes in length (and thus inertia), mass and strain are + mathematically predictable in our model, and we have thus derived general scaling + laws for these parameters in the electronic supplementary material. Mass had zero + effect on the natural frequency. Length ( L ) and strain in the horizontal + position ( εhor ) are related to the natural frequency ( fn ) as follows: fn ∝ + L−0.5 and fn ∝ 1 hor— 0:5 . These relations can also be applied to the altered + moment arms or distal axes models. For instance, a 6% longer tail, with 10% more + strain in horizontal equilibrium (4.4% strain instead of 4%), using the maximal + moment arms, would lead to a natural frequency of 0.79 × 1.06 −0.5 × 1.1 −0.5 = + 0.73 s −1 . The a These models only changed the moment arms within the 3rd – 5th + segments; segments 1 and 2 remained unchanged from the baseline model. moment arms + models cannot be combined with the distal axes models in this way, because these + models are based on measurements of morphological traits, instead of systematic + parameter variations. Their combinations are reported in electronic supplementary + material, table S2, and the above proportionality relations can be applied to + those combination effects as well.

+

modelrelative change in tail natural frequency from baseline (%)tail natural + frequency (s−1)corresponding walking speed (m s−1)baselinen.a.0.661.28mass +8% + mass −8% length +8% length −8% high-strain (+25%) low-strain (−25%)0 0 −3.8 +4.3 + −10.6 +15.50.66 0.66 0.63 0.69 0.59 0.761.28 1.28 1.23 1.33 1.14 1.48max. moment + arms min. moment arms ventral axesadorsal axesa+20.6 −14.3 +6.7 −13.30.79 0.56 + 0.70 0.571.54 1.09 1.36 1.11

+

Varying the moment arms had the largest overall effect on tail natural frequency, + followed by the models accounting for strain magnitude. Together, these models + accounted for uncertainty in the effective point of force application and + composition of the caudal interspinous ligaments. We have supplied a spreadsheet + as electronic supplementary material that interactively calculates natural + frequency and walking speed, depending on parameter changes input by the user.

+

4. Discussion

+

The primary goal of this study was to develop a reductionist method to estimate the + tail natural frequency of T. rex , and to combine + the results of this method with trackway data for a baseline estimate of PWS ( + table 1 ). Our secondary goal was to determine which morphological features of the + tail have the largest effect on its natural frequency, and these are represented + by all the other models in table 1 . Overall, results are most sensitive to + uncertainties regarding ligament composition and vertebral kinematics in the tail + of T. rex , while being minimally sensitive to + length (and thus moment of inertia), and not affected by mass.

+

Our model predicts several scaling laws which provide insight into the locomotion of + T. rex (derivations are provided in the + electronic supplementary material). Natural frequency scales with the inverse + square root of tail length ( table 1 ). This implies an ontogenetic decrease in + step frequency, but an increase in PWS proportional to L0.5 .

+

It has previously been proposed that the distal tail may act as a ‘dynamic + stabilizer’ [ 19 ]. Varying the distal rotational axes revealed that even though + the distal three segments accounted for only 14.5% of the total caudal mass, they + still had a meaningful effect on the natural frequency ( table 1 ). Our results + support the notion that the medio-distal tail plays an important role in overall + tail dynamics. Furthermore, the ventral axes model suggests that the chevrons + increase locomotor speed when the tail is in slight flexion, by increasing the + moment arm of the interspinous ligaments. Combining altered moment arms with + alternative rotational axes strengthened this effect (electronic supplementary + material, table S2). However, this table should be interpreted with caution + because the distal axes models do not represent axes that would have been used + throughout the whole step cycle. Instead, the rotational axes would migrate + ventrally during flexion, and dorsally during extension of the tail, which could + potentially be influenced by contraction of the epaxial and hypaxial musculature. + Thus, the bounds in electronic supplementary material, table S2 are + unrealistically wide, due to the assumptions underlying this table. Yet, to + acknowledge the uncertainties in dinosaur gait reconstruction, we used these as + the reported range in our result.

+

Varying the amount of strain imposed on the ligaments in the horizontal posture ( + εhor , expressed as a fraction) had a similar effect as varying the length: fn ∝ 1 + hor— 0:5 . This is analogous to increasing or decreasing the stiffness at a + constant moment of inertia. Ligaments that are subject to high levels of strain + have increased energy-saving properties [ 39 ]. Our high-strain model displayed a + lower natural frequency, implying that selective pressures for higher PWS may + result in less energy-efficient locomotion, and vice versa.

+

Our baseline model suggests a PWS of 1.28 m s −1 for T. + rex , which we interpret to be near its energetically + optimal walking speed. There are no extant analogues for tail-dependent obligate + bipedal locomotion, especially combined with such a heavy emphasis on elastic + storage. In fact, energetically optimal walking speed has only been measured in a + few species, so comparisons are limited to these studies. In the animal kingdom, + optimal walking speed is reported to be 1.0 m s −1 for ratites [ 3 ] (average of + the net and total MCOT) and elephants [ 5 ], 1.34–1.42 m s −1 for humans [1,2] and + 1.25 m s −1 for horses [ 4 ], and these speeds are closely related to their + respective PWS. PWS for giraffes [ 38 ] (reported as ‘semi-selected’) and + migratory gnus and gazelles [ 48 ] have also been determined to lie close to this + range. This is a remarkably close distribution, especially given the wide range in + body size, shape and locomotor modes. Most of our estimates of T. + rex walking speed fall within this range, with two of them + only slightly exceeding it.

+

Very little work has been done to directly investigate PWS of dinosaurs. Limb + natural frequencies have been used to estimate ‘comfortable walking speeds’ of + several sauropod taxa [ 10 ], and dynamic simulations were used to estimate PWS of + Argentinosaurus [ 49 ]. To our knowledge, no + such work has been done on theropods. Until now, most estimates for (submaximal) + dinosaur walking speeds are based on dynamic similarity (DS). Based on a + regression of predominantly mammalian walking data, Alexander proposed an equation + to estimate walking speed from trackways [ 23 ]. Of importance is the relative + stride length, defined as the ratio between the stride length and hip height. This + necessitates inferral of the hip height of the trackmaker, leading to + uncertainties in the estimate. Despite qualitatively similar scaling predictions, + our proposed NFM predicts lower speeds than trackway estimates using DS + [23,24,32,45,50]. Such trackway estimates are not strictly energetically optimal, + but we would expect them to tend towards the preferred speed if the sample of + trackways is large enough. For large theropods, estimates tend to range between 2 + and 3 m s −1 , which is much higher than the narrow range reported for extant + animals. These higher speeds are commonly attributed to allometric effects + [23,24,32,45,50]. A large source of potential errors, however, lies in the + equation that relates relative stride length to walking speed, in which the step + frequencies are implicit [24,51]. Alexander himself noted the high variability in + his walking data, which leads to errors in speed estimates [ 51 ]. Not only is + this true for dinosaurs, but even for the walking speed of the animals (including + humans) on which the regression is based [ 10 ]. It has been shown that limb + natural frequencies provided better predictions for elephants and giraffes near + the preferred speed [ 10 ]. Near preferred speeds, DS consistently overestimated + walking speeds, but the accuracy of the predictions improved at speeds + substantially higher than the preferred speeds [ 10 ]. When compared to walking + speeds of extant animals in a variety of sizes and gaits, DS also appears to + overestimate the walking speed of bipedal dinosaurs. Neglecting tail dynamics may, + therefore, be one of the oversimplifications inherent to DS, when applied to the + inverted pendulum model of walking.

+

Recent amendments have been proposed to improve DS-based estimations, either by + focusing on swing phase dynamics [ 52 ], or by incorporating taxon-specific + morphological parameters to reduce uncertainty [ 53 ]. The latter are of course + difficult to obtain from a fossilized trackway. Our method can help in this + regard, by providing a range of plausible step frequencies for any given taxon. + With this goal in mind, it would be most effective to estimate the natural + frequencies of taxa that are wellrepresented in the trackway record. That way, it + would be possible to combine step length data from several trackways, increasing + confidence that the average result should tend towards preferred (and therefore + optimal) gait. Unfortunately, this was not possible in the present study, because + large theropod trackways with reasonably close taxonomic proximity to + T. rex are exceedingly rare [ 45 ]. Bipedal + dinosaurs seem to have preferred to walk at a relative stride length of 1.3, and + it has been suggested that this might have been energetically optimal [24,45]. + This relation implicitly relates footprint length to stride length, by first + estimating hip height from footprint length, and also assumes leg kinematics are + known. Instead, we preferred to relate footprint length directly to stride length + from a trackway that could be ascribed to T. rex + with reasonable certainty [ 45 ]. This ensures sufficient geometric similarity to + RGM.792000. The resulting step lengths differed by less than 4% from the suggested + preferred relative stride length. Whenever considering trackway data, + substrate-related uncertainties will prevent any straightforward interpretation + [53,54], although presumably, these would not affect tail natural frequency.

+

DS does not incorporate MCOT, and therefore cannot be used to estimate PWS of + dinosaurs. Therefore, when estimating dinosaur foraging costs, researchers have + used DS to scale the walking speed [ 29 ], or even kept it at a constant 2 m s −1 + for larger taxa [ 31 ]. However, methods to calculate MCOT are heavily dependent + on both speed and mass [ 28 ], so we suggest that any comparison of foraging costs + should use the PWS of the respective taxa as a starting point. This could be done + using the natural frequency of the legs [ 10 ], musculoskeletal simulations [ 49 + ], or NFM as we propose it.

+

To our knowledge, musculoskeletal simulation models have not been used to estimate + how MCOT varied with walking speed in T. rex , + although this has been done for Argentinosaurus [ + 49 ]. A simulation model requires estimation of joint rotational axes, body mass + and moment arms for both the musculature and the ligaments [27,30,34,41,49]. The + muscular reconstruction then provides further uncertainties in fibre composition + type and architecture, both of which would significantly influence power output. + Simulation studies have generally focused on maximal speed, and have contributed + to the consensus that a long flight phase for large theropods would be unlikely + [25,27,30,35]. However, the extensive muscular reconstructions often strongly + affect the biomechanical simulations [41,49]. The largest uncertainty in the + present study is related to the moment arms of the interspinous ligaments. This is + an encouraging finding, because their extent is well-bounded by the skeleton. + Essentially, one of the more certain parameters provides the most uncertainty. + Future research on in vivo vertebral kinematics, dynamics and ligament + compositions of crocodilian tails could reduce this uncertainty. Similarly + encouraging is that our results are minimally sensitive to estimates of + muscularity or inertial parameters in general. Indeed, when isolating mass as a + free parameter in our sensitivity analysis, we have shown that our predicted + speeds are unaffected by mass-estimates ( table 1 ). This is due to the assumption + that the ligaments could passively support the tail: adding muscle mass would make + the ligaments proportionately stiffer, leading to the same natural frequency (see + the scaling relationships in the electronic supplementary material). This + assumption mimics the adaptations that naturally would have occurred in the + connective tissue, to keep the tail horizontal as the animal gained mass + throughout its life. It has been shown that body mass does not affect PWS in + humans [ 1 ], and this is also in accordance with DS [ 23 ]. Minimal sensitivity + to muscular and inertial estimates implies that our inverse dynamic approach to + constructing a non-rigid tail could be incorporated into more sophisticated + hindlimb simulation models, without adding much to the overall uncertainty of the + result. This could have implications for maximal running speeds of large taxa like + T. rex : maximum running speed was shown to + be limited by peak stresses on the limbs [ 30 ], but a compliant tail may serve to + reduce these stresses.

+

Our method for walking speed estimation is meant to be a reductionist analysis, so + we have only incorporated the interspinous ligaments, since we expect their + mechanical effect to dominate overall tail dynamics. Natural frequencies play an + important role in animal locomotion [ 6 – 13 ], and NFM, therefore, provides a + reasonable starting point for investigating how the resonant properties of + dinosaur tails affect overall locomotion. However, in reducing the analysis to the + mechanics of a single structure, many of the complex interactions are ignored. For + instance, animals tune the frequencies of their segments through muscular + contractions [6,7]. Tyrannosaurus rex could have + contracted its epaxial and hypaxial caudal musculature to add rotational stiffness + to the tail, which could be beneficial at higher speeds, for instance during a + pursuit. Our ventral axes model demonstrates a possibility in this regard: if + T. rex were to keep its distal tail in slight + flexion, thereby migrating the rotational axes of the vertebrae to the + articulations with the chevrons, the overall natural frequency of the tail would + increase.

+

Tail musculature could also be employed to enforce beneficial phase relationships + between vertical hip and tail oscillations. During walking, vertical oscillations + between withers and heads in most ungulates are out of phase, which provides them + with an energetic benefit [36–38,55]. This is theorized to be modulated by natural + frequencies of the neck segment [37,38], but there is an active component to this + behaviour as well [ 36 ]. This phase relationship reduces the losses incurred + during the step-to-step transitions, which are identified as major sources of + energy losses in inverted pendulum models of walking [ 56 ]. The implications of + energy storage are also further complicated by the serial elasticity in + musculotendon complexes: tendon stiffness can substantially impact the mechanical + work done by the muscle fibres [ 57 ]. Finally, the cost of cyclical muscle + contractions could also affect metabolic optima, depending on the task + requirements [16,17]. Such complex interactions would undoubtedly affect PWS, but + require extensive reconstructions of muscular contractile properties. Muscular + parameters currently provide the biggest uncertainty in most musculoskeletal + simulations of dinosaurs [41,49,58], so any investigation of these interactions + would require careful consideration. Inclusion of our compliant tail model into + fully actuated hindlimb simulations of T. rex may + shed further light on how the unique tails of non-avian dinosaurs functioned + during locomotion. However, in such complex simulations, the intricacies of tail + dynamics may be overshadowed by the uncertainties regarding the contractile + properties of the muscles.

+

We set out to develop a method that does not require assumptions regarding + contractile properties, so we accept that this limits the predictive power of our + method. Given the overall uncertainties we must deal with when investigating the + locomotion of extinct animals, we consider the simplicity of the method to be one + of its strengths. Provided the neck or tail posture was supported passively, our + method could be used to arrive at an estimate of the PWS of any sufficiently + complete dinosaur, without the need to estimate contractile properties.

+

5. Conclusion

+

Gait reconstruction of dinosaurs has numerous inherent uncertainties, and therefore + it is important to compare results from different methods, in an attempt to find a + convergent point. We have proposed a method based on reconstructing the natural + frequency of the vertical swaying of the tail, which we refer to as the Natural + Frequency Method. This method requires relatively few assumptions, which are + furthermore different assumptions to other methods. Our results for preferred + walking speed of T. rex are lower than previous + estimations for large theropods, but more closely match the preferred walking + speeds of a variety of extant animals, regardless of gait pattern and body size. + This seems to call in question the oft-cited high walking performance of bipedal + dinosaurs due to their cursorial adaptations. Investigating vertebral kinematics + and ligament composition in extant archosaurian tails could further constrain + predictions based on our method, which could, in turn, be used to improve trackway + estimates and foraging costs. Using simple, yet well-established principles from + animal locomotion, the Natural Frequency Method provides an independent line of + evidence to explore the locomotion of non-avian dinosaurs.

+

Data accessibility. Supplementary texts, 7 and tables S1–S 2, video S1, a spreadsheet + to calculate all possible parameter effects of our models, custom MATLAB code + written for the analyses and 3D scans of the iliosacrum and caudal skeleton of + RGM.792000 are provided as electronic supplementary material.

+

Authors’contributions. P.A.v.B., A.J.K.v.S. and A.S. conceived the project. + A.J.K.v.S. designed the biomechanical model. P.A.v.B. performed the + reconstructions, measurements, coding. P.A.v.B., A.J.K.v.S. and A.S. wrote the + manuscript.

+

Competing interests. The authors declare no competing interests;

+

Funding. Nothing to declare.

+

Acknowledgements. We thank K. Lemaire, M. Bobbert and D. Kistemaker for their vital + role in early discussions. Further thanks to V. Vanhecke for 3D scanning, P. + Larson for providing excavation maps and D. Tanke for providing photographs. H. + Mallison and P. Manning are thanked for in-depth discussion and feedback on + earlier versions of this manuscript. Discussions with W. Sellers, K. Bates and F. + van Diggelen further helped us to contextualize our work. Lastly, we extend our + gratitude to two anonymous reviewers whose constructive feedback and insights + helped shape the final manuscript.

+

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