In a prey pursuit scenario under Hypothesis 2, the theropod has just pushed off with its swing leg, and is pivoting about its stance leg as it protracts the swing leg. The body and swing leg are rotating about their collective COM, directly above the stance foot. Total I y in this case includes the entire axial body (minus the hind legs), and the contribution of the swing leg to total I y of the system.

Using relative indices of agility, encompassing origins for relevant ilium-based muscles, tail-originating muscles ( Table 1 ), and mass moments of inertia, enables us to address action beyond yaw alone. Muscles of the leg on the outside of a turn normally involved in linear motion would change the body’s direction by linearly accelerating the body in that direction, while muscles for the leg on the inside of the turn exert less torque. Muscles involved in stabilizing the limbs and body, and providing contralateral braking and abduction, would come into play during rotation of the body ( Anderson et al., 1991 ). Mass moment of inertia is the most stringent mass-property limit on turning ability in long, massive dinosaurs ( Carrier, Walter & Lee, 2001 ; Henderson & Snively, 2003 ). This simplified approach is predictive, testable with more complex investigations (including specific torques of muscle-bone couples: Hutchinson, Ng-Thow-Hing & Anderson, 2007 ), and allows broad comparisons of overall turning ability.

To compare agility in theropods, we divided ilium area (a proxy for muscle cross-sectional area and maximal force production), and estimated m. caudofemoralis longus (CFL) cross-sections, by I y (rotational inertia in yaw about the body’s center of mass (COM)). We also incorporated scaling of moment arm size in a separate analysis to better compare absolute turning performance in the theropods. We restrict our comparisons to proxies of agility at given body masses, rather than estimating absolute performance, because a generalized predictive approach enables us to compare many taxa. Viable paths for testing our results include musculoskeletal dynamics of turning involving all hind limb muscles, as undertaken by Rankin, Rubenson & Hutchinson (2016) for linear locomotion in ostriches, or simpler approaches such as Hutchinson, Ng-Thow-Hing & Anderson (2007) calculations for turning in Tyrannosaurus . However, the dynamics of turning are complicated to pursue even in extant dinosaurs ( Jindrich et al., 2007 ), and estimating absolute performance in multiple extinct taxa would entail escalating numbers of assumptions with minimal comparative return. We therefore focus here on relative metrics of turning performance, based as much as possible on direct fossil data.

The blue line shows the position of the greatest depth from the caudal ribs to the ventral tips of the chevrons, and greatest inferred width of the m. caudofemoralis longus. (B) The inferred region of muscle attachment on the ilium (modified from Brochu, 2003 ) is outlined in red, for scaled area measurement in ImageJ. (C) The initial reconstructed radius (blue) of m. caufofemoralis longus (CFL) is 0.5 times the hypaxial depth of the tail (blue line in A), seen in anterior view of free caudal vertebra 3 and chevron 3. The maximum lateral extent of CFL is here based on cross-sections of adult Mallison, Pittman & Schwarz, 2015 ). Note that the chevron in c is modified to be 0.93 of its full length, because it slopes posteroventrally when properly articulated ( Brochu, 2003 ). Bone images in (A) and (C) are “cartoonized” in Adobe Photoshop to enhance edges.

Reconstructions of(Field Museum FMNH PR 2081) in lateral view (A) and dorsal view (B) enable digitizing of dorsal, ventral, and lateral extrema where they cross the vertical red lines. The lateral view (A) is modified with the dorsal margin of the neck conservatively raised based on recent muscle reconstructions ( Snively & Russell, 2007a 2007b ). The hind leg (A and C) is outlined in green, and straightened (C) for digitizing. A red dot (A and B) specifies the center of mass of the axial body (minus the limbs) using this reconstruction. An equation for the volume of a given frustum of the body (D), between positions 1 and 2, assumes elliptical cross-sections.

This relative agility in theropods is testable by regressing estimated body mass ( Fig. 1 ) against indicators of agility, which incorporate fossil-based estimates of muscle force ( Fig. 2 ), torque, and body mass and mass moment of inertia ( I y ; Fig. 1 ). Given the same moment arm lengths, greater force relative to rotational inertia indicates the ability to turn more rapidly. Coupled with protracted juvenile growth periods ( Erickson et al., 2004 ), heightened agility would be consistent with the hypothesis that tyrannosaurids were predominantly predatory, and help to explain how late Campanian and Maastrichtian tyrannosaurids monopolized the large predator niche in the Northern Hemisphere.

At a gross level ( Trinkaus et al. 1991 ), muscle attachment size enables us to compare forces in fossil taxa, and to investigate relative agility. Muscle force is proportional to physiological cross-sectional area, and in turn on muscle volume, pennation angle, and dramatically on fiber length ( Bates & Falkingham, 2018 ), in addition to maximal isometric stress and activation level. Muscle anatomical cross-sectional area and hence volume vary proportionally with attachment size of homologous muscles (explained in detail under Methods). In fossil taxa, attachment size is a consistent, reliably preserved influence on muscle force. Relative muscle force is therefore a useful, replicable metric for comparative assessments of agility in fossil tetrapods. Estimates of theropod muscle force and the mass properties of their bodies can facilitate comparisons of turning ability in theropods of similar body mass.

Like other terrestrial animals, large theropods would turn by applying torques (cross products of muscle forces and moment arms) to impart angular acceleration to their bodies. This angular acceleration can be calculated as musculoskeletal torque divided by the body’s mass moment of inertia (=rotational inertia). Terrestrial vertebrates such as cheetahs can induce a tight turn by lateroflexing and twisting one part of their axial skeleton, such as the tail, and then rapidly counterbending with the remainder, which pivots and tilts the body ( Wilson et al., 2013 ; Patel & Braae, 2014 ; Patel et al., 2016 ). The limbs can then accelerate the body in a new direction ( Wilson et al., 2013 ). These tetrapods can also cause a larger-radius turn by accelerating the body more quickly with one leg than the other (pushing off with more force on the outside of a turn), which can incorporate hip and knee extensor muscles originating from the ilium and tail ( Table 1 ). Hence muscles originating from the ilium can cause yaw (lateral pivoting) of the entire body, although they do not induce yaw directly. Such turning balances magnitudes of velocity and lean angle, and centripetal and centrifugal limb-ground forces. When limbs are planted on the ground, the body can pivot with locomotor muscle alone. In either case, limb muscles actuate and stabilize their joints, positively accelerating and braking the body and limbs.

Materials and Methods

Comparing relative turning performance in tyrannosaurids and other theropods requires data on mass moment of inertia I y about a vertical axis (y) through the body’s COM, and estimates of leg muscle force and moment arms. I in this paper always refers to mass moment of inertia, not I as the common variable for area moment of inertia. To estimate mass, COM, and I y , we approximated the bodies of the theropods as connected frusta (truncated cones or pyramids) with superellipse cross-sections (Fig. 1). Superellipses are symmetrical shapes the outline of which (from star-shaped, to ellipse, to rounded rectangle) are governed by exponents and major and minor dimensions (Rosin, 2000; Motani, 2001; Snively et al., 2013).

Spreadsheet templates for calculations of dimensions, mass, COM, and rotational inertias are available as Supplementary Information. These enable the estimation of mass properties from cross-sectional and length dimensions, using Microsoft Excel-compatible software. Snively et al. (2013) provide coefficients and polynomial regression equations for super-elliptical frusta.

Mass moments of inertia: Hypothesis 1 (both legs planted) Mass moment of inertia for turning laterally, designated I y , was calculated about the axial body’s COM by summing individual I y for all frusta (Eq. (8), first term), and the contribution of each frustum to the total using the parallel axis theorem (Eq. (8), second term). (8) I y = ∑ i = 1 n ( π 4 ) ρ i l i r ¯ DV r ¯ LM 3 + m i r i 2 For calculating I y of an individual frustum, ρ i is its density, and l i is its anteroposterior length. The element π/4 is a constant (C) for an ellipse, with an exponent k of 2 for its equation. We modified C with superellipse correction factors for other shapes (Snively et al., 2013). The dimension r ¯ DV is the average of dorsoventral radii of the anterior and posterior faces of each frustum, and r ¯ LM are the average of mediolateral radii. The mass m i and COM of each frustum were calculated using the methods described above, and distance r i from the whole body’s COM to that of each frustum was estimated by adding distances between each individual frustum’s COM to that of frustum i.

Mass moments of inertia: Hypothesis 2 (pivoting about the stance leg) Here the body and leg are pivoting in yaw about a vertical axis passing through their collective COM COM body+leg , and the center of pressure of the stance foot. Here rotational inertia I y body+leg about the stance leg is the sum of the four right terms in Eq. (9). (9) I y body + leg = I y body + I y leg + m body r COM-to-body 2 + m leg r COM-to-leg 2 Term 1. I y body of the axial body about its own COM;

Term 2. I y leg of the swing leg about its own COM (assuming the leg is straight);

Term 3. The axial body’s mass m body multiplied by the square of the distance r COM-to-body from its COM to the collective COM of the body + swing leg (COM body + leg );

Term 4. The swing leg’s mass m leg multiplied by the square of the distance r COM-to-leg from its COM to the collective COM of the body + swing leg (COM body+leg ). We calculated I y body using Eq. (8). To calculate I y leg (Eq. (10)), we approximate the swing leg as extended relatively straight and rotating on its own about an axis through the centers of its constituent frusta. In Eq. (10), I y leg is the sum of I y frustum for all individual frusta of the leg, and I y frustum is in turn simply the sum of I x and I z of each frustum (Durkin, 2003). These are similar to the first term in Eq. (8), but with anteroposterior radii r AP instead of the dorsoventral radius of frusta of the axial body. (10) I y leg = ∑ i = 1 n ( π 4 ) ρ i l i ( r ¯ AP r ¯ LM 3 + r ¯ LM r ¯ AP 3 ) Equations (11) and (12) give distance r COM-to-body and r COM-to-leg necessary for Eq. (9); note the brackets designating absolute values, necessary to find a distance rather than a z coordinate. (11) r COM-to-body = | COM body+leg − COM body | (12) r COM-to-leg = | COM body+leg − COM leg | An Excel spreadsheet in Supplementary Information (theropod_RI_body+one_leg.xlsx) has all variables and equations for finding RI of the body plus leg.

Visualization of agility comparisons Although log transformation of mass is useful for statistical comparisons, plotting the raw data enables intuitive visual comparisons of tyrannosaur and non-tyrannosaur agility, and immediate visual identification of outliers (Packard, Boardman & Birchard, 2009). We plotted raw agility index scores against log10 body mass in JMP (SAS Institute, Cary, NC, USA), which fitted exponential functions of best fit to the data.

Statistical comparison of group differences: phylogenetic ANCOVA Phylogenetic ANCOVA (phylANCOVA) enabled us to simultaneously test the influence of phylogeny and ontogeny on agility in monophyletic tyrannosaurs vs. a heterogeneous group of other theropods. The phylANCOVA mathematically addresses phylogenetically distant specimens or size outliers that would require separate, semi-quantitative exploration in a non-phylANCOVA.

Determining strength of phylogenetic signal and appropriateness of phylogenetic regression To determine whether phylogenetic regression was necessary when analyzing theropod agility, we calculated Pagel’s λ (Pagel, 1999) for each trait examined. Phylogenetic signal was estimated using the R package {phytools} (Revell, 2010). We found that phylogenetic signal was high for all traits (λ agility force = 0.89; λ agility moment = 0.90; λ mass = 0.88), emphasizing the need for phylogenetically-informed regression and analysis of covariance.