Geometric scaling predicts a major challenge for legged, terrestrial locomotion. Locomotor support requirements scale identically with body mass (α M¹), while force generation capacity should scale α M^2/3 as it depends on muscle cross-sectional area. Mammals compensate with more upright limb postures at larger sizes, but it remains unknown how sprawling tetrapods deal with this challenge. Varanid lizards are an ideal group to address this question because they cover an enormous body size range while maintaining a similar bent-limb posture and body proportions. This study reports the scaling of ground reaction forces and duty factor for varanid lizards ranging from 7 g 37 kg. Impulses (force x time) scaled roughly as predicted by the inverted pendulum model (α M^0.99-1.34) while peak forces (α M^0.73-1.00) scaled higher than expected. Duty factor scaled α M^0.04 and was higher for the hindlimb than the forelimb. The proportion of vertical impulse to total impulse increased with body size, and impulses decreased while peak forces increased with speed. These results provide valuable data into how locomotor forces vary with body size and suggest how other, extinct, sprawling tetrapods may have dealt with the biomechanical challenges associated with generating sufficient locomotor forces at larger body sizes.

Ground reaction forces (GRFs) were collected for 30 varanid lizards from 12 species (table 1) nearly covering the entire possible range of varanid body sizes. Data were recorded from 266 individual stance phases across speeds ranging from 0.96 m s^-1 to 4.05 m s^-1, comprising 105 forelimb (figure 1b) and 112 hindlimb (figure 1c) trials. Forces were collected at 1,000 or 10,000 Hz using either a custom-built force-plate[12], a force-plate based on a Nano17 load cell (ATI Industrial Automation, Apex, NC, USA), or an IP65 gamma force-plate (ATI Industrial Automation, Apex, NC, USA). All animals (except V. komodoensis, which were zoo animals) were wild caught. All force-plates were calibrated using equal weights. Trials with aberrant gaits, or where the entire foot did not contact the force plate were excluded. Methods were approved under ethics SBS/195/12/ARC (QLD), ANA16104 (QLD), ANE1934 (QLD), ANE2054 (QLD), A2450 (QLD), ANE2054 (QLD), and RA/3/100/1188 (WA), and lizards were collected under permits WISP11435612 (QLD) and SF009075 (WA), 08-001092-5 (WA), and WA0001919 (QLD).

Dorsal and lateral video recordings were captured using two high-speed cameras (Fastec IL3-100; 1280 x 1024 pixels, Fastec Imaging Corporation, San Diego, CA, USA) recording at 250 frames per second. Custom MATLAB (Mathworks, Natick, MA, USA) programs and an internal trigger were used to synchronize forces and videos and identify strides. GRFs were analyzed during the stance phase, from footfall to toe-off, and data were smoothed using a fourth-order, zero-lag Butterworth low-pass filter. The polarity of forces was transformed so that positive values indicate ground support (GRF_Z), cranially-directed forces (GRF_X), and medially-directed forces (GRF_Y) relative to the body (figure 1e). Negative (GRF_X) forces indicate caudally-directed forces, and negative (GRF_Y) values indicate laterally-directed forces. Total force magnitude [1,25] was calculated as in Eq. 1.

Impulses were calculated as time integrals of the relevant force component. For cranial, caudal, medial, and lateral impulses, forces were first separated into positive and negative regions before integration of the absolute value.

 Average speed was estimated by quantifying the 2D motion of a single marker on the dorsal aspect of the pectoral or pelvic girdle using DeepLabCut, a machine-learning video tracking software [26]. Speeds were converted to relative speed based on the maximum sprint speed for each species (table 1). Relative speed [27] was used instead of dynamic speed (Froude number), because although dynamic speed is useful for identifying similar gaits among diverse species [23] and was well-correlated with relative speed in our dataset (figure S4), it has been shown to be of limited utility in predicting similar gaits in closely-related lizards [28] and it is not obvious which characteristic length would best capture the dynamics of sprawling locomotion.

All analyses were conducted using R 3.6.3 (R Development Core Team, 2020). The relationships between force parameters with body mass, relative speed, and limb (forelimb versus hindlimb) were assessed using mixed effects models using the lmer.R  function of the lme4 and lmerTest packages including subject as a random factor [29,30]. All continuous variables were log-transformed. In all cases, a full-interaction model was initially constructed. If the interaction terms were not significant, a reduced model with no interaction terms was used instead, provided that an ANOVA test found no significant difference between these models. When full models were used, non-significant variables were successively dropped from the model using the update.R function as long as each, successive model was not found to be significantly different from the previous one using an ANOVA test (table S1). Individual regressions for body mass and relative speed (table 2) were calculated using sim_slopes.R, and plotted using interact_plot.R [31]. Percentage differences of peak forces and impulses between the fore and hindlimbs were calculated as the ratio of the inverse logs from the regression intercept values.

To check for an influence of phylogenetic patterns, the slopes and confidence intervals of linear regressions for species means of each stride parameter were compared with phylogenetic independent contrasts of the same parameters (figure S1), calculated using the pic.R function [32] and the maximum-likelihood varanid tree built from 1030 bp of the NADH-2 gene [33]. In order to investigate the scaling of the position of the horizontal centre of mass, previously-published data from the two-scale method [34] (figure S5) were analysed (figure 1a).

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