Estimating ankle joint angle from skeletal geometry: a mechanical model of the calcaneal lever in terrestrial mammals

Background The ankle joint angle, typically measured between the tibia and metatarsus, shows only a small range of movement during the stance phase and remains relatively constant within species, but varies across taxa. This variation influences traits such as stride length, posture, and locomotor function. While joint angles are readily observable in living animals, they cannot be directly measured in extinct species, for which only skeletal remains are available. Therefore, estimating ankle joint posture from skeletal geometry is important for reconstructing locomotion in both extant and extinct mammals. In this study, we propose a mechanical model of the ankle extensor apparatus to estimate ankle joint angle from bones and test whether the muscle-lever system aligns consistently with skeletal features across taxa. Methods We developed a simplified mechanical model of the ankle extensor apparatus to calculate ankle extensor moment arm defined as the perpendicular distance from the ankle joint center to the muscle force line of action, which was assumed to be parallel to the tibia. To verify the Achilles tendon runs parallel to the tibia across taxa, dissections were performed on cadavers of 24 species in seven orders. We compared observed angle ( θ obs ) from 26 species of zoo-kept terrestrial mammals, covering various body mass and locomotor modes, with estimated angle ( θ est ) from skeletal specimens of the same species. θ obs was the mean tibia–metatarsus angle during the stance phase, recorded laterally with a high-speed camera. θ est was measured on reassembled skeletal specimens as the ankle joint angle that maximized the extensor moment arm in the model. Phylogenetic comparative methods, including phylogenetic ANOVA and PGLS, were applied to analyze relationships among θ obs , θ est , body mass, and locomotor mode based on a time-calibrated phylogeny. Results Dissections confirmed the Achilles tendon runs nearly parallel to the tibia across species. Stance phase ankle joint rotations were small. Therefore, θ obs could be considered as representative for each species. Over 85% of the studied species maintained their ankle joint angle at which the mechanical advantage of the calcaneal lever was greater than 0.9. No significant differences in the mechanical advantage of the calcaneal lever were found among locomotor modes or taxonomic orders. A strong positive correlation was observed between θ obs and θ est ( ρ = 0.70, p < 0.001). Conclusion Our mechanical model could estimate θ est from skeletal morphology that closely match θ obs during stance phase. Despite interspecific variation of θ obs , the mechanical advantage of the calcaneal lever remains within a narrow range, suggesting mechanical optimization of the ankle extensor apparatus across terrestrial mammals. This model informs postural reconstruction in extinct species.