Gait determinants of undulatory swimming via energy optimality principles

Animals with a slender body such as eels progress by propagating body waves; this swimming mode is called anguilliform. Though anguilliform swimming has been studied extensively, its optimality and gait selection principles remain unknown. Here, through model-based optimal gait analysis, we propose that three dimensionless characteristic parameters -- the stride length, muscle mechanical efficiency and environmental force anisotropy -- shape the gait, and they respectively determine a whole or a fraction of body undulation, a more C-shaped or S-shaped body wave, and the overall body undulation amplitude and swim direction. The body(muscle)-fluid properties and the intended swim speed influence the gait via these three. Observed biological gait features, including those of young animals and the counterintuitive ones, are shown to conform, qualitatively, to these principles, suggesting that their gaits may be explained by optimality with respect to the total of fluid and muscle damping power losses, with the latter occupying a small percentage. In addition, the prevailing feature of caudally increased body curvature may be a result of an adjustment to the gait due to the tapered body geometry and/or oncoming fluid vortices. Biological gaits could thus be understood as a composition of aforementioned gait features. Also explained by optimal gaits are the negative correlation between the Strouhal number and the stride length, and the wider range of Strouhal number of anguilliform swimming (0.31-0.93) than that of carangiform swimming (0.2-0.4). The neural signals for muscle activation are also calculated from the model-based analysis to predict control strategies underlying the observed gaits. Our results benefit both the understanding of biological locomotion and the gait planning for bio-inspired underwater vehicles.