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In nature, limbless locomotion is adopted by a wide range of organisms atvarious length scales. Interestingly, undulatory, crawling and inching/loopinggait constitutes a fundamental class of limbless locomotion and is oftenobserved in many species such as caterpillars, earthworms, leeches, larvae, and\emph{C. elegans}, to name a few. In this work, we developed a computationallyefficient 3D Finite Element (FE) based unified framework for the locomotion oflimbless organisms on soft substrates. Muscle activity is simulated with amultiplicative decomposition of deformation gradient, which allows mimicking abroad range of locomotion patterns in 3D solids on frictional substrates. Inparticular, a two-field FE formulation based on positions and velocities isproposed. Governing partial differential equations are transformed intoequivalent time-continuous differential-algebraic equations (DAEs). Next, theoptimal locomotion strategies are studied in the framework of optimal controltheory. We resort to adjoint-based methods and deduce the first-orderoptimality conditions, that yield a system of DAEs with two-point endconditions. Hidden symplectic structure and Symplectic Euler time integrationof optimality conditions have been discussed. The resulting discretefirst-order optimality conditions form a non-linear programming problem that issolved efficiently with the Forward Backwards Sweep Method. Finally, somenumerical examples are provided to demonstrate the comprehensiveness of theproposed computational framework and investigate the energy-efficient optimallimbless locomotion strategy out of distinct locomotion patterns adopted bylimbless organisms.

Adjoint-based optimal control of contractile elastic bodies. Application to limbless locomotion on frictional substrates