Elongate animals and robots use undulatory body waves to locomote through diverse environments. Geometric mechanics provides a framework to model and optimize such systems in highly damped environments, connecting a prescribed shape change pattern (gait) with locomotion displacement. However, the practical applicability of controlling compliant physical robots remains to be demonstrated. In this work, we develop a framework based on geometric mechanics to predict locomotor performance and search for optimal swimming strategies of compliant swimmers. We introduce a compliant extension of Purcell's three-link swimmer by incorporating series-connected springs at the joints. Body dynamics are derived using resistive force theory. Geometric mechanics is incorporated into movement prediction and into an optimization framework that identifies strategies for controlling compliant swimmers to achieve maximal displacement. We validate our framework on a physical cable-driven three-link limbless robot and demonstrate accurate prediction and optimization of locomotor performance under varied programmed, state-dependent compliance in a granular medium. Our results establish a systematic, physics-based approach for modeling and controlling compliant swimming locomotion, highlighting compliance as a design feature that can be exploited for robust movement in both homogeneous and heterogeneous environments.

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