Optimal swimming with body compliance in an overdamped medium

Abstract

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.

Figures (13)

• Figure 1: Concept of body compliance in a cable-driven three-link robophysical model and nematodes C. elegans. (a) The three-link cable-driven swimmer, mounted on a gantry, was immersed in a granular medium. (b) Three-link cable-driven swimmer robophysical model (skin off), with bilateral cables routed through pulleys and actuated by servo motors to produce in-plane bending and body compliance. (c) The nematode C. elegans has the compliant body walls and bilateral muscles along its body.
• Figure 2: Analytical model of the compliant three-link swimmer. Analytical three-link model with a body frame corresponding to a weighted average of the link positions and orientations. Each joint includes a motor connected in series with a spring. Insets illustrate both linear and nonlinear springs, which can be captured by the model.
• Figure 3: Tools from geometric mechanics for modeling and optimization. (a) The local connection vector field from geometric mechanics, which map joint velocities to body velocities and provide the foundation for displacement prediction. (b) The height function, the curl of the local connection vector field. The net displacement from a gait (purple) corresponds to the areas it encloses on the height function. The unit of the height function is (body length)/(radian$^2$), and its values are scaled by a factor of 100.
• Figure 4: Optimization flow for identifying optimal gaits under body compliance. First, the optimal emergent gait is identified by deriving the height function. Then, by incorporating the inverse body dynamics, the corresponding optimized gait is obtained.
• Figure 5: Cable-driven robophysical model design and joint compliance mechanism. Schematic of bilateral cable actuation at a single joint, where left and right cables tensioned to form the exact suggested joint angle (b), and left and right cable slacked to form a compliant region, so that the emergent joint angle can deviate from the suggested angle (c).