Mathematical models for the self-propulsion of articulated bodies in fluids at low Reynolds number can be interpreted geometrically in terms of connections in principal bundles. Visualization of the local curvature of a connection in this context can serve as a tool for motion planning. We use laboratory data from physical experiments to estimate the local curvatures of connections for two planar robotic systems — one canonical, one novel — propelling themselves through fluids at Reynolds numbers of order one. In each case, the estimated curvature agrees with that predicted by a trusted theoretical model, validating our approach as a new strategy for extracting curvature estimates from physical data when theoretical model are unavailable. For one of the two robotic systems in question, we also employ reinforcement learning to identify piecewise constant control inputs that optimize unidirectional translation. Our results show that optimal locomotion over long time scales is achieved through cyclic actuation that exploits curvature according to the geometric theory — and that exploits the nontrivial topology of the manifold of internal robot shapes — while optimal locomotion over short times scales requires adaptation for different initial conditions. We also investigate the self-propulsion of pitching hydrofoils at higher Reynolds numbers through physical experiments with freely swimming two- and four-hydrofoil arrays, resembling simplified fish schools, in a pool of water. These experiments highlight the influence of hydrodynamic coupling among individual foils on the overall speed and efficiency of such arrays. We modulate this coupling by varying the spacing among foils and by varying the manner in which individual foils pitch as functions of time, recording both swimming speed and power consumption to identify optimal cooperative pitching patterns.