a, Schematics of (i) cercariae swimming tail-first and (ii) proposed ‘T-swimmer’ model. (iii) Photograph of a macroscale, self-propelled, ‘T-swimmer’ robot (Supplementary Methods) designed to swim in a high-viscosity fluid (corn syrup, viscosity μ ≈ 8 Pa s) chosen to be dynamically similar (Rerobot ≈ 0.2) to swimming cercariae. The joint between the longitudinal links (red dot in (ii)) is active and actuated periodically with a given amplitude A and frequency f. The transverse joint is assumed to be a passive linear torsional spring (depicted as a black spiral) with stiffness Γtf to model the flexibility of the tail–fork joint in cercariae. The red and blue arrows in (iii) indicate the active and passive joints, respectively in the robot. Scale bar in (iii), 5 cm. b,c, Phase plots of the joint angles φtf and φt for a T-swimmer model (b) and robot (c) for a range of Γtf, showing the non-reciprocal nature of the swimming cycle, where Γtf is Γtf normalized by the torque scale μflc3. The arrows indicate the direction of phase trajectories. d,e, Plots of the average swimming speed (normalized by flc) for the T-swimmer model (d) and robot (e) as a function of Γtf for different actuation amplitudes A. Both the model (d) and robot (e) swim with the passive joint preceding the active joint, and the average swimming speed (normalized by flc) shows a single maximum at an O(1) value of Γtf, highlighting an optimal value of torsional stiffness for a given ‘T-swimmer’. The horizontal lines in e indicate measured swimming speeds for a robot with a free joint (dashed lines) and fixed joint (dotted lines), respectively. These speeds are an order of magnitude smaller than the peak values. Error bars correspond to standard deviations over different experiments. f, Snapshots of final positions after 60 s of swimming for T-swimmer robots with a range of Γtf (i)–(v) and frequency maintained at ≈0.4 Hz. The white dashed line denotes the starting point of the robots. The free (i) and fixed joint (v) robots show relatively small displacements (Supplementary Movie 7). Scale bar in (i), 5 cm.