Fig. 2: Robotic fish save energy by vortex phase matching (VPM).

a Relative power coefficient η shown as a function of the phase difference between the leader and the follower Φ and front-back distance D at left–right distance G = 0.31 BL. The dashed line (also in b) shows the functional relationship described in Eq. (2) that determines the theoretical phase relationship that maximally saves energy (Methods section). \({\Phi }_{0}^{* }\) is the optimal initial phase difference (fitted to the data points of maximum energy saving, as shown in b). The points marked by red square, blue circle and blue square indicate example cases depicted on panels c–e. b Location of maximal energy saving in the robotic trials. Point size and darkness denote the number of occurrences of each phase difference value at each front-back distance. c–e An illustration of important spatial configurations for vortex phase matching. Energy cost is related to how the follower moves its body relative to the direction of the induced flow of the vortices, in the opposite direction with Φ₀ = \({\Phi }_{0}^{* }\)+π (c) or in the same direction with Φ₀ = \({\Phi }_{0}^{* }\) (d, e). Followers interact with the induced flow of vortices with the same body phase at any front-back distance (within the range of hydrodynamic interactions), termed vortex phase matching. (d, e; Φ₀ = \({\Phi }_{0}^{* }\) describes the hydrodynamic interaction resulting in energy saving, see description in the text). As the front-back distance changes, the followers must dynamically adopt phase difference Φ, with respect to that of the leader.