Fig. 6. (A) A three-dimensional (3D) representation y_i+1(y_i, ₀) of the return map y_i+1(y_i) characterizes spring-mass running (system energy corresponds to v_X=5 m s^-1 at y_APEX=1 m; m=80 kg, l₀=1 m, k=20 kN m^-1) for different angles of attack ₀. For fixed angles of attack (slices in 3D), the corresponding return maps are shown on the left (y_i, y_i+1) plane. The red line depicts the return map for ₀=68°. Different return maps are possible if the angle of attack ₀ becomes dependent on the apex height y_i. An `optimal' control model with respect to stability would be a direct projection of any initial apex height y_i to a desired apex height y_CONTROL in the next flight phase, or y_i+1(y_i)=y_CONTROL=constant, as shown for apex heights of 1, 1.5 and 2 m (left plane). This corresponds to isolines on the 3D-surface y_i+1(y_i, ₀) indicating a dependency between the angle of attack ₀ and the initial apex height y_I, as shown for y_CONTROL=1, 1.5 and 2 m in (B). With careful selection of the retraction velocity _R and the retraction angle _R, the constant velocity leg retraction model can approximate the optimal control strategy.

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