Two explanations for the compliant running paradox: reduced work of bouncing viscera and increased stability in uneven terrain

Abstract

Economy is a central principle for understanding animal locomotion. Yet, compared with theoretical predictions concerning economy, animals run with compliant legs that are energetically costly. Here, we address this apparent paradox, highlighting two factors that predict benefits for compliant gaits: (i) minimizing cost of work associated with bouncing viscera; and (ii) leg control for robust stability in uneven terrain. We show that consideration of the effects of bouncing viscera predicts an energetic optimum for relatively compliant legs. To compare stability in uneven terrain, we introduce the normalized maximum drop (NMD), a measure based on simple kinematics, which predicts that compliant legs allow negotiation of relatively larger terrain perturbations without failure. Our model also suggests an inherent trade-off in control of leg retraction velocity (omega) for stability: low omega allows higher NMD, reducing fall risk, whereas high omega minimizes peak forces with terrain drops, reducing injury risk. Optimization for one of these factors explicitly limits the other; however, compliant legs relax this trade-off, allowing greater stability by both measures. Our models suggest compromises in leg control for economy and stability that might explain why animals run with compliant legs.

Figure 1.

Two factors that may account for the compliant legs and large stance angles used by running animals. (a,b) A model that includes mechanical work of the legs and hysteresis losses from bouncing ‘viscera’ (a), suggests that compliant legs are favourable for economy. (b) For infinite viscera stiffness (black line), equivalent to a point mass model, mechanical cost of transport (MCoT) increases with stance half-angle (Φ), and impulsive running with an infinite k_leg (Φ = 0) is energetically optimal. If the viscera dissipate energy (green lines), however, compliant legs become favourable for economy. (c,d) Compliant legs also provide robust stability in uneven terrain. Normalized maximum drop (NMD) estimates the maximum drop relative to leg length (ΔH_max/L_leg) before the leg misses stance entirely ((i) in c,d). For a fixed running speed, swing period and mean leg retraction velocity (, shown as dimensionless ) compliant legs ((ii) in c,d) have higher NMD than stiff legs ((iii) in c,d). Grey box in (b,d) indicates approximate Φ range used by animals (Farley et al. 1993).