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

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 (ω) for stability: low ω allows higher NMD, reducing fall risk, whereas high ω 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.