Integrated efficiency: a performance metric for oscillating appendages.*

Current widely-used propulsion efficiency metrics depend on a moving body’s overall dynamics. For example, the Froude efficiency uses the body speed, thrust force, and input power. However, in many cases one must evaluate the performance of a globally fixed propulsor, which these efficiency metrics do not address. For example, behavioral experiments often use tethered animals, where swimming speed is not defined. Numerical simulations frequently target fixed appendages to lower computational costs. Robotic platforms usually begin their development at the propulsor scale when studying potential locomotion strategies. Here we present a new efficiency metric (the integrated efficiency) which considers the force and velocity distributions along an appendage. The integrated efficiency can be computed for propulsors independently from the full-body swimming dynamics, yielding additional useful performance data. We perform experiments on bio-inspired flexible robotic paddles, showing (via the integrated efficiency) how the propulsor’s deformed shape during the beat cycle impacts performance in producing thrust and lift. The integrated efficiency can be used to improve animal propulsion studies and as a design parameter for future bio-inspired robots and devices.