Elastic-bound conditions for energetically optimal elasticity and their implications for biomimetic propulsion systems

Minimising the energy consumption associated with periodic motion is a priority common to a wide range of technologies and organisms. These include many forms of biological and biomimetic propulsion system, such as flying insects. Linear and nonlinear elasticity can play an important role in optimising the energetic behaviour of these systems, via linear or nonlinear resonance. However, existing methods for computing energetically optimal nonlinear elasticities struggle when actuator energy regeneration is imperfect: when the system cannot reuse work performed on the actuator, as occurs in many realistic systems. Here, we develop a new analytical method that overcomes these limitations. Our method provides exact nonlinear elasticities minimising the mechanical power consumption required to generate a target periodic response, under conditions of imperfect energy regeneration. We demonstrate how, in general parallel- and series-elastic actuation systems, imperfect regeneration can lead to a set of non-unique optimal nonlinear elasticities. This solution space generalises the energetic properties of linear resonance, and is described completely via bounds on the system work loop: the elastic-bound conditions. The choice of nonlinear elasticities from within these bounds leads to new tools for systems design, with particular relevance to biomimetic propulsion systems: tools for controlling the trade-off between actuator peak power and duty cycle; for using unidirectional actuators to generate energetically optimal oscillations; and further. More broadly, these results lead to new perspectives on the role of nonlinear elasticity in biological organisms, and new insights into the fundamental relationship between nonlinear resonance, nonlinear elasticity, and energetic optimality.