Modeling the dynamics of spring-driven oscillating-foil propulsion

In this paper we present a model for oscillating-foil propulsion in which springs are used to transmit forces from the actuators to the foil. The expressions for hydrodynamic force and moment on the foil come from classical, linear, unsteady aerodynamics, and these are coupled to linearized rigid-body mechanics to obtain the complete model for swimming. The model is presented as a low-order set of ordinary differential equations, which makes it suitable for the application of techniques from systems and control theory. The springs serve to reduce energy costs, and we derive explicit expressions for spring constants which are optimal in this sense. However, the use of springs can potentially lead to unstable dynamics. Therefore, we also derive a set of necessary and sufficient conditions for stability. A detailed example is presented in which energy costs for one actuator are reduced by 33%.