Identification and adaptive control of history dependent unsteady aerodynamics for a flapping insect wing

A history dependent formulation for the equations of motion of a ground-based robotic flapping wing, derived for an associated adaptive control strategy, was constructed to track observed flapping motions of insect flight. A general methodology was introduced in which lift and drag forces were represented in terms of history dependent integral operators to model and identify the unknown and unmeasurable aerodynamic loading on the flapping robot wing. First, computational fluid dynamics was used to predict the drag and lift forces for a flapping insect wing. The simulation data were then used to describe the adaptive control history. The resulting closed-loop system constitutes an abstract Volterra integral equation whose state consists of the finite-dimensional vector of generalized coordinates for the robotic system and an infinite-dimensional unknown function characterizing the kernel of the history dependent integral operator. Finite-dimensional approximations of the state equations were derived via quadrature formula and finite element methods. An adaptive control scheme based on passivity principles was derived for the approximate history dependent system. Lyapunov analysis guarantees stability of the closed-loop system, and that the tracking error and its derivative converge to zero. The novel control strategy introduced in this paper is noteworthy in that by introducing a history dependent adjoint operator in the state estimate equation, the analysis for convergence of the closed-loop history dependent equations closely resembles the analysis used for conventional ordinary differential equation (ODE) systems.