Adaptive control for flapping wing robots with history dependent, unsteady aerodynamics

This paper derives a history dependent formulation of the equations of motion of a flapping wing, ground-based robotic system and constructs an associated adaptive control strategy to track observed flapping motions in insect flight. A general methodology is introduced in which lift and drag forces are represented in terms of history dependent integral operators to model and identify the unknown and unmeasurable aerodynamic loading on the flapping wing robot. 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 are derived via quadrature formula and finite element methods. These approximations yield history dependent equations that evolve in euclidean space. An adaptive control scheme based on passivity principles is 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 ODE systems.