Stroke Plane Control for Longitudinal Stabilization of Hovering Flapping Wing Air Vehicles

Abstract

I N CONTRAST to the tailed flapping-wing micro air vehicles (MAVs), which possess inherent longitudinal static stability from the tail wing [1], unstable dynamic characteristics have been reported for the tailless flapping-wing MAVs [2–6]. Even though there exist flapping counterforces and torques [7,8] that provide each isolated degree of freedom (DOF) with stable damped dynamics, the coupled translational motion with pitching motion causes unstable longitudinal dynamics of the tailless flapping-wing MAVs [6,9,10]. Many studies have been conducted on the stabilization and control of flapping-wing MAVs [11–20]. Based on Doman et al. [11], Oppenheimer et al. [12] used the split-cycle constant-period frequency modulation with wing bias to control all the 6-DOF of an aircraft concept driven by two bimorph piezoelectric actuators. Bhatia et al. [20] used several sets of wing kinematic parameters for the stabilization of the vehicle in a gusty environment. Most previous studies have also controlled mathematically parameterized wing kinematics, which demand complex articulation mechanisms when they come to an actual hardware implementation [11–20]. Besides the controllability of the system, the control parameters of artificial flapping-wingMAVs need to consider actual implementation issues; a smaller number of control inputs is also intuitively preferable. This study presents the stabilization of all the six longitudinal flight state variables using two control parameters, which are the stroke plane angle and the wingbeat frequency. Within a limited range, the wingbeat frequency is one of the easily adjustable control parameters; virtually all the commercial toy flappers employ it as one of the control inputs. Instead of modulating the entire wing kinematics as proposed in [11–20], we propose to control the stroke plane angle for the stabilization of longitudinal flight dynamics. The effects of controlling the stroke plane angle and the wingbeat frequency to the flight dynamic stability are mainly characterized by establishing a flight simulator [21] of a hovering insect (i.e., the hawkmothManduca sexta). The flight trajectory of themodel vehicle is obtained by directly integrating the longitudinal nonlinear equations of rigid-body motion with time-varying wing inertia, which is coupled with a quasi-steady blade-element aerodynamic model. Linearization of the full nonlinear equations of the hovering flapping-wing MAV is conducted; the linear time-invariant system model is obtained by applying both small perturbation theory and cycle-averaging theory. The full-state feedback controller of the flapping-wing MAV is designed by minimizing a quadratic performance and is implemented on the longitudinal nonlinear equations of motion to confirm the effectiveness of the controller. Also, we found a very simple control strategy for a longitudinal stabilization: maintaining the stroke plane angle constant with respect to the global coordinates, based on an investigation in the designed optimal control inputs and consequent pitching dynamics of the flapping-wing MAV model. The effectiveness of this simple control strategy is verified using a high-fidelitymultibody simulation environment [1,6].