A Passive Dynamic Approach for Flapping-wing Micro-aerial Vehicle Control

This article outlines a new control approach for flapping-wing micro-aerial vehicles (MAVs), inspired both by biological systems and by the need for lightweight actuation and control solutions. In our approach, the aerodynamic forces required for agile motions are achieved indirectly, by modifying passive impedance properties that couple motion of the power stroke to the angle of attack (AoA) of the wing. This strategy is theoretically appealing because it can exploit an invariant, cyclical power stroke, for efficiency, and because an impedance-adjusting strategy should also require lower bandwidth, weight, and power than direct, intra-wingbeat control of AoA. We examine the theoretical range of control torques and forces that can be achieved using this method and conclude that it is a plausible method of control. Our results demonstrate the potential of a passive dynamic design and control approach in reducing mechanical complexity, weight and power consumption of an MAV while achieving the aerodynamic forces required for the types of high-fidelity maneuvers that drive current interest in autonomous, flapping-wing robotics. NOMENCLATURE c(r) Chord length of wing, at radius r cm(r) Mean chord position, at radius r COP Center of pressure on wing during flapping c̄w Normalized mean chord length CD Drag coefficient CL Lift coefficient CN Wing-normal force coefficient CT Wing-tangential force coefficient fD Drag force on a blade element FD Drag force on the entire wing fL Lift force on a blade element FL Lift force on the entire wing F∗ L Lift force when αr is optimal fw Flapping frequency (Hz) FN Aerodynamic force normal to wing FT Aerodynamic force tangential to wing kw Spring coefficient for wing rotation k̂w Non-dimensionalized spring coef. m Mass of the MAV r̂cop Normalized, radial distance to COP on wing Re Reynolds number Rw Span of a single wing, root to tip Sw Shape factor (non-dimensional) for the wing ur Air velocity, relative to wing ẑcop Normalized, spanwise distance to COP on wing α Geometric angle of attack, relative to body αr True angle of attack, relative to local airflow ν Kinematic viscosity ρ Density of air τk Wing-pitching torque due to spring forces τψ Wing-pitching torque due to aero forces φ Stroke angle of wing ψ Pitch rotation angle of wing ψo Offset angle for passive spring ωw Flapping frequency (rad/s) INTRODUCTION Many recent approaches to robot locomotion are inspired by the astonishing agility, efficiency and/or robustness of animal locomotion [1–3]. In this paper, we suggest a new approach for the control of underactuated, flapping-wing micro-aerial vehicles (MAVs) and examine its theoretical feasibility. Our motivation is to develop a method that (1) could realistically be implemented to provide a simple, low-weight actuation solution for a real robot, (2) exploits and modifies passive dynamics to steer. Throughout this paper, we assume a difference in time scale between fast flapping and the (slower) time-averaged control forces and torques that steer agile maneuvers. Because it takes several wingbeats for an insect or robot to complete a flapping-wing maneuver, we assume we can use the time-averaged aerodynamics over the course of a wingbeat to closely approximate the true forces over time. Bio-Inspired Flapping Flight Control Insects such as flies use two types of muscles in flight [4]. One set primarily provides power, employing cyclical high frequency, low variability motion. The second set operates indirectly and functions, primarily, to generate steering forces rather than to drive the gross velocity of the wing [5]. Although flies demonstrate amazingly agile maneuvers, as anyone who has attempted to catch one can verify, they employ a “remarkable economy of control” [6] in doing so. They are most certainly underactuated, with significant coupling between the forces and torques generated to control the six degrees of freedom of the body [6]. An underactuated animal or robot, however, can still be controllable, in the sense of finding some motion control plan to orient all six degrees of freedom into a particular configuration in finite time. We examine the feasibility of a similar, bioinspired approach, where a motor and transmission are used to drive a gross flapping motion in a harmonic motion, while the angle of attack (AoA) is indirectly determined via the impedance (stiffness, damping, and inertia) relationship between the wing spar and the wing itself. Although there is direct evidence that real insects employ active methods to flip the wing to control AoA [7], there is also evidence that passive wing pitch rotation is sufficient to generate lift and drag forces for flight [8]. We investigate a new approach where pitch rotation is controlled indirectly, through the mechanism of an adjustable impedance. Other researchers have already considered the case of an actively-powered flapping motion with passive but unrealistically instantaneous rotation of the pitch angle, ψ. This idealization assumes that there is no impedance to resist aerodynamic forces and that the wing flips over instantly until it hits a “hard stop” at some prescribed AoA [9, 10]. Such modeling approaches also assume constant velocity through the forward or backward portions of the wing stroke, φ, requiring a step change in velocity as the wing reverses direction. These design assumptions are impractical if not impossible to achieve in a real flight vehicle, and they do not allow for any variations in AoA. In this paper, we assume the wing “hinge” determining angle of attack has a finite impedance, which can be adjusted through the use of indirect control actuators. Specifically, we analyze a simplified case, where this impedance is a spring; that is, the mass of the wing and any damping at the wing hinge are both assumed to be negligible for this particular analysis. By adjusting the properties of this spring, one indirectly adjusts the relationship between stroke angle and angle of attack, and the question is: “How large are the subsequent steering forces that can be generated?” METHODS This section outlines the model and methods employed in quantifying the utility of this passive dynamic approach for controlling output forces and torque to maneuver the body of a small, flapping-wing vehicle. We assume that only the impedance properties of each wing may change over time, indirectly resulting in variations in the angle of attack throughout a wingbeat that result in some net force over time. In practice, this approach would best be used in conjunction with modulation of the frequency and/or waveform of the wing stroke. However, we intentionally examine an extreme case where the power stroke remains invariant, as a rigorous test of the range of aerodynamic forces that can be generated when the AoA of the wing is set “passively” by the tuned impedance, rather than being directly commanded throughout the stroke.