Haithem E. Taha

Wing Kinematics Optimization for Hovering Micro Air Vehicles Using Calculus of Variation

The weight and power constraints imposed on flapping-wing micro air vehicles necessitate optimal design of the flapping kinematics. To date, the approach adopted for kinematics optimization has been to assume specific functions for the Euler angles describing the wing motion with respect to the body. Then, optimization is performed on the parameters of these functions. In another approach, a number of instants over the flapping cycle are selected, and optimization is performed on the magnitude of the Euler angles at these instants. This latter approach provides more freedom for the variations of the Euler angles rather than confining them to certain patterns. Yet, in both approaches, finite-dimensional optimization is adopted and, as such, additional constraints are imposed. Considering that the problem is an infinite-dimensional optimization problem, we use in this work the calculus of variations to obtain true optimality. The combination of the quasi-steady aerodynamics and the calculus of variations ap...

The need for higher-order averaging in the stability analysis of hovering, flapping-wing flight

Because of the relatively high flapping frequency associated with hovering insects and flapping wing micro-air vehicles (FWMAVs), dynamic stability analysis typically involves direct averaging of the time-periodic dynamics over a flapping cycle. However, direct application of the averaging theorem may lead to false conclusions about the dynamics and stability of hovering insects and FWMAVs. Higher-order averaging techniques may be needed to understand the dynamics of flapping wing flight and to analyze its stability. We use second-order averaging to analyze the hovering dynamics of five insects in response to high-amplitude, high-frequency, periodic wing motion. We discuss the applicability of direct averaging versus second-order averaging for these insects.

Longitudinal Flight Dynamics of Hovering MAVs/Insects

The dynamics of flapping flight has been a research topic of interest with the objectives of better characterization of insect flight or realization of successful flight of microair vehicles (MAVs). In addition to the complex dynamics associated with flapping flight, the periodicity aspect of the flapping motion gives flapping flight dynamics a time-varying characteristic. Because hovering is usually performed at relatively high frequencies, with respect to the body’s natural frequencies, it is usually claimed that the body of a hovering insect feels only the cycle-averaged aerodynamic loads. In fact, this is a very common assumption in the analysis of flapping flight dynamics. Some research reports use this assumption on the basis of physical intuition [1–7]. Others use its rigorous mathematical representation [8–10]; namely, the averaging theorem. There still exists some debate about the range of validity of this assumption; see Taha et al. [11] for a discussion about the topic. If the averaging assumption is acceptable, or the dynamics of the insect under study are amenable to the averaging theorem, then one can convert the time-varying system into an autonomous one (averaged system). The stability of the averaged system is indicative of the stability of the original nonlinear time-periodic (NLTP) system. The inherent instability of any flying vehicle is mainly dictated by the aerodynamic loads due to the vehicle’s motion (stability derivatives). Several experimental [2,12] and computational [3–5,13] investigations have been performed to determine the cycle-averaged stability derivatives for hovering insects and MAVs. However, there have been few trials that aimed at deriving analytical expressions for the stability derivatives in terms of the system parameters. Cheng and Deng [14] derived the cycle-averaged stability derivatives for hovering insects in terms of their morphological parameters. However, because they did not adopt a specific aerodynamic model, they provided general expressions in terms of the cycle-averaged aerodynamic lift and drag coefficients that, in turn, need to be expressed in terms of the system parameters. Moreover, they did not account for the aerodynamic rotational effects [15], which may considerably affect pitch damping. Also, they considered only one trim configuration at hover that yielded four nonzero longitudinal stability derivatives out of the full nine. In this work, a formal derivation of the full longitudinal stability derivatives for hovering MAVs/insects is presented. A quasi-steady aerodynamic model that accounts for the dominant leading edge vortex contribution and the rotational effects is used. As such, the final expressions for the stability derivatives are provided directly in terms of the system parameters. For validation purposes, a numerical simulation for the aerodynamic model is performed over one cycle and a complex-step finite differencing is used to determine the stability derivatives of the hawkmoth using the kinematics and trim data considered by Sun et al. [4] in their direct numerical simulation of Navier–Stokes equations. The effects of trim configuration on the cycle-averaged stability derivatives are then determined. The averaging theorem is used to analytically assess the stability of this time-periodic system. Finally, a parametric study for the cycleaveraged stability derivatives and the eigenvalues of the averaged, linearized system using the symmetric flapping trim configuration (SFTC) is performed.

Unsteady Nonlinear Aerodynamics of Hovering MAVs/Insects

The Duhamel superposition principle, applied in unsteady linear aerodynamics, is extended to arbitrary nonlinear lift curves. The unsteady lift due to arbitrary wing motion is generated using the static nonlinear lift curve. A finite-state unsteady aerodynamic model that is based on a two-state approximation of Wagner function is developed. As such, it is suitable for dynamic stability analysis and control synthesis for the systems experiencing nonlinear unsteady flows such as wind turbine blades, rapidly maneuvering aircraft, highly flexible wings, and micro-air-vehicles. The proposed unsteady model is applied to flapping flight and validated through a comparison with direct numerical simulations of Navier-Stokes on hovering insects.

Geometric Control Approach to Longitudinal Stability of Flapping Flight

Flapping flight dynamics is appropriately represented by a multibody, nonlinear, time-varying system. The two major simplifying assumptions in the analysis of flapping flight stability are neglecting the wing inertial effects and averaging the dynamics over the flapping cycle. The challenges resulting from relaxing these assumptions naturally invoke the geometric control theory as an appropriate analysis tool. In this work, the full equations of motion governing the longitudinal flapping flight dynamics near hover are considered and represented in a geometric control framework. Then, combining tools from geometric control theory and averaging, the full dynamic stability of hovering insects is assessed.

Design Optimization of Flapping Ornithopters: The Pterosaur Replica in Forward Flight

There has been a recent interest to explore the shape and kinematics parameters of distinct pterosaurs from their fossil records. Clearly, far more evidence is needed to understand the nuances of dinosaurs flight. A multiobjective aerodynamic optimization problem of the wing kinematics and planform of a pterosaur replica ornithopter designed by Aerovironment is performed. Objective functions include minimization of the required cycle-averaged aerodynamic power and maximization of the propulsive efficiency. It is found that inclusion of the inertial power requirements is necessary for a physical and proper formulation of the optimization problem. Furthermore, the mere addition of the inertial power requirements is not enough to obtain reasonable results. Rather, one has to consider a partial (or even zero) elastic energy storage. The minimum power kinematic parameters closely match those of the previously designed pterosaur replica. Nevertheless, the obtained efficiency for such a design (minimum power) is...

Experimental-Based Unified Unsteady Nonlinear Aerodynamic Modeling For Two-Dimensional Airfoils

Unsteady force measurements of a plunging airfoil at different frequencies and mean angles of attack are used to construct frequency response models. The obtained frequency response models are then used to determine the linearized flow dynamics around each mean angle of attack, where an optimization-based linear system identification is performed to minimize the error between the predicted and measured frequency responses. Converting these models to state space form and writing the entries of the matrices as polynomials in the mean angle of attack, a global, unified unsteady model is developed. The developed reduced order model, represented in a state space form, is suitable for characterizing the nonlinear dynamical characteristics of the flow and the associated dynamics and control of a flying object.

Geometric control of a flapping plate

Control of flapping micro-air-vehicles (MAVs) is challenging because the system models are nonlinear and time-varying. Moreover, as internally actuated multi-body systems, flapping MAVs are inherently underactuated. With stringent weight and size constraints, the actuator mechanization must be as simple as possible, introducing a further challenge for control design. Geometric control and averaging theory can be used to design control laws for underactuated nonlinear systems. In this work, we consider control design for a flapping plate with three degrees of freedom and two actuators. The averaging theorem and geometric control methods are used to stabilize and control the system. The simple example demonstrates an algorithmic approach that could be used within a multi-disciplinary design optimization framework for the design of biomimetic vehicles and their gaits.

Control of underactuated mechanical systems using high frequency input

This paper describes an approach to closed-loop control of a class of underactuated mechanical systems using a high frequency periodic input whose amplitude is modulated through feedback. The approach makes use of the averaging theorem and the time scale separation between the high frequency forcing and the low frequency amplitude modulation. The paper begins with the special case of a two degree of freedom system with a single input. A closed-loop controller is developed that forces the time-averaged value of the unactuated coordinate to follow a desired (low frequency) trajectory. The method is then extended to systems with more degrees of freedom and multiple inputs and it is modified to enable low frequency trajectory tracking in the (averaged) actuated coordinates, as well as the unactuated coordinates. To illustrate, the approach is applied to the problem of position control for a three degree of freedom flapping plate in a uniform flow.

Coupled Unsteady Aero-Flight Dynamics of Hovering Insects/Flapping Micro Air Vehicles

The main objective of this paper is to provide a rigorous coupling between body flight dynamics and flow dynamics (unsteady aerodynamics) in hovering of insects and flapping-wing micro air vehicles...