Dipteran insect flight dynamics. Part 2: Lateral–directional motion about hover
Introduction
The computational workload an insect must perform to support flapping wing flight is still an active area of research. Experimental evidence has indicated physiological components involved in flight stabilization, such as tangential cells and descending neurons that translate optic flow estimates to flight motor commands. Despite the measurements made on the sensing and control components, the flight stabilization demands that these structures must support are unknown, but must be addressed via an analysis of the plant they are responsible for controlling (i.e., the insect dynamics). Recently, Taylor and Thomas, 2003a, Taylor and Thomas, 2003b used measurements of tethered locusts to estimate the stability and control properties associated with dipteran insect flapping flight, while Hesselberg and Lehmann (2007) and Hedrick et al. (2009) addressed the yaw dynamics of flapping flight. Aerodynamics investigations using computational fluid dynamics (CFD) (Sun and Xiong, 2005) have also addressed the longitudinal (motion in the insect's plane of symmetry) flapping flight. A concise understanding of how the control inputs available to a dipteran insect (in the form of kinematic variables) affect the long term motion of the insect (flight control) is expected to provide insight into why insects tend to use particular control inputs more readily than others and indicate how coupled control inputs can lead to particular motions. A large body of knowledge exists to provide for control analysis on state-space dynamics models (Ogata, 2002), meaning that a process that simplifies the relatively complex nonlinear dynamics of dipteran flapping wing insect flight is desirable. State-space models and the process to generate them are goals of this study. The simplified linear time invariant dynamics models may then be used to evaluate controllability and design control laws for hovering micro-air-vehicles near hover.
Faruque and Humbert (2010) analyzed the longitudinal dynamics of a flapping wing insect about hover using quasi-steady aerodynamics models (Dickinson, 1996) and Euler rigid body dynamics. The results indicated slow and fast subsidence modes, as well as an oscillatory pair that could be stabilized via halteres providing pitch rate feedback. The present study continues to examine the implication of passive aerodynamic stability mechanisms associated with flapping flight, this time for lateral–directional motion about hover. The goal of this study is to develop simplified dynamics models of flapping flight lateral motion about hover, which may be readily interpreted via traditional full scale aircraft dynamics and linear control analysis techniques. The underlying mechanics are again posed as nonlinear Euler rigid body dynamics paired with quasi-steady aerodynamics modeling including the effects of perturbations from the equilibrium. The results show that the quasi-steady aerodynamics model placed in context of state perturbations is able to predict dynamic behavior without the use of a more detailed and computationally intensive 3D CFD study such as Ramamurti and Sandberg, 2002, Ramamurti and Sandberg, 2007.
The organization of the paper is as follows. Section 2 reviews the kinematics, control inputs, aerodynamics model, aerodynamics averaging, and the method developed in Faruque and Humbert (2010) of extending the quasi-steady model to include perturbations from the hover point. Section 3 describes the system identification procedure using a nonlinear simulation environment encoding the perturbation velocities concept. Section 4 shows how the perturbation velocities concept can be used to develop estimates of the stability and control derivatives, and Section 5 presents identified models and shows how these results suggest passive aerodynamic mechanisms may reduce an insect's flight control requirements about hover.
Section snippets
Background
In this section, a review of quasi-steady aerodynamic theory and the governing equations for the analysis and simulation is presented.
System identification
To facilitate the identification of an equivalent linear system, the simulator developed in Faruque and Humbert (2010) was used in conjunction with Comprehensive Identification from FrEquency Responses (CIFER). The goal of this process was to identify a linear system that would allow predictions of the flapping wing flight control properties. Frequency sweeps (“chirp” signals) were applied to the inputs and to excite the lateral/directional dynamics of the insect. Based on the spectral
Wingstroke averaged forces and moments
In this section, the process of wingstroke averaging developed in Faruque and Humbert (2010) is applied to demonstrate how linearization of rigid body equations of motion forced by quasi-steady aerodynamics can provide estimates of the control input terms and also indicate physical mechanisms providing aerodynamic damping. The linear systems representing rigid body motion were derived with the understanding that we are concerned only with the low frequency rigid body modes describing the flight
Yaw dynamics
Consider the yaw dynamics case previously discussed in Hedrick et al. (2009), where a linear yaw dynamics model of the form was calculated, where a is formed from the wing's geometric and kinematic parameters as Here, the term formed by the force coefficient CF, geometric wing angle , and nondimensional flap speed has the mean value . Using parameters from the simulated insect, the FCT model predicts an aerodynamic
Conclusions
In this study, the quasi-steady aerodynamics model including perturbations (Faruque and Humbert, 2010) analyzed to determine reduced order control models that may be used to estimate the control properties associated with lateral–directional flight. The models were derived with the goal of describing the low order rigid body motion of an insect (not high frequency structural dynamics), so frequency-based system identification tools and wingstroke-based averaging were both used in deriving them.
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