Mathematical Modelling of Near-Hover Insect Flight Dynamicsxdeng/DSCC.pdf · MATHEMATICAL MODELLING...

MATHEMATICAL MODELLING OF NEAR-HOVER INSECT FLIGHT DYNAMICS

B. Cheng X. Deng

School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907

[email protected] ABSTRACT

Using a dynamically scaled robotic wing, we studied the aerodynamic torque generation of flapping wings during roll, pitch, and yaw rotations of the stroke plane. The total torque generated by a wing pair with symmetrical motions was previously known as flapping counter-torques (FCTs). For all three types of rotation, stroke-averaged FCTs act opposite to the directions of rotation and are collinear with the rotational axes. Experimental results indicate that the magnitude of FCTs is linearly dependent on both the flapping frequency and the angular velocity. We also compared the results with predictions by a mathematical model based on quasi-steady analyses, where we show that FCTs can be described through consideration of the asymmetries of wing velocity and the effective angle of attack caused by each type of rotation. For roll and yaw rotations, our model provided close estimations of the measured values. However, for pitch rotation the model tends to underestimate the magnitude of FCT, which might result from the effect of the neglected aerodynamics, especially the wake capture.

Similar to the FCT, which is induced by body rotation, we further provide a mathematical model for the counter force induced by body translation, which is termed as flapping counter-force (FCF). Based on the FCT and FCF models, we are able to provide analytical estimations of stability derivatives and to study the flight dynamics at hovering. Using fruit fly (Drosophila) morphological data, we calculated the system matrix of the linearized flight dynamics. Similar to previous studies, the longitudinal dynamics consist of two stable subsidence modes with fast and slow time constants, as well as an unstable oscillatory mode. The longitudinal instability is mainly caused by the FCF induced by an initial forward/backward velocity, which imparts a pitch torque to the

same direction of initial pitch velocity. Similarly, the lateral dynamics also consist of two stable subsidence modes and an unstable oscillatory mode. The lateral instability is mainly caused by the FCF induced by an initial lateral velocity, which imparts a roll torque to the same direction of initial roll velocity. In summary, our models provide the first analytical approximation of the six-degree-of-freedom flight dynamics, which is important in both studying the control strategies of the flying insects and designing the controller of the future flapping-wing micro air vehicles (MAVs).

INTRODUCTION Recent studies on the tuning dynamics of animal flight [1,

2] showed that during low-speed yaw turns (rotation about the vertical axis such as saccade), flapping wing fliers ranging in size from fruit flies to large birds are subject to substantial passive damping through an aerodynamic mechanism termed flapping counter-torque (FCT). As an inherent property, FCT helps the flapping-wing fliers to slow down body rotation during rapid maneuvers and thus reduces the required active torque produced by asymmetries of wing motion. As a trade-off, however, flapping-wing fliers must overcome extensive aerodynamic damping (a result of FCT) to accelerate or to initiate a maneuver [1]. Not only was the passive damping found crucial during fast yaw rotations, simulation results [1] suggested that it is also present during roll and pitch maneuvers. In flying animals, measurements of body kinematics showed that most yaw turns are accompanied by substantial change in roll angular velocity [3]. Even at low speed maneuvering or hovering, most flapping wing flies perform banked turns which involve rolling. Furthermore, during escape or tracking flight, rapid reorientations of roll and pitch angles (causing reorientation of the net aerodynamic force vector) are essential for fliers to achieve fast maneuvers [4, 5].

1 Copyright © 2010 by ASME

Proceedings of the ASME 2010 Dynamic Systems and Control Conference DSCC2010

September 12-15, 2010, Cambridge, Massachusetts, USA

DSCC2010-4234

Downloaded 29 Sep 2012 to 128.46.184.175. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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decaying magnitudes. The unstable oscillatory mode corresponds to a near out-of-phase coupling of and (phase difference is larger than 125º); therefore, in large part of an oscillating cycle, the insects pitch up (down) while moving forward (backward), also refers to [19]). Intuitively, for the in-phase coupling of initial and , pitch FCT and the pitch torque generated by FCF ( ) act together to reduce the angular velocity , resulting in stable subsequent motions. For the out-of-phase coupling of and , however, the pitch torque induced by FCF ( ) acts in the same direction as , thus magnifying the pitch angular velocity and causing the instability (see Discussion). Slow subsidence (mode 3) corresponds to a damped ascending/descending motion resulting from the FCF in a vertical direction.

TABLE 3 DIMENSIONAL AND NON-DIMENSIONAL EIGENVALUES

Longitudinal Lateral Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6

-39.15 14.17 ± 30.62i -5.11 -143.02 9.12 ±

34.85i -43.30

-0.18 0.067 ± 0.14 i -0.024 -0.67 0.043 ±

0.16i -0.20

Eigenvalues (dimension of Hz) are non-dimensionalized by dividing the flapping frequency . The inverse of Re( ) and Re( ) indicate the dimensional and non-dimensional time constants of the corresponding mode.

Similar to the longitudinal dynamics, the lateral dynamics consists of two subsidence modes with relative fast (mode 4) and slow (mode 6) convergences and an unstable oscillatory mode (mode 5). The fast subsidence mode corresponds to a highly damped yaw rotation (the larger value of means less wing-beat time to reduce the initial disturbance, Table 3) with an out-of-phase coupling of and . The slow subsidence mode has a similar out-of-phase coupling of and (with difference magnitudes), but with less damped yaw rotation. The unstable oscillatory has a near in-phase coupling of and

. Intuitively, for the out-of-phase coupling of initial and , roll FCT and the roll torque generated by FCF (

) act together to reduce the angular velocity , resulting in stable subsequent motions. For the in-phase coupling of and , however, the roll torque induced by FCF ( ) acts in the same direction as , thus magnifying the roll angular velocity and leading to an unstable subsequent motion.

DISCUSSION

Effects of unsteady aerodynamics on flapping counter-force/torque production

In the FCT models (Eqns. 1 to 5), we considered only the translational aerodynamic force resulting from the delayed stall and ignored other aerodynamic mechanisms [20]. Previous works show that most of those unsteady aerodynamic mechanisms alter the aerodynamic force transients near stroke

reversals. For example, the Kramer effect generates rotational forces before or after stroke reversals, depending on the phase of wing rotation, but wake capture occurs immediately after the stroke reversals [15].

As indicated by the plots of instantaneous torques about roll, pitch, and yaw axes (Figs. 4, 6, and 8), the peaks of roll and yaw torques occur near the middle stroke where their moments of arm are maximized. Pitch torque, however, reaches peak between the stroke reversal and the middle stroke. Therefore the unsteady aerodynamic effects, especially the wake capture, are most likely to affect pitch torque transients at the two quarter strokes after stroke reversals (ventral to middle and dorsal to middle strokes, Fig. 6). The roll and yaw torques, however, might be less affected by those unsteady effects. This might explain the discrepancy between the measured and the predicted pitch FCT (solid and dotted green, Fig. 6), especially at the two quarter strokes after the stroke reversal.

Flapping counter-torque and stability derivatives during free flight In the current study, we assumed a constant angular velocity during rotations around each principal axis of the stroke-plane frame and measured the corresponding FCT over one wing stroke. In free flight, however, the insect is expected to experience a time-varying angular velocity on non-principal axes. Previous simulation results suggest that a rotation about a non-principal axis would also yield an FCT that linearly depends on both stroke frequency and angular velocity. However, it will be very difficult to obtain mathematical models for such rotations.

Furthermore, how FCTs depend on the accelerations is still undetermined. Although the FCT and FCF models can be used to estimate the first derivatives with respect to time (e.g., ), we are unable to estimate the second derivatives (e.g., ). For fixed and rotary aircraft, such stability derivatives are important under some flight conditions (short-period mode, for example), in which the time constant associated is so short that the acceleration becomes critical to aerodynamics [17, 21]. However, it is still reasonable to assume that the first derivatives dominate the aerodynamics in most flight conditions [17, 21].

It should also be pointed out that by applying the small perturbation theory to approximate aerodynamic torques, we also neglected the nonlinear terms with higher orders, such as

, , , , and . The estimation of these terms can be directly obtained in the FCF or FCT model. For example, it can be shown that and are equal to ρ L α and ρ L α , respectively.

Aerodynamic damping during non-hovering flight By assuming the hovering condition in our study, we

consider that the effect of body velocity on flapping-wing aerodynamic model is negligible; therefore the previous quasi-steady aerodynamic model [15] is applicable. However, in non-hovering flight, i.e., forward flight with high advance ratio, the



aerodynamic model needs revision to give better estimations. Unfortunately, the understanding of the flapping-wing aerodynamics in non-hovering flight is now very limited. With respect to the forward flight, Dickson and Dickinson (2004) studied the effect of advance ratio on revolving wings and modified the quasi-steady aerodynamic model. This result can be directly applied to estimate the FCF in forward and backward translations based on Eqn. 6, with the revised lift and drag coefficients (by taking account of tip velocity ratio ). Similar to forward flight, vertical and lateral flights are very likely to have non-negligible effects on the aerodynamics. For example, in rotary wing aircraft a vertical descent changes the aerodynamics by reducing the downward momentum produced, and it greatly affects the lift force [17, 21]. A large descending velocity can even lead the flight into a so-called vortex-ring state, which is characterized by unstable flow conditions with erratic lift variations. Therefore an understanding of the aerodynamics, as well as the measurements of the force coefficients during non-hovering flight conditions, is desirable for further investigation of insect flight dynamics. However, as an approximation of the flight dynamics, our models provided an applicable analytical tool, which is important to designing the controllers for the flapping-wing MAVs, as well as to understanding the control strategies adapted by flying insects. For example, in the aircraft controller design, the uncertainties resulted from the neglected aerodynamics can be simply viewed as a disturbance term, which can be compensated by appropriate robust control strategies.

ANNEX A A mathematical model of FCT for yaw rotation has been

described in previous studies [1, 2]. In this appendix, we provide a derivation of FCT models for roll and pitch rotations. For simplification, wing deviation is not considered in the current study. First, with a standard blade-element model, the lift and drag of a single flapping wing at a particular instant in the wing stroke cycle [15, 22] are:

L , (A1) D , (A2)

where is the magnitude of the wing tip velocity at non-dimensional time , is the non-dimensional second moment of wing area, and are instantaneous lift and drag coefficients [15] as functions of effective angle of attack , is fluid/air density, is wing length, and is mean chord length. Roll, pitch, and yaw torques around the center of gravity are then calculated as:

cos , (A3)

sin , (A4)

, (A5)

where is stroke position and is the normalized center of pressure on the wing. We assume that the center of pressure is located at 70% along the wingspan at any instant of time [12].

Note that roll and pitch torques are determined only by lift force ( L), but yaw torque is determined only by drag force ( D). Each moment of arm is determined by the orthogonal distance between the center of pressure and the corresponding principal axis ( , , and ). Notably, it can be shown that equations A3-A5 describe the torque productions regardless of the location of center of gravity, as long as we assume that it is below the wing base and along the yaw axis .

Roll rotation Rotation around roll axis xs adds a downward velocity

(always normal to the stroke plane) to the left wing while increasing its effective angle of attack . On the contrary, however, it adds an upward velocity to the right wing and reduces its effect angle of attack. Therefore we write the lift of the left wing and the right wing as:

sin , (A6)

sin , (A7)

where is the geometric angle of attack determined by wing kinematics, is the angle between the stroke plane and total wing velocity , and and are magnitudes of wing tip velocity as a result of wing flapping and roll rotation, respectively:

and cos , (A8 and A9)

where is the non-dimensional angular velocity of the wing, and are wing-flapping amplitude and frequency, is

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arctan . (A10)

The magnitude of wing velocity can then be written as:

. (A11)

Collectively, the roll torque of the left and right wings are calculated as:

sin cos

, (A12)

sin cos . (A13)

We then approximate as:

| . (A14)

The total roll torque of a wing pair is:

|sin cos Φ , (A15)



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Note that the above equations also apply to the forward translation.

Lateral translation Similar to forward translation, the instantaneous drag

acting on a wing during lateral motion can be written as:

ρSCD α 2 , (B8)

where is the tip velocity ratio in lateral translation, which is defined as:

1 , (B9)

where = 0 and 1 for the left and right wings, respectively. We can see in Fig. A1B that during a lateral motion to the right (left), the net drag of a wing pair along the pitch axis is always pointing to the left (right) at any instant of time (i.e., an enhanced drag to the left (positive ) and decreased drag to the right (negative ) during downstroke at ventral half of the stroke plane, Fig. A1B). Collectively, we have

∆ Φ α sin , (B10)

and the stroke averaged value is:

∆ Φ α sın . (B11)

Moreover, the change of lift because of lateral motions is:

∆ ρ CL α sın . (B12)

Vertical translation A vertical translation changes the effective angle of attack

at a wing section by:

arctan , (B13)

where is the angle between total wing velocity, and the stroke plane is the spanwise location of the wing section. The magnitude of total wing velocity is:

. (B14)

Thus the sectional lift force during ascent is given by:

d sin

. (B15)

Furthermore, the change of the sectional lift force resulting from vertical velocity is:

Δd sin ,

sin |

. (B16)

Integrate the above equation over an entire wingspan, and we have the net lift change of a wing pair:

Δ Φ

|sin . (B17)

The stroke averaged value is:

Δ Φ

|sın . (B18)

ACKNOWLEDGMENTS This work was supported in part by NSF Grant#0545931.

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