Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Q1 16 17 Q5 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 Q2 58 59 60 61 62 63 64 65 66
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight$ Imraan A. Faruque, J. Sean Humbert University of Maryland, Department of Aerospace Engineering, College Park, MD 20742, USA
H I G H L I G H T S
A transformation is introduced that preserves the topology of recorded wing motions. The transformation is shown to be related to roll–yaw coupling in flapping flight. When applied to recorded insect flights, each trajectory shows proverse coupling. Proverse roll–yaw coupling could reduce the feedback demands of flapping flight.
art ic l e i nf o
a b s t r a c t
Article history: Received 22 January 2014 Received in revised form 23 June 2014 Accepted 25 July 2014
Whether the remarkable flight performance of insects is because the animals leverage inherent physics at this scale or because they employ specialized neural feedback mechanisms is an active research question. In this study, an empirically derived aerodynamics model is used with a transformation involving a delay and a rotation to identify a class of kinematics that provide favorable roll–yaw coupling. The transformation is also used to transform both synthetic and experimentally measured wing motions onto the manifold representing proverse yaw and to quantify the degree to which freely flying insects make use of passive aerodynamic mechanisms to provide proverse roll–yaw turn coordination. The transformation indicates that recorded insect kinematics do act to provide proverse yaw for a variety of maneuvers. This finding suggests that passive aerodynamic mechanisms can act to reduce the neural feedback demands of an insect's flight control strategy. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Insect Dynamics Roll Yaw Moment
1. Introduction Flapping wing flight in biological systems represents a dramatically more maneuverable form of flight than any man-made example. The simultaneous presence of aerial maneuverability and robust tracking in unknown environments in the insect world is made further more remarkable when one considers the incredibly limited neural processing available to an insect's flight control strategy. Many insects carry less than 100 mg worth of neural material distributed throughout the insect, of which 2/3 is devoted to visual processing (Egelhaaf et al., 2002), implying that their flight control strategies must be well-tuned to their flight dynamics so as to reduce the neural feedback demands. One possible means to reduce the demands on a flight control feedback ☆ Portions of this work were funded under Air Force Grant FA86511020004 and under Army Grant W911NF-08-2-0004. Special thanks to Martin F. Wehling for graciously hosting my AFRL rotation. E-mail address: [email protected] (I.A. Faruque).
system is to leverage passive aerodynamic mechanisms at this scale. Similar mechanisms to reduce feedback demands have been identified in cockroaches to reduce their neural control requirements by taking advantage of mechanical properties of the leg mechanism (Sponberg and Full, 2008). Whether this remarkable flight performance is because insects leverage inherent physics at this scale or employ specialized neural feedback mechanisms is an open research question. An understanding of the balance between aeromechanical design and neural feedback could unlock decisive improvements in micro-aerial vehicle flight control. An aeromechanical flight platform that provides complementary (proverse) roll–yaw coupling via inherent passive aerodynamic mechanisms would require one less regulator in the feedback circuit. Conversely, an aeromechanical flight platform that provides antagonistic (adverse) yaw requires regulation of the two states separately. Because MAV flight controllers operate with very stringent size, weight, and power demands, reducing the computational demands of a controller is vital to successful flight, and a mechanical platform with
http://dx.doi.org/10.1016/j.jtbi.2014.07.026 0022-5193/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91
I.A. Faruque, J. Sean Humbert / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
low feedback control demands has a dramatic advantage to one that requires multiple complex control loops. Despite the critical advantage, the inherent complexity of small-scale flapping flight aerodynamics has obscured a direct analysis of both biologically relevant and engineered wing kinematic perturbation strategies. Although the detailed aerodynamic mechanisms involved in small-scale flight are quite complex (Ramamurti and Sandberg, 2007), recent efforts have allowed extraction of reduced-order flight dynamics models, either for single degree of freedom experimental cases (Hesselberg and Lehmann, 2007a,b) direct analytic methods (Doman et al., 2010), or more general computationally (Sun et al., 2007) and spectrally derived models (Faruque and Humbert, 2010). As study progresses from hovering models to forward flight models, theoretical analysis and experimental measurements indicate that coupling increases (Dickson and Dickinson, 2004; Faruque et al., 2012), and the effect this coupling has on a control strategy is unknown. This study addresses the balance of control requirements through consideration of the roll–yaw coordination required in maneuvering free flight, using both theoretical and experimental examples. Section 2 reviews the description of wing motions, introduces a transformation between recorded kinematic time histories and biologically relevant control inputs, and develops the aerodynamic basis for roll–yaw coupling using an empirically derived insect aerodynamics model (Sane and Dickinson, 2002). Both simplified wing motions and experimentally measured kinematic inputs are evaluated using the aerodynamics model in relation to lift and drag-based roll–yaw coupling in Section 3. Finally, the synthetic and experimentally measured wing motion time histories for a variety of maneuvering flight sequences are transformed and transformed onto a manifold representing proverse yaw to evaluate the degree to which insects harness aerodynamic roll–yaw coupling to reduce their neural feedback demands.
idealized planar stroke motion, and a coordinate axes set aligned with this plane the stroke plane axes P ¼ fp^ x ; p^ y ; p^ z g. Define R ¼ fr^ x ; r^ y ; r^ z g a set of axes that move along with the right wing, with r^ z ¼ p^ z and r^ y to extend toward the wing tip as in Fig. 1b. Similarly, define L ¼ f^l x ; ^l y ; ^l z g for the left wing, with ^l y being extending inboard along the left wing spanline. The additional definition of the geometric angle with respect to the stroke plane as wing pitch angle αg provides the notation necessary to describe the orientation of two rigid wings at an instant in time. Both experimental kinematics measured in Section 2.1.2 and simplified synthetic kinematics introduced in Section 2.3 were used for analysis.
2.1.2. Experimental apparatus An experimental test rig shown in Fig. 2 was used to make detailed measurements of freely maneuvering flies. The experimental apparatus consists of three Vision Research Phantom V710 high speed video cameras and lighting array, orthogonally mounted about a 10 in. 10 in. 8 in. a Plexi-Glass test section. Camera calibration was accomplished with direct linear transformation (Hedrick, 2008). The 11 DLT coefficients identified in this calibration (Abdel-Aziz and Karara, 1971) allowed reconstruction with 0.1–0.2 pixel errors, indicating that lens distortion was a minor factor in the experimental setup and that the pinhole camera model assumed in the DLT method provided an acceptable model. Background subtraction was used to identify profiles and coefficients and generate a visual hull. Regions of volume pixels, or “voxels,” are then identified as wing or body using intensity segmentation, and a principal component analysis is performed
2. Background and approach 2.1. Kinematics measurement 2.1.1. Coordinate definitions The description of the insect flapping motion requires a family of axes centered at the insect wing hinge. Approximating the wings as rigid bodies, measured insect kinematics exhibit a roughly planar flap motion which is represented using 2–3–2 Euler angles. Define by reference to Fig. 1a a set of stability axes S ¼ fs^ x ; s^ y ; s^ z g passing through the insect center of mass G, the stroke plane angle β as the angle about the pitch axis to an
Fig. 2. Freely flying insects in the flight chamber were imaged at 7002 Hz to study straight and level flight.
Fig. 1. Axes and angle definitions. (a) Stroke plane axes/angle β, body hovering angle ξ; (b) stroke angle ϕr and R axes.
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
I.A. Faruque, J. Sean Humbert / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 Q3 59 60 61 62 63 64 65 66
3
Fig. 3. Flight sequences studied include a (a) straight and level forward flight sequence, (b) level left turn, (c) climb initiation, and (d) yaw motion (sideslip). Top views shown at 70 Hz, side view shown at 140 Hz.
on the wings and body to identify the stroke and deviation angles of each wing. Because the wing pitch angles are not well identified by principal component analysis, minimization of a Plücker line cost function was used to establish a wing pitch angle (Kostreski, 2012). Detailed free flight wing kinematics recording was made for Drosophila flies, including the maneuvers shown in Fig. 3. 2.2. Curve-fitted insect aerodynamics model Insect flapping wing aerodynamics is in general an unsteady, three-dimensional flow field and requires detailed computational fluid dynamics modeling (Sane and Dickinson, 2002; Sane, 2003; Ramamurti and Sandberg, 2007). However, remarkable agreement to both experimental and computational results has been demonstrated via experimentally derived lift and drag curve fits, which are normally presented in a “quasi-steady” form (Ellington, 1984; Dickinson et al., 1954). The experimentally derived models are amenable to simulation and mathematical analysis, which will allow conclusions to be made about the control properties of the system. The insect aerodynamics model in quasi-steady form models lift as LðtÞ ¼
1 ρSjut ðtÞj2 r^ 22 C L ½αðtÞ; 2
C L ½αðtÞ ¼ 0:225 þ 1:58 sin ð2:13αg 7:2Þ:
ð1aÞ ð1bÞ
where the instantaneous lift force L is written as a function of the air density ρ, the wing area S, tip velocity ut, nondimensional
2 second moment of area r^ 2 , and an experimentally determined lift curve slope C L ½αðtÞ (Dickinson, 1996; Dickinson et al., 1954; Sane and Dickinson, 2002). The wing angle of attack αðtÞ is modeled as a combination of some nominal angle of attack due to wing kinematic motion and angle of attack perturbations due to insect motion throughout the environment v ð2Þ α ¼ sgnϕ_ J αg J þ arctan z ; vx
where vz and vx are components of body rotational and translational velocities in the wing frame, respectively, as defined in Faruque and Humbert (2010). Drag forces are similarly parametrized with a drag curve slope C D ½αðtÞ: C D ½αðtÞ ¼ 1:92 1:55 cos ð2:04αg 9:82Þ:
ð3Þ
2.3. Homeomorphism The analysis of the coupled moments generated by a set of kinematics has previously been approached via direct numerical perturbation (Vance et al., 2013, 2010) or simulated dynamics, in part because direct visualization of the coupling is not straightforward. In this section, a transformation of coordinates will allow the kinematic coupling to be visualized much more easily. A 1D example of wing pitch is used here, extension to the full set of kinematics follows a similar procedure. If S is the L2 -integrable space of wing pitch time history αðtÞ A ½ π ; π for a wingstroke over time t A ½0; T=2, where T is
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
I.A. Faruque, J. Sean Humbert / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
the wingstroke period, then we may define a transformation f : S-A A ½0; T=2 as the following: 1 T ð4Þ α0 ¼ αðtÞ þ α t þ 2 2
αud ðsÞ ¼
1 T αðtÞ α t þ 2 2
ð5Þ
f ½αðtÞ combines a time delay with a rotation, as seen in Fig 4. Because f ½αðtÞ is bijective, continuous, and has a continuous inverse, it is a homeomorphism and the topological properties of S are preserved in A. Intuitively, this transformation maps a full wingstroke to composite input variables over a half wingstroke. Although the topological properties of S are retained, aerodynamic coupling in A has a particularly appealing structure, which will be shown in Section 2.4. The effect of the homeomorphic transformation may be demonstrated using simplified (planar) synthetic kinematics. The synthetic kinematics use a sine wave for stroke angle ϕ and triangle wave for wing pitch angle αg, and deviation angle assumed zero for a planar kinematics assumption. For the example kinematics, shown in Fig. 5, the transformation illustrates how fore/aft asymmetry in wing pitch angle αg ðtÞ is purely a change in αud , and has no effect on α0. 2.4. Aerodynamic basis for roll–yaw coupling The insect aerodynamics model can provide an understanding for the physical basis of roll–yaw coupling by comparing the relative signs and magnitudes of aerodynamic drag in each quarter of wingstroke under a given perturbation. Consider a positive roll perturbation p, which increases the angle of attack on the right wing and decreases the angle of attack on the left wing by an amount αp. The yaw moment due to set of wing kinematics is due to the difference in left vs. right wing drag coefficients during advancing vs. retreating strokes, as shown in Fig. 6. A yaw moment having the same sign as the roll rate would represent proverse yaw. Drag-based roll–yaw coupling is proverse during the advancing stroke and adverse during the retreating stroke. For symmetric kinematics, as in Fig. 6(a), the proverse and adverse yaw magnitudes are equivalent, and no coupling is observed. Due to the nonlinearity in the drag curve slope, the asymmetric case in Fig. 6(b) provides an advancing wing moment that is larger than
α(t)
τ
Rγ
α0(t) αud(t)
Fig. 4. Transformation of a wing pitch signal incorporates a rotation and a delay.
the retreating moment, and the integrated effect of the wingstroke is a proverse roll–yaw coupling. A nondimensional quantity n(t) quantifying the instantaneous yaw moment N(t) due to a kinematic pair in the fore-aft wingstroke may be defined as T T C D;l α t þ ; NðtÞ p nðtÞ ¼ ½C D;r ðαðtÞÞ C D;l ðαðtÞÞ C D;r α t þ 2 2
ð6Þ where t is an instant in the advancing (down) stroke.
3. Results and discussion 3.1. Roll yaw coupling In transformed phase space ðα0 ; αud Þ, the region of proverse yaw takes a particularly attractive form, as it lies on a manifold having a saddle point shown in Fig. 7. For axes centered at ðα0 ; αud Þ ¼ ð451; 0Þ, proverse roll–yaw coupling is found in quadrants I and III, with the magnitude being proportional to the distance from the origin. (The coupling illustrated in Fig. 6(b) is Quadrant I coupling.) A cyclic wingstroke forms a path on this manifold, and an insect could maximize favorable roll–yaw coupling by keeping its wingstroke time history within this region of coupling.
Fig. 6. When using symmetric kinematics in (a) the proverse roll–yaw coupling generated during the forestroke has equal magnitude as the adverse coupling during the aft stroke, but a different choice of wing pitch kinematics may provide proverse coupling, as in (b). (a) symmetric kinematics and (b) positive αud.
Fig. 5. Simplified synthetic kinematics in raw (a) and transformed (b) coordinates show that fore/aft wing pitch asymmetry is decoupled in the transformed coordinates.
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
I.A. Faruque, J. Sean Humbert / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 7. Roll–yaw coupling in the transformed phase space has the form of a saddle point (a), with proverse coupling contained in two quadrants (b).
3.2. Magnitude
120
NðtÞ ¼ 1=2ρSjut ðtÞj2 r^ 2 ΔC D r a ; 2
Nm ¼
ρSr^ 22 r a
Z
T
T=2 0
Z
T=2
Nm ¼ C 1 0
Φ2 ω2 R2 sin ðωtÞ2 n dt
n½1 cos ð2ωtÞ dt;
60 30 0 120
ð7Þ
where ΔC D ¼ C D;r C D;l during the advancing stroke and ΔC D ¼ C D;l C D;r during the retreating stroke, and ra is the effective point of application of the aerodynamic force. ut is a roughly sinusoidal variation, which provides a means to theoretically estimate the yaw moment. For ϕ ¼ Φ cos ðωtÞ, ut ¼ RΦω sin ðωtÞ, and the mean aerodynamic moment over a wingstroke is Z T=2 1 1 Nm ¼ ρSr^ 22 ra Φ2 ω2 R2 sin ðωtÞ2 n dt ð8Þ T=2 0 2
Measured Ensemble
90 Fore
Although the instantaneous sign of the passive aerodynamic coupling can be determined via comparison of two time instants as in Section 2.4, the integrated effect of the wingstroke must include the effects of instantaneous wing velocity in addition to the instantaneous difference in lift and drag coefficients. Under a right roll perturbation the instantaneous yaw moment N(t) is
90 Aft
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
5
60 30 0
0
0.1
0.2 0.3 Time, t/T
0.4
0.5
Fig. 8. A stereotypical set of fore and aft stroke wing kinematics was formed from the ensemble average (red) of the 35 consecutive digitized wingstrokes during a maneuver, such as the wing pitch during forward flight shown here. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
ð9Þ lift production concerns, while pitch angle bias affects only the and
C1 ¼
ρ
ΦωÞ ra
2 Sr^ 2 ðR
2T
αud term, as seen in Fig. 5.
2
ð10Þ
Eq. (10) allows one to estimate the wingstroke-averaged yaw moment as an inner product between a stereotypical stroke motion and the recorded wing pitch angle over the last half stroke. Such a result may be advantageous in onboard control laws, as it reduces the time that is required to compute wingstroke-averaged quantities to half a wingstroke period, potentially both increasing the speed and reducing the memory requirements of the operation. C1 is a constant involving geometric and stroke parameters which can be pre-computed and stored for the wingstroke. For a flapping wing MAV, proverse or adverse roll–yaw coupling can be chosen during the design phase by appropriate choice of αud , as shown in Fig. 10. Because of the variety of mechanisms in MAV design, kinematic trajectories are normally constrained to perturbations about a specific form by mechanical limitations of each flapper. The transformation introduced in this paper provides an additional constraint that can be implemented to provide roll–yaw coupling. For our example mechanism generating sinusoidal stroke motion and triangle wave wing pitch motion, guidelines to contain wingstrokes within the proverse coupling regions may be computed. For this particular kinematic program, stroke amplitude Φ and wing pitch angle bias must be set. Stroke amplitude is most often set via
3.3. Measured insect wing kinematics For this study, freely flying Drosophila melanogaster were recorded from the three high speed cameras while demonstrating various maneuvers in forward flight, and the wing motions digitized. For each flight record, time synchronous averaging was applied to identify a stereotypical wingstroke from 35 wingstrokes (as defined by stroke angle reversal), as seen in Fig. 8. After stroke synchronization, the experimental insect wing time histories were transformed and nondimensional yaw moments were computed for each wingstroke using the insect aerodynamics model. Transformed coordinates of recorded wingstrokes in varying conditions are plotted in Fig. 9, which indicates that Drosophila often employ wing kinematics that exploit passive aerodynamic mechanisms to provide roll–yaw coupling. Drosophila are able to maintain or even increase this proverse roll–yaw coupling during increased lift force maneuvers such as climbing forward flight in Fig. 9d, or banked turns such as those in Fig. 9f. Of particular interest is a saccadic maneuver in which an insect conducts a rapid turn that involves only yaw motion and not roll, as seen in Fig. 9e. Even in this case, the insect kinematics were seen to provide a proverse roll–yaw coupling.
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 Q4106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
I.A. Faruque, J. Sean Humbert / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
40
20
20
20
0
−40
α , deg ud
40
αud, deg
ud
40
−20
0
0
20
40
60
−40
80
0
20
40 α , deg
60
−40
80
40
20
20
20
0 −20
α , deg ud
40
0
20
40 α , deg
20
60
80
−40
40 α , deg
60
80
60
80
0 −20
−20
0
0
0
40
αud, deg
ud
0
0
0
−40
Start Stroke End
−20
−20
α , deg
α , deg
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
α , deg
6
0
20
40
60
80
α , deg
0
0
−40
0
20
40 α , deg 0
Fig. 9. Measured wing trajectories for Drosophila show that the midstroke during forward flight provides roll–yaw coupling that would reduce neural feedback demands. (a) Forward flight; (b) forward flight; (c) forward flight; (d) forward flight climb; (e) saccade; (f) banked left turn.
Table 1 Roll–yaw coupling N m;α values for various flight trials show variable, but consistently positive (proverse), behavior. Mode Fwd flight Fwd flight Fwd flight Fwd climb Saccade Banked turn
N α (Nm/deg) 10
0.129 10 0.582 10 10 1.398 10 10 1.325 10 10 1.903 10 10 0.979 10 10
2
11 Acceleration ac (deg/s ) 1.236 5.598 13.44 12.73 18.30 9.410
Fig. 10. Roll–yaw coupling for synthetic kinematics show that the choice of stroke pattern can be used for both proverse and adverse coupling.
Table 1 provides values of N m;α for a variety of flight maneuvers, and a characteristic acceleration ac due to a 11 perturbation in angle of attack. In every case, the wingstroke-averaged values indicate proverse coupling. In the case of forward flight, the coupling indicated varied, showing higher values even during increased lift maneuvers such as climbing flight. Remarkably, coupling was indicated both between the saccade and banked turns, while the lowest coupling was found in simple forward flight trials.
4. Conclusions In this study, a curve-fitted insect aerodynamics model was used to identify a manifold representing roll–yaw coupling. A region was identified representing proverse yaw, which would act to reduce the number of degrees of freedom requiring neural feedback. This manifold allowed the generation of a class of kinematics that provide favorable roll–yaw coupling, which are of use to flapping wing micro-aerial vehicle design. The coupling has a nonlinear physical basis, but a means of to estimate the coupling with reduced computational and memory requirements was developed. Both simplified synthetic kinematics and experimentally measured wing motions were transformed onto the manifold. The synthetic kinematics showed that asymmetric upstroke/downstroke motions can used as a design variable to induce either proverse or adverse coupling. When experimentally measured Drosophila dynamics were transformed onto the manifold, the analysis indicated that the animals choose kinematic patterns that provide turn coordination during a variety of free flight behaviors. This finding suggests that flying insects do make use of passive aerodynamic mechanisms to reduce the neural feedback demands imposed by flight control. The indication that insects consistently use wing kinematics that provide proverse roll–yaw coordination is important to enabling flapping wing MAV flight at small scales.
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jtbi.2014.07.026. References Abdel-Aziz, I.Y., Karara, H.M., 1971. Direct linear transformation into object space coordinates in close-range photogrammetry. In: Proceedings of the Symposium on Close-Range Photogrammetry, pp. 1–18. Dickinson, M.H., 1996. Unsteady mechanisms of force generation in aquatic and aerial locomotion. Am. Zool. 36, 537–554.
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
I.A. Faruque, J. Sean Humbert / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Dickinson, M.H., Lehmann, F.O., Sane, S.P., 1954. Wing rotation and the aerodynamic basis of insect flight. Science 284, 1999. Dickson, W.B., Dickinson, M.H., 2004. The effect of advance ratio on the aerodynamics of revolving wings. J. Exp. Biol. 207, 4269–4281. Doman, D.B., Oppenheimer, M.W., Sigthorsson, D.O., 2010. Wingbeat shape modulation for flapping-wing micro-air-vehicle control during hover. J. Guid. Control Dyn. 33 (3), 724–739. Egelhaaf, M., Kern, R., Krapp, H.G., Kretzberg, J., Kurtz, R., Warzecha, A.-K., 2002. Neural encoding of behaviourally relevant visual-motion information in the fly. Trends Neurosci. 25 (2), 96–102. Ellington, C.P., 1984. The aerodynamics of hovering insect flight I: the quasi-steady analysis. Philos. Trans. R. Soc. Lond. Ser. B: Biol. Sci. (1934–1990) 305 (1122). Faruque, I., MacFarlane, K., Humbert, J.S., 2012. Reduced-order forward flight dynamics models for dipteran insects. In: Proceedings of the Guidance, Navigation, and Control Conference. Faruque, I.A., Humbert, J.S., 2010. Dipteran insect flight dynamics. Part 1: longitudinal motion about hover. J. Theoret. Biol. 264 (2), 538–552. Hedrick, T.L., 2008. Software techniques for two and three-dimensional kinematic measurements of biological and biomimetic systems. Bioinspir. Biomim. 3 (034001). Hesselberg, T., Lehmann, F.-O., 2007a. Turning behaviour depends on frictional damping in the fruit fly Drosophila. J. Exp. Biol. 210 (24), 4319–4334. http://dx. doi.org/10.1242/jeb.010389.
7
Hesselberg, T., Lehmann, F.O., 2007b. Corrigendum: animal flight dynamics II. Longitudinal stability in flapping flight. J. Exp. Biol. 210, 4139–4334. Kostreski, N.I., 2012. Automated kinematic extraction of wing and body motions of free flying diptera. In: Digital Repository at the University of Maryland, University of Maryland, College Park, MD. No. 1903/13546. Ramamurti, R., Sandberg, W.C., 2007. Computational investigation of the threedimensional unsteady aerodynamics of Drosophila hovering and maneuvering. J. Exp. Biol. 201, 881–896. Sane, S.P., 2003. The aerodynamics of insect flight. J. Exp. Biol. 206, 4191–4208. Sane, S.P., Dickinson, M.H., 2002. The aerodynamic effects of wing rotation and a revised quasi-steady model of flapping flight. J. Exp. Biol. 205, 1087–1096. Sponberg, S., Full, R.J., 2008. Neuromechanical response of musculo-skeletal structures in cockroaches during rapid running on rough terrain. J. Exp. Biol. 211, 433–446. Sun, M., Wang, J., Xiong, Y., 2007. Dynamic flight stability of hovering insects. Acta Mech. Sin. 23 (4), 231–246. Vance, J.T., Faruque, I.A., Humbert, J.S., 2010. The effects of differential wing stroke amplitude and stroke offset on insect body movements during perturbed flight conditions. In: 34th Annual Meeting of the American Society of Biomechanics, Providence, RI, August 18–21. Vance, J.T., Faruque, I.A., Humbert, J.S., 2013. Kinematic strategies for mitigating gust perturbations in insects. Bioinspir. Biomim. 8 (1), 016004.
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Q1 16 17 Q5 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 Q2 58 59 60 61 62 63 64 65 66
Contents lists available at ScienceDirect
Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight$ Imraan A. Faruque, J. Sean Humbert University of Maryland, Department of Aerospace Engineering, College Park, MD 20742, USA
H I G H L I G H T S
A transformation is introduced that preserves the topology of recorded wing motions. The transformation is shown to be related to roll–yaw coupling in flapping flight. When applied to recorded insect flights, each trajectory shows proverse coupling. Proverse roll–yaw coupling could reduce the feedback demands of flapping flight.
art ic l e i nf o
a b s t r a c t
Article history: Received 22 January 2014 Received in revised form 23 June 2014 Accepted 25 July 2014
Whether the remarkable flight performance of insects is because the animals leverage inherent physics at this scale or because they employ specialized neural feedback mechanisms is an active research question. In this study, an empirically derived aerodynamics model is used with a transformation involving a delay and a rotation to identify a class of kinematics that provide favorable roll–yaw coupling. The transformation is also used to transform both synthetic and experimentally measured wing motions onto the manifold representing proverse yaw and to quantify the degree to which freely flying insects make use of passive aerodynamic mechanisms to provide proverse roll–yaw turn coordination. The transformation indicates that recorded insect kinematics do act to provide proverse yaw for a variety of maneuvers. This finding suggests that passive aerodynamic mechanisms can act to reduce the neural feedback demands of an insect's flight control strategy. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Insect Dynamics Roll Yaw Moment
1. Introduction Flapping wing flight in biological systems represents a dramatically more maneuverable form of flight than any man-made example. The simultaneous presence of aerial maneuverability and robust tracking in unknown environments in the insect world is made further more remarkable when one considers the incredibly limited neural processing available to an insect's flight control strategy. Many insects carry less than 100 mg worth of neural material distributed throughout the insect, of which 2/3 is devoted to visual processing (Egelhaaf et al., 2002), implying that their flight control strategies must be well-tuned to their flight dynamics so as to reduce the neural feedback demands. One possible means to reduce the demands on a flight control feedback ☆ Portions of this work were funded under Air Force Grant FA86511020004 and under Army Grant W911NF-08-2-0004. Special thanks to Martin F. Wehling for graciously hosting my AFRL rotation. E-mail address: [email protected] (I.A. Faruque).
system is to leverage passive aerodynamic mechanisms at this scale. Similar mechanisms to reduce feedback demands have been identified in cockroaches to reduce their neural control requirements by taking advantage of mechanical properties of the leg mechanism (Sponberg and Full, 2008). Whether this remarkable flight performance is because insects leverage inherent physics at this scale or employ specialized neural feedback mechanisms is an open research question. An understanding of the balance between aeromechanical design and neural feedback could unlock decisive improvements in micro-aerial vehicle flight control. An aeromechanical flight platform that provides complementary (proverse) roll–yaw coupling via inherent passive aerodynamic mechanisms would require one less regulator in the feedback circuit. Conversely, an aeromechanical flight platform that provides antagonistic (adverse) yaw requires regulation of the two states separately. Because MAV flight controllers operate with very stringent size, weight, and power demands, reducing the computational demands of a controller is vital to successful flight, and a mechanical platform with
http://dx.doi.org/10.1016/j.jtbi.2014.07.026 0022-5193/& 2014 Elsevier Ltd. All rights reserved.
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91
I.A. Faruque, J. Sean Humbert / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
low feedback control demands has a dramatic advantage to one that requires multiple complex control loops. Despite the critical advantage, the inherent complexity of small-scale flapping flight aerodynamics has obscured a direct analysis of both biologically relevant and engineered wing kinematic perturbation strategies. Although the detailed aerodynamic mechanisms involved in small-scale flight are quite complex (Ramamurti and Sandberg, 2007), recent efforts have allowed extraction of reduced-order flight dynamics models, either for single degree of freedom experimental cases (Hesselberg and Lehmann, 2007a,b) direct analytic methods (Doman et al., 2010), or more general computationally (Sun et al., 2007) and spectrally derived models (Faruque and Humbert, 2010). As study progresses from hovering models to forward flight models, theoretical analysis and experimental measurements indicate that coupling increases (Dickson and Dickinson, 2004; Faruque et al., 2012), and the effect this coupling has on a control strategy is unknown. This study addresses the balance of control requirements through consideration of the roll–yaw coordination required in maneuvering free flight, using both theoretical and experimental examples. Section 2 reviews the description of wing motions, introduces a transformation between recorded kinematic time histories and biologically relevant control inputs, and develops the aerodynamic basis for roll–yaw coupling using an empirically derived insect aerodynamics model (Sane and Dickinson, 2002). Both simplified wing motions and experimentally measured kinematic inputs are evaluated using the aerodynamics model in relation to lift and drag-based roll–yaw coupling in Section 3. Finally, the synthetic and experimentally measured wing motion time histories for a variety of maneuvering flight sequences are transformed and transformed onto a manifold representing proverse yaw to evaluate the degree to which insects harness aerodynamic roll–yaw coupling to reduce their neural feedback demands.
idealized planar stroke motion, and a coordinate axes set aligned with this plane the stroke plane axes P ¼ fp^ x ; p^ y ; p^ z g. Define R ¼ fr^ x ; r^ y ; r^ z g a set of axes that move along with the right wing, with r^ z ¼ p^ z and r^ y to extend toward the wing tip as in Fig. 1b. Similarly, define L ¼ f^l x ; ^l y ; ^l z g for the left wing, with ^l y being extending inboard along the left wing spanline. The additional definition of the geometric angle with respect to the stroke plane as wing pitch angle αg provides the notation necessary to describe the orientation of two rigid wings at an instant in time. Both experimental kinematics measured in Section 2.1.2 and simplified synthetic kinematics introduced in Section 2.3 were used for analysis.
2.1.2. Experimental apparatus An experimental test rig shown in Fig. 2 was used to make detailed measurements of freely maneuvering flies. The experimental apparatus consists of three Vision Research Phantom V710 high speed video cameras and lighting array, orthogonally mounted about a 10 in. 10 in. 8 in. a Plexi-Glass test section. Camera calibration was accomplished with direct linear transformation (Hedrick, 2008). The 11 DLT coefficients identified in this calibration (Abdel-Aziz and Karara, 1971) allowed reconstruction with 0.1–0.2 pixel errors, indicating that lens distortion was a minor factor in the experimental setup and that the pinhole camera model assumed in the DLT method provided an acceptable model. Background subtraction was used to identify profiles and coefficients and generate a visual hull. Regions of volume pixels, or “voxels,” are then identified as wing or body using intensity segmentation, and a principal component analysis is performed
2. Background and approach 2.1. Kinematics measurement 2.1.1. Coordinate definitions The description of the insect flapping motion requires a family of axes centered at the insect wing hinge. Approximating the wings as rigid bodies, measured insect kinematics exhibit a roughly planar flap motion which is represented using 2–3–2 Euler angles. Define by reference to Fig. 1a a set of stability axes S ¼ fs^ x ; s^ y ; s^ z g passing through the insect center of mass G, the stroke plane angle β as the angle about the pitch axis to an
Fig. 2. Freely flying insects in the flight chamber were imaged at 7002 Hz to study straight and level flight.
Fig. 1. Axes and angle definitions. (a) Stroke plane axes/angle β, body hovering angle ξ; (b) stroke angle ϕr and R axes.
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
I.A. Faruque, J. Sean Humbert / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 Q3 59 60 61 62 63 64 65 66
3
Fig. 3. Flight sequences studied include a (a) straight and level forward flight sequence, (b) level left turn, (c) climb initiation, and (d) yaw motion (sideslip). Top views shown at 70 Hz, side view shown at 140 Hz.
on the wings and body to identify the stroke and deviation angles of each wing. Because the wing pitch angles are not well identified by principal component analysis, minimization of a Plücker line cost function was used to establish a wing pitch angle (Kostreski, 2012). Detailed free flight wing kinematics recording was made for Drosophila flies, including the maneuvers shown in Fig. 3. 2.2. Curve-fitted insect aerodynamics model Insect flapping wing aerodynamics is in general an unsteady, three-dimensional flow field and requires detailed computational fluid dynamics modeling (Sane and Dickinson, 2002; Sane, 2003; Ramamurti and Sandberg, 2007). However, remarkable agreement to both experimental and computational results has been demonstrated via experimentally derived lift and drag curve fits, which are normally presented in a “quasi-steady” form (Ellington, 1984; Dickinson et al., 1954). The experimentally derived models are amenable to simulation and mathematical analysis, which will allow conclusions to be made about the control properties of the system. The insect aerodynamics model in quasi-steady form models lift as LðtÞ ¼
1 ρSjut ðtÞj2 r^ 22 C L ½αðtÞ; 2
C L ½αðtÞ ¼ 0:225 þ 1:58 sin ð2:13αg 7:2Þ:
ð1aÞ ð1bÞ
where the instantaneous lift force L is written as a function of the air density ρ, the wing area S, tip velocity ut, nondimensional
2 second moment of area r^ 2 , and an experimentally determined lift curve slope C L ½αðtÞ (Dickinson, 1996; Dickinson et al., 1954; Sane and Dickinson, 2002). The wing angle of attack αðtÞ is modeled as a combination of some nominal angle of attack due to wing kinematic motion and angle of attack perturbations due to insect motion throughout the environment v ð2Þ α ¼ sgnϕ_ J αg J þ arctan z ; vx
where vz and vx are components of body rotational and translational velocities in the wing frame, respectively, as defined in Faruque and Humbert (2010). Drag forces are similarly parametrized with a drag curve slope C D ½αðtÞ: C D ½αðtÞ ¼ 1:92 1:55 cos ð2:04αg 9:82Þ:
ð3Þ
2.3. Homeomorphism The analysis of the coupled moments generated by a set of kinematics has previously been approached via direct numerical perturbation (Vance et al., 2013, 2010) or simulated dynamics, in part because direct visualization of the coupling is not straightforward. In this section, a transformation of coordinates will allow the kinematic coupling to be visualized much more easily. A 1D example of wing pitch is used here, extension to the full set of kinematics follows a similar procedure. If S is the L2 -integrable space of wing pitch time history αðtÞ A ½ π ; π for a wingstroke over time t A ½0; T=2, where T is
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
I.A. Faruque, J. Sean Humbert / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
the wingstroke period, then we may define a transformation f : S-A A ½0; T=2 as the following: 1 T ð4Þ α0 ¼ αðtÞ þ α t þ 2 2
αud ðsÞ ¼
1 T αðtÞ α t þ 2 2
ð5Þ
f ½αðtÞ combines a time delay with a rotation, as seen in Fig 4. Because f ½αðtÞ is bijective, continuous, and has a continuous inverse, it is a homeomorphism and the topological properties of S are preserved in A. Intuitively, this transformation maps a full wingstroke to composite input variables over a half wingstroke. Although the topological properties of S are retained, aerodynamic coupling in A has a particularly appealing structure, which will be shown in Section 2.4. The effect of the homeomorphic transformation may be demonstrated using simplified (planar) synthetic kinematics. The synthetic kinematics use a sine wave for stroke angle ϕ and triangle wave for wing pitch angle αg, and deviation angle assumed zero for a planar kinematics assumption. For the example kinematics, shown in Fig. 5, the transformation illustrates how fore/aft asymmetry in wing pitch angle αg ðtÞ is purely a change in αud , and has no effect on α0. 2.4. Aerodynamic basis for roll–yaw coupling The insect aerodynamics model can provide an understanding for the physical basis of roll–yaw coupling by comparing the relative signs and magnitudes of aerodynamic drag in each quarter of wingstroke under a given perturbation. Consider a positive roll perturbation p, which increases the angle of attack on the right wing and decreases the angle of attack on the left wing by an amount αp. The yaw moment due to set of wing kinematics is due to the difference in left vs. right wing drag coefficients during advancing vs. retreating strokes, as shown in Fig. 6. A yaw moment having the same sign as the roll rate would represent proverse yaw. Drag-based roll–yaw coupling is proverse during the advancing stroke and adverse during the retreating stroke. For symmetric kinematics, as in Fig. 6(a), the proverse and adverse yaw magnitudes are equivalent, and no coupling is observed. Due to the nonlinearity in the drag curve slope, the asymmetric case in Fig. 6(b) provides an advancing wing moment that is larger than
α(t)
τ
Rγ
α0(t) αud(t)
Fig. 4. Transformation of a wing pitch signal incorporates a rotation and a delay.
the retreating moment, and the integrated effect of the wingstroke is a proverse roll–yaw coupling. A nondimensional quantity n(t) quantifying the instantaneous yaw moment N(t) due to a kinematic pair in the fore-aft wingstroke may be defined as T T C D;l α t þ ; NðtÞ p nðtÞ ¼ ½C D;r ðαðtÞÞ C D;l ðαðtÞÞ C D;r α t þ 2 2
ð6Þ where t is an instant in the advancing (down) stroke.
3. Results and discussion 3.1. Roll yaw coupling In transformed phase space ðα0 ; αud Þ, the region of proverse yaw takes a particularly attractive form, as it lies on a manifold having a saddle point shown in Fig. 7. For axes centered at ðα0 ; αud Þ ¼ ð451; 0Þ, proverse roll–yaw coupling is found in quadrants I and III, with the magnitude being proportional to the distance from the origin. (The coupling illustrated in Fig. 6(b) is Quadrant I coupling.) A cyclic wingstroke forms a path on this manifold, and an insect could maximize favorable roll–yaw coupling by keeping its wingstroke time history within this region of coupling.
Fig. 6. When using symmetric kinematics in (a) the proverse roll–yaw coupling generated during the forestroke has equal magnitude as the adverse coupling during the aft stroke, but a different choice of wing pitch kinematics may provide proverse coupling, as in (b). (a) symmetric kinematics and (b) positive αud.
Fig. 5. Simplified synthetic kinematics in raw (a) and transformed (b) coordinates show that fore/aft wing pitch asymmetry is decoupled in the transformed coordinates.
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
I.A. Faruque, J. Sean Humbert / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Fig. 7. Roll–yaw coupling in the transformed phase space has the form of a saddle point (a), with proverse coupling contained in two quadrants (b).
3.2. Magnitude
120
NðtÞ ¼ 1=2ρSjut ðtÞj2 r^ 2 ΔC D r a ; 2
Nm ¼
ρSr^ 22 r a
Z
T
T=2 0
Z
T=2
Nm ¼ C 1 0
Φ2 ω2 R2 sin ðωtÞ2 n dt
n½1 cos ð2ωtÞ dt;
60 30 0 120
ð7Þ
where ΔC D ¼ C D;r C D;l during the advancing stroke and ΔC D ¼ C D;l C D;r during the retreating stroke, and ra is the effective point of application of the aerodynamic force. ut is a roughly sinusoidal variation, which provides a means to theoretically estimate the yaw moment. For ϕ ¼ Φ cos ðωtÞ, ut ¼ RΦω sin ðωtÞ, and the mean aerodynamic moment over a wingstroke is Z T=2 1 1 Nm ¼ ρSr^ 22 ra Φ2 ω2 R2 sin ðωtÞ2 n dt ð8Þ T=2 0 2
Measured Ensemble
90 Fore
Although the instantaneous sign of the passive aerodynamic coupling can be determined via comparison of two time instants as in Section 2.4, the integrated effect of the wingstroke must include the effects of instantaneous wing velocity in addition to the instantaneous difference in lift and drag coefficients. Under a right roll perturbation the instantaneous yaw moment N(t) is
90 Aft
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
5
60 30 0
0
0.1
0.2 0.3 Time, t/T
0.4
0.5
Fig. 8. A stereotypical set of fore and aft stroke wing kinematics was formed from the ensemble average (red) of the 35 consecutive digitized wingstrokes during a maneuver, such as the wing pitch during forward flight shown here. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)
ð9Þ lift production concerns, while pitch angle bias affects only the and
C1 ¼
ρ
ΦωÞ ra
2 Sr^ 2 ðR
2T
αud term, as seen in Fig. 5.
2
ð10Þ
Eq. (10) allows one to estimate the wingstroke-averaged yaw moment as an inner product between a stereotypical stroke motion and the recorded wing pitch angle over the last half stroke. Such a result may be advantageous in onboard control laws, as it reduces the time that is required to compute wingstroke-averaged quantities to half a wingstroke period, potentially both increasing the speed and reducing the memory requirements of the operation. C1 is a constant involving geometric and stroke parameters which can be pre-computed and stored for the wingstroke. For a flapping wing MAV, proverse or adverse roll–yaw coupling can be chosen during the design phase by appropriate choice of αud , as shown in Fig. 10. Because of the variety of mechanisms in MAV design, kinematic trajectories are normally constrained to perturbations about a specific form by mechanical limitations of each flapper. The transformation introduced in this paper provides an additional constraint that can be implemented to provide roll–yaw coupling. For our example mechanism generating sinusoidal stroke motion and triangle wave wing pitch motion, guidelines to contain wingstrokes within the proverse coupling regions may be computed. For this particular kinematic program, stroke amplitude Φ and wing pitch angle bias must be set. Stroke amplitude is most often set via
3.3. Measured insect wing kinematics For this study, freely flying Drosophila melanogaster were recorded from the three high speed cameras while demonstrating various maneuvers in forward flight, and the wing motions digitized. For each flight record, time synchronous averaging was applied to identify a stereotypical wingstroke from 35 wingstrokes (as defined by stroke angle reversal), as seen in Fig. 8. After stroke synchronization, the experimental insect wing time histories were transformed and nondimensional yaw moments were computed for each wingstroke using the insect aerodynamics model. Transformed coordinates of recorded wingstrokes in varying conditions are plotted in Fig. 9, which indicates that Drosophila often employ wing kinematics that exploit passive aerodynamic mechanisms to provide roll–yaw coupling. Drosophila are able to maintain or even increase this proverse roll–yaw coupling during increased lift force maneuvers such as climbing forward flight in Fig. 9d, or banked turns such as those in Fig. 9f. Of particular interest is a saccadic maneuver in which an insect conducts a rapid turn that involves only yaw motion and not roll, as seen in Fig. 9e. Even in this case, the insect kinematics were seen to provide a proverse roll–yaw coupling.
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 Q4106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
I.A. Faruque, J. Sean Humbert / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
40
20
20
20
0
−40
α , deg ud
40
αud, deg
ud
40
−20
0
0
20
40
60
−40
80
0
20
40 α , deg
60
−40
80
40
20
20
20
0 −20
α , deg ud
40
0
20
40 α , deg
20
60
80
−40
40 α , deg
60
80
60
80
0 −20
−20
0
0
0
40
αud, deg
ud
0
0
0
−40
Start Stroke End
−20
−20
α , deg
α , deg
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
α , deg
6
0
20
40
60
80
α , deg
0
0
−40
0
20
40 α , deg 0
Fig. 9. Measured wing trajectories for Drosophila show that the midstroke during forward flight provides roll–yaw coupling that would reduce neural feedback demands. (a) Forward flight; (b) forward flight; (c) forward flight; (d) forward flight climb; (e) saccade; (f) banked left turn.
Table 1 Roll–yaw coupling N m;α values for various flight trials show variable, but consistently positive (proverse), behavior. Mode Fwd flight Fwd flight Fwd flight Fwd climb Saccade Banked turn
N α (Nm/deg) 10
0.129 10 0.582 10 10 1.398 10 10 1.325 10 10 1.903 10 10 0.979 10 10
2
11 Acceleration ac (deg/s ) 1.236 5.598 13.44 12.73 18.30 9.410
Fig. 10. Roll–yaw coupling for synthetic kinematics show that the choice of stroke pattern can be used for both proverse and adverse coupling.
Table 1 provides values of N m;α for a variety of flight maneuvers, and a characteristic acceleration ac due to a 11 perturbation in angle of attack. In every case, the wingstroke-averaged values indicate proverse coupling. In the case of forward flight, the coupling indicated varied, showing higher values even during increased lift maneuvers such as climbing flight. Remarkably, coupling was indicated both between the saccade and banked turns, while the lowest coupling was found in simple forward flight trials.
4. Conclusions In this study, a curve-fitted insect aerodynamics model was used to identify a manifold representing roll–yaw coupling. A region was identified representing proverse yaw, which would act to reduce the number of degrees of freedom requiring neural feedback. This manifold allowed the generation of a class of kinematics that provide favorable roll–yaw coupling, which are of use to flapping wing micro-aerial vehicle design. The coupling has a nonlinear physical basis, but a means of to estimate the coupling with reduced computational and memory requirements was developed. Both simplified synthetic kinematics and experimentally measured wing motions were transformed onto the manifold. The synthetic kinematics showed that asymmetric upstroke/downstroke motions can used as a design variable to induce either proverse or adverse coupling. When experimentally measured Drosophila dynamics were transformed onto the manifold, the analysis indicated that the animals choose kinematic patterns that provide turn coordination during a variety of free flight behaviors. This finding suggests that flying insects do make use of passive aerodynamic mechanisms to reduce the neural feedback demands imposed by flight control. The indication that insects consistently use wing kinematics that provide proverse roll–yaw coordination is important to enabling flapping wing MAV flight at small scales.
Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jtbi.2014.07.026. References Abdel-Aziz, I.Y., Karara, H.M., 1971. Direct linear transformation into object space coordinates in close-range photogrammetry. In: Proceedings of the Symposium on Close-Range Photogrammetry, pp. 1–18. Dickinson, M.H., 1996. Unsteady mechanisms of force generation in aquatic and aerial locomotion. Am. Zool. 36, 537–554.
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
I.A. Faruque, J. Sean Humbert / Journal of Theoretical Biology ∎ (∎∎∎∎) ∎∎∎–∎∎∎
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Dickinson, M.H., Lehmann, F.O., Sane, S.P., 1954. Wing rotation and the aerodynamic basis of insect flight. Science 284, 1999. Dickson, W.B., Dickinson, M.H., 2004. The effect of advance ratio on the aerodynamics of revolving wings. J. Exp. Biol. 207, 4269–4281. Doman, D.B., Oppenheimer, M.W., Sigthorsson, D.O., 2010. Wingbeat shape modulation for flapping-wing micro-air-vehicle control during hover. J. Guid. Control Dyn. 33 (3), 724–739. Egelhaaf, M., Kern, R., Krapp, H.G., Kretzberg, J., Kurtz, R., Warzecha, A.-K., 2002. Neural encoding of behaviourally relevant visual-motion information in the fly. Trends Neurosci. 25 (2), 96–102. Ellington, C.P., 1984. The aerodynamics of hovering insect flight I: the quasi-steady analysis. Philos. Trans. R. Soc. Lond. Ser. B: Biol. Sci. (1934–1990) 305 (1122). Faruque, I., MacFarlane, K., Humbert, J.S., 2012. Reduced-order forward flight dynamics models for dipteran insects. In: Proceedings of the Guidance, Navigation, and Control Conference. Faruque, I.A., Humbert, J.S., 2010. Dipteran insect flight dynamics. Part 1: longitudinal motion about hover. J. Theoret. Biol. 264 (2), 538–552. Hedrick, T.L., 2008. Software techniques for two and three-dimensional kinematic measurements of biological and biomimetic systems. Bioinspir. Biomim. 3 (034001). Hesselberg, T., Lehmann, F.-O., 2007a. Turning behaviour depends on frictional damping in the fruit fly Drosophila. J. Exp. Biol. 210 (24), 4319–4334. http://dx. doi.org/10.1242/jeb.010389.
7
Hesselberg, T., Lehmann, F.O., 2007b. Corrigendum: animal flight dynamics II. Longitudinal stability in flapping flight. J. Exp. Biol. 210, 4139–4334. Kostreski, N.I., 2012. Automated kinematic extraction of wing and body motions of free flying diptera. In: Digital Repository at the University of Maryland, University of Maryland, College Park, MD. No. 1903/13546. Ramamurti, R., Sandberg, W.C., 2007. Computational investigation of the threedimensional unsteady aerodynamics of Drosophila hovering and maneuvering. J. Exp. Biol. 201, 881–896. Sane, S.P., 2003. The aerodynamics of insect flight. J. Exp. Biol. 206, 4191–4208. Sane, S.P., Dickinson, M.H., 2002. The aerodynamic effects of wing rotation and a revised quasi-steady model of flapping flight. J. Exp. Biol. 205, 1087–1096. Sponberg, S., Full, R.J., 2008. Neuromechanical response of musculo-skeletal structures in cockroaches during rapid running on rough terrain. J. Exp. Biol. 211, 433–446. Sun, M., Wang, J., Xiong, Y., 2007. Dynamic flight stability of hovering insects. Acta Mech. Sin. 23 (4), 231–246. Vance, J.T., Faruque, I.A., Humbert, J.S., 2010. The effects of differential wing stroke amplitude and stroke offset on insect body movements during perturbed flight conditions. In: 34th Annual Meeting of the American Society of Biomechanics, Providence, RI, August 18–21. Vance, J.T., Faruque, I.A., Humbert, J.S., 2013. Kinematic strategies for mitigating gust perturbations in insects. Bioinspir. Biomim. 8 (1), 016004.
Please cite this article as: Faruque, I.A., Sean Humbert, J., Wing motion transformation to evaluate aerodynamic coupling in flapping wing flight. J. Theor. Biol. (2014), http://dx.doi.org/10.1016/j.jtbi.2014.07.026i
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
