Analysis of Dynamics in Escaping Flight Initiation in Drosophila

Transcription

1 8 IEEE International Conference on Biomedical Robotics and Biomechatronics (BioRob 8) Analysis of Dynamics in Escaping Flight Initiation in Drosophila Francisco A. Zabala, Gwyneth M. Card, Ebraheem I. Fontaine, Richard M. Murray, and Michael H. Dickinson Division of Engineering and Applied Science California Institute of Technology Pasadena, California 95 Author for correspondence ( Abstract We present a mathematical reconstruction of the kinematics and dynamics of flight initiation as observed in high-speed video recordings of the insect Drosophila melanogaster. The behavioral dichotomy observed in the fruit flies flight initiation sequences, as a response to different stimuli, was reflected in two contrasting sets of dynamics once the flies had become airborne. By reconstructing the dynamics of unconstrained motion during flight initiations, we assess the fly s responses (generation of forces and moments) amidst these two dynamic patterns. Moreover, we introduce a 3D visual tracking algorithm as a tool to analyze the wing kinematics applied by the insect, and investigate their relation(s) to the production of these aerodynamic forces. Using this framework we formulate different hypotheses about the modulation of flight forces and moments during flight initiation as a way towards refining our understanding of insect flight control. I. INTRODUCTION Difficulties inherent to the miniaturization of unmanned aircrafts have evoked studies of biological mechanisms for the development of innovative means of perception, actuation, and control. Particular interest has been put into characterizing insect flight, where these three components seem to interact quite effectively. This reverse engineering feat has not been without challenges; roughly, we can classify these challenges into three major categories: ) modeling unsteady aerodynamics of flapping flight, ) mapping wing and body kinematic variations to the production of aerodynamic forces and moments, and 3) understanding mechano-sensorial connections that trigger such kinematic variations. Further difficulties are also faced at a higher stratus of insect flight control, in particular, recognizing patterns of behavior and understanding decision-making processes. Figure illustrates the authors view of the macroscopic components of insect flight control, and the general approaches followed for its study, as presented in literature [] [6]. While simplified aerodynamic models give us acceptable estimates of the insect s production of forces and moments, other simplifications and assumptions limit a comprehensive understanding of insect flight control. In the past, empirical assessments of ) have environment data Sensors perceived data Fly's Controller kinematic patterns Muscles Wings [Dickinson,Sane,Fry] [Taylor,Zbikowski,Hedrick] [Epstein,Straw] 3 aerodynamic forces & moments Body position attitude velocity acceleration Fig.. Macroscopic Components of Insect Flight Control.,, and 3 are the main areas where the study of flight control has focused. Last names correspond to most representative references included in this text. been made by studying two flight conditions, namely, hovering and stable forward flight. Under these conditions, plausible wing kinematic responses to small perturbations about the insect s desired operating point have been studied. This leads to three important questions: What occurs when the perturbations are not small? Does the insect consistently produce the same forces (and moments) to counteract these perturbations? Does the insect consistently use similar wing kinematics to produce particular forces (and moments)? In this study, we present the framework we are using to address these questions, and thus, refine our understanding of insect flight control. The dependence of our analyses on specific sensor information is an ongoing part of the work presented here [7]. Our approach consists of using high-speed video recordings of flight initiations of the fruit fly Drosophila melanogaster to extract kinematic information (i.e., rotational and translational positions, velocities and accelerations) [8], and to use those data to derive the (global) flight dynamics of the insect. Additionally, we introduce a novel visual tracking algorithm [9] that allows us to observe a subset of the wing kinematics during these take-offs, from which we can reconstruct the aerodynamic forces and moments produced by the insect. By combining these two sets of data, i.e., flight forces (and moments) produced by the wings, and global unconstrained motion dynamics, we seek to understand control principles governing insect flight. In Section II we give an overview of

2 the flight initiation phenomena in Drosophila, and we analyze the body and wing kinematics from video data. In Section III we present our reconstruction and analysis of the dynamics of flight initiation, and we offer a preliminary assessment of the modulation of aerodynamic forces and moments. Finally, Section IV presents conclusions and our intents for future work. II. KINEMATIC ANALYSIS OF FLIGHT INITIATION OF Drosophila A. Coordinate Frames, Transformations and Notation The following figures summarize the Coordinate Frames used throughout this paper: large instabilities that are rapidly mitigated [8] in a clear depiction of the insect s remarkable performance, robustness and stability. Fig. 4. Voluntary Flight Initiation. Prior wing elevation (-4.ms -.7ms), and simultaneous leg-extension and wing depression (-.3ms ms) lead to a stable controlled flight initiation (.8ms 8.5ms) Fig.. The two coordinate frames used in the paper. We employ standard aerodynamic coordinate frames to describe the motion of the insect in 3D space. We use earth-fixed (x f, y f, z f ) and bodycentered (x b, y b, z b ) frames as presented traditionally in aerodynamics literature []. In addition, we denote as roll, pitch and yaw the rotations about body-centered axes; bank, elevation and heading refer to attitude with respect to the earth-fixed coordinate frame. Fig. 3. Subset of wing kinematics. The top plane is meant to be parallel to the wing profile, while the bottom one is intended to be aligned with the direction of travel of the wing. The stroke position φ (not shown) refers to the angle of travel of the wing along the stroke plane. The main thrust for this study comes from the identification of two modes in the flight initiations of Drosophila [], [], one of which results in Fig. 5. Escape Flight Initiation. Leg extension begins prior to wing elevation (-3.ms ms), this leads to flight initiation without coordinated leg-extension and wing-depression (ms), which in turn results in tumbling flight (7.3ms 8.ms+). These transient instabilities motivate a new interpretation of the insect s flight control mechanisms beyond typical a priori assumptions of small deviations around an operating point. Moreover, the contrasting nature of the two modes of flight initiation entices an explicit comparison between them using control theory criteria. The first identified mode of flight initiation in Drosophila consists of a voluntary response where early wing elevation leads to a slower and more stable flight initiation. Contrastingly, strong visual stimuli cause a fly to generate a rapid (< 5ms) leg extension to propel itself off the ground without the aid of the coordinated wing motion observed in voluntary takeoffs. The lack of proper wing depression leads to tumbling flight where the fly translates faster, but also rotates rapidly about the body axes [8]. Figures 4, and 5 illustrate these phenomena. Using three orthogonally aligned, high-speed video cameras we captured flight initiations sequences (see [8] for details). From these data, we were able to extract body and wing kinematics; the results are summarized in the next subsections.

3 x 4 x 4 x p [deg.s! ].5 q [deg.s! ].5 r [deg.s! ].5 u [m.s! ].5 v [m.s! ].5 w [m.s! ].5!.5!.5!.5!.5!.5!.5!!!!.5!.5!.5. x 7. x 7. x p [deg.s ].5!.5 q [deg.s ].5!.5 r [deg.s ].5!.5 u [m.s ]!8 v [m.s ]!8 w [m.s ]!8 A! ! ! B! ! ! Fig. 6. Entire sequence for kinematics ([A] rotational, [B] translational) during flight initiation of a couple of fruit flies. Black (dark gray) trace shows a voluntary take-off while magenta (light gray) corresponds to an escape response. 3 x 4 3 x 4 3 x 4 3 x 4 3 x 4 3 x 4 p [deg.s! ]! q [deg.s! ]! r [deg.s! ]! p [deg.s! ]! q [deg.s! ]! r [deg.s! ]!.5 x 7.5 x 7.5 x 7.5 x 7.5 x 7.5 x p [deg.s ].5!.5 q [deg.s ].5!.5 r [deg.s ].5!.5 p [deg.s ].5!.5 q [deg.s ].5!.5 r [deg.s ].5!.5!.5!.5!.5!.5!.5!.5 A C u [m.s! ].!. v [m.s! ].!. w [m.s! ].!. u [m.s! ].!. v [m.s! ].!. w [m.s! ].!.!.4!.4!.4!.4!.4!.4!.6!.6!.6!.6!.6! u [m.s ]! v [m.s ]! w [m.s ]! u [m.s ]! v [m.s ]! w [m.s ]!!9!9!9!9!9! B! ! ! D! ! ! Fig. 7. Aligned data for all flies where each trace corresponds to a flight initiation sequence. B. Body Kinematics Figure 6 shows the kinematics of two (different) flies during flight initiation with similar time periods, (i.e., no alignment on the data has been performed). The black (darker grey) trace corresponds to a fly performing a voluntary take-off, while the magenta (lighter grey) corresponds to a fly executing an escape response. The data comprise rotational rates about the three axes of rotation (p, q, r), rotational accelerations (ṗ, q, ṙ), translational velocities (u, v, w) and translational accelerations ( u, v, ẇ). We observe that besides obvious amplitude differences the responses had very similar patterns. This consistency was found in several pairs of flies (i.e., one escape one voluntary) with similar time lengths. This motivated further our interest in assessing dynamic responses for initiating flight under these two conditions. For studying the dynamic responses specifically during take-off, we needed to align the data at an initial time t = ms. We used the acceleration spike in the vertical direction to discriminate flight initiation; t lift-off = t max( ẇ ). The aligned kinematic data (about the Center of Mass (COM) for all flies are shown in Figure 7, where each trace corresponds to a flight initiation video sequence.

4 In Figure 7 we show the kinematics for voluntary (A,B) and escape (C,D) flight initiations. Vertically, we have the rotational (A,C) and translational (B,D) velocities (top) and acceleration (bottom). Since trajectories have different lengths, a single fly is meant to be representative (different colored trace) of the group. Our observations from these plots include,. In voluntary take-offs, roll accelerations have (on average, indicated by ( ) similar magnitude than yaw, and pitch accelerations ṗ q ṙ. However, in escape take-offs, we empirically derive the proportionality q, ṙ.3 ṗ.. Large rotations (particularly about the x-axis) are observed in the escape responses (C) [8], p voluntary << p escape. These particular phenomena have important implications in the dynamics of the fly during flight initiation, this is discussed in our dynamic analysis (Section III). C. Wing Kinematics Intuitively, we can deduce that flies do not follow a purely ballistic trajectory during flight initiations. Thus, we are very much interested in extracting from video data, the actual kinematics of the wings, and then use standard aerodynamic models of flapping flight to study the actual modulation of aerodynamic forces and moments. In particular, we are interested in three parameters, the angle of attack, the stroke position, and the stroke position rate (see Figure 3 for illustration, and [6] for definitions of these parameters). Fig. 8. To automatically track Drosophila, a geometric model of the fly is matched to the reconstructed projection rays of the image silhouette from three camera views. Previous studies of aerobatic maneuvers in insects have required laborious manual methods to capture kinematic data from high speed video usually, by clicking a set of points to match a given template [8], [3], [4]. Fontaine et al have recently developed a technique to automatically track the 3D motion of Drosophila from multiple camera views [9]. This technique is an extension of previous visual tracking systems for studying motion kinematics in nematodes Frame 55 Fig. 9. Representative tracking results during flight initiation. Estimated location of model is plotted at various intervals to illustrate the gross trajectory of the fly from of 3 camera views. and zebrafish [5]. The algorithm consists of four components to achieve reconstruction of the fly s motion. First, an accurate geometric model of the fly is constructed whose compact parameterization encodes relevant measurements of the fly s body and wing motion. A dynamic model is utilized to provide prior knowledge about the fly s motion and provide accurate prediction in the presence of misleading visual measurements and self-occlusions. To fit the geometric model to the images, the projection rays of the image silhouette are reconstructed from 3 camera views and the distance between the projection rays and corresponding model points are minimized using an iterated Kalman filter (see Figure 8). Finally, due to the symmetric shape of the fly s body about it s head/tail axis, constraints are applied to the model s configuration, due to the unobservability of rotations about this symmetric axis, (i.e. the roll angle). Using the method described above, we have accurately captured the motion of free flying Drosophila during flight initiation (Figure 9). In particular, we have extracted the desired kinematic parameters for reconstructing aerodynamic forces and moments. Roughly, this allows us to get a handle on what portion of the dynamics correspond to the fly. III. DYNAMIC ANALYSIS OF FLIGHT INITIATION As we have mentioned in the previous section, after extracting the kinematics of body movement and wing motion, we reconstruct flight forces and moments to assess the dynamics of flight initiation. A. Mechanics of Unconstrained Motion Applying Newton s second law, the system of equations F T (t) = m V(t) + ω(t) (mv(t)) () M T (t) = [I] ω(t) + ω(t) ([I] ω(t)),

5 describes in body-fixed coordinates the forces and moments about the 6DOF in which an object can move in 3D space []. Boldface notation is used to indicate vectorial quantities, and the dot ( ) is used to denote derivatives with respect to time. F T is the total force (including weight) observed (i.e., reaction force), M T is the net moment about the origin which is located at the estimated COM, m is the mass of the fly (m = µg), V and ω are the translational and rotational velocity vectors respectively, and [I] is the inertia tensor. Figure illustrates the reaction forces of the flies during voluntary (A,B) and escape (C,D) take-offs. Traces for all flies are included, and a particular fly has been selected (black trace) to represent the overall dynamic behavior. The forces are decomposed as follows, the top traces (A) and (C) represent the x-y-z components of F = m V, where the superscript is used for notation purposes only. Similarly, middle traces correspond to F = ω (mv). The bottom traces show (F T ). (B,D) portray the magnitude of the total force ( F T ). To validate our estimates of this particular method, we compared these results (B,D) with those presented in recent studies. First, we consider our initial forces (at t = ; averaged to 97.3 ± 5.µN for voluntary, and 73. ±.µn for escape; mean±s.d.; n =, 3 respectively). This spike in the total force produced comes from the extension of the legs, which are reported to produce 74µN on average by Zumstein et al [6]. However, these measurements were made in escaping flies only, which accounts for a portion of our underestimation. Flight forces yielded better estimates to those in literature where it is found that the maximum force produced throughout the wingstroke of Drosophila is 3.µN. We use a similar approach for analyzing the moments produced during flight initiation based on an approximation of the shape of the fly by a cylinder (diameter =.7mm, length =.5mm [3]). Recall that the moment of inertia tensor for a cylinder comprises three principal components I xx = (mr ), I yy = I zz = (ml ) + 4 (mr ), while the six off-diagonal terms I xy = I yx = I xz = I zx = I yz = I zy =. Solving () for the moment about x: M x = I xx ω x + (I y I z ) q.r, where (I yy I zz ) =, which means that the moment about the x-axis is only proportional to the rotational acceleration about this axis. Moreover, given the symmetry of the fly, consider the values in [3] for radial and longitudinal lengths of Drosophila (L =.5mm, and R =.7mm). We have, I yy = I zz = I xx.6 I xx. () Now, when we look at this relation of the components of the inertia tensor (3), and consider the symmetry of the initial forces during escape flight initiation, we expect the fly to rotate (at least initially) approximately.6 times more about the x-axis than about the other two (by the definition of moment of inertia). Numerically, from actual kinematics, we observe that the ratio. p.3, considering t < 5ms ; q + r the ( ) notation is used to indicate the individual mean. An interesting phenomena is observed in the moments of the fly, which similar to the forces are almost proportional to accelerations (in this case rotational); the moment about the x-axis, despite fast rotational acceleration is small. Thus, the fly seems unable to produce enough counter-torque to rapidly mitigate these rotations. However, we observe large oscillations in the moments about the two other axes, and we hypothesize that the mechanism for stabilizing roll so rapidly (as illustrated in the motion kinematics) corresponds to a relation with the moments produced about the other two-axes (which exhibit a strong relative symmetry) [7]. The system described by () is nonlinear, as each equation presents coupling between the velocities. However, in Figure we observe that the terms corresponding to this coupling are rather small (ω (mv) and ω ([I] ω)). So, decoupling the system of equations is plausible, F T (t) = m V(t) + D V(t) (3) M T (t) = [I] ω(t) + E ω(t), where D V(t) << m V(t) and E ω(t) << [I] ω(t). Different system identification methods can be used to construct a model of this system [7]. However, we are interested in understanding the left hand side of the equations in (3). Particularly, we can rewrite the forces as, F T (t) = F A (t) + F L (t) + W, where (4) F A (t) = f (α(t), φ(t),...). W is the animal s weight vector, F A (t) R 3 is the aerodynamic force; f : S R is the mapping of the n-dimensional space S = {α(t), φ(t),...} (of n different wing kinematic parameters); and F L (t) is the force produced by the legs. In the plots in Figure, we notice that F T (t) = Gδ(t), where G R and δ(t) is just an impulse (finite) at t =, represents a reasonable approximation. However, determining F A (t) is a non-trivial task, thus we resort to the visual tracking algorithm introduced in Section II, and reverse-engineer this relation. As we would like to distinguish between the effects of leg forces versus aerodynamic forces, we performed

6 A [N] [N]!! x!4 x!4! x! [N] [N]!! x!4 x!4! x! [N] [N]!! x!4 x!4! x! !) "*"+,!&-( B & '(#!!& % $ #!!67$'"%#8!et al,!!!!9::;!<7$=5!!)./!et al, 3!!!!'".*%!+%.+3%!! " #! #" $! $" %! %" &!!"#$%!&$'( C [N] [N]!! x!4 x!4! x! [N] [N]!! x!4 x!4! x! [N] [N]!! x!4 x!4! x! D!) "*"+,!&-( & '(#!!& % $ #!!67$'"%#8!et al,!!!!9::;!<7$=5!!)./!et al, 3!!!!'".*%!+%.+3%!! " #! #" $! $" %! %" &!!"#$%!&$'( Fig.. Reconstructed forces for voluntary (A,B) and escaping (C,D) flies. Solid lines on (B) and (D) represent results reported in literature. The value at 74µ is comparable only to escaping flies (D). some initial assessment of the dynamics in terms of how fast they converge to an empirically defined stability point (for this task, only the first 3ms of the flies trajectory were available). Assuming that the forces are decreasing monotonically, we let τ represent a heuristic measure for stability, defined as τ : f (x(t), ẋ(t)) < f + ɛ, t > τ, (5) where f is a function describing the system dynamics, f is the mean value of the function for all time, and ɛ a fixed parameter bounding the function, in this case the magnitude of the standard deviation (of each trajectory) was used. Corroborating intuition, we observed that τ was slower for escape responses (larger perturbations). However, it gives us a measure of how fast the transient effects induced by leg-extension dieoff. B. Modulation of Aerodynamic Forces and Moments Novel approaches such as dynamically scaled robotic models in combination with the use of highspeed cameras, have permitted the refinement of the quasi-steady model to account for unsteady aerodynamic effects present in flapping flight [3], [4], [8]. We use this model to compute aerodynamic forces to assess the performance of our visual tracking algorithm described above. Thus, we compute F L,tr = ρ C tr,n(α w (t)) U cp(t) S w, and (6) F D,tr = ρ C tr,p(α w (t)) U cp(t) S w. Preliminary results are shown in Figure for two flies taking-of voluntarily. Since we are only using the simplest (quasi-steady) aerodynamic model, and not accounting for the unsteady aerodynamics, we expect to underestimate the total aerodynamic force. We use dashed lines to represent the means of the wing (red) and the body (blue) forces. We see that our estimates (based on the quasi-steady model (6)) for two flies were below the reconstructed forces from the body kinematics. However, we also emphasize that we were not accounting for the large force produced by leg extension at t = ms. Devising a controller inspired by flying insects becomes complicated by the fact that identical production of aerodynamic forces and moments can be

7 x!4 x!4 F total [N].5 F total [N].5 A B Fig.. Estimation of translational aerodynamic forces using wing kinematic parameters extracted from two individual flies performing a voluntary take-off C L,tr,C D,tr.5 C L,tr,C D,tr Angle of Attack [deg] Angle of Attack [deg] Fig.. Estimation of lift and drag coefficients from the wing kinematics from the parameters extracted using our visual tracking algorithm achieved by different kinematic patterns of the wings and body; thus, solutions to the control problem are not unique and may not reflect what insects actually do. Given the aerobatic dexterity of flying insects, we are lead to believe that the not-necessarily unique solution chosen by the insect is optimal with respect to non-trivial criteria. Thus, before making any conclusions about the system, we need to collect a larger dataset which will lead to natural conclusions about the correlation of these kinematic parameters to production of flight forces and moments. IV. CONCLUSIONS AND FUTURE WORK Drosophila melanogaster (fruit fly) has been observed to perform high-speed turns (saccades) and return to a stable forward trajectory in a matter of milliseconds (a detailed study is presented in [8]). Additionally, during flight initiations elicited by visual stimuli, this insect tends to tumble rapidly about all axis of rotation, but is also able to stabilize its attitude rapidly [8]. These remarkable aspects of insect flight control (i.e., stability, performance, and robustness), which exceed counterparts in any manmade system, entice a detailed analysis of the insect s flight mechanics. In this paper we describe our framework for studying the dynamics of flight initiation in Drosophila. By reconstructing the kinematic data of motion we derived the forces and moments acting on the system and observed that the overall dynamics uncouple. However, we did not incorporate the notions of delay, noise, and assumed perfect sensory data. Under these conditions, we showed that this system is deterministic (from a mechanical point of view), since knowledge about the present state determines future behavior. Thus, we proceeded to reconstruct the reaction forces and moments to the observed dynamics. Our simple estimation of aerodynamic forces based on the quasisteady model was acceptable, as by taking the mean value for comparison is not representative due to the large forces produced by the legs. Our current understanding of the dynamics of flight initiation is limited by our inability to uniquely map kinematic variations of aerodynamic parameters to the production of flight forces and moments. The incorporation of an automated software for tracking into the study of insect flight control is a great benefit, since not only are we able to automate data gathering

8 tasks, but we can also observe parameters (e.g., wing kinematics) that are otherwise unobservable. Understanding how aerodynamic forces and moments are produced is essential to fully assess dynamic behavior of the system. Refinements to our visual tracking algorithm will allow observations of escape flight initiation, thus permitting a contrast between wing kinematics under the two different take-off modes. REFERENCES [] G. K. Taylor and A. L. R. Thomas, Dynamic flight stability in the desert locust Schistocerca gregaria, J. Exp. Biol., vol. 6, pp , 3. [] T. Hedrick and T. Daniel, Flight control in the hawkmoth Manduca sexta: the inverse problem of hovering, J. Exp. Biol., vol. 9, pp , 6. [3] C. P. Ellington, The aerodynamics of hovering insect flight. i-iv, Philos. Trans. Roy. Soc. London B, Biolog. Sci., vol. 35, pp. 8, 984. [4] M. Epstein, S. Waydo, S. B. Fuller, W. Dickson, A. Straw, M. H. Dickinson, and R. M. Murray, Biologically inspired feedback design for Drosophila flight, in Proc. IEEE American Control Conference (ACC 7), New York, USA, July 7, pp [5] R. Zbikowski, S. A. Ansari, and K. Knowles, On mathematical modelling of insect flight dynamics in the context of micro air vehicles, J. Exp. Biol., vol. 6, pp , 3. [6] W. B. Dickson, A. D. Straw, C. Poelma, and M. H. Dickinson, An integrative model of insect flight control, in Aerospace Sciences Meeting and Exhibit (ACC 7), Nevada, USA, Jan. 6, pp [7] G. M. Card and M. H. Dickinson, An angular dependence on visual stimuli during flight initiation in Drosophila, in progress. [8] G. Card and M. Dickinson, Performance trade-offs in the flight initiation of Drosophila, J. Exp. Biol., vol., pp , 8. [9] E. Fontaine, A 3d visual tracking algorithm for studying flight initiation in Drosophila melanogaster, in progress. [] W. F. Phillips, Mechanics of Flight. Hoboken, NJ: John Wiley & Sons, 4. [] S. Hammond and M. O Shea, Ontogeny of flight initiation in the fly Drosophila melanogaster: implications for the giant fiber system, J. Exp. Biol., vol., pp , 8. [] J. R. Trimarchi and D. M. Schneiderman, Different neural pathways coordinate Drosophila flight initiations evoked by visual and olfactory stimuli, J. Exp. Biol., vol. 98, pp. 99 4, 995. [3] S. N. Fry, R. Sayaman, and M. H. Dickinson, The aerodynamics of free-flight maneuvers in Drosophila, Science, vol. 3, pp , Apr. 3. [4], The aerodynamics of hovering flight in Drosophila, J. Exp. Biol., vol. 8, pp , 5. [5] E. Fontaine, D. Lentink, S. Kranenbarg, U. K. Muller, J. L. van Leeuwen, A. H. Barr, and J. W. Burdick, Automated visual tracking for studying the ontogeny of zebrafish swimming, J. Exp. Biol., vol., pp , 8. [6] N. Zumstein, O. Forman, U. Nongthomba, J. C. Sparrow, and C. J. H. Elliott, Distance and force production during jumping in wild-type and mutant Drosophila melanogaster, J. Exp. Biol., vol. 7, pp , 4. [7] F. A. Zabala, System identification of flight initiation in Drosophila, in progress. [8] M. H. Dickinson, F.-O. Lehmann, and S. P. Sane, Wing rotation and the aerodynamics of insect flight, Science, vol. 84, pp , June 999.