A robust technique for determining the angle of attack of insect wings from film of free flight has to date proved elusive. This report describes the development of two new methods – the Strips and Planes techniques – which were designed to overcome some of the limitations experienced in previous studies. The accuracy and robustness of these novel methods were tested extensively using simulated hawkmoth wing outlines generated for a realistic range of wing positions and torsion. The results were compared with those from two existing methods – the Symmetry and Landmarks procedures. The performance of the latter technique will be strongly species-dependent; it could not be successfully applied to hawkmoth flight because of practical difficulties in obtaining suitable landmarks.

The Planes method was the least successful of the remaining techniques, especially during those phases of the wingbeat when the orientations of the two wings relative to the camera viewpoint were similar. The Symmetry and Strips methods were tested further to investigate the effects on their performance of introducing additional camber or wing motion asymmetry. The results showed clearly that the Strips method should be the technique of choice wherever the axis of wing torsion is close to the longitudinal axis of the wing. The procedure involves the experimenter matching a model wing divided into chordwise strips to the true wing outline digitized from high-speed film. The use of strips rather than the points digitized in previous studies meant that the analysis required only one wing outline to be digitized. Symmetry of motion between the left and right wings is not assumed.

The implications of requiring human input to the Strips method, as opposed to the strictly numerical algorithms of the alternative techniques, are discussed. It is argued that the added flexibility that this provides in dealing with images which have typically been recorded in suboptimal conditions outweighs the introduction of an element of subjectivity. Additional observations arising from the use of the Strips analysis with high-speed video sequences of hawkmoth flight are given.

A common problem in biomechanical studies is the need to reconstruct three-dimensional motions from two-dimensional film images. Ideally, this would involve the analysis of sequences recorded simultaneously from two or more viewpoints but, in practice, the experimenter is often restricted to single-view images (van den Berg, 1994). The latter situation has been encountered in a wide range of studies which has included the analysis of fish head movements (van den Berg, 1994), the motion of insect wings during free flight (e.g. Ellington, 1984b; Azuma and Watanabe, 1988; Dudley and Ellington, 1990) and the flight path of ground-nesting wasps (Voss and Zeil, 1995). The nature of these subjects means that the recording conditions are often suboptimal, and conventional image-processing algorithms may not be flexible enough to cope reliably with the resulting images. Human input to the analysis is less objective and considerably slower, but it may be a more robust technique; the difficulty of developing an automated analysis system which can match the performance of a human operator has been shown by Voss and Zeil (1995).

This report describes the development and testing of a number of techniques for addressing one such image-analysis problem: the determination of the angle of attack of a beating insect wing during free flight. This problem not only remains one of the most pressing in the field of insect flight studies (Ennos, 1989), but it also requires the wider issue of the relative merits of human analysis or strict adherence to numerical algorithms to be considered. The image sequences to be analysed were high-speed video sequences (1000 frames s−1) of the hawkmoth Manduca sexta L. in free flight at speeds between hovering and 5 m s*. These films were obtained as part of an investigation into the changes in wing kinematics that accompany changes in flight speed (Willmott and Ellington, 1997).

‘Angle of attack’ has been used as a general term encompassing a number of ways of describing the orientation of a chordwise strip of a beating wing. More specifically, the angle can be measured between the chord line at a given radial position and either the horizontal (the geometric angle of attack) or the projection of the chord line onto the stroke plane. Each of these can later be converted into more aerodynamically relevant measures such as the angle of incidence: the angle between the chord line and the local airflow velocity. In the Materials and methods and Discussion sections of this study, the term angle of attack is inclusive of the angle between the chord line and any predetermined plane. The angle of attack relative to the stroke plane was used throughout the detailed comparison of methods reported in the Results section.

The first detailed studies of insect flight kinematics used tethered individuals (e.g. Weis-Fogh, 1956; Nachtigall, 1966; Vogel, 1967; Zarnack, 1972). Under such conditions, two or more views of the flying insect can be recorded simultaneously, allowing the wingtip kinematics as well as the angle of attack to be analysed using stereophotographic techniques. Detailed information on the patterns of angles of attack have been obtained in this way for a number of dipterans (e.g. Nachtigall, 1966). The relationship between tethered and free-flight performance is not known, however, and several recent studies have investigated free-flight kinematics directly. The difficulties of obtaining controlled free flight and in predicting its precise location and orientation (Dudley and Ellington, 1990) have led to the requirement for a wider field of view which has, in practice, restricted filming to a single view (e.g. Ellington, 1984b; Dudley and Ellington, 1990; Wakeling and Ellington, 1997; Willmott and Ellington, 1997). Reconstructing the three-dimensional wing motion and angles of attack from these two-dimensional images has required the development of novel techniques.

A suitable method for analysing the wingtip kinematics from such sequences, assuming symmetry of motion between the two wings, was developed by Ellington (1984b), and its accuracy and robustness have been demonstrated for flight in a range of insect groups (Ennos, 1989; Dudley and Ellington, 1990; Wakeling and Ellington, 1997; Willmott and Ellington, 1997). The current techniques for determining the angle of attack of wings from single-view images are less advanced. Where clearly identifiable ‘landmarks’ can be identified on the images, simple geometry can be used to estimate the wing shape and twisting (e.g. Vogel, 1967; Bilo, 1971, 1972). Even where such landmarks exist, however, they may not provide a good coverage of the whole wing surface or be visible from a single viewpoint throughout the wingbeat.

Dudley and Ellington (1990) developed a new approach which used the digitized margins of the two wings, or wing couples, to provide information on the spanwise changes in the wing twisting. This technique invoked the same symmetry assumption as Ellington’s (1984b) wingtip kinematic analysis, and problems arose when the corresponding portions of the two wings were parallel to each other. No data exist for the accuracy and robustness of the existing methods when used with realistic wing shapes, but it appears likely that neither will be sufficiently adaptable or reliable to be applied successfully to a wide range of insect species.

The purpose of the present study was to develop alternative methods for angle-of-attack analysis which avoided the problems of uneven coverage, parallel wing margins and/or the need to assume symmetry of the wing motions. Two of these techniques – termed the Strips and Planes methods – are described here, and their performance with a wide range of simulated hawkmoth wing outlines are compared with those of the existing approaches. The Strips method is shown to be the most suitable for use in insect flight studies, and further observations concerning its use with real hawkmoth images are given. The results of the latter analysis are presented in a companion paper (Willmott and Ellington, 1997).

Description of the methods

Four possible methods for angle-of-attack analysis were considered. Two of these – the Symmetry and Planes methods – require both wing margins to be digitized on each frame. The remaining approaches – the Strips and Landmark methods – need data from only one wing or wing couple. Each technique will be described briefly; a detailed account of the Symmetry method can be found in Dudley and Ellington (1990). It is assumed in all the techniques that the wingtip positions relative to the wingbases in three-dimensional space have previously been determined.

The Symmetry method

Dudley and Ellington’s (1990) method uses the known two-dimensional coordinates of corresponding points on the two wing margins to provide a three-dimensional reconstruction of the near wing margin. Symmetry of the near-and far-wing kinematics is assumed; the plane of symmetry for the wingbeat is constructed from the equations of lines joining each wingtip at points in the cycle close to supination and pronation (Ellington, 1984b; Fig. 1A).

Fig. 1.

The Symmetry method. (A) Determination of the plane of symmetry of the wing motion. The arrowed arcs indicate the wingtip paths for the two wings. The symmetry plane (shaded) is calculated from the equations of the lines n and f which join the wingtip positions for points close to pronation with those close to supination for the near and far wings, respectively. (B) The location of the far-wing point P4 corresponding to a point P3 on the near-wing margin is defined as the intersection between the far-wing margin and a line drawn through P3 and parallel to a line joining the two wingtips (P2 and P1). The farwing margin in the vicinity of P4 is modelled as a straight line section between the two points (circled) which are immediately adjacent to the intersection.

Fig. 1.

The Symmetry method. (A) Determination of the plane of symmetry of the wing motion. The arrowed arcs indicate the wingtip paths for the two wings. The symmetry plane (shaded) is calculated from the equations of the lines n and f which join the wingtip positions for points close to pronation with those close to supination for the near and far wings, respectively. (B) The location of the far-wing point P4 corresponding to a point P3 on the near-wing margin is defined as the intersection between the far-wing margin and a line drawn through P3 and parallel to a line joining the two wingtips (P2 and P1). The farwing margin in the vicinity of P4 is modelled as a straight line section between the two points (circled) which are immediately adjacent to the intersection.

Both wing outlines are digitized from the frame to be analysed, and the gradient of the line joining the near and far wingtips (points P1 and P2, respectively) is calculated. A line with the same slope is constructed through each point P3 on the near-wing margin (Fig. 1B). The intersection of this line with the far-wing outline is taken as the corresponding point P4 on the far wing. The location of P3 in the direction perpendicular to the image plane is calculated from the coordinates of points P1P4. The positions of the leading- and trailing-edge points are found by interpolation at regular intervals along the wing. The angle of attack at each spanwise location is determined from the line joining the corresponding leading- and trailing-edge points.

The Planes method

The Planes method also requires digitization of both wing margins, but the analysis procedure is designed to be able to cope with parallel wing margins. Under the symmetry assumption, the images of the two wings can be taken to represent a single wing viewed from two different orientations. The information which they contain is therefore complementary, and a three-dimensional reconstruction can be undertaken in a manner analogous to the analysis of true stereophotographic paired images.

The positions of the leading- and trailing-edge points (PL and PT, respectively) at a given spanwise location are determined using the following procedure. First, each wing margin is digitized separately in the film-based y*,z*coordinate system (see Dudley and Ellington, 1990; Fig. 2A). At this stage, only the positions of the wingtips and wingbases are known accurately in three-dimensional space, but the third coordinate (in the x*direction perpendicular to the film plane) is calculated for all the points under the assumption that the near- and farwing margins lie within the respective xz″ planes shown in Fig. 2A. The plane for each wing is constructed perpendicular to the longitudinal axis for that wing, with the x″ axis horizontal and with the origin at the desired spanwise position on the longitudinal axis. Only the leading- and trailing-edge points which correspond to PL and PT (whose positions on the wing margins are not known at this stage) will be accurately located in three-dimensional space by this process. The distortion of the remaining points on the margin will depend upon the orientation of the longitudinal axis relative to the camera viewpoint and will, therefore, be different for the two wings. The x*,y*,z* coordinates are then transformed to give the location of each point in the appropriate x″,z″ coordinate system. Finally, if the two xz″ planes and the respective wing outlines are overlaid, with their origins superimposed, the positions of PL and PT are indicated by the intersections of the two leading edges and of the two trailing edges (Fig. 2B). The coordinates of these two points are transformed back into a gravitational system where the angle of attack is determined.

Fig. 2.

The Planes method. The fate of points on the wing margin at a given spanwise position during the Planes analysis. (A) Construction of the xz″ planes perpendicular to the longitudinal axis of each wing. The left (open squares, solid line) and right (filled triangles, dashed line) wing margins are shown in the y*z* plane (Dudley and Ellington, 1990), as seen from a viewpoint behind and above the insect. The wings are close to supination, each wingtip lying farther from the viewer than its corresponding wingbase. For each wing, a plane xz″ has been constructed perpendicular to the longitudinal axis with the x″ axis horizontal and the origin at the desired spanwise location on the longitudinal axis. The digitized points on each wing margin are assumed to lie in the plane for that wing, and the x″,z″ coordinates for each point are calculated accordingly. For clarity, only a limited region of each plane is shown. The exact locations in these planes of the required leading- and trailing-edge points (represented by open and filled circles, respectively) are unknown at this stage of the process. (B) The wing margins once the two xz″ planes have been superimposed. The intersections of the two leading edges and the two trailing edges give the positions of the leading- and trailing-edge points PL and PT, respectively. With the wings in this orientation, there are actually two intersections for each edge of the wing but, from consideration of where on the margins these points are located, it is clear which of the two alternatives is correct.

Fig. 2.

The Planes method. The fate of points on the wing margin at a given spanwise position during the Planes analysis. (A) Construction of the xz″ planes perpendicular to the longitudinal axis of each wing. The left (open squares, solid line) and right (filled triangles, dashed line) wing margins are shown in the y*z* plane (Dudley and Ellington, 1990), as seen from a viewpoint behind and above the insect. The wings are close to supination, each wingtip lying farther from the viewer than its corresponding wingbase. For each wing, a plane xz″ has been constructed perpendicular to the longitudinal axis with the x″ axis horizontal and the origin at the desired spanwise location on the longitudinal axis. The digitized points on each wing margin are assumed to lie in the plane for that wing, and the x″,z″ coordinates for each point are calculated accordingly. For clarity, only a limited region of each plane is shown. The exact locations in these planes of the required leading- and trailing-edge points (represented by open and filled circles, respectively) are unknown at this stage of the process. (B) The wing margins once the two xz″ planes have been superimposed. The intersections of the two leading edges and the two trailing edges give the positions of the leading- and trailing-edge points PL and PT, respectively. With the wings in this orientation, there are actually two intersections for each edge of the wing but, from consideration of where on the margins these points are located, it is clear which of the two alternatives is correct.

The Strips method

The central theme of the Strips approach is to match a computer-generated simulation of the wing outline to the true outline of the wing as digitized from an individual frame of the high-speed film. The wing model is created by dividing the untwisted wing planform into a number of chordwise strips (Fig. 3A), each of which can be rotated independently about the longitudinal axis of the wing. The simulated edges of each strip are calculated for an identical viewpoint to that of the camera in the film image, and they are then superimposed onto the true outline.

Fig. 3.

The Strips method. (A) The starting images are the strips comprising the planform of the untwisted wing (top) and the true wing margin as digitized from the video image (bottom). The dashed line indicates the orientation of the longitudinal axis of the wing. (B) The composite image after the two longitudinal axes have been brought into coincidence and the strips of the model wing have been rotated until they match the digitized outline. For clarity, a few of the strips have been omitted.

Fig. 3.

The Strips method. (A) The starting images are the strips comprising the planform of the untwisted wing (top) and the true wing margin as digitized from the video image (bottom). The dashed line indicates the orientation of the longitudinal axis of the wing. (B) The composite image after the two longitudinal axes have been brought into coincidence and the strips of the model wing have been rotated until they match the digitized outline. For clarity, a few of the strips have been omitted.

The wingtip position relative to the wingbase must be entered to allow the two longitudinal axes to be matched, and then the orientation of each strip in turn is adjusted by the operator until the leading and trailing edges coincide with the real wing margin. Once all of the strips have been fitted successfully, the angles of attack of the individual strips can be calculated.

In the present study, the procedure described above was implemented in LabVIEW (National Instruments, Austin, TX, USA) on a Macintosh Quadra 650. The package generated a graphical user interface displaying the digitized outline both of the true wing and of 25 simulated strips, as well as control buttons which permitted the user interactively to change the angles of attack of the individual strips. Wingtip position and the apparent viewpoint of the operator were selected prior to the alignment of the simulated wing with the real outline.

The basic principle of matching the observed wing shape to the outline of an adjustable model wing is similar to that used by Nachtigall (1979). The use of strips provides two sources of information which relate to the orientation of the wing at that radial position: the tangents to the leading- and trailingedge sections, and the apparent width of the wing chord. The combination of these two indicators is, in theory, sufficient for a unique three-dimensional reconstruction of the outline using only one wing or wing couple. The symmetry assumption is not required.

The Landmarks method

This method involves digitizing the coordinates of clearly identifiable ‘landmarks’ on the wing margin, and then the use of simple geometry to determine the three-dimensional position of each point. The landmarks may be natural features, such as the intersection of two wing veins, or they may be artificially added spots or lines (Azuma and Watanabe, 1988). The true distance of each mark from the wingbase is measured directly for the untwisted wing, either before or after the flight sequences. In an individual frame, the position of each landmark in turn is digitized and its apparent distance from the wingbase calculated. The third coordinate is found from the known length of the line between the points by Pythagoras’ theorem.

The positions of the leading and trailing edges at a given spanwise position must be determined from the adjacent landmarks by interpolation and/or extrapolation. The angle of attack of the wing is calculated from the resulting estimates for the location of these points.

Testing the different methods

The Landmarks method differs from the other techniques in its use of isolated points rather than complete margins. Its performance depends upon the nature and size of the marks, as well as on their distribution over the surface of the wing. Generalizations are hard to make since the landmarks will be unique for each species, and so the accuracy must be checked for every study under the appropriate experimental conditions. Such an investigation was undertaken as part of the present study, but the Landmarks approach proved to be impractical for use with Manduca sexta because of the difficulty of obtaining suitable landmarks. The wing veins are largely hidden by the wing scales, and the wing markings neither show up sufficiently clearly on high-speed videos nor provide a good coverage of the wing surface. Attempts were made, therefore, to add a number of artificial landmarks at regular intervals along the leading and trailing margins of the wing couple. Marking materials included enamel paints, white correction fluid, silver paints and a range of fluorescent powders and paints.

Three problems were encountered with all of these materials. First, the presence of wing scales and their tendency to be shed during flight meant that the landmarks were often lost fairly rapidly after marking. Second, none of the materials adhered readily to either the wing cuticle or the scales; in order to mark both wings successfully, the moths had to be rendered inactive by chilling or CO2 anaesthesia to such an extent that the possibility of damage to their neuromuscular system could not be ruled out. Finally, spots large enough to be followed throughout the wingbeat cycle were so heavy that the distribution of wing mass (as measured by the method of Ellington, 1984a) was significantly changed.

The Landmarks method was not tested further in this study.

Accuracy and repeatability of the wing margin methods

Preliminary testing of the robustness of the three remaining methods indicated that the conditions under which each failed to give satisfactory results could not easily be generalized. Instead, a systematic approach was adopted in which the accuracy and repeatability of the methods were measured using simulated hawkmoth wing margins generated for 36 combinations of stroke plane orientation, wing position within the stroke plane and wing twist. The latter was modelled with the angle of attack relative to the stroke plane varying linearly between preselected values at the wingtip and wingbase. The range of parameters values, which are shown in Table 1, was sufficiently wide that most, if not all, of the potential problem areas would be encountered. Near- and far-wing outlines for each case were generated on a Macintosh Quadra 650 using a Mathematica (Wolfram Research, Inc., Champaign, IL, USA) program which transformed a digitized planform view of an untwisted hawkmoth wing couple into three-dimensional coordinates for any designated combination of wing parameters and camera viewpoint. The viewpoint throughout this study was the same as that used during the filming of the free-flying hawkmoths (Willmott and Ellington, 1997): behind the moth, slightly to the left of the body axis, and at an elevation of 21 °. The range of values for the kinematic parameters described in Table 1 covered most of the combinations recorded during flight.

Table 1.

The selected range of values for the three kinematic parameters

The selected range of values for the three kinematic parameters
The selected range of values for the three kinematic parameters

For each combination, three sets of wing outlines were generated which differed only in a random error term to simulate digitizing error. The magnitude of this added term was between −0.69 and +0.69 % of the wing length, which corresponded to an error of one pixel for the digitized wing length of approximately 150 pixels that was typical for the high-speed filming. Each triplet of wing outlines was analysed using each of the three methods, and the angles of attack with respect to the stroke plane were calculated at nine evenly spaced positions between 0.1R and 0.9R from the wingbase, where R is the wing length.

The mean and standard deviation of the differences between the expected and calculated angles were determined at all nine spanwise positions, but the results were condensed by dividing the wing into only three segments (inner, middle and outer) and averaging the angles of attack for 0.1–0.3R, 0.4–0.6R and 0.7–0.9R. These divisions are used in the following discussion, but the same conclusions would be reached from the original data set.

The effects of camber change or asymmetry between near- and far-wing motions on the performance of the methods

The coordinate-generation program also permitted investigation of the influence of other realistic sources of error on the accuracy of the angle-of-attack estimates. In particular, the simulated wing outlines could be modified to introduce either a camber change or an asymmetry between the orientations of the two wing axes. Increasing camber was modelled by reducing the width of the strips by a selected percentage. Asymmetry between the near and far wings could be generated by adding a small discrepancy between the sweep angles ϕ (as defined in Willmott and Ellington, 1997) of the two wings.

Performance of the wing margin methods

Comparative results from testing the three methods with the parameter combinations described in Table 1 are given in Tables 24. Table 2 contains the mean errors and the standard deviations of the three estimates for angle of attack with respect to the stroke plane for each wing section. Tables 3 and 4 present the same information but using shading to emphasize the differences in magnitude among the mean errors and the standard deviations, respectively.

Table 2.

Performance of the three analysis methods under the conditions given in Table 1

Performance of the three analysis methods under the conditions given in Table 1
Performance of the three analysis methods under the conditions given in Table 1
Table 3.

Mean errors of the three analysis methods (from Table 2) shaded to emphasize their magnitude

Mean errors of the three analysis methods (from Table 2) shaded to emphasize their magnitude
Mean errors of the three analysis methods (from Table 2) shaded to emphasize their magnitude
Table 4.

Standard deviations for the three analysis methods (from Table 2) shaded to emphasize their magnitude

Standard deviations for the three analysis methods (from Table 2) shaded to emphasize their magnitude
Standard deviations for the three analysis methods (from Table 2) shaded to emphasize their magnitude

The Planes method fared worst with regard both to accuracy (mean errors) and repeatability (standard deviations of those errors). At extreme sweep angles, there was often no intersect for either the leading or trailing edge, whilst at intermediate angles the magnitude of the mean error and/or the standard deviation was regularly greater than 5 °. The results from the Symmetry method were better, but many of the errors were still in the 2–5 ° or 5–10 ° bands. There were no clear trends between accuracy and any of the wing parameters, but the errors tended to be smaller at the more extreme sweep angles and, in particular, at the highest stroke plane angle, 50 °. Both of these conditions resulted in far wings which had few, if any, sections which were parallel either to sections of the near wing or to the line joining the two wingtips. An example of a successful analysis is shown in Fig. 4A; Fig. 4B shows one in which the margins are parallel, resulting in poor accuracy and repeatability. Fig. 4 also demonstrates the outcome of a Planes analysis for each of these two scenarios, illustrating the types of conditions under which this approach also succeeded and failed. It is evident from the latter case that the Planes method does not avoid the problem of parallel wing margins when the orientations relative to the camera of the near- and far-wing planes are very similar.

Fig. 4.

Examples of wing positions that give accurate results and of those that give poor results with both the Symmetry and Planes techniques. The wing outlines are drawn as seen from the camera viewpoint. The results for the Symmetry method show the near and far wing margins represented by a series of points and a line which has been constructed through the circled point P3 and which is parallel to the line joining the two wingtips (see Fig. 1B). For the Planes method, the near- and far-wing xz″ planes have been superimposed (see Fig. 2B) with the near-wing margin denoted by the open squares and the farwing margin by the filled triangles.(A) Accurate results are obtained when the orientations of the two longitudinal axes relative to the camera viewpoint are sufficiently different that the line drawn through P3 in the Symmetry method and the far-wing margin intersect at a steep angle, as do the two leading edges and the two trailing edges in the Planes method. The exact locations of the wing margin points defined by these intersections are then relatively insensitive to digitizing error or to slight asymmetry between the near and far wings.(B) Performance is poor when large sections of the two digitized wing margins are parallel to each other and to the line joining the wingtips. Under these conditions, the point-defining intersections occur at shallow angles. Multiple intersections may arise, and the locations of the desired leading- and trailing-edge points are very sensitive to the errors mentioned above.

Fig. 4.

Examples of wing positions that give accurate results and of those that give poor results with both the Symmetry and Planes techniques. The wing outlines are drawn as seen from the camera viewpoint. The results for the Symmetry method show the near and far wing margins represented by a series of points and a line which has been constructed through the circled point P3 and which is parallel to the line joining the two wingtips (see Fig. 1B). For the Planes method, the near- and far-wing xz″ planes have been superimposed (see Fig. 2B) with the near-wing margin denoted by the open squares and the farwing margin by the filled triangles.(A) Accurate results are obtained when the orientations of the two longitudinal axes relative to the camera viewpoint are sufficiently different that the line drawn through P3 in the Symmetry method and the far-wing margin intersect at a steep angle, as do the two leading edges and the two trailing edges in the Planes method. The exact locations of the wing margin points defined by these intersections are then relatively insensitive to digitizing error or to slight asymmetry between the near and far wings.(B) Performance is poor when large sections of the two digitized wing margins are parallel to each other and to the line joining the wingtips. Under these conditions, the point-defining intersections occur at shallow angles. Multiple intersections may arise, and the locations of the desired leading- and trailing-edge points are very sensitive to the errors mentioned above.

The results of the Strips analysis were excellent. This result must be qualified slightly by the fact that the axis of rotation used in generating the wing outlines corresponded to the axis assumed by this method of analysis, whereas the real wing twisting is likely to be more complicated. Despite this reservation, the method clearly has good resolution and is much less susceptible to digitizing error than are the other two approaches. Moreover, the available evidence suggests that the assumption of a rotation axis close to the longitudinal axis is reasonable (see Discussion).

The Planes method was rejected at this point as being insufficiently robust, and quantification of the effects of camber change and wingbeat asymmetry was carried out only for the remaining two approaches.

The effects of camber change and asymmetry of wing motion

From the original 36 combinations of wing position and twist, seven were selected for which the mean error from each approach was always less than 5 °, and in most cases less than 2 °. Any subsequent reduction in performance could then be attributed to the effects of camber change or wing asymmetry and not to inherent problems with the method. The seven combinations gave a reasonable range of stroke planes and wing positions.

Two levels of increased camber were tested, corresponding to reductions in chord length of 5 and 10 %. As a guide to the camber these values represent, arcs of uniform curvature corresponding to chord reductions of 5 and 10 % have height-to-width ratios of 1:7 and 1:3.4, respectively. The asymmetry introduced into the wing positions in the final test was 5 °, with the sweep angle for the near wing 2.5 ° lower than the nominal value, and for the far wing 2.5 ° greater. An unchanged level of digitizing error (up to 0.69 % of R) was used throughout.

Tables 57 show the mean errors and standard deviations for the above cases. The accuracy of both analysis methods worsened with the transition from 0 to 5 % chord reduction, and then from 5 to 10 % reduction. The deterioration was most marked for the Strips method, in which errors of over 2 ° become common; some situations (e.g. 60/30−20/10) generated errors of over 5 ° for all three segments. However the accuracy of the Strips method and its robustness to digitizing error, as indicated by low variation between the three replicates for each parameter combination (Table 7), were still superior to those of the Symmetry approach.

Table 5.

Performance of the Symmetry and Strips methods with added camber and asymmetry

Performance of the Symmetry and Strips methods with added camber and asymmetry
Performance of the Symmetry and Strips methods with added camber and asymmetry
Table 6.

Mean errors for the Symmetry and Strips methods with added camber and asymmetry (from Table 5) shaded to emphasize their magnitude

Mean errors for the Symmetry and Strips methods with added camber and asymmetry (from Table 5) shaded to emphasize their magnitude
Mean errors for the Symmetry and Strips methods with added camber and asymmetry (from Table 5) shaded to emphasize their magnitude
Table 7.

Standard deviations for the Symmetry and Strips methods with added camber and asymmetry (from Table 5) shaded to emphasize their magnitude

Standard deviations for the Symmetry and Strips methods with added camber and asymmetry (from Table 5) shaded to emphasize their magnitude
Standard deviations for the Symmetry and Strips methods with added camber and asymmetry (from Table 5) shaded to emphasize their magnitude

The errors from the Symmetry analysis which arose when there was asymmetry between the wings were markedly larger than those in the symmetrical condition, especially for the outer two wing segments where a number of ‘no valid intersection’ cases occurred. Similar trends were seen in the standard deviations. Asymmetry between the wings does not affect the Strips method because it requires only the coordinates of the near wing outline. The wingtip position was slightly inaccurate here, but slight corrections to the long-axis orientation can be made by the operator before the strips are rotated. This technique, therefore, has the additional benefit of providing a check on the results obtained from the wingtip kinematic analysis. This independence from the effects of asymmetry is clear in the same low mean errors and standard deviations as were seen for the symmetrical wings.

Evaluation of the different methods

The testing provided a good indication of the accuracy and robustness of the various analytical methods and of the behaviour of each in response to a range of potential sources of error. Some general conclusions concerning the relative merits of the methods can now be drawn.

The Planes approach was the first to be rejected: in general, the means and standard deviations for its errors were the highest of the methods tested, and under certain conditions it generated no answer at all. The theoretical approach has potential; for wings whose transformed outlines intersect at sufficiently large angles, the results are very accurate; however, a robust implementation of the method has yet to be found.

The results for the Symmetry method confirmed that this approach breaks down when the wing margins are parallel (Dudley and Ellington, 1990), a regular occurrence for at least limited stretches of the wing outlines. The initial introduction of realistic wing planforms, and a level of digitizing error appropriate for high-speed video analysis, resulted in errors which were unacceptably large and variable. Furthermore, the analysis is very susceptible to any asymmetry between the wings. This was seen when an asymmetry was introduced into the sweep angles, but a similar increase in the proportion of missed and inappropriate intersections would arise from differences in wing twist or from asymmetry in any of the other kinematic parameters. This problem arises from the need to use both wing outlines, which itself follows from the use of digitized points on the near wing margin. The two-dimensional coordinates for each point do not provide sufficient information for a three-dimensional reconstruction, and hence the location of the ‘equivalent’ point on the far-wing margin is required. The same requirement for the use of both margins is at the heart of the Planes method.

These problems are avoided in the Strips method by the use of wing margin sections, which provide two complementary sources of information: the chord width, and the tangents to the leading and trailing edges at that position. The chord width was the most powerful indicator of angle of attack but the tangents were valuable for fine tuning of the angles, especially when the chord width had been reduced by camber.

The accuracy of the method worsened slightly when camber was increased, but the errors were only 5 ° on average for a chord reduction of 10 %. The latter represents a substantial increase in camber, especially since the original wing planform already included the camber inherent in the wing structure. Wing asymmetries do not affect the analysis. The Strips method was clearly more accurate and robust than the other approaches and was judged to be the most suitable for use with the hawkmoth images.

The Landmarks method has great potential owing to its simplicity, absence of assumptions and requirement for only one wing outline. Two major constraints on its use, however, are the nature of the landmarks and the proportion of the wing which they cover. For opaque insect wings, either natural wing markings may be used or artificial landmarks could be added. The former may not be small or distinct enough to allow digitization with sufficient accuracy, and their distribution across and along the wing is unlikely to be regular. For artificial marks, practical difficulties with adhesion will be compounded by the need to keep marks small both to improve digitizing accuracy and to limit changes in the mass distribution of the wing. For clear wings, vein intersections might prove to be suitable landmarks. However, their coverage of the wing is also irregular and they are not always visible even in high-speed cine films (e.g. Ellington, 1984b; J. M. Wakeling, personal communication). These are serious obstacles to the landmark approach, and its accuracy cannot be tested without details of the nature, size and distribution of usable landmarks.

In conclusion, this investigation has shown that the Strips method is the most accurate and robust of the methods examined here, and it should be the method of choice wherever the assumption of rotation about a common and known axis is acceptable. The control of wing twisting is more complicated than this, of course, but this model is a reasonable approximation to the wing motion in a number of insect groups, most notably the Odonata and Diptera. Many mechanisms for control of wing twist have been identified in both groups, but the strong leadingedge spar which is characteristic of these Type C wings (Wootton, 1990) runs parallel and close to the longitudinal axis, and their twisting is not complicated by wing coupling. The wings of Drosophila melanogaster, for example, were found to rotate about an axis close to the longitudinal axis (Zanker, 1990). The composite aerofoil of the hawkmoths also has a rigid anterior support, formed from the subcostal and radial veins, which runs close to the longitudinal axis. However, wing motion is complicated by the wing coupling and by the interaction between the abdomen and the anojugal region of the hindwing. Nevertheless, the high-speed film of Manduca sexta taken during the kinematic study (Willmott and Ellington, 1997) suggests that the assumption of rotation about the longitudinal axis is reasonable. The axis for any ‘internal torsion’ (Wootton, 1993) is not so clear, but this source of wing twisting does not appear to be pronounced in hawkmoths. The Strips method was, therefore, adopted for use in the analysis of the real flight sequences. Some additional comments on this approach arising from its practical use are given below.

Observations on the use of the Strips method with real flight sequences

The method worked well for frames at most stages of the wingbeat and across the whole range of flight speeds (0–5 ms−1). The strips at spanwise positions from 0.3R to 0.9R could readily be matched to the true outline, and the spatial and temporal changes in angle of attack relative to the stroke plane corresponded well with visual estimates from the high-speed video. Two problems became apparent during the analysis: matches could not be obtained during parts of some upstrokes, and the innermost strips were difficult to analyse. During early to mid-upstroke, the width of the real wing outline sometimes exceeded that of the model’s strips, even when the latter were perpendicular to the camera axis. Some of this discrepancy may have arisen from a slight posterior movement of the hindwing relative to the forewing. The video sequences show that this effect is small even when it does occur, and it would only provide an explanation for strips up to and including 0.5R. A more likely source of the problem is the slight longitudinal flexion observed during this phase of the wingbeat (Willmott and Ellington, 1997). The small decrease in projected wing length can be corrected for, in the Strips analysis, by an equivalent reduction in the scaling factor which normalises the coordinates. This modification brings the real and model wingtips into coincidence, but it also results in a slight decrease in the width of the model strips. A reduction of 1 % or less in the distance between the wingtip and wingbase can prevent matches during wingbeat phases where the chords are approximately perpendicular to the camera, as they were during the early upstroke in the hawkmoth study.

The difficulty in analysing the innermost strips arises from the structure of the anojugal region of the hindwing and from its interaction with the moth’s abdomen. The region contains folds which allow the surface area to vary during the wingbeat as the orientation of the longitudinal axis changes. In addition, the most proximal flap lies flat against the abdomen, and its movements lag behind those of the more distal regions; this generates torsion in the anojugal region which is not centred about the longitudinal axis. These two effects prevent accurate analysis of these inner strips, and in practice only the strips from 0.3R outwards were used.

During parts of the wingbeat, the near-wing longitudinal axis was approximately parallel to the viewing axis, resulting in very short image wing lengths and greater relative digitizing errors. Under these conditions, the far-wing outline was used instead. The only modification required to generate the predicted model outline for the far wing was a reflection of the wing in the sagittal plane. The ability to choose the best wing for analysis is another advantage of this method since, whatever the camera orientation, there will always be awkward frames.

Slight errors in the sweep and elevation for the wingtip can result from the kinematic analysis. The magnitude of these errors was small, however (Willmott and Ellington, 1997), and they were corrected for by slight adjustments which brought the true and predicted wingtip positions into coincidence.

Throughout the analysis, the width of the wing chord proved to be the most important indicator in matching the wing outlines. For any wing orientation, except those perfectly perpendicular to the camera axis, there are at least two angles of attack which give the correct width, but the appropriate solution is usually obvious. The location of the V-shaped cleft between the fore- and hindwings was also useful in differentiating between possible angles; this strip was oriented first, providing a good starting point for the strips on either side of it. The tangents to the leading and trailing edges were essential, however, in determining the correct solution when the chord was approximately perpendicular to the viewing axis and the two potential angles of attack with respect to the stroke plane were therefore relatively close in magnitude.

The Strips technique for wing-orientation analysis was suitable for use in the study of hawkmoth flight. It performed well for most phases of the wingbeat, with the only major exception being the early to mid-upstroke. This difficulty can be understood in the light of the observed wing behaviour, demonstrating a further important benefit of this approach: the requirement for input from the experimenter. The latter introduces an element of subjectivity into the analysis, but this is more than offset by the clear picture of wing behaviour with which the operator is presented. If slight problems arise, then they can be identified better by this interactive process than by the rigidly objective mathematical solutions of the alternative methods. Even where the technique fails, the results can be used to investigate how the wing behaviour is violating the assumptions of the approach.

The Strips method is an accurate and robust technique which should be suitable for determining angles of attack in any species where there is little or no longitudinal bending of the wing. It is best suited to studies of isolated wings, but its performance has been shown here to be acceptable for wing couples that are sufficiently tightly joined.

The authors are grateful to Dr J. M. Wakeling for his help during the development of the LabVIEW software for the Strips method and to Dr A. L. R. Thomas for stimulating discussions. This work was supported by grants from the BBSRC (A.P.W.), the SERC and the Hasselblad Foundation (C.P.E.).

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