Video analysis of an arrow in flight: Murakami 2022

A group of Japanese engineers wrote a programme that reads the velocity and attitude of an arrow in flight from high-speed video by itself, and checked it against the by-eye reading used until now. The comparison covers arrows launched with compressed air, and so without flexing, at initial speeds between 40 and 60 metres per second: on the horizontal velocity component the discrepancy from the manual reading stays within 0.9 per cent, on the vertical within 2.0, and around 60 metres per second, the typical speed in competition, the discrepancy falls to about 0.2 per cent. The angle of attack came out below 1 degree and the angular velocity below 0.5 radians per second. The working time is the biggest difference: for velocity it goes from about twenty minutes (adding up four analysts) to two, for attitude from about four hours to four minutes.

With the programme in hand the authors look at three things. The first is the dispersion of an archer: in an indoor range, at 55 metres, an athlete shot thirty-six arrows, eighteen with each of two A/C/E shafts of almost identical mass (17.73 and 17.99 grams), with mean speeds of 59.67 and 59.21 metres per second and standard deviations of 0.34 and 0.40. The vertical spread of the impacts was 107 and 108 millimetres, already at 55 metres, that is, wider than the 122 millimetre ten at 70 metres. The robot, over the same period, held a velocity standard deviation of 0.13 metres per second and a vertical spread of 34 millimetres; after a modification, with a different X10 shaft, the standard deviation fell to 0.019 metres per second and the vertical spread to 9 millimetres, which at 70 metres would correspond to staying inside the X ring.

The second thing is the link between the two measures: the variation of the vertical velocity component at launch accounts for the vertical spread on the target, and in the robot's data the relation matches the one predicted from the flight time. In the archer's data the match is poorer, and the authors put this down to the greater variability of the horizontal component. The third concerns points: with the compressed-air launcher, at the same initial speed (57 metres per second), the bulge point slowed the arrow more than the streamlined one, a sign that the boundary layer along the shaft was turbulent rather than laminar; the behaviour stayed the same when the speed was changed. The maximum angles of attack computed were 0.46 and 0.51 degrees with the streamlined point and 0.44 with the bulge point.

The flight tests were run in the 50 metre indoor range of the Edogawa gymnasium in Tokyo, with the air conditioning switched off to come as close as possible to still air; the distance from the shooting position to the target was 55 metres, the longest available. Three monochrome high-speed cameras stood 1.5, 2 and 2.5 metres from the shooting position, a fourth in colour one metre in front of the target; the framing was set to contain two arrow lengths. Filming was at 3,000 frames per second with a 20 microsecond exposure, at resolutions of 1280 by 720 pixels (0.83 to 1.1 millimetres per pixel) and, on one of the cameras, 2048 by 1952 (0.69 millimetres per pixel); the scale was obtained by filming a calibration bar. For every shot the deviation of the impact from the centre of the target was also measured. The measurement campaigns date from August and December 2020 and June and July 2021.

Shots were made in three ways: a compressed-air launcher (3.0 to 7.5 megapascals, with the nock removed so the nozzle could be inserted into the shaft), a robot and an archer. The arrows were Easton A/C/E and X10 shafts with 4.4 centimetre Diamond vanes or Spin-Wing vanes, with a streamlined or a bulge point, of masses between 14.5 and 21.2 grams.

The image analysis is written in MATLAB. The velocity of the centre of mass is obtained by subtracting a background derived from the sequence itself, removing salt-and-pepper noise with a median filter, extracting the outline, binarising and filling in the missing pixels, then locating point and tail with the Harris corner detector; the centre of mass is placed using the ratios measured with the arrow at rest. The attitude is obtained another way, because pixel coordinates are not enough for very small angles: the image in which the centre of mass passes the middle of the frame is taken as the reference, the images of the other ten pairs of frames are rotated, their two-dimensional Fourier transforms are compared, and the rotation that minimises the difference between the spectra is sought, interpolated with a parabola. The by-eye reading used as the reference was the mean of four or five analysts, and the error bars in the comparison graphs are their standard deviations. The time course of the angle of attack was then computed with a numerical programme starting from the camera data and using aerodynamic coefficients obtained earlier in a magnetic suspension wind tunnel.

The limit the authors put first is the missing dimension: the cameras film the vertical plane, and an arrow shot from a bow flexes and rotates about its own axis horizontally too. Seeing all of it would require stereo filming from two directions, which is left as future work. The effect of this gap shows in the data: the length of the X10 shaft launched by the robot, measured frame by frame, came out up to about 2 centimetres shorter than the length measured with the arrow at rest, with a regular pattern over time from which the authors estimate a flexing frequency of about 75 hertz.

The reference against which the programme is validated is the by-eye reading of four or five people, that is, a comparison between two methods rather than with an independent value; and the validation was done only on the arrows launched with compressed air, which move as rigid bodies. For the arrows shot from a bow there is no equally solid check. The estimate of the angle of attack over time is not a measurement either: it is the result of a calculation starting from the initial conditions read from the videos, and the two nearby cameras give somewhat different curves, which the authors attribute to the sensitivity of the attitude to the initial angular velocity and list as a point to improve. With the bulge point the length read from the images came out slightly short, because of the difficulty of locating a rounded tip, and one of the four cameras, the oldest, gave reliable speeds but not reliable angular velocities.

The figures on dispersion come from one archer, thirty-six arrows in one day, indoors and without wind: they do not describe the variability of archers in general, nor that of the same athlete in competition. The study was carried out with a precision engineering company as part of research into criteria for assessing the accuracy of archery equipment, and the robot whose progress it reports is part of that collaboration. The text is in Japanese, with the abstract and captions in English.

The number worth remembering is the vertical spread of one athlete over thirty-six arrows, indoors and without wind: 107 millimetres at 55 metres, measured as a standard deviation from the mean of the impacts. The ten at 70 metres is a circle 122 millimetres across, and the spread grows with distance: with that variability, indoors, the competition ten cannot be held. The robot, under the same conditions, was at 34 millimetres and after a tune-up at 9.

The interesting part is why. The study shows that the vertical spread of those arrows is almost entirely accounted for by the variation of the vertical velocity component at the moment of launch: not by wind, not by the target, but by how differently one arrow leaves compared with another. In the robot's shots the link between the two matches what the flight time predicts; in the archer's shots the relation is noisier, because the horizontal component varies as well.

For choosing arrows there is a second finding: the two A/C/E shafts used by the athlete, of almost identical mass, gave mean speeds differing by about half a metre per second, with the heavier shaft slower. And the shape of the point counts: at the same speed, the bulge point slowed the arrow more than the streamlined one, because the boundary layer turned turbulent instead of staying laminar. This was measured with the compressed-air launcher rather than with a bow, so it is a result about the aerodynamics of the arrow, not about what happens during the launch.

Nothing in this study concerns shooting technique: the archer is there as a source of variability, and the contribution is a tool that makes it possible to measure one's own equipment quickly, at the range, instead of spending hours reading videos by hand.