HK1136030B - Methods and systems for identifying the launch positions of descending golf balls - Google Patents

Description

Method and system for determining a launch position of a falling golf ball

Technical Field

The present invention relates to a method and system for determining a launch location of a falling golf ball. The method and system of the present invention may be used, for example, in the case of golf shots made on golf courses and other golf establishments.

Disclosure of Invention

According to the present invention there is provided, on the one hand, a method for determining a launch position of a falling golf ball from among a plurality of launch positions from which the ball was launched, and on the other hand, a system for determining a launch position of a falling golf ball from among a plurality of launch positions from which the ball was launched, comprising the steps or means of measuring the parameters of the fall of the falling ball to obtain at least the time of fall of the falling ball and a measurement dependent on its angle of elevation of fall, the step or means of calculating an estimate of the flight duration of the falling ball as a function of the measurement dependent on its angle of fall, the step or means of measuring the time interval for each launch position between the launch of the ball from that position and the time of fall of the falling ball, the step or means of comparing the time interval measured for each launch position with the calculated estimate of the flight duration of the falling ball, a step or means of determining which shot positions have a time interval closely matching the calculated estimate, and a step or means of using the shot positions where there is a close match to determine the position at which the falling ball was shot.

A measurement of the falling ball dependent on the angle of elevation of the fall can be obtained by measuring the component velocity of its falling trajectory.

Reference to the "fall" of a golf ball refers to the end portion of the flight trajectory of the ball, to distinguish it from any portion of its subsequent bounce or roll trajectory. Likewise, "dropped golf ball" is used to refer to a golf ball that is flying near the end of its flight trajectory and preferably within the last 10% of the trajectory. The end of the flight trajectory is the point where the ball hits a "target" some distance from the initial launch location. The launch location may typically be, for example, one of several "tee boxes" of a driving range.

The present invention is applicable to all golf shots but is particularly applicable to shots having a net aerodynamic force (i.e., the vector addition of lift and drag) greater than 1.0 meter per second (m/s)²) But more preferably more than 10m/s²The ball is hit.

It is an object of the present invention to provide a method and system for determining a golf shot that does not rely on flight prediction or trajectory tracking and does not require special markers or electronic tags or the like for the golf ball.

The flight prediction method requires expensive ball-launch measurement equipment capable of measuring the spin component of the ball, and sometimes requires a specialized ball. Flight predictions are also subject to a large amount of non-systematic errors caused by high winds and/or random variations in the aerodynamic properties of the ball, which change due to surface degradation. The flight prediction method can be improved by measuring the ball-drop position and the flight duration. However, since most of the information on the flight history of the ball is destroyed at the time of landing, a system relying on only measurement of the landing position and time is insufficient. Systems for determining individual golf shots using flight predictions of the ball and measurements of the landing position are described in US-B-6179720 and US-A-2007/0167247.

Orbital tracking methods (e.g., video tracking or radar tracking systems) are very expensive, require large data processing equipment and may not work reliably when multiple balls are present simultaneously within the flight space. Typically, one or more cameras or radar tracking devices lock onto the flight path of one ball at a time and track the ball throughout the flight from initial impact to at least the final landing site. This means that the camera or radar has to capture data that lasts almost a few seconds, whereas in the present invention the measurement of the ball trajectory only needs a few milliseconds just after the initial impact and just before landing on the outfield. GB-a-2294403 describes a method of determining a golf shot by video tracking in conjunction with spin measurement.

The use of a tag device such as an embedded RFID chip to identify individual balls is expensive and the mechanical means of collecting the balls to decode the RFID data is cumbersome and unreliable. Systems for determining individual golf shots using RFID tags are described in JP- ｃA-8224331 and US-B-6607123.

The measurement of the fall parameters and the emission parameters may utilize electromechanical, electroacoustic, electromagnetic, electro-optical, doppler microwave radar, ultrasonic doppler, high speed video cameras or any other technology that can provide electronic measurements suitable for signal processing.

Typically, the drop parameter can be measured as the ball approaches a 'target' located at the court outside of the range. The target may be a marked circular, oval or rectangular area or the like, or may be a beautified golf course that mimics a real golf course. Various alternative target designs may be used. For example, earthworks or structures similar to a large arrow target, hydrological elements or large sand pits in which a ball lands and disappears without popping, and various other forms. Alternatively, the measurement may be taken when a falling ball is about to fall on any part of the overall outfield or a substantial part thereof. Preferably, but not limitatively, the descent parameters are measured within a short distance (for example, the last 5%, or more preferably the last 2% of the flight distance) from the flight end of the ball.

Limiting the extent of the measurement drop range reduces the cost of the measurement equipment and improves measurement reliability. In a preferred embodiment, the fall parameter is measured within less than 2 meters of the landing surface or ground level, wherein the device itself is less than 2 meters above ground level. The height of the perception device defined at the outfield makes it less noticeable on the landscape of the outfield, which is very satisfactory.

A preferred apparatus for measuring fall parameters at each target uses at least four "detection planes". The detection plane comprises a fan-shaped beam having a very small angular field of view in a direction perpendicular to the detection plane and a wide field of view in the plane of the detection plane. The device is arranged to sense the angular position (within the fan beam) and the instant at which the ball passes through the detection plane. Typically, the detection planes are arranged in two co-facing planes that are parallel and offset from each other, but oriented in different directions, so that the two-dimensional position of the ball can be derived by triangulation as it passes through the common field-of-view plane. The time difference between the moments when the ball passes through the detection plane and the offset separation distance between the planes including the center of the field of view provide a measure of the ball's velocity vector and its instantaneous position in space. Preferably, but not limitatively, the field of view of all detection plane sensors is horizontal or almost horizontal and positioned close to the surface of the external field surface.

It is desirable to measure parameters of a launched ball by a device that is low cost and has reliability and wide "shot acceptability". In this context, "acceptability of hits" means the ability of the measuring device to measure all types of hits as they leave the launch point at various speeds and directions. Preferably, the initial flight parameters are measured on a short length of the initial trajectory of the ball (e.g., no more than 5 meters, but more preferably less than 2 meters) so that balls hit from adjacent tee divisions do not interfere with the measurements. However, any device that measures the initial launch parameters of a golf ball in the presence of other flying golf balls may be used. The signal processing means is required to determine the individual tee-off areas within the driving range and to record data on the extent, azimuth and elevation of the individual tee-off points relative to the individual targets on the golf apparatus.

The impact time of a ball at a serving sector is measured by sensing the impact acoustic energy using one or more microphones. If desired, a coarse measurement of the launch velocity and direction of the ball can be obtained by analysis of the signals from several microphones, which may be configured as a phased array. Alternatively, the time of impact may be measured by optical means, for example by detecting the passage of the ball through one or more optical detection planes. The optical device may be configured to measure the time of impact and optionally at least one of ball launch velocity, launch azimuth and launch elevation of each hit ball.

Although a conventional golf ball can be used in one form of the invention, it is beneficial to use a golf ball having a reflective surface. The benefits of a reflective golf ball are twofold. First, the ball is more easily detected by the optical sensor device, especially at long distances. Second, at night or under conditions of low ambient light, the ball is more easily seen by the human eye when illuminated by a light source close to the golfer. This allows a significant reduction in illumination brightness and intensity, which is environmentally desirable.

The major disadvantages of retroreflective golf balls include increased manufacturing costs and the potential for degradation of the retroreflective surface with frequent use. It is therefore an object of the present invention to provide a method and system for determining a golf shot suitable for a normal, unmodified golf ball, and also to provide a method and system designed to operate with a reflective golf ball.

It is also an object of the present invention to provide a method and system that is substantially unaffected by differences in aerodynamic performance of different types and states of golf balls. Although the diameters and masses of the different types of golf balls are very similar, their dimple patterns are significantly different and this adds substantial differences in flight length and flight duration under the same launch conditions. This is illustrated in a paper describing the USGA study of Golf ball flight prediction (Quintavaila, S.J.2002.A general Applicable Model for the Aerodynamic Beihaior of gold balls. in Science and gold IV, ed.E.Thain, 356-348. London: Routege). In this paper, Quintavalla uses a hybrid type ball showing variations in flight length and flight time due to differences in dimple patterns only, with flight length variations of up to 25 meters (where the average flight length is 238 meters) and flight duration variations of up to 1.0 second (where the average flight duration is 6.3 seconds). Other dimple patterns not included in this study may well exceed these differences, and the effect of surface degradation will be more severe as the degradation becomes severe.

It is therefore apparent that the effect of aerodynamic lift and drag is different for golf balls having different surface characteristics. However, for short approach shots (up to 25 meters, for example), lift and drag are very weak compared to gravity, so the trajectory is very close to a parabola, with the drop angle and drop velocity equal to the launch angle and launch velocity (on a horizontal striking surface). The 'bird's eye view of a short approach ball shows that the ball always flies along a substantially straight line without significant side-to-side steering (i.e., without a slice ball and a hook ball). On the other hand, this is because the effects of sidespin and/or crosswind on the flight of the golf ball at low speeds are negligible. It has been found that if the speed of fall of the ball is measured, the time to impact the ball can be found (assuming the ball is launched on a particular horizontal plane (e.g. on the ground plane)). Furthermore, if the azimuth direction and velocity of the ball are also measured, the initial impact position can be accurately calculated. In this way, it is possible to determine who has made the shot regardless of the ball's dimple pattern or general surface conditions. This illustrates the principles of the invention and is equally applicable to short-range balls.

However, the practice field is sometimes layered and therefore cannot be assumed to launch the ball from ground level. In the case of a short tall ball launched at a very large angle, the error in the measurement of its descent parameters makes the estimation of the hit from the upper or lower layers unreliable. For this reason, it is preferable to measure the impact time of all shots. This additional information allows the location and time of impact of the close-proximity golf shot to be reliably determined. The impact time also records how many balls were hit on each section. This monitors the ball for use by the consumer and helps prevent the ball from being stolen.

As launch speed and flight distance increase, lift and drag also increase. The forward speed during flight decreases rapidly so that the falling portion of the track is significantly shorter than the rising portion. Thus, the falling elevation angle is greater than the launch elevation angle and the falling velocity is less than the launch velocity. Flight simulations over a wide range of shots and wind conditions show that the flight duration of any shot can be estimated very accurately for a particular flight length, using only the information of the angle of fall. When a ball lands on a target, an optical or radar sensor or the like measures the velocity of its falling component just before it lands. The central computer then estimates the flight duration and searches to match it to the interval following the strike time of each of the most recently struck balls.

In many cases, the process of matching an estimated stroke time to a real stroke time is all that is required to determine a golf shot. However, it is often necessary to consider more than one estimated strike time and the true strike time for a reliable determination. Thus, in order to determine the initial launch location of a particular fallen golf ball, it is sometimes necessary to attempt to match a set of fall parameters (for a particular shot) with the initial launch parameters of one of a plurality of "possible shots" that happen to have nearly simultaneous impact times.

For the sake of clarity of the description herein, it is desirable to denote the parameters associated with "possible shots" by the subscript K, where K takes a value of 1 to K, and K is the total number of "possible shots" determined by the determination process, and is never greater than the number of golfers using the facility.

It was found that the estimated flight duration Edur of the kth possible shot can be estimated very accurately by the following equation_k：

Edur_k＝C1_k+C2_k×β (1)

In the above equation: edur_kIs the expected flight duration in seconds; c1_kAnd C2_kIs a known constant dependent on the flight distance from the kth serving sector to the drop position; and β is the measured falling angle of the ball at the falling position in degrees.

Other forms of calculation methods may be used. For example, the right hand side of equation (1) may include additional terms depending on other descent parameters, on the average wind speed along the flight direction, and on the air density. Optionally, the calculation may be performed using a look-up table or other form of algorithm.

In one form of the invention, Edur is determined by a calculation that is entirely dependent on the descent elevation angle β and the flight distance_kThe value of (c). More preferably, Edur_kDepending on the falling elevation angle, the flight distance and at least one of the following parameters: absolute speed of descent, azimuth of descent, wind speed in the direction of ball flight, and air density.

Except for Edur_kTwo other parameters are sometimes used in the present invention. These are, the expected directions Edir_kAnd a predicted deceleration ratio Edec_k. However, Edur_kEstimated using the descent parameters alone or in combination with the wind parameters, the predicted direction Edir_kAnd a predicted deceleration ratio Edec_kIs calculated from the launch and fall parameters in combination with the wind parameters.

The predicted direction or azimuth Edir of the kth possible shot can be obtained by the following equation_k：

Edir_k＝αF_k+C3_k×(αF_k-αL_k) (2)

In the above equation, C3_kIs a constant, α L_kIs the azimuthal emission angle measured at the kth emission sector and α F_kIs the azimuth angle of the position of the falling ball relative to its initial position at the kth launch sector. The angle Edir is usually measured clockwise from a fixed reference direction_k、αL_kAnd alpha F_k。

In one form of the invention, C3 is used for all k values_kThe value of (b) is preferably 1.0. More preferably, C3_kDepending on the flight distance and at least one of the following parameters: crosswind intensity and direction, flight bias, total wind speed and direction, launch elevation, launch velocity, flight duration, and air density.

Term (α F) without sidespin and crosswind, and exactly matching serving partition_k-αL_k) Is zero, that is, the ball travels along a straight line (in the bird's eye view) from the serving point to the drop position. When the lateral rotation and/or lateral wind makes the ball deviate from the straight line, the simulation shows that even the lateral rotation and lateral wind toolsEquation (2) still provides a very accurate prediction of the direction of the falling azimuth with very large variations.

Edir_kThe comparison of the different values of (f) with the true azimuth direction of the drop (obtained by measuring the drop parameters) provides a way to determine the correct serving sector for a particular dropped ball.

Edec_kIs a dimensionless parameter related to the overall deceleration of the horizontal velocity of the ball and is defined as the ratio of the horizontal velocity of the ball at launch to the horizontal velocity of the ball at drop. This ratio depends mainly on the flight duration and distance. For geodesic balls, the ratio tends to unity since the change in horizontal velocity is negligible. For a remote ball, the ratio is typically 3.0 or greater. Preferably, Edec is obtained by the following equation_k：

Edec_k＝C4_k+C5_k×Dur_k (3)

In the above equation, C4_kAnd C5_kIs constant and Dur_kIs a "possible duration" value corresponding to one of a plurality of serve zones. Thus, Edec_kAnd Dur_kEach having a plurality of values, and each Edec_kComparison with the deceleration ratio obtained by the ratio of the true launch velocity and the drop velocity provides a method of determining the correct serving sector.

C4_kAnd C5_kThe value of (d) is preferably dependent on the flight distance and initial launch angle of the shot achieved; but more preferably depends on the flight distance, launch angle, wind speed, wind direction and air density.

Parameter Edur_k、Edir_kAnd Edec_kEach having a different error distribution that can be determined by analyzing the actual measurement data when the actual measurement data is obtained. Often, a correct match of the expected and actual parameters can be obtained with 100% certainty.This will usually be the case when only a few golfers are using the arrangement according to the invention and the striking time for their shots is a few seconds apart. However, this will also occur frequently during busy hours. These "100% determined" records of shots plus records of prevailing environmental and wind conditions can be used to improve shot determination algorithms and accumulate error distribution data.

The preferred method of determining the serving sector corresponding to a particular falling ball is to first base this determination method on only the K possible flight durations Dur_kOne of (K ═ 1, 2.. K) and the corresponding estimated duration Edur_kThe match between them. The number K may be determined depending on the flight distance and wind conditions, and may be selected so as to include only all shots struck within a achievable duration. Alternatively, the default value of K can be set by including all shots hit during the first 10 seconds. K possible flight durations Dur_kIs equal to the time difference (t)_D-t_k) Wherein t is_DIs the time at the moment of measurement of the fall and t_kIs the impact time of the shot at the kth tee-off. In some cases there will only be one close match, with mismatches for other shots up to 3 (sigma) sigma or more. In this case, it can be safely assumed that a single match is correct.

When there are two real impact times and Edur_kIn the case where the degree of match is within 3sigma (or some other margin of error), it is preferable to use three event probabilities P (Dur)_k)、P(Dir_k) And P (Dec)_k) Are matched, three probabilities P (Dur)_k)、P(Dir_k) And P (Dec)_k) From the error distribution. P (Dur)_k) Is defined as the true flight duration falling within Edur_kNear ± (Edur)_k-Dur_k) Complementary probabilities of probabilities within (completion of probability). Because of Edur_kIs the most likely value of duration (obtained by analyzing a large number of previous shot samples), so the cumulative distribution function F (Edur)_k) Is 0.5, and F (Dur)_k) Will depend on whether it is greater or less than Edur_kAnd take a value above or below 0.5, respectively. We define P (Dur) as follows_k)：

P(Dur_k)＝1-2×|F(Dur_k)-0.5| (4)

When Edur_kAnd Dur_kWhen the values of (c) are very close, the probability P (Dur)_k) Tending to one, conversely, when the difference in these values is 3sigma or more, P (Dur)_k) Tending to zero.

P (Dir) is defined in the same manner from the directional error distribution and the deceleration ratio error distribution_k) And P (Dec)_k). The correct match is assumed to be the shot with the greatest joint probability, which can be P (Dur)_k，Dir_k) Or P (Dur)_k，Dec_k) Or more preferably P (Dur)_k，Dir_k，Dec_k)。

It should be noted that there is some correlation between duration and deceleration, and so can be shown for Dur_kAnd Dec_kNot an independent event. However, the error between the estimated and the real parameters is very small and is due to non-systematic measurement errors and variations of non-measured parameters, such as backspin and ball roughness. Thus Edur_kIs independent of Edec_kAnd hence the joint probability P (Dur)_k，Dec_k) It is dependent on the product being equal to a single probability.

In a hypothetical example, if Edur_k、Edir_kAnd Edec_kIs a normal distribution with 1 sigma values of 0.1 second, 0.7 degrees and 0.1, and:

Edur₂4.32 seconds Dur₂4.32 seconds

Edir₂81.5 degree Dir₂85.0 degree

Edec₂＝2.2 Dec₂＝2.4

Then: p (Dur)₂)＝1 P(Dir₂)＝0 P(Dec₂)＝0.046

In the above example, the expected duration of the ball falling perfectly matches the real hitting time of the second possible hit (k 2), so P (Dur)₂) Equal to 1.0. However, since the difference between the expected and true directions is much greater than 3sigma, P (Dir)₂) Is zero. Thus, the joint probability that the second candidate shot is a correct match is zero, and thus this shot does not match.

Drawings

The method and system according to the invention will now be described, by way of example, with reference to the accompanying drawings, in which:

FIG. 1 is a logical block diagram of a system for providing a shot determination device at a golf facility in accordance with the present invention;

FIG. 2 is a view showing the general shape of a golf shot when viewed from the side;

FIG. 3 is a graph of flight distance versus backspin for a representative golf shot;

fig. 4(a) and 4(b) are graphs of flight duration T versus drop angle β for a simulated golf shot.

FIG. 5 is a schematic plan view of a driving range showing the orbital path of two representative golf shots;

FIG. 6 is a plot of azimuth error as a function of crosswind velocity for a simulated golf shot;

FIG. 7 is a graph of simulated deceleration rate of a golf shot as a function of flight duration;

FIG. 8 is a plan view of a target and ball drop measurement device according to the present invention; and

FIG. 9 is a side view of the target and ball drop measurement device shown in FIG. 8.

Detailed Description

Coordinate axes X, Y and Z are shown for ease of orientation, where appropriate in some of the figures referenced in the description below. In this regard, the Z-axis is vertical and points upward, the Y-axis is horizontal and points in the launch direction (i.e., generally along the line of flight of the struck golf ball) and the X-axis is perpendicular to Y and Z and points in a direction from left to right as viewed in the launch direction.

Fig. 1 is a block diagram summarizing a top level system of a golf facility (golf facility) according to one aspect of the present invention, wherein several golfers hit golf balls to substantially the same area and sensing devices are provided to determine the initial tee (tee) location of each shot.

Block 1 represents the first input, which is at random time t_nFrom having the coordinate x_n、y_nAnd z_nA sequence of n balls hit at random serve positions. Blocks 2 and 3 represent second inputs comprising various "disturbance" or non-measured inputs such as the angular velocity ω of rotation of the ball, the tilt τ of its axis of rotation, its roughness, the wind speed and direction at successive times along its flight path, and the air density ρ, which varies with atmospheric pressure and temperature.

Box 4 represents the gravity and the air lift and drag experienced by the ball throughout its flight, all of which determine the shape and duration of the flight. Block 5 represents a launch analyzer that measures the linear parameters of each ball after impact, including vector velocity, position coordinates at impact and time of impact. Typically, the emission analyzer does not measure the rotation parameter because such measurement is difficult and requires expensive equipment. The data from the emission analyzer is fed to a central computer 6.

The object sensor 7 also sends data to the computer 6. The data measured by the target sensor 7 includes the vector velocity and the position coordinates of the falling ball. Since not all balls in the n ball strike sequences 1 reach the target, the sample of balls measured by the target sensor 7 is a subset m of the n ball strikes (m < n). Alternatively, the target sensor can extend over (extended across) the entire external field so that all balls that reach the external field can be measured. Optionally, data from one or more anemometers 8 can be sent to the computer.

The computer 6 processes the data from the various inputs to determine which tee position (and hence which golfer) corresponds to each of the m balls measured by the target sensor 7. When the golfer successfully lands the ball on the target, the computer 6 sends a score indication to the reading device 9. The reading device 9 may be a central device serving all users of the facility. For example, once a golfer has completed a round of training in a driving bay (driving bay), he or she can collect printed data showing the score for each shot. Additionally, separate score indication devices can be provided for each section, which can be simple devices based on audible tones, LED light indicators, etc., or more advanced devices such as touch screen displays.

In a preferred embodiment, the system is provided with two or more means for displaying the score or other information. The outfield flag marks the center of each of several targets and can be provided with three different colored LED beacons for each flag, positioned and focused such that the LED signals are readily visible to the golfer at the tee-off. A simple form of scoring may be used, for example scoring hits falling within 10%, within 5% or within 2% of the target corresponding to red, white or blue beacon lights (beacon light), respectively, as 1 point, 2 points or 3 points. The lights from the beacon are preferably pulsed on and off to be more conspicuous or to allow for gradual extinguishment over a few seconds after the scoring stroke is completed. A beacon mounted flag at the target area would provide one indicating means and a similar second indicating means could be provided at each serving sector. These second indicating means can be an audio signal or a suitable coloured LED indicator or the like which is activated at the serving sector only when a scoring shot is struck from that sector.

One benefit of a simple sounder or LED indicator is that it is very low cost and very robust and therefore less prone to being stolen or vandalized than more sophisticated devices. However, golfers often prefer the display and touch screen devices of computers and the like and such displays may be permanently installed. Alternatively, it can be beneficial to set these displays as portable units, as opposed to setting them as permanently fixed devices. Such portable units may be purpose-built displays that are selectively rented by golfers during their game or may be the users own laptop or handheld devices, such as Personal Digital Assistants (PDAs) or "smart phones" or the like. Modern PDAs and laptops are often equipped with Bluetooth (Bluetooth)^TM) Short range data communication, so that appropriate application software can be installed on the user's personal mobile computer so that it can communicate with the shot determination and scoring system, which must also be bluetooth enabled. Any suitable wireless communication standard may be used, including radio frequency and infrared technologies. In addition to stroke scoring, the application software can provide many other services such as online reservations and payments, scoring by multiple golfers, contest enrollment, stroke analysis, personal performance history, and the like.

Typically, golfers at such establishments are provided with electronically readable "sports cards" or similar devices that can contain a user identification code. All of the serving sectors (or at least some) of the device are equipped with motion card readers. When a motion card is read on a particular launching zone, the launch sensor and readout device associated with that particular launching zone is activated. The golfer can choose to use only the basic readout indicator or to use a portable computer. The golfer's portable computer must be programmed to interface data and control instructions unique to the identification code contained within his or her sports card. Some users of the facility can, if desired, hit the ball out of the serving sector without using a sports card, but in this case do not initiate a data connection associated with their serving sector.

The target sensor 7 may also be configured to measure the rebound of a ball falling on the target. Because rebound is primarily dependent on randomly varying ground conditions such as surface irregularities and impact absorption, these measurements are not used in the determination of shots. However, because the additional data provides an indication of the true bounce of the golf ball after its flight trajectory, and these enable the range of possible final travel of the ball (i.e., bounce and roll) to be obtained, it is very useful to the golfer. Moreover, the predictions of subsequent bounce and roll can be calibrated quite accurately for a particular terrain.

Fig. 2 is a view showing a general shape of a golf shot (golf shot) viewed from the side. The flight trajectory 20 comprises the flight path of the ball from its initial launch point 21 to its first bounce point 22. It should be noted that the flight trajectory is asymmetric, with the initial launch elevation angle ε being smaller than the drop angle β (both angles relative to the horizontal). This asymmetry (β > epsilon) is almost always true for a real golf shot, but is almost symmetric for a short chip shot (e.g., 25 meters or less) trajectory.

Figure 3 shows a plot of the percent flight distance of a typical driving ball as a function of backspin under windless conditions at a 12 degree launch angle and a launch velocity of 57 m/s. Values for flight length and flight duration are calculated using standard formulas and models known to approximate the lift and drag coefficients that simulate the performance of a real golf ball. As shown, the maximum range occurs when the backspin is about 4500rpm, but the flight distance reduction is less than 5% for a wide range of backspins from below 3000rpm to above 7000 rpm. The time of flight T and the fall angle β are shown at three points in the figure, namely at 2000rpm, 4500rpm and 8000 rpm. This shows that as the backspin increases, the time of flight T and the angle of descent β also increase.

There is a large correlation between flight duration and angle of descent. This follows from the fact that the longer the ball stays in the air, the higher must be the climb in the air and will therefore have a steep fall, compared to other balls flying the same distance. In the absence of wind and with all balls on the device having very matched aerodynamic properties, the time of flight T of any ball can be determined very accurately by measuring the drop angle β alone. Typically, variations in wind and surface roughness (which affect aerodynamic performance) will also affect T.

Surface roughness can be both intentionally created (e.g., molding a golf ball indentation pattern onto a surface during manufacturing) and accidentally created (e.g., roughness caused by cutting, abrasion, surface contamination, etc.). The relationship between surface roughness and aerodynamic performance is very complex, but for simplicity we assume here that the air resistance and lift increase with increasing surface roughness. The aerodynamic forces exerted on the ball are generally proportional to the square of the velocity of the ball through the surrounding air.

One aspect of the invention is based on the recognition that, first, wind, spin and surface roughness have similar effects on the flight of a ball. Headwind has the effect of increasing the lift and drag exerted on the ball because they increase the speed of the ball relative to the surrounding air, whereas tailwind has the opposite effect. Thus, the common effect of high backspin, upwind and high surface roughness is an increase in flight duration T and an increase in the falling angle β for a particular range distance. Conversely, low swirl, downwind and low surface roughness equate to shorter flight duration and reduced drop angle.

Fig. 4(a) and 4(b) are graphs of time of flight T as a function of β, where the data was obtained by simulation.

In chart 4(a), all hits fly 91.4 meters (100 yards) upwind, which varies randomly from 3.5m/s to 8.5m/s and averages 6 m/s. The emission angle varied randomly from 27.0 to 33.4 degrees and the initial convolution also varied randomly with an average of 2710 Revolutions Per Minute (RPM). For each shot, the launch velocity was adjusted to provide 91.4 meters of flight, with an average launch velocity of 36.7 m/s.

The data of graph 4(b) were obtained using the same values for random emission angles but with the assistance of downwind with a variation of average 6 m/s. In the case of downwind, the average launch velocity required to obtain a flight of 91.4 metres is reduced to 33.6m/s, with a corresponding average backspin of 2500 RPM. The graph shows that upwind significantly increases the flight duration and also makes the data more diffuse relative to downwind.

It is evident from the graphs of fig. 4(a) and 4(b) that the relationship between T and β is in each case almost linear. Thus, a best fit line through each set of data will provide a very accurate estimate of T as a function of β. Therefore, we can form a simple formula to obtain an estimate Edur of the predicted flight duration as a function of β_kWhere K takes a value from 1 to K and K is the total number of "possible shots" determined by the determination process.

Edur_k＝C1_k+C2_k×β (1)

Equation (1), which may be expressed in the above form or in an alternative form, nevertheless still provides a very accurate estimate Edur of the flight duration independent of the roughness of the ball, the prevailing wind and the imparted spin_k。

Constant C1_kAnd C2_kCan be determined as a function of the flight distance from the kth serving sector to the falling position, but, in particular for long flight distances, it isThey are preferably determined by several parameters including flight distance, wind speed, wind direction, launch angle, launch velocity and air density.

Preference is given to selecting C1 which is characteristic for a small range of values of beta_kAnd C2_kAnd in particular C1 specific to a small range of descent phase wind speeds measured by at least one anemometer_kAnd C2_k. Upwind produces the largest error (relative to no wind conditions and downwind) because this increases aerodynamic effects and increases flight duration for a particular flight length. This is evident in the graphs of fig. 4(a) and 4(b) corresponding to upwind and downwind, respectively.

The wind measured at the fixed point is rarely steady but generally has a variation of gusts typically 1.3 to 1.6 times the average wind speed and of duration two or three seconds (that is, less than half the flight duration of a typical club shot). Further, the Taylor's fixed Turbulence Hypothesis (Taylor's freozen Turbulence dynamics) teaches that wind Turbulence is transmitted in the direction of and at the velocity of the total mean wind. Thus, if the average wind speed is 5m/s, a gust of a certain intensity and duration at point a will repeat at point B, located 10 meters downwind of point a, approximately two seconds later. This "fixed turbulence" has a significant effect on the wind affecting the flight of the ball. The speed of the golf ball is much greater than the average wind speed at the driving range. Thus, the ball quickly flies through the gust, balancing the high and low values of the gust along the flight path (average out). Thus, in a gust, the instantaneous wind speed experienced by the ball changes several times throughout the flight, but the average wind speed during the first half of the ball's flight distance is typically almost the same as the average wind speed during the second half of the flight distance. This is especially the case with wind gusts that are upwind. Thus, to simulate the effect of upwind on ball flight, it is reasonable to assume that the balls experience a constant wind speed throughout their flight.

Using simulations, we can obtain C1 corresponding to different flight distances_kAnd C2_kTo a suitable value. Table 1 shows the results of the simulation study,wherein the upwind is randomly varied from 3.5m/s to 8.5m/s and wherein the emission angle and the backspin are randomly varied to produce a spread beta value. The launch velocity is adjusted to provide a fixed flight distance.

TABLE 1

Distance of flight (code)	Duration min/max (seconds)	Beta minimum/maximum (degree)	C1k(second)	C2k(second/degree)	Delta (millisecond)
						50	2.7/3.3	35.7/47.8	0.94	0.048	27
100*	4.4/5.9	45.9/63.0	0.97	0.076	70
						200	4.3/6.4	24.8/49.7	2.2	0.085	86

^*Data are obtained from FIG. 4(a)

It should be noted that, in general, C1_kAnd C1_kBut to generate the data of table 1, we simulated a shot with exact flight distance as shown and a constant blowing upwind averaging 6m/s, so in this case, for each of the three flight distance simulations, C1_kAnd C1_kWith a fixed value.

The last column of table 1 provides the standard deviation of the error of the estimate in milliseconds. The error is very low and only requires a measurement of beta and an approximate measurement of the normally blowing wind conditions. No information on the emission speed, the convolution or the emission angle is required. To obtain these estimation accuracy benefits, it is necessary to measure the β value very accurately, since the constant C2_kIs typically about 40 to 90 milliseconds per degree. So that a measurement error of 1.0 degree of β would itself produce an estimation error of 40 to 90 milliseconds. Thus, preferably, the error in the measured value of the descent parameter for measuring the descent elevation angle of the ball is less than 1.0 degree standard deviation and more preferably less than 0.5 degree standard deviation.

Since the error in the flight duration estimate is usually very small, the impact position of a falling ball can in many cases be determined by calculating this parameter only. This enables a very fast determination.

Fig. 5 is a plan view of a typical golf facility having a tee box arranged at an arc 50 and a remote target 51 located on the outfield. In this example, the target may be 100 to 200 meters or more from the transmitting sector, so that a considerable number of dextro and levo spheres can be generated. In the assumed scenario, two golfers at adjacent bays 52 and 53 hit the ball simultaneously at nearly the same launch velocity, elevation angle and backspin so that their hits fly the same distance and land at the same time, but the hit ball at the tee bay 52 takes a slice spin whereas the hit ball at the tee bay 53 does not have a sidespin and also does not have a crosswind affecting the flight path.

Solid line 54 shows the azimuthal trajectory of the shots from the tee section 52 and solid line 55 shows the azimuthal trajectory of the shots from the tee section 53. The shot 54 falls at a point 56 within the target 51 whereas the shot 55 falls at a point 57 to the right of the target. However, if the ball 55 is struck with a side-twist, as indicated by the dotted line 58, it can fall at the exact same point (56) as the ball 54.

A central computer (not shown) has to determine the serving sector from which the ball falling at point 56 came. In this case, both serve sectors are selected as possible sectors, with the serve sector 52 being assigned k 2 and the serve sector 53 being assigned k 1. However, the real impact time Dur₂And Dur₁Corresponding to Edur₂However, the difference in the measurements of the azimuth provides a means to select the tee section 52 and reject the tee section 53.

At the tee-off zone, the initial launch azimuth α L of each shot is measured and the drop azimuth α of any ball that reaches the target is measured at the respective target_des. All angles are measured according to a fixed direction and measured in a clockwise or counterclockwise rotation. In fig. 5 we show the angle measured rotationally in a clockwise direction according to the-X axis. For each impact point and each drop position, there is an additional angle, which we call the final angle α F, which is the assumed straight line between the impact point and the drop position (forThe shot 54 in fig. 5 is shown as dashed line 59).

Because the shot 55 is straight, the initial launch angle, final angle, and drop angle are all equal. For the shot 54, the slice spin causes the azimuth trajectory to curve in the shape of a generally circular arc, so that the angles are all different. The azimuth orbit is not exactly circular, especially if the crosswind affects the flight path, but in most cases we find (by simulation) that (α F- α L) is almost equal to (α F- α L)_desα F) corresponding to a circular arc.

We can use this relationship to measure α F_kAnd alpha L_kTo accurately estimate the azimuth direction Edir of the ball_k. Edir from a particular tee box for a particular shot_kValue sum alpha_desAn exact match between them indicates that the particular tee section correctly matches the particular shot with a high probability. Referring again to FIG. 5, because Edir is the estimated value of serve sector 52₂Value and measured value alpha_desExactly matched, so the central computer can correctly determine that the tee box 52 is the source of the shot 54 and Edir₁With the measured value alpha_desAnd not matched.

In general, Edir is obtained by the following equation_k：

Edir_k＝αF_k+C3_k×(αF_k-αL_k) (2)

By simulation we found that the coefficient C3 is made_kA value equal to 1.0 provides a substantially good estimate for the majority of possible golf shots. More preferably, C3_kDepending on the flight distance and at least one of the following parameters: crosswind magnitude and direction, flight bias, total wind speed and direction, launch elevation, launch velocity, flight duration, and air density.

Optionally, one anemometer 60 is positioned near the tee-off section 50 and a second anemometer 61 is positioned at the far end of the outfield. The anemometer preferably uses a 2-axis sound speed sensing means (sonic sensing means) that senses very short term changes in wind speed and direction. Data from the anemometer, which may be recorded at a sampling rate of 10 samples per second or greater, is used to calculate the approximate wind speed and direction throughout the external field by means of interpolation and extrapolation. The anemometer is preferably mounted on a mast, about 10 meters high or at the expected average height of a golf shot. In other arrangements, only one 2-axis anemometer is used to obtain a rough estimate of the intensity and direction of the current wind, or several anemometers, which may be of the 2 or 3-axis type, are used to obtain a more accurate estimate. In particular, it can be equipped with a purpose-built anemometer in which the path length of the sound speed measurement extends up to several meters rather than several centimeters as is common on commercially used devices.

FIG. 6 is a simulated Edir shot of a distance ball_kIs plotted as a function of crosswind speed. Each shot in this sample flies 220 meters (240 yards) and is shot at random elevations and backspin. In addition, each shot has a large (random) side spin component, which is sufficient to cause an average of 27 meters of deviation in the absence of wind. The crosswind changes from-4 m/s to +12m/s (where negative crosswinds contribute to the yaw bias and vice versa). A best fit line 62 passes through the data points and shows the general trend of the error. The error is negligible in the case of low crosswinds, but increases gradually with increasing crosswinds. The incremental error is less than 0.25 degrees for every 1m/s increase in crosswind. We find that the standard deviation of the error relative to the best fit line 62 is less than 0.25 degrees. This is almost 1/100 (1/100) of the average angular deviation caused by only imparted sidespin^th). It is therefore evident that equation (2) in combination with the crosswind correction (if measured) provides a very accurate estimate Edir even in the presence of high crosswind and crosswind_k。

To ensure a very low probability of mismatch, it is preferable to include the true impact time and the estimated flight duration Edur_kThe difference between them is less than 3-sigma or even less than 4-sAny serve partition of igma as a possible match. This will sometimes result in two or more serve sectors as possible matches even when the 3-sigma has a value of around one tenth of a second. Then, we use Edir_kTo select between the two or more serving sectors. By including the third parameter in the matching process that matches the drop velocity and possibly launch velocity of the shot, a higher certainty of a correct match can be provided. We have found that the ratio of the horizontal velocity of a ball at launch to the horizontal velocity of a ball at drop (herein we call this a 'deceleration ratio') can be accurately estimated even when the ball has greatly decelerated along its flight length.

Fig. 7 is a graph of a simulated golf shot showing the change in deceleration rate with respect to flight duration. The launch angle, backspin and breeze of all shots in fig. 7 were random values, with the launch velocity adjusted so that the flight distance was 220 meters (240 yards). The variation range of the initial emission parameters is as follows: speed 69.4 to 77.2m/s, slew 1010 to 3140RPM, transmit elevation 12 to 14 degrees. In addition, the wind along the flight direction varies within a range of ± 1.3m/s (± 3 mph). These data represent very long drive-out distances, representing professional golfers or very high-level amateur golfers.

From figure 7 we see that there is a very good correlation between the deceleration rate and the flight duration, which varies from 5.2 seconds to 6.8 msec. Line 70 is the best fit line through the data. The formula of this fit line shown below is given as the kth 'possible flight duration' Dur_kThe deceleration ratio Edec of the kth 'possible serve sector' of the function of_kIs estimated.

Edec_k＝C4_k+C5_k×Dur_k (3)

Constant C4_kAnd C5_kMainly depending on the launch elevation, actual flight distance, wind speed, wind direction and altitudeAnd (4) air tightness.

For a relatively large range of emission conditions, we have found that Edec_kThe error distribution between the real data in fig. 7 has a standard deviation of only 3.2%. For most shots, Edec_kCloser to the real data. This is particularly true for shots with relatively low launch velocities, as the deceleration ratio tends to be unity there and the error becomes negligible. In this way, it is possible to provide a fairly accurate measurement of launch horizontal velocity, launch elevation angle and fall horizontal velocity, at Edec_kThe difference between possible matches to the deceleration ratio at multiple tee-off bays provides a very reliable method of determining a golf shot. The Edec can be further increased if there are measurements of wind speed and wind direction_kTo the accuracy of (2).

The shot determination process relies on an estimation of the true impact time of one of the multiple shots from the tee and additionally (if required), the true launch direction and/or true launch horizontal velocity, the probability of matching the measured drop parameter of the ball. To estimate this probability, data from previous shot samples are first analyzed to derive Edur_k，Edir_kAnd Edec_kThey are the values of the three relevant parameters with the highest probability density. Then, the results were higher or lower than Edur_k、Edir_kAnd Edec_kThe resulting distribution of (a) provides the standard deviation values for the three corresponding errors from which the probability can be estimated. Edur is determined for each possible combination of fall parameters, wind parameters and launch parameters_k，Edir_kAnd Edec_kAnd their standard deviation, are not practical, but data analysis can form a model so that values for any particular set of conditions can be determined by interpolation from a look-up table or other computational method.

Due to Edur_k，Edir_kAnd Edec_kThe difference between the real parameters of ' close to possible shot ' (shot) ' is very small, the error distribution may be symmetric and normal, and thus the standard is usedThe formula can yield this probability.

For example, the probability that the hit time for the kth shot is the true time to hit a particular falling ball in a set of possible hits can be derived from the following equation:

P(Dur_k)＝1-2×|F(Dur_k)-0.5| (4)

wherein P (Dur)_k) Is Dur_kThe cumulative probability function of (2). The direction and probability of deceleration can be obtained using similar methods.

Referring now to fig. 8 and 9, the flag pole 80 marks the center of a circular target 81, which circular target 81 is located at the center of the golf tee shot range and is typically at a distance of 50 to 250 meters from a row of several tee shots (not shown). Several such targets are arranged along the length of the pitch and may be of different sizes and shapes. The boundaries of the targets may be marked so that a golfer at the tee box can see or the targets 81 may simply be designated as a circular area on the outfield at a fixed radius from the flag pole 80.

The two sensor units 82 and 83 are rigidly fixed slightly above ground level by a brace 90 and are positioned to the right of the target and preferably near the far right hand boundary of the golf course, where a golf ball would not normally land. A reflector structure 84 extends along the length of the far left hand side of the range (or a lesser length as desired) on the opposite side of the target, again generally without a golf ball falling thereon.

The positions of the sensor unit and reflector structure may be interchanged or, if desired, both sides of the target may be equipped with sensor units and reflector structures.

The sensor units each comprise an upper light emitter 91 and a lower light emitter 92 mounted above and below a cooperating light receiver 93. The light receivers of the sensor units 82 and 83 each have an angled horizontal field of view, indicated by dashed lines 85 and 86, respectively, sufficient to cover the target 81 (e.g., about 45 degrees) but with an offset axis direction, as shown. The light emitters of sensor units 82 and 83 have a horizontal light emission area that extends at least beyond the field of view of their cooperating light receivers. The two light emitters 91 and 92 in each sensor unit have vertical light emitting areas which are only slightly divergent but sufficient to illuminate two separate retro-reflective strips 94 and 95, which retro-reflective strips 94 and 95 are connected to the side of the reflector structure facing the light emitters. Preferably, the vertical separation HR of the retro-reflective strips 94 and 95 is equal to the vertical separation HE of each pair of light emitters 91 and 92, and is in the range of 10 to 50 centimeters, without limitation.

Preferably, the light emitters are energized cyclically, so that in a half cycle the two upper light emitters 91 are on and the two lower light emitters 92 are off, and vice versa in the other half cycle. During each half cycle, light reflected from a pair of light emitters is reflected by retro-reflective strips 94 and 95 and received by a light receiver 93, the light receiver 93 preferably having a large aperture and high gain condenser. The light receiver 93 focuses the reflected light into one or more linear light sensor arrays (not shown) located within the light receiver subsystem.

The retro-reflective bands 94 and 95 have a uniform vertical width in the range of 20 to 30 mm, for example 25 mm, which is slightly smaller than the diameter of a golf ball, and the vertical aperture of the light emitter is arranged to be the same height as the retro-reflective bands 94 and 95, for example 25 mm.

When the light emitter 91 is switched on, a fan beam of light, typically infrared light, illuminates a portion of the retro-reflected bands 94 and 95 and includes two light paths indicated by dotted lines 96 and 97. When the golf ball enters the light path 96, a portion of the light from the retro reflector 94 is interrupted and this is detected in both sensor units 82 and 83. In each light receiver, at least one photosensor array pixel will detect the light interruption. It should be noted that the at least one pixel will still receive light from the retro reflector 94 through the optical path 97, so the signal in the at least one pixel changes by-6 db, and this is sufficient to reliably detect the presence of a golf ball in the optical path. The angular position of the golf ball interrupting the light with respect to the two sensor units can be determined by detecting the respective positions of the pixels. The instantaneous height of the golf ball interrupting the light is determined by the height of the light path 96. Thus, knowing the angular position of the golf ball relative to the two sensor units and its height, its three-dimensional instantaneous position can be determined. As the ball passes through all four light paths 96 to 99, its velocity vector can be measured. It should be noted that intermediate optical paths 97 and 98 are not required to measure the ball's velocity vector, but nonetheless intermediate optical paths 97 and 98 are present because it is impractical to focus the fan beam from the light emitter 91 only to the retro-reflector 94 but not to the retro-reflector 93, and similarly when it is impractical to focus the fan beam from the light emitter 92 only to the retro-reflector 95.

The flag pole 80 is provided with a ball striking indicating means 87 including three different colored light beacons which are turned on according to the detected proximity of the golf ball to the flag pole. This provides a means of determining a 'score' depending on the accuracy and distance of the different golf shots.

The improved version of the sensor unit and retro-reflective strip of figures 8 and 9 may be used to measure launch parameters at the launch zone. In this case, it is preferred to position the fan beam and the field of view of the light sensor in a vertical plane.

Claims (18)

1. A method for determining the position of a falling golf ball from among a plurality of launch positions from which the ball was launched, comprising measuring the descent parameters of the falling ball to derive at least the descent time of the falling ball and a measurement dependent on the descent elevation angle of the falling ball, calculating an estimate of the flight duration of the falling ball as a function of the measurement dependent on the descent elevation angle, measuring the time interval between launching the ball from that position and the descent time of the falling ball for each launch position, comparing the time interval measured for each launch position with the calculated estimate of the flight duration of the falling ball to determine which launch position's time interval closely matches the calculated estimate, and employing launch positions for which there is a close match to determine the position from which the falling ball was launched.

2.A method according to claim 1, wherein the measured descent parameters comprise a measure of a descent velocity component of the descending ball to obtain a measure dependent on the angle of descent in elevation.

3. A method as claimed in claim 1 or 2, wherein the estimate of flight duration is calculated from a substantially linear function dependent on the measured value of the falling elevation angle.

4. A method according to claim 3, wherein the substantially linear function is the sum of at least two terms, the first term being a constant and the second term being the product of the constant and a measurement dependent on the elevation of fall.

5. The method of claim 4, wherein the constants of the first and second terms each depend on the flight distance and at least one of the following parameters: wind speed, wind direction, launch angle, air density, falling speed of a falling ball, and falling azimuth of a falling ball.

6. A method according to claim 1 or claim 2, wherein a measure of the azimuth of descent of the descending ball is derived from the measured descent parameters, the step of determining the position at which the descending ball is launched comprising the step of identifying for such determination which of the launch positions of the launched balls there is a close match, the identification being based on, for each of the launched balls, the degree of match between the measure of the azimuth of descent of the descending ball and its calculated estimate, and wherein the estimate is calculated as a function of the measure of the azimuth of launch of the target launched ball and the measure of the azimuth of the descending ball relative to the launch position of the target launched ball.

7. A method according to claim 6, wherein the calculated estimate of the azimuth of descent of the descending ball is the sum of a measure of the azimuth of the descending ball relative to the launch position of the target launched ball and a term dependent on the difference between that azimuth and the measure of the launch azimuth of the target launched ball.

8. A method according to claim 1 or claim 2, wherein a measure of the horizontal velocity of a descending ball is derived from the measured descent parameters, the step of determining the position at which the descending ball is launched comprising the step of identifying for such determination which of the launch positions of the launched balls there is a close match, such identification being based, for each of the launched balls, on the extent of the match between: (a) a calculated ratio of a measured value of the horizontal velocity of the ball at launch to a measured value of the horizontal velocity of the ball falling, and (b) a function of a value dependent on the predicted flight duration of the target launch ball from its launch location.

9. A method as claimed in claim 1 or 2, wherein the measurement of the descent parameters of the descending ball is made less than 2 metres above ground level.

10. A system for determining the position of a falling golf ball from among a plurality of launch positions from which the ball was launched, comprising means for measuring the parameters of the fall of the falling ball to derive at least the time of fall of the falling ball and a measure dependent on the angle of elevation of the fall of the falling ball, means for calculating an estimate of the duration of flight of the falling ball as a function of the measure dependent on the angle of elevation of the fall, means for measuring the time interval between the launch of the ball from that position and the time of fall of the falling ball with respect to each launch position, the time interval measured for each launch location is compared to a calculated estimate of the flight duration of the falling ball, means for determining which shot positions have a time interval closely matching the calculated estimate, and means for using the shot positions where there is a close match to determine the position at which the falling ball was shot.

11. A system according to claim 10, wherein the measured descent parameters comprise a measure of a descent velocity component of the descending ball to obtain a measure dependent on the angle of descent in elevation.

12. A system as claimed in claim 10 or 11, wherein the estimate of flight duration is calculated from a substantially linear function dependent on the measured value of the falling elevation angle.

13. The system of claim 12, wherein the substantially linear function is a sum of at least two terms, a first term being a constant and a second term being a product of the constant and a measurement dependent on the elevation of the descent.

14. The system of claim 13, wherein the constants of the first and second terms each depend on the flight distance and at least one of the following parameters: wind speed, wind direction, launch angle, air density, falling speed of a falling ball, and falling azimuth of a falling ball.

15. A system according to claim 10 or 11, wherein a measure of the descent azimuth of the descending ball is derived from the measured descent parameters, and determining the position at which the descending ball was launched comprises identifying for such determination which of the launch positions of the launched balls there is a close match, such identification being based on: for each of these launched balls, a degree of match between the measured value of the falling azimuth of the falling ball and its calculated estimate, and wherein the estimate is calculated as a function of the measured value of the launch azimuth of the target launched ball and the measured value of the azimuth of the falling ball relative to the launch position of the target launched ball.

16. A system according to claim 15, wherein the calculated estimate of the azimuth of descent of the descending ball is the sum of a measure of the azimuth of the descending ball relative to the launch position of the target launched ball and a term dependent on the difference between that azimuth and the measure of the launch azimuth of the target launched ball.

17. A system according to claim 10 or 11, wherein a measure of the horizontal velocity of a descending ball is derived from the measured descent parameters, and determining the position at which the descending ball was launched comprises, for such determination, identifying between launch positions of launched balls which there is a close match, such identification being based, for each of the launched balls, on the extent of the match between: (a) a calculated ratio of a measured value of the horizontal velocity of the ball at launch to a measured value of the horizontal velocity of the ball falling, and (b) a function of a value dependent on the predicted flight duration of the target launch ball from its launch location.

18. A system according to claim 10 or 11, wherein the measurement of the descent parameters of the descending ball is made less than 2 metres above ground level.