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When is a goal not a goal? Mathcad measures up

19 June 2012

Disputed goals are a major headache in professional football. This case study looks at one particular type where a ball, bouncing on the ground after its impact with the crossbar, spins back into play. An experimental rig was used to derive the critical parameters for this behaviour and mathematical modelling confirms how easily such events can occur in practice. Dr Ken Bray reports

Graphic 1

The definition of a goal has been clear ever since football codified its rules in the mid-19th century: for a goal to be scored, all of the ball must cross all of the line. So in Graphic 1, C is a goal whereas A and B are not.

Most disputed goals result from defenders illegally returning the ball into play after it has crossed the line, but significant numbers occur without any human intervention at all. Perhaps the most famous example is Geoff Hurst’s second goal at the 1966 World Cup final between England and West Germany.

His shot cannoned off the crossbar and was judged to have bounced behind the goal line before rebounding back into play. There have been other controversial examples over the years. After Hurst’s, the highest-profile incident occurred at the 2010 World Cup in the second round match between England and Germany (again!), when Frank Lampard’s shot struck the crossbar and bounced a good half-metre behind the line.

On this occasion the goal was disallowed by the match officials, but hundreds of millions of viewers around the world saw it repeated in high-definition, slow motion replays.

Why such incidents occur is perplexing at first sight. In Graphic 2, the ball hits the bar and because the ball is travelling towards the rear of the goal, surely it will continue in that direction after striking the ground? This ignores the considerable backspin the ball acquires in its contact with the bar.

Graphic 2

If this spin is great enough the ball's horizontal velocity can be reversed when it bounces on the ground and if it doesn't bounce too far behind the line it can clear the crossbar in the recoil, and bounce out. This event is so fleeting, only two tenths of a second from bar to ground, that it’s hardly surprising that referees make snap decisions which are often incorrect.

We set out at the University of Bath to investigate this behaviour. Can a ball be made to reverse its velocity under experimental conditions and what are the important parameters that govern the detailed behaviour?

In a previous case study I described how Mathcad helped us to understand the dynamics of the swerving free kick and hinted that instead of using human subjects in the research we would employ a programmable ball launcher in future work. A ball launcher was absolutely essential in our study of disputed goals.

No professional, not even one with the skills of a David Beckham, could repeatedly fire shots at an experimental crossbar with sufficient accuracy to ensure that all of the critical events would occur.

Graphic 3 gives an impression of our experimental setup. In the foreground is the ball launcher which can be pre-programmed to deliver a range of shots at the required speeds and impact angles.

Graphic 3

The actual rig consisted of a short section of steel tubing the same diameter as a crossbar mounted in a rigid frame which was bolted to the ground. A strip of artificial grass was used to mimic the surface of the soccer pitch and the crossbar was mounted at exactly the regulation height (8ft) above the ground.

To film the impacts a high-speed digital camera was situated to the side of the rig so that its axis was centred on the field of view and orthogonal to the plane of the shots. Over 50 impact events covering a range of speeds and launch angles were recorded. An extract from the full data set (see Graphics 4 & 5) shows two impacts in “freeze frame” format.

One is clearly a goal, the ball bouncing too far behind the goal line to spin back. The other shows a velocity reversal with the ball rebounding back into play. Just how we turned these images into data files where accurate positional coordinates, ball velocities and spin rates could be obtained is explained in the section which follows.

High-speed video can be enormously helpful in developing sporting excellence in a coaching context but it is practically useless if quantitative measures are required. Such questions as "how fast is that athlete running" or "what forces are acting on the ball" can only be answered if the movement field is in some sense calibrated.

Graphic 5

This is sometimes done by introducing devices such as measuring rods in the digital camera’s view, but there is no guarantee that the camera reproduces its images linearly right across the field and so a more sophisticated technique is needed for accurate work.

Let's suppose a ball is moving with real-world coordinates (x,y) given in metres relative to a fixed origin. A camera will record an image which can later be analysed on a computer screen. Suppose its position in this virtual world is (u,v) where u and v are conventionally measured in pixels. By looking at a mapping* of the real-world coordinates through the camera lens onto the screen it can be shown that (u,v) and (x,y) are related by the following equations:

u=(L
1x+L2y+L3)/(1+L7x+L8y)  (1)
v=(L4x+L5y+L6)/(1+L7x+L8y)  (2)

This mapping is known as the Direct Linear Transformation (DLT) and the parameters L
1 – L8 as the DLT coefficients. If these numbers were known, for example if they were absolute constants for the particular camera, then by measuring u and v and knowing L1 – L8 it would be possible to back-calculate for x and y using the above equations. Sadly, life’s not that simple. The DLT parameters are specific to the camera's position, how much it has been zoomed, the aperture chosen and so on. So they need to be determined for each particular case.

The way this is done is to place a calibration object with known coordinates (x,y) in the field of view. Its apparent position (u,v) on the screen can then be measured in pixels. We now have two equations in eight unknowns and very quickly see that adding another calibration point would give a further two equations. In fact four calibration points do the trick; eight independent equations, eight unknowns so the problem is solved and L1 – L8 are known for that particular experiment.

Of course, we don't stop there. A minimum of four calibration points would not produce very accurate results and the aim is to spread the calibration throughout the entire movement field. This can be done in a variety of ways but one common method is to use a vertical pole with marker points placed very accurately along its length (see Graphic 6).

Then by stepping the pole through a series of known positions and photographing these using the camera we can build up a calibration grid. This introduces lots of redundancy into the problem. A hundred calibration points would generate 200 equations from which all different combinations of eight equations in the above format can be chosen and solved. Simple averaging then gives accurate values of the DLT coefficients.

The computation is quite intensive but Mathcad’s contributed functions for handling simultaneous equations make very short work of the process. All that remains is to film a sequence of ball impacts against the crossbar and the ground. For these, the ball is appropriately marked so that its spin can be tracked throughout the flight.

Graphic 6

There follows a long, tedious sequence of digitising, during which the will to live can occasionally be questioned. However, long careful digitising sessions are good for the souls of aspiring engineers and (occasionally) their supervisors.

Finally, having determined the true coordinates of the ball as described above, trajectories can be fitted to the data. Here again, Mathcad comes into its own whether your preference is for polynomials or splines. Pre and post-impact velocities together with the ball's angular velocity quickly follow using Mathcad's differentiation routines.

Graphic 7 shows an extract from the processed data where the ball’s position is tracked through a short sequence illustrating a goal, a bounce where the forward velocity is arrested and finally one where the horizontal velocity is reversed.

What these experiments tell us
The results confirm the importance of spin acquired during the ball’s impact with the crossbar and show that under the right conditions a ball can easily cross the line and rebound back into play, to the consternation of both the players and the match officials.

Graphic 7

In our work, the critical zone – the region behind the goal line from which a spinning ball would bounce out – was very narrow, about 35cm, not much greater than the ball’s diameter of 22cm.

This was because our surface was very hard, leading to a mean coefficient of restitution (the e-value measuring the elasticity of the impact) of 0.75. It is generally accepted that natural playing surfaces have much reduced e-values compared with synthetics, typically as low as 50% of the synthetic value for really soggy turf. So we set out to use our data to model crossbar impacts with e-values typical of natural surfaces.

In doing this we used a simple model of the ball’s impact with the surface. Both were assumed to be rigid, the elasticity of the bounce being modelled by using appropriate e-values.  By considering the linear and angular momenta before and after impact it is possible to formulate simple equations relating pre and post-impact velocities and spins.  So for the horizontal and vertical velocity components:

V
h1=(3Vh-2rs)/5                (3)

Vv1=eVV                           (4)

In these equations, V and V
1 are the pre and post-impact velocities, the suffixes h and v denoting the horizontal and vertical components.  The coefficient of restitution is e, r is the ball’s radius and s the spin acquired during the ball’s impact with the crossbar. A glance at (3) shows that the horizontal velocity will be reversed if s>3Vh/2r and a quick calculation confirmed that this was so for certain values of these quantities in the experiments.

Typical spin rates and recoil velocities following the ball’s impact with the crossbar were taken from our measurements: these were 75rad/sec and 17m/sec respectively. An e-value of 0.45 (60% of our measured values) was used to reflect the generally softer playing surfaces encountered in practice.

The ball was assumed to be projected downwards at a range of angles following impact with the bar. The post-bounce conditions could then be determined from (3) and (4) and the ball’s trajectory after the bounce determined assuming that it subsequently moved under gravity alone.

Graphic 8

Mathcad is an indispensable tool in this kind of exercise: the ability to ask “what if” type questions and to evaluate outcomes rapidly is an enormous benefit compared with spreadsheeting, for example.

Graphic 8 shows the results. The ball’s behaviour has been modelled from the point where the horizontal velocity is first arrested (the red trajectory) to the point where the velocity reversal would lead to a second impact with the crossbar as the ball bounced out. Impacts in the green zone would lead to a conventional goal; the ball’s velocity is modified but not reversed.

Within the blue zone the ball reverses but the ground impact positions are too far from the goal line to enable the ball to escape by clearing the crossbar. Red is the danger area. Under the assumed conditions, balls bouncing here would be returned into play.

The e-values used (60% of our experimental findings) show that the critical impact zone extends to 54cm behind the goal line. This is in very good agreement with Frank Lampard’s 2010 World Cup goal described earlier.  
 
The case for goal line technology
Following the farrago of England’s non-goal at the 2010 World Cup, FIFA, the organising body for international football, seems to have had a change of heart concerning the role technology might play in this important area.

At the time of writing two possible solutions are under intensive investigation: England’s Hawk-Eye system relies on visual tracking of the ball and its rival, the Danish product GoalRef, on an electronic device in the ball’s interior.

Both must report the ball’s position relative to the goal line with high precision and it seems that a choice will be made by FIFA in Kiev, immediately following the final game of the Euro 2012 Tournament.

The most likely timing for its introduction would then seem to be the 2013-14 season for the English Premier League; but given that Major League Soccer starts earlier in 2013, the United States may well pioneer the system. Watch out America!

Of course if either system does its job it doesn’t matter whether the goal resulted from a crossbar impact or not. But suppose the evaluations are inconclusive. Then it might be profitable to think about mitigation measures to eliminate crossbar impacts as an important source of disputed goals. This could be done by making crossbar impacts “spin neutral” and we at Bath have some ideas on the matter. Let’s see what happens.
 
Dr Ken Bray is a theoretical physicist and a Senior Visiting Fellow in the Faculty of Engineering and Design at the University of Bath, UK. He has made a special study of the mathematics of football and the factors affecting the ball's flight, and has lectured, broadcast and published widely for both academic and general audiences. He has used Mathcad extensively in his researches, for example to calculate solutions to multiple differential equations covering the three-dimensional flight path of a football. He has also gone on record as saying that as much as 30 percent of a footballer’s technique is down to an intuitive understanding of maths and science (although they shouldn’t go anywhere near a computer!)
His book How to Score: Science and the Beautiful Game is published by Granta
 


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