# The Football Positioning System

It’s NFL playoff time, which means that sports fans will be treated to the sight of the most high-stakes farce in sports, namely the ritual of “bringing out the chains” to determine whether a team has gained enough yards for a first down. We’ve all seen this: the play is whistled dead, a referee un-stacks the pile of players, picks up the ball, and puts it down more or less where the player was stopped. Then he tosses the ball into the middle of the field, to a second referee, who tries to replicate the spot closer to the center of the field. Then a guy on the sideline carrying a big stick (connected by a ten-yard chain to another stick held by another guy) tries to put the end of the stick at the same position as the ball.

Three plays later, the spotting procedure is repeated, and then the sticks are bought out to the center of the field, the chain is stretched taut, and they measure the position of the ball to the nearest millimeter. Because, of course, there’s absolutely no error in placing the sticks.

The whole ritual is preposterous, and anybody with the slightest scientific inclination has to wonder: “Isn’t there a better way of doing this?” So, what would be required to do a better job of this?

There are a bunch of ways you might choose to approach this problem, but let’s take the Global Positioning System as a model. GPS works by using a network of satellites orbiting the earth in well-defined orbits. Each satellite contains an atomic clock, and it constantly broadcasts the time according to that clock. Your GPS receiver picks up the time signals from at least three satellites, and compares the times. The signals from satellites that are farther away take longer to arrive, and thus the difference between times gives you the distance from any given satellite to your position on Earth. Calculating that distance for three satellites specifies your position on the face of the earth, assuming that you know the orbits of the satellites very well.

GPS works well enough to prevent even the most directionally challenged person from getting lost, so it ought to be good enough for football. We could imagine setting up receivers at the four corners of the field, putting a small chip in the ball that sends out a signal, and using the time delay between the signals reaching each of the four receivers to determine the exact position of the ball on the field.

(Actually, it might be easier to put both transmitters and receivers on the four corners, and some passive resonant circuit in the ball, then look for the echo at the receivers. The math works out the same either way, so it doesn’t make much difference.)

So, what would such a football positioning system require in terms of timing accuracy? Well, an American football field is 300 feet long and 160 feet wide, and radio waves travel roughly one foot per nanosecond. If we imagine putting the ball exactly in the center of the field, we can use the Pythagorean Theorem to find the distance to each corner, which is 170 feet. If we placed the ball at the precise center of the field, a signal from the ball would arrive at each detector simultaneously, 170 ns after it was emitted.

Now, let’s imagine moving the ball one foot (about the length of the ball) closer to one end zone or the other. The distance from the closer receivers to the ball is now 169.12 feet, and from the more distant receivers, it’s 170.88 feet. The signal from the ball to the corners of the field would arrive at one end zone about 1.8 nanoseconds later after the other.

So, getting the position of the ball on the field to within one length of the ball would require a system that could detect differences of about one nanosecond in the arrival times of pulses from the ball. Other positions on the field are pretty comparable. I calculated positions for a few different cases, and the smallest difference I got was for a ball exactly on the goal line, at the center of the field. Moving that ball back one foot changes the difference between arrival time by a hair under a nanosecond (unless I made a horrible math error, which is possible).

That’s challenging, but might be doable. Of course, the length of the ball is pretty coarse by the standards of farcical distance measurements in football, so you’d like a system that could do much better. And that starts to get tough– getting the position to a centimeter would require time resolution of about three one-hundredths of a nanosecond. Getting transmitters and receivers that are stable to that precision is difficult proposition.

This is, of course, an incredibly naive way of approaching the problem, and I’m sure somebody with more knowledge of signal processing and the workings of GPS than I have could do better. But it should serve to give you some idea of what would be required to eliminate the ritual bringing out of the chains in football games.

1. #1 jl
January 5, 2009

The only problem is that you’d also need a way to signal when the ball-carrier’s knee (or elbow, or whatever), touches the ground, so that you don’t have the case of a player being down and then extend the ball forward.

2. #2 Mark B.
January 5, 2009

I suppose RFID tags are already encrypted with an identity signature to prevent spoofing? Otherwise this would introduce some fun new ways to cheat.

3. #3 RPM
January 5, 2009

Assuming you got the system working to an accuracy of less than a ball-length, would you put a transmitter/receiver on each tip of the ball?

4. #4 Anon
January 5, 2009

http://www.nytimes.com/2009/01/01/sports/football/01chains.html

You might find it interesting–same issue that you bring up, different solutions.

5. #5 MRW
January 5, 2009

I heard a story about someone actually doing this on NPR a week or two ago. Right now, they’ve got a 30-ft. accuracy, but they claim mm-scale should be possible. Here’s a version from Wired:

6. #6 dominich
January 5, 2009

Why not use the Hawk-Eye system as deployed in cricket and tennis, multiple cameras track the ball and calculate trajectories and ground contact. Fixing the point where play stops would be trivial for such a system.

7. #7 Steve
January 5, 2009

My brother looked at this for an electrical engineering project. You get other problems involving echo signals from the stadium walls, etc.

It’s a lot trickier than it looks. Your best bet would be a camera based system – the ref places the ball, the cameras record it’s position, and you work from there. A few high res cameras would be much more accurate – although you still have problems with people placing the ball accurately enough, that’s pretty much always going to be a judgement call.

8. #8 dominich
January 5, 2009

Ooops, twitchy posting finger.

… No need for any modifications to the ball.

9. #9 Joe Shelby
January 5, 2009

Biggest problem with the camera approach (i.e., the same “motion capture” as used in movies today to create characters like Gollum and skeleton pirates) is that unlike tennis where the ball is always in the open, the ball in football is almost always buried in some guy’s arms. It would be very rare in tennis for a player to be able to block the ball from enough cameras to get an inaccurate reading.

In football, however, it often comes down to only one camera being able to accurately see it drop (judging a fumble vs clean down vs incomplete pass, a very common referee review situation), but one camera isn’t enough for accurate triangulation when location matters.

10. #10 G Barnett
January 5, 2009

Hmmm, a network of sensors, spaced 1-5 cm apart (depending on desired accuracy) buried beneath the field might work by tracking passive sensor chips at each end of the ball. Of course, they’d have to be deep enough and robust enough to handle both the pounding the field takes AND any inclement weather….

January 5, 2009

#1: The only problem is that you’d also need a way to signal when the ball-carrier’s knee (or elbow, or whatever), touches the ground, so that you don’t have the case of a player being down and then extend the ball forward.

You could synch it up with the referees’ whistles. Add something to the whistles that generates a signal when the play is blown dead, and spot the ball there.

#7: My brother looked at this for an electrical engineering project. You get other problems involving echo signals from the stadium walls, etc.

Well, you just, um, make the stadium walls absorptive? Use stealth technology?

I’m not trying to claim this as THE solution to the problem, but it’s sort of fun to think about.

#9:Biggest problem with the camera approach (i.e., the same “motion capture” as used in movies today to create characters like Gollum and skeleton pirates) is that unlike tennis where the ball is always in the open, the ball in football is almost always buried in some guy’s arms. It would be very rare in tennis for a player to be able to block the ball from enough cameras to get an inaccurate reading.

Exactly.

The laser-based things mentioned in the Times article have a similar problem. Projecting a mark on the field itself sounds attractive, but there are all those people to get in the way.

12. #12 Nathan
January 5, 2009

I like the randomness induced by the silly placement technique, much as I like the randomness induced by having non-machine umpires in baseball. It keeps things interesting, and makes for good stories.

I wouldn’t mind high-def cameras along all the sidelines for those in/out of bounds calls, though.

13. #13 Clark
January 5, 2009

Re #11, #1:

It seems to me that the whistle signaling the end of a play is the only thing out there less accurate than the placement of the ball. Sometimes plays appear to be ‘dead’ for 1 or 2 full seconds before anyone even thinks about blowing a whistle. Without the ability to lock down both measurements to similar uncertainties, it doesn’t make much sense to do it to just one of them.

Similarly: I’ve noticed that it seems to be just about required that refs place the nose of the ball (it can be either end) on one of the yard lines. (I watch a lot of college FB and this seems to be the case >90% of the time. I watch virtually no NFL, so maybe they don’t have this same problem.) It’s amazing how often there is a run up the middle, 9 players in a pile, and when it is all said and done, the ball ends up with its nose exactly on the 37 yard line.

14. #14 Kevin W. Parker
January 5, 2009

the play is whistled dead, a referee un-stacks the pile of players, picks up the ball, and puts it down more or less where the player was stopped. Then he tosses the ball into the middle of the field, to a second referee, who tries to replicate the spot closer to the center of the field. Then a guy on the sideline carrying a big stick (connected by a ten-yard chain to another stick held by another guy) tries to put the end of the stick at the same position as the ball.

Not that it makes a huge difference to your overall point, but in my experience they skip the middle step, not moving the ball to the appropriate position along the hashmarks until after the measurement is done.

I look at it about the same way I look at counting votes when the difference is in the statistical noise: yeah, it’s nonsense, but you have to at least seem to have a rational procedure.

15. #15 Phil
January 5, 2009

The problem with football is you can’t always see the ball, so multiple cameras are sort of moot. Try spotting the ball in a quarterback sneak.
How about imbedding sensors into the lines on the field?

16. #16 Bob
January 5, 2009

The whistle would be useless. In both collegiate and pro football, officials refrain from blowing many plays dead simply because replay could rule a fumble, etc. The end of the play is not always marked by a whistle. Sometimes the actual end of play occured well before the whistle.

You’d have to train refs to blow a whistle as soon as the knee touches the turf. This would be faulty if the player happened to be losing the ball the instant beforehand — no doubt a focused ref would screw this up more often than not just like baseball umps screw up close calls at bases when there is no sound cue which they ordinarily use to determine the arrival of the ball due to a soft-toss, loud stadium, etc.

Your idea could work if you required an additional on-field official whose only role was to push a silent transmitter when the ball carrier’s knee, etc, were ‘down’. This transmitter would be used for placement but not for determining the official end of play. This official’s only job would be to observe the limbs of ballplayers. Unfortunately, he’d also have to be very fast. Maybe he could wear a jet pack and somehow hover over the action without getting in the way so he could monitor long pass plays and changes of possession.

January 5, 2009

The whistle would be useless. In both collegiate and pro football, officials refrain from blowing many plays dead simply because replay could rule a fumble, etc. The end of the play is not always marked by a whistle. Sometimes the actual end of play occured well before the whistle.

Those cases are pretty rare, and generally end up going to replay anyway. It would be a simple matter to combine a position signal with the replay system, and have the referee determine a frame at which the play was dead, and then have the system extract the appropriate position of the ball.

The only real problem would be with the “forward progress” rule, but even there, they blow the whistle when they think the runner has been stopped. You could easily just change the rule to be that the ball is spotted where it was when the whistle blew, or even use an electronic monitoring system to spot the ball at the farthest point it reached before the whistle blew.

18. #18 Ian Durham
January 5, 2009

I like the randomness induced by the silly placement technique, much as I like the randomness induced by having non-machine umpires in baseball. It keeps things interesting, and makes for good stories.

As a long, long, long-suffering native Buffalonian and lifelong Red Sox fan I have witnessed my share of disputed calls. But I still that too much technology takes the human element out of the game. In baseball, arguing with the umps is a part of baseball lore. It would be a dull game if we no longer saw managers sweeping dirt onto homeplate just to tick off an ump.

As for football, if you take away the chains, could you really legitimately call it a game of inches anymore? There wouldn’t be any suspense since you’d know instantly whether the guy got the first down or not. I think we could improve spots since the refs frequently mess that up, but I think that could be improved through instant replay and by giving the booth referee authority to overrule field referee.

19. #19 WRMartin
January 5, 2009

Lasers. High-powered IR lasers.
Players would learn to get out of the way of the ball when the whistle blew or be cut in two. Then, watch those ratings skyrocket! ;)

20. #20 Mary
January 5, 2009

Hmm. Whever I want to measure distances, I do it with interferometery… If we could figure out how to do that with football, it would become a game of nanometers.

21. #21 Charlie Dodgson
January 5, 2009

This smells a lot like differential GPS. The fanciest setups claim millimeter accuracy — but the ball-with-receiver would probably be heavy enough (and lumpy enough) to seriously mess with a long pass…

22. #22 Matt Springer
January 5, 2009

This is probably a ludicrous suggestion, but how about dead reckoning via accelerometer? The navy uses it in submarine navigation without sonar. Now you’d need very accurate accelerometers that can survive being smashed around by huge guys, but on the plus side each play is short and thus there’s not a lot of time for the t^2 error to propagate.

Or forget acceleration, how about Doppler radar? Put a nice dipole wire inside the ball, and measure integrated velocity to determine the position. cm/sec accuracy is pretty easy to achieve, and ought to be adequate.

23. #23 Brian
January 5, 2009

Or, we could just wise up and start getting into rugby, thus eliminating the need for such shenanigans.

24. #24 dave
January 5, 2009

“(Actually, it might be easier to put both transmitters and receivers on the four corners, and some passive resonant circuit in the ball, then look for the echo at the receivers. The math works out the same either way, so it doesn’t make much difference.)”

If you can get away with measuring round-trip time instead of one-way, you get a factor of two improvement in distance resolution for free. If your achievable resolutions is similar in scale to what you need, this could make the difference between useable and not-useable.
(My day job involves processing radar signals, and sometimes its easier to just use 1.5e8m/s as the speed of light than to add another factor to compensate for the two-way trip.)

January 5, 2009

This is silly. The amount of variability introduced into the system by the judgment call as to when the ball is dead–either by the player being downed, going out of bounds, or having forward progress stopped–vastly outweighs the errors introduced by the ball-spotting and chain-measurement system.

26. #26 Tercel
January 6, 2009

A camera based system would work fine, it would just require a ref to look at a screen and draw a dot on the ball. This would tell the computer where in the image the ball is. And the cameras they film football with are already well instrumented to calibrate pixel positions to real field positions. This is how the 1st down line magic works, so its already in place.

Maybe you could even have that camera that flies around on cables above the field drop down automatically and place the ball at the new spot location after the ref tells the computers where to put it. How sweet would that be?

For the record I do image processing professionaly, so I love this stuff.

27. #27 CCPhysicist
January 6, 2009

Pay attention. They blow the whistle about one second after the play has ended. It signifies that everyone should have realized it is over and stopped fighting.

I’ve always thought that a system of lasers on tracks would be ideal. Line up one after the ball is spotted and the other tells you where you have to go. A pair of cameras on that same line, well above the field, from both sides, could be used to judge forward progress. However, that confuses sport with science. The game is meant to be messy.

I’d be happy if the referee (not to mention the booth referee) actually knew the rules.

28. #28 Tercel
January 6, 2009

Yeah, that’s a good point. I’m a nerd, but sometimes I wish they’d just play the game.

Usually, those times are when the inaccuracy is working against me…

:)

29. #29 Rich
January 6, 2009

Forget rfid, and they should eliminate the chains also, just go with the official’s estimate of where forward progress was, and get the game moving.

January 6, 2009

Great post! Considering how perfectly simple the act of slamming bodies into each other is, the two-sticks-and-a-chain method seems to fit. Now how about fixing that BCS system.

31. #31 LHD
January 6, 2009

I’m disappointed that no one picked up on the truly profound suggestion in Chad’s comment #11: