Built on Facts

Football and Error

Over at Uncertain Principles, Chad is talking football. There’s this pesky problem of spotting the ball at the end of the play. In a game where fractions of an inch can make or break the end result, too often the issue is determined by a more or less random guess by the referee of where the ball stopped. Instant replay has helped the issue, but not come anywhere close to fixing it. It’s too imprecise and often made less useful because there’s enormous football players diving for the ball and thus obscuring it from the cameras.


There’s good suggestions. DGPS, radar, optical tracking, and lots of other things have been proposed. I threw in dead reckoning by accelerometer as a left field suggestion, though it’s probably among the least practical. The idea is that you put an accelerometer in the ball and integrate the acceleration twice with respect to time to determine the final position. Submarines (and I think maybe some aircraft) do this, though uncertainty builds up and the position becomes more and more uncertain with time.

How uncertain? Well, let’s take a look at the equation which gives you distance traveled if you’re accelerating with some uniform acceleration a for a time t. (In football the acceleration is far from constant, but you can treat it as the limiting case of many very small accelerations over the course of the play.) The equation is:


There’s two sources of error. The first is error involving uncertainty in the measured acceleration. If your measured value is a little lower than the real value, your calculated distance will be shorter than the real distance. Same thing with time. If you get the acceleration perfectly right but incorrectly measure the time the same kind of error will result. We can put approximate boundaries on this error by calculating how fast the distance function varies with slightly different accelerations and times, and multiplying by the error in those measurements. We do this with a little calculus. Calling the distance error delta d, we get:


That’s more formal than we need. In our football application it will be very easy to measure the time of the motion very precisely. The accelerations involved are large, but not so large that a modern clock circuit can’t easily measure intervals small enough so that the acceleration is constant for all practical purposes. That means the second term under the square root is zero. And that means the square and square root undo each other, leading to:


We would like delta d to be small, maybe something close to 1cm. The accelerometer need only function over very short time intervals, because so long as the ball is not covered by the bodies of the players it can be tracked by optical means or something similar. Only during the final hit that stops the play is the position really at issue. The time t should be in the single second range or so. I propose that adequate accuracy might be reachable even with an accelerometer that’s only good to within an error of 0.01 m/s^2 or so. Maybe even looser tolerances would do. That’s still pretty precise, especially when the hits are so hard and produce such large accelerations. And you don’t want the sensor to distort the handling characteristics of the ball in any way.

As such my impractical suggestion probably remains impractical. But it might at least be an interesting thing for a project for a clever undergraduate engineer looking for a project for class!


  1. #1 Max Fagin
    January 6, 2009

    “But it might at least be an interesting thing for a project for a clever undergraduate engineer looking for a project for class!”

    Figures. You propose this just AFTER I complete an engineering class where the whole point was to find and solve a simple engineering problem.

  2. #2 Matthias
    January 6, 2009

    A while ago there were somewhat similar considerations over here in Germany considering a tracking system for Soccer.

    There the main Problem was: Has or has not the ball crossed the line (see this about the infamous Wembley goal of ’66). IIRC they didn’t consider accelerometers, but putting something inside the ball was actually considered, so it should be feasible without distorting the characteristics of the ball.

  3. #3 yud
    January 6, 2009

    For the curious, here’s a product brochure for a commercially available inertial navigation system: MK 39 MOD 3A Ring Laser.

    The military loves inertial nav systems, because you can’t rely on GPS on the battlefield.

  4. #4 andyb
    January 6, 2009

    The fact that the ball is likely to be spinning (faster than an aircraft or submarine) and that the mass of the device will have to be small will make the challenge intractable, I think.

  5. #5 Matt Springer
    January 6, 2009

    Multiple accelerometers would probably do the trick, allowing you to subtract out spin. But fitting all that into a small pigskin without going over the 15 oz limit would be very tricky.

  6. #6 TBRP
    January 6, 2009

    I think this has been attempted before, but I don’t think they used any external tracking to synch up the IMU to reduce accumulated error.

  7. #7 Tercel
    January 6, 2009

    Just pointing out that the sort of laser accelerometer that they use for submarine navigation is staggeringly sensitive, and would reduce accumulated error problems to insignificance in football. Of course, they are also so fragile, big, etc. that you couldn’t actually put one in a football. The small accelerometers that you’d use for football are not quite so accurate, but they are still pretty good.

    Those laser accelerometers are really neat pieces of optical technology, actually. If you’re an optics dork like me.

  8. #8 Comrade PhysioProf
    January 6, 2009

    As I pointed out at Chad’s place, this is silly. The amount of variability introduced into the system by the inescapable 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.

  9. #9 Matt Springer
    January 6, 2009

    Don’t be too sure, a good real-time tracking system can in and of itself fix the forward progress and out of bounds calls. Both of those are purely positional. Just note where the ball leaves the field of play. It wouldn’t help for deciding when a player is downed, but instant replay could be adapted to work with a tracking system to at least fix things in the vast majority of cases.

  10. #10 Comrade PhysioProf
    January 6, 2009

    Do you guys even watch any fucking football? The ball is dead the instant when (1) the player’s foot touches the out of bounds line, (2) the player is downed, and (3) forward progress is stopped. Most plays end with the player being downed. This is a very difficult thing to call and cannot be automated. And your not gonna have an instant replay review of every fucking play that ends on a player being downed, or every game will last six fucking hours.

    There is absolutely no need for a more accurate measuring system for spotting the ball or measuring the first down! I can’t believe we’re even having this discussion! If you damn physicists wanna pop woodies about all kinds of laser doohickeys and satellites and nanoseconds and shit, how about the fucking baseball strike zone? Now THAT you fuckers should go to fucking TOWN on!!

  11. #11 Donalbain
    January 7, 2009

    OK.. here it is!

    Magnets! REALLY strong electromagnets. With switches.
    Magnets in the ball AND in the ground

    When the ball hits the ground, the magnets somehow get switched on. Then the ball can’t move. And they can set up that line up thing they do before turning off the magnets.

    Please note that I have not thought about this at all.

  12. #12 Oldfart
    January 8, 2009

    Comrade PhysioProf: You need to invest in an upgrade of your adjective bank. You seem to have run out and are using the same adjective over and over again. It’s pretty boring even if your comments are otherwise germane.

  13. #13 Bob Sykes
    January 8, 2009

    In every sport, there are three teams on the field, one of them being the officials. Each team makes errors of all kinds, which is one of the attractions of sports. The idea that one team must be perfect and the others may make errors is absurd. The only thing the players can reasonably expect is that the officials be consistent. It is up to the players to learn how the officials are calling the game, what fouls are being ignored, the size and location of the strike zone, etc.

    It follows that I am a fierce opponent of officials using instant replay. Instant replay destroys the flow of the game. Suck it up and play!. For a long time, baseball was perfect because there was no instant replay. Instant replay is like aluminum bats and worse than the designated hitter.

    I do like instant replay. But it should restricted to the amusement and incitement of the fans. I would allow it to be used to judge the performance of the officials in their end of year reviews.

  14. #14 IBY
    January 8, 2009

    How does delta d/delta a goes to 1/2a^2?