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I'm the lead programmer on a FIRST robotics team, and this year's competition is about throwing Frisbees. I was wondering if there was some sort of "grand unified equation" for Frisbee trajectory that takes into account the air resistance, initial velocity, initial height, (lift... It has to fly at max 50') etc. and solves for the required angle to reach a specific distance (angle will be the only manipulated variable).

Basically, I would like to acquire data from an ultrasonic rangefinder, the encoders that determine the speed of our motors, the angle of our launcher, the rotational force (should be pretty constant. We'll determine this on our own) and the gravitational constant, and plug it into an equation in real time as we're lining up shots to verify/guesstimate whether or not we'll be close. Probably other variables, as I was informed on StackOverflow. There is no wind.

If anyone has ever heard of such a thing, or knows where to find it, I would really appreciate it! (FYI, I have already done some research, and all I can find are a bunch of small equations for each aspect, such as rotation and whatnot. It'll ultimately be programmed in C++.)

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You've picked a very challenging problem. Once you acquire your data, could you please share it with us? I see several articles on this topic (biosport.ucdavis.edu/research-projects/…, web.mit.edu/womens-ult/www/smite/frisbee_physics.pdf), do any of them roughly cover what you are interested in? – emarti Jan 9 at 4:56
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Even if we came up with a formula, it would have all sorts of hard-to-measure parameters (like the shape of the disk) I suspect. Ultimately it seems you will have test throws and you can use that data to calibrate. Is that right? In that case it seems you should be looking for algorithms to efficiently root find with an unknown but sample-able function (the distance past the target) of a variable (the angle). The algorithm should also be robust against sample errors (fix the angle and do 10 trials - you may very well observe a spread in the rangefinder measurements). – Chris White Jan 9 at 6:06
I trust that the robot with the most fun wins. Anything else would violate the Spirit of the Game. – Larry OBrien Mar 25 at 23:59

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It turns out (to your luck) I read a paper about this very subject two days ago. Simulation of a spin-stabilised sports disc where they compare some experiments, wind tunnel data, and simulations to determine the accuracy of the equations and the ideal launch angles. Since they determine ideal launch angles to find the maximum distance (and minimum deviation from straight-flight path), the equations can also be used to find the angle for any specified distance.

Their results from the simulations match the experimental data fairly well. The experimental data from the wind-tunnel can be found in Frisbee(TM) Aerodyanmics and the related PhD thesis by the author.

In hindsight, I should have joined an experimental aerodynamics group and done my thesis on throwing Frisbee's...

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