Best launch angle and velocity: HANDS DOWN..

[Symph]

STEMpunk’s shooter, optimized with applied mathematics:

Ever wonder what the single greatest, most spectacular, ABSOLUTE BEST angle and velocity to shoot your game piece at is?

Well, for us, it’s 39 degrees at 42.5 feet per second.

(From ~43.3in above the ground. Giving us a 13.5 ft thick strip of the playing field to make the shot on.)

If you want to optimize your robot like we did ours, for 2014 or any other year involving a goal and a projectile, I would encourage you to read on and find out how got these numbers.

A week into the build season our robot was well past the first basic stages of design. Our team, having decided on the layout of our drivetrain (utilizing an unusual application of the mechanum drive) and our shooter, (a virtually one piece rig functioning as an all in one, passing, shooting, ground/ air pickup mechanism) needed to know now, what results our shooter should produce to classify as a “good shooter”.
The algorithms we contrived to give us this definition, if utilized properly, should afford most teams with the capability to find that one combination of angle and velocity giving them the single greatest, most spectacular, ABSOLUTE BEST strip of the playing field to make the shot from. You can find our results above. This is how we did it:

(See "MathProofCondensed.docx" attachment.)

No doubt you will want someone who enjoys rigorous mathematics suiting these calculations to your own robot, but it’s worth the work, as this not only draws a bold line between a "bad shooter" and a "good shooter", but also marks the difference between a "good shooter" and a "perfect shooter".

[Ether]

Originally Posted by Symph

Well, for us, it’s 39 degrees at 42.5 feet per second.

That's an incomplete specification. You need to state what height above the floor that is specified at.

[Symph]

Thanks, fixed it.

[Ether]

Uh-oh. Back to the drawing board:

Y_max = ½*a*t²

t_max = sqrt(2*Y_max/a) = 0.58

[Symph]

I have revised the entire proof, sorry about the inconvenience, it should be sound now. However, considering the fact that I was thinking precisely that same thing last time I submitted this, I would encourage you to feel free to check my math again!
What say you, Ether?

[Ether]

Originally Posted by Symph

I have revised the entire proof, sorry about the inconvenience, it should be sound now. However, considering the fact that I was thinking precisely that same thing last time I submitted this, I would encourage you to feel free to check my math again!
What say you, Ether?

You forgot to edit your original post. It still says "42.5 feet per second" and "13.5" ft scoring range.

[Symph]

Originally Posted by Ether

You forgot to edit your original post. It still says "42.5 feet per second" and "13.5" ft scoring range.

Looks like my original post is now too old for me to edit.

[matthewdenny]

According to the model we are using there is a strange occurrence:

From a height of 3.5'
And an angle of 35 deg.
And a ball speed of 35 fps

There is a pretty decent window.

Not quite perfect, but it would have been a good design point to start at.

[Ether]

Originally Posted by matthewdenny

According to the model we are using

Is it a simple parabolic model (like Gabe is using)? Or does it include air drag?

For a simple parabolic trajectory with those launch parameters, the max height of the top of the ball is 10.8 feet*, which exceeds the top of the goal.

* assuming you meant that the 3.5' launch height was the top of the ball, rather than the center

[Chris is me]

Great work on the kinematics, though the pedant in me objects to absolute terms like "best" and "ideal". There are a variety of factors at play driving the shot of a ball.

Now, for extra credit, using reasonable assumptions for energy transfer, figure out how much spring potential energy your team's system needs to make this shot.

One thing I didn't see, but I'm too busy to compute - for what range of distances is the ball able to go in the goal?

[Ether]

Originally Posted by Chris is me

One thing I didn't see, but I'm too busy to compute - for what range of distances is the ball able to go in the goal?

Check out post#13.

[Ether]

I put together a few observations about parabolic trajectories in a short 2-page paper here.

I'll be adding more as time permits.

[matthewdenny]

Some follow-up questions, if I may?

- are you using a linear or quadratic air drag model?

- how did you validate the model? i.e. How did you determine the proper value for the drag coefficient for this year's game piece?


I used a terminal velocity of 36fps, which someone on here had obtained. To double check it I translated it to a Cd of .58, which given the calculated Reynolds number, corresponded to a NASA page relating drag on a sphere to Cd.

Once I had Cd, I used .01sec intervals starting from the launch position, Calculate Vx, Vy, and Drag Force, I use Drag force and gravity to adjust the numbers for the new angle of motion and Velocity, and repeat to trace the arc.

Here is my Spread Sheet. See the sheets for 'Flight Calc' and 'Flight Path' to get all the details. (You can check out the other sheets too if the interest you)

[Ether]

Originally Posted by matthewdenny

Here is my Spread Sheet. See the sheets for 'Flight Calc' and 'Flight Path' to get all the details. (You can check out the other sheets too if the interest you)

Nice job on the spreadsheet. Thanks for posting it.

Originally Posted by matthewdenny

I used .01sec intervals starting from the launch position, Calculate Vx, Vy, and Drag Force, I use Drag force and gravity to adjust the numbers for the new angle of motion and Velocity, and repeat to trace the arc.

May I offer a suggestion? The way you implemented the integration appears to have accuracy/stability issues, which could easily be fixed.

All the plots below are based on the parameters in your spreadsheet (launch speed = 35 ft/sec, launch angle = 35 deg,
g = 32.2 ft/sec², terminal velocity = 36 ft/sec, ball weight =3 lbf etc):

Plot1 shows your results compared to the true* flight path.

Plot2 shows your model with the time step changed from 0.01 to 0.005. Notice the improvement.

Plot3 shows your model with the time step changed to 0.003. It is beginning to diverge.

Plot4 shows your model with the time step changed to 0.002. It is very much off track.

Plot5 shows what is possible with a carefully-implemented Euler method at 0.01 timestep

Plot6 shows Euler with a 0.002 timestep.

Plot7 shows Trapezoidal integration of X and Y at 0.01 timestep. Notice you can't even see the red.

Trapezoidal integration is easy to do in Excel for problems of this type. I can write a short paper explaining how to do it if anyone is interested.


*mathematically "true" based on the model used. I computed the true flight path using second-order Runge-Kutta at 0.01 time step. With RK2, smaller time steps are not necessary for this application as they yield virtually identical results: See PlotRK

[Ether]

By the way, switching gears for a moment from math to real-world engineering, here's a comparison between the parabolic trajectory and the air_drag trajectory, assuming terminal velocity of game piece is 36 ft/sec and air drag is proportional to the square of velocity.

Does anyone else have test data they'd be willing to share, that could be used to confirm the air drag?