03-19-2011 09:29 PM

boomergeek

Re: paper: MINIBOT acceleration solution

Quote:

Originally Posted by Ether For physics and/or math students who may be interested, I just posted a short write-up showing how to setup and solve the differential equation for the MINIBOT accelerating from a dead stop up the pole. An analytical solution is given so you can just plug numbers in using a calculator (if you have the patience), or better yet a spreadsheet. The model ignores friction, which may be a significant factor, but the physics (and math) is nonetheless interesting and useful for gaining insight and rough approximation. http://www.chiefdelphi.com/media/papers/2470

03-20-2011 12:54 AM

sanddrag

Re: paper: MINIBOT acceleration solution

Ether, what would it take to get you to come over here and join my team? You mathematical contributions to the Chief Delphi forums this season have been fantastic.

Regarding your model, and the (neglected) friction, I've learned it does play a BIG factor. Tonight I did some testing and visually saw much longer acceleration times than your model predicts. Anyhow, your previous calculations inspired me to try out a few things of my own, so last night and this morning I built up a minibot with the following specs:

~2.6-2.7 lbs
2 motors, direct drive
0.42 inch rollers
Approximately 6.5lbs normal force to the pole (magnets).
Mostly charged battery (had been charging for an hour or two) measuring 14.5 Volts when pulled off the charger.

I have friction in bushings supporting the end of the rollers, and friction of a limit switch against the pole. My best time was 1.6 seconds.

One thing I found interesting was that on a battery which measured 11.98 Volts (no load), the minibot didn't have enough force to accelerate itself at all. On this rather dead battery, with a thumb on one roller (for as long and hard as I could without burning myself), battery voltage was dropping to 9.5ish under load. So, the lesson here is that if your minibot battery measures 12 Volts, it's dead.

I predict the teams on Einstein will have their deployments perfected, and the winner will be the team who can get their minibot the lightest, and tuned to operate at its peak performance point.

03-20-2011 09:24 AM

Ether

Re: paper: MINIBOT acceleration solution

Quote:

Originally Posted by sanddrag Ether, what would it take to get you to come over here and join my team? You mathematical contributions to the Chief Delphi forums this season have been fantastic. Regarding your model, and the (neglected) friction, I've learned it does play a BIG factor.

03-20-2011 01:30 PM

boomergeek

Re: paper: MINIBOT acceleration solution

Thanks for the update.

I used 7400 for freewheel speed based on Richard's dynamometer results and then I adjusted the friction number until it approximately matches the video evidence of our minibot...
it predicts 2 lbs of friction!!

I'll try some experiments the week with a weight scale and the pole.

The other thing we might try is a time measurement of a distance measured freefall of the unbraked minibot on the pole as compare it to the speed of freefall without a pole and consider the difference to be the loss of acceleration due to friction. (I like pulling out my 400 fps camera).

I also started toying with the idea of tweaking your model to compare a rampbot to a vertical bot. I haven't got it quite figured out yet.

03-20-2011 01:31 PM

Ether

Re: paper: MINIBOT acceleration solution

Here's another idea for measuring MINIBOT rolling friction.

Release the bot from the top of the pole and time (with a stopwatch) how fast it takes to fall a distance X. Do this several times and average the results, throwing away any outliers. Make sure the motor leads are open-circuit, not shorted.

Then use this formula to calculate the friction force:

f = w - 2*X*M/(t^2)


f is friction force in Newtons

w is MINIBOT weight in Newtons

M is the mass of the MINIBOT in kilograms

t is the average time in seconds

X is the distance it fell in meters

The above formula comes directly from F=M*a, assuming constant applied force:

F = M*a

(w-f) = M*a

a = (w-f)/M

v = integral(a*dt) = a*t = (w-f)/M*t  (assuming V = 0 at start)

x = integral(v*dt) = integral(a*t*dt) = (1/2)*a*t^2 (assuming x=0 at start)

x= (1/2)*[(w-f)/M]*t^2

solve for f to get the formula

03-20-2011 01:33 PM

Ether

Re: paper: MINIBOT acceleration solution

Quote:

Originally Posted by boomergeek The other thing we might try is a time measurement of a distance measured freefall of the unbraked minibot on the pole

03-20-2011 01:46 PM

Ether

Re: paper: MINIBOT acceleration solution

Quote:

Originally Posted by boomergeek I used 7400 for freewheel speed based on Richard's dynamometer results and then I adjusted the friction number until it approximately matches the video evidence of our minibot... it predicts 2 lbs of friction!!

03-20-2011 02:30 PM

boomergeek

Re: paper: MINIBOT acceleration solution

"Make sure the motor leads are open-circuit, not shorted."

If you do... make sure you have created a method of slowing down your minibot before it crashes to pieces on the tower platform...

"Did you use .09 or .094 for the stall torque?"

I used .09.


I usually don't trust the stopwatch for such a short race- too easy to have hidden bias.

A basic 30 fps video camera is easy to get stop and start points estimated to within 30-60 ms based on multiple playbacks and just counting time by single stepping through frames.
And the data is all recorded so someone else can check the calculation.

YMMV... If we can get within 15-25%, I would be VERY happy.


Thanks, Ether, for providing this... The students will have fun with this one.
We are working in New Hampshire...

I noticed you are retired and in "US", which FRC teams did you use to work with?

03-21-2011 09:54 PM

Ether

Re: paper: MINIBOT acceleration solution

Quote:

Originally Posted by boomergeek A basic 30 fps video camera is easy to get stop and start points estimated to within 30-60 ms based on multiple playbacks and just counting time by single stepping through frames. And the data is all recorded so someone else can check the calculation

03-22-2011 09:02 AM

JesseK

Re: paper: MINIBOT acceleration solution

If only these equations didn't require some backwards-thinking Lambert-W function to solve them for [t]. Ether, would it be possible to use your equation-solving software to do so? Wolfram-Alpha keeps putting the Lambert-W function in the results and I can't find a generic approximation that doesn't spiral out of control. nsolve should work in Mathematica, but on WolframAlpha it's just like solve.

03-23-2011 11:36 AM

JesseK

Re: paper: MINIBOT acceleration solution

Minor ramblings...

Last night I found the Newtonian approximation (uses Newton's method) for solving the Lambert-W. [result] = W(z) is the notation, and z is given via solving x(t) for t. Turns out, for a small enough z (|z| < 0.01 in most cases) Newton's method converges to the correct answer (with w0=0) in 1 iteration. The answer is ... get this ... z. Since our z is exp(-B^2*x/D), and B^2/D is LARGE, making the denominator of z HUGE, the Lambert evaluation = exp(-B^2*x/D). It typically mucks up when there's too much weight, way more motors than needed, or ridiculously improperly geared (making z > 0.01).

Maybe I'll update my tools post-season. Really the only thing that's useful from it is to see the time to goal as a function of Gearing, so a very close gearing estimation can be made for exactly how a team wants to play the game. It may be useful to generate a set of bounds for software since PWM output correlates directly to the power of a motor. In the direct-drive minibot's case, t is a function of 2 major variables -- diameter and weight (assuming constant friction that decreases available torque) -- so finding the optimal diameter for a weight would be easy if it were set up.

Oh, and to relate the equations to drive train (no minibot), replace "weight" with "weight*sin(theta)" where theta is the angle of the ramp to climb in radians. Theta = 0 for a flat surface. This explains a minor flaw in the direct t=a*I rotary equations I was using that prevents me from even attempting to do ramp calculations.

Still working on a way to at least characterize friction. The chart shows a theoretical climb of 1.6 seconds for our minibot, yet the reality is about 2.6. Really it's a question of whether friction robs the robot of acceleration torque, maximum speed, or both (and if both, what proportions?).

03-23-2011 12:12 PM

Ether

Re: paper: MINIBOT acceleration solution

The desired engineering result for LogoMotion is to find the shaftDiameter which minimizes the time required to reach a specified height x. This can be done by using numerical methods on the solution x=f(time,shaftDiameter) presented in the paper. In the example spreadsheet, Excel's "Solver" does this quite easily with just a few iterations. A macro could be written to do the iterations, but it's almost as easy to adjust the shaftDiameter manually to converge on a close-enough solution.

A proposed method for characterizing friction is presented here. If anyone runs the test and finds that friction vs velocity is strongly nonlinear, then it would be worthwhile to explore finding solutions to the resulting nonlinear DE.

03-24-2011 09:23 PM

boomergeek

Re: paper: MINIBOT acceleration solution

We set up a camera today. The yard stick was not in perfect position so I had to estimate some of the distances,
The times were 29 frames/sec on an MPEG4 camera.
(Dartfish software licensed to us free last year would have estimated all the measurements automatically-I miss it.)

time (sec) Distance(m)
0 0
0.034483 0.00635
0.068966 0.01524
0.103448 0.03175
0.137931 0.0508
0.172414 0.0762
0.206897 0.1016
0.241379 0.13208
0.275862 0.1651
0.310345 0.2032
0.344828 0.24765
0.37931 0.29845
0.413793 0.3556
0.448276 0.4191
0.482759 0.4826
0.517241 0.5588
0.551724 0.635
0.586207 0.7112

2.3 lb direct drive minibot

Your tool estimates about 6-10 Newtons of force without much velocity sensitivity if I plugged it in properly.


We may decide to try lo lower the normal force and measure again.

03-24-2011 11:08 PM

Ether

Re: paper: MINIBOT acceleration solution

Quote:

Originally Posted by boomergeek Your tool estimates about 6-10 Newtons of force without much velocity sensitivity if I plugged it in properly.

03-26-2011 01:44 PM

boomergeek

Re: paper: MINIBOT acceleration solution

Quote:

Originally Posted by Ether Thanks for posting this. Very interesting! See attached screenshot. The friction has a lot of "noise" (to be expected) but it appears to be independent of velocity. So I fit a least-squares line instead of a quadratic. I'd say the friction is darn close to 6 Newtons. Arghh. Won't let me attach screenshot. I'm going to upload it with the paper attachments. OK, here's a link to the screenshot: http://www.chiefdelphi.com/media/papers/download/2999

03-27-2011 12:34 AM

Ether

Re: paper: MINIBOT acceleration solution

Quote:

Originally Posted by boomergeek I think the largest remaining error is the phenomena of sliding friction that occurs when the power is first applied.

04-03-2011 04:49 PM

vamfun

Re: paper: MINIBOT acceleration solution

Quote:

Originally Posted by Ether Good catch. I was thinking about that just the other day. The model assumes there is enough friction between the "wheel" (drive shaft) and the pole so that the wheel never slips. But if your shaft diameter is small enough and your normal force isn't high enough, the stall torque of the Tetrix can "burn rubber" (like a dragster). The model does not account for this. I could add another user input: coefficient of friction, and include that to model wheel slip. But not only would that number be fuzzy (so many factors can cause it to vary), it would also make the system nonlinear and not solvable analytically. Numerical methods would have to be used. Of course, once you say goodbye to the analytical solution and start using numerical methods, you can introduce all sorts of clever nonlinear refinements. Maybe a fun project for the off season.

04-13-2011 07:47 PM

boomergeek

Re: paper: MINIBOT acceleration solution

We finished our last regional and did not implement our secret weapon:
(although we did tell the Boston Regional judges about it)
Rubber on rubber (1.2 CoF sliding friction) for the first 3 inches of vertical climb before switching to rubber to steel (0.3-0.6 CoF sliding friction).
We projected that would save us 0.2 to 0.3 seconds and still keep normal force low.

The rubber on rubber is minibot to hostbot ramp that runs vertically parallel to the tower pole and then switches onto the pole before the minibot crosses the 18" line.