Re: paper: MINIBOT acceleration solution
 

Thanks Again!

We've been doing more testing to analyze the model accuracy.
I think the largest remaining error is the phenomena of sliding friction that occurs when the power is first applied.
The model suggest that the robot should move its first half inch within 0.05 seconds of applying the power.
We see it taking about .3 seconds based on video analysis.

It's approximately 0.2-0.3 seconds whether we turn the motors on after the mininbot is attached to the pole or if we start the motors up prior to attaching to the pole.

After accounting for the time of this initial sliding friction, the model you produced accurately predicts the measured time of our minibot!
I'd be curious if anyone else would be willing to share their results.


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I would think a ramp bot could be better able to minimize this initial sliding friction and also immediately start doing "work" (some of which that will be turned into vertical momentum) as soon as it turns on. The pole bot has to wait until it reaches the pole to have any of its work being used for creating vertical momentum, Thus, I'm a firm believer that theoretically, a ramp bot can obtain a better time than a pole bot.

Re: paper: MINIBOT acceleration solution
 

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.

Re: paper: MINIBOT acceleration solution
 

It is important to have the wheel slip modeled if you want to get the optimum wheel radius and normal force. I have done the modeling a couple of times... once for my own excel dynamic model and I also posted the numerical solver as a modified version of Team 1640 Drivetrain model on my blog here for others to use.
This has the spec motor curves built in but by playing with the parameters, you can easily make it look like the dyno model.

I measured the coefficient of drag as .22 for our minibot model using a high speed camera and a drop test. The coefficient of drag is defined as Drag_force/Normal_force. This model used Tetrix kit bearings. Typically the friction force is 6 to 7 newtons for our 2.5lb minibot. We always measure it with a newton scale prior to each run.


A excellent approximation to the climb time can be made by using your variable definitions.

The steady state climb speed v_ss = D/B (should be the same as peak_pwr/W)
System time constant tau = 1/B
Dist = pole height.
Then
time_to_climb = Dist/v_ss + tau

This falls out of the exponential solution when the time > 5 tau which is typically the case for the minibot. To include the effects of drag... simply replace W with W + Drag.

Glad to see you are spending some time to document the physics for others. I have been a little lax this year on my posts.

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.