paper: MINIBOT acceleration solution

[JesseK]

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?).