The key to an elegant solution to a physics problem is finding the right system to analyze, and the right coordinate system to perform the analysis. The torque coming from the motor feeds in to this via D. The results I found here seem to indicate that if you have "too much torque", you don't get wheels spinning out against the floor, but rather the module continuing down and lifting the robot (presumably until the module rotates past 90 degrees, when it will eventually spin freely in the air). Of course, many modules will be designed with "stops" to keep this from happening, which will introduce a considerable force not accounted for in my analysis.
As an example of an inelegant solution, one of my earlier attempts found me proving through a number of trigonometric identities that:
Quote:
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cos (θ - Ф) + sin (θ - Ф) tan θ = sec θ cos Ф
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I've learned over the years that when you prove something like this, it's time to look for a different coordinate system. This is why I analyzed forces perpendicular to B rather than vertical and horizontal components.