7/31/10
For those familiar with calculus, an alternate set of parametric equations for bump-free mecanum roller profile is derived. This derivation may be easier to visualize than the previous presentation given in "mecanum 'bumpless' roller profile RevC.pdf".
7/31/10
mecanum roller profile update: Here's a slightly optimized version of the algorithm given in the paper "mecanum 'bumpless' roller profile RevC.pdf":
Given R (wheel radius) and r (roller radius at midpoint), proceed as follows:
set D = R-r
set U = 2*D^2
set W = R*sqrt(2)
set Smax = sqrt(2*(R^2-D^2))
Now vary the parameter "S" from zero to Smax (for a half-curve) or from -Smax to +Smax (for a full curve) and perform the following calculations (in the order listed) for each S:
G = sqrt(2*U + S^2)
T = W/sqrt(U + S^2)
X = (S/2)*(T-1)
Y = (G/2)*(T-1)
where "Y" is the radius of the roller at a distance "X" from the roller's midpoint measured along the roller's axis.
7/26/10
mecanum roller profile update: I analyzed the problem completely differently (using calculus) from what is presented in file "mecanum 'bumpless' roller profile RevC.pdf", and came up with a different set of equations which I could not solve analytically. So I solved them numerically and got results identical to those from the XY data table algorithm presented in that paper. So now I am inclined to believe that the very slight deviation of the XY data from a perfect parabola is real. For most purposes the parabola will work fine. If your CAD program supports importing XY data then do that for greatest accuracy.
7/25/10
I created a simple console application to calculate the parabolic profile for a "bump-free" mecanum roller. User enters the mecanum wheel radius and the radius of the roller at the center; program outputs the equation and XY data for the parabolic profile.
7/25/10
in file "mecanum 'bumpless' roller profile RevC.pdf", I still have not been able to demonstrate analytically that a parabola is an exact theoretical solution, but for all the test cases I have tried so far a least-squares parabola fits the data almost perfectly (within a couple thousandths of an inch).
7/25/10
note that equation (6) in file "Mecanum Kinematic Analysis 100531.pdf" shows that there is a proper inverse kinematic solution for the mecanum wheel speeds in a vehicle where the wheels are at the corners of a rectangle (need NOT be a square). The physical meaning of this is that the wheels will operate just as well in a rectangle as in a square.
7/25/10
although not explicitly stated in file "omni vs mec chart 100623_.gif", it is assumed that the mecanum rollers are mounted at the standard 45-degree angle.
7/25/10
Although not explicity stated in file "omni vs mec forces 100627_.pdf", it is assumed that the mecanum rollers are mounted at the standard 45-degree angle, and the omni wheels are mounted at 45 degrees.
7/25/10
file "roller profile parabola vs ellipse.pdf" takes an XY data table generated using the algorithm derived in file "mecanum 'bumpless' roller profile RevC.pdf" and then uses least-squares to fit a parabola and then an ellipse to the data. The graphs show that the parabola fits the data almost perfectly.