Ok after some help from Graham S. we have come up with the equations for the system above, after some testing I have found the following:

1) the equations work at least for quadrants 1 and 4

2) The gear reduction needed on each arm is pretty large, about 100:1 according to my calculations, gearheads of that ratio cost upwards of 1,000 dlls.

3) The resolution is variable depending on the position of the arms, having a resolution of about 0.015" when both arms are horizontal, and a resolution of about 0.021" when both arms point in a 45ยบ angle, when the tool is closest to the pivot point of the first arm, the resolution comes down almost to 0.002", all this considering a 4'x8' working envelope, and a 100:1 gear reduction.

4) With that kind of reduction, and using a 10x microstep on a 200 step per revolution stepper with mach 3 at 45KHz, the arms would turn at about 10 rpm, which again would make the max feedrate of the tooltip variable because of the distance from the tip to the centers of rotation.

5) Making a rigid enough member and joint for this configuration is no easy task.

6) The implementation of the kinematic formulas on mach 3 works on an end of move basis, they are not calculated on a step by step basis, this means that if you make a move from (5,6) to (5,-6) instead of getting a straight vertical line, you would actually get a semicircle starting and ending on those points. This makes the generation of the toolpath a bit difficult because you have to segment the toolpath in your CAD program in order to get acceptable results, this is not too troublesome in a 2 axis toolpath, but it its on a 3d contour.

7) manual jogging works the same way as in point 6)

There are a ton of issues as you can see that would make this machine not an easy task to build and operate, but it is still a feasible proyect, i would like to build one of

these, the only things stoping me from doing this is the following:

1) Formulas being calculated on an end point basis and not step by step.

2) Not being able to use a bigger reduction to get acceptable resolution and keeping feedrate acceptable (both for plasma use at least)

I guess number 2 would not be a problem if using a Grex.

Number 1 is a bit more trouble since a new calculating algorithm has to be coded into mach 3, although art has expressed interest in implementing this, I am sure its not a priority. So until that day comes, i will put this build on hold for now.

Best Regards

Fernando