Drag knife compensation algorithm

[ftkalcevic]

I'm writing a bit of code to generate some G-Code for a drag knife I've installed in my cnc router. I can do the straight stuff and turn corners fine (similar to this http://www.summa.be/pages/nfo_tangentialordrag.html).

What I am wondering about, is cutting circles and curves. If the knife has an offset, will the cut circle be smaller than the one traced? If so, how do I calculate the compensation amount for the circle? I assume it is a function of the offset and the circle diameter?

Thanks,
Frank

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[ftkalcevic]

I think I worked this out...

If I keep the knife point on the circle at a tangent, then it is a simple case of using the pythagoras theorem to calculate the compensated radius...

compenstated_radius = sqrt( radius ^ 2 + knife_offset ^ 2 )

[Kiwi]

That looks OK to me.
Your next problem is to move the cutter to the start of the arc will also need some trig.

[ftkalcevic]

My circles are actually holes, so it is easy enough to start in the center, move to the edge, then swivel to point in the right direction.

[SBC Cycle]

I think I worked this out...

If I keep the knife point on the circle at a tangent, then it is a simple case of using the pythagoras theorem to calculate the compensated radius...

compenstated_radius = sqrt( radius ^ 2 + knife_offset ^ 2 )

I believe the curve you are looking for is a cycloid. Search google for "brachistochrone". Your method will possibly work for a complete circle, but it will fail if you move from a line to an arc to a line. Think of a truck making a tight 90 degree turn with a long trailer. This is an interesting mathematical problem.

[Kiwi]

I believe the curve you are looking for is a cycloid. Search google for "brachistochrone".

I googled "brachistochrone" and "cycloid", but I'm unable to see the relationship to a toolpath.

This is how I see a toolpath for a keyhole would be.
Open to corrections.

[SBC Cycle]

Examine this drawing carefully and you will see the error of the arc offset method. A simple arc offset will only approximate the path the centerline of the tool should follow (represented by the points). It may be more than sufficient for the purpose of representing a circle while cutting vinyl, but it also makes for inefficient toolpath.

Also open to corrections.

[Kiwi]

Examine this drawing carefully and you will see the error of the arc offset method........Also open to corrections.

Sorry to question your post, BUT is your drawing to scale?
I am unable to duplicate your drawing.
I'm sure a trailing blade does not give a perfect shape. I imagine the blade will act like a fish tail in some areas. I guess this goes with the benefits of a trailing blade setup.

[SBC Cycle]

Sorry to question your post, BUT is your drawing to scale?
I am unable to duplicate your drawing.
I'm sure a trailing blade does not give a perfect shape. I imagine the blade will act like a fish tail in some areas. I guess this goes with the benefits of a trailing blade setup.

Please, don't apologize! I very well could be wrong. When I get a moment I will start over with a fresh BobCAD file and try to explain how I got my results. I love this kind of problem. Some people are crazy over Sodoku, I get my kicks on math puzzles.