Month: October 2010

Adaptive sampling drop-cutter

Inspired by this post on the pycam forum and by this 1993 paper by Luiz Henrique de Figueiredo (or try another version) I did some work with adaptive sampling and drop-cutter today.

The point based CAM approach in drop-cutter, or axial tool-projection, or z-projection machining (whatever you want to call it) is really quite similar to sampling an unknown function. You specify some (x,y) position which you input to the drop-cutter-oracle, which will come back to you with the correct z-coordinate. The tool placed at this (x,y,z) will touch but not gouge the model. Now if we do this at a uniform (x,y) sampling rate we of course face the the usual sampling issues. It's absolutely necessary to sample the signal at a high enough sample-rate not to miss any small details. After that, you can go back and look at all pairs of consecutive points, say (start_cl, stop_cl). You then compute a mid_cl which in the xy-plane lies at the mid-point between start_cl and stop_cl and, call drop-cutter on this new point, and use some "flatness"/collinearity criterion for deciding if mid_cl should be included in the toolpath or not (deFigueiredo lists a few). Now recursively run the same test for (start_cl, mid_cl) and (mid_cl, stop_cl). If there are features in the signal (like 90-degree bends) which will never make the flatness predicate true you have to stop the subdivision/recursion at some maximum sample rate.

Here the lower point-sequence (toolpath) is uniformly sampled every 0.08 units (this might also be called the step-forward, as opposed to the step-over, in machining lingo). The upper curve (offset for clarity) is the new adaptively sampled toolpath. It has the same minimum step-forward of 0.08 (as seen in the flat areas), but new points are inserted whenever the normalized dot-product between mid_cl-start_cl and stop_cl-mid_cl is below some threshold. That should be roughly the same as saying that the toolpath is subdivided whenever there is enough of a bend in it.

The lower figure shows a zoomed view which shows how the algorithm inserts points densely into sharp corners, until the minimum step-forward (here quite arbitrarily set to 0.0008) is reached.

If the minimum step-forward is set low enough (say 1e-5), and the post-processor rounds off to three decimals of precision when producing g-code, then this adaptive sampling could give the illusion of "perfect" or "correct" drop-cutter toolpaths even at vertical walls.

The script for drawing these pics is: http://code.google.com/p/opencamlib/source/browse/trunk/scripts/pathdropcutter_test_2.py

Here is a bigger example where, to exaggerate the issue, the initial sample-rate is very low:

waterline with bullcutter

Update: Here is another example with the CL-points coloured differently. At each z-height the innermost loop is with the ball-cutter, next is the bullcutter, and the outermost loop is calculated for a cylindrical cutter. The points are coloured based on which test (vertex, facet, edge) produced them. Vertex-test points are red. Facet-test points are green. The edge-test is further subdivided into (1) a test for horizontal edges (orange), (2) a test for contact with the cylindrical shaft of the cutter (magenta), and (3) the general edge-push function (light blue for ball/bull, pink for cyl). If/when I get the cone-cutter done the cutter-location algorithms in opencamlib should be complete (at least for the moment...), and I can move on to more interesting high-level algorithms.

three_cutter_waterline

waterline_cutters

This figure shows one of the first times I got the push-cutter/waterline algorithm working for bullcutter (filleted endmill, bull-nose cutter, toroidal cutter, a dear child has many names...).

The thin cyan lines are edges of a triangle. The outer cyan spheres are valid cutter locations (CL-points) for a cylindrical endmill. The innermost yellow CL-points are for a spherical (or ball-nose) endmill. Between these two point-sets the new development is the magenta points, which are CL-points for a bull-nose cutter.

The algorithm works by pushing the cutter at a specified Z-height along either the X-axis or the Y-axis into contact with the triangle. There are three sub-functions for handling the case where the cutter makes contact with a vertex, the triangle facet, and an edge. The edge-contact case is the non-trivial (read "hard") one. The approach I am using is based on the offset-ellipse, courtesy of the freesteel blog. Pushing a toroid into contact with an edge/line is equivalent to pushing the cylindrical "core" of the bullcutter into contact with an edge that has been 'inflated' to a cylinder with a radius equal to the bullcutter corner radius. Slicing this cylinder/tube with a z-plane gives us an ellipse, and the sought cutter-location lies on the offset of this ellipse. I should make some diagrams and post longer/better explanation later (I wonder if anyone reads these 🙂 ).

The bullcutter is important not only in itself, but also because it is the offset of a cylindrical cutter. When we want to do z-terrace roughing with a cylindrical cutter, and specify a stock-to-leave value, we do it by calculating the toolpath with cylcutter->offsetCutter() which is a bullcutter, and then actually machining with the cylindrical cutter. That will achieve the desired stock to leave (to be removed later by a finish operation).

Links - 2010 Oct 21

View of hard disk with windirstat

Where does all that hard drive space disappear? Windirstat will draw you a picture (http://windirstat.info/ or on Linux http://kdirstat.sourceforge.net/)

This is the 78G windows-partition on my laptop. The size of pagefile.sys and hiberfil.sys probably scale with the amount of RAM (I have 4G). Matlab is just huge (and bloated?). I wonder if an 18G windows-folder is about normal for win-7 users, or if I have a lot of temporary stuff and installer-files etc. hiding in there?
harddisk