Suppose you are moving a tool of some unknown, but static, geometry around a workpiece that also has unknown and static geometry, and you have a thing rigged up that stops motion when they touch, preventing damage, and tells you. And you have a high-resolution positional feedback system that tells you what the position of the tool is, relative to its starting point; and there are no other important sources of motion or unknown geometry in the system. What can you learn from this?
Well, if you have only one degree of freedom, you can tell the initial distance, or angle or whatever, from the tool to the workpiece. Or the distance at any later time, which comes to the same thing when you always know how far you’ve come.
If you have two translational degrees of freedom, you can trace out a curve in the plane that is sort of the Minkowski sum of the tool and the workpiece; if the point of the tool is not too large, this is an approximation of the workpiece’s shape. In particular, if the tool is convex, you will never see tighter convex curvature on this workpiece approximation than the tightest curvature on the tool. This can allow you to bound the error due to the tool being non-pointlike.
In three translational degrees of freedom a similar property holds, but now you’re tracing out a Minkowski-sum surface rather than just a curve.
But how can you separate out the contributions from the tool and the workpiece?
One piece of information comes from recognizing motifs in the surface: every point and edge in the workpiece surface manifests as a copy of part of the tooltip shape, like the camera bokeh in a photo, but the Minkowski sum is more similar to grayscale morphological dilation than like the convolution with a bokeh. So in theory from a purely information-theoretical perspective you ought to be able to recognize that these repeated motifs are generated by the tooltip and infer enough about the tooltip shape to give you a more random-looking workpiece landscape, though with holes in it where the tooltip didn’t fit. But another approach comes from having more degrees of freedom.
Suppose you have redundant degrees of freedom. The simplest example here is having two translational degrees of freedom, plus the ability to rotate the tool around the axis perpendicular to them. This enables you to sort of measure the shape of the tool, by rotating it and measuring it against the same position on the workpiece. If you’re measuring it against a flat on the workpiece, you can use this to find the shape of the convex hull, though only up to some constant radius offset. If you measure against a needlelike point on the workpiece, you can use it to find the shape of the tool to high accuracy, again up to some constant radius offset. Combining this with the workpiece motifs, which tell you how big the radius should actually be (at least if the workpiece has a few asperities on it somewhere), you should be able to infer quite precisely what the tooltip shape is, and thus a great deal about the workpiece shape.
This extends into three translational dimensions as well, if you have two separate rotational degrees of freedom.
A sort of intermediate case here is where you don’t have four or five degrees of freedom, but the degrees of freedom you do have admit multiple solutions to inverse-kinematics problems. Consider an XZC* setup with a tool that can scan back and forth past the center of a turntable, and can also be raised and lowered. This only has three degrees of freedom, only enough to bring the end effector into contact with any position on or in the workpiece, but for every point that isn’t on the center axis of the turntable, you can establish contact in two separate ways, with relative orientations 180° apart. These two orientations give two separate Minkowski-sum surfaces; the difference between them tells you something about the difference between the tooltip and a half-turn-rotated version of the tooltip.
This isn’t necessarily enough information to tell you anything interesting; it’s impossible to distinguish any two half-turn-symmetric tooltips that are capable of producing the same Minkowski-sum surface. By contrast, a redundant rotational degree of freedom enables you to distinguish any two tooltips that differ by anything other than a circular dilation, or in the 3-D case, a cylindrical or toroidal dilation.
This is a particularly interesting problem to me because, when you’re cutting or forming any kind of workpiece, there is usually tool wear, and avoiding tool wear requires tradeoffs that may be unappealing for other reasons, like lower material removal rate or more expensive tool materials. So to the extent that it’s possible to automatically and continuously compensate for that, it may be an extremely worthwhile ability to develop, providing order-of-magnitude advantages in speed/precision tradeoffs. Probably the right way to do this is to start by cutting or forming one or more reference points on the workpiece to have favorable geometry in a way that is resilient to tooltip shape errors, such as a sharp-edged circular hole or a sharp cone, and then using those reference points periodically thereafter to measure the tooltip. The reference points can be eliminated at the end of the process, or they can be placed on things that aren’t part of the workpiece proper (but are rigidly fixed in relation to it), or they can be placed in places where their shape doesn’t matter.
* The X and Z axes can also be rotational rather than purely translational; it makes no difference in this case.