The standard representations for Computer Aided Design include volumetric, boundary and CSG models. Current advanced modelers, can produce process plans for specific machines in order to manufacture the object. We believe that the process plan and associated information (e.g., the tool path, the tool to be used, its speed, etc.) provide a strong basis for analyzing the manufacturing and inspection steps with respect to tolerances.
A tolerance specification on the shape geometry must be transformed into the corresponding tolerance on the machining operation and vice versa. This in turn can be used to select an appropriate manufacturing process, given knowledge of the manufacturing accuracy of the process. This yields direct methods for deciding on sensing strategies both to monitor the manufacture of the part, as well as for post-manufacturing inspection. These sensing strategies are derived from an analysis of where the toolpath is most likely to deviate from the tolerance specification.
These must all be done as efficiently as possible; in particular, it must be:
The keys to our approach are:
In order to structure the analysis process, we focus here on NC milling, and use the toolpath as the basis upon which design and manufacturing tolerance and sensor measurements will be compared. Much as operational semantics allows the meaning of a high level program to be defined in terms of the particular architecture upon which it executes, so can the CAD specification of a part be defined in terms of the machining operations which produce its shape.
Given the CAD geometry for a part, a tolerance specification, and a class of
NC mill to be used, generic knowledge about such mills can be used to
generate a desired toolpath with its associated tolerance (call it ).
Once a specific mill is selected, the nominal toolpath from
together with the accuracy of the mill determine the actual toolpath (call this
). These two toolpaths allow us to determine a great deal about the
efficiency and uncertainty regions of the process.
First, if is true, then we know that the tolerance
should, in principle, be achieved. If
is large, then the
selected machine may be too precise, and therefore, too expensive. If the
boundary of
is close to that of
, this signals places where
sensing may be necessary to guarantee the inclusion relation. This also gives
insight into how accurate the sensing needs to be. Even if
is not
contained in
, this approach allows us to estimate what percentage
of milled parts will be out of spec, and thus an informed decision can be
made whether to tighten the accuracy of the machine, or where to sense with
high probability of part error. Thus, the toolpath representation allows
insight into design, manufacture and inspection in a common framework.
For this approach to work well requires a clear and efficient implementation technique, and we propose the interval spline for this purpose. The use of interval Bezier curves for a complete description of approximation errors was proposed by Sederberg and Farouki [9] (see paper for details). The basic idea is to extend splines to polynomials whose coefficients are intervals with well defined arithmetic operations. Such splines define a region in space rather than a curve. This notion captures very nicely the semantics of a tolerance specification. We have developed interval curves for both 2D and 3D.
For curves in 2D, an interval is a set of 3 points (corresponding to the
nominal point and two bounds). The spline interpolation is done (on 6
consecutive points) separately on each of the 3 corresponding curves (see
Figure 2). Note
that since the initial 3 points are on a line, evaluation at any parameter
yields 3 points on a line.
As indicated above, the determination of inclusion of one interval spline
within another is an important question. Figure 3 shows the case where
inclusion is true.
We have developed a technique to answer this question (see Appendix A). Moreover, if one interval contains another, then the area of the difference of the two intervals is also determined.
In 3D, we assume that the uncertainty around a point is described by an ellipse (in the plane normal to the curve). The problem becomes how to determine if one ellipse is inside another. We have developed an algebraic solution to this problem (see Appendix B).