Out-of-Plane Swept Geometries

In Section 4.1.4 we discussed some of the relative advantages and disadvantages of parametric vs. topological constraint modication operations. Where applicable, topological operations give us the ability to introduce entirely new classes of design solutions, along with new design parameters. Consider the vibratory bowl feeder example in Section 3.2, where one of the parameter sets used to determine the feeder-lter function was the shape of the supporting track. As we saw above, the coupling between the support transition boundaries and the supporting track polygon vertices is very pronounced and non-linear. In trying to terminate a given part motion path on the CS by having it encounter an unsupported region, it would be tempting to simply place such a region in front of the path by generating the appropriate track geometry instead of wrestling with the constraints imposed by the existing track geometry. In fact, we can consistently dene such a function, which we refer to as the cutout function.

Figure 4.10

(a)

illustrates a polygonal part in the plane. We begin by selecting

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Figure 4.9: Apparent inversion of a support transition boundary on the surface of the CS.

4.2: Design Functions

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(a) (b) (c)

Figure 4.10: Generating a support track cutout.

an (xy) conguration of the part where we wish it to transition from a supported to an unsupported state. Next, we specify the direction in which we want the part to fall by dening an oriented line through the part

cg

. This line represents the axis about which the part will rotate out of the (xy) plane at the given position.

To generate the track contour that will allow this out-of-plane motion of the part we generate a shaped cutout from the track corresponding to the part contour on the unsupported side of the fall axis, as shown in Figure 4.10

(c)

. We can think of the cutout function as using the part as a sort of \can opener" by rotating the part about the fall axis to sweep out a portion of the track

(b)

.⁶

The above description of the cutout function deals exclusively with the part and track geometries, whereas we have been focusing our attention on the motion constraint representations to describe function. What does the cutout function look like in terms of the support transition boundaries on the CS in conguration space?

Figure 4.11 shows the result of adding a cutout at an (xy) point on a CS facet. In the ideal case, the unsupported region generated by the cutout would appear simply as a single point or small region on the contact facet surface at the desired (xy) conguration. In reality, however, a number of additional non-local unsupported regions are also introduced on the CS surface. These additional unsupported regions arise from the simple fact that a hole generated for a part to drop through in one orientation does not, in general, prevent parts in other orientations from falling

6As described in Section 5.7, the implemented cutout function generates a convex approximation to the cutout contour described here. In addition, to preserve the genus-zero topology of the supporting track polygon, the cutout is connected to the outer track contour where necessary.

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Chapter 4: Design through as well. The location, number, size, and shape of these additional regions are determined by the location and shape of the track cutout contour, which in turn is determined by the chosen (xy) conguration and fall axis orientation for the cutout. As expected, the topological modications created by the cutout function add complexity as well as exibility to the design task by introducing a number of new polygon vertices. Controlling this additional complexity while at the same time taking advantage of the ability to introduce an unsupported region at any arbitrary location on the surface of the CS is the major challenge of utilizing the cutout function as a design tool.

We noted in Section 4.1.1 that the shape created by sweeping out a part along a desired path in conguration space satised the necessary conditions, but not the sucient conditions for the desired motion constraints along that path. Does the cutout function above satisfy both the necessary and sucient conditions for produc-ing the desired motion constraints, and if so, why? We recall from Section 4.1.1 that the problem with the swept shape generation approach was the fact that the volume swept out by the xed object on one portion of the specied path could eliminate shape features that were necessary to constrain the object along another portion of the path, as illustrated in Figures 4.2 and 4.3. If we imagine the rotation of the part out of the (xy) plane as a path in a higher dimensional conguration space, then only the point corresponding to the beginning of the path is contained within the lower dimensional (xy) slice of that space. As a result, the support constraints provided by the track at the (xy) position where the out-of-plane path begins are not aected by any other point along the remainder of that path. By this argument, the shape generated by the swept out-of-plane motion provides the intended motion constraints, i.e. support constraints, at the selected (xy) point. We are, how-ever, ignoring an important additional factor { we neglect the remainder of the part motion outside the (xy) plane. In particular, we could imagine a case where the half of the part on the supported side of the fall axis is wider than the unsupported half used to generate the cutout contour. In this case, a part may rotate out of the plane as desired, but become caught in the cutout rather than falling o the track.

In this research we are assuming that the parts are thin enough compared to their (xy) dimension that we could, if necessary, cut an additional narrow slot along the fall axis whose length exceeded the widest cross-section of the part. This slot would be swept out by the part as it reached a vertical orientation and then slid downward in the ^;z direction. Although an inelegant solution, such a slot would allow the part to slide o the track while at the same time yet be narrow enough so as not to compromise the support characteristics of parts in other orientations in the plane.

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Figure 4.11: Multiple unsupported regions generated on the CS by the cutout func-tion applied to a single (xy) congurafunc-tion (indicated by the \P" in the top gure).

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