(+)
Potentially a great deal of exibility in exploring dierent design possibilities encompassing many classes of designs (i.e. many design spaces).(+)
The ability to \jump" between designs that are distant in terms of design parameters, or are not contained within the same design space.(-)
Dicult (often impossible) to perform consistently in the context of inverting motion constraints.(-)
A very limitedset of viable design techniques exists for a few application domains.(-)
Adding and removing design variables, or otherwise changing the topology of the design space, can make it dicult to close in on a viable design. Jumping around within a design space may cause us to miss potentially valid designs, and generally makesa methodical search of possible designs dicult to perform.Adding design variables invariably increases the dimension of the design space, also adding to the overall complexity of the design task.
As we can see, the strengths and weaknesses of parametric and topological modi-cations are more or less complementary in nature. Basically, we would like as much exibility as possible while at the same time control the complexity that we must deal with. This suggests that a good design strategy would be to combine the best of both approaches wherever possible. Specically, we will want to take advantage of the exibility of topological modications early on in a design to \jump" among possible nominal design topologies until we nd what appears to be a promising class of designs. Then, when we are near what we hope is feasible design, we may more methodically explore the local parameter space for possible solutions. Section 4.4.5 describes one methodology appropriate for designing vibratory bowl feeders.
116
Chapter 4: Design4.1.5 Interactive vs. Automated Design
In our discussion so far we have assumed that the modications necessary to generate a design are to be carried out by a human designer working interactively with a rep-resentation for visualizing, and set of tools for manipulating, the motion constraints that describe function. What about automated design? We are able to automatically generate representations of function in terms of motion constraints given geometry and other parameters as described in Chapter 2 and detailed in Chapter 5. These constraints are mathematically precise, and because they are computer generated they are also computationally accessible. Therefore, it would seem that we should be able to manipulate these representations automatically using such tools as appar-ent inversion in order to perform design.
The kinematic and dynamic constraints that we consider explicitly in the mo-tion constraint representamo-tion are only part of a much larger set of potential design constraints, i.e. cost, machinability, maintainability, etc.. Automating these por-tions of the design task will almost certainly result in the generation of many useless designs.4 The human designer, on the other hand, can keep more of constraints in mind while using this tool. In this research we focus on the interactive/iterative design paradigm because it provides us with more exibility and is generally a more tractable approach to design. We will therefore focus on the role of the computer as a tool for automating the task of generating and displaying the explicit represen-tations of motion constraints, and allowing us to interactively manipulate both the constraints and the parameters dening them using such tools as apparent inversion.
This emphasis on interactive design will also require us to produce an implementa-tion that is fast and ecient in computing the necessary representaimplementa-tions. Later, as we gain insight into the relationships between parameters and constraints, and the more general relationship between function and motion constraints, these tools will also be useful in developing the additional representations and algorithms necessary to support automated and semi-automated design, which we will briey discuss in Section 6.2.3.
4.2 Design Functions
Our goal is to have consistent modications of motion constraints mapped into the appropriate parametric modications. Recalling the space of design parameters dis-cussed in Section 4.1.2, variational modications to design parameters may be viewed as a path between states in design space. In this context, we view design as one or more functions mapping desired changes to motion constraints, expressed as paths
4By the same token, a system that automatically generates large sets of design alternatives, even though many are infeasible for one reason or another, may have value as an aid to human designers.
See Ulrich 75] for examples of this approach.
4.2: Design Functions
117
imposed by a designer on a representation of motion constraint features, into para-metric changes represented as a path in design space. More specically,design func-tions are mappings from user inputs that (i) select, and (ii) modify the appropriate design parameters in the context of motion constraint modications.
Although the above description implies that design functions are intended purely for modication of existing parameters, we may expand the notion of design func-tions to include those topological modicafunc-tions to motion constraints that may be performed consistently. In this section we will examine examples of both paramet-ric and topological design functions. We will rst consider parametparamet-ric functions to implement apparent inversion for two forms of motion constraint representations:
contact facets and planar support constraints. We will then describe a topological design operator for generating classes of support polygon geometries and discuss why it may be implemented consistently.
4.2.1 Apparent Inversion of Motion Constraints
There are three distinct functional components of apparent inversion from motion constraints:
1.
Parameter selection:
Select parameter(s) that are to be varied from the constraint representations.2.
Input mapping:
Map the designer's intended modication to the selected parameters into a path in (xy) conguration space.53.
Parametric mapping:
Map the (xy) path to a path in design space, specif-ically to the selected parameter(s), by inverting the corresponding constraint expressions.More precisely, these operations may be expressed in terms of the following functions:
F
selectPCS(xy) ~FA=B!s (4:3)
where PCS(xy) is a point selected from the surface of the facet ~FA=B, and s is a parameter or set of parameters from the facet's describing equation. For the input mapping, we have:
F
input(Pinput)!PCS(xy) (4:4)
5Input mapping is an artifact of the type of input device used. Typically the input will in the form of an oset in the screen coordinates of a cursor via a mouse. It is necessary to map such a two variable input into a motion in the three-dimensional conguration space. If a three d.o.f.
input device is used, then the input mapping function may be unnecessary.
118
Chapter 4: Design where Pinput is an input path in whatever space the designer interacts with the constraint representations, and PCS(xy) is the corresponding path in conguration space. And nally we have: which applies the input path in conguration space to the selected parameters, which are assumed here to consist of polygon vertices.The above functions are designed to operate on parameters describing polygon geometry in terms of (xy) vertices, and motion constraints represented in (xy) conguration space. In this research we have developed and implemented specic apparent inversion functions for two constraint representations: the contact facets of the CS and the support transition boundaries on the surface of the CS. These parameters and representations were chosen because of their prominence in deter-mining the function of vibratory bowl feeders and compliant assemblies described in the previous chapter. Similar functions could also be generated to operate on other design parameters, such as dynamics and material properties, as well as ma-nipulating parameters within the context of other constraint representations, such as the forward projections including discrete paths and bounded energy regions. Such additional design functions have not been detailed or implemented in this report.