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Chapter 2: Representing Function tween planar and non-planar motions. Finally, after characterizing the above set of motion constraints, we examined various means by which those constraints could be combined within the conguration space representation.The purpose of this chapter was to develop the representations that will serve as the foundation upon which we may build a set of tools that will allow us to perform both analysis and design of functionally useful shapes. To make these representations and visualization techniques more concrete, in the next chapter we will introduce a set of four examples: peg-in-hole assembly, vibratory bowl feeders, assembly pallets and xtures, and another vibratory feeder known as APOS. These examples have been chosen because they span the set of constraint representations developed here, as well as to highlight similarities among and dierences between the various forms of functional constraints.
Visualization and Application Domains
Chapter 3
In this chapter we will examine four application domains introduced in Figure 1.8:
compliant assembly, vibratory bowl feeders, assembly xtures, and the APOS vibra-tory feeding system. We will use the motion constraint representations developed in the previous chapter to visualize, reason about and analyze the functional charac-teristics of examples from each of these domains in terms of motion constraints.
In Section 2.3.1 we referred to the similarities between the surface of a CS facet and a \real" surface that produced reaction forces and torques in response to applied forces. In Section 2.3.2 we referred to features on the surface of the CS using terms such as valleys, ridges, and peaks that convey images of multiple features combining to form what amounts to a landscape in conguration space. The intent of this visual imagery is to convey an intuitive feel for some of the structure imbedded in the CS and how these constraints act to guide motions of a point representing the motion of a physical object.
A point in conguration space, whose (xy) components correspond to the posi-tion of the reference point of, and the component to the orientation of, the moving object can be thought of as the point of action through which external and reaction forces act to constrain the motion of the object. All interactions between shapes of both the moving and stationary objects are combined so as to be local to this point. If we imagine that point as a ball bouncing or sliding across the surface of the CS constraints, then we have a powerful metaphor with which to visualize how con-straints interact. For example, we have discussed energy in terms of non-kinematic constraints, or bounds, that determine where a point, or ball, may travel in the presence of an externally applied motive force such as weight due to gravity. This energy imposes on the conguration space a sense of up and down that immediately implies a sense of where the ball will and will not go based on its interaction with the CS surface. The curved surfaces of individual CS facets guide the motion of the ball along curved trajectories imposed by the ever-changing surface normal along the
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Chapter 3: Visualization and Application Domains facet, while valleys between CS facets guide and constrain the ball along their oor.Intuition regarding the behavior of rolling and bouncing balls can prove quite useful in the more abstract domains of mathematical constraint surfaces and conguration space.
With these visual metaphors for constraint surfaces in mind, we return to the question of what to do with them{ how do we represent function? First of all, forward projections, like simulations, provide a visual verication of motion in the presence of motion constraints. With the conguration space representation, however, we have in addition to a verication of where the motionwill go a sense of where the motion might have gone, or might go, as a result of perturbations to one or more system parameters. For example, in examining the motion of a discrete path across a facet surface, we also have in the surface of the facet itself the family of potential motions sharing the same contact { we know at a glance where else that motion could and could not go. As another example, consider the case of an energy-bounded forward projection for a dropped object intersecting the surface of the CS. Were we to expand the range of the cone encompassing reachable states, say by increasing the value of the coecient of restitution e from 0:8 to 0:9 through a change of materials, then we would immediately be able to determine what new regions of conguration space would now be reachable and what new constraints might interact with the new motions. This sort of \what if" visualization of dierent scenarios is critical for determining the robustness of a system as well as determining what changes or new features might be required of or desirable in a new design.1
There is a considerable amount of geometric detail contained within the motion constraints as shown in Figure 2.4, in some ways perhaps too much detail. Since the CS is a mathematicallyprecise embodimentof the complete set of motion constraints generated by two interacting objects, all of the corresponding kinematic constraint information is available. The question becomes, then, how can we recognize and abstract what we need from what is not necessary? Of course, what is and what isn't necessary depends on the application in question. If we wish to verify a motion or set of motions, as described above, then the detailed quantitative information contained in the CS may be necessary. If, on the other hand, our goal is to abstract functional characteristics for a class of constraints or class of motions across dierent specic examples, then such detail could prove unnecessary and even distracting.
For this purpose, we will develop on functional metaphors that seek to describe, in qualitative terms, the topology of the motion constraints (both kinematic and non-kinematic) that best characterize a particular function. We should stress that the role of these metaphors will be to complement, rather than replace, the motion
1We refer to the visualization of constraints for a particular set of parameters asstatic constraint visualization. Another form of visualization to be discussed in the next chapter has to do with visualizing the couplingbetween constraints as parameters are varied, which we will refer to as dynamic constraint visualization.