There are a number of other mappings of constraints to conguration space that, depending on the application, can be useful in representing function in terms of motion. The following three constraint mappings are due to Brost 13] and were
15One caveat to this approach has to do with the fact that interactions with the CS surface that correspond to contacts among slices at dierent heights would impose out of plane torques to the object that would be outside the scope of the (xy ) representation. Special care would have to be taken to ensure that the distribution of forces among the slices in contact was consistent.
Techniques analogous to the support transition boundaries of Section 2.4.3 might also be useful in representing the eects of these torques.
2.5: Mapping Constraints into Motion Space
63
X Y
Figure 2.14: Support transition boundaries intersected with the surface of the CS.
64
Chapter 2: Representing Function applied to the tasks of analyzing and planning pushing and dropping motions of planar polygons:
Sticking regions due to friction.
Given a constant applied force, the contact congurations on the surface of the CS that would result in no motion due to friction are identied and marked so that they may be avoided in constructing backprojections from a goal conguration.
State transition cones.
Given an applied force, local forward projections may be generated and displayed at discrete points on the surface of the CS to indicate the \ow" eld of constrained motions. This representation is partic-ularly useful in capturing and representing uncertainty in motion parameters in terms of cones of possible motions from each discrete point.
Arbitrary Constraints.
Some object features should be avoided during a manipulationoperation because they are particularly delicate, or maybe coated with an adhesive or other material which should not be brought into contact with other objects except in a certain predened conguration. The constraint facets corresponding to contacts with these features are labeled as o limits. Someother potentially useful constraint mappings derivedfrom the energy bounded forward projections discussed earlier include:
Topographic potential energy map.
The plane corresponding to an e = 1 bounded energy forward projection represents a constraint for a single energy level. It is not dicult to imagine generating curves on the CS surface corre-sponding to slices of the CS at dierent energy levels. Such a family of contours would be equivalent to a topographical map of the CS surface, and would pro-vide a global picture of the \hills and valleys" in the set of motion constraints.The \valleys" in particular are interesting to us since they correspond to local minima in which the moving object could come to rest for certain motions.
CS intersections with vibratory impact forward projection bound-aries.
Similar to the topographical boundaries in the previous example, curves representing the set of congurations where the vibratory forward projection boundaries intersect the CS surface serve to partition the CS surface into reach-able and unreachreach-able congurations for a given set of initial conditions. As the amplitude or frequency of vibration is varied, these boundaries would give a global picture of the changes in system behavior accompanying changes in these parameters. As in the case of the support constraint boundaries, the resulting intersection curves would not obscure surface details of the CS.The above representations in (xy) conguration space capture motion con-straints in terms of geometric structures that include parametric surfaces, planes,
2.6: Summary
65
and space curves. These structures are useful both as computationally accessible constraint representations suitable for manipulation by algorithms, and as visual representations for display and interpretation by humans. This second application is particularly attractive since we will need to understand the nature of the constraints we wish to impose before we can attempt to develop algorithms that generate them automatically.
2.6 Summary
In this chapter we explored the representation of function in terms of motion con-straints. We examined what we mean when we speak of functional constraints on motion, and underscored the need for a mathematically precise representation for these constraints within conguration space. We described a zero-velocity state space (conguration space) to capture object motions, and in so doing focus our attention on motion instead of shape as a language in which to describe the func-tionality of object interactions. Since, in general, the conguration space can be quite large, we limited our discussion to objects whose motions were constrained to lie in a plane. The result was a three dimensional conguration space whose axes are (xy). We considered two broad classes of motion constraints: kinematic and non-kinematic. Kinematic motion constraints arise from interactions between ob-ject shapes, and may take the form of individual contact surfaces or supersets of contact constraints in conguration space. Non-kinematic motion constraints arise from the forces derived from the mechanics of contact, as well as externally applied forces and gravity elds. The mechanics of contact we considered included sliding friction, represented geometrically as the friction cone in conguration space, as well as elastic and inelastic collisions between objects. From these mechanics we were able to construct forward projections of motions that further partitioned congura-tion space into regions of reachable and unreachable states. Two kinds of forward projection that we considered in detail were the exact integration of motions for the cases where we had a detailed model of the dynamics, and bounded energy motion constraints for those cases where the dynamics could not easily be characterized. In both cases, we likened forward projections to a timeless superset of simulations of object motions under the specied constraints. A third special case of forward pro-jection we considered was the support constraint, the stability of which was viewed as a constraint on the potential energy in a gravity eld of the moving object's center of gravity while resting on a at surface. Those regions of conguration space where the potential energy of the object could be reduced by means of a rotation out of the plane were deemed to be unsupported. The support constraint was also given as an example of a simplication whereby constraints on higher dimensional motions could be represented in a lower dimensional conguration space as transitions
be-66
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.