How the work of Bourdieu informs of classed self/identity

From a theoretical point of view, this result has the following interest. Con-sider the one-step compliant motion planning problem in 3D amidst precisely known polyhedral obstacles. This problem may be addressed via 3D backprojections in 3

[CR] have shown that deciding containment in such a D backpro'ection is NP-hard.

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In particular, such backprojections can have an exponential number of faces. How-ever, in the previous theorem we demonstrated a special class of 3D backprojections that have only O(n 4) faces, along with an efficient algorithm for deciding contain-ment. This special class of backprojections arises in the presence of model error.

Specifically, they arise when C is R2, J is one-dimensional, and no motion 'is per-mitted across J. In this case, the non-holonomic constraints that keep the robot ithin one slice ess ent'ally disallow the kind of fanning out and branching that [CR] discovered in R3. Thus, our polynomial-time algorithm identifies a tractable subclass of the 3D motion planning problem with uncertainty. This subclass is also interesting in that it arises naturally 'in planning with model uncertainty.

6.4.2 1[ssues in the Critical Slice Method

The critical slice method represents a theoretical algorithm. It has not been implemented in LIMITED. It was described here to give some characterization for bounds on planning with model error. In particular, it gives a precise, ombina-torial description for the 3D backprojection in R2 x S, and an exact algorithm for deciding containment. The containment algorithm drectly addresses the ques-tion of planning guaranteed strategies since a backchaining preimage planner can be constructed by approximating preimages using backprojections. The termina-tion conditermina-tion for such a planner 'is when the start re 'on is contained within a backprojection.

Most 'important, the ctical slice method attempts to put the slice techniques used 'in LIMITED on a firm mathematical footing. 4 It provides a principled way-a specific method-for choosing wich slices to consider, a bound on how many slices are required, and a conservative algorithm for deciding containment.

Much work remains however:

0 We have only addressed deciding the containment problem in a precise com-binatorial fashion. Generalize to computing set-differences and to deciding their distinguishability-that is, deciding G vs. H distinguishability-using the critical slice approach.

0 J is one dimensional 'in our discussion. Generalize the critical slice method to multidimensional model error.

0 This analysis addresses the complexity of verifying an EDR strategy, but does not speak to the complexity of the search. What 'is the complexity of finding a strategy or determining that none exists? This issue will be attacked in a later section, by developing a combinatorial description of the non-directional backpro'ection.

4Note that slice methods have been studied 'in other domains. See, for example, Lozano-Perez, Schwartz and Yap, Erdmann].

Derive bounds on deciding containment after relaxing the no-pushing restric-tion and allowing morestric-tion across J.

Let us say a few words about the last point. Suppose now that can rotate passively when pushed. Hence motion across J is possible, and projection regions must be propagated across slices. For example, a forward projection can begin in free-space in one slice, contact an obstacle edge generated by B, rotate across J into another slice, and fly off the edge into free-space in that slice. Hence forward projections must be propagated across slices. This process was described above in sec. 61. The obvious question is: What 'is the complexity of propagating the projection regions across slices? The complexity of one step of the propagation is not difficult to derive. For example, consider the forward projection. There are O(n) obstacle edges in the forward projection 'in a planar slice. For each edge, a constant tme quasi-static analysis is performed to determine whether pushing against that edge can cause rotation of B, that is, motion across J. See fig. 38.

If so the forward pr 'ection must be propagated along that algebraic surface into an adjacent slice. This can result in a propagated tart region of size O(n) in the ad'acent slice. This start region is used to compute a new forward projection 'in that

planar slice. See figs. 39 and 40. This propagated forward projection must then be unioned with any other forward projections within that slice. See fig. 43, which

is a detail of fig. 42. When does the propagation process terminate? A correct termination condition is: Terminate propagation when any propagated start region lies within an existing forward projection.

Now while the complexity of each of these steps is known, it 'is not clear how long it takes for the propagation process to terminate. In particular, results of [CR]

suggest that 3D forward projections may even have exponential size.' Experimental evidence the backprojector of [Erdmann]-concurs. Furthermore, when propaga-tion is permitted, more slices may be required. For example, it is conceivable that a path within the forward projection may break contact and fly off into a slice which is between the chosen critical values. In other words, propagation may increase the number of critical values. The additional critical values can occur as follows. The plane sweep algorithm 'is only correct when the velocity cone is smaller than the friction cone on any edge (see chapter VI). Hence we will assume it 'is convex. Then contact can be broken when the inner product of an extremal vector 'in the velocity cone by an outward-facing edge normal 'is positive. Hence the zero-crossings of this dot-product are potentially critical values; there are O(n) such values. While this is a start the complexity of computing projections when pushing can cause motion

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across J requires further study.

5 [CR] provide an exponential lower bound for the size of the forward projection in 3 am'dst polyhedral obstacles. It remains to determine the applicability of their proof in the non-holonomic (model error) case.

This completes theinformal dscussion of the one-step EDR planner in LIMITED.

Later in the thesis we will dscuss the details of the plane-sweep algorithm and how LIMITED implements the EDR theory to compute multi-step strategies. In the next section, we will discuss a number of theoretical and practical 'issues relating to the construction and implementation of the one-step EDR planning algorithms in LIMITED.

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We show four slices of generalized configuration space, at a = 60,120, and 180. The projections take into account possible rotation of under pushing.

Here the weak backprojections across slices are shown. The "spikes' represent regions from which jamming of the gears must occur.

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