The Augmented Caging Graph

2.6 The Contact Space Caging Graph

2.6.2 The Augmented Caging Graph

The following corollary follows directly from Theorem 2.6.4.

Corollary 2.6.5 The caging graphGhas two properties:

1. Each connected component of U¤c discretizes into a connected subgraph of G containing all critical points of dps1, s2qlying in the connected component ofU¤c and no others.

2. Every pair of sublevel sets ofdps1, s2qinU that meet at a saddle point,v, discretizes into two subgraphs of Gthat meet at the corresponding node,v, ofG.

The tunnel curves lie in the free c-space, F, with their endpoints located on the submanifold S. Each tunnel curve starts at a non-immobilizing local minimum ofπin S, follows aσ-decreasing path inF until reaching S at a lowerσ value, and then continues withinS until reaching a point corresponding to a node ofG. The tunnel curves are constructed as follows.

Tunnel curves construction: Let pq₀¹, σ¹₀qbe a local minimum of the functionπpq, σq σ in S, which is notan immobilizing grasp ofB. For instance, Figure 2.9 depicts such a non-feasible local minimum, where one finger is located at a vertex while the other finger is located at an interior point of an opposing edge of B. Starting atpq₀¹, σ₀¹q, at least one finger will be able to locally move away from B in a straight line toward the other finger (see proof of Theorem 2.6.7). Retract this finger while holding the other finger fixed on the object’s boundary, until the finger hits a new edge ofB(if the retracting finger hits the stationary finger, this gives an escape point discussed below). At this stage both fingers contact the object. Slide both fingers simultaneously along their respective object edges while minimizing σ (i.e. squeeze both fingers), until reaching the unique minimum of the inter-finger distance along the current object edges. This defines the tunnel-curve’s other endpoint.

Figure 2.10 shows the tunnel curve, which starts with a single finger retracting and continues with a squeezing of both fingers to a local minimum. If the fingers meet during the closing process, their contact space point is located on the diagonal, ∆, where σ 0. In this case the current contact- space rectangle contains two escape nodes at its corners. Set the tunnel curve’s other endpoint at the closest escape node along the diagonal ∆.

Based on the construction procedure, the tunnel curves are defined as follows.

Definition 2.6.6 Let S be the double-contact submanifold parametrized by contact space U. A tunnel curvestarts at a non-immobilizing local minimum ofπpq, σq σinS, moves with decreasing σin the free c-space,F, then continues withinS to the endpoint located at a node ofGas described above.

Remark: The contact space scheme [25] also augments thecrawling graph with special transition edges analogous to the tunnel curves. These special transition edges are added when the two fingers reach along the object’s boundary a corner point ofB, where one finger can break away from the boundary towards the opposing finger. While the number of such transition edges is significantly larger than the number of tunnel curves, these edges offer a means to augment the crawling graph with curves that ensure sub-level equivalence with the ambient free c-spaceF.

LetT be set of all tunnel curves in F. The following theorem asserts that the union SYT (and hence the unionUYT) is sublevel equivalent to the free c-spaceF.

Theorem 2.6.7 Let pq₀, σ₀q be an immobilizing grasp of B, and let pq_esc, σ_escq be the maximal puncture point associated with pq0, σ0q. The union of the double-contact submanifold S with the

Figure 2.9: A polygonal objectBwith a handle-like feature. Selected grasps are shown.

Figure 2.10: Contact space contours of dps₁, s₂q and selected grasps for the object of Figure 2.9.

Portions ofU¤dare shown shaded; the two disjoint regions (which are connected in the free c-space) are problematic. The dotted line represents atunnel curvewhich does not lie in contact spaceU.

Figure 2.11: A portion of the augmented caging graph for the object of Figure 2.9, corresponding to the contact space region shown in Figure 2.10. The tunnel curve edge is depicted as a thick red line.

tunnel curves,SYT, issublevel equivalentto the connected component ofF¤σcontainingpq0, σ0q forσPrσ0, σescs.

A proof of Theorem 2.6.7 appears in the appendix. Each tunnel curve starts at a local minimum of dps1, s2q in U which is a non-feasible equilibrium grasp of B. This start point is a node of G.

The tunnel curve then moves in the free c-spaceF while monotonically decreasing the inter-finger distance, until establishing a new two-finger grasp at a lower finger opening. The tunnel curve continues with a squeezing motion along the object’s current edges, until reaching the unique local minimum ofdps1, s2qalong the two object edges (Figure 2.10). The resulting endpoint is also a node ofG. Each tunnel curve can therefore be thought of as ahandleattached to contact spaceU at two points which are nodes of the caging graphG. This interpretation leads to the following definition of theaugmented caging graph.

Definition 2.6.8 Let G be the caging graph of a polygonal object B. The augmented caging graph,denoted GT, is the graphG augmented with edges corresponding to the tunnel curves inF. Figure 2.11 shows a portion of the augmented caging graph, G_T, for the object of Figure 2.9, corresponding to the contact space region shown in Figure 2.10. The tunnel curve edge depicted in Figure 2.11 connects a non-feasible local minimum ofdps₁, s₂qwith another node of G, located at an immobilizing grasp of B. The original caging graph, G, is sublevel equivalent to contact space U (and hence to the double-contact submanifold S) according to Theorem 2.6.4. The unionSYT

is sublevel equivalent to the free c-space F according to Theorem 2.6.7. Since G_T GYT, the augmented caging graphG_T is sublevel equivalent to the free c-spaceF.