First, the critical points of the cage representation in the hand configuration space appear as the critical points of the finger distance function in the contact space. The critical points of the toe distance function are shown to be the same as the critical points of the cage.
Motivation
Review of Existing Literature
Grasping
Caging
Some works consider lattice regions created by constraining the relative position of the fingers in higher dimensional regions. Given two initial finger placements on the boundary of the polygon, they presented an algorithm which finds 2D regions in which the third finger can be placed.
Caging Basics
With these methods, it is difficult to ensure that the sampling methods will accurately represent the underlying geometry of the problem. This motivates the definition of hand shape, which is the position of the fingers relative to each other.
Current Uses and Limitations of Caging
Most cage algorithms involve two-dimensional objects and/or two pointed (or disc) fingers. Automatic attachment of quasi-two-dimensional objects (such as those created by a laser cutter or water jet) can be a useful application of two-dimensional algorithms.
Thesis Organization and Contributions
This method considers topology of the configuration space of the hand in terms of level sets of an interfinger distance function. Second, it provides a detailed analysis of the properties of the interfinger distance function in contact space.
Preliminaries and Problem Definition
Initial cage set - the largest set of cages from which the fingers are guaranteed to return to the initial immobilization grip by holding the object in the cage during the toe tightening process. Maximum cage set - the largest cage set containing the initial cage set such that the fingers can be squeezed into a limited number of possible immobilizing grips while keeping the object in the cage during the finger squeezing process.
Contact Space Formulation of Caging
Starting from the immobilizing grip, pq0, σ0q, the value of σ at the first piercing point can be seen as the minimum value, σ1, such that there exists a sublevel path aσ1 which starts at pq0, σ0q and ends at an escape point or reaches a neighboring immobilizer. chap. The value of σ at the last puncture point, denoted σesc, can be seen as the minimum value of σ, so there is a sublevel path aσesc between pq0, σ0q and an escape point.
Contact Space Representation of the Caging Sets
Two-finger Equilibrium Grasps in Contact Space
Tangentp1ps1qat the vertex can be any vector orthogonal to the vectors of the generalized contact normal at the vertex. As σ increases in intervals rσ1, σ1 s, two locally distinct connected components of the sublevel setF¤σ meet at the puncture point and become a single component forσ¥σ1.
The Inter-finger Distance Function in Contact Space
Rectangular Decomposition of Contact Space
The critical points of dps1, s2qinU, excluding the diagonal∆ and using the endpoints of the degenerate critical points connected to the parallel edges, are isolated points on the bounding lines of the rectangles of the contact space. Thus, excluding the diagonal ∆, all critical points of dps1, s2q include at least one vertex of B, and therefore must lie on the bounding lines of the rectangles of the contact space.
The Contact Space Caging Graph
The Caging Graph
Proof The isolated critical points of dps1, s2qinU lie on bounding lines of the rectangles of the contact space according to Proposition 2.5.5. These critical points span line segments with endpoints on bounding lines in the contact space rectangles.
The Augmented Caging Graph
The tunnel curve continues with a pressure movement along the object's current sides, until the unique local minimum of dps1, s2q is reached along the two object sides (Figure 2.10). The tunnel curve edge depicted in Figure 2.11 connects a non-feasible local minimum of dps1, s2q to another node of G, located at an immobilizing grip of B.
The Contact Space Algorithm
The lowest indexed local maximum in X associated with the puncture represents the puncture point grip of the local array of cages surrounding the immobilization grip. The highest indexed local maximum associated with the piercing in X represents the maximum grip of the piercing point of the maximal set of cages through which object B can escape to infinity.
Algorithm Walk Through
This is because once the search algorithm detects a saddle, it continues to search for lower value nodes in the unexplored part of the previously partitioned sublevel set of the augmented trellis graph, GT. Finally, all other local maxima associated with the drill represent the drill point grips associated with all intermediate lattice groups surrounding the immobilizing grip.
Graphical Depiction of Caging Set as Two Capture Regions
The upper range of finger capture is limited by γ2 and γ11, as well as by the part of the object boundary that penetrates this region. Any two-finger placement in the shaded grasp regions with finger opening σ¤σ1 will retreat to an immobilizing grasp at the center of the object.
Caging Set Computational Example
A plot of dps1, s2q at the nodes of X developed by the algorithm is shown in Figure 2.18. The neighboring immobilizing grips and the series of drilling grips calculated by the algorithm are shown in physical space in Figure 2.16.
Extensions of the Caging Algorithm
Critical catches in the closed list X are shown with the same symbols as in Figure 2.16. Starting from an immobilizing grip inside the object hole, an increasing finger opening will eventually reach a piercing grip beyond which the two fingers can join inside the hole (Figure 2.19(a)).
Summary and Extensions
However, proper computation of the cage sets requires augmentation of U with tunnel curves, which form additional edges of the cage graph G. However, contact space can potentially be used to define a generalized cage graph that computes the cage sets associated with two disc fingers.
Preliminaries and Problem Statement
The initial cage set - the largest cage set from which the fingers are guaranteed to return to the initial immobilizing grip while holding the object caged during the pressing process. The intermediate cage set—any cage set containing the initial cage set such that all fingers end up at a finite number of possible immobilizing grips while holding the object during a pinch.
Contact-Space Reformulation of Caging
Eventually the distance between fingers reaches a critical value at which the object can escape the cage formed by the two fingers. The standard definition of an escape is the ability to move the two-fingered hand with a fixed finger opening arbitrarily far from the object.
The Caging Graph
Location of Important Grasps
Note that because contacts that both lie on faces of the polyhedron do not form frictionless equilibrium grips (since parallel faces are excluded), all these points lie on the boundaries of the contact space polycora. Nodes: The nodes of G correspond to all critical points of the distance function between the fingers in U.
The Caging Algorithm
Analysis of the Closed List
For such objects, we need to perform a rigorous characterization of the critical points of dpsq PU, augmenting G with a (usually small) number of additional edges. While the node representing this puncture is a saddle of the functiondpsqin contact space,U, it appears as a local maximum inX.
An Example
The lowest indexed puncture-related local maximum in X represents the puncture point associated with the local cage set. The highest indexed puncture-related local maximum in X represents the maximum puncture point associated with the maximum cage set.
Characterization of Nodes of G
This happens because once the search algorithm discovers a saddle, it continues to search for lower-valued nodes in the unexplored part of the previously decoupled sublevel set of the cage graph, G. The following characterizations of immobilizing and leaky grasp will be used to explain the above to analyze four cases.
A Catalog of Immobilizing and Puncture Grasps
Each contact represents a local minimum in the inter-finger distance when the opposing finger is fixed. Each contact represents a saddle in the distance between fingers when the opposite finger is fixed.
Sublevel Equivalence
- Sublevel Equivalence of U and G
- The Sublevel Equivalence of U and F
- Sublevel Equivalence at False Immobilizing Grasps
- Sublevel Equivalence at False Puncture Grasps
- Tunnel Curve Construction
If neither condition holds, the topology of sublevel sets of U does not change atq0. Under misunderstanding, the changes in the topology of F¤c and U¤c are qualitatively different, but the sub-level equivalence remains.
Summary
Motivational Example
Thus, allowing the hand shape to vary from σa to σb does not yield a cage on B. Also shown is a decomposition of free space around a convex polygon, B, dividing it into regions, Ri. Restricting the hand shape to this region of S will prevent the hand from leaving the initial triad,T2,3,5.
Robust Caging Definition
The black crowd is the result of immobilizing grips where all three fingers rest on the edges of B.
Single Triad Caging
Thus, the escape region denoted by E associated with Tp,q,r is the union of six individual transition escape regions Tp,q,r. Similarly, the cage region labeled C associated with Tp,q,r is the intersection of six single transition cage regions of Tp,q,r.
Multi-Triad Caging
Thus, hand shapes in the escape area associated with this transition do not allow the hand to escape from U. The cage region associated with justTpqris consists of handshape region Hpqralong with the constraint that the initial placement lies in Ipqr.
Divisions of Shape Space
Puncture Manifolds
Four of the equilibria that generate these manifolds are shown by dashed lines in Fig. Corresponding points in S are shown as colored circles in fig. 4.7. variations of q), but for some of the values of σnearσ, fi is unable to cross bj for any values of q.
Immobilizing Manifolds
However, for perturbations that reduce any (or all) of the toe distances, f3 will not be able to exceed bj. Thus, this configuration straddles the boundary between the transition and cage single rescue regions in S.
Other Grasps
This hand shape also lies in the interior of the single transition escape region, Et, associated with f3 crossing bj. Similar arguments (not presented here) can be made about other types of grips in which two fingers touch the object, such as grips where both fingers touch the edges of B, or grips where one finger lies at a vertex of B, while the other on lies an edge but not at the perpendicular projection of the vertex on that edge.
A Test of Caging Status
Test for Feasibility
If this distance is non-negative, then there exist finger placements f1 and f2 such that f3 lies in free space, and thus a feasible placement for hand shape σ exists. Since a, b, etc are all functions of the geometry vanei,ej,ek, andσ, this can be easily evaluated for any value ofσ.
Test for Escape
Example
This provides tests to determine if a point is a cage or escape point, and whether it is feasible or not feasible. The cage area associated with these triads is quite complicated and is shown in Fig.
Computational Complexity
Conclusion
In addition, methods to find simple geometric regions within the robust cage region (eg, the largest inscribed sphere) would be useful. The two-finger, three-dimensional algorithm is an extension of the two-dimensional algorithm to three dimensions.
Opportunities for Future Work
When σ varies in the open interval between adjacent critical values of π, the sublevel sets F¤c tpq, σq PF : πpq, σq ¤quares topologically equivalent (homeomorphic) to each other. Let eπin be a critical point F.1 The type of a critical point is characterized by the behavior.
