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# Two Finger Caging of Concave Polygon - PowerPoint PPT Presentation

Two Finger Caging of Concave Polygon. Peam Pipattanasomporn Advisor: Attawith Sudsang. Outline. Objectives & Basic Concepts Maximal Cage Problem Minimal Cage Problem Discussion & Conclusion. Objectives & Basic Concepts. Definition of Caging.

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### Two Finger Caging of Concave Polygon

Peam Pipattanasomporn

• Objectives & Basic Concepts

• Maximal Cage Problem

• Minimal Cage Problem

• Discussion & Conclusion

### Objectives & Basic Concepts

• Object is caged when it cannot escape to infinity w/o penetrating obstacles.

• Our system:

• Rigid Object, represented with simple polygons.

• 2 Point Fingers.

• On a plane, 2D problem.

• “Determine sets of configurations that can cage the object with two fingers.”

• Characterize ALL maximal cages & minimal cages.

• Rimon & Blake’s: Two 1-DOF finger caging

• Largest cage that leads to a certain immobilizing grasp.

• Topological change of Free (configuration) space.

• Transform the Configuration space into a Search graph.

• All largest possible cages.

• Not cage that leads to a specified immobilizing grasp.

• System of 7-DOF

• 3-DOF rigid object orientation/position

• 2x2-DOF positions of the fingers

• However, whether the object is caged is independent from choice of coordinates (3-DOF ambiguity.)

• Fix the rigid object’s orientation/position.

• 2x2-DOF positions of the fingers (u, v).

• Analyze motion of fingers relative to the object.

• Object is not caged when two fingers are at the same point.

### Maximal Cage Problem

• A connected set containing every configuration (u, v) that can cage the object.

• A maximal cage is associated with ONE critical distance d+.

• Least separation distance between fingers that allows object to escape.

• d+(u,v)

• Different d+impliesDifferentmaximal cage.

• Characterize all Maximal Cages.

• Set Description

• Describe configurations in a maximal cage.

• By a configuration in the maximal cage and its d+.

• Point Inclusion

• Which maximal cage a configuration (u, v) is in?

• If so, what is d+ of the maximal cage?

Determining d+(u, v)

• To characterize a maximal cage, we need:

• A configuration (u,v) inside a maximal cage.

• d+ of such configuration.

• How to determine d+(u,v), least upper-bound separation distance that allows the object to escape?

• Consider an escape motion starting from (u,v).

u

v

d+(u, v)

d+(u, v)

• Consider all possible escape motions starting from (u, v) for least separation distance.

• Infinitely many motions.

8

P

Solution Overview

• R4 Config’ SpaceFinite Graph

• A Fingers’ MotionA Path in the Graph

• Configuration (u, v)State P, (u,v)  P

• Separation distanceTransition distance

u

v

Upper-bound separation distance  Upper-bound Transition Distance

R4 Config’ SpaceFinite Graph

A Fingers’ MotionA Path in the Graph

Configuration (u, v)State P, (u,v)  P

Separation distanceTransition distance

d+(u, v) d+P

To determine d+ of a configuration is to determine d+ of a state.

Solution Overview

• States

• Partition R4  Configuration Pieces Pi (States)

• States’ Representatives:

• Each representative is a certain configuration (u, v) inP, d+P = d+(u, v).

• Finding d+ of all representatives (d+P for all P) is sufficient to characterize all maximal cages.

Configuration Space Partitioning***

• Configuration that squeezes to the same pair of edges is in the same configuration piece.

• State  Configuration Piece

• State can be referred by an edge pair: {ei, ej}

ei

ej

• From any (u, v) in a piece P:{ei, ej}, there exists a “non-increasing separation distance” finger motion from (u, v) to a local minimum of P.

• FACT: Each piece partition this way is associated with at most ONE maximal cage.

• FACT: If a configuration in piece is in a maximal cage, then its local minimum is as well.

• Use the state’s local minimum as state’s representative.

• Consequently: Computing d+ of all representatives is sufficient for characterizing all maximal cages.

• Two nearby pieces P, Q in R4 is linked with PQ.

• Represents Fingers’ Motion from local minimum of P to that of Q with least upper-bound separation distance.

• Transition distance [PQ] = Least upper-bound separation distance of such Motion.

• Concatenating a series of transitions from P to a piece associated with {ek, ek} (k is a constant) to obtain an Escape Path.

• An Escape Path implies An Escape Motion.

d+ of Piece

• d+P is obtained from an Escape Path with least upper-bound transition distance.

• Use Dijkstra’s Algorithm to solve this problem.

• With an upper-bound fact:

• d+P ≤ max(d+Q, [PQ])

• d+P ≤ d+Q + |PQ|

• Start from any {ek, ek}

• O(n2) states. (n = # edges)

• Partitioning requires O(1) for each state  O(n2).

• Dijkstra’s Algorithm takes: O(n2 lg n + t), t = number of transitions.

• Only “basic transitions” should be included in the graph.

• At most 3 basic transitions for each distinct pair of edge ei and vertex v.

• Link between edges sharing v (ej, ek).

• Link between an edge w/ v as an end point and em.

• x is a projection of v on ei

• = |v – x|

• Transition: Sliding fingers from one local minimum to the other.

• Candidates: fingers’ motion on edges.

• v must be included in the motion.

• Transit between pieces at (v, x) is minimal.

• Recall: “Piece’s Property”

• Possible non-basic transition (a).

• Replace such with sequence of basic transitions w/ equal (or less) upper-bound separation distance.

• Require a ray-shoot for em .

• O((√k) lg n) for each ray-shoot query.

• Ray-shoot algorithm require O(n2) pre-computation time.

• (k = # simple polygons.)

• By Hershberger & Suri.

• Total time required: O(n2 (√k) lg n)

• O(n2 (√k) lg n) for pre-computation

• O(n2 lg n) for d+ propagation w/ Dijkstra’s.

• O((√k) lg n) for maximal cage query.

• If d+ of local minimum of P (d+P) is known.

• Given (u, v) in piece P.

• If |u-v| < d+P , (u, v) is in a maximal cage.

• Squeeze (u,v) toan edge pair tofind (u,v)’s containing piece P.

• O((√k) lg n)

### Minimal Cage

• Greatest separation distance that allows object to escape.

• d-(u,v)

• Characterize all Minimal Cages.

• Set Description

• Describe configurations in a minimal cage.

• By a configuration in the minimal cage and its d-.

• Point Inclusion

• Which minimal cage a configuration (u, v) is in?

• If so, what is d- of the minimal cage?

• Configuration that stretches to the same pair of vertices is in the same piece.

• A piece P is associated with a vertex pair: {vi, vj}(the local maximum)

• Every (u, v) in P can move to the local maximum of P with non-decreasing separation motion.

• After the graph construction

• Piece - pair of vertices

• Transitions - basic transitions

• Solve all d- with Dijkstra’s Algorithm in the same manner.

### Discussion & Conclusion

• Combinatorial Search Algorithm.

• n = # vertices, k = # simple polygons

• O(n2 √k lg n) pre-computation time (characterize all maximal/minimal cages.)

• O(√k lg n) optimal cage query time.

• In characterizing all Maximal Cages.

• Partition free space (R2) into ‘r’ Convex Regions.

• Pieces are cartesian product of a pair of convex regions.

• O(n2 +r2 lg r), pre-computation time

• O(lg n), maximal cage query time.

• Can be applied to characterizing all maximal cages in 3D.