Two Finger Caging of Concave Polygon

1 / 47

# 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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Two Finger Caging of Concave Polygon' - bebe

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Two Finger Caging of Concave Polygon

Peam Pipattanasomporn

Outline
• Objectives & Basic Concepts
• Maximal Cage Problem
• Minimal Cage Problem
• Discussion & Conclusion

### Objectives & Basic Concepts

Definition of Caging
• 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.
Objectives
• “Determine sets of configurations that can cage the object with two fingers.”
Objectives
• Characterize ALL maximal cages & minimal cages.
Previous Work
• Rimon & Blake’s: Two 1-DOF finger caging
• Largest cage that leads to a certain immobilizing grasp.
• Topological change of Free (configuration) space.
Our Work
• Transform the Configuration space into a Search graph.
• All largest possible cages.
• Not cage that leads to a specified immobilizing grasp.
Configuration Space
• 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.)
Configuration Space
• 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

Maximal Cage
• A connected set containing every configuration (u, v) that can cage the object.
• A maximal cage is associated with ONE critical distance d+.
Critical Distance d+
• Least separation distance between fingers that allows object to escape.
• d+(u,v)
• Different d+impliesDifferentmaximal cage.
Problem Definition
• 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)
• Consider all possible escape motions starting from (u, v) for least separation distance.
• Infinitely many motions.
5

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
Graph Construction
• States
• Partition R4  Configuration Pieces Pi (States)
Graph Construction
• 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

Piece’s Property
• 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.
Piece Property
• 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.
Piece Property
• Use the state’s local minimum as state’s representative.
• Consequently: Computing d+ of all representatives is sufficient for characterizing all maximal cages.
Transitions
• 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.
Transition Concatenation
• 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.
Reduction to Shortest Path Prob.
• 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}
Running Time Analysis
• 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.
Basic Transitions
• 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
Transition Distance
• = |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”
Basic Transitions are Sufficient
• Possible non-basic transition (a).
• Replace such with sequence of basic transitions w/ equal (or less) upper-bound separation distance.
Basic Transitions
• 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.
Running Time Analysis
• 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.
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

Critical Distance d-
• Greatest separation distance that allows object to escape.
• d-(u,v)
Problem Definition
• 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?
Grouping Configurations
• 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.
Characterize Minimal Cages
• After the graph construction
• Piece - pair of vertices
• Transitions - basic transitions
• Solve all d- with Dijkstra’s Algorithm in the same manner.

### Discussion & Conclusion

Algorithm
• 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.
(4.2) Improvement
• In characterizing all Maximal Cages.
• Partition free space (R2) into ‘r’ Convex Regions.
• Pieces are cartesian product of a pair of convex regions.
Improvement
• 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.