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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

Two Finger Caging of Concave Polygon

Peam Pipattanasomporn

Advisor: Attawith Sudsang

outline
Outline
  • Objectives & Basic Concepts
  • Maximal Cage Problem
  • Minimal Cage Problem
  • Discussion & Conclusion
definition of caging
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
Objectives
  • “Determine sets of configurations that can cage the object with two fingers.”
objectives1
Objectives
  • Characterize ALL maximal cages & minimal cages.
previous work
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
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
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 space1
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
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
Critical Distance d+
  • Least separation distance between fingers that allows object to escape.
  • d+(u,v)
  • Different d+impliesDifferentmaximal cage.
problem definition
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
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 v1
d+(u, v)
  • Consider all possible escape motions starting from (u, v) for least separation distance.
  • Infinitely many motions.
solution overview
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

solution overview1
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
Graph Construction
  • States
    • Partition R4  Configuration Pieces Pi (States)
graph construction1
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 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
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
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 property1
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
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
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+ of Piece
  • d+P is obtained from an Escape Path with least upper-bound transition distance.
reduction to shortest path prob
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])
  • Instead of:
    • d+P ≤ d+Q + |PQ|
  • Start from any {ek, ek}
running time analysis
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
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
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
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 transitions1
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 analysis1
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
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)
critical distance d1
Critical Distance d-
  • Greatest separation distance that allows object to escape.
  • d-(u,v)
problem definition1
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
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
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.
algorithm
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
(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
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.
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