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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


  • Objectives & Basic Concepts

  • Maximal Cage Problem

  • Minimal Cage Problem

  • Discussion & Conclusion

Objectives basic concepts

Objectives & Basic Concepts

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.


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


  • 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).



D u v
d+(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




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



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}



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.


  • 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.


  • 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.


  • 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.