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Two Finger Caging of Concave Polygon

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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

- 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?

- Set Description

- 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

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

5

8

P

- R4 Config’ SpaceFinite Graph
- A Fingers’ MotionA Path in the Graph
- Configuration (u, v)State P, (u,v) P
- Separation distanceTransition distance

u

v

Upper-bound separation distance Upper-bound Transition Distance

R4 Config’ SpaceFinite Graph

A Fingers’ MotionA Path in the Graph

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

Separation distanceTransition distance

d+(u, v) d+P

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

- 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 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 PQ.
- Represents Fingers’ Motion from local minimum of P to that of Q with least upper-bound separation distance.
- Transition distance [PQ] = 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+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, [PQ])

- Instead of:
- d+P ≤ d+Q + |PQ|

- 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?

- Set Description

- 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.