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  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 + implies Different maximal 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

Upperbound Separation Distance

d + (u, v)

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

Solution Overview  R 4 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 P5 8

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  d + (u, v)  d + P  To determine d + of a configuration is to determine d + of a state.

Graph Construction  States  Partition R4  Configuration Pieces P i (States)

Graph Construction  States’ Representatives:  Each representative is a certain configuration (u, v) in P, 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: {e i, e j } e i e j e j

Piece’s Property  From any (u, v) in a piece P:{e i, e j }, 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 R 4 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 {e k, e k } (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])  Instead of:  d + P ≤ d + Q + |P  Q|  Start from any {e k, e k }

Running Time Analysis  O(n 2 ) states. (n = # edges)  Partitioning requires O(1) for each state  O(n 2 ).  Dijkstra’s Algorithm takes: O(n 2 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 e i and vertex v.  Link between edges sharing v (e j, e k ).  Link between an edge w/ v as an end point and e m.  x is a projection of v on e i

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 e m.  O((√k) lg n) for each ray- shoot query.  Ray-shoot algorithm require O(n 2 ) pre- computation time.  (k = # simple polygons.)  By Hershberger & Suri.

Running Time Analysis  Total time required: O(n 2 (√k) lg n)  O(n 2 (√k) lg n) for pre-computation  O(n 2 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) to an edge pair to find (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: {v i, v j } (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(n 2 √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 (R 2 ) into ‘r’ Convex Regions.  Pieces are cartesian product of a pair of convex regions.

Improvement  O(n 2 + r 2 lg r), pre-computation time  O(lg n), maximal cage query time.  Can be applied to characterizing all maximal cages in 3D.