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OUTSTANDING PROBLEMS IN GEOMETRIC CONSTRAINT SOLVING FOR CAD. Meera Sitharam, University of Florida Partially supported by NSF grants CCR 99-02025, EIA 00-96104. ORGANIZATION. CAD motivation and state of the art Suite of Formal Problems Our contribution-- FRONTIER

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Outstanding problems in geometric constraint solving for cad l.jpg

OUTSTANDING PROBLEMS IN GEOMETRIC CONSTRAINT SOLVING FOR CAD

Meera Sitharam,

University of Florida

Partially supported by NSF grants

CCR 99-02025, EIA 00-96104


Organization l.jpg
ORGANIZATION CONSTRAINT SOLVING FOR CAD

  • CAD motivation and state of the art

  • Suite of Formal Problems

  • Our contribution-- FRONTIER

  • Unsolved Problems


Cad motivation 1 4 l.jpg
CAD MOTIVATION CONSTRAINT SOLVING FOR CAD1/4

Variational constraint representation and feature hierarchy


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CAD MOTIVATION CONSTRAINT SOLVING FOR CAD2/4

Another Assembly constraint representation and subassembly

hierarchy


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CAD MOTIVATION CONSTRAINT SOLVING FOR CAD3/4

A geometric (variational) constraint representation with feature hierarchy is:

  • Generated declaratively.

  • Easily updated and maintained.

  • Minimal, complete.


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CAD MOTIVATION CONSTRAINT SOLVING FOR CAD4/4

The Catch: implicit representation. How to

  • Want explicit geometric realization(s):

    • Navigate conformation of each feature consistent with subfeatures.

    • Derive implied geometric properties/invariants.

    • Eliminate inconsistencies in requirements.

  • Independently manipulate features and interface with other representations.


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STATE OF THE ART CONSTRAINT SOLVING FOR CAD1/3

  • 2 dimensions :Small, simple, no feature hierarchy, stand- alone.

  • 3 dimensions : 2d views; CSG; history of sweeps, extrusions; parametric constraint solving

Hoffman et al (EREP), Bruderlin et al, Bronsvoort et al, Kramer et al,

Michelucci et al,Owen et al (D-cubed), Latham, Middleditch et al


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STATE OF THE ART: 3 Dimensions CONSTRAINT SOLVING FOR CAD2/3

Pictures of 2d views of 3d part


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STATE OF THE ART: 3D CONSTRAINT SOLVING FOR CAD3/3

D-cubed's pipe routing


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GOAL CONSTRAINT SOLVING FOR CAD


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FORMAL BASIC PROBLEM CONSTRAINT SOLVING FOR CAD1/7

Input1:Primitive geometric objects:

(id, type) (type chosen from repertoire)


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FORMAL BASIC PROBLEM CONSTRAINT SOLVING FOR CAD2/7

Input2:Geometric constraints:

(object1, object2, .., objectk, type)

(type chosen from repertoire)

constraint types include some inequalities


Formal basic problem 3 7 l.jpg
FORMAL BASIC PROBLEM CONSTRAINT SOLVING FOR CAD3/7


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FORMAL BASIC PROBLEM CONSTRAINT SOLVING FOR CAD4/7

  • Input3 : Feature hierarchies:

  • (more than one) partial order or DAG of subsets of objects

  • partial realization (output) information for the nodes of DAG.


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FORMAL BASIC PROBLEM CONSTRAINT SOLVING FOR CAD5/7


Formal basic problem 6 7 l.jpg
FORMAL BASIC PROBLEM CONSTRAINT SOLVING FOR CAD6/7


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SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD1/12

  • Existence:of realization

  • Conformation: One conformation (if it exists)for each node in feature hierarchy, represented as a rigid transformation applied to each child's conformation.


Formal basic problem 7 7 l.jpg
FORMAL BASIC PROBLEM CONSTRAINT SOLVING FOR CAD7/7

  • For conformation, need to solve polynomial system over the reals.

d2=((x2-x1)2 + (y2-y1)2

Problem classification

Red: Algebraic; Blue: Combinatorial; Purple: Mixture


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SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD2/12

  • Generic, parameter-freeversion of existence

  • Approached combinatorially using only the geometric constraint graph, object and constraint types.


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SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD3/12

  • Generic answer holds

  • For all but a small set of forbidden parameter values that satisfy discriminant/resultant (in)equalities.


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SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD4/12

  • Generic Classification:some information on how many conformations exist?

  • finitely many (rigid or wellconstrained)

  • infinitely many(flexible or underconstrained)

  • none(inconsistently overconstrained)


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SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD5/12

  • Navigation:A well-defined set of conformations for each node in feature hierarchy, represented as a set of transformations applied to each child's set of conformations?

  • Meaning of well-defined: complete in some formal sense, systematically navigable.

  • Invariant:Does a given geometric property hold for all conformations?


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SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD6/12


Suite of formal problems 7 12 l.jpg
SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD7/12


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SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD8/12


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SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD9/12


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SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD10/12

  • Generic Overconstraint correction: a well-defined set of removable constraint-sets for each node in feature hierarchy.


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SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD11/12

  • Generic underconstraint navigation: a well-defined set of addable constraint-sets for each node in feature hierarchy.


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SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD12/12

  • Combinatorial complete generic solution: Big open question. Gives rise to a combinatorial theory of rigidity. Whiteley et al.

  • Laman's theorem: complete combinatorial classification for 2D points and distances. Simple dof analysis.


Our contributions 1 12 l.jpg
OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD1/12

  • (1) Formalizing decomposition problem and performance measures.


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OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD2/12

A Decomposition-Recombination plan (DR-plan) for an input constraint system G, consistent with an input feature hierarchy F is a DAG:

  • nodes are subsets of primitive objects of G such that their induced subsystems are well-over-constrained 1

  • nodes include the nodes of F

  • each leaf/source is a primitive object in S;

  • each root/sink represents a maximal well-over-constrained subsystem of G1

1 more generally, they possess atmost a specified number of degrees of freedom


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OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD3/12


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OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD4/12

  • Other performance measures on DR-planners

  • An optimal DR-planner minimizes the maximum fan-in (size of the largest subsystem in DR-plan)


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OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD5/12

  • (2) Partial-generic characterization of DR-plan based on degree of freedom analysis of constraint graphs: minimal dense subgraph usually corresponds to well-over-constrained subsystem.

  • Algorithm for construction of DR-plan: using

  • network flows to iteratively find the minimal dense subgraphs in current graph

  • graph transformations that repeatedly simplify them.


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OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD6/12


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OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD7/12


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OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD8/12

Optimal DR Planning problem (Partial-generic version)

  • Already finding smallest well-constrained graph is NP-complete. Polynomial time algorithms known for special cases. Approximation status unknown.


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OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD9/12

  • (3) Towards a more complete generic solution


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OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD10/12


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OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD11/12

  • (4) Decomposition gives partial-generic solution to:

  • Existence

  • Classification

  • Overconstraint Correction

  • Generic underconstraint Navigation

  • Dealing with mixed representations, multiple input feature hierarchies

  • (5) Plus additional work on equation and conformation management gives:

  • Well-constrained Conformations

  • Well-constrained Navigation

  • Easy updates of constraint repertoire

  • Easy updates of constraint representation, feature hierarchy and realizations

  • Online constraint solving


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OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD12/12

  • (6) Software architecture and implementation


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REITERATING UNSOLVED PROBLEMS CONSTRAINT SOLVING FOR CAD1/3

  • Isolation of Conformation: Chirality, Semi-global constraints, Symmetries, Forces.

  • Efficiently solving polynomial systems for rigid transformations : physically based semi-numerical algorithms are welcome.

  • Invariant problem.

  • Inverse problem of finding minimal constraint representation


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REITERATING UNSOLVED PROBLEMS CONSTRAINT SOLVING FOR CAD2/3

  • Underconstrained Conformation and Navigation: in addition to addable constraint sets, need forbidden parameter regions.


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REITERATING UNSOLVED PROBLEMS CONSTRAINT SOLVING FOR CAD3/3

  • Complete generic solution to original problems-- combinatorial geometry, geometric graphs.

  • Approximation algorithm for Optimal DR-plan problem, even the partial-generic version based on dof analysis.

  • Complexity of existence problem

    NP-hard; not known to be in NP; in DNPR (partial algebraic version of NP); not known to be DNPR-hard.

  • Algebraic description of generic describe the semi-algebraic set of forbidden parameter values when generic solution does not hold


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