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# OUTSTANDING PROBLEMS IN GEOMETRIC CONSTRAINT SOLVING FOR CAD - PowerPoint PPT Presentation

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

Meera Sitharam,

University of Florida

Partially supported by NSF grants

CCR 99-02025, EIA 00-96104

• CAD motivation and state of the art

• Suite of Formal Problems

• Our contribution-- FRONTIER

• Unsolved Problems

Variational constraint representation and feature hierarchy

Another Assembly constraint representation and subassembly

hierarchy

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

• Generated declaratively.

• Easily updated and maintained.

• Minimal, complete.

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.

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

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

Pictures of 2d views of 3d part

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

D-cubed's pipe routing

FORMAL BASIC PROBLEM CONSTRAINT SOLVING FOR CAD1/7

Input1:Primitive geometric objects:

(id, type) (type chosen from repertoire)

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 CONSTRAINT SOLVING FOR CAD3/7

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.

FORMAL BASIC PROBLEM CONSTRAINT SOLVING FOR CAD5/7

FORMAL BASIC PROBLEM CONSTRAINT SOLVING FOR CAD6/7

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 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

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.

SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD3/12

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

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)

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?

SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD6/12

SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD7/12

SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD8/12

SUITE OF FORMAL PROBLEMS CONSTRAINT SOLVING FOR CAD9/12

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.

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.

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 CONSTRAINT SOLVING FOR CAD1/12

• (1) Formalizing decomposition problem and performance measures.

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

OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD3/12

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)

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.

OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD6/12

OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD7/12

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.

OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD9/12

• (3) Towards a more complete generic solution

OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD10/12

OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD11/12

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

• Existence

• Classification

• Overconstraint Correction

• Dealing with mixed representations, multiple input feature hierarchies

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

• Well-constrained Conformations

• Easy updates of constraint repertoire

• Easy updates of constraint representation, feature hierarchy and realizations

• Online constraint solving

OUR CONTRIBUTIONS CONSTRAINT SOLVING FOR CAD12/12

• (6) Software architecture and implementation

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

REITERATING UNSOLVED PROBLEMS CONSTRAINT SOLVING FOR CAD2/3