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OUTSTANDING PROBLEMS IN GEOMETRIC CONSTRAINT SOLVING FOR CADPowerPoint Presentation

OUTSTANDING PROBLEMS IN GEOMETRIC CONSTRAINT SOLVING FOR CAD

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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

Meera Sitharam,

University of Florida

Partially supported by NSF grants

CCR 99-02025, EIA 00-96104

ORGANIZATION CONSTRAINT SOLVING FOR CAD

- CAD motivation and state of the art
- Suite of Formal Problems
- Our contribution-- FRONTIER
- Unsolved Problems

CAD MOTIVATION CONSTRAINT SOLVING FOR CAD1/4

Variational constraint representation and feature hierarchy

CAD MOTIVATION CONSTRAINT SOLVING FOR CAD2/4

Another Assembly constraint representation and subassembly

hierarchy

CAD MOTIVATION CONSTRAINT SOLVING FOR CAD3/4

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

- Generated declaratively.
- Easily updated and maintained.
- Minimal, complete.

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.

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

GOAL CONSTRAINT SOLVING FOR CAD

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

- Generic answer holds
- 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
- 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

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

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

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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