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OUTSTANDING PROBLEMS IN GEOMETRIC CONSTRAINT SOLVING FOR CAD

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

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  1. OUTSTANDING PROBLEMS IN GEOMETRIC CONSTRAINT SOLVING FOR CAD Meera Sitharam, University of Florida Partially supported by NSF grants CCR 99-02025, EIA 00-96104

  2. ORGANIZATION • CAD motivation and state of the art • Suite of Formal Problems • Our contribution-- FRONTIER • Unsolved Problems

  3. CAD MOTIVATION 1/4 Variational constraint representation and feature hierarchy

  4. CAD MOTIVATION 2/4 Another Assembly constraint representation and subassembly hierarchy

  5. CAD MOTIVATION 3/4 A geometric (variational) constraint representation with feature hierarchy is: • Generated declaratively. • Easily updated and maintained. • Minimal, complete.

  6. CAD MOTIVATION 4/4 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.

  7. STATE OF THE ART 1/3 • 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

  8. STATE OF THE ART: 3 Dimensions 2/3 Pictures of 2d views of 3d part

  9. STATE OF THE ART: 3D 3/3 D-cubed's pipe routing

  10. GOAL

  11. FORMAL BASIC PROBLEM 1/7 Input1:Primitive geometric objects: (id, type) (type chosen from repertoire)

  12. FORMAL BASIC PROBLEM 2/7 Input2:Geometric constraints: (object1, object2, .., objectk, type) (type chosen from repertoire) constraint types include some inequalities

  13. FORMAL BASIC PROBLEM 3/7

  14. FORMAL BASIC PROBLEM 4/7 • Input3 : Feature hierarchies: • (more than one) partial order or DAG of subsets of objects • partial realization (output) information for the nodes of DAG.

  15. FORMAL BASIC PROBLEM 5/7

  16. FORMAL BASIC PROBLEM6/7

  17. SUITE OF FORMAL PROBLEMS 1/12 • 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.

  18. FORMAL BASIC PROBLEM 7/7 • 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

  19. SUITE OF FORMAL PROBLEMS 2/12 • Generic, parameter-freeversion of existence • Approached combinatorially using only the geometric constraint graph, object and constraint types.

  20. SUITE OF FORMAL PROBLEMS 3/12 • Generic answer holds • For all but a small set of forbidden parameter values that satisfy discriminant/resultant (in)equalities.

  21. SUITE OF FORMAL PROBLEMS 4/12 • Generic Classification:some information on how many conformations exist? • finitely many (rigid or wellconstrained) • infinitely many(flexible or underconstrained) • none(inconsistently overconstrained)

  22. SUITE OF FORMAL PROBLEMS 5/12 • 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?

  23. SUITE OF FORMAL PROBLEMS 6/12

  24. SUITE OF FORMAL PROBLEMS 7/12

  25. SUITE OF FORMAL PROBLEMS 8/12

  26. SUITE OF FORMAL PROBLEMS9/12

  27. SUITE OF FORMAL PROBLEMS 10/12 • Generic Overconstraint correction: a well-defined set of removable constraint-sets for each node in feature hierarchy.

  28. SUITE OF FORMAL PROBLEMS 11/12 • Generic underconstraint navigation: a well-defined set of addable constraint-sets for each node in feature hierarchy.

  29. SUITE OF FORMAL PROBLEMS 12/12 • 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.

  30. OUR CONTRIBUTIONS 1/12 • (1) Formalizing decomposition problem and performance measures.

  31. OUR CONTRIBUTIONS 2/12 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

  32. OUR CONTRIBUTIONS 3/12

  33. OUR CONTRIBUTIONS 4/12 • Other performance measures on DR-planners • An optimal DR-planner minimizes the maximum fan-in (size of the largest subsystem in DR-plan)

  34. OUR CONTRIBUTIONS 5/12 • (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.

  35. OUR CONTRIBUTIONS 6/12

  36. OUR CONTRIBUTIONS 7/12

  37. OUR CONTRIBUTIONS 8/12 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.

  38. OUR CONTRIBUTIONS 9/12 • (3) Towards a more complete generic solution

  39. OUR CONTRIBUTIONS 10/12

  40. OUR CONTRIBUTIONS 11/12 • (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

  41. OUR CONTRIBUTIONS 12/12 • (6) Software architecture and implementation

  42. REITERATING UNSOLVED PROBLEMS 1/3 • 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

  43. REITERATING UNSOLVED PROBLEMS 2/3 • Underconstrained Conformation and Navigation: in addition to addable constraint sets, need forbidden parameter regions.

  44. REITERATING UNSOLVED PROBLEMS 3/3 • 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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