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July 11, 2006 RNC-7-20061
Vadim Shapiro
On Tolerancing and Metrology of Geometric (Solid) Models
Vadim ShapiroMechanical Engineering & Computer Sciences
University of Wisconsin - Madison
July 11, 2006 RNC-7-20062
Vadim Shapiro
Outline
• Practical motivation: data quality • What is the problem?• Detour: mechanical tolerancing• Attempts at possible solutions:
– Perturbations– Interval & Partial Solids– Epsilon-regularity– Tolerant complexes
July 11, 2006 RNC-7-20063
Vadim Shapiro
Outline
• Practical motivation: data quality• What is the problem?• Detour: mechanical tolerancing• Attempts at possible solutions:
– Perturbations– Interval & Partial Solids– Epsilon-regularity– Tolerant complexes
July 11, 2006 RNC-7-20064
Vadim Shapiro
Courtesy EDS (UG) Corporation
• construction (geometric design)
• drawing, rendering, annotation
• mass properties, mechanisms
• sections, interference, meshing (almost)
• NC machining, manufacturing planning
• etc.
1970-1980’s: Computer-Aided Everything
July 11, 2006 RNC-7-20065
Vadim Shapiro
1900’s-now: Automation, Collaboration, & Interoperability
• Computer model is the master model• Produced in large quantities• Transferred, exchanged, and translated
• Emerging issue: “data quality”
July 11, 2006 RNC-7-20066
Vadim Shapiro
Structure Problem: Void
• This rounded, square feature does not plunge deep enough into this model. It traps a “pocket of air” in this corner. The faces of this void have areas between 0.0062 and 0.013 mm2. There is a microscopic face in the lower left corner with an area of 0.000044 mm2.
Courtesy of Doug Cheney, CAD Interoperability Consultant, ITI TranscenData
July 11, 2006 RNC-7-20067
Vadim Shapiro
Realism Problem: Crack
• This linear protrusion is defined from a profile on side wall. Because of a draft angle on the side wall the protrusion has a crack underneath. The angle between the two bottom faces is 1.0 deg.
Courtesy of Doug Cheney, CAD Interoperability Consultant, ITI TranscenData
July 11, 2006 RNC-7-20068
Vadim Shapiro
Accuracy: Edge Endpoint Gaps
• These five edges are all connected at a single vertex. The largest gap between their endpoints is 0.008 mm. Several dissimilar types of surfaces intersect here (counterclockwise starting on left side): two complex blends, one simple round, a plane, and a cylindrical surface.
Courtesy of Doug Cheney, CAD Interoperability Consultant, ITI TranscenData
July 11, 2006 RNC-7-20069
Vadim Shapiro
Accuracy: Edge Endpoint Gaps
• Surfaces from an industrial design system were imported then stitched together to form this solid. All of these highlighted edges are connected at a single vertex. The gaps between their endpoints are as large as 2.02 mm. These gaps are tolerated by this CAD system in order to complete the sewing operation.
Courtesy of Doug Cheney, CAD Interoperability Consultant, ITI TranscenData
July 11, 2006 RNC-7-200610
Vadim Shapiro
Accuracy: Edge Face Gaps
• Each of these blend surfaces has an edge at least 0.001 mm off of its underlying surface. Both of these are at the intersection of a blend with a planar face. These edges lie on the planar surfaces but not precisely on the blends.
Courtesy Doug Cheney, CAD Interoperability Consultant, ITI TranscenData
July 11, 2006 RNC-7-200611
Vadim Shapiro
Realism Problem: Pinched Face
• A single, planar surface defines this right, inside face. Between the upper and lower portions this face is pinched down to 0.023 mm. These edges are not connected at this location.
Courtesy of Doug Cheney, CAD Interoperability Consultant, ITI TranscenData
July 11, 2006 RNC-7-200612
Vadim Shapiro
Problem Resolution (Poor): Pinched Face
• One possible resolution that does not require feature changes is to relax the CAD system’s modeling tolerance so that this intersection is recognized. While these edges are connected at a single vertex, there are gaps as large as 0.027 mm between their endpoints. This resolution is not recommended. Courtesy of Doug Cheney, CAD Interoperability Consultant, ITI TranscenData
July 11, 2006 RNC-7-200613
Vadim Shapiro
Not a “robustness” issue per se• All models were considered valid in some system
where they were created
• Some models become invalid in some systems after transfer
• Some models in some systems may be inconsistent with the engineering intent
• Focus on validity of boundary representations of solids; parallel problems apply to other reps
July 11, 2006 RNC-7-200614
Vadim Shapiro
How important is the problem?
• US Automotive industry: $1 billion per year (Source: Frechette 1996, National Institute of Standards and Technology report)
• Much more globally today• New industries
– Repair and healing (e.g. ITI TranscenData)– Translation (e.g. STEP Tools, Proficiency, …)
• International Standards (in preparation)– STEP ISO 10303-59 Part 59 (product data quality)– SASIG (global automotive industry) PDQ, 2004 draft
July 11, 2006 RNC-7-200615
Vadim Shapiro
Example from SASIG
July 11, 2006 RNC-7-200616
Vadim Shapiro
July 11, 2006 RNC-7-200617
Vadim Shapiro
… and so on …
for a total of 77 geometric quality criteria! (plus additional non-geometric)
July 11, 2006 RNC-7-200618
Vadim Shapiro
Outline
• Practical motivation: data quality • What is the problem?• Detour: mechanical tolerancing• Attempts at possible solutions:
– Perturbations– Interval & Partial Solids– Epsilon-regularity– Tolerant complexes
July 11, 2006 RNC-7-200619
Vadim Shapiro
Key Issue
Implemented data structures and algorithms rely on real numbers and do not correspond to the assumed exact theories of geometric and solid modeling
Two inter-related problems:
Robustness: design data structures and algorithms that “work” with exact theories
Tolerancing & metrology: formulate theories that support and tolerate inaccurate models and computations with real numbers
July 11, 2006 RNC-7-200620
Vadim Shapiro
Exact Theory I: Regular Sets and Regularized Set Ops
• Regularized set operations are used in CAD systems to specify many solid constructions: additive, subtractive operations.
• Dual model: open regular sets ( take interior of closure)
interior Closure of interior
Closed regular setNon-regular 2D set
Regularization
Two i nt ersec tingSolids A, B
Intersection A B ∩
Regularized intersection
A
B
July 11, 2006 RNC-7-200621
Vadim Shapiro
Exact Theory II: Manifolds and Boundaries
• Boundary representations (of regular sets) • Orientable manifolds• Data structures
– Abstract cell complex K– Geometric embedding in E3
– Assumed to be exact
Boundary representations are used in CAD systems to store and archiveresults of all operations (including set operations)
Fi
Fk
Fi
Fk
July 11, 2006 RNC-7-200622
Vadim Shapiro
Exact Theory III:Point Membership Classification (PMC)
• Validity: single most important computation• Set operations rely on PMC• Requires closure and interior• Boundary construction relies on PMC• PMC on boundary representation uses Jordan curve theorem
∈ Boundary of Sx∈ Outside Sxout
∈ Inside Sxin
onfalse
∈ SxtrueOtherwise
Two in te r fer ing So l i ds A, B
Intersection A B∩
Regularized intersection A * B∩
July 11, 2006 RNC-7-200623
Vadim Shapiro
What really happens• Approximate computations
– Floating point– Finite resolution– Subdivision methods
• No exact closure • No sets are closed regular• No exact set operations
• Answers are correct only within some distance ε
• In principle, if the input were exact, could answer correctly for any ε>0 … given enough time …
Kettner et al 2004
July 11, 2006 RNC-7-200624
Vadim Shapiro
Relation to Geometric Robustness• Many excellent surveys
– Hoffmann 89, 01, Yap 97, Michelucci 98, Schirra 98, …
• Popular techniques– Perturbations– Certified computations, Filters– Exact computations on demand– Intervals, tolerances
• Challenges– Exactness of input, round-off– Degree, proliferation– Consistency, lack of transitivity
• Different (but related) problem
July 11, 2006 RNC-7-200625
Vadim Shapiro
Inaccuracy by design• Results of expensive computations must be archived
– High algebraic degree, e.g. intersection of two bi-cubicsurfaces S(u,v) is degree 324
– Precision grows exponentially in degree and depth
• Round-off is unavoidable
• Imprecise, sampled, or transferred data
• Incomplete representation spaces– Algebraic varieties vs rational parameterizations– Set operations
Hoffmann & Stewart, 2005
SolidWorks → STEP → SolidWorks
July 11, 2006 RNC-7-200626
Vadim Shapiro
Real boundary representations• Contain gaps, cracks, self-intersections
• “Tiny” faces, edges may disappear under reduced precision
• Geometric embeddings inconsistent with combinatorial structure
• Subject to all problems of non-robustness
• Jordan curve theorem does not hold
• PMC test can fail (leading to invalid models, system failures …)
July 11, 2006 RNC-7-200627
Vadim Shapiro
The problem • Recognize errors and inaccuracies as given fact of life
• What is the meaning of models with inaccuracies ?
• How do we specify (tolerate) and inspect (measure) such models?
• How do we represent and compute on such models?
• Precision of algorithms is important, but secondary issue
July 11, 2006 RNC-7-200628
Vadim Shapiro
Outline
• Practical motivation: data quality • What is the problem?• Detour: mechanical tolerancing• Attempts at possible solutions:
– Perturbations– Interval & Partial Solids– Epsilon-regularity– Tolerant complexes
July 11, 2006 RNC-7-200629
Vadim Shapiro
Rough and brief history of mechanical tolerancing
Geometric dimensioning & tolerancing (zones, material conditions, containment), finite set of symbols, measures
1970s
Formal definition of semantics (first edition 1994), standardized metrology algorithms
1990s -present
Parametric +/- tolerances, idealized form, notes1950s
Tolerances mentioned on drawings1930s-1940s
Gaging and interchangeability, first notions of accuracyand consistency … mass production around 1900s
1700-1800s
Skilled artisans manufacture to precision, custom fit, small batches, no notion of accuracy or measurement
< 1700s
Geometric models as manufactured objects: where are we now?
July 11, 2006 RNC-7-200630
Vadim Shapiro
from
P.J. Booker
A history of engineering drawings
July 11, 2006 RNC-7-200631
Vadim Shapiro
Functional Gauging of parts for assembly
What do these measure?
July 11, 2006 RNC-7-200632
Vadim Shapiro
Least Material Condition
Tolerance semantics relies on Zones
Maximum Material Condition
Datums
July 11, 2006 RNC-7-200633
Vadim Shapiro
Are these measurable? … computable?
July 11, 2006 RNC-7-200634
Vadim Shapiro
Lessons from mechanical tolerancing
• Inaccuracy and Tolerances can be good
• LMC, MMC – idealized notions that derive from and include nominal “exact” object
• Other tolerances (size, form, position) are specified with respect to LMC/MMC
• Inspection: do not need to know the nominal exact object!
• Algorithms for deciding whether a given object belongs to the LMC/MMC interval
• Not always decidable – theoretically
• ... But we build cars anyway …
July 11, 2006 RNC-7-200635
Vadim Shapiro
Outline
• Practical motivation: data quality • What is the problem?• Detour: mechanical tolerancing• Attempts at possible solutions:
– Perturbations– Interval & Partial Solids– Epsilon-regularity– Tolerant complexes
July 11, 2006 RNC-7-200636
Vadim Shapiro
Semantics of tolerancing and metrology
• Specify point set model that tolerates errors near the boundaries? (tolerancing)– All models are invalid in exact sense– But most models are “valid enough” in their native system …
what does this mean?
• How do we inspect and validate a given boundary representation? (metrology)– Invalid model used to be valid under some conditions … what
are they?
• From robustness point of view, there are 2 choices:– Perturbation semantics– Interval semantics
July 11, 2006 RNC-7-200637
Vadim Shapiro
Perturbation semantics• Representation R is invalid• But there exists a perturbation of R that is valid and
represents model M• Perturbation approaches rely on existence of M’s to
construct perturbed R’s• Problems:
– Perturbations propagate globally– M may or may not exist – Choice of M is arbitrary (exponentially many choices?)
• Apply perturbation semantics to a given “invalid” boundary representations?
July 11, 2006 RNC-7-200638
Vadim Shapiro
Example: perturbation semantics• Assume that input data is “well-formed”• Unique Quasi-NURBS set using Whitney extension theorem• Bounds on distance, normals• Use its properties to develop algorithms and proofs
• No longer NURBs• Fixed combinatorial structure• Does not preserve constraints• Stringent assumptions on input
[Andersson, L.-E., Stewart, N. F. and Zidani, 2005][Hoffmann & Stewart, 2005][Stewart & Zidani, 2006]
July 11, 2006 RNC-7-200639
Vadim Shapiro
Geometric repair is a perturbation
SolidWorks → STEP → SolidWorks after repair
SolidWorks → STEP → Pro/Engineer -- EXACTLY, after repair
0.001 mm thickeness
1000 mm
Perturbations semantics is limited … or dangerous
July 11, 2006 RNC-7-200640
Vadim Shapiro
Interval semantics
• Idea: do not fix it, find the containing set interval
• Extension of interval arithmetic
• Valid models are not sets, but set intervals
• The exact set is the limit as the interval shrinks
• Draw on “robust” approaches that compute and reason in terms of intervals and zones
July 11, 2006 RNC-7-200641
Vadim Shapiro
Set intervals: examples• B-spline, Bezier, curves and surfaces are limits of polyhedral enclosures
• Beacon et al, 1989: inner, outer, boundary segments
• Segal, 1990; Jackson 1995: tolerant zones for boundary representations
July 11, 2006 RNC-7-200642
Vadim Shapiro
Set Intervals: Interval Solids
Sakkalis, Shen, Patrikalakis, 2001
Sakkalis & Peters, 2003
• Motivated by interval arithmetic to computer intersection curves
• Approximation of the exact solid
• Boundary is ambient isotopic to the exact
• Perturbation semantics
July 11, 2006 RNC-7-200643
Vadim Shapiro
Set Intervals: Partial Solids (Edalat & Lieutier, 1999)
• Point Membership Classification PMC: E3 → {true, false} is not computable in domain theoretic sense (not continuous function)
• Redefine PMC as
• PMC is continuous with Scott topology• A solid is an ordered sequence of set pairs (Inside, Outside)• The maximal element is open regular set (interior of exact solid)
Inside = interior closure (Inside)• Define Boolean set operations (not regularized) • Can be approximated by a nested sequence of rational polyhedra• Solids are Hausdorff computable• Set operations are computable, but not Hausdorff computable
⊥ ∈ Otherwisex∈ Outside Sxfalse
∈ Inside Sxtrue
July 11, 2006 RNC-7-200644
Vadim Shapiro
How to tolerance a set interval?• Specify a set interval that tolerates errors near the boundary• Should include some measure ε of tolerance• If the ε → 0, should get an exact regular solid • The interval should contain all sets that are valid within ε
• How do we inspect and validate specific representations, and boundary representations in particular?
• ε-regular sets and intervals
July 11, 2006 RNC-7-200645
Vadim Shapiro
ε- “Topological” Operations
• Classical topological operations are cases where ε = 0• Many (but not all) theorems generalize• Similar to (but different from) to morphological operations (dilation, erosion)
July 11, 2006 RNC-7-200646
Vadim Shapiro
Regular and ε-Regular Sets
• For any set X, i0(X) ⊆ X ⊆ k0(X)
• “Regularization”– Grow interior by ε-closure kε
– Shrink closure by ε-interior iεiεk0(X) ⊆ X ⊆ kεi0(X)
• As ε → 0i0k0(X) ⊆ X ⊆ k0i0(X)
• Usually do not know X, but only interval [X-, X+]
July 11, 2006 RNC-7-200647
Vadim Shapiro
ε-Regular Set Interval [X-, X+]
iε (X+) ⊆ X- ⊆ X+ ⊆ kε (X-)
Qi & Shapiro, 2005
July 11, 2006 RNC-7-200648
Vadim Shapiro
Properties of ε-regular interval [X-, X+]
• The Hausdorff distance between sets is at most ε
• The Hausdorff distance between complements is at most ε
• The Hausdorff distance between boundaries is at most ε
• Any set within the interval is ε-regular
• Any sub-interval is ε-regular
July 11, 2006 RNC-7-200649
Vadim Shapiro
ε−regular intervals – what for?
• Specify tolerant solid models– Define a (Boolean?) algebra of ε-regular intervals– Requires ε-regularized set operations
• Formulate problems in data transfer– Avoid repair whenever possible– Increasing tolerances does solve some problems!
• Reconcile different level of details– FE meshing versus small features
• Validate other representations, e.g. boundary– Does it define a ε-regular interval?
July 11, 2006 RNC-7-200650
Vadim Shapiro
Is this a boundary of an ε-regular set ?
Abstract complex K Orientable manifold Can be realized in Ed
as |K|
Depends on the size of tolerances
July 11, 2006 RNC-7-200651
Vadim Shapiro
Is this a boundary of an ε-regular set ?
Abstract complex K Orientable manifold Can be realized in Ed
as |K|
Too small
July 11, 2006 RNC-7-200652
Vadim Shapiro
Is this a boundary of an ε-regular set ?
Abstract complex K Orientable manifold Can be realized in Ed
as |K|
Too large? Wrong topology
July 11, 2006 RNC-7-200653
Vadim Shapiro
Is this a boundary of an ε-regular set ?
Abstract complex K Orientable manifold Can be realized in Ed
as |K|
So large, topology is correct, but “destroys” K
July 11, 2006 RNC-7-200654
Vadim Shapiro
Is this a boundary of an ε-regular set ?
Abstract complex K Orientable manifold Can be realized in Ed
as |K|
Just right
July 11, 2006 RNC-7-200655
Vadim Shapiro
What is a tolerant boundary representation?
• Assume combinatorial structure – Abstract complex K (vertices, edges, faces)– Orientable 2-cycle – Can be realized in E3 as |K|
• With every cell ci ∈ K associate a zone Zi, – Defined by either known error, or accuracy of algorithm
• When is union of zones U(Zi) a thickening of |K|?– implies homotopy equivalence between U(Zi) and |K|.
• … Then induce ε-regular interval …– Need generalization of Jordan-Brower separation theorem
• … Use zones instead of the imprecise or unknown geometry
July 11, 2006 RNC-7-200656
Vadim Shapiro
Nerve Theorem• Collection of sets {Xi}, union of sets UXi
• Associate vertex (0-simplex) with every set Xi
• A simplex (Xi, Xj, … Xn) is in the nerve N{Xi} if intersection IXi is not empty
• Theorem: If every intersection IXi is contractible then N{Xi} is homotopy equivalent to UXi
July 11, 2006 RNC-7-200657
Vadim Shapiro
When is the union of zones is homotopyequivalent ( ) to exact boundary
e1
e2
v
e2
v e1
Ze2
Zv Ze1
Ze2
Zv Ze1
Ze2
Zv Ze1
Ze2
Zv Ze1
Ze2
Zv Ze1
Ze2
Zv Ze1
Ze2
Zv Ze1
Ze2
Zv Ze1
July 11, 2006 RNC-7-200658
Vadim Shapiro
Earlier heuristic approaches:neither necessary nor sufficient
Segal, 1990 (polyhedral modeler)• Implies isomorphism between the two nerves• Intersections must be connected• Other implementation-specific informal rules
Jackson, 1995 (commercial solid modeler)• Connected components of intersections must be contractible• Required intersections are not indicated• Additional size/containment conditions
July 11, 2006 RNC-7-200659
Vadim Shapiro
… to be continued
• Nerve defines a set of validity conditions • Reduce/collapse the nerve to obtain special cases• Need algorithms
– Contractibility test– Collapsibility test – Difficult in general, use known properties
• …• How to induce thickening?• PMC (and other algorithms) on thickening?
July 11, 2006 RNC-7-200660
Vadim Shapiro
Summary
• Precision versus (in)accuracy• Mathematical theory to include inaccuracy• Language for specifying inaccuracy (tolerancing)• Algorithms for inspecting, testing (metrology)
• Do we need new data structures and algorithms?
• Validity versus consistency• Relation to mechanical tolerancing?
July 11, 2006 RNC-7-200661
Vadim Shapiro
Thank you
Supported in part by
• National Science Foundation grants DMI-0500380, DMI-0323514• National Institute of Standards & Technology (NIST)• General Motors Corporation