On Tolerancing and Metrology of Geometric (Solid) Models Translation Errors - Vadim Shapiro at U of...

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

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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

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Outline

• Practical motivation: data quality• What is the problem?• Detour: mechanical tolerancing• Attempts at possible solutions:


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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

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1900’s-now: Automation, Collaboration, & Interoperability

• Computer model is the master model• Produced in large quantities• Transferred, exchanged, and translated

• Emerging issue: “data quality”

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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

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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.


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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.


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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.


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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

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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.


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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

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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

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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

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Example from SASIG

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… and so on …

for a total of 77 geometric quality criteria! (plus additional non-geometric)

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Outline



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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

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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

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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

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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∩

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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

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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

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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

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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 …)

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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

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Outline



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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?

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from

P.J. Booker

A history of engineering drawings

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Functional Gauging of parts for assembly

What do these measure?

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Least Material Condition

Tolerance semantics relies on Zones

Maximum Material Condition

Datums

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Are these measurable? … computable?

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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 …

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Outline



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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

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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?

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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]

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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

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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

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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

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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

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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

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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

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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)

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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+]

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ε-Regular Set Interval [X-, X+]

iε (X+) ⊆ X- ⊆ X+ ⊆ kε (X-)

Qi & Shapiro, 2005

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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

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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?

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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

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as |K|

Too small

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as |K|

Too large? Wrong topology

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as |K|

So large, topology is correct, but “destroys” K

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as |K|

Just right

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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

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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

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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

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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

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… 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?

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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

On Tolerancing and Metrology of Geometric (Solid) Models Translation Errors - Vadim Shapiro at U of...

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