On Tolerancing and Metrology of
Geometric (Solid) Models
Vadim Shapiro
Mechanical Engineering & Computer Sciences
University of Wisconsin - Madison
July 11, 2006
RNC-7-2006
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Vadim Shapiro
Outline
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Practical motivation: data quality
What is the problem?
Detour: mechanical tolerancing
Attempts at possible solutions:
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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:
–
–
–
–
Perturbations
Interval & Partial Solids
Epsilon-regularity
Tolerant complexes
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Vadim Shapiro
1970-1980’s: Computer-Aided Everything
Courtesy EDS (UG) Corporation
• construction (geometric design)
• drawing, rendering, annotation
• mass properties, mechanisms
• sections, interference, meshing
(almost)
• NC machining, manufacturing planning
• etc.
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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.
Courtesy of Doug Cheney, CAD Interoperability Consultant, ITI TranscenData
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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.
Courtesy of Doug Cheney, CAD Interoperability Consultant, ITI TranscenData
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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.
Courtesy of Doug Cheney, CAD Interoperability Consultant, ITI TranscenData
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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.
Courtesy of Doug Cheney, CAD Interoperability Consultant, ITI TranscenData
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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
•
•
•
•
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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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
Regularization
Closure
of interior
interior
Non-regular 2D set
Closed regular set
A
B
Two i nt er sec tin g
So li ds A, B
Intersection
A ∩B
Regularized
intersection
• 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)
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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
Fk
Fk
Fi
Fi
Boundary representations are used in CAD systems to store and archive
results of all operations (including set operations)
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Exact Theory III:
Point Membership Classification (PMC)
in
true x ∈ S
false Otherwise
•
•
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out
on
x ∈ Inside S
x ∈ Outside S
x ∈ Boundary of S
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
Two in te r f er in g
So li ds A, B
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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
Kettner et al 2004
• 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 …
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Relation to Geometric Robustness
• Many excellent surveys
– Hoffmann 89, 01, Yap 97, Michelucci 98, Schirra 98, …
• Popular techniques
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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-cubic
surfaces S(u,v) is degree 324
– Precision grows exponentially in degree and depth
• Round-off is unavoidable
Hoffmann & Stewart, 2005
SolidWorks → STEP → SolidWorks
• Imprecise, sampled, or transferred data
• Incomplete representation spaces
– Algebraic varieties vs rational parameterizations
– Set operations
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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
•
•
•
•
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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Rough and brief history
of mechanical tolerancing
< 1700s
Skilled artisans manufacture to precision, custom fit, small
batches, no notion of accuracy or measurement
17001800s
Gaging and interchangeability, first notions of accuracy
and consistency … mass production around 1900s
1930s1940s
Tolerances mentioned on drawings
1950s
Parametric +/- tolerances, idealized form, notes
1970s
Geometric dimensioning & tolerancing (zones, material
conditions, containment), finite set of symbols, measures
1990s present
Formal definition of semantics (first edition 1994),
standardized metrology algorithms
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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Tolerance semantics relies on Zones
Datums
Maximum Material Condition
Least Material Condition
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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
•
•
•
•
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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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
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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
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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
0.001 mm
thickeness
1000 mm
SolidWorks → STEP → Pro/Engineer -- EXACTLY, after repair
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
• Motivated by interval arithmetic to
computer intersection curves
• Approximation of the exact solid
• Boundary is ambient isotopic to
the exact
• Perturbation semantics
Sakkalis & Peters, 2003
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Set Intervals: Partial Solids
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(Edalat & Lieutier, 1999)
Point Membership Classification PMC: E3 → {true, false} is not computable
in domain theoretic sense (not continuous function)
Redefine PMC as
true x ∈ Inside S
false x ∈ Outside S
x ∈ Otherwise
⊥
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
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How to tolerance a set interval?
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•
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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
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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 iε
iεk0(X) ⊆ X ⊆ kεi0(X)
• As ε → 0
i0k0(X) ⊆ X ⊆ k0i0(X)
• Usually do not know X, but only interval
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[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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Is this a boundary of an ε-regular set ?
Abstract complex K
Orientable manifold
Can be realized in Ed
as |K|
Too small
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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|
Too large? Wrong topology
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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|
So large, topology is correct,
but “destroys” K
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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|
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 homotopy
equivalent ( ) to exact boundary
e1
v
e2
v
e1
e2
Zv
Ze1
Zv
Ze2
Zv
Ze1
Zv
Ze2
Ze2
Ze1
Ze2
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Zv
Ze1
Ze2
Zv
Ze1
Zv
Ze2
Ze1
Ze2
Ze1
Zv
Ze1
Ze2
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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
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•
•
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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?
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Thank you
Supported in part by
•
•
•
National Science Foundation grants DMI-0500380, DMI-0323514
National Institute of Standards & Technology (NIST)
General Motors Corporation
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