Geometric Tolerance Analysis Methods
for Imperfect-Form Assemblies
Scott Pierce
M.I. Technologies, L.L.C.
Duluth, GA
David Rosen
George W. Woodruff School of Mechanical Engineering
Georgia Institute of Technology
Outline
• Introduction and Background
– Motivation
– Our Approach: The Generate and Test Method
• Development of the Tolerance Analysis Module
– The Variational Modeling Environment
– Simulation of Mating Between Imperfect-Form Components
• Case Study
• Conclusions and Future Work
Motivation
• The principal objective of this
research is the development of a
new, computer-aided approach
to tolerance analysis
– The purpose of tolerance
analysis is to define the
relationships between tolerance
values, product functionality
and manufacturing cost
– In particular, we are interested
in analysis of geometric
tolerances that control form
and orientation
A Very Simple Example: Two
Squares in a Slot
Motivation
A More Complex Example: The High-Speed Stapling Mechanism
• A compound slider mechanism
composed of several
components
• Multiple mating surfaces
• Alignment between the driver,
bender and bonnet affects
functionality of the mechanism
• Manufacturer has quality
assurance data and process
experience that define “typical
manufacturing errors”
Bonnet
Driver Slider
Bender and Driver
Slider Reciprocate in Bonnet and
Driver Reciprocates in Bender
Driver
Bender
How can the experience-based process knowledge be
incorporated into this complex tolerance analysis
problem?
Our Approach: Generate-and-Test
Step 1: Generate “As Manufactured” Component Models
Our Approach: Generate-and-Test
Step 2: Test the Effects of Manufacturing Errors by Simulating Mating Between
As-Manufactured Components and Measuring Attributes of Functionality
Our Approach: Generate-and-Test
• This Generate-and-Test Process is
Repeated for a Series of Error
Geometries and Magnitudes That are
Representative of the Proposed
Manufacturing Processes
• Information Gained from this
Analysis is Used to Guide the
Tolerance Selection Process
Development of the Tolerance Analysis Module
• The Variational Modeling
Environment
• Simulation of Mating Between
Imperfect-Form Components
The Variational Modeling Environment
• We
have built a CAD environment that uses a NURBS surface
representation to construct models of imperfect-form variants of
prismatic components
• The ACIS geometry engine is used
as the modeling core
• Constructed as a set of C++ classes
and extensions to the ACIS api’s
• Basic capabilities of the
modeling environment allow:
– Creation of prismatic parts
– Use of Boolean operations
to generate more complex
prismatic geometry
– Application of rigid-body
transformations
The Variational Modeling Environment
• Variational modeling
capabilities allow:
– Definition of pointset
classes that define variant
surfaces
– Fitting of NURBS surfaces
to a pointset to within a
specified fitting tolerance.
– Replacing nominally planar
faces of prismatic
components with variant
NURBS surfaces.
The Variational Modeling Environment
Verification:
Table 3.1: Measurement of Fitting Accuracy of Nurbs Surfaces To Pointset
Measurements For A Series Of Milling Errors.
Manufacturing Error
The Variational Modeling
Module supports modeling of
as-manufactured component
variants to a resolution that is
significant to tolerance
analysis.
Typical machining errors for
end milling are on the order
of 0.01 mm.
Side-Milling Cutter
Deflection - 0.25 mm.
Depth of Cut
Side-Milling Cutter
Deflection - 1.2 mm.
Depth of Cut
Side-Milling Cutter
Deflection - 0.5 mm.
Depth of Cut
Side-Milling Cutter
Deflection - 1.2 mm.
Depth of Cut
Transverse Cutter
Deflection of a Slot
Bottom Surface- Slot Cut
By Side Milling With
Very Light Cut
Tooth-to-Tooth Runout
in Upmilling Using an
Arbor Cutter (Note: This
surface has a highfrequency periodic error)
Surface
Length
(mm.)
Surface
Maximum
Average Maximum
Width Deviation of the Fitting
Fitting
(mm.) Machining Data Error
Error
from a Perfect
(mm.)
(mm.)
Plane (mm.)
20.3
15.2
0.0458
0.00002
0.00004
20.3
15.2
0.0902
0.00002
0.00004
50.0
25.4
0.0695
0.00002
0.00008
50.0
25.4
0.1144
0.00000
0.00006
70.0
40.0
0.0038
0.00000
0.00005
24.0
22.0
0.0860
0.00001
0.00004
Simulation of Mating Between Imperfect-Form Components
• Simulation of mating between surfaces that can be
represented analytically is a well understood problem.
• Simulation of mating between non-analytic, freeform
surfaces is a much more difficult problem.
Following a formulation originally
proposed by Turner, we have chosen to
formulate the mating problem as a
mathematical programming problem of
the form:
Minimize Z = total distance from perfect fit
s.t. non-interference between components
Simulation of Mating Between Imperfect-Form Components
• How should “perfect fit” be defined?
– For perfect-form, planar surfaces perfect fit means that faces are
coplanar and that outward-facing normal vectors point in opposite
directions.
– Coplanarity implies that the distance between any point on one
surface and the corresponding closest point on the other surface is
zero.
• This leads to the idea that for imperfect-form surfaces we should
try to minimize the distance between any point on one surface and
the corresponding closest point on the other surface
Simulation of Mating Between Imperfect-Form Components
• We have chosen to use
sampling grids to perform
distance measurements and
interference detection between
surfaces
• We find that the use of
sampling grids is much more
computationally efficient than
the use of Boolean intersections
• Grid density can be adjusted so
that the resolution is fine
enough to represent any
significant surface feature.
Non-Mating But Potentially
Interfering Pair
Mating Pair
Mating Pair
Simulation of Mating Between Imperfect-Form Components
Using this sampling grid approach, we can formulate the mating problem as a
constrained optimization problem:
Find: x = (roll, pitch, yaw, x, y, z) = six degrees of freedom of the movable body.



  M mF nF


2
1
     d Fij ( x )  
Minimize: Z   M

  F  1 i 1 j  1


m
n



F
F



 F1


s. t.: d Fij  0
1
2

Non-Mating But Potentially
Interfering Pair
F = 1…N, i = 1…mF, j = 1…nF
M = number of mating face pairs
Mating Pair
N = total number of mating face pairs, both
mating and potentially interfering
mF = number of gridpoints in the u-parameter
direction for the given face pair
nF = number of gridpoints in the v-parameter
direction for the given face pair
d Fij ( x )= minimum signed distance from gridpoint ij
to the mating surface.
Mating Pair
Simulation of Mating Between Imperfect-Form Components
• Selection of a Solution
Method:
• Finding the minimum distance
between a grid point and the
mating surface requires the
solution of a point-projection
problem
• Point projections are the most
computationally-intensive part
of the mating simulation
Grid Point
Closest Point on
Mating Surface
d Fij
Simulation of Mating Between Imperfect-Form Components
Selection of a solution
method (continued):
We used published
measurements of end-milled
surfaces to generate test
surfaces
We explored the topography of
the solution space generated by
mating these surfaces
We found the solution space to
be nonlinear
We found the boundaries of the
feasible region to be nonlinear
and in some cases non-convex
Simulation of Mating Between Imperfect-Form Components
• We examined several potential solution methods including:
– Successive linearization:
– We have shown that both the objective and the constraints are
highly nonlinear. Successive linearization will generally not
converge well under these conditions
– Generalized reduced gradient methods
– Capable of handling nonlinear problems
– Requires solution of a prohibitively large number of point
projection problems
Simulation of Mating Between Imperfect-Form Components
• We have chosen to modify the formulation of the mating
problem to use the penalty function approach:


2
  N mF nF 

non 
1
int erfering


int
erfering
      WF d ij  x 
Minimize: Z   N

W
d
x



INT ij

  F  1 i 1 j  1  

  mF nF 
 F1

where:
WF = mating/non-mating face switch
WINT = interference weighting factor (=100)
The penalty function formulation converts the constrained
formulation into an unconstrained problem, allowing the use of
unconstrained optimization algorithms


2 
  
  


1
2
Simulation of Mating Between Imperfect-Form Components
To test potential solution algorithms we used two test problems:
End-milling cutter deflection
with perfect-fit
End-milling cutter deflection
with four different surfaces
“Correct” objective = 0.1484
Simulation of Mating Between Imperfect-Form Components
• We tested three different solution algorithms for use with
the penalty-function formulation:
• Method 1: Simulated Annealing With Downhill Simplex:
– Analogous to an annealing process, there is a finite probability of
accepting an “uphill” move
P( X 2 )  C exp Z 2  Z1  / T 
Where Z2 > Z1
– This probability is reduced as the “temperature” is reduced
– In theory, allows a more thorough search of the solution space so
that local minima are avoided
In tests where we purposely introduced local minima into the solution
space, simulated annealing was not very successful in avoiding them.
Simulation of Mating Between Imperfect-Form Components
• Method 2: Randomized Hooke-Jeeves pattern search
– Direct search method does not require the calculation of numerical
gradients
– Explores the region around a test point for the steepest descent
direction, then moves in that direction until descent stops
– When a downhill move cannot be found the step size is reduced
and the exploration is repeated
– Very robust in the presence of nonlinearities
Simulation of Mating Between Imperfect-Form Components
• Method 3: Quasi-Newton Method
–
–
–
–
–
Gradient-based method
Uses a quadratic approximation to the objective function
Use first-order information to approximate the Hessian
Use line searches to generate the step size
We use the Broyden-Fletcher-Goldfarb-Shanno form of the quasiNewton method.
– We used a line search that starts with a quadratic approximation,
then reverts to a golden section search when convergence slows.
Simulation of Mating Between Imperfect-Form Components
0.4
Correct Answer:
Simulated Annealing
Objective = 0.1484
Best Objective (mm.)
0.35
0.3
Hooke-Jeeves
0.25
Best Answer from
This Test:
0.2
BFGS
0.15
BFGS Objective =
0.1484
0.1
0
200
400
600
800
1000
1200
Objective Evaluations
Convergence of all three solution methods for the four
surface cutter deflection example
Simulation of Mating Between Imperfect-Form Components
0.5
Best Objective (mm.)
0.45
0.4
0.35
Simulated Annealing
0.3
0.25
0.2
Hooke-Jeeves
0.15
0.1
0.05
BFGS
0
0
500
1000
1500
Objective Iterations
Convergence of all three solution methods for the
perfect-fit cutter deflection example (correct objective
value = 0)
Simulation of Mating Between Imperfect-Form Components
0.02
Best Objective (mm.)
0.018
Hooke-Jeeves
Algorithm
Alone
0.016
0.014
Hybrid Algorithm When
BFGS is Active
0.012
0.01
0.008
Hybrid Algorithm After
“Fallback” to Hooke-Jeeves
0.006
BFGS Algorithm
Alone
0.004
0.002
0
0
50
100
150
200
250
300
350
400
Objective Evaluations
Use of the hybrid BFGS/Hooke-Jeeves algorithm for
the perfect-fit problem (correct objective value = 0)
Tolerance Analysis Module - Summary
• Allows construction and manipulation of “asmanufactured” variant models
• Simulates assembly of imperfect-form component variants
• We now have a testbed that can be used to demonstrate the
generate-and-test approach to tolerance analysis
Case Study
• The case study is built
upon a simplified version
of the high-speed stapling
mechanism
• The components that have
the most influence on the
quality of the stapling
process are included in the
study:
– Driver
– Bender
– Bonnet
BONNET
DRIVER
BENDER
PAGES
TO BE
STAPLED
WIRE
Case Study
Attribute of Functionality - Z-axis Rotation
+Y
+X
DIRECTION OF FORCE
ON STAPLE LEG
Z-Axis Rotation Attribute: Maximum possible difference between
bender Z-rotation and driver Z-rotation
Functional Limit: If Z-axis rotation exceeds 1.7 mrad the staple
will buckle
Case Study
In order to control the attributes of functionality we assign
geometric tolerances of form and orientation:
Bender
What tolerance values should be assigned in order to ensure
that the mechanism will function?
Case Study
• Step 1:
Group the surfaces of the
stapling mechanism into
four groups
• All surfaces within a group
would be manufactured in
a single setup, therefore
they share a common level
of precision
Case Study
•
•
Step 2:
Generate “as-manufactured”
models of higher precision (higher
cost) and lower precision (lower
cost) variants of each surface group
All error data comes from
published measurements or results
of end-milling simulations.
Case Study
• Step 2 (continued):
• Each component variant was measured using a functional gauging routine
• The results of each measurement were the tolerance values that the
particular variant would meet
OUTER BOTTOM
FACE FLATNESS
(mm.)
OUTER NEGATIVE Y
FACE
PERPENDICULARITY
(mm.)
GROOVE AT
HIGHER
PRECISION,
OUTER
SURFACES AT
HIGHER
PRECISION
GROOVE AT
LOWER
PRECISION,
OUTER
SURFACES
AT LOWER
PRECISION
0.05
0.12
0.11
0.19
Case Study
• Mating simulation was performed for every combination of
higher/lower precision surface groups
• The mechanism components were mated at a series of positions
through the stapling process
• Functional attributes were measured
• The results were used to construct a 24 full-factorial analysis for
each functional attribute
Case Study
Analysis of Variance for the Z-axis rotation attribute:
DRIVER
SINGLE
FACTOR
EFFECT
BENDER BENDER
GROOVE OUTER
BONNET
-0.49713 -39.21363
-8.803
BENDER
GROOVE/
BENDER BENDER
BENDER DRIVER/ GROOVE/ OUTER/
BONNET BONNET BONNET
OUTER
DRIVER/ DRIVER/
BENDER BENDER
GROOVE OUTER
TWO FACTOR
INTERACTION
EFFECT
-0.49713
-1.01513
0
-0.72031
-0.4569
-2.40009
-0.56839
Significant effects: Bender Groove, Bonnet
Setting both of these surface groups to higher precision results in a maximum Zaxis rotation of 1.92 mrad.
This is still above the functional limit of 1.7mrad, so the tolerance on the bender
groove needs to be tightened further if possible
Case Study
Combining the results of the Z-axis rotation study with results from a study on a
second functional attribute, we selected geometric tolerance values that strike a
balance between the need for precision and manufacturing cost.
Conclusions
• I have described the development of an environment for
computer-aided tolerance analysis.
• Allows the inclusion of experience-based manufacturing
information through the use of the generate-and-test
method of tolerance analysis.
• Development of an effective algorithm for simulation of
mating between imperfect-form, non-analytic surfaces was
key to the generate-and-test method.
• Through the case study, I have shown that this method can
be used as an aid in the selection of geometric tolerances of
form and orientation.
Possibilities for Future Work
• Link mating simulation with a kinematic analysis in order
to bring force balance information into the picture.
• Extend to non-prismatic geometry.
• Apply the non-analytic surface mating methods in areas
other than tolerance analysis (e.g. design of components
whose perfect-form geometry is non-analytic).