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Exact Constraint Design Using Tolerance Analysis Methods
Danny Smith
Brigham Young University
15 June 2001
Special Acknowledgements to:ADCATS ResearchNSF Grant DMI 0084880
ADCATS 2001
ADCATS 2001
Presentation Outline
Background Constraint Analysis and Screw Theory Tolerance Analysis
Variation-based Constraint Analysis of Assemblies (VCAA) Method
Case Studies Conclusion
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Why Analyze for Constraints?
Key Definitions:
Degrees of Freedom Exact-constraint Overconstraint Underconstraint
O
tx ,x
tz ,z
ty ,y
Z-axis
Y-axis
X-axis
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Common Assembly Problems
Overconstraint or Redundant DOF
Underconstraint or Idle DOF
CrankSpherical
Joints
Slider
Link
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Current Constraint Methods
Kinematic Constraint Pattern Analysis [Blanding 1999]
Geometric Constraint Solving [Hoffmann and Vermeer 1995]
Screw Theory-Based Constraint Analysis [Adams 1998], [Konkar 1993], and [Adams and
Whitney 2001]
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Screws – Twists and Wrenches
Twist Wrench
T x y z x y zv v v W f f f m m mx y z x y z
Please see [Adams and Whitney 2001] for details
v
Screw Axis P
r
O
Z
Y
X
O
Z
Y
X
f
m
Screw AxisP
r
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Fundamental Principles
Reciprocity of twists and wrenches
Screw coordinate representation
Virtual coefficient Solve for twistmatrix
and wrenchmatrix
T W 0
T
1 1 1 1 1 1
2 2 2 2 2 2
x y z x y z
x y z x y z
v v v
v v v
W
f f f m m m
f f f m m mx y z x y z
x y z x y z
1 1 1 1 1 1
2 2 2 2 2 2
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Screw Theory Steps
1. Locate mating features on assembly using transformation matrices.
2. Form Twist matrices for each mating feature
3. Use screw algorithms and linear algebra to solve for Resultant Twist and Resultant Wrench matricesPlease see [Adams and Whitney 2001] for details
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DOF Analysis Example
Individual feature screw representation
Algorithms Resultant Twistmatrix
and Wrenchmatrix Interpretation
2
2
4
X
Y
Z
x
x
x
y
yy
zz
f2f1
Taken From [Adams and Whitney 2001]
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Example (Cont.)
2
2
4
X
Y
Z
x
x
x
y
yy
zz
f2f1
Assembly DOF and Constraint Solution
F 1
1 0 0 2
0 1 0 2
0 0 1 0
0 0 0 1
F 2
0 1 0 2
1 0 0 6
0 0 1 0
0 0 0 1
TR esu ltan t 0 0 1 2 2 0
W R esu ltan t
0 1 0 0 0 2
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0Taken From [Adams and Whitney 2001]
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Tolerance Analysis Background
Dimensional, Kinematic, and Geometric Variation
Direct Linearization Method (DLM)
Vector Loops Global Coordinate Method
(GCM)
+
R + RR
U1
U2
U
A
R
Please see [Chase 1999] and [Gao 1993] for complete details
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Direct Linearization Method
Manufactured or Independent variables Assembly or Dependent variables Geometric Feature variables
a
b
f
c
1
A x B u F 0
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Vector Loops and GCM
A Matrix Independent Variable
Sensitivity Matrix B Matrix
Dependent Variable Sensitivity Matrix
F Matrix Geometric Feature Variable
Matrix
a
b
f
c2
1
2
c1
Y
X
0.025
0.030
0.010 3, 4
1
2
Sensitivities are determined by the GCM
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Development of the Variation-based Constraint Analysis of Assemblies
(VCAA) Method
Variation analogies Velocity Force and moments
GCM connection Employs screw theory Solves for under- and
overconstraints
underconstraint
information[B]
overconstraint
information[F]
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VCAA for Underconstraints
[B] Tcolumn Tjoint i
Wintermediate-joint i Wintermediate-partj TResultant-partj
DLM ToleranceAnalysis
Transpose and Switch
Associate DependentVariables to Joint Types
ReciprocalOperation
ReciprocalOperation
Union MatricesFor Each Part
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VCAA for Overconstraints
DLM ToleranceAnalysis
Transpose
[F] Wcolumn Wjointi
Tintermediate-jointi
Tintermediate-part j or
Tintermediate-loopk
WResultant-part j or
WResultant-loopk
Associate GeometricFeature Variables to Joint Types
ReciprocalOperation
ReciprocalOperation
Union MatricesFor Each Part or Loop
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Case Studies of VCAA
Case 1 - One-way Clutch Assembly in 2-D Case 2 - Stacked Blocks Assembly in 2-D Case 3 - Crank Slider Assembly in 3-D
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Case 1 - One-Way Clutch Assembly
Transmits torque in one rotational direction
Assembly formed from Roller, Hub, and Ring
Pressure Angle 1 is the key dimension
a
b
f
c2
1
2
c1
Y
X
0.025
0.030
0.010 3, 4
1
2
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Case 1 - Sensitivity Matrices
Sensitivity Matrices Calculated using GCM
B
b 1 2
1 39 075 0
0 4 8101 0
0 0 0
0 0 0
0 0 0
0 1 1
.
.
F
1 2 3 4
0 0 1 2 2 1 9 0 1
1 0 9 9 2 5 1 1 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
.
.
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Case 1 - Underconstraint Analysis
Form Joint Twists for each joint from [B]
Perform intermediate steps See [Adams 1998]
Evaluate Resultant Twist for each part to identify underconstraint information
CylinderSlider
ParallelCylinder
Revolute
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Case 1 - Underconstraint Solution and Results
Resultant Twists for each part show any underconstrained degrees of freedom
T
T
T
R esu ltan t R o lle r
R esu ltan t R in g
R esu ltan t H u b
0 0 1 0 0 0
0 0 0 0 0 0
0 0 1 3 9 0 7 5 4 8 1 0 5 0. .
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Case 1 - Overconstraint Analysis
Form Joint Wrenches for each joint from [F]
Perform intermediate steps See [Adams 1998]
Evaluate Resultant Wrench for each part to identify overconstraint information
0.025
0.030
0.010 3, 4
1
2
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Case 1 - Overconstraint Solution and Results
Resultant Wrench for each set shows any overconstrained degrees of freedom
W R esu ltan t - F u ll -S e t
0 0 0 1 0 0
0 0 0 0 1 0
0 0 1 0 0 0
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Case 2 – Stacked Blocks Assembly
Theoretical assembly for tolerance analysis
Assembly formed from Base, Block, and Cylinder
Vertical placement A of cylinder is key dimension
Three Vector Loops needed
A
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Case 2 - Sensitivity Matrices
Sensitivity Matrices Calculated using GCM
B 1
1 2 3 4
0 0 9 6 4 6 7 0 0 0 1 8 7 1 8 1 0 0 4 8 0 0
1 0 2 6 3 4 7 1 0 0 6 6 2 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 1 1 0 0
A D E H L . . .
. .
B 2
1 2 3 4
0 0 0 0 9 6 4 6 7 0 0 1 0 0 4 8 4 0 6 0 0
0 0 1 0 2 6 3 4 7 0 0 0 3 9 0 5 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 1 0
A D E H L . . .
. .
B 3
1 2 3 4
0 0 0 0 9 6 4 6 7 0 9 6 4 6 7 0 0 4 0 6 0 1 0 6 7 5
0 0 0 0 2 6 3 4 7 0 2 6 3 4 7 0 0 3 9 0 5 2 8 1 2 5
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 1 1
A D E H L . . . .
. . . .
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Case 2 - Sensitivity Matrices
Sensitivity Matrices Calculated using GCM
F 1
1 2 3 4 5 6 7
1 1 0 2 6 3 4 7 0 2 6 3 4 7 1 0 0
0 0 0 9 6 4 6 7 0 9 6 4 6 7 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
. .
. .
F 2
1 2 3 4 5 6 7
0 0 0 0 1 0 2 6 3 4 7 0
0 0 0 0 0 0 9 6 4 6 7 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
.
.
F 3
1 2 3 4 5 6 7
0 0 0 0 0 0 2 6 3 4 7 0 2 6 3 4 7
0 0 0 0 0 0 9 6 4 6 7 0 9 6 4 6 7
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
. .
. .
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Case 2 - Underconstraint Analysis
Form Joint Twists for each joint from [B]
Perform intermediate steps See [Adams 1998]
Evaluate Resultant Twist for each part to identify underconstraint information
Y
X
cylslider2
cylslider1
edgeslider2
edgeslider1edgeslider3
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Case 2 - Underconstraint Solution and Results
Resultant Twists for each part shows any underconstrained degrees of freedom
TR esu ltan t cy lin d er 0 0 1 1 8 7 1 8 6 6 2 0 0. .
TR esu ltan t b lo ck 0 0 0 0 0 0
TR esu ltan t b ase 0 0 0 0 0 0
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Case 2 - Overconstraint Analysis
Form Joint Wrenches for each joint from [F]
Perform intermediate steps See [Adams 1998]
Evaluate Resultant Wrench for each part to identify overconstraint information
0.020
0.050
0.080
0.080
0.020
0.0500.050
1
7
6
2
4
5
3
Y
X
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Case 2 - Overconstraint Solution and Results
Resultant Wrench for each set show any overconstrained degrees of freedom
W R esu ltan t
0 2 7 3 1 1 0 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 1 0 0 0
.
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Case 3 – Crank Slider Assembly
Assembly formed from Base, Crank, Link, and Slider
Slider Position U is the key dimension
One Vector Loop needed
X ZY
U
CRANK
BASE
LINK
SLIDER
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Case 3 - Sensitivity Matrices
Sensitivity Matrices Calculated using GCM
B
1 2 3 4 5 6
1 7 6 7 8 0 8 5 7 9 1 3 9 6 7 7 0 3 11 1 5 2 4 4 6 8 1
1 0 4 3 5 7 8 4 8 5 3 9 8 5 1 3 3 9 7 1 6 4 2 6 0 7 8 4 4 3 6 0 4 0
1 4 0 4 1 9 8 4 8 5 3 6 7 2 0 1 0 2 4 7 1 5 4 1 9 4 3 5 5 0
0 9 2 3 9 0 0 1 5 6 6 0 0 3 4 9 2 0 8 7 2 1 0
0 3 5 3 6 0 7 0 7 1 0 6 9 8 4 0 0 6 2 2 3 0 4 8 9 4 0
0 1 4 6 4 0 7 0 7 1 0 6 9 8 4 1 0 7 0 0 6 0 0
U. . . . .
. . . . . .
. . . . .
. . . .
. . . . .
. . . .
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Case 3 - Sensitivity Matrices
Sensitivity Matrices Calculated using GCM
F
1 2 3 4 5
6 7 8 9 1 0 1 2 3
0 0 0 9 2 3 9 0 3 4 9 2 0 1 5 6 6
0 7 0 7 1 0 7 0 7 1 0 3 5 3 6 0 6 2 2 3 0 6 9 8 4
0 7 0 7 1 0 7 0 7 1 0 1 4 6 5 0 7 0 0 6 0 6 9 8 4
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 4 8 9 4 0 8 7 2 1 0 0 0 0 0
0 0 8 7 2 1 0 4 8 9 4 1 0 0 0 3 9 7 1 6 4
1 0 0 0 1 0 3 9 7 1 6 4 0
0 0 0 0 0 1 0 0
0
. . .
. . . . .
. . . . .
. .
. . .
.
0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
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Case 3 - Underconstraint Solution and Results
Resultant Twists for each part show any underconstrained degrees of freedom
TR esu ltan t b ase 0 0 0 0 0 0
TR esu ltan t lin k 6 3 0 8 6 2 4 1 4 2 1 1 2 0 7 11 7 1 2 5 9 6 9 5 8 8 3 8. . . . .
TR esu ltan t s lid er 0 0 0 0 0 0
X ZY
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Case 3 - Overconstraint Solution and Results
Resultant Wrench for each set shows any overconstrained degrees of freedom
W R esu ltan t lin k
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
W R esu ltan t s lid er
0 1 0 0 0 0
0 0 1 0 0 0
X ZY
CRANK
BASE
LINK
SLIDER
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Conclusions
VCAA Method connects Constraint Analysis and Tolerance Analysis
Based on Screw Theory and the Global Coordinate Method
The VCAA Method can extract twist and wrench matrices directly from the vector model
Can perform a constraint analysis and a tolerance analysis simultaneously
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Bibliography
Adams, Jeffrey D. Feature Based Analysis of Selective Limited Motion in Assemblies. Master of Science Thesis, Massachusetts: Massachusetts Institute of Technology, 1998.
Adams, Jeffrey D.; Whitney, Daniel E. “Application of Screw Theory to Constraint Analysis of Mechanical Assemblies Joined by Features.” In Journal of Mechanical Design: Transactions of the ASME, Vol. 123, pp. 26-32, March 2001.
Blanding, Douglass L. Exact Constraint: Machine Design Using Kinematic Principles. New York: ASME Press, 1999.
Chase, Kenneth W. Dimensioning & Tolerancing Handbook, ed. Paul J. Drake, Jr., New York: McGraw Hill, “Multi_Dimensional Tolerance Analysis.”, 1999.
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Bibliography (cont.)
Chase, Kenneth W.; Gao, Jinsong; Magelby, Spencer; Sorensen, Carl. “Including Geometric Feature Variations in Tolerance Analysis of Mechanical Assemblies.” In IIE (Institute of Industrial Engineers) Transactions, Chapman & Hall Ltd., pp. 795_807, 10 Oct 1996.
Gao, Jinsong; Chase, Kenneth; Magleby, Spencer. “Generalized 3-D Tolerance Analysis of Mechanical Assemblies with Small Kinematic Adjustments.” In IIE (Institute of Industrial Engineers) Transactions, Chapman & Hall Ltd, pp. 367_377, 4 April 1998.
Gao, Jinsong; Chase, Kenneth; Magleby, Spencer. “Global Coordinate Method for Determining Sensitivity in Assembly Tolerance Analysis” in Proceedings of the ASME International Mechanical Engineering Conference and Exposition, Anaheim, California, 1998
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Bibliography (cont.)
Gao, Jinsong. Nonlinear Tolerance Analysis of Mechanical Assemblies. A Doctor of Philosophy Dissertation, Provo, Utah: Brigham Young University, August 1993.
Hoffmann, Christoph; Vermeer, Pamela. Computing in Euclidean Geometry (2nd Edition), ed. Du, Ding-Zhu; Hwang, Frank, Singapore: World Scientific Publishing Co. Pte. Ltd., “Geometric Constraint Solving in 2 and 3.”, pp. 266-298, 1995.
Konkar, Ranjit. Incremental Kinematic Analysis and Symbolic Synthesis of Mechanisms. Doctor of Philosophy Dissertation, Palo Alto, California: Stanford University, June 1993.
Konkar, Ranjit; Cutkosky, M. “Incremental Kinematic Analysis of Mechanisms.” In Journal of Mechanical Design, Vol. 117, pp. 589-596, December 1995.
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Bibliography (cont.)
Roth, Bernard. “Screws, Motors, and Wrenches that Cannot be Bought in a Hardware Store.” In Robotics Research, Chapter 8, pp 679-693, 1984.
Waldron, K. J. “The Constraint Analysis of Mechanisms.” In The Journal of Mechanisms, Volume 1, pp 101-114. Great Britain: Pergamon Press, 1966.
