Including GD&T Tolerance Variation
in a Commercial Kinematics
Application
Jeff Dabling
Surety Mechanisms & Integration
Sandia National Laboratories
Research supported by:
Summary
Variation Propagation
 Obtaining Sensitivities
 Variation/Velocity Relationship
 Equivalent Variational Mechanisms in 2D
 EVMs in 3D
 Example in ADAMS

3 Sources of Variation in Assemblies
DA
+ A
A
q
q+
Dq
R
R +DR
U
U +DU
Dimensional and
Kinematic
q
R
A
U
U + DU
Geometric
R
DLM Vector Assembly Model
CL
RL
Gap
Open Loop
RT
i
e
Plunger
a
b
h
Base
RL
r
u
q

Arm
Closed Loop
g
Pad
Reel
How Geometric Variation Propagates
3D cylindrical
slider joint
Y
X
Flatness
Tolerance
Zone
Z
Nominal
Circle
Translational
Variation
Rotational
Variation
Cylindricity
Tolerance
Zone
Flatness
Tolerance
Zone
View looking down the cylinder axis
View normal to the cylinder axis
The effect of feature variations
in 3D depends upon the joint
type and which joint axis you
are looking down.
3D Propagation of Surface Variation
K Kinematic Motion
F Geometric Feature Variation
F
F
K
K
y
K
x
K
Cylindrical Slider Joint
K
x
F
z
F
z
K
y
K
F
Planar Joint
Variations Associated with Geometric
Feature – Joint Combinations
(Gao 1993)
Geom
Joints Tol
Planar
R x Rz
Rx Rz
Rx R z
R x Rz
Rx Rz
Rx R z R xRzT y
Revolute
R x Rz
Rx Rz
Rx Rz
Rx R z
R x Rz
Rx Rz
Rx R z
Rx Rz
T x Tz
Tx T z
Cylindrical
R x Rz
R x Rz
Rx Rz
Rx R z
R x Rz
Rx Rz
Rx R z
Rx Rz
T x Tz
Tx T z
Prismatic
R xR yR z R xR yRz
Spherical
R xRy Rz R xR yR z Rx RyR z R xR yR z
T x T yTz
CrsCyl
Ty
Ty
ParCyl
Ty R x
T y Rx
E
P T y Rx
CylSli C
P T y Rx
Pt
PntSli
Ty
P
S
SphSli
Ty
P
T y Rx
EdgSli
Tx T yTz
Ty
T y Rx
Ty R x
Ty R x
Ty
Ty R x Ty R x
Ty R x
T y Rx
T x TyT z Tx T yT z Tx TyT z
Rx
Rx
Rx
Ty R x
Ty R x
Ty R x
Ty R x
Ty R x
Ty R x
Ty R x
Ty R x
Ty R x
Ty R x
Ty R x
Ty R x
Ty R x
Ty R x
Ty R x
Ty R x
Ty
Ty
Ty R x
Ty
Ty
Ty R x
Ty R x
Ty
Ty R x
Ty
Ty
Ty
Ty
Ty
Ty
Ty
Ty
Ty
Ty
Ty
Ty
Ty
Ty
Ty
Ty
Ty
Including Geometric Variation

Variables used have nominal values of zero

Variation corresponds to the specified tolerance value

Rotational variation due to flatness
variation between two planar surfaces:
 Flatness Tolerance Zone 
D  tan 1 

 Characteristic Length 

Translational
variation:
variation
Translational Variation  
due
a
2
to
flatness
Rotational = ±D
Variation
Flatness
Tolerance = a
Zone
Translational =±a/2
Variation
Flatness
Tolerance =
Zone
a
Geometric Variation Example


Translational: additional vector with
nominal value of zero. (a3, a4)
Rotational: angular variation in the joint
of origin and propagated throughout the
remainder of the loop. (1, 2)
f
.01
U2
q
R
H
.02
A
.01
U1
H x  U1 cos(0) + R1 cos(0) + H cos(90 + 1+2) + R2 cos(90 + f + 1+ 2) +
.01
(a3, a4)
R3 cos(90 + f + 1+2 ) +a3cos(90 + 1+2) +a4cos(90 + 1+2) +
f
R2
U2
U 2 cos(180 + f + 1+2) + A cos(270 + 1+ 2)  0
q
R3
H
A
R1
U1
(1, 2)
Sensitivities from Traditional 3D
Kinematics
Sandor,Erdman 1984:

3D Kinematics using 4x4 transformation matrices [Sij] in a
loop equation
[ S ]  [ S ][ S ][ S
][ S ]  [ I ]
00

01
( n 1) n
n0
Uses Derivative Operator Matrices ([Qlm], [Dlm]) to eliminate
need to numerically evaluate partial derivatives
[ Sij (qm )]
qm

23
 [ Sij (qm )][Qlm ]
Equivalent to a small perturbation method; intensive
calculations required for each sensitivity
Sensitivities from
Global Coordinate Method
(Gao 1993)
Uses 2D, 3D vector equations
 Derives sensitivities by evaluating effects of
small perturbations on loop closure equations

Length Variation Rotational Variation
H x
 cosa
Li
H x
  3Y   2 Z
fi
H y
H y
Li
 cos 
fi
 1Z   3 X
H z
 cos 
Li
H z
  2 X   3Y
fi
Hq x
Hq x
Li
Hq y
Li
Hq z
Li
0
f i
1
0
Hq y
0
Hqx
 3
f i
f i
 2
(taken from Gao, et. al 1998)
Variation – Velocity Relationship
(Faerber 1999)
Tolerance sensitivity solution
da 2 
da 2 
 dr 
 dr 
1


 1 
da 3 
1

   B A dr2   Si , j  dr2 
da 4 
 dr 
 dr 
3


 3
 dr4 
 dr4 
24
r3
23
r4
r2
22
r1
21
Velocity analysis of the equivalent mechanism
 2 
 r 
 1 
 3 
1
    B A  r2   J i , j
 4 
 r 
 3
 r4 


 2 
 r 
 1 
 r2 
 r 
 3
 r4 
 
r3
r2
r4
When are the sensitivities the same?
2D Equivalent
Variational Mechanisms
24




2D Kinematic
Joints:
Add
dimensional
variations to a
kinematic model using kinematic
elements
Parallel
Converts
analysis
Planar
Cylinders
Edge Slider kinematic
Cylinder
Slider to
variation analysis
Equivalent Variational Joint:
Extract tolerance sensitivities from
velocity analysis
Parallel(no
Even works for static assemblies
Edge Slider
Planar Cylinder Slider Cylinders
moving parts)
(Faerber 1999)
r3
r3
23
r4
r2
r2
22
r1
21
Kinematic Assembly
Static Assembly
r4
3D Equivalent
Variational Mechanisms
3D Kinematic Joints:
Rigid (no motion)
Equivalent Variational Joints:
Prismatic
Revolute
Parallel Cylinders
Cylindrical
Spherical
Planar
Edge Slider
Cylindrical Slider
Point Slider
Spherical Slider
Crossed Cylinders
Geometric Equivalent
Variational Mechanisms
f
f
f
f
Z
Y
f
f
Z
f
Cylindrical
d
f
Parallel Cylinders
f
f
f
f
f
Spherical
Planar
d
f
f
R
Edge Slider
d
f
R1
R2
f
f
d
f
R
Cylindrical Slider
f
f
f
X
f
R2
Revolute
Prismatic
f
X
R1
f
f
Rigid
f
f
f
f
d
Y
f
f
Point Slider
Spherical Slider
Crossed Cylinders
Example Model: Print Head
Geometric EVM
A
Pro/E model
f2
a2
Inset A
f3
j
f3
h
q1
a3
g
i
Inset B
k
f
e
d
B
f
c
b
Z
a
X
e
c
f1
a1
d
Print Head Results
Results from Global Coordinate Method:
A
C 0
f1  0

F  1
f3  0
B
D
E
G I J
0
0
0
0
0.2410
0.2410
0.2410
0
 0.0602  0.0602  0.0602  1
 0.2410  0.2410  0.2410 0
1
0
0
0
K
1
0
0  0.2410
0 0.0602
0  0.2410
L
0
0

1

0
Results from ADAMS velocity analysis:
A
C 0
f1  0

F  1
f3  0
B
D
E
G I J
0
0
0
0
0.2410
0.2410
0.2410
0
 0.0602  0.0602  0.0602  1
 0.2410  0.2410  0.2410 0
1
0
0
0
K
1
0
0  0.2410
0 0.0602
0  0.2410
L
0
0

1

0
3D GEVM in ADAMS
Research Benefits





Comprehensive system for including geometric
variation in a kinematic vector model
More efficient than homogeneous transformation
matrices
Allows use of commercial kinematic software to
perform tolerance analysis
Allows static assemblies to be analyzed in addition to
mechanisms
Ability to perform variation analysis in more widely
available kinematic solvers increases availability of
tolerance analysis
Current Limitations
Implementing EVMs is currently a manual
system, very laborious
 Manual implementation of EVMs can be very
complex when including both dimensional and
geometric variation
 Difficulty with analysis of joints with
simultaneous rotations

Questions?