Feature Based Analysis of Selective
Limited Motion in Assemblies
Jeff Adams
ADCATS 2000
Brigham Young University
Jeff Adams
ADCATS 2000, BYU
Project Motivation
• Need computational methods to support “Top-Down”
assembly design philosophy
• Desire to take advantage of adjusting the position of parts
during assembly to reduce variation. Methods are needed
to calculate the location, direction, and amount of possible
adjustment in an assembly.
• Locational overconstraint of compliant parts can lead to
stored energy in the assembly. Methods are needed to
detect the location and directions of overconstraint.
Jeff Adams
ADCATS 2000, BYU
Topics
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•
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•
•
Top down design philosophy
Overview of Motion Limit Analysis (MLA)
Sketch of screw theory
Application of screw theory to constraint analysis
Examples
Jeff Adams
ADCATS 2000, BYU
Top Down Assembly Design
•
•
•
•
Establish key characteristics (KCs)
Construct Datum Flow Chain (DFC)
Choose assembly features to physically realize the DFC
Use MLA to calculate areas of adjustment and
overconstraint
• Use control theory to optimize assembly sequence,
location and amount of adjustment. Moving toward an
analysis that suggests an assembly feature design
Jeff Adams
ADCATS 2000, BYU
Assembly Design Overview
• Assembly design theory has three elements
– constraint defines how parts are located with
respect to each other
– assembly features on parts define where parts
are located with respect to each other
– tolerances on feature size and location define
how accurately parts are located with respect to
each other
• The Datum Flow Chain creates a top-down model
that supports all three elements
Jeff Adams
ADCATS 2000, BYU
Datum Flow Chain (DFC)
• A DFC is a directed acyclic graph that defines the
nominal relationships between assembled parts as well
as assembly fixtures, tooling, and equipment such as
X, z
robots
C
A
Y, Z,
x, y
(6)
B
• A DFC characterizes a nominal design by identifying
the part mates that convey dimensional control and by
identifying the hierarchy that determines which parts
or fixtures define the locations of which other parts
• DFCs also contain information on the type of mating
feature and the type of motion constraint applied by
that feature
Jeff Adams
ADCATS 2000, BYU
Motion Limit Analysis Overview
• Motion Limit Analysis (MLA) uses the mathematics
of screw theory to model the ability of mechanical
assembly features to allows or constrain rigid body
motions in six degrees of freedom.
• If rigid body motion is allowed, the direction and
quantitative amount of motion will be calculated.
• The ability to calculate rigid body motions of a part
is important for enabling in-process adjustment
during assembly to precisely establish key assembly
dimensions.
Jeff Adams
ADCATS 2000, BYU
Sketch of Screw Theory
Chasle’s Theorem: Any motion of a rigid body can be
reproduced as a rotation of the body about a unique line in
space and a translation along that same line.
T = [wx wy wz vx vy vz]
w = angular velocity
v=wxr
A d
F=

O
1


w = (Aw)T
r = dT
v=wxr
Jeff Adams
A = rotation matrix, d = displacement vector
ADCATS 2000, BYU
Motion Limit Vector
Motion limits are defined by a vector.
A Motion Limit Vector (MLV) is a 6x1 vector that
describes the three numerical limits on translational
motion in three independent directions, and the three
numerical limits on rotational motion about axes aligned
with the same independent directions.
MLVs are defined for each assembly feature (inherent
property for positive and negative motions).
 Px 
P 
 y 
 0 
 P x 
 0 


 180 
The main purpose of MLA is to combine the effects of
several sets of MLVs associated with the features that are
being used to connect one part to others, and calculate the
net pair of MLVs that describe the resultant motion
properties of the part as a whole.
Jeff Adams
ADCATS 2000, BYU
Examples of Mating Features
{dofs} and
[Twistmatrix]
Plate pin in through-hole
x
y
y
d
z
Plate pin in slotted-hole
x
{z}
w
v
{z Ly}
y
l
y
z
Round peg in throughhole
x
y
y
w
0

v1 
v 2 
{z Lz}
w
0

v1 
v 2 
0
0
 
0
 
0
0
 
 
0
P 
 y
0
 
0
0
 
  
0
0
 
 Pz 
 
0
0
 
  
0
0
 
0
 
0
0
 
 
 0 
N 
 y
 0 
 
 0 
 0 
 
  
 0 
 0 
 
N z 
 
 0 
 0 
 
  
Type of mating feature
Plate pin in oversized-hole
x
d1
Type of mating feature
Positive and
Negative
Motion Limit
Vectors
y
d2
y
{z Lx Ly}
w
0

 0
0
v1 

v 2 
z
Thin rib on plane surface
x
y
z
{x z Lx Ly}
w x
w
 z
0

0
18 and counting ...
z
Jeff Adams
{dofs} and
[Twistmatrix]
ADCATS 2000, BYU
v1 
0

v2 

v3 
Positive and
Negative
Motion Limit
Vectors
 Px 
P 
 y
0
 
0
0
 
  
N x 
N 
 y
 0 
 
 0 
 0 
 
  
 Px 
P 
 y 
 0 


 P x 
 0 


  
 Nx 
N 
 y 
 0 


 N x 
 0 


  
Use of Screw Theory to Check
Mobility and Constraint
• Create library of elementary features
• Each such feature has a twist matrix representation
– Don’t need geometry!
– Each row in the twist matrix represents an
unconstrained degree of freedom
• Make constraining joints between parts by using one or
more elementary features in combination
• Combine effects of all features and check degree of
mobility and constraint using twist intersection algorithm
by Konkar
Jeff Adams
ADCATS 2000, BYU
Combination of Mating Features in an Assembly
Questions:
• Are there are any motions between parts? and, if any,
• What kind of motions?
• What is the quantitative amount of each motion?
Jeff Adams
ADCATS 2000, BYU
For each RDoF, and
then for each ISA ...
Resultant
Twist matrix Ttot
Rotation allowed
by each feature:  i
Analysis of
Translational
DoFs
Motion Limit Analysis
Overview
Resultant
MLVs for
each part
Jeff Adams
Rotational
DoFs
Find the max allowed
rotation/translation
motions
Find the max
rotation/translation
components in the PCF
ADCATS 2000, BYU
Rotational DoFs
N mating features
N twist matrices
2N MLVs
Max rotation
around ISA
max = min( i)
Each max is
broken in x y z
components
Search for the
max possible
rotation axis
Screw Theory for Constraint Analysis
Poinsot’s Principle: Any set of forces and couples applied to a body can be
reduced to a single force acting along a specific line in space, and a
pure couple acting in a plane perpendicular to that line.
A wrench is a screw describing the resultant force and moment of a force
system acting on a rigid body.
W = [fx fy fz mx my mz]
f = Fi
i = 1, … , n
Mi = ri  Fi i = 1, … , n
m = Mi i = 1, … , n
Jeff Adams
r
ADCATS 2000, BYU
Analysis of Constraints
• Each twist has a reciprocal called a wrench expressed in
part center coordinates as [Fx Fy Fz Mx My Mz]
• It represents all the forces and torques that the feature can
transmit to a mating part
• Where motion is allowed, no force or torque can be
transmitted, and vice versa
• The intersection of all wrenches acting on a part shows the
amount of constraint on the part provided by those features
• If a part is constrained in some direction, then every
feature can provide that constraint
Jeff Adams
ADCATS 2000, BYU
Analysis of a Combined Feature Made
from Engineering Features
• Analysis results:
• Motion about Z is possible
• The rotation center is about f1
• The amount of rotation can be
calculated if peg and slot
dimensions are known and
slight clearance is assumed
• Mathematical Result
• T = [0 0 1 2 -2 0]
• Rotation occurs about Z axix
at point [2,2,0]
Jeff Adams
ADCATS 2000, BYU
Constraint Analysis Results
•Overconstraint of a displacement in Y
and a rotation about Z
•Overconstraint of displacement in Z
0
0
Wrench = 
0

0
Jeff Adams
1 0 0 0 2
0 1 0 0 0

0 0 1 0 0

0 0 0 1 0
•Overconstraint of rotation about X
•Overconstraint of rotation about Y
An error in the distance along the Y
axis between the two features could
cause difficulty in assembling the parts.
The other three constraints come about
because both features contain a planar
mate.
ADCATS 2000, BYU
Second Example
• Analysis results:
• No motion is possible, assembly
is fully constrained
• Overconstraint exists about X
and Y
• Overconstraint exists along Z
• Mathematical Result:
• Twistmatrix is null
• W = [0 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
Jeff Adams
ADCATS 2000, BYU
Assembly Using Features on the Parts
Splice
Str3
(6)
Aft
Skin
(6)
(6)
Fwd
Skin
FORWARD SKIN
(5)
(1)
(6)
SPLICE STRINGER
Plus
Chord
Str4-11
AFT SKIN
(a)
ASSEMBLY LEVEL DATUMS
PART LEVEL DATUMS
MATING FEATURE (SLOT)
MATING FEATURE (HOLE)
Jeff Adams
ADCATS 2000, BYU
Str1-2
Other Uses for Screw Theory
• Evaluating local constraints during assembly sequence
analysis
• Finding unstable subassemblies
• Determining if a DFC constrains a KC
• Determining how much adjustability there is in a Type-2
assembly
• Determining if KCs conflict
Jeff Adams
ADCATS 2000, BYU
Conclusions and Future Work
• A sufficient set of assembly features has been modeled to provide
mathematical models of connections between parts in an assembly.
• Motion Limit Analysis has the ability to combine the motion
characteristics of several assembly features, to yield two motion limit
vectors describing the motion of the part as a whole.
• Work is being done to apply MLA to compliant parts using a simple
FEA model.
• Combining MLA with other algorithms such as assembly sequence
analysis, DFC analysis, control theory based tolerance analysis, and a
CAD system has the potential to yield a comprehensive tool to
support the “Top Down” approach to assembly design.
Jeff Adams
ADCATS 2000, BYU
Additional References
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J. D. Adams, "Feature Based Analysis of Selective Limited Motion in Assemblies," M.S. Thesis,
Mechanical Engineering. Cambridge: MIT, February 1998.
R. Mantripragada, “Assembly Oriented Design: Concepts, Algorithms and Computational Tools,” in
Ph.D. Thesis, Mechanical Engineering. Cambridge: MIT, 1998.
R. Konkar and M. Cutkosky, “Incremental Kinematic Analysis of Mechanisms,” Journal of
Mechanical Design, Vol. 117, December 1995, pp. 589-596.
R. Mantripragada and D. Whitney, “Modeling and Controlling Variation Propagation in Mechanical
Assemblies Using State Transition Models,” IEEE Transactions on Robotics and Automation, vol 15,
no 1, Feb. 1999.
R. Mantripragada and D. Whitney, “The Datum Flow Chain: A Systematic Approach to Assembly
Design and Modeling,” Research in Engineering Design, vol 10, 1998, pp 150-165.
Whitney, D.E., R. Mantripragada, J.D. Adams, and T.W. Cunningham, “Use of Screw Theory to
Detect Multiple Conflicting Key Characteristics,” ASME Design Engineering Technical Conferences,
Las Vegas, Sept. 1999.
Whitney, D.E., R. Mantripragada, J.D. Adams, and S.J. Rhee, “Designing Assemblies,” Research in
Engineering Design, (1999) 11:229-253..
Jeff Adams
ADCATS 2000, BYU