Tolerance Analysis of Assemblies Using

Kinematically-Derived Sensitivities

Paul FaerberMotorola Corporation – Lawrenceville, GA

Presented by:Jeff Dabling

BYU – Graduate Student

• Differences (Kinematic)• Dimensions not allowed to vary• Requires sensitivity to velocity

inputs• Multiple Position Analysis

Similarities• Both use vector loops• Both use kinematic joints• Both use sensitivities

• Differences (Tolerance)• Dimensions allowed to vary

• Requires sensitivity to dimensional variation

• Single position analysis

Assembly

NetworkGraph

Skeleton Diagram

VectorLoops

VariationAnalysis

IncrementPosition (ifmechanism)

DimensionalVariation

Inputs

Mechanism

NetworkGraph

Skeleton Diagram

VectorLoops

Velocity andAcceleration

Analysis

IncrementMechanism

PositionKinematicInputs

Tolerance Analysis Kinematic Analysis

Research Objectives

• Model static assemblies with a kinematic modeler• Extract tolerance sensitivities from a kinematic

solver• Perform tolerance analysis on a mechanism in

multiple positions• Assess difficulty of adding tolerance analysis to

commercial kinematic software

Tolerance Analysis of Mechanisms

Tolerance models for eachposition of interest

1

2 4

3

Current Method:

2

1

4

3

Solid ModelNew Position

Solid ModelNew Position

Solid ModelNew Position

OriginalSolid Model

1

2 4

3

1

2 4

3

Merging Kinematic and Tolerance Analyses

MultiplePositions

Tolerance models for each position

SensitivitiesKinematic/ Dynamic Model

Kinematic Analysis

2

1

4

3

r2

r3

r4

r1

3

2

1

4

4-bar mechanism Vector loop equation

Resulting velocity equations

Matrix formulation

r e r e r e r ei i i i1 2 3 4

1 2 3 4 0( ) ( ) ( ) ( )

2 2 2 3 3 3 4 4 4

2 2 2 3 3 3 4 4 4

0

0

r r r

r r r

sin sin sin

cos cos cos

3

4

12

2 2 4 4 2

3 4 3 3 4

2 2 3 3 2

4 4 3 3 4

2

B A

r

rr

r

(cos sin cos sin )

(cos sin cos sin )(cos sin cos sin )

(cos sin cos sin )

A B2

3

40

Br r

r r

3 3 4 4

3 3 4 4

sin sin

co s co s

A

r

r

2 2

2 2

sin

cos

Kinematic Solution

Outputs Input

Tolerance Analysis

2

1

4

3

r2

r3

r4

r1

3

2 1

4

3

2

4

4-bar mechanism Vector loop equations

Linearized equations

r e r e r e r ei i i i1 2 3 4

1 1 2 1 2 3 1 2 3 4 0( ) ( ) ( ) ( )

co s co s co s co s ( sin sin sin )

( s in sin ) sin

sin sin sin sin ( co s co s co s )

( co s co s ) co s

1 1 2 2 3 3 4 4 2 2 3 3 4 4 2

3 3 4 4 3 4 4 4

1 1 2 2 3 3 4 4 2 2 3 3 4 4 2

3 3 4 4 3 4 4 4

0

0

dr dr dr dr r r r d

r r d r d

dr dr dr dr r r r d

r r d r d

Tolerance Analysis, continued

4-bar mechanism Matrix Formulation

Tolerance Analysis Solution (non-statistical)

A

d

dr

dr

dr

dr

Bd

d

2

1

2

3

4

3

4

0

A r r r

r r r

2 2 3 3 4 4 1 2 3 4

2 2 3 3 4 4 1 2 3 4

sin sin sin cos cos cos cos

cos cos cos sin sin sin sin

B

r r r

r r r

3 3 4 4 4 4

3 3 4 4 4 4

sin sin sin

cos cos cos

dr4 dr4

d3

d4

B A

d2

dr1

dr2

dr3

Si,j

d2

dr1

dr2

dr3

1

Outputs

Inputs

2

1

4

3

r2

r3

r4

r1

3

2 1

4

3

2

4

Estimated Tolerance Accumulation4-bar mechanism

r2

r3

r4

r1

3

2 1

4

3

2

4

d dr

drr

drr

drr

dr

dr

drr

drr

drr

dr

33

22

2 3

11

2 3

22

2 3

33

2 3

44

2

44

22

2 4

11

2 4

22

2 4

33

2 4

44

2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

RSS

RSS

Observations

r2

r3

r4

r1

3

2

1

4 3

2

4

3

42

2 2 4 4 2

3 4 3 3 4

2 2 3 3 2

4 4 3 3 4

2

J

r

rr

r

ij

(co s sin co s sin )

(co s sin co s sin )(co s sin co s sin )

(co s sin co s sin )

d

dB A

d

dr

dr

dr

dr

S

d

dr

dr

dr

dr

i j

3

4

1

2

1

2

3

4

2

1

2

3

4

,

Kinematic Analysis Solution

Tolerance analysis solution (non-statistical)

2 x 1

2 x 5

Kinematic Analysis of an Equivalent Variational Mechanism

r2

r3

r4

r1

3

2

1

4

r e r e r e r ei i i i1 2 3 4

1 2 3 4 0( ) ( ) ( ) ( )

co s co s co s co s sin sin sin

sin sin sin sin co s co s co s

r r r r r r r

r r r r r r r1 1 2 2 3 3 4 4 2 2 2 3 3 3 4 4 4

1 1 2 2 3 3 4 4 2 2 2 3 3 3 4 4 4

0

0

A

r

r

r

r

B

2

1

2

3

4

3

40

Br r

r r

3 3 4 4

3 3 4 4

sin sin

cos cos

3

4

1

2

1

2

3

4

2

1

2

3

4

B A

r

r

r

r

J

r

r

r

r

i j

,

Ar

r

2 2 1 2 3 4

2 2 1 2 3 4

sin co s co s co s co s

co s sin sin sin sin

Kinematic analysis solutionMatrix formulation

Resulting velocity equations

Vector loop equation

(Dimension ri not constant)

Comparisons

Tolerance analysis solution (non-statistical)

Kinematic analysis of equivalent mechanism solution

d S d S dr S dr S dr S dr

d S d S dr S dr S dr S dr

3 1 1 2 1 2 1 1 3 2 1 4 3 1 5 4

4 2 1 2 2 2 1 2 3 2 2 4 3 2 5 4

, , , , ,

, , , , ,

4

3

2

1

2

,

4

3

2

1

2

1

4

3

r

r

r

r

J

r

r

r

r

AB ji

4

3

2

1

2

,

4

3

2

1

2

1

4

3

dr

dr

dr

dr

d

S

dr

dr

dr

dr

d

ABd

dji

3 1 1 2 1 2 1 1 3 2 1 4 3 1 5 4

4 2 1 2 2 2 1 2 3 2 2 4 3 2 5 4

J J r J r J r J r

J J r J r J r J r, , , , ,

, , , , ,

Are the sensitivities the same?

Transformation to Relative AnglesRelative rotations Differential rotations Psuedo-velocities

Kinematic analysis solution

Tolerance analysis solution (after substitution)

dt

drJ

dt

drJ

dt

drJ

dt

drJ

dt

dJ

dt

d 45,1

34,1

23,1

12,1

21,1

3

dt

drJ

dt

drJ

dt

drJ

dt

drJ

dt

dJ

dt

d 45,2

34,2

23,2

12,2

21,2

4

d

dtJ

d

dtJ

dr

dtJ

dr

dtJ

dr

dtJ

dr

dt

31 1

21 2

11 3

21 4

31 5

41 ( ), , , , ,

d

dtJ J

d

dtJ J

dr

dtJ J

dr

dtJ J

dr

dtJ J

dr

dt4

2 1 1 12

2 2 1 21

2 3 1 32

2 4 1 43

2 5 1 54 ( ) ( ) ( ) ( ) ( ), , , , , , , , , ,

3

dt

d

dt

d

dt

d 233

dt

d

dt

d

dt

d 344 344 ddd

233 ddd 233

344

• Uses kinematic elements to represent dimensional variations in a kinematic model of the assembly

• ri are kinematic inputs

• ri are proportional to dimensional tolerances

Equivalent Variational Mechanisms

r2

r3

r4

• Vector loops identical for both types of analyses

• Use tolerance analysis techniques to develop vector loops for assemblies

• Use these vector loops as a starting point in developing EVM

Equivalent Variational Mechanisms

Mechanism

NetworkGraph

Skeleton Diagram

VectorLoops

Velocity andAcceleration

Analysis

IncrementMechanism

Position

Kinematic Analysis

EquivalentVariational

Inputs

KinematicInputs

Tolerance Analysis

Assembly

NetworkGraph

Skeleton Diagram

VectorLoops

VariationAnalysis

IncrementPosition (ifmechanism)

DimensionalVariation

Inputs

EquivalentVariationalMechanism

The Stack Blocks Assembly

Frame

BlockCylinder

Gap

p

b

c

a

r

dn

e

Creating Vector Loop Assembly Models

Gap

Frame

Cylinder

Block

Loop 2

Loop 1

Loop 3

Kinematic JointsNetwork Graph

Cylindrical DRF

FeatureDatum Cylinder

Block

Frame

Rectangular DRF

Part and Feature Reference Frames

Closed Loop #1 Closed Loop #2 Open Loop #1

Dimensional Variations

± d ddt

Skeleton modelEquivalent jointAngular dimension

ddt

dl

Skeleton modelEquivalent JointIndependent variation

l ± dl

dldt

Slider Joint

Each vector represents a link in the EVM

Angular Variations

Linear Variations

Stacked Blocks Model Dimensional Variations

Equivalent Kinematic Joints

PlanarJoint

Joint SkeletonModel

EquivalentJoint

CylindricalSlider Joint

EdgeSliderJoint

ParallelCylinders

Jointcenter of curvature

Stacked Blocks Model Dimensional Variations Kinematic Variations

Stacked Blocks Model (completed) Dimensional Variations Kinematic Variations Fixed Joints Pin Joints

EVM Stack Blocks Assembly

EVM Modeling Techniques

• Extracting Sensitivities from Kinematic Solver• Unit velocities are applied to each

independent joint, one at a time. Resulting dependent variables represent the row of the tolerance sensitivity matrix corresponding to that joint.

Independent variables:r2, r3, r4, 2

Dependent variables:2, 3, 4

r2 = 1

r3 = 0

r4 = 0

4

3

2

1

2

5,24,23,22,21,2

5,14,13,12,11,1

4

3

r

r

r

r

JJJJJ

JJJJJ

22

3

4

Variation Results for Stack Blocks

a b c e n g q r theta

dgap -0.306 0.306 -1 -1.046 1 0 1.231 -3.495 -0.197

Gap Sensitivities

dgap S dxi j j ( ),2

dgapda db dr de dn

dg dq dr d theta

( . ) ( . ) ( ) ( . ) ( )

( ) ( . ) ( . ) ( . )

0 3 0 6 0 3 0 6 1 1 0 4 6 1

0 1 2 3 1 3 4 9 5 1 9 7

2 22

2 2 2

2 2 2 2

Tolerance Analysis Using Equivalent Variational Mechanisms

• Variations in link lengths are allowed by including slider elements in each link.

• Sensitivities are used to form RSS expressions used in statistical tolerance analysis.

• Kinematic modeler moves to the next location, and the process is repeated.

Dependent variables:2, 3, 4

Independent variables:r2, r3, r4, 2

Modeling Techniques for Mechanisms

• Apply independent velocities as a reciprocating time function• Each new time step (second) is a new analysis point at the nominal

link-lengths

• Controlling the period of the function controls the resolution of the analysis, which affects the accuracy of the analysis

• Cosine Function• Full magnitude at time = 0

• Period equal to the time step

r

a

-a

period r= acos(bt)

period =2b

amplitude = a

r

Variation Results for Four-bar Mechanism

Tolerance Sensitivities

-4.50

-4.00

-3.50-3.00

-2.50

-2.00

-1.50

-1.00

-0.500.00

0.50

1.00

0.00E+00 1.00E+02 2.00E+02 3.00E+02 4.00E+02

Angular Position of Input Link

Se

nsi

tivi

ty

dalpha3/dtheta2

dalpha4/dtheta2

dalpha3/dr2

dalpha4/dr2

dalpha3dr3

dalpha4/dr3

dalpha3/dr4

dalpha4/dr4

r2

r3

r4

r1

3

21

4

3

2

4

Output Variation

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 60 120 180 240 300 360

Input Angle

Var

iati

on

dalpha3

dalpha4

Sensitivities vary with position(r1, 1 can also vary)

Greatest sensitivity at 270º

Contributions

• Defined relationship between kinematic and tolerance analyses

• Developed method for creating and analyzing equivalent variational mechanisms (EVM)• Equivalent 2-D kinematic joints presented• Demonstration of method on static assemblies, as well as

kinematic

• Demonstrated using commercial kinematic software, ADAMS• Method for extracting kinematic sensitivities• Method for returning the model to its nominal dimensions at each

time step

Recommendations for Future Research

• Investigate relationship between the higher kinematic derivatives (acceleration and jerk) and the higher statistical moments (skewness and kurtosis)

• Integrate with commercial kinematics CAD applications– Develop a user interface

– Study degree of freedom problems

• Extend into three-dimensional assemblies

• Include form tolerances in this method