Position Error in Assemblies and Mechanisms

Statistical and Deterministic Methods

By: Jon Wittwer

Outline

Position Error of Part FeaturesPosition Error in AssembliesDirect LinearizationDeterministic MethodsStatistical MethodsSummaryQuestions

Position Error of Part Features

x

y

A B0.06

B

A

A Norm al Dis tribution About a Target Mean

0

5

10

15

20

25

30

35

40

45

24.94 24.96 24.98 25 25.02 25.04 25.06

1-D Statistical ErrorTarget (Nominal) Dimension: 25.00 inTolerance: ±.03 in

Process Standard Deviation: = ±.01 inYield: 99.73%

3

24.97 25.03

2-D Position Tolerance

x

y

A B0.06

B

A

Tolerance Zone

0.06

IdealPosition

ActualPosition

Assuming Both x and y are normally distributed…

2-D Statistical Position Error

Contours of Equal Probability: CIRCLE

Case 1: If x = y

XY

Frequency Distribution

Tolerance Zone

Yield: 98.889%

R 3

R 4

R 6

IdealPosition

2-D Statistical Position Error

Contours of Equal Probability: ELLIPSE

If x ≠ y

XY

Frequency Distribution

Tolerance Zone

3

4

6

Yield: 65%

Position Error in Assemblies

x

y

Position Error in Assemblies

x

y

r2

r3

r4

P

a3

b3

r1

Pbarr

rrrr

4332

1432 0Closed Loop:

Open Loop:

Position Error in Assemblies

The x and y position error of the Coupler Point (P) are no longer independent.

Position Error in Assemblies

x

y

r1

Position Error in Assemblies

x

y

r1

Px

Py

Position Error in Assemblies

Methods

Deterministic (Worst-Case):• Involve fixed variables or constraints

that are used to find an exact solution.

Probabilistic (Statistical):• Involve random variables that result

in a probabilistic response.

Direct Linearization (DLM)

01432 rrrr

Closed Loop:

0sinsinsinsin

0coscoscoscos

11443322

11443322

rrrr

rrrrhx:hy:

Taylor’s Series Expansion:

]0[}]{[}]{[}{ UBXAH

{X} = {r1, r2, r3, r4} :primary random variables{U} = {3, 4} :secondary random variables }]{[}{ 1 XABU

Solving for Assembly Variation

Open Loop:

Px =

Pbar 432

)90sin(sinsin

)90cos(coscos

333322

333322

bar

bar

}]{[}]{[}{ 1 XSXAEBCP

}]{[}]{[}{ UEXCP

Taylor’s Series Expansion:

Solving for Position Variation:Sensitivity Matrix

Py =

Worst-Case vs. Statistical

jij XSP

2)( jij XSP

Worst Case:

Statistical (Root Sum Square):

Deterministic Methods:1. Worst-Case Direct

Linearization:• Uses the methods just discussed.

2. Vertex Analysis: • Finds the position error using all

combinations of extreme tolerance values.

3. Optimization:• Determines the maximum error using

tolerances as constraints.

Analogy for Worst-Case Methods

Tolx

Toly

Ideal Position

Ideal Position: Center of Room

Tolerance Zone: Walls

Analogy: Vertex AnalysisFinds Corners of the Room

Ideal Position

Tolx

Toly

Analogy: Worst-Case DLMFinds Walls of the Room

Ideal Position

Tolx

Toly

Analogy: OptimizationFinds way out of the room

Ideal Position

Tolx

Toly

Deterministic Results

Statistical Methods1. Monte Carlo Simulation

• Thousands to millions of individual models are created by randomly choosing the values for the random variables.

2. Direct Linearization: RSS• Uses the methods discussed previously.

3. Bivariate DLM• Statistical method for position error

where x and y error are not independent.

Bivariate Normal Position Error

2

2

2

2

2

2

2

i

i

i

Xi

y

i

xxy

Xi

yyy

Xi

xxx

X

P

X

PV

X

PV

X

PV

Variance Equations

The partial derivatives are the sensitivitiesthat come from the [C-EB-1A] matrix

yxy

xyx

VV

VV

Variance Tensor

21

22

r

Finding Ellipse Rotation:Mohr’s Circle

Vxy

Vx

Vy

Vxy1: Major Diameter2: Minor Diameter

2

V1=12

V2=22

Statistical Method Results

Coupler Point Error

Max. Perpendicular

Max. Normal Error

Maximum Normal Error vs. Crank Angle

0.04

0.042

0.044

0.046

0.048

0 60 120 180 240 300 360

Crank Angle (degrees)

Err

or

(in

ch

es

)

Benefits of Bivariate DLM

1. Accurate representation of the error zone.

2. Easily automated. CE/TOL already uses the method for assemblies.

3. Extremely efficient compared to Monte Carlo and Vertex Analysis.

4. Possible to estimate the yield for a given tolerance zone.

5. Can be used as a substitute for worst-case methods by using a large sigma-level

Summary

2-D Position error is not always a circle.Accurate estimation of position error in assemblies must include correlation.Where it is feasible, Direct Linearization is a good method for both worst-case and statistical error analysis.

Questions