1. Position Error in Assemblies and Mechanisms Statistical and Deterministic Methods By: Jon Wittwer

2. Outline Position Error of Part Features Position Error in Assemblies Direct Linearization Deterministic Methods Statistical Methods Summary Questions

3. Position Error of Part Features x y A B0.06 B A

4. 1-D Statistical Error Target (Nominal) Dimension: 25.00 in Tolerance: ±.03 in Process Standard Deviation:  = ±.01 in Yield: 99.73% 33 24.97 25.03

5. 2-D Position Tolerance x y A B0.06 B A Tolerance Zone 0.06 Ideal Position Actual Position Assuming Both x and y are normally distributed…

6. 2-D Statistical Position Error Contours of Equal Probability: CIRCLE Case 1: If  x =  y X Y Frequency Distribution Tolerance Zone Yield: 98.889% R 3  R 4  R 6  Ideal Position

7. 2-D Statistical Position Error Contours of Equal Probability: ELLIPSE If  x ≠  y X Y Frequency Distribution Tolerance Zone 33 44 66 Yield: 65%

8. Position Error in Assemblies x y

9. x y r2r2 r3r3 r4r4 P a3a3 b3b3 r1r1 Closed Loop: Open Loop:

10. Position Error in Assemblies The x and y position error of the Coupler Point (P) are no longer independent.

11. Position Error in Assemblies x y r1r1

12. x y r1r1 PxPx PyPy

13. x y

14. 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.

15. Direct Linearization (DLM) Closed Loop: hx:hx: hy:hy: Taylor’s Series Expansion: {X} = {r 1, r 2, r 3, r 4 }:primary random variables {U} = {  3,  4 }:secondary random variables

16. Solving for Assembly Variation Open Loop: P x = Taylor’s Series Expansion: Solving for Position Variation: Sensitivity Matrix P y =

17. Worst-Case vs. Statistical Worst Case: Statistical (Root Sum Square):

18. 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.

19. Analogy for Worst-Case Methods Tol x Tol y Ideal Position Ideal Position: Center of Room Tolerance Zone: Walls

20. Analogy: Vertex Analysis Finds Corners of the Room Ideal Position Tol x Tol y

21. Analogy: Worst-Case DLM Finds Walls of the Room Ideal Position Tol x Tol y

22. Analogy: Optimization Finds way out of the room Ideal Position Tol x Tol y

23. Deterministic Results

24. Statistical Methods 1. 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.

25. Bivariate Normal Position Error Variance Equations The partial derivatives are the sensitivities that come from the [C-EB -1 A] matrix Variance Tensor 2121 2222 rr

26. Finding Ellipse Rotation: Mohr’s Circle V xy VxVx VyVy  1 : Major Diameter  2 : Minor Diameter 22 V1=12V1=12 V2=22V2=22

27. Statistical Method Results

28. Coupler Point Error Max. Perpendicular

29. Max. Normal Error

30. 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

31. 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.

32. Questions