Position Error in Assemblies and Mechanisms

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. 0.06 B A A y B x Position Error of Part Features

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

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

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

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

8. y x Position Error in Assemblies

9. r2 P r3 b3 a3 r4 r1 Closed Loop: Open Loop: Position Error in Assemblies y x

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

11. y r1 x Position Error in Assemblies

12. y r1 x Position Error in Assemblies DPy DPx

13. y x Position Error in Assemblies

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: hy: Taylor’s Series Expansion: {X} = {r1, r2, r3, r4} :primary random variables {U} = {q3, q4} :secondary random variables

16. Solving for Assembly Variation Open Loop: Px = Py = Taylor’s Series Expansion: Sensitivity Matrix Solving for Position Variation:

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

18. Deterministic Methods: • Worst-Case Direct Linearization: • Uses the methods just discussed. • Vertex Analysis: • Finds the position error using all combinations of extreme tolerance values. • Optimization: • Determines the maximum error using tolerances as constraints.

19. Analogy for Worst-Case Methods Ideal Position: Center of Room Tolerance Zone: Walls Ideal Position Toly Tolx

20. Analogy: Vertex Analysis Finds Corners of the Room Ideal Position Toly Tolx

21. Analogy: Worst-Case DLM Finds Walls of the Room Ideal Position Toly Tolx

22. Analogy: Optimization Finds way out of the room Ideal Position Toly Tolx

23. Deterministic Results

24. Statistical Methods • Monte Carlo Simulation • Thousands to millions of individual models are created by randomly choosing the values for the random variables. • Direct Linearization: RSS • Uses the methods discussed previously. • Bivariate DLM • Statistical method for position error where x and y error are not independent.

25. Variance Tensor qr 2s2 2s1 Bivariate Normal Position Error Variance Equations The partial derivatives are the sensitivities that come from the [C-EB-1A] matrix

26. V1=s12 V2=s22 2q 2s1: Major Diameter 2s2: Minor Diameter Finding Ellipse Rotation:Mohr’s Circle Vxy Vx Vy Vxy

27. Statistical Method Results

28. Max. Perpendicular Coupler Point Error

29. Max. Normal Error

30. Benefits of Bivariate DLM • Accurate representation of the error zone. • Easily automated. CE/TOL already uses the method for assemblies. • Extremely efficient compared to Monte Carlo and Vertex Analysis. • Possible to estimate the yield for a given tolerance zone. • 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