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Figure 4.1: Schematic of the remote positioner with basic dimension labels.

4.0 Problem Description

The remote positioner is a mechanical linkage that positions point P in two-dimensional space. Point P is meant to remain in a fixed location while the angular orientation of Part 5 varies. This model demonstrates the use of two open loops with two closed loop constraints.

This model includes 14 independent variables, and solves for six kinematic variables. In addition, two open loops allow us to solve for additional assembly variations.

Table 4.1: Manufactured Variables (Independent).

 Variable Name Basic Size Initial Tolerance (+/-) 1 90.00[[ring]] .02[[ring]] A 22.000 in .005 in B 10.400 in .005 in C 22.000 in .005 in D 5.200 in .003 in E 9.0067 in .004 in 2 30.00[[ring]] .02[[ring]] 3 42.60[[ring]] .02[[ring]] F 12.900 in .005 in G 49.300 in .010 in H 12.900 in .005 in I 49.300 in .010 in 4 42.60[[ring]] .02[[ring]] J 22.000 in .005 in

4.1 Design Requirements

Table 4.2: Assembly Variables (Dependent).

 Variable Name Basic Size Upper Spec. Limit(USL) Lower Spec. Limit(LSL) q1 60.0[[ring]] -- -- q2 120.0[[ring]] -- -- q3 0.0[[ring]] -- -- q4 47.4[[ring]] -- -- q5 132.6[[ring]] -- -- q6 47.4[[ring]] -- -- X1 0.0 in .1 in -.1 in Y1 0.0 in .1 in -.1 in 1 0.0[[ring]] -- -- X2 0.0 in -- -- Y2 0.0 in -- -- 2 0.0[[ring]] .05 in (.26[[ring]]) -.05 in (-.26[[ring]])

Remarks>> X1 and Y1 are the Cartesian coordinate locations of point P relative to Ground and are used to calculate the position variation. 2 is the variation in the angular orientation of Part 5 relative to Part 1 and is used to calculate the parallelism variation. 1, X2, and Y2 can also be solved for from the open loops, but they are not necessary to estimate the parallelism and position assembly variations.

4.2 Modeling Considerations

• In modeling this problem it is very important to make sure the correct variations are included in each loop. For example, to calculate the variation in the position of point P, the variation in 1 must be included in the open loop for that specification. The open loop for calculating the parallelism of I relative to A should not include 1. The open loop endpoints will be located at the same coordinates, but in the position loop the endpoint will be associated with the Ground, and in the parallelism loop, the endpoint will be associated with

Part 1.

• When manufactured angles and kinematic angles occur at the same joint, the placement of the Datum Reference Frames (DRFs) determines which vectors are used to define the dependent angles. For example, placing the DRF for Part 2 at the joint between A and B, and the DRF for Part 3 at the joint between C and D tells the analyzer that q2 is the angle formed by B and C. Changing the Part 3 DRF to the joint between F and G tells the analyzer that q2 is the angle between B and F. This distinction becomes important in cases when design specifications are applied to dependent angles. In the case of the remote positioner, it is not that important, since no design specifications were applied to the dependent angles. The joint connecting Part 2 and Part 5 has the same kind of situation. Locating the DRF for Part 5 at the joint between it and Part 4 tells the analyzer that q6 is formed by H and I.

• In order to create the most robust model, avoid placing part DRFs at joints with manufactured angle variations. For example, if Part 2 uses the point between B and I as its DRF in manufacturing, for modeling purposes it's better to locate its DRF at another joint and put a feature datum between B and I. The same applies to Part 3 and Part 5.

• CATS does not automatically account for loose clearance fits in revolute joint. In order to model the variation caused by the gap between the pin and the hole of a revolute joint, a true position geometric tolerance of the same magnitude as the expected clearance is applied to the joint.

4.3 Design Goal

The object of this problem is to calculate the variation in the position of point P relative to Ground, as well as the parallelism between Part 1 and Part 5 and optimize the tolerances to meet the parallelism specification limits.

4.4 Part Names and DRFs

Figure 4.2: Diagram showing the location of the part DRFs.

4.5 Kinematic Joints

Seven joints are required to model the remote positioner. All of them are revolute joints in the physical device. However, with all joints free to rotate, the system is indeterminate, and CATS cannot solve for the variations. Therefore, one joint must be designated the input angle and its rotational degree of freedom removed. This is done by either replacing that revolute joint with a rigid joint (in AutoCats) or by "turning off" that joint's rotational degree of freedom (in any of the workstation-based analyzers). For this problem, the input angle (1) was at joint 1, so joint 1 was modeled as a rigid joint.

Figure 4.3: Kinematic joint diagram.

Table 4.3: Kinematic Joints of the Remote Positioner.

 Joint Number Part One Part Two Joint Type 1 Ground Part 1 rigid 2 Part 1 Part 2 revolute 3 Part 2 Part 3 revolute 4 Part 3 Ground revolute 5 Part 3 Part 4 revolute 6 Part 4 Part 5 revolute 7 Part 5 Part 2 revolute

4.6 Network Diagram, Vector Loops, and Design Specifications

Two closed loops are necessary to constrain the remote positioner assembly. A location specification relative to joint 1 has been applied to point P, as well as a parallelism specification relative to length A (Part 1). Therefore two open loops are also needed, one for each design specification.

Figure 4.4: Network diagram and open and closed loops for the remote positioner.

Remarks>> Open loops are analyzed in the same manner as closed loops. They are more sensitive to modeling errors than closed loops are, so correct placement of loop endpoints and part DRFs is critical when calculating variations with open loops.

The direction of open loops is important when gap and position specifications are used. CATS assumes the first part is fixed in space and the parts "downstream" all rotate relative to it. This arises due to the non-commutative property of matrix multiplication. To generate the correct open loop direction, create the final endpoint (the moving endpoint) first and the starting endpoint (fixed endpoint) second.

The allowable position specification variation is given as a diameter.

4.7 Geometric Tolerances

True position geometric tolerances have been applied to the seven joints to account for clearance variations. Each position tolerance is modeled as two orthogonal, independent vectors.

Figure 4.5: Geometric tolerance diagram.

Remarks>> Applying position tolerances to the joints in this assembly is not completely accurate. In this case, the position tolerance is not related to the position of the holes (or pins). Instead, it is being used as a way to approximate the variations that occur in the assembly due to the small clearances between the pins and holes.

4.8 Sensitivity Matrices

Constraint Sensitivities

A Matrix

 1 A B C D E 2 X1 0 -1.0000 -.50000 1.0000 1.0000 0.0000 0 Y1 0 0.0000 -.86603 0.0000 0.0000 1.0000 0 1 1 0.0000 0.0000 0.0000 0.0000 0.0000 0 X2 0 0.0000 0.0000 0.0000 0.0000 0.0000 0 Y2 0 0.0000 0.0000 0.0000 0.0000 0.0000 0 2 0 0.0000 0.0000 0.0000 0.0000 0.0000 -1

A Matrix (continued)

 3 F G H I 4 J X1 0 0.0000 0.0000 0.0000 0.0000 0 0.0000 Y1 0 0.0000 0.0000 0.0000 0.0000 0 0.0000 1 0 0.0000 0.0000 0.0000 0.0000 0 0.0000 X2 0 -.73610 0.0000 .73610 0.0000 0 0.0000 Y2 0 -.67688 -1.0000 .67688 1.0000 0 0.0000 2 1 0.0000 0.0000 0.0000 0.0000 0 0.0000

B Matrix

 q1 q2 q3 q4 q5 q6 X1 0.0000 -9.0067 -9.0067 0.0000 0.0000 0.0000 Y1 22.000 27.200 5.2000 0.0000 0.0000 0.0000 1 1.0000 1.0000 1.0000 0.0000 0.0000 0.0000 X2 0.0000 0.0000 0.0000 -8.7317 -58.032 -49.300 Y2 0.0000 0.0000 0.0000 9.4957 9.4957 0.0000 2 0.0000 1.0000 0.0000 1.0000 1.0000 1.0000

F Matrix

 1 1 2 2 3 3 4 X1 1 0 1 0 1 0 1 Y1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 0 X2 1 0 0 0 1 0 0 Y2 0 0 0 0 0 1 0 2 0 0 0 0 0 0 0

F Matrix (continued)

 4 5 5 6 6 7 7 X1 0 0 0 0 0 0 0 Y1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 X2 0 1 0 1 0 1 0 Y2 0 0 1 0 1 0 1 2 0 0 0 0 0 0 0

C Matrix

 1 A B C D E 2 X1 58.307 -1.0000 -.50000 0.0000 0.0000 0.0000 49.3 Y1 -5.2 0.0000 -.86603 0.0000 0.0000 0.0000 22 1 1 0.0000 0.0000 0.0000 0.0000 0.0000 1 X2 0 -1.0000 -.50000 0.0000 0.0000 0.0000 49.3 Y2 0 0.0000 -.86603 0.0000 0.0000 0.0000 22 2 0 0.0000 0.0000 0.0000 0.0000 0.0000 1

C Matrix (continued)

 3 F G H I 4 J X1 0 0.0000 0.0000 0.0000 0.0000 0 1.0000 Y1 0 0.0000 0.0000 0.0000 -1.0000 -22 0.0000 1 0 0.0000 0.0000 0.0000 0.0000 -1 0.0000 X2 0 0.0000 0.0000 0.0000 0.0000 0 1.0000 Y2 0 0.0000 0.0000 0.0000 -1.0000 -22 0.0000 2 0 0.0000 0.0000 0.0000 0.0000 -1 0.0000

D Matrix

 q1 q2 q3 q4 q5 q6 X1 58.307 0.0000 0.0000 0.0000 0.0000 0.0000 Y1 16.8000 0.0000 0.0000 0.0000 0.0000 -22.000 1 1.0000 0.0000 0.0000 0.0000 0.0000 -1.0000 X2 58.307 0.0000 0.0000 0.0000 0.0000 0.0000 Y2 16.8000 0.0000 0.0000 0.0000 0.0000 -22.000 2 0.0000 0.0000 0.0000 0.0000 0.0000 -1.0000

G Matrix

 1 1 2 2 3 3 4 X1 1 0 1 0 0 0 0 Y1 0 1 0 1 0 0 0 1 0 0 0 0 0 0 0 X2 1 0 1 0 0 0 0 Y2 0 1 0 1 0 0 0 2 0 0 0 0 0 0 0

G Matrix (continued)

 4 5 5 6 6 7 7 X1 0 0 0 0 0 -1 0 Y1 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 X2 0 0 0 0 0 -1 0 Y2 0 0 0 0 0 0 -1 2 0 0 0 0 0 0 0

Tolerance Sensitivities

-B-1A Matrix

 1 A B C D E 2 q1 -1 .11103 .05551 -.11103 -.11103 2.52E-18 0 q2 1 -.08479 -.00303 .08479 .08479 -.04545 0 q3 -1 -.02624 -.05249 .02624 .02624 .04545 0 q4 -1 .08479 .00303 -.08479 -.08479 .04545 1 q5 1 -.08479 -.00303 .08479 .08479 -.04545 -1 q6 -1 .08479 .00303 -.08479 -.08479 .04545 1

-B-1A Matrix (continued)

 3 F G H I 4 J q1 0 0.0000 0.0000 0.0000 0.0000 0 0.0000 q2 0 0.0000 0.0000 0.0000 0.0000 0 0.0000 q3 0 0.0000 0.0000 0.0000 0.0000 0 0.0000 q4 -1 .02756 .01865 -.02756 -.01865 0 0.0000 q5 1 .04373 .08666 -.04373 -.08666 0 0.0000 q6 -1 -.07128 -.10531 .07128 .10531 0 0.0000

-B-1F Matrix

 1 1 2 2 3 3 4 q1 -0.11103 2.52e-18 -0.11103 2.52e-18 -0.11103 2.52e-18 -0.11103 q2 0.08479 -0.04545 0.08479 -0.04545 0.08479 -0.04545 0.08479 q3 0.02624 0.04545 0.02624 0.04545 0.02624 0.04545 0.02624 q4 -0.08479 0.04545 -0.08479 0.04545 -0.10507 0.0268 -0.08479 q5 0.08479 -0.04545 0.08479 -0.04545 0.10507 -0.13211 0.08479 q6 -0.08479 0.04545 -0.08479 0.04545 -0.08479 0.15077 -0.08479

-B-1F Matrix (continued)

 4 5 5 6 6 7 7 q1 2.52e-18 0 0 0 0 0 0 q2 -0.04545 0 0 0 0 0 0 q3 0.04545 0 0 0 0 0 0 q4 0.04545 -0.02028 -0.01865 -0.02028 -0.01865 -0.02028 -0.01865 q5 -0.04545 0.02028 -0.08666 0.02028 -0.08666 0.02028 -0.08666 q6 0.04545 1.8e-17 0.10531 1.8e-17 0.10531 1.8e-17 0.10531

C-DB-1A Matrix

 1 A B C D E 2 X1 0 5.4737 2.7369 -6.4737 -6.4737 1.47E-16 49.3 Y1 -2.1e-14 2.22E-16 -5.6E-16 -2.2E-16 -2.2E-16 -1.0000 1.07e-14 1 1 .02624 .05249 -.02624 -.02624 -.04545 8.88e-16 X2 -58.307 5.4737 2.7369 -6.4737 -6.4737 1.47E-16 49.3 Y2 5.2 2.22E-16 -5.6E-16 -2.2E-16 -2.2E-16 -1.0000 1.42e-14 2 -8.9e-16 .02624 .05249 -.02624 -.02624 -.04545 8.88e-16

C-DB-1A Matrix (continued)

 3 F G H I 4 J X1 0 0.0000 0.0000 0.0000 0.0000 0 1.0000 Y1 22 1.5682 2.3168 -1.5682 -3.3168 -22 0.0000 1 1 .07128 .10531 -.07128 -.10531 -1 0.0000 X2 0 0.0000 0.0000 0.0000 0.0000 0 1.0000 Y2 22 1.5682 2.3168 -1.5682 -3.3168 -22 0.0000 2 1 -.04545 -.02624 -.04545 -.02624 -1 0.0000

G-DB-1F Matrix

 1 1 2 2 3 3 4 X1 -5.4737 1.47e-16 -5.4737 1.47e-16 -6.4737 1.47e-16 -6.4737 Y1 -2.2e-16 8.88e-16 -2.2e-16 8.88e-16 -4.4e-16 -3.3168 -2.2e-16 1 -0.02624 -0.04545 -0.02624 -0.04545 -0.02624 -0.15077 -0.02624 X2 -5.4737 -5.4737 -5.4737 1.47e-16 -6.4737 1.47e-16 -6.4737 Y2 -2.2e-16 8.88e-16 -2.2e-16 8.88e-16 -4.4e-16 -3.3168 -2.2e-16 2 -0.02624 -0.04545 -0.02624 -0.04545 -0.02624 -0.15077 -0.02624
`                                                   `

G-DB-1F Matrix (continued)

 4 5 5 6 6 7 7 X1 1.47e-16 0 0 0 0 -1 0 Y1 -1 -4e-16 -2.3168 -4e-16 -2.3168 -4e-16 -3.3168 1 -0.04545 -1.8e-17 -0.10531 -1.8e-17 -0.10531 -1.8e-17 -0.10531 X2 1.47e-16 0 0 0 0 -1 0 Y2 -1 -4e-16 -2.3168 -4e-16 -2.3168 -4e-16 -3.3168 2 -0.04545 -1.8e-17 -0.10531 -1.8e-17 -0.10531 -1.8e-17 -0.10531

4.9 Predicted Assembly Variation

Table 4.4: Independent Variable Tolerances and Control Factors

 Dim.Name +/- Tol. Std. Dev. Cp Dk Cpk Sk Wt. Factor Tol. Basic Fixed 1 .02deg. .0067 1 0.25 0.75 0 0 0 Yes A .005 in .0017 1 0.25 0.75 0 3 1 No B .005 in .0017 1 0.25 0.75 0 2 1 No C .005 in .0017 1 0.25 0.75 0 3 1 No D .003 in .0010 1 0.25 0.75 0 2 1 No E .004 in .0013 1 0.25 0.75 0 2 1 No 2 .02deg. .0067 1 0.25 0.75 0 0 0 Yes 3 .02deg. .0067 1 0.25 0.75 0 0 0 Yes F .005 in .0017 1 0.25 0.75 0 2 1 No G .010 in .0033 1 0.25 0.75 0 4 1 No H .005 in .0017 1 0.25 0.75 0 2 1 No I .010 in .0033 1 0.25 0.75 0 4 1 No 4 .02deg. .0067 1 0.25 0.75 0 0 0 Yes J .005 in .0017 1 0.25 0.75 0 3 1 No

Table 4.5: Kinematic Assembly Variables (No Geometric Tolerances)

 Variable Name Degree of Freedom +/- Assembly Variations (ZASM = 4.500) Worst Case RSS Case Six-Sigma q1 Rotation ([[ring]]) .11860 .08271 .11028 q2 Rotation ([[ring]]) .09444 .06541 .08721 q3 Rotation ([[ring]]) .06500 .04419 .05892 q4 Rotation ([[ring]]) .17160 .08290 .11054 q5 Rotation ([[ring]]) .25879 .13371 .17828 q6 Rotation ([[ring]]) .29596 .15601 .20801

Table 4.6: Geometric Tolerances

 Feat. Joint Part Name Feature Type Tolerance Band Char. Length 1 1 Part 1 True Position .001 in N/A 2 2 Part 1 True Position .001 in N/A 3 3 Part 3 True Position .001 in N/A 4 4 Part 3 True Position .001 in N/A 5 5 Part 4 True Position .001 in N/A 6 6 Part 4 True Position .001 in N/A 7 7 Part 5 True Position .001 in N/A

Table 4.7: Kinematic Assembly Variables (Geometric Tolerances Included)

 Variable Name Degree of Freedom +/- Assembly Variations (ZASM = 4.500) Worst Case RSS Case Six-Sigma q1 Rotation ([[ring]]) .13133 .08326 .11069 q2 Rotation ([[ring]]) .10936 .06593 .08761 q3 Rotation ([[ring]]) .07322 .04442 .05910 q4 Rotation ([[ring]]) .18992 .08337 .11089 q5 Rotation ([[ring]]) .28597 .13426 .17870 q6 Rotation ([[ring]]) .32295 .15655 .20842

Table 4.8: Six-Sigma Percent Rejects (Parallelism Specification)

 Spec. Name Spec. Type Nominal Dimension (+/-) Computed Variation With Geometric Tolerances Without Geometric Tolerances P Parallelism 0.0000 .03725 Z Rej. Z Rej. ZASM = 4.500 USL 0.05 Upper Tail 6.04 7.8e-4 6.05 7.3e-4 (Rejects in PPM) LSL -0.05 Lower Tail -6.04 7.8e-4 -6.05 7.3e-4

Remarks>> The variation in the assembly due to the small gaps in the pin joints is relatively insignificant. For tolerance allocation, the effects of the gaps (represented by the true position geometric tolerances) will be ignored.

Table 4.9: Six-Sigma Percent Contributions To Parallelism Of P
(No Geometric Tolerances)

 Variable Name Variance Six-Sigma G 2.191e-7 38.80 I 2.191e-7 38.80 F 2.509e-8 4.44 H 2.509e-8 4.44 3 2.407e-8 4.26 4 2.407e-8 4.26 B 1.360e-8 2.41 E 6.530e-9 1.16 other 8.026e-9 1.43

Table 4.10: Six-Sigma Percent Rejects (True Position)

 Spec. Name Spec. Type Nominal Dimension (+/-) Computed Variation With Geometric Tolerances Without Geometric Tolerances P True Pos. 0.0000 X: .10398 Y: .08715 Z Rej. Z Rej. ZASM = 4.500 X-rad. .100 Upper Tail N/A 59.4 N/A 55.2 (Rejects in PPM) Y-rad. .100 Lower Tail N/A 59.4 N/A 55.2

4.10 Tolerance And Nominal Allocation

Weight Factor Tolerance Allocation

Table 4.11: Six-Sigma Weight Factor Tolerance Allocation
(Parallelism, No Geometric Tolerances)

 Assembly Specs. Nom. USL LSL +/- ZASM Parallelism (P) 0.0000 0.0500 -0.0500 4.500
 Dimension Name Specified Values Allocated Values Nom. +/-Tol. Nom. +/-Tol. STDEV % Cont. 1 ([[ring]]) 90.0000 .02000 90.0000 .02000 .00889 1.9e-30 * A (in) 22.0000 .00500 22.0000 .00545 .00242 0.40 B (in) 10.4000 .00500 10.4000 .00364 .00162 0.70 C (in) 22.0000 .00500 22.0000 .00545 .00242 0.40 D (in) 5.2000 .00300 5.2000 .00218 .00097 0.06 E (in) 9.0067 .00400 9.0067 .00291 .00129 0.34 2 ([[ring]]) 30.0000 .02000 30.0000 .02000 .00889 1.9e-30 * 3 ([[ring]]) 42.6000 .02000 42.6000 .02000 .00889 2.36 * F (in) 12.9000 .00500 12.9000 .00364 .00162 1.30 G (in) 49.3000 .01000 49.3000 .01454 .00646 45.39 H (in) 12.9000 .00500 12.9000 .00364 .00162 1.30 I (in) 49.3000 .01000 49.3000 .01454 .00646 45.39 4 ([[ring]]) 42.6000 .02000 42.6000 .02000 .00889 2.36 * J (in) 22.0000 .00500 22.0000 .00545 .00242 0.00 Assem. Total Nom. +/-Var. Nom. +/-Var. STDEV 100.00 P (in) 0.0000 .03720 0.0000 .05000 .01111 Min./Max. -.03720 .03720 -.0500 .0500 * Fixed Nom./Tol.
 Before Optimization After Optimization Rejects Z PPM Z PPM Upper Tail 6.05 7.3e-4 4.5 3.4 Lower Tail -6.05 7.3e-4 -4.5 3.4 Total Rejects 1.5e-3 Total Rejects 6.8

Proportional Scaling Tolerance Allocation

Table 4.12: Six-Sigma Proportional Scaling Tolerance Allocation

(Parallelism, No Geometric Tolerances)

 Assembly Specs. Nom. USL LSL +/- ZASM Parallelism (P) 0.0000 0.0500 -0.0500 4.500
 Dimension Name Specified Values Allocated Values Nom. +/-Tol. Nom. +/-Tol. STDEV % Cont. 1 ([[ring]]) 90.0000 .02000 90.0000 .02000 .00889 1.9e-30 * A (in) 22.0000 .00500 22.0000 .00686 .00305 0.63 B (in) 10.4000 .00500 10.4000 .00686 .00305 2.51 C (in) 22.0000 .00500 22.0000 .00686 .00305 0.63 D (in) 5.2000 .00300 5.2000 .00412 .00183 0.23 E (in) 9.0067 .00400 9.0067 .00549 .00243 1.20 2 ([[ring]]) 30.0000 .02000 30.0000 .02000 .00889 1.9e-30 * 3 ([[ring]]) 42.6000 .02000 42.6000 .02000 .00889 2.36 * F (in) 12.9000 .00500 12.9000 .00686 .00305 4.63 G (in) 49.3000 .01000 49.3000 .01372 .00610 40.41 H (in) 12.9000 .00500 12.9000 .00686 .00305 4.63 I (in) 49.3000 .01000 49.3000 .01372 .00610 40.41 4 ([[ring]]) 42.6000 .02000 42.6000 .02000 .00889 2.36 * J (in) 22.0000 .00500 22.0000 .00686 .00305 0.00 Assem. Total Nom. +/-Var. Nom. +/-Var. STDEV 100.00 P (in) 0.0000 .03720 0.0000 .05000 .01111 Min./Max. -.03720 .03720 -.0500 .0500 * Fixed Nom./Tol.
 Before Optimization After Optimization Rejects Z PPM Z PPM Upper Tail 6.05 7.3e-4 4.5 3.4 Lower Tail -6.05 7.3e-4 -4.5 3.4 Total Rejects 1.5e-3 Total Rejects 6.8

Remarks>> Comparison of the two tables above shows some of the differences between weight factor and proportional scaling allocation. Proportional scaling adjusts all non-fixed dimension tolerances proportional to their original tolerances. Weight factor allocation distributes the assembly variable variance pool among the non-fixed dimension tolerances proportional to their weight factors. If all weight factors are set equal, weight factor allocation becomes equivalent to proportional scaling.

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