Remote Positioning Mechanism 4-13
REMOTE POSITIONING MECHANISM
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 (±)g1 / 90.00˚ / .02˚
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
g2 / 30.00˚ / .02˚
g3 / 42.60˚ / .02˚
F / 12.900 in / .005 in
G / 49.300 in / .010 in
H / 12.900 in / .005 in
I / 49.300 in / .010 in
g4 / 42.60˚ / .02˚
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)
f1 / 60.0˚ / -- / --
f2 / 120.0˚ / -- / --
f3 / 0.0˚ / -- / --
f4 / 47.4˚ / -- / --
f5 / 132.6˚ / -- / --
f6 / 47.4˚ / -- / --
DX1 / 0.0 in / .1 in / -.1 in
DY1 / 0.0 in / .1 in / -.1 in
Dq1 / 0.0˚ / -- / --
DX2 / 0.0 in / -- / --
DY2 / 0.0 in / -- / --
Dq2 / 0.0˚ / .05 in (.26˚) / -.05 in (-.26˚)
Remarks> DX1 and DY1 are the Cartesian coordinate locations of point P relative to Ground and are used to calculate the position variation. Dq2 is the variation in the angular orientation of Part 5 relative to Part 1 and is used to calculate the parallelism variation. Dq1, DX2, and DY2 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 g1 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 g1. 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 f2 is the angle formed by B and C.
Changing the Part 3 DRF to the joint between F and G tells the analyzer that
f2 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 f6 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 (g1) 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 Type1 / 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
g1 / A / B / C / D / E / g2X1 / 0.0000 / -1.0000 / -.50000 / 1.0000 / 1.0000 / 0.0000 / 0.0000
Y1 / 0.0000 / 0.0000 / -.86603 / 0.0000 / 0.0000 / 1.0000 / 0.0000
q1 / 1.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
X2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
Y2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
q2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / -1.0000
A Matrix (continued)
g3 / F / G / H / I / g4 / JX1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
Y1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
q1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
X2 / 0.0000 / -.73610 / 0.0000 / .73610 / 0.0000 / 0.0000 / 0.0000
Y2 / 0.0000 / -.67688 / -1.0000 / .67688 / 1.0000 / 0.0000 / 0.0000
q2 / 1.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
B Matrix
f1 / f2 / f3 / f4 / f5 / f6X1 / 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
q1 / 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
q2 / 0.0000 / 1.0000 / 0.0000 / 1.0000 / 1.0000 / 1.0000
F Matrix
X1 / 1.0000 / 0.0000 / 1.0000 / 0.0000 / 1.0000 / 0.0000 / 1.0000
Y1 / 0.0000 / 1.0000 / 0.0000 / 1.0000 / 0.0000 / 1.0000 / 0.0000
q1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
X2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 1.0000 / 0.0000 / 0.0000
Y2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 1.0000 / 0.0000
q2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
F Matrix (continued)
a4 / a5 / a5 / a6 / a6 / a7 / a7X1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
Y1 / 1.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
q1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
X2 / 0.0000 / 1.0000 / 0.0000 / 1.0000 / 0.0000 / 1.0000 / 0.0000
Y2 / 0.0000 / 0.0000 / 1.0000 / 0.0000 / 1.0000 / 0.0000 / 1.0000
q2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
C Matrix
g1 / A / B / C / D / E / g2X1 / 58.307 / -1.0000 / -.50000 / 0.0000 / 0.0000 / 0.0000 / 49.300
Y1 / -5.2000 / 0.0000 / -.86603 / 0.0000 / 0.0000 / 0.0000 / 22.000
q1 / 1.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 1.0000
X2 / 0.0000 / -1.0000 / -.50000 / 0.0000 / 0.0000 / 0.0000 / 49.300
Y2 / 0.0000 / 0.0000 / -.86603 / 0.0000 / 0.0000 / 0.0000 / 22.000
q2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 1.0000
C Matrix (continued)
g3 / F / G / H / I / g4 / JX1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 1.0000
Y1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / -1.0000 / -22.000 / 0.0000
q1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / -1.0000 / 0.0000
X2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 1.0000
Y2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / -1.0000 / -22.000 / 0.0000
q2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / -1.0000 / 0.0000
D Matrix
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
q1 / 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 / 16.800 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / -22.000
q2 / 1.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / -1.0000
G Matrix
a1 / a1 / a2 / a2 / a3 / a3 / a4X1 / 1.0000 / 0.0000 / 1.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
Y1 / 0.0000 / 1.0000 / 0.0000 / 1.0000 / 0.0000 / 0.0000 / 0.0000
q1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
X2 / 1.0000 / 0.0000 / 1.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
Y2 / 0.0000 / 1.0000 / 0.0000 / 1.0000 / 0.0000 / 0.0000 / 0.0000
q2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
G Matrix (continued)
a4 / a5 / a5 / a6 / a6 / a7 / a7X1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / -1.0000 / 0.0000
Y1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / -1.0000
q1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
X2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / -1.0000 / 0.0000
Y2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / -1.0000
q2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
Tolerance Sensitivities
-B-1A Matrix
g1 / A / B / C / D / E / g2f1 / -1.0000 / .11103 / .05551 / -.11103 / -.11103 / 2.52E-18 / 0.0000
f2 / 1.0000 / -.08479 / -.00303 / .08479 / .08479 / -.04545 / 0.0000
f3 / -1.0000 / -.02624 / -.05249 / .02624 / .02624 / .04545 / 0.0000
f4 / -1.0000 / .08479 / .00303 / -.08479 / -.08479 / .04545 / 1.0000
f5 / 1.0000 / -.08479 / -.00303 / .08479 / .08479 / -.04545 / -1.0000
f6 / -1.0000 / .08479 / .00303 / -.08479 / -.08479 / .04545 / 1.0000
-B-1A Matrix (continued)
g3 / F / G / H / I / g4 / Jf1 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
f2 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
f3 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000 / 0.0000
f4 / -1.0000 / .02756 / .01865 / -.02756 / -.01865 / 0.0000 / 0.0000
f5 / 1.0000 / .04373 / .08666 / -.04373 / -.08666 / 0.0000 / 0.0000
f6 / -1.0000 / -.07128 / -.10531 / .07128 / .10531 / 0.0000 / 0.0000
-B-1F Matrix