mathematical modeling of assembly coordinate frames –each part has a base coordinate frame...
Post on 21-Dec-2015
214 views
TRANSCRIPT
![Page 1: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/1.jpg)
Mathematical Modeling of Assembly
• Coordinate frames– each part has a
base coordinate frame
• Relationships between parts are expressed as 4x4 matrix transforms
![Page 2: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/2.jpg)
Matrix Math
• 4x4 matrices relate adjacent frames
• Matrix contains rotational part and translational part
• Translation occurs first, so rotation does not change position of new frame
![Page 3: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/3.jpg)
Basic Translation and Rotation
![Page 4: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/4.jpg)
Watch Transform Ordering!
![Page 5: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/5.jpg)
Composite Transforms
![Page 6: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/6.jpg)
Nominal Mating of Parts
![Page 7: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/7.jpg)
Example - Pin & Hole Mating(pin translated)
![Page 8: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/8.jpg)
Example - Pin & Hole Mating(pin rotated)
![Page 9: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/9.jpg)
Example - Pin & Hole Mating(feature on first part)
![Page 10: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/10.jpg)
Example - Pin & Hole Mating(feature on second part)
![Page 11: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/11.jpg)
Example - Pin & Hole Mating(Assembling two parts)
![Page 12: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/12.jpg)
Part Location Variation
• Varied location of Part B calculated from nominal location of Part A
• Uses same math as nominal model!
![Page 13: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/13.jpg)
Chaining together parts
![Page 14: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/14.jpg)
Variation Analysis
• Error types:– Change in process average (mean shift)– Process variation around average
(variance)
![Page 15: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/15.jpg)
Precision vs. Accuracy
![Page 16: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/16.jpg)
Desired Distribution
• LSL = lower specification limit• USL = upper specification limit
![Page 17: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/17.jpg)
Error Accumulation
• Worst case tolerancing:– assume all errors at extremes– errors accumulate linearly w/ # of
parts– deterministic, not statistical
• Statistical tolerancing– assume errors distributed randomly
between limits– errors accumulate as sqrt of # of parts
• if mean is equal to nominal dimension!
![Page 18: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/18.jpg)
Error Accumulation
• Sums of zero-mean errors accumulate as sqrt(N), because + and - errors cancel
• Sums of non-zero-mean errors accumulate as N, because there are no cancellations
![Page 19: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/19.jpg)
How do Non-zero-mean errors occur?
• Example:– operator stops material removal as
soon as part enters tolerance zone
![Page 20: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/20.jpg)
Example - Stapler Variations
![Page 21: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/21.jpg)
(small) Error Transform
![Page 22: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/22.jpg)
Using Error Transform
![Page 23: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/23.jpg)
Using Error Transform
![Page 24: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/24.jpg)
Using Error Transform - Part 2
![Page 25: Mathematical Modeling of Assembly Coordinate frames –each part has a base coordinate frame Relationships between parts are expressed as 4x4 matrix transforms](https://reader030.vdocument.in/reader030/viewer/2022032704/56649d625503460f94a452a3/html5/thumbnails/25.jpg)
Using Error Transform - Part 2