UNCERTAINTY ANALYSIS AND QUALITY ASSURANCE FOR COORDINATE MEASURING SYSTEM SOFTWARE
Table Of Contents
TABLE OF CONTENTS 1.
INTRODUCTION .................................................................................................... 1 1.1
Coordinate Measuring Systems .......................................................................... 1 1.2
The Problem of Uncertainty................................................................................ 4 1.3
The Problem of Software Uncertainty ................................................................ 6 1.4
The NIST ATEP-CMS software testing program............................................. 11 2.
BACKGROUND INFORMATION AND LITERATURE SURVEY................... 14 2.1
NIST work ........................................................................................................ 14 2.2
NIST Algorithm Information............................................................................ 17 2.3
NPL work.......................................................................................................... 21 2.4
Other documentation......................................................................................... 24 3.
LEAST SQUARES FITTING ................................................................................ 26 3.1
The Problem of Least Squares Fitting .............................................................. 26 3.2
The Need for Least Squares Fitting .................................................................. 27 3.3
Issues involved in solving Least Squares Fitting Problems.............................. 27 3.4
An Approach to Least Squares Fitting.............................................................. 28 3.5
Implementation of the Approach ...................................................................... 28 3.6
Results of the Implementation .......................................................................... 29 3.7
An Application of Least Squares Fitting .......................................................... 29 4.
MINIMUM ZONE FITTING ................................................................................. 39 4.1
The Problem of Minimum Zone Fitting............................................................ 39 4.2
The Need for Minimum Zone Fitting ............................................................... 40 4.3
Issues involved in solving Minimum Zone Fitting Problems........................... 42 4.4
An Approach to Solving Minimum Zone Fitting ............................................. 43 4.5
Implementation of the Approach ...................................................................... 44 4.6
Reference vs. Commercial Algorithm Performance......................................... 46 4.7
Results of the Implementation .......................................................................... 48 5.
MAXIMUM INSCRIBED AND MINIMUM CIRCUMSCRIBED FITTING...... 49 5.1
The Problem of Maximum Inscribed and Minimum Circumscribed Fitting.... 49 5.2
The Need for Maximum Inscribed and Minimum Circumscribed Fitting........ 51 5.3
Issues involved in solving Maximum Inscribed and Minimum Circumscribed Fitting Problems...................................................................................................... 52 5.4 An approach to Maximum Inscribed and Minimum Circumscribed Fitting .... 53 5.5 Implementation of the Approach ...................................................................... 59 5.6 Reference vs. Commercial Algorithms for Maximum Inscribed and Minimum Circumscribed Fitting ............................................................................................. 59 5.7 Results of the Implementation .......................................................................... 61 6. LEAST SQUARES FITTING OF COMPLEX SURFACES................................. 62 6.1 The Problem of Least Squares Fitting of Complex Surfaces............................ 62 6.2 The Need for Fitting of Complex Surfaces....................................................... 63 6.3
Issues involved in solving Least Squares Fitting of Complex Surfaces Problems ................................................................................................................................. 64 6.4
An Approach to Solving Least Squares Fitting of Complex Surfaces Problems ................................................................................................................................. 68 6.5 Implementation of the Approach ...................................................................... 69 6.6
Results of the Implementation .......................................................................... 77 iv 7.
OTHER ISSUES AND FUTURE WORK ............................................................. 78 7.1
Other Issues Involved in CMS Fitting Software............................................... 78 7.2
Future Work...................................................................................................... 79 8. Conclusions............................................................................................................. 81
REFERENCES ........................................................................................................... 87
Thesis Abstract
Thesis Overview
1. INTRODUCTION 1.1
Coordinate Measuring Systems Coordinate measuring systems (CMSs) are installed in factories, research and medical labs, as well as many other industrial and scientific facilities. The definition of a CMS is, "…any piece of equipment which collects coordinates (points) and calculates and displays additional information using the measured points," [8]. To find the dimensions of a part, a CMS measures point locations on the object’s surface. This coordinate data is then processed to determine the part’s dimensions and the types and locations of variations in the surface. Note that the raw coordinate data generally must be interpreted before the information gathered is of any real use. Specifically, once the coordinate data points are collected from the surface of the part by the CMS hardware, the information is processed by software, which usually performs a geometric fit to the gathered data. This fitting software, which is usually integrated as part of the CMS, uses the coordinate data to, for instance, determine a part’s location, orientation, concentricity, or deviation of the part from the corresponding perfect geometry. The software can apply appropriate processing of the data to determine if a part is within tolerances defined in specifications. Since a part is measured through only a sampling of points, its true surface can never be known exactly; instead, an approximation of the surface is known based on a finite sampling of coordinate points. The software will often be required to compute a “substitute geometry” based on the imperfect data. This substitute geometry is a perfect, theoretical, mathematical shape fit to the points. For example if a CMS samples points on a surface that is 2 nominally cylindrical, then the software can compute a fit to find the “best” perfect cylinder that is represented by the imperfectly measured points on the imperfect physical surface. Just how this substitute feature is determined can be complicated and is discussed in this paper. CMSs are used to measure everything from pistons and cylinders to gears and screw threads to airplane wings and car doors. Sometimes their uses go beyond manufactured parts to include, for example, bones and vertebrae alignment in the medical field. Many different types of coordinate measuring systems are in use today including theodolites, photogrammetry, optical systems, and coordinate measuring machines. Though such variation exists among CMSs, the software packages that they normally come equipped with are similar and share some basic problems and issues. Examples of several different types of systems are given1 in figure 1.1. Shown in the pictures are: 1) CMM: a measuring system with the means to move a probing system and capable of determining spatial coordinates on a workpiece surface, 2) theodolite: a small telescope mounted and moving on two graduated circles, one horizontal, the other vertical, while its axes pass through the center of the circles. The data points are found using triangulation, and 3) photogrammetry: this system works by taking pictures of the object being measured with a digital camera then inputting the image into the software to determine its part information. 1
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