Constrained Least-Squares Fitting for Tolerancing and Metrology

2019 ◽  
pp. 255-275
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Constrained Least Squares ◽  
Least Squares Fitting

Optimality Conditions for Constrained Least-Squares Fitting of Circles, Cylinders, and Spheres to Establish Datums

2018 ◽  
Vol 18 (3) ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Optimality Conditions ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Necessary Condition ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Candidate Solution ◽  
Cylinders And Spheres ◽  
New Algorithms

This paper addresses the combinatorial characterizations of the optimality conditions for constrained least-squares fitting of circles, cylinders, and spheres to a set of input points. It is shown that the necessary condition for optimization requires contacting at least two input points. It is also shown that there exist cases where the optimal condition is achieved while contacting only two input points. These problems arise in digital manufacturing, where one is confronted with the task of processing a (potentially large) number of points with three-dimensional coordinates to establish datums on manufactured parts. The optimality conditions reported in this paper provide the necessary conditions to verify if a candidate solution is feasible, and to design new algorithms to compute globally optimal solutions.

Convexity and Optimality Conditions for Constrained Least-Squares Fitting of Planes and Parallel Planes to Establish Datums

2018 ◽  
Vol 19 (1) ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Optimality Conditions ◽  
Global Optimum ◽  
Objective Functions ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Optimum Solution ◽  
Set Of Points ◽  
Theoretical Issues ◽  
Global Optimum Solution

This paper addresses some important theoretical issues for constrained least-squares fitting of planes and parallel planes to a set of points. In particular, it addresses the convexity of the objective function and the combinatorial characterizations of the optimality conditions. These problems arise in establishing planar datums and systems of planar datums in digital manufacturing. It is shown that even when the set of points (i.e., the input points) are in general position, (1) a primary planar datum can contact 1, 2, or 3 input points, (2) a secondary planar datum can contact 1 or 2 input points, and (3) two parallel planes can each contact 1, 2, or 3 input points, but there are some constraints to these combinatorial counts. In addition, it is shown that the objective functions are convex over the domains of interest. The optimality conditions and convexity of objective functions proved in this paper will enable one to verify whether a given solution is a feasible solution, and to design efficient algorithms to find the global optimum solution.

Optimality Conditions for Constrained Least-Squares Fitting of Circles, Cylinders, and Spheres to Establish Datums

2017 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Optimality Conditions ◽  
Necessary Conditions ◽  
Three Dimensional ◽  
Necessary Condition ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Candidate Solution ◽  
Cylinders And Spheres ◽  
New Algorithms

This paper addresses the combinatorial characterizations of the optimality conditions for constrained least-squares fitting of circles, cylinders, and spheres to a set of input points. It is shown that the necessary condition for optimization requires contacting at least two input points. It is also shown that there exist cases where the optimal condition is achieved while contacting only two input points. These problems arise in digital manufacturing, where one is confronted with the task of processing a (potentially large) number of points with three-dimensional coordinates to establish datums on manufactured parts. The optimality conditions reported in this paper provide the necessary conditions to verify if a candidate solution is feasible, and to design new algorithms to compute globally optimal solutions.

Convexity and Optimality Conditions for Constrained Least-Squares Fitting of Planes and Parallel Planes to Establish Datums

2017 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Optimality Conditions ◽  
Feasible Solution ◽  
Global Optimum ◽  
Objective Functions ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Optimum Solution ◽  
Theoretical Issues ◽  
Global Optimum Solution

This paper addresses some important theoretical issues for constrained least-squares fitting of planes and parallel planes to a set of input points. In particular, it addresses the convexity of the objective function and the combinatorial characterizations of the optimality conditions. These problems arise in establishing planar datums and systems of planar datums in digital manufacturing. It is shown that even when the input points are in general position: (1) a primary planar datum can contact 1, 2, or 3 input points, (2) a secondary planar datum can contact 1 or 2 input points, and (3) two parallel planes can each contact 1, 2, or 3 input points, but there are some constraints to these combinatorial counts. In addition, it is shown that the objective functions are convex over the domains of interest. The optimality conditions and convexity of objective functions proved in this paper will enable one to verify whether a given solution is a feasible solution, and to design efficient algorithms to find the global optimum solution.

Toward a New Mathematical Definition of Datums in Standards to Support Advanced Manufacturing

2018 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Measurement Techniques ◽  
Digital Technologies ◽  
Advanced Manufacturing ◽  
Digital Measurement ◽  
Constrained Least Squares ◽  
Iso Standards ◽  
Least Squares Fitting ◽  
Mathematical Definition ◽  
Definition Of

Recent advances in the digitization of manufacturing have prompted ASME and ISO standards committees to reexamine the definition of datums. Any new definition of datums considered by the standards committees should cover all datum feature types used in design, and support both traditional metrological methods and new digital measurement techniques. This is a challenging task that requires some careful compromise. This paper describes and analyzes various alternatives considered by the standards committees. Among them is a new mathematical definition of datums based on constrained least-squares fitting. It seems to provide the best compromise and has the potential to support advanced manufacturing that is increasingly dependent on digital technologies.

On Algorithms and Heuristics for Constrained Least-Squares Fitting of Circles and Spheres to Support Standards

2018 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Theoretical Work ◽  
Two Dimensions ◽  
International Standards ◽  
Three Dimensions ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Practical Algorithms ◽  
Coordinate Metrology ◽  
Set Of Points

Constrained least-squares fitting has gained considerable popularity among national and international standards committees as the default method for establishing datums on manufactured parts. This has resulted in the emergence of several interesting and urgent problems in computational coordinate metrology. Among them is the problem of fitting inscribing and circumscribing circles (in two-dimensions) and spheres (in three-dimensions) using constrained least-squares criterion to a set of points that are usually described as a ‘point-cloud.’ This paper builds on earlier theoretical work, and provides practical algorithms and heuristics to compute such circles and spheres. Representative codes that implement these algorithms and heuristics are also given to encourage industrial use and rapid adoption of the emerging standards.

On Algorithms and Heuristics for Constrained Least-Squares Fitting of Circles and Spheres to Support Standards

2019 ◽  
Vol 19 (3) ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Theoretical Work ◽  
Two Dimensions ◽  
International Standards ◽  
Three Dimensions ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Practical Algorithms ◽  
Coordinate Metrology ◽  
Set Of Points

Constrained least-squares fitting has gained considerable popularity among national and international standards committees as the default method for establishing datums on manufactured parts. This has resulted in the emergence of several interesting and urgent problems in computational coordinate metrology. Among them is the problem of fitting inscribing and circumscribing circles (in two dimensions) and spheres (in three dimensions) using constrained least-squares criterion to a set of points that are usually described as a “point-cloud.” This paper builds on earlier theoretical work, and provides practical algorithms and heuristics to compute such circles and spheres. Representative codes that implement these algorithms and heuristics are also given to encourage industrial use and rapid adoption of the emerging standards.

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