Minimum centroid neighborhood for minimum zone sphericity

The minimum zone sphericity tolerance is derived from the ANSI and ISO standards for roundness and has extensive applications in the tribology of ball bearings, hip joints and other lubricated pairs. The worst-case proposed in this paper provides theoretical evidence that the minimum zone center of the two (circumscribed and inscribed reference) spheres with minimum radial separation containing the sampled spherical surface is included in a spherical neighborhood centered in the centroid of radius 2π-2EC, where EC is the sphericity error related to the centroid, which can be determined in closed form. Such linear estimating (about 20% of EC from the centroid, i.e., about one order of magnitude lower than the sphericity tolerance to be assessed) can be used to locate the sphere center with a given tolerance and as a search neighborhood for minimum zone center-based algorithms, such as metaheuristics (genetic algorithms, particle swarm optimization, etc.). The proposed upper bound has been experimentally assessed, using a genetic algorithm (GA) with parameters previously optimized for roundness and extended to three dimensions, which has overcome most of all available datasets from the literature that have been tested with center-based minimum zone algorithms by different authors. The optimum dataset size on artificially generated datasets is also discussed and it is speculated to allow the extension of the proposed upper bound to partial (or incomplete) spherical features

Bayesian Hierarchical Model for Combining Two-resolution Metrology Data

This dissertation presents a Bayesian hierarchical model to combine two-resolution metrology data for inspecting the geometric quality of manufactured parts. The high- resolution data points are scarce, and thus scatter over the surface being measured, while the low-resolution data are pervasive, but less accurate or less precise. Combining the two datasets could supposedly make a better prediction of the geometric surface of a manufactured part than using a single dataset. One challenge in combining the metrology datasets is the misalignment which exists between the low- and high-resolution data points. This dissertation attempts to provide a Bayesian hierarchical model that can handle such misaligned datasets, and includes the following components: (a) a Gaussian process for modeling metrology data at the low-resolution level; (b) a heuristic matching and alignment method that produces a pool of candidate matches and transformations between the two datasets; (c) a linkage model, conditioned on a given match and its associated transformation, that connects a high-resolution data point to a set of low-resolution data points in its neighborhood and makes a combined prediction; and finally (d) Bayesian model averaging of the predictive models in (c) over the pool of candidate matches found in (b). This Bayesian model averaging procedure assigns weights to different matches according to how much they support the observed data, and then produces the final combined prediction of the surface based on the data of both resolutions. The proposed method improves upon the methods of using a single dataset as well as a combined prediction without addressing the misalignment problem. This dissertation demonstrates the improvements over alternative methods using both simulated data and the datasets from a milled sine-wave part, measured by two coordinate measuring machines of different resolutions, respectively

Decision support system for form verification of manufactured parts.

The form verification of manufactured parts is a process composed of a set of operations that are expensive and yet add no value to the product. Yet, the resources used to inspect the parts add a small but significant amount of noise that can affect the outcome of the process. For this reason, this research provides guidelines to effectively perform the inspection process by suggesting new mathematical models and approaches that can be used for the creation of a decision support system that can assist in the verification of the accuracy of machined parts.This research proposes two approaches to improve the robustness of the mathematical models from the noise induced by the inspection process. The Dynamic Angle Approach (DAA) and the Free Form Orientation approach (FFO) presented here focus on finding the parameters of the axes and origin of the form that counteract the inaccuracies of the inspection equipment.In summary, this research suggests formalized methods for feature extraction, sampling, path planning, and form fitting, although the last mentioned received the most attention. It is believed that this comprehensive, integrated analysis will lead to the development of a decision support system.The proposed approaches and mathematical models were verified using measurements from features that were perfectly aligned with the coordinate system of the inspection equipment and from features that were intentionally misaligned. The results showed that the models were accurate and robust enough to estimate the parameters and zone of error of the form features and they performed better than existing models.The main goal of this research is to develop procedures that are simple to implement but at the same time are robust enough to provide reliable information that help the metrologist to make accurate decisions about the inspected parts. Form features such as spheres, cylinders, cones, frustums, and torus forms are commonly used to design complex parts. However, the procedures to verify most of these form features have not been developed yet by the national standards. Therefore, this research proposes new mathematical models that combine the concepts of analytic geometry and optimization to provide optimal solutions