Statistical modeling and design for CMM-type data locating known two-dimensional geometries

This research was motivated by a problem from a car-manufacturer that needed to compare/analyze two hood assembly fixtures. The consistency of the car hood placement was examined for both fixtures. A “constant width” gap between the hood and the car's body is preferred. A “wide” gap at one point and a “narrow” gap at another constitutes a bad quality placement of the car hood, and will affect customer satisfaction.; 3-D data were taken using a Coordinate Measuring Machine (CMM) with 12 (fixed) probe paths to the hood. For each fixture, ten “12-dimensional” vectors were taken and the original goal was to compare the 2 fixtures and decide which one gives more consistent/better placement of the hood. As a first step in statistical modeling and analysis for this type of problem, we consider a 2-dimensional idealization ignoring the 4 “Up-Down” measurements and replacing the complicated real hood geometry with simple ideal geometries of approximately the same overall size. Then the data become ten “8-dimensional” vectors.; A “rough” analysis done by the car-manufacturer analyst was to take ten “8-dimensional” vectors for both fixtures A and B and make 8 comparisons of sample variances (one probe path at a time). This ignores geometry effects/physical dependence and answers the wrong question (how variable are the measurements, versus how variable is hood placement).; In this dissertation, we consider statistical analysis specifically focused on the issue of “Where is the hood?” (as opposed to “What is the distribution of a Y_i?”). Our main contribution is in the realm of study planning, where the object is to choose a set of probe paths that provide optimal precision for estimating the position of a single “hood” placement. We propose a figure of merit for comparing alternative designs (data collection plans) and compare several algorithms for optimizing this criterion. Our comparisons of algorithms are across design sizes, “hood” geometries and nominal locations. We conclude that the design sizes, geometries, and nominal locations affect the optimum design while the candidate probe paths doesn't. The paths in an optimum/best design will generally be located closest to the “extremes” of the object boundary or will be aimed near “corners” of the object.