ISO definition for form tolerances requires the form error of a geometry to be less than some specific limit. Most inspectors measure form tolerances as the minimum zone solution which minimizes the maximum deviations between the measured points and the estimated feature. Current Coordinate Measuring Machines (CMMs) measure the form error based on the least square method, which minimizes the sum of the squared errors between the measured points and the estimated feature, possibly resulting in some overestimations on the errors.
The form evaluation algorithm for the conical geometry developed in this thesis computes the minimum zone solution based on the ISO standards, which is inherently a nonlinear optimization problem with local minimums. The developed algorithm consists of two phases. The first phase is to find good initial values for the cone geometry. Disregarding the angular components of the measured data points, 3-D data points are mapped into 2-D data points. From the 2-D points, least square line can be fitted and the tolerance zone with respect to the line can be calculated. Comparing the tolerance zone, the cone axis which has the smallest tolerance zone can be obtained. With these information, initial values for the cone geometric factors can be fixed. The second phase is to search geometric factors for the minimum zone cone with the initial values from the first phase. The problem is formulated as a constrained nonlinear optimization problem, which finds two nearest coaxial cones locating all the measured data points between the two cones, which satisfies the ISO minimum zone criteria. At each phase, the nonlinear constrained optimization problem is solved by sequential quadratic programming (SQP).
Using the adequate initial values from the first phase of the present algorithm, the minimum zone solution can be efficiently and conveniently obtained. Lots of numerical examples are presented in this thesis, which show that this cone evaluation algorithm meets the ISO requirements and that the present algorithm is reliable and efficient for the geometric evaluation of cone.
