Minimum-Zone Form Tolerance Evaluation Using Rigid-Body Coordinate Transformation

Abstract

This paper presents an optimization approach for the accurate evaluation of minimum-zone form tolerances from discrete coordinate measurement data. The approach minimizes the minimum-deviation objective function defined as the difference between the maximum and minimum distances of the measured coordinate data from the reference feature. The objective function is formulated as a function of rigid-body coordinate transformation parameters and involves fewer independent parameters than the existing tolerance evaluation algorithms. As a result, improved convergence efficiency and numerical stability are achieved. A standard direct search algorithm, the downhill simplex search algorithm, is employed to minimize the objective function. The least-squares estimates are employed as good initial conditions to facilitate convergence to the global solutions. A new method, named as the Median Technique, is implemented to well center the circularity measured data and well align the cylindricity measured data in order to provide valid least-squares estimates based on the Limacon approximation. Results from simulation and comparative studies have shown that the proposed method evaluates minimum-zone form tolerances with reliable accuracy.