Roundness Error And Cylindricity Error Evaluation Based On The Hausdorff Distance

Roundness error and cylindricity error are the important parts of the geometric error evaluation. Whether roundness error and cylindricity error can be evaluated accurately and efficiently or not will directly influence the performance and life of the mechanical products. The essence of roundness error and cylindricity error evaluation is the process of comparing the practical measured elements with the ideal elements, which is calculating the maximum deviation between the practical measured elements and the ideal elements to measure the similarity of them. Hausdorff distance, as an often used similarity measurement between the geometric shapes, becomes larger when the deviation is larger and the similarity is lower; on the contrary, the Hausdorff distance is smaller. Therefore, an evaluation method of roundness error and cylindricity error based on the minimum directed Hausdorff distance is proposed in this paper.Firstly, the definition and the classification of the Hausdorff distance are introduced, as well as the basic knowledge of Bezier curve/surface, B-spline curve/surface and NURBS curve/surface. Based on these, an algorithm of calculating the minimum distance between a point and a curve is proposed. The algorithm turns the minimum distance computation problem into the problem of solving the nonlinear equation, which is solved by hybrid method of the projection polyhedron algorithm and the Newton method by two steps.Secondly, the four cases of Hausdorff distance between curves are discussed and the corresponding constraint equations are given. The candidate points of EA and EB are directly obtained by the algorithm of calculating the minimum distance between a point and a curve; for the candidate points of I type, the two-dimensional projection polyhedron algorithm is used to isolate all the roots, and the accuracy of the solutions is improved by the multivariate Newton method. As a result, the solutions of Hausdorff distance between curves are realized.Finally, an evaluation method of roundness error and cylindricity error based on the minimum directed Hausdorff distance is proposed. The computation of the roundness error and cylindricity error is turned into the computation of the minimum directed Hausdorff distance based on this new method. A mathematical programming model to exact the minimum directed Hausdorff distance under transformation is proposed and the linearization method of the programming model is provided. Optimization results of roundness error and cylindricity error are obtained by linear programming method, which satisfy the principle of the least condition.