Fast Evaluation of Minimum Zone form Errors of Freeform NURBS Surfaces

Abstract

Nowadays freeform components are more and more commonly employed in precision engineering and it is crucial to evaluate their form qualities. The widely adopted PV parameters for form errors need to be evaluated in the sense of minimum zone, which is a non-differentiable optimization problem, and very difficult to be solved. Currently NURBS has become a ubiquitous data format in computer aided design and manufacturing. However, little can be found concerning the minimum zone fitting between the NURBS surfaces and measured data, due to the mathematical complexity of this problem. A fast evaluation strategy is proposed in this paper. It is implemented in three stages. The NURBS model is decomposed into Bezier patches first and the iterative closest point matching is implemented. And then the relative position is refined by the orthogonal distance least squares fitting. Finally the minimum zone fitting is carried out by the primal-dual interior point method. The solutions are recursively updated by line search until the Karush-Kuhn-Tucker conditions are satisfied. Numerical experiments proved that the proposed method is capable of minimum zone fitting for freeform NURBS surfaces with very high accuracy and efficiency.