Fast registration of 3D point clouds with offset surfaces in precision grinding of free-form surfaces

Because of its high sensitivity to misalignment, precision grinding of free-form surfaces with micron accuracy requires accurate registration of the surface measurement point cloud. Registration of point clouds obtained with a coordinate measuring machine (CMM) is generally an iterative process of finding optimal coordinate transformation between the CMM frame and the model frame of the workpiece by minimizing the point-to-surface distances with probe radius compensation. For free-form surfaces, frequent calculation of point-to-surface distances consumes very much time, and a trade-off has to be made between the efficiency and the accuracy. This paper presents a method for fast registration of free-form surface point clouds based on the point-to-triangle distance which involves only Delaunay triangulation of a two-dimensional dataset, and the surface normal is quickly calculated from cross product. Probe radius compensation is realized by registering the probe center points with the offset surface. We prove that it is equivalent to registering the probe contact points with the nominal surface through theoretical analysis. The registration problem is then formulated as sequential linear least-square problems with properly defined ball constraints. To validate the method, numerical simulations are presented to show the accuracy of the point-to-triangle distance. The registration algorithm also shows excellent robustness against misalignment of tens of millimeters/degrees. Finally measurement, registration, and grinding of a free-form optical surface are experimentally demonstrated. The surface error obtained after registration is used for compensatory grinding which reduces it to micron level.