Mathematical Model for Touching of Spherical Probe and Complex Measured Surface at CMMs

To measure complex surfaces, a wide application was made of Coordinate Measuring Machines that use the contact method. The first task in the measurement is to determine the coordinates of the sensor contact point and the surface. The touch sensor locks the coordinates of the center of the spherical probe at the moment of contact with the surface. However, the coordinates of the contact point on the probe's sphere are based on the calculation of the nominal surface. Two basic methods of calculation are known. The first method is to find the normal to the nominal surface that passes through the center of the sphere. This method is the simplest and is used to compensate the probe radius when measuring elementary surfaces or with a small number of control points. The second method uses the normals to the equidistant surface obtained at the centers of the spherical probe. This method is used when scanning a surface or in the absence of a CAD model. The accuracy of determining the point of contact is influenced by the accuracy of the size and shape of the spherical probe. Therefore, a new mathematical model is proposed that takes into account the geometry of the sphere of the probe. The idea is that the probe area is calibrated to fit the shape and creates its 3D model. For this it is proposed to use the Scanning probe microscopy, in particular, an atomic-force microscope. The calculation of the contact point is realized by a numerical algorithm according to which the point on the sphere is located with the minimum distance to the nominal surface in the direction of the normal. The presented algorithm has good convergence even when using a large number of points on the sphere. The check of the developed algorithm for measuring the rolling bearing parts in comparison with the standard method showed a 9% decrease in the measurement error.