Instrumentation and Surface Modeling for the Measurement of Disks, Circular- and Cylindrical-Strips

Abstract

This paper presents an instrument for the measurement of the 3-D surface geometry of disks and circular strips. Furthermore, tensor product surface approximations and analysis techniques are implemented, which enable a mode analysis of the data. The measurement device consists of a motorized rotary stage and a set of six tactile sensors which measure the displacement of the surface during rotation. The data from all six sensors are acquired synchronously, yielding six tracks of data lying on a circular field with a constant number of samples per track. This delivers data in polar coordinates. Each tactile sensor is individually calibrated, and a linearizing polynomial determined. This improves the achievable accuracy of the displacement measurements. It should be noted that the selection of six sensors is arbitrary and the methods presented work for any number of sensors. Furthermore, the sensors may be repositioned should a smaller separation be required to obtain additional resolution when measuring smaller devices. A data analysis method is implemented that enables the computation of individual surface mode models. The analysis tool is also applicable to cylindrical objects, as demonstrated. A tensor product approximation procedure for surfaces of revolution is presented. The method uses complex anisotropic moments, yielding both the magnitude and phase of specific surface modes. A covariance propagation analysis is derived for the tensor product approximation. This method is used in conjunction with a Kolmogorov-Smirnov test to determine how many surface modes are required to obtain a satisfactory fit. The results for a set of four test objects are presented, showing that a standard deviation of the approximation residual of σ ≈ 10 μm is obtained for objects with diameters of Φ = 200 mm