2. Measurement Process Characterization
2.3. Calibration
2.3.4. Catalog of calibration designs
2.3.4.4. Roundness measurements

## Multiple-trace roundness designs

High precision measurements High precision roundness measurements are required when an object, such as a hemisphere, is intended to be used primarily as a roundness standard. The method outlined on this page is appropriate for either a turntable-type instrument or a spindle-type instrument.
Measurement method The measurement sequence involves making multiple traces of the roundness standard where the standard is rotated between traces. Least-squares analysis of the resulting measurements enables the noncircularity of the spindle to be separated from the profile of the standard. The reader is referred to the publication on the subject (Reeve) for details covering measurement techniques and analysis.
Method of n traces The number of traces that are made on the workpiece is arbitrary but should not be less than four. The workpiece is centered as well as possible under the spindle. The mark on the workpiece which denotes the zero angular position is aligned with the zero position of the spindle as shown in the graph. A trace is made with the workpiece in this position. The workpiece is then rotated clockwise by 360/n degrees and another trace is made. This process is continued until n traces have been recorded.
Mathematical model for estimation For i = 1,...,n, the ith angular position is denoted by $$\theta_i = \frac{360(i-1)}{n} \mbox{deg} \,\, .$$
Definition of terms relating to distances to the least squares circle The deviation from the least squares circle (LSC) of the workpiece at the $$\theta_i$$ position is $$\alpha_i$$.

The deviation of the spindle from its LSC at the $$\alpha_i$$ position is $$\beta_i$$.

Terms relating to parameters of least squares circle For the jth graph, let the three parameters that define the LSC be given by $$R_j, \, a_j, \, b_j$$ defining the radius R, a, and b as shown in the graph. In an idealized measurement system these parameters would be constant for all j. In reality, each rotation of the workpiece causes it to shift a small amount vertically and horizontally. To account for this shift, separate parameters are needed for each trace.
Correction for obstruction to stylus Let $$Y_{ij}$$ be the observed distance (in polar graph units) from the center of the jth graph to the point on the curve that corresponds to the $$\theta_i$$ position of the spindle. If K is the magnification factor of the instrument in microinches/polar graph unit and $$\delta$$ is the angle between the lever arm of the stylus and the tangent to the workpiece at the point of contact (which normally can be set to zero if there is no obstruction), the transformed observations to be used in the estimation equations are: $$Z_{ij} = K \, \mbox{cos} (\delta) \, Y_{ij} \, .$$
Estimates for parameters The estimation of the individual parameters is obtained as a least-squares solution that requires six restraints which essentially guarantee that the sum of the vertical and horizontal deviations of the spindle from the center of the LSC are zero. The expressions for the estimators are as follows: \begin{array} $$\widehat{\alpha}_i = \sum_{j=1}^n \sum_{k=1}^n t_{k-i+j+n} \, Z_{kj} \\ \widehat{\beta}_i = \sum_{j=1}^n \sum_{k=1}^n t_{k-i+1+n} \, Z_{kj} \\ \widehat{R}_j = \sum_{k=1}^n Z_{kj} \\ \widehat{a}_j = \frac{2}{n} \sum_{k=1}^n Z_{kj} \, \mbox{cos} (\theta_k) \\ \widehat{b}_j = \frac{2}{n} \sum_{k=1}^n Z_{kj} \, \mbox{sin} (\theta_k) \,\, , \\ \end{array} where \begin{array} \( t_m = \frac{n-3}{n^2} ; \,\, m=1 \\ t_m = -\frac{1}{n^2} \left( 1 + 2 \mbox{cos} (\theta_m) \right) ; \,\, 2 \le m \le n \,\, . \end{array} Finally, the standard deviations of the profile estimators are given by:  {\large s}_{\alpha_i} = {\large s}_{\beta_i} = \frac{\sqrt{n-3}}{n} {\large s} \, ; \,\, (i = 1, \, \ldots, \, n) \, .  Computation of standard deviation The computation of the residual standard deviation of the fit requires, first, the computation of the predicted values,  \widehat{Z}_{ij} = \widehat{\alpha}_{i+j-1} - \widehat{\beta}_i + \widehat{R}_j + \widehat{a}_j \, \mbox{cos}(\theta_i) + \widehat{b}_j \, \mbox{sin}(\theta_i) \,\, .  The residual standard deviation with \( \nu = n^2 - 5n + 6$$ degrees of freedom is $${\large s} = \sqrt{\frac{1}{n^2 - 5n + 6} \sum_{j=1}^n \sum_{i=1}^n \left( Z_{ij} - \widehat{Z}_{ij} \right) ^2} \,\, .$$