## 1. Introduction

## 2. Materials and Methods

- Three-point inverse kinematic algorithm (TPIK): a calibration procedure that references the local coordinates of each sensor into one global coordinate system (G). This produces system parameters whose values are based on a numerical solution of mixed non-linear equations. Further discussion to follow in Section 2.2.
- Three-point forward kinematic algorithm (TPFK): a closed form solution that algebraically fits the transformed data points through the circumference of a circle. Further discussion to follow in Section 2.1.

#### 2.1. Three-Point Forward Kinematics (TPFK) Algorithm—Recovery of Circle Geometry

#### 2.2. Three-Point Inverse Kinematics (TPIK) Algorithm—Calibration Procedures

_{1}as shown in Figure 1) should remain unchanged during the calibration procedure.

#### 2.3. Roundness Index Based on TPIK and TPFK Algorithms

## 3. Results and Discussion

#### 3.1. Validation of the TPIK Algorithm by Mathematical Simulation

_{rs}value could be used as an indicator to evaluate the difference between the numerical solution and the true solution if the solution exists and is unique. While the existence and uniqueness of a real solution for both the TPIK algorithm and the optimization method was not investigated, the researcher purposely limited this paper to the numerical comparison. The numerical values of $S$ in Table 2 are rounded to 3 decimal points and displayed for each trial when the optimization algorithm converges or stops. For example, trials 1, 4, 5, 8, and 9 in Table 2 produced acceptable results because they converged quickly with very small E

_{rs}values. While trials 6, 7, and 10 failed to produce acceptable results due to their non-convergence and very large E

_{rs}values. On the other hand, trials 2 and 3 did not converge, however they produced adequate final values because E

_{rs}were considerably small when compared to trials 6, 7, and 10. These studies also show that the numerical solution in trial 8, which has the fastest convergence and smallest E

_{rs}, had the least difference from the true solution.

#### 3.2. Validation of the TPFK Algorithm by Mathematical Simulation

_{3p}} values for both radii agree closely with the true radii values.

#### 3.3. Validation of the TPIK Algorithm by CAD Simulation

_{rs}value. The effect of the measurement resolution was studied using the initial estimations to set the true values. This was carried out for n = {6,5,4,2} in trials 4–8, which confirms that high measurement resolution was not always necessary to attain an acceptable solution. Similarly, a low-resolution measurement may successfully converge to a solution, but is far off from the true values. For example, trial 5 was rounded to n = 5 decimal places, and the optimization had successfully converged to the true values with a small E

_{rs}. The simulation was repeated using the same initial estimation with n = {4,3,2}. Interestingly, trial 7 was simulated for n = 3 (trial 5) and had not only converged quicker than that of n = 5 (trial 3), but also had a smaller E

_{rs}value. However, the solution for trial 7 was farther off from the true value when evaluated against trial 5.

#### 3.4. Validation of the TPFK Algorithm by CAD Simulation

_{e}ranged between 38.000 mm to 39.500 mm and with all of their true centers fixed at (10.010, 38.788) mm. The TPFK algorithm was implemented to compute the “fit” of their radii and centers based on the displacement measurement and the sensor location.

#### 3.5. Study of the Roundness Index Based on the Mathematical Model

#### 3.6. Study of Roundness Index Based on CAD Model

^{o}by using the dimensioning tools in 2D drawing. Subsequently, the {a, b, ${R}_{3p}$} were measured at every increment using the TPFK algorithm with results in Table 8.

#### 3.7. Validation of the TPIK and TPFK Algorithms Based on the Empirical Study

_{1}, S

_{2}, S

_{3}} were used to measure the displacements from the initialized. coordinates into the surface of the bearing. Researchers applied adhesive tape of known thickness on the bearing surface to create multiple radii for both calibration and inspected part experiments.

_{d}measurements are summarized in Table 9. Likewise, following the calibration procedures an inspected part with radius of ${R}_{e}=41.03\text{}\mathrm{mm}$ was created by adding a 1.03 mm thick tape on a section on the bearing. Following this, the corresponding displacements were measured and results are shown in Table 10. A summary of all measurements is shown in the error plot in Figure 13.

_{rs}= 0.0071 mm. The final values were applied in the TPFK algorithm to calculate the radius of the inspected part using the data in Table 10. The estimated {a, b, R

_{3p}} values were {25.72, 44.75, 40.78} mm, respectively. This indicates that the error in measuring the inspected part radius was $er=\left|{R}_{3p}-{R}_{e}\right|=0.25$ mm or 0.61% percentage error, which represents acceptable agreement between values given the simplicity of the experiment setup used in this study.

## 4. Conclusions and Future Work

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Umbach, D.; Jones, K.N. A few methods for fitting circles to data. IEEE Trans. Instrum. Measure.
**2003**, 52, 1881–1885. [Google Scholar] [CrossRef] - Safaee-Rad, R.; Tchoukanov, I.; Smith, K.C.; Benhabib, B. Three-dimensional location estimation of circular features for machine vision. IEEE Trans. Robot. Autom.
**1992**, 8, 624–640. [Google Scholar] [CrossRef] - Traband, M.T.; Medeiros, D.J.; Chandra, M.J. A statistical approach to tolerance evaluation for circles and cylinders. IIE Trans. (Inst. Ind. Eng.)
**2004**, 36, 777–785. [Google Scholar] [CrossRef] - Landau, U.M. Estimation of a circular arc center and its radius. Comput. Vis. Gr. Image Process.
**1987**, 38, 317–326. [Google Scholar] [CrossRef] - Thomas, S.M.; Chan, Y.-T. A simple approach for the estimation of circular arc center and its radius. Comput. Vis. Gr. Image Process.
**1989**, 45, 362–370. [Google Scholar] [CrossRef] - Thom, A. A statistical examination of the megalithic sites in Britain. J. R. Stat. Soc. Ser. A
**1955**, 118, 275–295. [Google Scholar] [CrossRef] - Robinson, S.M. Fitting spheres by the method of least squares. Commun. ACM
**1961**, 4, 491. [Google Scholar] [CrossRef] - Lin, S.; Jusko, O.; Härtig, F.; Seewig, J. A least squares algorithm for fitting data points to a circular arc cam. Measurement
**2017**, 102, 170–178. [Google Scholar] [CrossRef] - Hopp, T.H. Computational metrology. Manuf. Rev.
**1993**, 6, 295. [Google Scholar] - Schwenke, H.; Knapp, W.; Haitjema, H.; Weckenmann, A.; Schmitt, R.; Delbressine, F. Geometric error measurement and compensation of machines—an update. CIRP Annals.
**2008**, 57, 660–675. [Google Scholar] [CrossRef] - Phillips, S.D.; Borchardt, B.; Abackerli, A.J.; Shakarji, C.; Sawyer, D.; Murray, P.; Rasnick, B.; Summerhays, K.D.; Baldwin, J.M.; Henke, R.P.; et al. The validation of CMM task specific measurement uncertainty software. In Proceedings of the ASPE 2003 Summer Topical Meeting “Coordinate Measuring Machines”, Charlotte, NC, USA, 25–26 June 2003; pp. 25–26. [Google Scholar]
- Pegna, J.; Guo, C. Computational metrology of the circle. In Proceedings of the Computer Graphics International (Cat. No. 98EX149), Hannover, Germany, 26–26 June 1998; pp. 350–363. [Google Scholar]
- Rusu, C.; Tico, M.; Kuosmanen, P.; Delp, E.J. Classical geometrical approach to circle fitting—Review and new developments. J. Electron. Imaging
**2003**, 12, 179–194. [Google Scholar] - De Guevara, I.L.; Muñoz, J.; De Cózar, O.D.; Blázquez, E.B. Robust fitting of circle arcs. J. Math. Imaging Vis.
**2011**, 40, 147–161. [Google Scholar] [CrossRef] - Brandon, J.A.; Cowley, A. A weighted least squares method for circle fitting to frequency response data. J. Sound Vib.
**1983**, 89, 419–424. [Google Scholar] [CrossRef] - Muralikrishnan, B.; Raja, J. Computational Surface and Roundness Metrology; Springer Science & Business Media: Berlin, Germany, 2008. [Google Scholar]
- Zelniker, E.E.; Clarkson, I.V.L. A statistical analysis of the Delogne—Kåsa method for fitting circles. Digit. Signal Process.
**2006**, 16, 498–522. [Google Scholar] [CrossRef] - Hopp, T.H. The Sensitivity of Three-Point Circle Fitting; NIST Report NISTIR-5501; National Institute of Standards and Technology: Gaithersburg, MD, USA, 1994. [Google Scholar]
- MathWorks. Matlab. 2019. Available online: https://www.MathWorks.com (accessed on 1 April 2019).
- SolidWorks. SolidWorks. 2019. Available online: https://www.SolidWorks.com (accessed on 13 March 2019).

**Figure 1.**Three-point method: Measurement of the circle geometry by using three arbitrary points on an arc $\left\{{P}_{1},{P}_{2},{P}_{3}\right\}$ using global reference G located at ${s}_{1}.$ The sensors are arranged in parallel to reduce system complexity and set-up error.

**Figure 2.**Calibration master: (

**a**) consists of three concentric arcs $\left\{{R}_{c1}{R}_{c2}{R}_{c3}\right\}$ and one shared center ${C}^{c}$. (

**b**) Manufacturability can be achieved by machining a thick cylinder. (

**c**) A radial bearing insert will allow free rotation about a fixed center. Sensors are arranged vertically to allow a measurement of one radius at a time.

**Figure 3.**Roundness: measurement of roundness by using N number arc-segments. (

**a**) Every arc-segment I is measured from three points {${P}_{1,i},{P}_{2,i},{P}_{3,i}\}$ and has radius ${R}_{3p,i}$ obtained by the three-point forward kinematic algorithm (TPFK) and three-point inverse kinematic algorithm (TPIK) algorithms. (

**b**) The best-fit radius and center of the inspected part by using all the segmentations.

**Figure 4.**Sampling patterns: measurement of roundness with total of M number of sampled points. (

**a**) Non-overlapping arc-segmentations, i.e., M = 3N. (

**b**) Overlapping arc-segmentations with one common point, i.e., M = 2N + 1. (

**c**) Overlapping arc-segmentations with two common points, i.e., M = N − 2. Any two segments should not share same three points to avoid redundancy.

**Figure 5.**Mathematical simulation of calibration procedures: the circles are constructed for $\left({a}_{c},{b}_{c}\right)=\left(10,40\right)\text{}\mathrm{mm}$ and two radii $R=\left\{38,36\right\}\text{}\mathrm{mm}$ by using two parametric functions $t={\mathrm{cos}}^{-1}\left(\left({a}_{c}-x\right)/R\right)$ and $y={b}_{c}-R\mathrm{sin}\left(t\right)$.

**Figure 6.**Validation of the TPFK/TPIK algorithms by CAD simulation: (

**a**) calibration master assembly. (

**b**) Inspected part assembly.

**Figure 7.**2D-CAD drawing of the assembly with annotations. The global reference is fixed at the tip of sensor 1 with y+ direction to the right and x+ direction downward.

**Figure 9.**Mathematical representation of the inspected part with perturbation: (

**a**) perturbed parts and (

**b**) rotation of the inspected part at two angle, θ = 0° and θ = 10°.

**Figure 10.**Exponential responses of the roundness index $\mathsf{\beta}$ simulated for the sampling increment $\mathsf{\theta}$, ratio of nominal radius to fluctuation amplitude ${\mathrm{R}}_{\mathrm{e}}/{\mathrm{R}}_{\mathrm{p}}$, and fluctuation frequency $\mathsf{\omega}$. (

**a**) Roundness index vs. fluctuation ratio at frequency 50 rad/s. (

**b**) Exact radius to nominal radius ratio vs. fluctuation ratio at frequency 50 rad/s. (

**c**) Roundness index vs. fluctuation amplitude at frequency 5 rad/s. (

**d**) Exact radius to nominal radius ratio vs. fluctuation amplitude at frequency 5 rad/s.

**Figure 11.**Roundness index based on the CAD model with ${R}_{e}=42.500$ mm, ${R}_{p}=0.425\text{}\mathrm{mm}$, $\omega \left(t\right)=20\frac{\mathrm{rad}}{\mathrm{s}},a=7.873513\text{}\mathrm{mm},\text{}\mathrm{and}\text{}b=43.48817$ mm. (

**a**) 3D-CAD model. (

**b**) 2D-CAD drawing.

**Figure 13.**Displacements measurement obtained for the calibration master and an inspected part. Each displacement was measured three times.

**Table 1.**Parametric simulation of the displacements between sensor coordinates and the calibration master.

${\mathit{R}}_{\mathit{c}2}=38.00\mathbf{mm}$ | ${\mathit{R}}_{\mathit{c}3}=39.00\mathbf{mm}$ | ||
---|---|---|---|

Sensor | Simulated | Sensor | Simulated |

${y}_{12}$ | 3.33939 mm | ${y}_{13}$ | 2.30385 mm |

${y}_{22}$ | 2.00000 mm | ${y}_{23}$ | 1.0000 mm |

${y}_{32}$ | 3.33939 mm | ${y}_{23}$ | 2.30385 mm |

**Table 2.**TPFK algorithm based on the mathematical simulation: solution of the S$\left\{{x}_{2o},{x}_{3o},{y}_{2o},{y}_{3o},{a}_{c},{b}_{c}\right\}$ obtained for several initial estimations.

Test | Initial Guess $[{\mathit{x}}_{2\mathit{o}},{\mathit{x}}_{3\mathit{o}},{\mathit{y}}_{2\mathit{o}},{\mathit{y}}_{3\mathit{o}},$ ${\mathit{a}}_{\mathit{c}},{\mathit{b}}_{\mathit{c}}]$ | # Iterations to Convergence | $\mathit{E}\mathit{r}\mathit{s}$ $1\times {10}^{-10}\mathbf{mm}$ | ${\mathit{x}}_{2\mathit{o}}\mathbf{mm}$ | ${\mathit{x}}_{3\mathit{o}}\mathbf{mm}$ | ${\mathit{y}}_{2\mathit{o}}\mathbf{mm}$ | ${\mathit{y}}_{3\mathit{o}}\mathbf{mm}$ | ${\mathit{a}}_{\mathit{c}}\mathbf{mm}$ | ${\mathit{b}}_{\mathit{c}}\mathbf{mm}$ |
---|---|---|---|---|---|---|---|---|---|

True solution is obtained from the parametric equations: | 10.000 | 20.000 | 0.000 | 0.000 | 10.000 | 40.000 | |||

Trial 1 | [8,19,1,1,9,39] | 23 | 0.000534 | 9.998 | 19.998 | 0.000 | 0.000 | 9.999 | 40.000 |

Trial 2 | [12,18,2,1,4,35] | 95 (stopped) | 538.63 | 19.998 | 20.000 | 0.001 | 0.000 | 9.999 | 40.000 |

Trial 3 | [8,13,2,3,12,43] | 95 (stopped) | 342.70 | 10.195 | 20.000 | 0.001 | 0.000 | 9.999 | 40.000 |

Trial 4 | [15,25,2,2,15,50] | 52 | 10.085 | 10.017 | 19.998 | 0.000 | 0.000 | 9.999 | 40.000 |

Trial 5 | [20,25,2,2,20,50] | 16 | 0.000538 | 10.000 | 19.998 | 0.000 | 0.000 | 9.999 | 40.000 |

Trial 6 | [20,25,2,2,5,5] | 117 (stopped) | 0.56 × 10^{10} | 16.977 | 40.307 | 64.830 | −9.854 | 27.854 | 29.399 |

Trial 7 | [1,1,1,1,1,1] | 106 (stopped) | 2.05 × 10^{10} | 9.5110 | 22.302 | −61.161 | 10.345 | 26.507 | −25.103 |

Trial 8 | [15,25,−2,−2,15,50] | 15 | 0.000865 | 10.000 | 19.998 | 0.000 | 0.000 | 9.999 | 40.000 |

Trial 9 | [20,25,−2,−2,20,50] | 45 | 0.000141 | 9.999 | 19.998 | 0.000 | 0.000 | 9.999 | 40.000 |

Trial 10 | [8,19,1,−1,9,39] | 90 (stopped) | 526 × 10^{13} | 12.097 | 26.698 | 74.837 | −1.913 | 15.937 | 37.914 |

**Table 3.**TPFK algorithm based on mathematical simulation: computation of the {a, b, R

_{3p}} of the inspected part by using true and computed ${S}_{p}$ values.

True Radius R | $\mathbf{True}\text{}\mathbf{Center}\text{}\mathbf{of}\text{}\mathbf{Inspected}\text{}\mathbf{Part}\text{}\left(\mathit{a},\mathit{b}\right)=\left(10,40\right)\mathbf{mm}$ | ||||||||
---|---|---|---|---|---|---|---|---|---|

Calculation Based on True Values S _{p} = {10,20,0,0} | Calculation Based on Trial 5 in Table 2 S _{p} = {10,19.998,0,0} | ||||||||

${\mathit{y}}_{1}$ | ${\mathit{y}}_{2}$ | ${\mathit{y}}_{3}$ | $\mathit{a}$ | $\mathit{b}$ | ${\mathit{R}}_{3\mathit{p}}$ | $\mathit{a}$ | $\mathit{b}$ | ${\mathit{R}}_{3\mathit{p}}$ | |

38.000 | Values from Table 1 | 10.0000 | 40.0108 | 38.0108 | 9.999 | 40.0033 | 38.0033 | ||

39.000 | Values from Table 1 | 10.0000 | 39.9956 | 38.9956 | 9.999 | 39.9879 | 38.9879 |

**Table 4.**TPIK algorithm based on CAD simulation: displacements measured between sensor coordinates and the calibration master for two radii by using direct CAD dimensioning tools rounded to n = 6 decimal places.

${\mathit{R}}_{\mathit{c}2}=38.000000\mathbf{mm}$ | ${\mathit{R}}_{\mathit{c}3}=39.000000\mathbf{mm}$ | ||
---|---|---|---|

Sensor | Measurement (mm) | Sensor | Measurement (mm) |

${y}_{12}$ | 2.130124 | ${y}_{13}$ | 1.094501 |

${y}_{22}$ | 1.963025 | ${y}_{23}$ | 0.963052 |

${y}_{32}$ | 2.138668 | ${y}_{23}$ | 1.103195 |

**Table 5.**TPIK algorithm based on CAD simulation studied for several initial estimations $[{x}_{2o},{x}_{3o},{y}_{2o},{y}_{3o},a,b]$, and measurement resolutions, n.

Trial # | Initial Guess | n | ${\mathit{x}}_{2\mathit{o}}\left(\mathbf{mm}\right)$ | ${\mathit{x}}_{3\mathit{o}}\left(\mathbf{mm}\right)$ | ${\mathit{y}}_{2\mathit{o}}\left(\mathbf{mm}\right)$ | ${\mathit{y}}_{3\mathit{o}}\left(\mathbf{mm}\right)$ | $\mathit{a}\left(\mathbf{mm}\right)$ | $\mathit{b}\left(\mathbf{mm}\right)$ | Ers 10^{−7} (mm^{2}) | # Iterations |
---|---|---|---|---|---|---|---|---|---|---|

- | True * | 10.000 | 20.000 | −1.175 | −0.014 | 10.010 | 38.788 | - | - | |

1 | [10,20,−2,−1,11,39] | 6 | 10.010 | 20.000 | −1.1750 | −0.014 | 10.010 | 38.788 | 3.0458 | 108 (stopped) |

2 | [10,20,−2,0,9,39] | 6 | 10.012 | 20.002 | −1.175 | −0.014 | 10.012 | 38.788 | 2.1151 | 107 (stopped) |

3 | [12,21,2,1,11,38] | 6 | 9.048 | 17.536 | 72.354 | −3.990 | 17.060 | 36.135 | 50 | 94 (stopped) |

4 | [10,20,−1.175,−0.014,10.01,38.788] | 6 | 10.010 | 20.000 | −1.175 | −0.014 | 10.010 | 38.788 | 3.045 | 99 (stopped) |

5 | 5 | 10.010 | 20.000 | −1.175 | −0.014 | 10.010 | 38.788 | 1.2075 | 85 | |

6 | 4 | 10.006 | 20.000 | −1.175 | −0.014 | 10.010 | 38.788 | 43.69 | 97 (stopped) | |

7 | 3 | 9.927 | 19.987 | −1.152 | 0.027 | 9.927 | 38.811 | 1.1 | 8 | |

8 | 2 | 10.583 | 21.148 | −1.331 | −0.010 | 10.574 | 38.629 | 11 | 54 |

Part True R _{e} | Probes Measurement (mm), n = 3 | Calculated (mm) | Error (mm) | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{y}}_{1}$ | ${\mathit{y}}_{2}$ | ${\mathit{y}}_{3}$ | $\mathit{a}$ | $\mathit{b}$ | ${\mathit{R}}_{3\mathit{p}}$ | $\left|\mathit{a}-{\mathit{a}}_{\mathit{e}}\right|$ | $\left|\mathit{b}-{\mathit{b}}_{\mathit{e}}\right|$ | $\left|\mathit{R}-{\mathit{R}}_{\mathit{e}}\right|$ | |

Group 1: True part center location is fixed at (a$=\left(10.010,38.788\right)mm$_{e},b_{e}) | |||||||||

38.000 | Values from Table 3, n = 3 | 10.0092 | 38.7851 | 37.9971 | 0.001 | 0.003 | 0.003 | ||

38.250 | 1.871 | 1.713 | 1.880 | 10.0092 | 38.7831 | 38.2451 | 0.001 | 0.005 | 0.005 |

38.500 | 1.612 | 1.463 | 1.621 | 10.0093 | 38.7845 | 38.4965 | 0.001 | 0.003 | 0.004 |

38.750 | 1.353 | 1.213 | 1.362 | 10.0094 | 38.7895 | 38.7515 | 0.001 | 0.002 | 0.002 |

39.000 | Values from Table 3, n = 3 | 10.0113 | 38.7836 | 38.9956 | 0.001 | 0.004 | 0.004 | ||

39.250 | 0.836 | 0.713 | 0.845 | 10.0095 | 38.7809 | 39.2429 | 0.001 | 0.007 | 0.007 |

39.500 | 0.577 | 0.463 | 0.586 | 10.0096 | 38.7964 | 39.5084 | 0.000 | 0.008 | 0.008 |

Group 2: Part center location is arbitrarily (unrestricted positioning) | |||||||||

38.000 | 8.268 | 9.303 | 10.742 | 5.5273 | 45.8615 | 37.9977 | - | - | 0.002 |

38.000 | 2.040 | 2.963 | 4.277 | 5.9495 | 39.5931 | 38.0215 | - | - | 0.022 |

38.000 | 2.408 | 2.244 | 2.422 | 10.000 | 39.0798 | 38.0108 | - | - | 0.011 |

38.000 | 1.067 | 0.903 | 1.081 | 10.000 | 37.7388 | 38.0108 | - | - | 0.011 |

38.000 | 1.540 | 1.376 | 1.554 | 10.000 | 38.2118 | 38.0108 | - | - | 0.011 |

38.000 | 1.364 | 0.660 | 0.310 | 11.9549 | 37.4382 | 38.0035 | - | - | 0.004 |

38.000 | 0.185 | 0.120 | 0.398 | 9.6352 | 36.9458 | 38.0026 | - | - | 0.003 |

**Table 7.**Calculation of Roundness for ${R}_{p}$ = 38/40 mm, ${R}_{e}=38\text{}\mathrm{mm}$, and frequency $\omega =5$ rad. The system parameters were set to ${x}_{1o}=0,\text{}{x}_{2o}=10.000\text{}\mathrm{mm},{x}_{3o}=20.000\text{}\mathrm{mm},{y}_{1o}=0,\text{}{y}_{2o}=0\text{}\mathrm{mm},\text{}\mathrm{and}\text{}{y}_{3o}=0$.

Inc. i | $\mathit{\theta}$ (°) | $\mathbf{True}\text{}\mathbf{Part}\text{}\mathbf{Center}\text{}\mathbf{is}\text{}\mathbf{Fixed}\text{}\mathbf{and}\text{}\mathbf{Rotates}\text{}\mathbf{about}\text{}\mathbf{a}\text{}=\text{}10\text{}\mathbf{mm}\text{}\mathbf{and}\text{}\mathbf{b}\text{}=\text{}20\text{}\mathbf{mm}\text{}\mathbf{by}\text{}\mathbf{Sampling}\text{}\mathbf{Increment}\text{}\mathbf{of}\text{}\mathit{\delta}\mathit{\theta}=$ 20° $,\text{}\mathbf{and}\text{}\mathbf{Frequency}\text{}\mathit{\omega}=5\text{}\mathbf{rad}.$ | |||||
---|---|---|---|---|---|---|---|

Probes Measurement (mm), n = 3 | Calculated (mm) | ||||||

${\mathit{y}}_{1}$ | ${\mathit{y}}_{2}$ | ${\mathit{y}}_{3}$ | ${\mathit{a}}_{3\mathit{p},\mathit{i}}$ | ${\mathit{b}}_{3\mathit{p},\mathit{i}}$ | ${\mathit{R}}_{3\mathit{p},\mathit{i}}$ | ||

1 | 20 | 3.366 | 2.349 | 2.004 | 10.14 | 40.445 | 37.975 |

2 | 40 | 3.337 | 2.3440 | 2.024 | 9.943 | 40.337 | 38.003 |

3 | 60 | 3.314 | 2.307 | 1.987 | 9.883 | 39.452 | 38.024 |

4 | 80 | 3.351 | 2.325 | 1.981 | 10.097 | 39.859 | 37.989 |

… | … | … | … | … | … | … | … |

12 | 320 | 3.354 | 2.355 | 2.024 | 10.041 | 40.590 | 37.986 |

13 | 340 | 3.315 | 2.318 | 2.004 | 9.860 | 39.731 | 38.023 |

14 | 360 | 3.333 | 2.310 | 1.975 | 10.008 | 39.519 | 38.006 |

Average | $\overline{a}=9.9994$ | $\overline{b}=39.9821$ | ${R}_{a}$= 37.9992 |

**Table 8.**Computation of roundness index using CAD data. The system parameters used were ${x}_{2o}=10.000\text{}\mathrm{mm},{x}_{3o}=20.000\text{}\mathrm{mm},{y}_{2o}=-1.175\text{}\mathrm{mm},\text{}\mathrm{and}\text{}{y}_{3o}=-0.014\text{}\mathrm{mm}$.

$\mathbf{\Delta}\mathit{\theta}$ | True Part Center is Fixed and Rotated about a = 7.873513 and b = 43.48817 | |||||
---|---|---|---|---|---|---|

Probes Measurement, n = 3 | Calculated | |||||

${\mathit{y}}_{1}$ | ${\mathit{y}}_{2}$ | ${\mathit{y}}_{3}$ | $\mathit{a}$ | $\mathit{b}$ | ${\mathit{R}}_{3\mathit{p}}$ | |

0 | 1.357 | 2.445 | 3.169 | 5.4487 | 52.8828 | 51.8131 |

2 | 1.598 | 2.622 | 2.941 | 5.9349 | 63.4376 | 62.1237 |

4 | 1.904 | 2.610 | 2.619 | 7.8732 | 62.9325 | 61.5343 |

6 | 2.118 | 2.413 | 2.379 | 9.3908 | 51.5730 | 50.3386 |

8 | 2.141 | 2.122 | 2.335 | 9.6553 | 40.5330 | 39.5875 |

10 | 1.972 | 1.875 | 2.488 | 9.1946 | 34.3121 | 33.6218 |

12 | 1.697 | 1.792 | 2.758 | 8.4051 | 32.6860 | 32.1086 |

14 | 1.433 | 1.906 | 3.033 | 7.3904 | 35.1336 | 34.5015 |

16 | 1.296 | 2.164 | 3.198 | 6.2524 | 41.9387 | 41.1208 |

18 | 1.357 | 2.445 | 3.169 | 5.4487 | 52.8828 | 51.8131 |

20 | 1.598 | 2.622 | 2.941 | 5.9349 | 63.4376 | 62.1237 |

- | - | - | - | - | - | - |

358 | 1.296 | 2.164 | 3.198 | 6.2524 | 41.9387 | 41.1208 |

Average | $\overline{a}=$7.7273 | $\overline{b}=$46.1588 | ${R}_{a}$= 45.1944 |

**Table 9.**Displacements of the calibration master measured at two radii by using displacement sensors with ±0.01 mm accuracy.

${\mathit{R}}_{\mathit{c}2}=40.00\mathbf{mm}$ | ${\mathit{R}}_{\mathit{c}3}=42.13\mathbf{mm}$ | |||
---|---|---|---|---|

Sensor | Avg./SD (mm) | Measurement (mm) | Avg./SD (mm) | Measurement (mm) |

S_{1} | ($\overline{{y}_{12}}$, sd_{12}) | (14.37, 0.06) | ($\overline{{y}_{13}}$, sd_{13}) | (11.67, 0.10) |

S_{2} | ($\overline{{y}_{22}}$, sd_{22}) | (3.88, 0.02) | ($\overline{{y}_{23}}$, sd_{13}) | (1.80, 0.03) |

S_{3} | ($\overline{{y}_{32}}$, sd_{32}) | (15.04, 0.15) | ($\overline{{y}_{33}}$, sd_{13}) | (12.17, 0.10) |

**Table 10.**Displacements of the inspected part whose nominal radius is priory known ${R}_{e}$ = 41.03 mm, measured with an accuracy of ± 0.01 mm.

${\mathit{R}}_{\mathit{e}}=41.03\mathbf{mm}$ | ||
---|---|---|

Sensor | Avg./SD (mm) | Measurement (mm) |

S_{1} | ($\overline{{y}_{1}}$, sd_{1}) | (13.10, 0.03) |

S_{2} | ($\overline{{y}_{2}}$, sd_{2}) | (2.85, 0.01) |

S_{3} | ($\overline{{y}_{3}}$, sd_{3}) | (13.69,0.11) |

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).