An Adaptive Multi-Population Approach for Sphericity Error Evaluation in the Manufacture of Hemispherical Shell Resonators

The performance of a hemispherical resonant gyroscope (HRG) is directly affected by the sphericity error of the thin-walled spherical shell of the hemispherical shell resonator (HSR). In the production process of the HSRs, high-speed, high-accuracy, and high-robustness requirements are necessary for evaluating sphericity errors. We designed a sphericity error evaluation method based on the minimum zone criterion with an adaptive number of subpopulations. The method utilizes the global optimal solution and the subpopulations’ optimal solution to guide the search, initializes the subpopulations through clustering, and dynamically eliminates inferior subpopulations. Simulation experiments demonstrate that the algorithm exhibits excellent evaluation accuracy when processing simulation datasets with different sphericity errors, radii, and numbers of sampling points. The uncertainty of the results reached the order of 10−9 mm. When processing up to 6000 simulation datasets, the algorithm’s solution deviation from the ideal sphericity error remained around −3 × 10−9 mm. And the sphericity error evaluation was completed within 1 s on average. Additionally, comparison experiments further confirmed the evaluation accuracy of the algorithm. In the HSR sample measurement experiments, our algorithm improved the sphericity error assessment accuracy of the HSR’s inner and outer contour sampling datasets by 17% and 4%, compared with the results given by the coordinate measuring machine. The experiment results demonstrated that the algorithm meets the requirements of sphericity error assessment in the manufacturing process of the HSRs and has the potential to be widely used in the future.


Introduction
The hemispherical resonator gyroscope (HRG) is the most accurate inertial sensor in the world [1][2][3][4][5].In recent years, the demand for HRGs has significantly increased in aerospace and deep space exploration [6][7][8][9].The hemispherical shell resonator (HSR) is the core component of the HRG and its main working area is the hemispherical thin-walled spherical shell structure.During the manufacturing process, the inevitable sphericity error can cause uneven mass distribution and elastic modulus in thin-walled spherical shells, which can affect the accuracy of HRGs.In addition, shaping and manufacturing the HSR requires multiple processes, such as rough machining, precision grinding, and precision polishing [10,11].Sphericity error is a critical technical indicator used to assess the quality of thin-walled spherical shells during each process.Different processes have varying requirements for sphericity error.Accurately evaluating the sphericity error in thin-walled spherical shells is crucial for achieving the high-precision manufacturing of HRGs [12].
Sensors 2024, 24 Form error measurement equipment, such as a coordinate measuring machine (CMM), typically saves the sampled shell contour data as a large dataset in XYZ coordinate form [20].
The International Organization for Standardization (ISO) and the American Society of Mechanical Engineers (ASME) standards encompass four criteria for assessing sphericity error: the least squares criterion, maximum inscribed criterion, minimum circumscribed criterion, and minimum zone criterion [21,22].In order to control the mass distribution of the thin-walled spherical shells, the minimum zone criterion is the most suitable method for evaluating the sphericity error of the HSRs.However, the ISO and ASME standards lack explicit methodologies for implementing the minimum zone criterion [23][24][25].Furthermore, evaluating this dataset based on the minimum zone criterion involves non-differentiable and unconstrained problems [26][27][28].Meeting the requirements for speed, accuracy, and robustness in the HSR manufacturing process presents a significant challenge for algorithms that evaluate sphericity error based on the minimum zone criterion.
In order to address the challenge of sphericity error evaluation based on the minimum zone criterion, Chen et al. [29] established four concentric sphere models by solving systems of linear algebraic equations and selected the one with the minimum sphericity error as the final evaluation result.This method laid the foundation for approaches based on selecting sample points [12].Fan et al. [30] proposed a minimum potential energy method inspired by this method.At the same time, Fei Liu et al. [31] designed a method for evaluating sphericity error using chord relationships among sample points.As this type of method requires searching and traversing all sampling points to select combinations that meet the minimum zone criterion, when the volume of data is relatively small, the method of selecting sampling points exhibits high accuracy and robustness.However, when measuring and evaluating the HSRs, the required amount of data increases dramatically, leading to a pronounced surge in the computational cost and decreasing robustness for the algorithms based on selecting sampling points.Therefore, these methods are unsuitable for evaluating sphericity errors in the manufacturing process of the HSRs.
With the development of heuristic algorithms [32][33][34][35], they are increasingly being used for sphericity error evaluation.Jiang et al. [36] proposed using a cuckoo search algorithm for evaluating sphericity error, while Lei et al. [37] introduced a geometric optimization search-based sphericity error evaluation algorithm.Compared with the method of selecting sample points, the solution accuracy and robustness of the heuristic algorithm are less dependent on the amount of sampled data, and the solution results depend more on the design of the algorithm itself.Moreover, heuristic algorithms are directly applied to the assessment of the HSR's sphericity errors without the integration of specialized search strategies informed by domain knowledge.In that case, there is a risk of compromising their solution's accuracy and robustness.After incorporating domain-specific search strategies, these techniques can potentially be used to assess sphericity errors in the manufacturing process of the HSRs.Furthermore, heuristic algorithms based on multi-populations have recently experienced notable advancements.The collaboration of diverse subpopulations enhances heuristic algorithms' search capabilities [38][39][40].However, no researcher has yet applied this method to the field of sphericity error assessment.
The manufacturing process of the HSRs involves multiple processes, and the field of inertial navigation has an immense demand for the most accurate inertial sensor.Therefore, in the HSRs' manufacturing process, the evaluation of their sphericity error exhibits characteristics of large data volume, high speed, high precision, and robustness.Current methods for assessing sphericity error do not meet the demands of this scenario.After analyzing the distribution characteristics of sphericity error, we propose a novel adaptive multi-population method to effectively assess high-precision sphericity error in the manufacturing process of the HSRs.The main contributions of this study can be summarized as follows: 1.
For the specific needs of the HSR production, this study explores the spatial distribution characteristics of the sphericity error.The sphericity error gradient has been observed to be larger, and the characteristics are more concise at locations farther from Sensors 2024, 24, 1545 3 of 23 the ideal sphere center.In contrast, the sphericity error changes are complicated in the region close to the ideal sphere center.

2.
Based on the in-depth analysis of the sphericity error distribution characteristics, we design an adaptive multi-population cooperative search algorithm.The algorithm guides the subpopulation individuals to search through the global optimal solution and subpopulation optimal solution, and periodically reorganizes the subpopulations and eliminates the inferior subpopulations.This not only achieves fast convergence at the beginning of the search but also enables a detailed search later when the region near the ideal sphere center is approached.

3.
The proposed algorithm's accuracy and robustness are verified through numerous experiments, proving that it can effectively meet the needs of HSR production.This algorithm can significantly improve the accuracy of the existing form error measurement equipment in data processing.The non-uniform mass distribution of a thin-walled spherical shell significantly impacts the HSR's Q factor and vibration characteristics.Compared to other criteria given by ASME and ISO standards [21,22], such as the least squares criterion, maximum inscribed criterion, and minimum circumscribed criterion, evaluating the sphericity error based on the minimum zone criteria is more suitable for ensuring the quality control of the HSRs [11,41].Figure 1 shows the schematic diagram of the hemispherical resonator gyroscope and provides a real picture of the hemispherical shell resonator.

Methods
tion characteristics of the sphericity error.The sphericity error gradient has been served to be larger, and the characteristics are more concise at locations farther f the ideal sphere center.In contrast, the sphericity error changes are complicate the region close to the ideal sphere center.2. Based on the in-depth analysis of the sphericity error distribution characteristics design an adaptive multi-population cooperative search algorithm.The algor guides the subpopulation individuals to search through the global optimal solu and subpopulation optimal solution, and periodically reorganizes the subpop tions and eliminates the inferior subpopulations.This not only achieves fast con gence at the beginning of the search but also enables a detailed search later when region near the ideal sphere center is approached.3. The proposed algorithm's accuracy and robustness are verified through nume experiments, proving that it can effectively meet the needs of HSR production.algorithm can significantly improve the accuracy of the existing form error meas ment equipment in data processing.

Mathematical Model of Minimum Zone Criteria
The non-uniform mass distribution of a thin-walled spherical shell significantly pacts the HSR's Q factor and vibration characteristics.Compared to other criteria g by ASME and ISO standards [21,22], such as the least squares criterion, maximum scribed criterion, and minimum circumscribed criterion, evaluating the sphericity e based on the minimum zone criteria is more suitable for ensuring the quality contr the HSRs [11,41].Figure 1 shows the schematic diagram of the hemispherical reson gyroscope and provides a real picture of the hemispherical shell resonator.For the thin-walled spherical shell profile dataset collected by the form error m urement equipment, the method of assessing the sphericity error based on the minim zone criteria is to find two concentric spheres encompassing the entire measured pr and to minimize the difference in radius between the two concentric spheres.The s ricity error objective function based on the minimum zone criterion can be expresse Equation (1).In this process, the central task is to find an ideal spherical center  * minimizes the sphericity error S of the objective function.

𝑆 = min max‖𝐩 − 𝐩 ‖ 𝟐 − min‖𝐩 − 𝐩 ‖ 𝟐
where  represents the center coordinate of the sphere, and  represents the s sampled point coordinates, where i is the number of sampled points.For the thin-walled spherical shell profile dataset collected by the form error measurement equipment, the method of assessing the sphericity error based on the minimum zone criteria is to find two concentric spheres encompassing the entire measured profile and to minimize the difference in radius between the two concentric spheres.The sphericity error objective function based on the minimum zone criterion can be expressed as Equation (1).In this process, the central task is to find an ideal spherical center p * 0 that minimizes the sphericity error S of the objective function.

S = min max∥p
where p 0 represents the center coordinate of the sphere, and p i represents the set of sampled point coordinates, where i is the number of sampled points.

Sphericity Error Spatial Distribution Characteristics
It is widely acknowledged that the sphericity error varies at different rates depending on the position [42,43].To demonstrate this, we use the spherical contour sampling dataset from reference [29] and apply the least squares method to obtain the initial spherical center.
Sensors 2024, 24,1545 And the optimal solution is located close to the least squares spherical center.Therefore, we plot the variations of sphericity error along the X, Y, and Z directions of the least squares spherical center, as shown in Figure 2. The graph reveals that the decrease in sphericity error rate is steeper farther away from the least squares solution, and there are potential locally optimal solutions around the ideal solution.The potential locally optimal solutions exist because the sphericity error depends on the sampling points, and abrupt changes in these points can lead to sudden shifts in the sphericity error.
search algorithm and employed clustering methods to initiate subpopulations, perio cally recombining the subpopulations.Guided by both the global optimal solution a the optimal solution within each subpopulation, most subpopulations swiftly converg to the vicinity of the ideal spherical center in the early stages of the search.During search process, we designed an elimination mechanism, where subpopulations with adequate individuals that cannot be clustered are eliminated, and population individu converge to the dominant subpopulations.In the later stages of the search, while subpo ulations converge to the ideal spherical center vicinity, the number of subpopulatio gradually decreases while the number of individuals in the remaining subpopulations creases, thus achieving a detailed search near the ideal spherical center.In the manuf turing process of the HSRs, we can use the above mechanism to achieve a sphericity er assessment that meets the requirements of speed, precision, and robustness.

Proposed Method
The algorithmic framework designed for this study is illustrated in Figure 3. Initia a large population is randomly split into various subpopulations for the purpose of i tialization.Subsequently, each subpopulation conducts evolution and search process with periodic updates made to both the global optimal solution and the optimal soluti specific to the subpopulation.Following this, the subpopulations are recombined bas on clustering methods, utilizing the optimal solution of each subpopulation as the cent From Figure 2, it is evident that in the vicinity of the least squares spherical center, the variation in sphericity error is complex.Conversely, when moving away from the least squares spherical center, the gradient of sphericity error variation is substantial, indicating a pronounced downward trend.
Capitalizing on this distinctive characteristic, we formulated a multi-population search algorithm and employed clustering methods to initiate subpopulations, periodically recombining the subpopulations.Guided by both the global optimal solution and the optimal solution within each subpopulation, most subpopulations swiftly converged to the vicinity of the ideal spherical center in the early stages of the search.During the search process, we designed an elimination mechanism, where subpopulations with inadequate individuals that cannot be clustered are eliminated, and population individuals converge to the dominant subpopulations.In the later stages of the search, while subpopulations converge to the ideal spherical center vicinity, the number of subpopulations gradually decreases while the number of individuals in the remaining subpopulations increases, thus achieving a detailed search near the ideal spherical center.In the manufacturing process of the HSRs, we can use the above mechanism to achieve a sphericity error assessment that meets the requirements of speed, precision, and robustness.

Proposed Method
The algorithmic framework designed for this study is illustrated in Figure 3. Initially, a large population is randomly split into various subpopulations for the purpose of initialization.Subsequently, each subpopulation conducts evolution and search processes, with periodic updates made to both the global optimal solution and the optimal solution specific to the subpopulation.Following this, the subpopulations are recombined based on clustering methods, utilizing the optimal solution of each subpopulation as the center.Meanwhile, inferior subpopulations are adaptively eliminated.The search stops when the remaining subpopulation falls below a specified threshold, and the global optimal solution is the output of the center of the concentric spheres.
ensors 2024, 24, x FOR PEER REVIEW 5 of 23 Meanwhile, inferior subpopulations are adaptively eliminated.The search stops when the remaining subpopulation falls below a specified threshold, and the global optimal solution is the output of the center of the concentric spheres.

Initialization Process (1) Initializing the Search Space
Establishing an appropriate search space R is crucial for enhancing search efficiency The minimum zone solution is commonly considered to be contained within a sphere.The center of the sphere corresponds to the least squares solution, and the radius is determined by the least squares sphericity error [44].Therefore, the search space R, initial spherica center ( ,  ,  ), and initial sphericity error  are initialized based on the least squares method.
By solving Equation (3), the least squares spherical center coordinates  ( ,  ,  ) can be obtained.Then, the initial sphericity error  can be defined based on the minimum zone criterion.
After obtaining the initial spherical center and the initial sphericity error  ( ,  ,  ) , the spatial position of the minimum zone spherical center  ( ,  ,  ) can be determined by the formulation presented in Equation (4).Equation (4) serves as a representation of the search space R and defines the boundary conditions. (4) (2) Initializing the Subpopulations

Initialization Process (1) Initializing the Search Space
Establishing an appropriate search space R is crucial for enhancing search efficiency.The minimum zone solution is commonly considered to be contained within a sphere.The center of the sphere corresponds to the least squares solution, and the radius is determined by the least squares sphericity error [44].Therefore, the search space R, initial spherical center (x 0 , y 0 , z 0 ), and initial sphericity error S 0 are initialized based on the least squares method.
A linear system of equations can be constructed from a set of sampled point coordinates p i = [x i , y i , z i ] T ∈ S, where i = (1, 2, . . . ,n), as shown in Equation ( 2), where 3) can be obtained by transforming the system of equations into regular equations.
By solving Equation ( 3), the least squares spherical center coordinates O LS (x LS , y LS , z LS ) can be obtained.Then, the initial sphericity error S 0 can be defined based on the minimum zone criterion.
After obtaining the initial spherical center and the initial sphericity error O LS (x LS , y LS , z LS ), the spatial position of the minimum zone spherical center O MZ (x MZ , y MZ , z MZ ) can be determined by the formulation presented in Equation (4).Equation ( 4) serves as a representation of the search space R and defines the boundary conditions.
After obtaining the initial population Pop, K individuals P ai = P a1 , P a2 , .., P aK are randomly selected as cluster centers.The Euclidean distance ∥Pop i − P ai ∥ 2 between each individual Pop i in the initial population Pop and the cluster center P ai is calculated.Then, each individual Pop i in the population is assigned to the subpopulation subPop1, subPop2, . .., subPopK.corresponding to the nearest cluster center P ai .By using clustering methods for subpopulation initialization, the initialized subpopulations are distributed in non-overlapping local regions of the search space R.This step completes the initialization of the algorithm.And the pseudocode describing the subpopulation initialization based on clustering methods is as Algorithm 1 follows.Assign Pop i to the subpopulation subPopi corresponding to the nearest cluster center P ai ; 6: end for 7: Return subPop1, subPop2, . .., subPopK

Search Mechanism (1) Subpopulation Evolution Mechanism
Before each subpopulation starts searching, it undergoes a preliminary independent evolution to ensure the diversity of the subpopulation.The evolutionary search process is as follows.
Firstly, a tournament selection method is used to select individuals.Then, two individuals are randomly selected with equal probability and compared based on their sphericity error.The individual with the smaller sphericity error is selected to enter the next generation subpopulation, until the new subpopulation size reaches the original size.The optimal solution of the subpopulation is directly retained for the next generation.
Secondly, two parents are selected with equal probability for uniform crossover operation, generating two new offspring.The offspring with the smaller sphericity error is selected to be retained in the next generation subpopulation.
Finally, we can use Gaussian mutation as a means of mutating individuals in the population.We select individuals from outside the optimal solution of the subpopulation with a given probability and apply perturbations based on Gaussian mutation.The individuals of the subpopulation are perturbed based on the Gaussian mutation probability shown in Equation (5).Before perturbation, the d-dimensional component size of the subpopulation j is subPopK i jd , which after perturbation becomes subPopK i+1 jd .In Equation ( 5), σ represents the standard deviation, i represents the generation number, and j represents the individual number within the subpopulation.
Sensors 2024, 24, 1545 7 of 23 (2) Subpopulation Search Mechanism We introduce the concepts of the subpopulation optimal solution (pbest) and the global optimal solution (gbest) in the traditional particle swarm algorithm.The pbest refers to the optimal solution within a specific subpopulation, while the pbest is the optimal solution among all subpopulations.During the search process, we update the global optimal solution (gbest) and the subpopulation optimal solution (pbest) in real time.The detailed search strategy includes position and velocity update formulas; the position update formula is shown in Equation ( 6), and the velocity update formulas are shown in Equations ( 6) and (7).In the search process, we will execute this part of the content several times until it reaches the predetermined number of times N iter .
In the equations, c 1 represents the inertia factor, while c 2 , c 3 , and c 4 are learning factors.The variable rand refers to a uniformly distributed random number between 0 and 1. V i j denotes the velocity of an individual in the population that migrates to another subpopulation, while Xbest j represents the optimal historical position of an individual in the population.The pbest is the optimal historical solution among individuals in a subpopulation, while gbest is the global optimal solution among all subpopulations.The subPopK i j represents an individual within a particular subpopulation, where i denotes the generation, and j denotes the index of the individual within the subpopulation.
In addition, Equation ( 8) is used to limit the velocity between Vmin and Vmax, ensuring the stability and accuracy of the search process.Vmin and Vmax represent the minimum and maximum values of the velocity, respectively.

Adaptive Reconstruction and Elimination Mechanism
After completing the search process, we employed the K-means clustering method to restructure all the subpopulations.We selected the optimal subpopulation solutions pbest1, pbest2, . .., pbestK from each subpopulation subPop1, subPop2, . .., subPopK as the cluster centers.Then, we calculated the Euclidean distance ∥Pop i − pbestK∥ 2 between all the individuals Pop i and optimal solutions of each subpopulation pbest1, pbest2. .., pbestK.Finally, each individual Pop i was assigned to the subpopulation (subPop1, subPop2. .., subPopK) with the closest distance to its corresponding cluster center.As shown in Algorithm 2.
After recombining the subpopulations using clustering methods, each subpopulation is distributed into separate local spaces within the search space R without overlapping.Any subpopulations that do not acquire adequate individuals following reclustering are classified as inferior.Typically, population individuals converge toward the ideal solution.However, two situations can lead to the production of inferior subpopulations: one is the erroneous search direction, and the other is the failure to compete with other subpopulations, even though the search direction of this subpopulation is correct.Figure 4 is a two-dimensional schematic diagram that shows the reasons for producing inferior subpopulations after the recombination of subpopulations.Each color represents a different subpopulation.
To address this issue, we have developed an elimination mechanism to remove subpopulations that cannot cluster enough individuals.If a reclustered subpopulation's number of individuals falls below the threshold, it is considered inferior and will be eliminated.The search stops after a certain proportion of inferior subpopulations have been eliminated.
By using an adaptive subpopulation set, the global optimum can be found efficiently and accurately.
To address this issue, we have developed an elimination mechanism to remove s populations that cannot cluster enough individuals.If a reclustered subpopulation's n ber of individuals falls below the threshold, it is considered inferior and will be el nated.The search stops after a certain proportion of inferior subpopulations have b eliminated.By using an adaptive subpopulation set, the global optimum can be fo efficiently and accurately.4 is not obtained from actual experiment but is a simulation presentation using a two-dimensional view to explain m clearly the two reasons that lead to the creation of inferior subpopulations).

Algorithm 2: Adaptive Reconstruction and Elimination Mechanism Input:
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The First Part of the Simulation Experiments
Three groups of simulation datasets were generated to verify algorithm accuracy in different manufacturing processes for the HSRs with known sphericity errors.Table 1 presents each simulation dataset's sphericity errors, radii, and number of simulated samples.The sphericity error in the simulation data was controlled using the following method [42].The points located on the surface of the concentric sphere are referred to as control points.Based on the coefficient equation of the sphere in polar coordinates, as shown in Equation ( 9), ten control points were randomly generated on the inner and outer surfaces of a concentric sphere with radii of R and R + S 0 , respectively.Thus, three groups of simulation sampling coordinates P i , i = 1, 2, 3, . . ., 12 are generated.The 20 control points can constrain the sphericity error of the simulation datasets for the difference S 0 between the radii of the inner and outer spheres.This approach provided reliable data to verify the algorithm's accuracy under different manufacturing processes.
The initial total number of populations was set to 300, and the initial number K of subpopulations was 10.The subpopulation was eliminated when the number of individuals in the subpopulation was less than 3.The threshold of population elimination number was 3, and the search was finished when the number of subpopulations was less than 3.Then, we could calculate the above simulation data 10 times independently and calculate the mean and standard deviation of the results for each dataset.
The datasets within group A had identical sphericity error values of 0.00005 mm and a radius of 50 mm, with varying numbers of sampling points at 50, 100, 500, and 2000, respectively.Figure 5 shows the distribution of group A's dataset on the sphere surface.These datasets were processed using the least squares method and our algorithm, and the resulting analysis is presented in Table 2.The datasets in group B had the same sphericity error of 0.00001 mm and the sampling point number of 100.Their radii were 25 mm, 50 mm, 75 mm, and 100 mm, respectively.The distribution of the group B dataset on the sphere surface is shown in Figure 6.The data of group B were processed using the least squares method and our algorithm, and the processing results are shown in Table 3.The datasets in group B had the same sphericity error of 0.00001 mm and the sampling point number of 100.Their radii were 25 mm, 50 mm, 75 mm, and 100 mm, respectively.The distribution of the group B dataset on the sphere surface is shown in Figure 6.The data of group B were processed using the least squares method and our algorithm, and the processing results are shown in Table 3.The datasets in group B had the same sphericity error of 0.00001 mm and the sampling point number of 100.Their radii were 25 mm, 50 mm, 75 mm, and 100 mm, respectively.The distribution of the group B dataset on the sphere surface is shown in Figure 6 The data of group B were processed using the least squares method and our algorithm and the processing results are shown in Table 3.The datasets in group C had the same radius of 50 mm and the sampling point number of 500.Their sphericity errors were 0.00002 mm, 0.00004 mm, 0.00006 mm, and 0.00008 mm, respectively.Figure 7 shows the distribution of group C's simulation dataset on the sphere surface.The data of group C were processed using the least squares method and our algorithm, and the processing results are shown in Table 4.The datasets in group C had the same radius of 50 mm and the sampling point number of 500.Their sphericity errors were 0.00002 mm, 0.00004 mm, 0.00006 mm, and 0.00008 mm, respectively.Figure 7 shows the distribution of group C's simulation dataset on the sphere surface.The data of group C were processed using the least squares method and our algorithm, and the processing results are shown in Table 4.The datasets in group C had the same radius of 50 mm and the sampling point number of 500.Their sphericity errors were 0.00002 mm, 0.00004 mm, 0.00006 mm, and 0.00008 mm, respectively.Figure 7 shows the distribution of group C's simulation dataset on the sphere surface.The data of group C were processed using the least squares method and our algorithm, and the processing results are shown in Table 4.  From the above experimental results, it can be seen that our algorithm can accurately evaluate the spherical sphericity error when dealing with a simulation dataset of different sphericity errors, different radii, and different sampling points.Compared with the least squares method, our algorithm has about 10% accuracy improvement, and the standard deviation of the algorithm solution is in the order of 10 −9 mm.

The Second Part of the Simulation Experiments
Additionally, to assess the robustness and evaluation accuracy of the algorithm, we generated six groups of simulated data with known sphericity errors using the aforementioned method from the first part of the simulation experiment.
The difference in each group of simulated data lies in the number of sampling points, with each simulating 50, 100, 500, 1000, 1500, and 2000 sampling points, respectively.Each group of simulated data contains 100 spheres with different radii and ten types of sphericity errors.The radii of the simulated data are uniformly distributed in the range of [1 mm, 100 mm].The sphericity errors are uniformly distributed in the range of [1 × 10 −5 mm, 1 × 10 −4 mm].Each group consists of 1000 datasets, for a total of 6000 datasets.
We conducted simulation experiments based on these six groups, a total of 6000 datasets.And we performed statistical analysis on the deviation value D between the experimentally obtained sphericity error result  and the ideal sphericity error value  .We used Formula (10) to calculate the deviation value.
The results, presented in Table 5, show that each group's experimental results  have a mean deviation from the ideal sphericity error  of approximately 10 −9 mm.It is noticeable that the number of sampling points has a negligible effect on the experimental  From the above experimental results, it can be seen that our algorithm can accurately evaluate the spherical sphericity error when dealing with a simulation dataset of different sphericity errors, different radii, and different sampling points.Compared with the least squares method, our algorithm has about 10% accuracy improvement, and the standard deviation of the algorithm solution is in the order of 10 −9 mm.

The Second Part of the Simulation Experiments
Additionally, to assess the robustness and evaluation accuracy of the algorithm, we generated six groups of simulated data with known sphericity errors using the aforementioned method from the first part of the simulation experiment.
The difference in each group of simulated data lies in the number of sampling points, with each simulating 50, 100, 500, 1000, 1500, and 2000 sampling points, respectively.Each group of simulated data contains 100 spheres with different radii and ten types of sphericity errors.The radii of the simulated data are uniformly distributed in the range of [1 mm, 100 mm].The sphericity errors are uniformly distributed in the range of [1 × 10 −5 mm, 1 × 10 −4 mm].Each group consists of 1000 datasets, for a total of 6000 datasets.
We conducted simulation experiments based on these six groups, a total of 6000 datasets.And we performed statistical analysis on the deviation value D between the experimentally obtained sphericity error result S E and the ideal sphericity error value S I .We used Formula (10) to calculate the deviation value.
The results, presented in Table 5, show that each group's experimental results S E have a mean deviation from the ideal sphericity error S I of approximately 10 −9 mm.It is noticeable that the number of sampling points has a negligible effect on the experimental results, indicating the algorithm's capability to access the sphericity error effectively under varying process requirements.The standard deviation of D for each group's experiment results is also approximately 10 −9 mm, which demonstrates the algorithm's high level of robustness.And the time required for each sphericity error assessment is less than 1 s.Extensive experiments have provided evidence that the algorithm can effectively satisfy the requirements of precision, speed, and robustness for sphericity error assessment in the manufacturing process of the HSRs.This suggests that there is potential for high-precision sphericity error assessment in the manufacturing process of the HSRs.

Comparison Experiments
In addition, to further verify the accuracy of the algorithm, we conducted experiments using the dataset provided by the authoritative literature in the field of sphericity error research.We then compared our results with the works in the literature that cite the same dataset.The three datasets include the surface sampling data of the hemispherical part and the simulation data generated on two concentric spheres.We then performed the experiments ten times independently based on each dataset and calculated the mean and standard deviation of the experimental results.
Dataset 1 is given by reference [42], consisting of 100 coordinate points with a sphericity error of 1.0 mm.The sphericity error of this dataset has been verified by several works [34,42] and the data distribution is shown in Figure 8.Ten independent experiments were conducted based on dataset 1.The mean value of sphericity error in the experimental results was 1.00000000035 mm, and the standard deviation of the results was 2.4 × 10 −10 mm.The detailed experimental results are presented in Table A1 in Appendix A, while Table 6 shows a comparison with existing literature results.
Sensors 2024, 24, x FOR PEER REVIEW 13 of 23 level of robustness.And the time required for each sphericity error assessment is less than 1 s.Extensive experiments have provided evidence that the algorithm can effectively satisfy the requirements of precision, speed, and robustness for sphericity error assessment in the manufacturing process of the HSRs.This suggests that there is potential for highprecision sphericity error assessment in the manufacturing process of the HSRs.

Comparison Experiments
In addition, to further verify the accuracy of the algorithm, we conducted experiments using the dataset provided by the authoritative literature in the field of sphericity error research.We then compared our results with the works in the literature that cite the same dataset.The three datasets include the surface sampling data of the hemispherical part and the simulation data generated on two concentric spheres.We then performed the experiments ten times independently based on each dataset and calculated the mean and standard deviation of the experimental results.
Dataset 1 is given by reference [42], consisting of 100 coordinate points with a sphericity error of 1.0 mm.The sphericity error of this dataset has been verified by several works [34,42] and the data distribution is shown in Figure 8.Ten independent experiments were conducted based on dataset 1.The mean value of sphericity error in the experimental results was 1.00000000035 mm, and the standard deviation of the results was 2.4 × 10 −10 mm.The detailed experimental results are presented in Table A1 in Appendix A, while Table 6 shows a comparison with existing literature results.
Dataset 2 is given by the reference [45] and consists of 384 coordinate points.All 384 coordinate points were obtained from real spherical sampling using the birdcage method, and were evenly distributed on 12 lines on the sphere.The distribution of dataset 2 is shown in Figure 9.For dataset 2, ten groups of replicate experiments were each performed.The results of the ten experiments are shown in Table A2 in Appendix A; the mean experimental result is 0.015384870588 mm, with a standard deviation of 6.4 × 10 −11 mm.
A comparison with the results in the literature is shown in Table 7.Compared with the results in reference [45], although both algorithms gave the same results after rounding to the same significant figures, our algorithm provided more significant figures.It exhibited a higher uncertainty, indicating that the present algorithm will have a significant advantage in the mass production of HSRs.
Sensors 2024, 24, x FOR PEER REVIEW 14 of 23 Dataset 2 is given by the reference [45] and consists of 384 coordinate points.All 384 coordinate points were obtained from real spherical sampling using the birdcage method, and were evenly distributed on 12 lines on the sphere.The distribution of dataset 2 is shown in Figure 9.For dataset 2, ten groups of replicate experiments were each performed.The results of the ten experiments are shown in Table A2 in Appendix A; the mean experimental result is 0.015384870588 mm, with a standard deviation of 6.4 × 10 −11 mm.
A comparison with the results in the literature is shown in Table 7.Compared with the results in reference [45], although both algorithms gave the same results after rounding to the same significant figures, our algorithm provided more significant figures.It exhibited a higher uncertainty, indicating that the present algorithm will have a significant advantage in the mass production of HSRs.
A comparison with the results in the literature is shown in Table 8.The accuracy of the present algorithm is improved by about 15 nanometers compared to the results of reference [13].Given the stringent geometric and positional tolerances required for HSRs, their measurement accuracy needs to reach the nanometer level.In this high-precision measurement scenario, the accuracy improvement of the algorithm in this paper is highly significant.

Table 7.
Comparison with results based on dataset 2. (Rounding is as given in the respective papers; units: mm).
A comparison with the results in the literature is shown in Table 8.The accuracy of the present algorithm is improved by about 15 nanometers compared to the results of reference [13].Given the stringent geometric and positional tolerances required for HSRs, their measurement accuracy needs to reach the nanometer level.In this highprecision measurement scenario, the accuracy improvement of the algorithm in this paper is highly significant.
Sensors 2024, 24, x FOR PEER REVIEW 14 of 23 Dataset 2 is given by the reference [45] and consists of 384 coordinate points.All 384 coordinate points were obtained from real spherical sampling using the birdcage method, and were evenly distributed on 12 lines on the sphere.The distribution of dataset 2 is shown in Figure 9.For dataset 2, ten groups of replicate experiments were each performed.The results of the ten experiments are shown in Table A2 in Appendix A; the mean experimental result is 0.015384870588 mm, with a standard deviation of 6.4 × 10 −11 mm.
A comparison with the results in the literature is shown in Table 7.Compared with the results in reference [45], although both algorithms gave the same results after rounding to the same significant figures, our algorithm provided more significant figures.It exhibited a higher uncertainty, indicating that the present algorithm will have a significant advantage in the mass production of HSRs.
A comparison with the results in the literature is shown in Table 8.The accuracy of the present algorithm is improved by about 15 nanometers compared to the results of reference [13].Given the stringent geometric and positional tolerances required for HSRs, their measurement accuracy needs to reach the nanometer level.In this high-precision measurement scenario, the accuracy improvement of the algorithm in this paper is highly significant.

Practical Application Experiments
Finally, we applied the algorithm to sphericity error detection in the manufacturing process of the HSRs.In a controlled environment, we used a Hexagon coordinate measuring machine to measure the inner and outer contours of the thin-walled spherical shell of the HSR sample.The sampling diagram is shown in Figure 11, while the corresponding data for the inner and outer contours are given in the Tables A6 and A7 in Appendix A.

Practical Application Experiments
Finally, we applied the algorithm to sphericity error detection in the m process of the HSRs.In a controlled environment, we used a Hexagon coo uring machine to measure the inner and outer contours of the thin-walled of the HSR sample.The sampling diagram is shown in Figure 11, while the data for the inner and outer contours are given in the Tables A6 and A7 in A  In terms of outer contour sampling, we employed a method similar to the one used for inner contour, as shown in Figure 13.We measured five sections on the outer contour, obtaining 130 coordinate data, and reconstructed the outer contour data accordingly.
Ten repeated experiments were conducted on the sampled outer contour data, and the results are shown in Table A5 in Appendix A. The average value of the ten experiments is 0.002799199921 mm, with a standard deviation of 2.1 × 10 −11 mm.The CMM provided a sphericity error evaluation of 0.0029 mm.And the algorithm's accuracy improved by approximately 4% over the CMM result in the experiment based on the outer contour data, as shown in Table 10.In terms of outer contour sampling, we employed a method similar to the one used for inner contour, as shown in Figure 13.We measured five sections on the outer contour, obtaining 130 coordinate data, and reconstructed the outer contour data accordingly.
Ten repeated experiments were conducted on the sampled outer contour data, and the results are shown in Table A5 in Appendix A. The average value of the ten experiments is 0.002799199921 mm, with a standard deviation of 2.1 × 10 −11 mm.The CMM provided a sphericity error evaluation of 0.0029 mm.And the algorithm's accuracy improved by approximately 4% over the CMM result in the experiment based on the outer contour data, as shown in Table 10.In terms of outer contour sampling, we employed a method similar to the one used for inner contour, as shown in Figure 13.We measured five sections on the outer contour, obtaining 130 coordinate data, and reconstructed the outer contour data accordingly.
Ten repeated experiments were conducted on the sampled outer contour data, and the results are shown in Table A5 in Appendix A. The average value of the ten experiments is 0.002799199921 mm, with a standard deviation of 2.1 × 10 −11 mm.The CMM provided a sphericity error evaluation of 0.0029 mm.And the algorithm's accuracy improved by approximately 4% over the CMM result in the experiment based on the outer contour data, as shown in Table 10.

Figure 2 .
Figure2.The sphericity error variation curves along the X, Y, and Z directions are presented, w the origin located at the least-squares spherical center.The original data used for drawing w obtained from reference[29].

Figure 2 .
Figure2.The sphericity error variation curves along the X, Y, and Z directions are presented, with the origin located at the least-squares spherical center.The original data used for drawing were obtained from reference[29].

Figure 3 .
Figure 3. Schematic diagram of the algorithm flow.

Figure 3 .
Figure 3. Schematic diagram of the algorithm flow.

Figure 4 .
Figure 4. Schematic diagram of the reasons for the formation of inferior subpopulations.Each c in the diagram represents a subpopulation (it should be noted that Figure4is not obtained from actual experiment but is a simulation presentation using a two-dimensional view to explain m clearly the two reasons that lead to the creation of inferior subpopulations).

Figure 10 .
Figure 10.Schematic diagram of the distribution of dataset 3.

Figure 9 .
Figure 9. Schematic diagram of the distribution of dataset 2.

Figure 10 .
Figure 10.Schematic diagram of the distribution of dataset 3.

Figure 10 .
Figure 10.Schematic diagram of the distribution of dataset 3.

Figure 13 .
Figure 13.Schematic diagram of sampling and data reconstruction of the hemispherical shell resonator outer contour (a) Actual image of the HSR during measurement.(b) The schematic illustrates the distribution of measurement points on the surface of the HSR.

Figure 12 .
Figure 12.Schematic diagram of sampling and data reconstruction of the HSR's inner contour.(a) Actual image of the HSR during measurement.(b) The schematic illustrates the distribution of measurement points on the surface of the HSR.

Figure 12 .
Figure 12.Schematic diagram of sampling and data reconstruction of the HSR's inner contour.(a) Actual image of the HSR during measurement.(b) The schematic illustrates the distribution of measurement points on the surface of the HSR.

Figure 13 .
Figure 13.Schematic diagram of sampling and data reconstruction of the hemispherical shell resonator outer contour (a) Actual image of the HSR during measurement.(b) The schematic illustrates the distribution of measurement points on the surface of the HSR.

Figure 13 .
Figure 13.Schematic diagram of sampling and data reconstruction of the hemispherical shell resonator outer contour (a) Actual image of the HSR during measurement.(b) The schematic illustrates the distribution of measurement points on the surface of the HSR.

Algorithm 1: Initialize the Subpopulations Based on Clustering
i -Cluster centers P a1 , P a2 , .., P aK Output: -Subpopulations subPop1, subPop2, . .., subPopK 1: Generate the initial population Pop within the search space R; 2: Randomly select K individuals P a1 , P a2 , .., P aK as cluster centers; 3: for each Pop i in Pop: 4: Calculate the Euclidean distance ∥Pop i − P ai ∥ 2 between Pop i and each cluster center P ai ; 5: Return the global optimal solution gbest.
Figure 4. Schematic diagram of the reasons for the formation of inferior subpopulations.Each color in the diagram represents a subpopulation (it should be noted that Figure4is not obtained from the actual experiment but is a simulation presentation using a two-dimensional view to explain more clearly the two reasons that lead to the creation of inferior subpopulations).

Table 2 .
Sphericity error evaluation results of group A data. (Unit: mm).

Table 3 .
Sphericity error evaluation results of group B data (unit: mm).

Table 3 .
Sphericity error evaluation results of group B data (unit: mm).

Table 3 .
Sphericity error evaluation results of group B data (unit: mm).

Table 4 .
Sphericity error evaluation results of group C data.(Unit: mm).

Table 4 .
Sphericity error evaluation results of group C data.(Unit: mm).

Table 5 .
Statistical results of deviations between experimental results and ideal values for each group of datasets (unit: mm).

Table 5 .
Statistical results of deviations between experimental results and ideal values for each group of datasets (unit: mm).

Table 6 .
Comparison with results based on dataset 1. (Rounding is as given in the respective papers; units: mm).

Table 6 .
Comparison with results based on dataset 1. (Rounding is as given in the respective papers; units: mm).

Table 8 .
Comparison with results based on dataset 3. (Rounding is as given in the respective papers; units: mm).

Table 8 .
Comparison with results based on dataset 3. (Rounding is as given in the re units: mm).

Table 9 .
Comparison of experimental results based on the inner contour sampling data (unit: mm).

Table 9 .
Comparison of experimental results based on the inner contour sampling data (unit: mm).

Table 9 .
Comparison of experimental results based on the inner contour sampling data (unit: mm).

Table 10 .
Comparison All data that support the findings of this study are included within this article.The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
of experimental results based on the outer contour sampling data (unit: mm).AlgorithmSpherical Center Coordinate (X, Y, Z)

Table A1 .
Results of ten experiments based on dataset 1 (unit: mm).

Table A2 .
Results of ten experiments based on dataset 2 (unit: mm).

Table A3 .
Results of ten experiments based on dataset 3 (unit: mm).

Table A4 .
Results of ten experiments based on the inner contour sampling data (unit: mm).

Table A5 .
Results of ten experiments based on the outer contour sampling data (unit: mm).

Table A6 .
Original data sampling from the HSR's inner contour (unit: mm).

Table A7 .
Original data sampling from the HSR's outer contour (unit: mm).