Research on multi-sensor measurement system and evaluation method for roundness and straightness errors of deep-hole parts

: Precision deep-hole parts are widely used in various fields of industrial production and their machining quality has a great impact on fatigue limit, geometric accuracy, and stability of products. Since roundness and straightness errors are essential technical indexes to evaluate the machining quality of deep-hole parts, accurate measurement and effective evaluation of them are of great significance to ensure the performance of related products. A multi-sensor integrated device that can measure two kinds of shape errors simultaneously was developed based on laser displacement sensor, two-dimensional position-sensitive detector, angle sensor, and laser distance sensor. Aiming at the problem of roundness error evaluation, the solution process of the control points of the minimum zone circle was optimized by calculating the distance between points and searching according to the polygon removal rule. Besides, the rotating projection method was used to evaluate the straightness error effectively. Eventually, the effectiveness of the measuring device and the shape error evaluation method was verified by experimental research.

equipment such as laser displacement sensor, capacitance sensor, and charge-coupled device (CCD) camera to obtain data information without touching the measured surface. For example, the diameter and roundness error measuring system was developed by Mekid and Vacharanukul [3] based on the corresponding relationship between displacement and light intensity; the cylindrical non-contact capacitance probe was used by Ma et al. [4]; the deep-hole inner surface parameter comprehensive measurement device was designed by Akio et al. [5], consisting of laser emitter, DC motor, measurement unit, and optical element for detecting posture; the adjustable five-dimensional measuring system was established by Ma et al. [6]; the measuring device consisting of two high-resolution single-lens reflex cameras was proposed by Zatočilová et al. [7]; additionally, the system designed by Jurevicius and Vekteris [8][9][10] was based on the phase difference coding of modulated optical signals emitted by two photodiodes. All of them are successful typical applications of non-contact mode in the field of shape error measurement and have been improved to varying degrees regarding measurement rationality and accuracy. However, these current applications are only for hole parts with small diameter or small ratio of length to diameter, and only one measured parameter such as roundness or straightness errors can be obtained in a measurement process. Therefore, there are still few measuring methods which are really suitable for deep-hole parts and can accurately obtain the roundness and straightness errors at the same time.
Similarly, the error evaluation of the measurement point set is the key to obtain the shape error value.
There are four basic methods for evaluating roundness error: least square circle (LSC), minimum circumscribed circle (MCC), maximum inscribed circle (MIC), and minimum zone circle (MZC).
Considering that the four methods have different reference circles, how to acquire the ideal reference circle is essential to ensure the evaluation accuracy. Li and Shi [11] proposed the concept of minimum zone line and used it as the control line of MZC. Meanwhile, the MCC and MIC reference circles of the measurement point set were obtained [12][13] according to the three relations of relative diameters. By establishing convex hull, Gadelmawla [14] employed the size relationship of arc radius to remove invalid points in the measurement point set in turn and circularly searched in the remaining point set until the control points of MCC, MIC, and MZC were searched out. Lei et al. [15] set a circular area near the center of LSC and obtained MCC, MIC, and MZC center points using the proposed polar coordinate transformation algorithm. Goch and Lubke [16] applied the improved Chebyshev algorithm to approximate geometric elements and acquired the maximum internal element and the minimum external element. Besides, Li and Shi [17] revealed that the control points of MCC, MIC, and MZC are interrelated, and their control points can be solved mutually.
Since the axis of deep-hole parts is a spatial straight line, its straightness error evaluation belongs to the spatial straightness error evaluation. Among many evaluation methods, the two-point connection method is the simplest while its accuracy is the worst; least square method (LSM) is simple and fast.
Both of them can meet the general precision requirements. However, they are not suitable for precision machining. There is no unified method for solving the minimum area, though the error value evaluated by the minimum zone method (MZM) is the smallest and unique. For example, Dhanish et al. [18] adopted the mode of iterative calculation of measurement point set and searched for control point combinations; the minimum parallelepiped envelope method was used by Huang et al.; Cho and Kim [19] proposed a linearization algorithm based on data envelopment analysis. Moreover, spatial straightness error evaluation is a nonlinear optimization problem. To address this issue, Ding et al. [20][21] combined semi-definite programming and Chebyshev approximation theory; Calvo et al. [22] applied a calculation method of coordinate vector integration. For the spatial problem, projection can play the role of "dimension reduction", and the problem of spatial straightness error can be transformed into the problem of solving the MCC on the plane by projecting the measurement point along the projection axis to the plane perpendicular to it. Endrias et al. [23] continuously rotated the projection axis according to the three situations (three points, four points, and five points in contact between the minimum containment zone and the measurement point set) until the control points conforming to the minimum zone was searched.
For shape error evaluation, no one method is universal. The data point sets obtained by different measuring devices have their own applicable evaluation methods [24]. Therefore, in order to obtain the accurate and effective error values, it is necessary to equip the newly developed measuring device with the suitable error evaluation method.
In this paper, the measurement and evaluation of roundness and straightness errors of deep-hole parts are deeply investigated, and a convenient roundness and straightness error measuring device is developed to measure two kinds of shape errors simultaneously. Furthermore, the improved MZC method is applied to evaluate roundness error, and the rotating projection method based on the principle of the minimum region is employed to solve the straightness error.

Measuring principle
Roundness error refers to the radius difference between the two smallest concentric circles containing the actually measured contour in the same section. As illustrated in Fig.1, M c represents the actually measured contour, Zc denotes the smallest contained concentric circle, and the radius difference fc indicates the roundness error. Straightness error is defined as the diameter of the smallest cylindrical surface containing the measured axis. It is extracted from the fitting center trajectories of several sections of the measured cylindrical surface. In Fig.2, L denotes the measured extraction axis, S represents the measured section, Zs refers to the smallest containment cylinder of the measured extraction axis, and its diameter fs is the straightness error. Therefore, the measurement of roundness error is the process of determining the actually measured contour while that of straightness error is the process of extracting the actual axis of deep-hole parts. According to the measuring principle of roundness and straightness errors, a multi-sensor integrated measuring device has been developed by effectively combining laser displacement sensor, angle sensor, position-sensitive detector (PSD), and laser distance sensor. The schematic diagram of the measuring device is displayed in Fig.3.
Initially, the laser displacement sensor is driven by the driving components to scan the contour point by point in the measured section and conduct equal angle sampling. This sampling method can not only equalize the angles between adjacent sample points in the same section but also ensure the same number of sample points in each measured section, making it convenient for subsequent evaluation. Additionally, the displacement value measured by the laser displacement sensor is transformed into the rectangular coordinate value of the corresponding measurement point according to the geometric relationship. The schematic diagram of transformation from displacement value to coordinate value is illustrated in Fig.4 where ri denotes the measured displacement value of the i-th measurement point, θi refers to the included angle between the laser beam and the x-axis, a represents the calibrated distance between the laser emission hole and the rotation axis, and b indicates the eccentricity between the rotation axis and the Since PSD is fixed on the measuring device, the center of its photosensitive surface is located on the axis of the device. Then, the laser distance sensor after leveling and straightening is placed on one side and its emitted laser is taken as the reference axis of the inner hole. In the measured section, PSD and laser distance sensor are combined to record the displacement offset of the device, and the angle sensor reflects the attitude variation. Thus, the coordinate values of the point set in the measurement coordinate system are transformed into those in the absolute coordinate system. As illustrated in where a and b represent the horizontal and vertical displacement offsets recorded by PSD, respectively.
After the coordinate values of the point set in the absolute coordinate system are obtained, the least square circle method is used to solve the center of the measured section. Depending on locate walking components, multiple measured sections can be obtained by moving the device back and forth.
Roundness error can be calculated by a relevant evaluation algorithm in each measured section. Then, the center of each measured section is connected in turn to obtain the axis of deep-hole parts and realize the solution of straightness error.

Roundness Error-Improved MZC Method
The MZC method is to evaluate the roundness error using the MZC center of the point set, and the MZC refers to two concentric circles that meet the following conditions: a) all the measurement points are on or between the two concentric circles; b) the radius difference of them should be the smallest. The Generally, solving the MCC and MIC needs to construct the outer convex hull point set and the inner convex hull point set respectively [25], so as to simplify the calculation and obtain the measurement points that meet the requirements. However, the construction process is extremely complicated, and the non-uniqueness of the inner convex hull point set would cause great uncertainty to the evaluation results.
Therefore, based on the distance between measurement points and the polygon removal rule, the search process of control points is optimized, and the control points of MCC and MIC are determined quickly and accurately without solving convex hull point set. The specific evaluation process of the improved MZC method for evaluating roundness error is described as follows: (1) The MCC control points.
The distance between each point in the point set and Ooch is successively calculated to obtain the point P1 farthest from Ooch and the straight line P1Ooch. Then, we search the points P2 and P3 on both sides of the line P1Ooch with the largest distance from it, as presented in Fig.6(a). The points P2, P3, and P1 together form a triangle. For three points Pi-1 (xi-1, yi-1), Pi (xi, yi) and Pi+1 (xi+1, yi+1), the angle θ corresponding to Pi could obtained by Eq. (4). And whether the formed triangle is obtuse triangle could be judge.
When it belongs to the obtuse triangle, the longest side is taken as the diameter, and the midpoint of it is taken as the center of circumscribe circle. If it contains all the measurement points, the circle is the MCC we expected. Assuming that some measurement points are located on or outside the circle, the two points farthest from the longest side on both sides of it are solved, forming a quadrilateral with two endpoints of the longest side. For each point of the quadrilateral, the radius of the circle passing through it and its adjacent points is calculated. After the point with the maximum radius is removed, the remain points constitute a new triangle. constitute a quadrilateral P1P2P4 P3. In the quadrilateral, the radius of arc determined by three adjacent points is calculated successively, and the middle point corresponding to the maximum radius value is removed to form a new triangle. where, The above steps are repeated until the circumscribe circle can contain all the measurement points.
Then, the circumscribe circle is the MCC. In Fig.6(d), the triangle vertices corresponding to the circle represent the control points of MCC.
For the same measurement point set Pi (xi, yi), i=1, 2, ⋯, n, the center point Op of the point set is calculated by Eq. (3). Then, the distance between each point in the point set and Op are solved in turn to obtain the point P1 with the shortest distance from Op and the straight line P1Op. As presented in Fig.7(a), the points P2 and P3 with the shortest distance from the line P1Op on both sides are searched, and they form a triangle together with the point P1. Besides, whether the triangle is obtuse is judged according to

Eq. (4).
When it belongs to the obtuse triangle, the longest side is taken as the diameter, and the midpoint of it is taken as the center of circumscribe circle. If all the measurement points are on or outside the circumscribe circle, the circle is the MIC we expected. In a certain calculation process, the triangle P1P2P4 is an obtuse triangle and some points are inside the circumscribe circle (Fig.7(b)); then, the two points P1 and P5 with the smallest distance from the longest side P2P4 on both sides are solved. Besides, the points P1、P5、P2 and P4 constitute a quadrilateral P1P2P5P4, as exhibited in Fig.7(c). In the quadrilateral, the radius of arc determined by adjacent three points is calculated in turn, and the middle point corresponding to the minimum radius value is removed to form a new triangle. The above steps are repeated until all the points are on or outside the circumscribe circle. Then, the circle is the MIC. In Fig.7(d), the triangle vertices corresponding to the circle represent the control points of MIC.
In Fig.8(a) where k1 denotes the slope of P1P2, k2 denotes the slope of P3P4, x1mid and y1mid denote the abscissa and ordinate of the midpoint of P1P2, respectively, and x2mid and y2mid denote the abscissa and ordinate of the midpoint of P3P4, respectively. For all the 3 v-4 cases, whether the concentric circle is constructed in each case is calculated with all the points in Cmax and Cmin. As indicated in Fig.8(b)

Straightness Error-Rotating Projection Method
For straightness error evaluation, the projection method can transfer the solution process from threedimensional space to two-dimensional plane, reducing the amount of calculation and the difficulty of solution. On the two-dimensional plane, the minimum zone containing all measurement points is analyzed to evaluate the straightness error, and the points located on the boundary of the minimum zone are called the control points.
The traditional projection method projects the measurement points to the vertical plane perpendicular to the least square midline, transforming the spatial straightness problem into the solution of the plane MCC. There are significant disadvantages in this process. For example, only by projecting to multiple planes, the distribution of measurement points can be fully reflected, and projection suppression points would be produced, leading to the omission of control points. Therefore, the rotating projection method proposed by Chen Hui et al. [26] based on the principle of the minimum zone is applied to the straightness error evaluation, verifying that this method has good solution speed and operation accuracy.
The evaluation process of the rotating projection method is detailed as follows. Firstly, by rotating the coordinate system, the least square midline of measurement point set passes through the origin of This process is repeated until dmax = dc. Moreover, the zone determined by the control line is the smallest zone containing the plane measurement points, and the axis straightness error is f = 2dmax.

Measurement and evaluation system
Based on the measuring principle of roundness and straightness errors, the mechanical structure of the measuring device, measuring units such as sensors and driving components such as servo motors, were explored, and a new measuring device was developed. The measuring device is mainly composed of measuring unit, driving unit, and locate walking unit. Its overall structure is displayed in Fig.9. The measuring unit consists of LK-H050 laser displacement sensor from KEYENCE, 6-axis WT901C485 angle sensor, PSD 2D-9 position-sensitive detector from TEM Messtechnik GmbH, and industrial laser distance sensor. Besides, the driving unit includes ECM-A3H-CY0401 servo motor from Delta and PA050-C-0108-A0825 reducer. The locate walking unit is composed of three supporting arms with a circumferential interval of 120 degrees to ensure that the axis of the measuring device and the axis of deep-hole parts are located on the same straight line. Moreover, the distance between the supporting arm and the center of measuring device is slightly larger than the radius of the measured hole when the compression spring in the unit is in a free state. Thus, the compression spring is always in a compressed state when the measuring device works, ensuring that the universal ball at the end of the supporting arm is always in close contact with the wall surface. According to this feature, the developed device is suitable for measuring deep-hole parts with a diameter of 150-160mm. A three-dimensional model and a solid model of the measuring device are exhibited in Fig.10 and Fig.11, respectively. (1) Output the signal to perform pulse control on servo motor such as start-stop, clockwise, and counterclockwise rotation; (2) Control the sensors to complete the collection, storage and transmission of measurement data; (3) Obtain roundness and straightness errors by performing the coordinate transformation, section center calculation, roundness evaluation, and straightness evaluation on the sensor measurement data. Fig.12 The computer software platform.

Calibration experiment of a laser displacement sensor
There is an offset distance a and an eccentricity value b between the laser emission hole and the rotating axis (Fig.13). Therefore, the distance between each sampling point and the rotating center is not equal to the output value of the laser displacement sensor, making parameters a and b exert a huge influence on the measurement results. It is necessary to eliminate the errors caused by parameters a and b in the measurement process by calibrating them.  Parameters a and b were calibrated on the section passing through the laser and perpendicular to the rotating axis. The calibration principle of parameter a is presented in Fig.15. The laser displacement sensor was rotated to face the calibration plane A, the position of the sensor was adjusted to make the laser perpendicular to the plane, and the output value r1 of the sensor was recorded. Then, the sensor was rotated to face the calibration plane B, the position of the sensor was adjusted to make the laser perpendicular to the plane, and the output value r2 was recorded. According to the geometric relationship, the offset distance a between the rotating axis and the laser emitting surface of the laser displacement sensor can be calculated by Eq. (9). 1 2 ( ) 2 D r r a − − = (9) where D denotes the calibration distance of U-shaped gauge block.  Fig.15 The calibration principle of parameter a. Fig.16 The calibration principle of parameter b.
The calibration principle of parameter b is illustrated in Fig.16. Specifically, the laser displacement sensor was rotated to face the calibration plane A, the position of the sensor was adjusted within the angle of ±15 between the laser and the plane, and the minimum output value r1min of the sensor was recorded. Since The U-shaped gauge blocks were changed with different calibration distances, and parameters a and b were calibrated five times, so as to reduce the calibration error. Among them, the calibration distance of U-shaped gauge blocks can be measured by CMM. The calibration results of the parameters a and b are provided in Table 1 and Table 2, respectively. The average value of five measurement results is selected to obtain a =31.345mm and b =1.038mm.

Measurement and evaluation experiment of roundness and straightness errors
To verify the effectiveness of the proposed measuring device and evaluation method, the roundness and straightness errors of the measuring ring gauge with an inner diameter of 155mm, the thickness of 36mm, nominal roundness error of 2.5μm, and nominal straightness error of 3μm were measured and evaluated using the CMM and the developed system, respectively. The results were compared and analyzed.
The inner hole of the measuring ring gauge was equally divided into 13 sections every 3mm along the thickness direction. The first and last sections were removed, with a total of 11 sections. Besides, the inner hole surface was scanned and equiangularly sampled using the Hexagon Global Advantage CMM.
The number of sampling points in each section is 100. Afterward, the shape error was evaluated according to the least square method equipped by the CMM. The measuring ambient temperature is 20.2ºC, and the relative humidity is 35%. Furthermore, three groups of experiments were performed on three measuring ring gauges of the same model to eliminate the experimental contingency. The measurement and evaluation results are listed in Table 3.  Particularly, the right side of the measuring ring gauge was taken as the measuring zero plane, and the measuring device was slowly moved until the laser line emitted by laser displacement sensor was in the measuring zero plane. Then, the installation position of the angle sensor was flattened by the measuring probe of CMM, and the angle sensor was zeroed and calibrated at this position. The motor was rotated to make the lower bottom of the mounting bracket for fixing the laser displacement sensor face upwards, and the measuring probe of CMM and the motor were used for fine adjustment to keep the lower bottom of the mounting bracket horizontal. Furthermore, the laser distance sensor located on the backside of the measuring device and fixed on the lifting platform was adjusted to allow the laser line emitted by the sensor vertically to irradiate the center of PSD photosensitive plane. Starting from the measuring zero plane, the measuring device was moved forward by 3mm every time, and 11 measurement sections were obtained and the same as those in the CMM measuring experiment. The measurement experimental scene with the developed device is exhibited in Fig.17. The sampling frequency of the laser displacement sensor was set to 1kHz, and the storage period was 1000 times of the sampling period, that is, one second recorded the data of one point. Since the number of measuring points in each section is 100, the motor speed should be set to 0.6 rpm. When measuring, in the first section, the horizontal direction was taken as the starting position, counterclockwise rotation was performed, and sampling was conducted every 3.6°. To prevent the missing of measurement points caused by the backlash of the reducer during the change of the rotation direction, each sampling was rotated by an additional 15°, and the redundant sampling points were removed during data processing.

Measuring probe of CMM
On the second section, starting from the end position of the first measurement process, the sampling started after rotating clockwise by 15°; it also rotated by an additional 15° at the end. By analogy, counterclockwise and clockwise rotation was alternately conducted to avoid the winding of cables during the measurement process, realizing the measurement of 11 sections. The information collected by the sensors was transmitted to the PC through the data acquisition device. Finally, the roundness and straightness errors were evaluated by the improved MZC method and the rotating projection method.
Similarly, three measuring ring gauges were tested, as indicated in Table 3.  Fig. 18, we can see that the measurement result curve of the developed device is much closer to the nominal value curve than the CMM measurement result curve, that is to say, the developed system shows a better measurement and evaluation effect. Therefore, the validity and accuracy of the measuring device and the evaluation method adopted are verified.

Conclusion
According to the measuring principle of roundness and straightness errors of deep-hole parts, an integrated measuring device composed of laser displacement sensor, angle sensor, PSD, and laser distance sensor is developed, which is suitable for measuring deep-hole parts with a diameter of 150-