Vibration errors compensation method based on self-feature registration for the 3-D dynamic measurement of metallic sealing ring forming surface

The formed surface quality of metallic sealing ring of aero-engine affects the aircraft service performance directly. However, existing inspection methods, such as the final destructive inspection and the line laser scanner section profile measurement, only evaluate the formed quality from a 2-D view, that is, single or multiple radial formed section profiles. The lack of geometric information of 3-D surface is not conducive to the comprehensive monitoring of forming quality and process planning. Therefore, based on the line laser scanners, this paper mainly proposes a vibration errors compensation method based on self-feature registration. Aiming at the problem of rigid transformation of the measurement profile caused by random vibration during the rotary motion of metallic sealing ring, the feature of measurement profile in stationary scene (MPSS), that is, the medial axis, is used as the reference for the correct pose of measurement profile. The principle of finding the correct pose of measurement profile in rotary motion scene (MPRMS) is to minimize the distance between the medial axes. Next, based on the rotary motion information of metallic sealing ring and the geometric information of measurement system, a 3-D reconstruction matrix is built, so as to convert each measurement profile to the base coordinate system in turn, and finally a 3-D dynamic measurement method for the metallic sealing ring forming surface is built. The effectiveness of the proposed method is verified through simulation experiment and real measurement.


Introduction
Metallic sealing ring is a key part widely used in aeroengine system, 1 its surface formed quality directly affects the sealing effect and plays an important role in the safe service of aircraft.To meet the sealing needs of different cases, metallic sealing rings have a variety of section forms, mainly divided into C-rings, W-rings (inner and outer openings), and multi-wave-rings.Due to the complex section, small size, overall ring closed, etc., metallic sealing rings are difficult to inspect.In the past decades, the formed result of metallic sealing ring heavily relied on the final destructive inspection, which involves randomly sampling from the same batch of products, cutting the rings radially, and scanning the section to obtain inspection results.The lack of inspection means in the forming process makes it difficult to find forming defects in time.In recent years, optical measurement has been widely used by virtue of its flexibility, portability, and high measurement accuracy. 2In the work, 3 the non-destructive, on-machine measurement of the inner and outer radial section profiles of the metallic sealing ring is realized by means of two line laser scanners.However, the current works only evaluate the formed quality of metallic sealing ring from a 2-D view, that is, single or multiple radial formed section profiles, the lack of geometric information of 3-D surface is not conducive to the comprehensive monitoring of forming quality.Therefore, the 3-D measurement of the metallic sealing ring forming surface is very significant to provide support for process planning, and contributes to lower scrap rates and reduced production costs.
Although the measurement data of the on-machine measurement based on laser scanners is section profile point cloud, the metallic sealing ring rotates around its own axis during the forming process.Therefore, 3-D dynamic measurement of surface can be achieved with the help of the self-rotary motion.The 3-D dynamic measurement of surface mainly involves two aspects: 3-D reconstruction and dynamic measurement.The former works are represented by multi-view registration, where the measurement point clouds from different views are aligned to the common coordinate system, so as to achieve object reconstruction, such as aero-engine blade, 4 wind turbine blade, 5 but the sensor and the object are often stationary when performing fixed point measurements in different views.The latter is to measure the profile of object during its motion (some scenes for sensor motion).In motion scene, the decrease of measurement data precision and the relative random vibration between measurement object and sensor present new challenges, and dynamic measurement has thus become a technical challenge.For improving the measurement data precision, Pan et al. 6 built an online dynamic wheel size measurement method based on multisensory data fusion, the measurement mode is through multiple sets of data, with the shape-constant characteristics of a wheel body, which can remove measurement noise and fill up the missing data.Liu et al. 7 proposed an on-machine measurement method for outof-round of wheel profile, a dynamic measurement model is built to separate wheel profile from the measurement data.In the dynamic measurement of rail profile, the random vibration between the laser sensor and the rail can cause the optical plane to be not parallel to the section profile of the rail, resulting in measurement profile distortion.Wang et al. 8 analyzed the triple primitive of the geometric difference between the positive and distorted section profiles, by constructing the coordinate set from the distorted profile projection to the standard profile, the self-compensation of the iterative external camera parameters attached to the predefined measurement coordinate system is realized, so as to ensure the light plane are parallel.Sun et al. 9 obtained the rail section profiles by a multi-line structured-light vision sensor, each profile can be segmented into two curves, one represents the rail waist is used to fit the rail longitudinal axis and establish an auxiliary plane perpendicular to the direction, the deviation rectifying process is to project the rail section profiles onto the auxiliary plane, to recover perpendicularity between the light plane and the rail.In addition to distortion, the random vibration associated with the motion also cause rigid transformation of the measurement profile (i.e.rotation and translation).Lim et al. 10 utilized the overlaps of the dual lasers after projection and measured the height difference between the objects to calculate the vibration errors.However, the method achieves vibration errors compensation in the vertical direction only.In summary, the work on dynamic measurement of geometric profile is less, and the lack of a dynamic measurement method under the rotary motion of ring part makes it impossible to accurately measure the 3-D surface.Therefore, this paper carries out the vibration errors compensation for the problem of rigid transformation of the measurement profile caused by random vibration during the rotary motion of metallic sealing ring, in order to realize 3-D dynamic measurement of surface.
In this paper, a vibration errors compensation method is proposed, so as to rigidly transform the measurement profile to the correct pose.Vibration errors compensation can be transformed into a registration problem, which is divided into two steps: (1) Determine the correct pose of the measurement profile in the motion scene; (2) Calculate the rigid transformation between the current pose and the correct pose of the measurement profile.However, the correct pose of measurement profile is unknown and the rigid transformation cannot be calculated.One method is to assume that the measurement profile in the stationary scene (MPSS) is the reference for the correct pose of the measurement profile in the rotary motion scene (MPRMS), then, the MPRMS is transformed to the reference by registration.In the above method, the assumption of correct pose does not provide evidence, the principle of finding the correct pose of the MPRMS is to minimize the distance between the measurement profiles, which may not be true.Therefore, how to define the correct pose and find the correct pose of MPRMS is the key to solve the above problems.In the section profile of metallic sealing ring, there is a special feature: the medial axis.As shown in Figure 1, the medial axis has the property of equal distance from each point on the line to the inner and outer profiles.The 3-D CAD model of metallic sealing ring is obtained by rotating the 2-D section profile in fixed axis, in the forming process, the roundness is controlled by the flexible restraint rollers, which is easier to control than the section shape, and when there are shape differences between the section profiles of the actual formed part, align the medial axes of measured profiles is more in line with the fact.Therefore, the medial axis of MPSS is used as the reference for the correct pose of MPRMS, and the principle of finding the correct pose of MPRMS is converted to minimizing the distance between the medial axes.After completing the vibration errors compensation, a 3-D reconstruction matrix is constructed based on the rotary motion information of metallic sealing ring and the geometric information of measurement system, and each measurement profile in the measurement coordinate system is sequentially transformed to the base system.Finally, a 3-D dynamic measurement method for metallic sealing ring forming surface is built.The main contribution and significance are as follows: (1) Based on the laser scanners, a 3-D dynamic measurement method for metallic sealing ring forming surface is proposed, which supports the comprehensive monitoring of the forming quality, and provides a reference for the 3-D dynamic measurement of ring part surface under the rotary motion.(2) A vibration errors compensation method based on self-feature registration is proposed.The Voronoi diagram is used to extract the medial axis of the section profile, the medial axis of the MPSS is used as the reference for the correct pose of measurement profile.The principle of finding the correct pose of the MPRMS is to minimize the distance between the medial axes, which effectively solves the problem of measurement profile transformation due to the random vibration associated with the rotary motion of the metallic sealing ring.(3) The validity of the above methods are verified by simulation experiments, the real measurements provide a feasible example for comprehensively monitoring the forming quality of metallic sealing ring.
The rest of this paper is organized as follows.In Section 2, the measurement system and problem are introduced.In Section 3, the vibration errors compensation method and the 3-D reconstruction method of surface is described in detail.Section 4 presents the simulation experiments, real measurement and corresponding results discussion, and Section 5 concludes this paper.

Measurement system and problem
The schematic diagram of on-machine measurement system of metallic sealing ring is shown in Figure 2. The two laser scanners measure from the inner and outer sides of the radial section profile of the ring, and their optical planes are aligned to ensure that the measured section is the same.During the forming process, the metallic sealing ring rotates around its own axis, and the 3-D dynamic measurement of surface can be realized by the self-rotary motion.However, the random vibration during the rotary motion of the ring leads to rigid transformation of the measurement profile in measurement coordinate system, the measurement profile deviates from the correct pose, that is, it generates rotation and translation, resulting in vibration errors.Therefore, it is necessary to compensate for vibration errors, in other words, the measurement profile is transformed to the correct pose.
In the following, the vibration errors analysis is given.Define the base coordinate system o Ba -x Ba y Ba z Ba and the measurement coordinate system o Me -x Me y Me z Me , as shown in Figure 2. The axis of the metallic sealing ring is located on the z Ba axis of the base coordinate system o Ba -x Ba y Ba z Ba , and the axis of the metallic sealing ring is ensured to be co-linear with the z Ba axis by means of flexible clamps and restraining rollers.The x Me -z Me plane of the measurement coordinate system o Me -x Me y Me z Me , the radial measurement section of the metallic sealing ring and the y Ba -z Ba plane of the base coordinate system are coplanar.The y Ba axis is co-linear with the z Me axis.During the forming process, the metallic sealing ring is rotated around the z Ba axis, and its radial section profiles pass through the x Me -z Me plane in turn, while the effect of random vibration associated with the rotary motion will be directly reflected in the x Me -z Me plane.
Decompose random vibration into three translational axes and three rotational axes.The translational vibration of x Ba axis will cause the measurement profile to be non-radial, the measurement profile in x Me -z Me plane will be distorted and the translation in z Me axis direction will be generated.The rotation vibration of x Ba axis will cause the measurement profile to rotate in x Me -z Me plane.The translational vibration of y Ba axis will cause the measurement profile to translate in z Me axis direction.The rotation vibration of y Ba axis will cause the measurement profile to be non-radial, the measurement profile in x Me -z Me plane will be distorted.The translational vibration of z Ba axis will cause the measurement profile to translate in x Me axis direction, and the rotation vibration of z Ba axis is the same as the rotation direction of the metallic sealing ring.
In summary, the rotation vibration of x Me -z Me plane, the translational vibration of y Ba axis and the translational vibration of z Ba axis will lead to a rigid transformation of the measurement profile in the measurement coordinate system, that is, the translation in two directions and the rotation in x Me -z Me plane.This paper focuses on compensating the vibration errors in the three directions mentioned above, and finally develops a 3-D dynamic measurement method for the metallic sealing ring forming surface.

Method overview
The 3-D dynamic measurement method for the metallic sealing ring forming surface mainly includes: (1) Vibration errors compensation; (2) 3-D reconstruction of surface.Assume that K section profiles are measured during a one-cycle of metallic sealing ring, denoted as Profile 0, Profile 1, ..., and Profile K-1.Profile 0 is collected in stationary scene, while the remaining K-1 profiles are collected in rotary motion scene.The 3-D dynamic measurement method is shown in the Figure 3, take Profile 0 and Profile 1 as examples, the basic steps are as follows: Step 1: Vibration errors compensation based on selffeature registration.
Step 1.1: Profile registration.Based on rigid registration, Profile 1 is rigidly transformed to Profile 0 to achieve optimal alignment between the two.
Step 1.2: Feature extraction.Extract the self-features of Profile 0 and Profile 1 based on Voronoi diagram, that is, medial axes.
Step 1.3: Feature-based fine registration.The transformation matrix is obtained by rigidly registering the medial axis of Profile 1 to the medial axis of Profile 0, and apply it to Profile 1, Profile 1 is transformed to correct pose.
Step 2: 3-D reconstruction of surface.Based on the rotary motion information of metallic sealing ring and the geometric information of measurement system, a 3-D reconstruction matrix is constructed, and transform the Profile 1 compensated for vibration errors in the measurement coordinate system to the base coordinate system.
Step 3: Perform the above steps on the measured Profile 2, ..., and Profile K-1, to achieve the 3-D dynamic measurement of surface of metallic sealing ring.

Vibration error compensation
In the section profile of metallic sealing ring, there is a self-feature called medial axis, the medial axis has the property of equal distance from each point on the line to the inner and outer profiles, when the thickness of each radial section of the metallic sealing ring is different, align the medial axis of each measured profile is more in line with the fact.Therefore, the medial axis of the MPSS is used as the reference for correct pose.The principle of finding the correct pose of the MPRMS is to minimize the distance between the medial axes.The vibration errors compensation based on self-feature registration mainly includes two parts: registration and self-feature extraction.In vibration errors compensation, a total of two registrations are performed.The first registration is profile registration, and the second registration is feature-based registration.The purpose of profile registration is to provide initial pose for feature-based registration to obtain accurate results.Due to the data form of measurement profile is point cloud, the registration is point-to-point, and both registrations use the iterative closest point (ICP) algorithm. 11,12ICP is a classical rigid registration algorithm, which alternates between closest point query in the target set and minimization of distance between corresponding points, and is guaranteed to converge to a locally optimal alignment. 13efine the point cloud of MPSS as Q Me 0 = fq Me 00 , . . ., q Me 0N g, the point cloud of MPRMS as P Me 1 = fp Me 10 , . . ., p Me 1M g, where, Me represents that the point cloud is located in the measurement coordinate system, M, N represents the number of points.The rigid transformation from P Me 1 to Q Me 0 is represented as: min R, t where D i (R, t) = min q Me 0 2Q Me 0 jjRp Me 1i + t À q Me 0 jj, R is a rotation matrix, t is a translation vector, I SO d ð Þ is an indicator function for the special orthogonal group SO(d), which requires R to be a rotation matrix: Equation ( 1) is solved by alternating between the correspondence step and alignment step.In the correspondence step, find the closest point in the alignment step, update the transformation by minimizing the distance between the corresponding points: the alignment step can be solved in closed form by singular value decomposition (SVD).Finally, the rigid transformation matrix T R is expressed as: In the self-features extraction, select the medial axis of the section profile of the metallic sealing ring for selffeature.The extraction methods of medial axis mainly include Topological Thinning, Distance Transform and Voronoi diagram. 14Among them, the Topological Thinning and Distance Transform are mainly used for the extraction of medial axes of objects represented by pixels or voxels.Voronoi diagram is a geometric structure that divides spatial regions, take Q Me 0 = fq Me 00 , . . ., q Me 0N g as an example, the Voronoi region of the points is: where, jjx À q Me 0i jj is the Euclidean distance between two points, q Me 0k is any point other than q Me 0i .In this way, the plane is divided into convex polygon that are not always finite, the boundaries of these polygons define the boundaries between regions with a distance equal to two generating points, the polygon vertices (points with equal distances to two or more, usually three generating points) are Voronoi vertices. 14Because the Delaunay triangulation is the dual of the Voronoi diagram, 15 the calculation of Voronoi diagram is realized by computing the Delaunay triangle and then computing the dual of the result.Therefore, Delaunay triangulation is performed on the point set, and then the Voronoi vertices are get.By utilizing the property that the distance between each point on the medial axis and the profiles on both sides is equal, the medial axis point set is extracted from the Voronoi vertices.Therefore, the medial axis is extracted based on the Voronoi diagram.
As shown in Figure 4, the basic steps are as follows: Step 1: Profile division.Based on the maximum and minimum points in the x M axis direction of the measurement coordinate system, Q Me 0 = fq Me 00 , . . ., q Me 0N g is divided into inner profile point cloud and outer profile point cloud, denoted as A = fa 1 , . . ., a Na g and B = fb 1 , . . ., b Nb g.
Step 3: Extract medial axis.Firstly, extract the union of A={a 1 , . . .,a Na } and B={b 1 , . . .,b Nb } at the maximum and minimum points in the x M axis direction and z M axis direction, denoted as Xmin, Xmax, Zmin and Zmax.Secondly, extract the points V (1) ={v (1) , . . .,a Na } V ð1Þ = fv is considered as a point on the medial axis.e is used to determine whether the distance between d Ai and d Bi is similar, t 0 is the distance threshold, and is used to determine whether v ð1Þ i is located between the inner and outer profiles.The point set of medial axis is denoted as V ð2Þ = fv ð2Þ 1 ; . . .; v ð2Þ Nv ð2Þ g.

The 3-D reconstruction of surface
After completing the vibration errors compensation, each measurement profile is still located in the measurement coordinate system.Therefore, based on the rotary motion information of metallic sealing ring and the geometry information of measurement system, a 3-D reconstruction matrix is built, and the measurement profiles are transformed into the base coordinate system to obtain the 3-D surface.
As shown in Figure 5, the rotary motion information is the rotation angle of the metallic sealing ring relative to the initial position.When the kth measurement profile is collected, the rotation angle is expressed as: the geometric information of the measurement system is the distance l between the origin o Me of measurement coordinate system and the origin o Ba of base coordinate system, which can be obtained by calibration.Therefore, according to the coordinate transformation method 16 and Figure 5, the 3-D reconstruction matrix is: where ( Me k x i , Me k y i , Me k z i ) is the ith point in the kth measurement profile after vibration errors compensation in the measurement coordinate system, ( Ba k x i , Ba k y i , Ba k z i ) is its coordinate in the base coordinate system.

Simulation experiments
Coordinate measuring machine (CMM) is a highprecision inspection method for precision surface, 17 but the metallic sealing ring has an irregular section and an overall annular closed structure, which severely limits the reachability of the probe.Therefore, the 3-D measurement of surface cannot be realized.Due to the lack of 3-D surface measurement methods for metallic sealing ring, the accuracy of the proposed method cannot be directly verified in real measurement, so the dynamic measurement of 3-D surface is simulated.The simulation experiment settings are as follows: (1) The measurement objects are as follows: A standard multiwave-ring with diameter of 820 mm and section profile within 4:280mm38:610mm, a standard W-ring (outer opening) with diameter of 745 mm and section profile within 4:400mm36:220mm, a standard W-ring (inner opening) with diameter of 450 mm and section profile within 4:450mm35:400mm, all three are virtual 3-D digital models, in order to verify the effectiveness of the proposed method in the dynamic measurement of 3-D surface of ring part with irregular section.(2) In the simulation, the metallic sealing ring rotates around its own axis, based on the self-rotary motion, 1000 section profile point clouds are collected radially and equidistantly by the on-machine measurement system, and random rigid transformations are applied to the point clouds (except the first point cloud, the first point cloud is obtained in stationary scene) to simulate the vibration errors, the random translation applied in the x Me direction is between [24.2 mm, 4.2 mm], the random translation applied in the z Me direction is between [22.7 mm, 2.7 mm], and the random rotation applied in the x Me -z Me plane is between [2108,108], in order to more realistically reproduce the random vibration under real measurement.(3) To verify the accuracy of the proposed method, only vibration errors exist in the point clouds, therefore, the 3-D digital models are the standard measurement results under the simulation experiment, and the accuracy verification is realized by comparing the distance deviation of the processed results with the 3-D digital models.
where, N three is the point number of 3-D point cloud.
The evaluation results are shown in Table 1, the  simulation results show that the processing results of the proposed method are very close to the standard measurement results.

Real measurement
In previous work, 3 the precise measurement of the section profile is achieved.In simulation experiment, the accuracy of the vibration errors compensation method and the 3-D reconstruction of surface method has been fully verified.Therefore, the accuracy of real measurements has also been proven.
Next, the effect of the proposed method under experimental conditions is shown by real measurements, so as to provide a feasible case for comprehensive monitoring of the forming quality of metallic sealing ring: (1) In order to simulate the rotary motion in the forming process of metallic sealing ring, an on-line inspection test equipment for continuous rolling forming of metallic  sealing ring is built, as shown in Figure 7, the equipment can simulate the rotary motion in the forming process of metallic sealing ring with a diameter of 2502850 mm.
(2) The real measurement object is a multi-wave-ring with a diameter of 820 mm, it is placed on the test equipment and positioned and clamped by corresponding clamps, the rotation speed of the multi-wave-ring is set to 12 r/min, its section profile is measured by the onmachine measurement system.(3) The on-machine measurement system is equipped with two laser scanners of the same model, the z Me axis resolution is 1.5 mm, the x Me axis resolution is 2048 point/profile, the z Me axis measurement range is 15 mm, and the x Me axis measurement range is 25 mm, the profile frequency, exposure time, and idle time of the laser scanners satisfy the following equation the profile frequency is: where, the three parameters can be adjusted according to the measurement requirements, and the maximum value of the profile frequency can be set to 1000 Hz theoretically.(4) In the rotation of the metallic sealing ring for one cycle, 10 section profiles are collected (i.e.Profile 0, Profile 1, Profile 2, ., Profile 9).( 5) Because the measurement section is irregular, there are serious outliers, missing and noise in the measurement point cloud, therefore, before the vibration errors compensation and 3-D reconstruction of surface, the outliers removal, missing part repair and noise correction are performed.The 3-D point cloud of the multi-wave-ring is obtained through the proposed method, align the point cloud with the corresponding CAD model, and calculate the deviation d i of each point within the point clouds to form a comprehensive monitoring of forming quality.Figure 8 shows the 3-D reconstruction result and forming deviation, the calculation results of the forming deviation are shown in Table 2.In summary, the proposed method realizes the 3-D dynamic measurement of metallic sealing ring forming surface and provides a support for comprehensive monitoring of the forming quality.

Conclusion and future works
In this paper, a 3-D dynamic measurement method for the metallic sealing ring forming surface is proposed.Based on the developed on-machine measurement system, the 3-D dynamic measurement of metallic sealing ring surface during the forming process is achieved, which supports the comprehensive monitoring of the forming quality, and provides a reference for the 3-D dynamic measurement of ring part surface under the rotary motion.Vibration errors compensation and 3-D reconstruction of surface are the main components of this method.
In vibration errors compensation, the medial axis of metallic sealing ring section profile is selected for the self-feature.The medial axis of MPSS is used as the reference for the correct pose of measurement profile, the principle of finding the correct pose of MPRMS is to minimize the distance between the medial axes, which effectively solves the problem of measurement profile transformation caused by random vibration associated with rotary motion.In terms of 3-D reconstruction of surface, a 3-D reconstruction matrix is built based on the rotary motion information of metallic sealing ring and the geometric information of measurement system, and each measurement profile is sequentially transformed to the base coordinate system to realize the 3-D reconstruction of ring part surface.The effectiveness and accuracy of this method have been verified through simulation experiment.Finally, the real measurement is performed, so as to provide a feasible case for comprehensive monitoring of the forming quality of metallic sealing ring, which contributes to lower scrap rates and reduced production costs.
In the future, the research will be carried out on the robustness of the vibration error compensation method

Figure 2 .
Figure 2. On-machine measurement system schematic diagram and vibration errors analysis.

Figure 3 .
Figure 3.The 3-D dynamic measurement method for the surface of metallic sealing ring.

ð1Þ 1 ;
. . .; v ð1Þ Nv ð1Þ g in V = fv 1 ; . . .; v Nv g where the x M coordinate is within Xmin and Xmax, the z M coordinate is within Zmin and Zmax.Finally, use kd-tree to search for the nearest neighbor points A i and B i in A = fa 1 ; . . .; a Na g and B = fb 1 ; . . .; b Nb g of point v ð1Þ i in V ð1Þ = fv ð1Þ 1 ; . . .; v ð1Þ Nv ð1Þ g, and record the corresponding Euclidean distances as d Ai and d Bi .When d Ai À d Bi j j 4e and d Ai + d Bi 42t 0 , v ð1Þ i Àsinu k 0 lsinu k sinu k cosu k 0 Àlcosu k The 3-D point clouds of metallic sealing rings are obtained through the proposed method.Align the point clouds with the corresponding 3-D digital models, and calculate the deviation d i of each point in the point clouds.The schematic diagram of the simulation and the deviation chromatogram of the point clouds are shown in the Figure 6.Use the MEAN and s of each deviation as evaluation indicators for the proposed method:

Figure 6 .
Figure 6.Simulation experiment and results: (a) Simulation measurement.(b) Measurement point cloud under the simulation.(c) Processing result.(d) Processing result of standard multi-wave-ring.(e) Processing result of standard W-ring (outer opening).(f) Processing result of standard W-ring (inner opening).(g) Local amplification of standard multi-wave-ring processing results.(h) Local amplification of standard W-ring (outer opening) and (i) Local amplification of standard W-ring (inner opening).

Figure 8 .
Figure 8.The 3-D reconstruction and forming deviation: (a) 3-D reconstruction.(b) Differences between measurement profile and CAD model point cloud (the green point cloud is the measurement profile, the red point cloud is the CAD model), and (c) Forming deviation.

Table 1 .
The proposed evaluation results.
Figure 7. On-line inspection test equipment for continuous rolling forming of metallic sealing ring.

Table 2 .
The forming deviation of the measured object.