Geometry error compensation using multiply laser sensors: a case study

Abstract

Purpose

A multi-laser sensors-based measurement instrument is proposed for the measurement of geometry errors of a differential body and quality evaluation. This paper aims to discuss the aforementioned idea.

Design/methodology/approach

First, the differential body is set on a rotation platform before measuring. Then one laser sensor called as “primary sensor”, is installed on the intern of the differential body. The spherical surface and four holes on the differential body are sampled by the primary sensor when the rotation platform rotates one revolution. Another sensor called as “secondary sensor”, is installed above to sample the external cylinder surface and the planar surface on the top of the differential body, and the external cylinder surface and the planar surface are high in manufacturing precision, which are used as datum surfaces to compute the errors caused by the motion of the rotation platform. Finally, the sampled points from the primary sensor are compensated to improve the measurement accuracy.

Findings

A multi-laser sensors-based measurement instrument is proposed for the measurement of geometry errors of a differential body. Based on the characteristics of the measurement data, a gradient image-based method is proposed to distinguish different objects from laser measurement data. A case study is presented to validate the measurement principle and data processing approach.

Research limitations/implications

The study investigates the possibility of correction of sensor data by the measurement results of multiple sensors to improving measurement accuracy. The proposed technique enables the error analysis and compensation by the geometric correlation relationship of various features on the measurand.

Originality/value

The proposed error compensation principle by using multiple sensors proved to be useful for the design of new measurement device for special part inspection. The proposed approach to describe the measuring data by image also is proved to be useful to simplify the measurement data processing.

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1. Introduction

Measuring plays an important role in manufacturing industries. Various measurement equipment and instruments have been developed in past years (Pottmann and Wallner, 1999; Katsuki et al., 2011; Tianci et al., 2022; Sommer et al., 2022; Guo et al., 2021). According to sensors, the existing measuring instruments can be categorized into tactile and non-contact measurements. The tactile measuring instruments like Coordinate Measuring Machines (CMM) have been widely applied in engineering domains, which are high in both reliability and measurement accuracy within the context of dimensional inspection, but they are limited in measuring speed, measurement point density and the measurement surface complexity and are unsuitable for on-machine measuring and automatic measuring since they have a large space requirement and high requirements in work environment, and many researches were conducted for the error compensation of CMMs. In general application case, CMMs are applied on the good operation environment without vibration and noise. To reduce the influences of vibration as an environmental factor on measurement accuracy, a reference part-based measurement model are developed and added into a CMM library and an auto-learning algorithm was proposed to reduce the required process time (Mohammadi et al., 2020, 2022).

As a contrast, non-contact measuring instruments like optical sensors generally have an adequate integration and a lower cost. A major drawback is low in measurement reliability and unknown in measurement uncertainties. They usually are dependent on the surface properties and measuring strategy. As a result, most of optical measuring instruments like white light interferometer, structured light, and focus variation can only be used for measurements orthogonal to the surface for roughness measurement or measurement of small distances (Seokbae et al., 2002). Fortunately, different from general non-contact sensors, laser sensors can achieve high speed and accuracy and have been widely applied in industry, such as online measurement on welding robot for seam tracing (Rout et al., 2022). A laser sensor-based instrument for additive manufacturing was developed, in which a measure path is planned to digitize the part to ensure the shape deviation (Bordron et al., 2019). Since laser sensors, integrated with a light source, are larger in size and unsuitable to measure small diameter hole, few of works are conducted on the laser measurement technology applied in internal surface measurements like holes. Recently, a great progress is made in laser profile sensors and a number of points can be sampled by laser sensor at a single time.

In this study, four holes and a spherical surface on the differential body need to be measured for quality evaluation. Due the complexity of the structure of workpiece and high requirement in measurement accuracy it has been measured by CMM. It is expensive in both cost and time. In addition, to improve measurement accuracy the spherical surface and the holes of the differential body need to be measured simultaneously. To address this problem, in our scheme, the spherical surface and the holes of the differential body are scanned and partially sampled by single laser sensor when the differential body, driven by a rotation platform, rotates for one revolution around its centerline. Although some surfaces of those holes are sampled by the laser senor the number of sampled points is much more than those of CMM. Theoretically, six points is enough to determine the centerline of the cylinder for a hole. Practically, the holes are created on a milling machine by cutting with high precision and the surface profiles of each individual hole formed by cutting edge of cutter are continuous. However, considering the effects of outliers and noise the number of sampled points also is very important to improve measurement accuracy.

Actually, a lot of work had been conducted in geometric construction domains (Zhang et al., 2022; Singh et al., 2020). The cell decomposition model was proposed to reconstruct arbitrary surfaces as irregular polygonal meshes from point cloud data (Li and Cripps, 2007). In the volume modeling algorithm, point cloud is enclosed as a volume. It is divided into several children or sub-volumes and then the reconstruction is conducted on each of them, respectively. In surface oriented methods, a mesh can be reconstructed from point cloud directly, three points of the point cloud are firstly selected to create a facet, and a new point is continuously selected and inserted into the facet by using Delaunay Triangulation algorithm until the reconstruction is completed (Dong et al., 1996). To compensate probe radius errors for the measurement of aero-engine blades, an error separation-based compensation approach was proposed by using genic algorithm (Zhengqing et al., 2020). The existing construction algorithms can be used to reconstruct arbitrary geometries from the measurement data but most of them are low in reliability and robust. Therefore special algorithm has to be studied in practice. Inspired by references (Bookstein, 1979; Ellis et al., 1992; Gander et al., 1994), in the study, the geometrical profiles and characteristics of surfaces are taken into account in reconstruction to improve the robust and fitting precision. In order to capture surface data, the workpiece has to be rotated and the errors resulted in the rotating process are unavoidable. In our scheme, a novel error compensation method is proposed to address this problem. Although error compensation technology had been studied for decades (Lee et al., 2022; Hu et al., 2022; Li et al., 2022), no universal approach is found. Therefore, specific technique has to been developed in engineering application.

Different from the traditional error compensation algorithm, in this paper, a special error compensation strategy is proposed. In which the workpiece is measured by two laser sensors simultaneously that are controlled by single controller, and error compensations are performed by using the relationship among the geometries of the sampled surfaces. One sensor is responsible for measurement and another is responsible for capturing data from the high precision geometries for calibration and composition purpose.

The main features of the proposed measurement instrument are that the primary sensor is mounted inside the differential body so that four-hole surfaces and the spherical surface can be “seen” by the senor when it is rotated. First the data sampled by the secondary sensor from the geometries with high manufacturing accuracy (a planar and a cylinder surfaces) are reconstructed, and at arbitrary rotation orientation the errors caused by the rotation platform can be computed as offsets of reconstruction geometry and the ideal geometry. Further, considering that the errors caused by the rotation platform are common for two sensors and the two sensors are sampling simultaneously the offsets can be applied to correct the corresponding sampled points of the primary sensor for compensation.

The present paper proceeds as follows. Section 2 describes the principle of measurement instrument. In section 3, the principle of error compensation of the measurement instrument is presented. In section 4, data processing approach is described in detail. The reconstruction of cylinder and ellipse are described in section 5. An experiment with case study is given in section 6, which is followed by conclusions in section 7.

2. Principle of measurement instrument

In this study, the geometric relations among the four holes and a spherical surface are concerned in the transmission performance of differential products. In the manufacturing stage, the four holes and the spherical surface of the differential body have to be measured for the quality evaluation. Traditionally, the task is conducted by contact measurement instrument (Coordinate Measurement Machine, simplified CMM), which is slow in measuring speed and expensive in cost, and only a small number of parts were randomly inspected. The traditional measurement process can be described as follows:

First, more than six points are sampled from the cylinder surface of a hole by using CMM. The centerline of the hole is estimated as a segment connected by centers of two circles, each of them is constructed by three measured points, respectively. Then the six points are sampled from the spherical surface that intersects with holes to estimate the center of the sphere. After that, the offsets of centerline of the holes are computed and compared with the required values of manufacturing specification for evaluation.

The major drawbacks of the existing approach include: (1) The data points sampled from the surfaces of both holes and sphere are not performed simultaneously so that errors will be resulted from the repositioning of the probe. (2) The acquisition of data is slow with point by point. Therefore, only a few points are sampled in practice, and it also results in low in measurement accuracy. As a contrast, non-contact sensors are high in data acquisition speed; nonetheless the acquisition quality remains lower and depends on the measuring strategy. Few works were conducted to have non-contact sensors to be applied in high accuracy measurement.

Considering that the characteristics of the differential body to be measured, as shown in Figure 1, in this paper, a novel approach is proposed to implement both measurement and error compensation. Note that the boss on the top of the differential body is used for assembly, and it is manufactured with high precision, which can be found from the technical drawing of differential bodies. In fact, the side and bottom of the boss also are used as datum surfaces in assembly process. Therefore, in this scheme, two laser profile sensors are selected and they are controlled by single controller. That is, two laser sensors can measure simultaneously. In proposed approach, one of the sensors is responsible for measurement of four holes and the spherical surface, and another is responsible for capturing data from the high-accuracy surfaces. The primary sensor is mounted in intern of the differential body so that four-hole surfaces and the spherical surface can be “seen” by it when the differential body rotates around the fixed axis. At the same time, the sampled data of boss surfaces from secondary sensor are used to reconstruct both the cylinder surface and the planar surface. Based on the reconstruction results, the errors generated from rotating process at arbitrary orientations can be estimated by the offsets of the reconstructed geometries and ideal geometries. After that, the offset value at each orientation is respectively added into the corresponding sampled data of the primary sensor for compensation.

As shown in Figure 1, a differential body driven by the rotation platform is rotating with a constant speed, and the data are sampled by the two laser sensors simultaneously. To get the required measurement accuracy, a 2D profile laser sensor, called high-accuracy 2D laser displacement sensor, are selected. Figure 2 shows the parameters of the laser sensor. The laser sensor can capture 800 points at a time. The measurement accuracies of the sensor are 1.0 micrometer in depth, 10 micrometer in width. The reference distance is 80 mm for setup, the motion range $\pm 23$ mm along Z and 25 mm–-39 mm along X (reference distance 32 mm), As shown in Figure 2(b).

3. Principle of error compensation

Although the laser sensors on the measurement instrument are high in precision and the rotation platform itself has high accuracy. In the measurement process, errors are mainly caused by the rotation platform. The errors are composed of the error of axial runout and the error of radial runout, and both of them should be compensated to improve accuracy. In this case, the boss surfaces have high manufacturing accuracy and they can be used as datum to estimate the offset errors of the rotation platform. Further, the offsets can be used for error compensation.

3.1 Compensation of axial error

Note that the axial runout error of the workpiece caused by the rotation process varies at measurement points due to the motion of the rotation platform. As a result, the points sampled from the bottom surface of the boss are not coplanar. To compensate the axial error of runout in the rotation process of the rotation platform, the idea is to find out the ideal planar surface by using total sampled points of the bottom surface of the boss by using planar fitting method as the first step. Then, a straight line is fitted by using the sampled data that are obtained at one time sample when the rotation platform rotates to a certain direction. The distance between the line and the planar surface can be used as a measurement of the axial error at the direction. The implementation of the algorithm can be described as follows:

To find the plane that fits best those sampled points by minimizing the sum of the distances (perpendicular to the plane) between the plane and the points. First, the algorithm of least squares regression is applied to compute the equation of a plane that minimizing the sum of the vertical distances between the points and the plane. Assume that the equation of the plane is represented as

(1)

z = a x + b y + c

where a, b and c represent the constant coefficients of the plane equation. They are determined by solving equation (1), which means that(2)

S = \sum_{i = 1}^{n} {(a x_{i} + b y_{i} + c - z_{i})}^{2}

Solving for a, b and c

Considering that equation (2) represents a linear system, the derivative appears linearly, i.e. ∂S/∂a = 0, ∂S/∂b = 0, and ∂S/∂c = 0. The matrix equation for a, b and c is formulated as(3) ${\begin{cases} 2 \sum_{i = 1}^{n} (a x_{i} + b y_{i} + c - z_{i}) x_{i} = 0 \\ 2 \sum_{i = 1}^{n} (a x_{i} + b y_{i} + c - z_{i}) y_{i} = 0 \\ 2 \sum_{i = 1}^{n} (a x_{i} + b y_{i} + c - z_{i}) = 0 \end{cases}$

Let $X = \sum_{i = 1}^{n} x_{i}; X^{2} = \sum_{i = 1}^{n} x_{i}^{2}; X Y = \sum_{i = 1}^{n} x_{i} y_{i}$ . Equation (3) can be further represented by transformation matrix as(4) $[\begin{array}{l} a \\ b \\ c \end{array}] = {[\begin{array}{c} X^{2} & X Y & X \\ X Y & Y^{2} & Y \\ X & Y & n \end{array}]}^{- 1} [\begin{array}{c} X Z \\ Y Z \\ Z \end{array}]$

To compute the straight line that fits best those sampled points. The sampled points are obtained by the secondary sensor at a single sampling time when the rotation platform stops at a special position. Assume that a straight line can be represented as $z = d x + e y - f$ , its constant coefficients d, e and f can be respectively determined by minimizing the following equation. $F (d, e, f) = {\sum_{i - 1}^{n} (z_{i} - d x_{i} - e y_{i} - f)}^{2} .$ Since the system ∂F/∂d = 0, ∂F/∂e = 0 and ∂F/∂f = 0 is a linear system of equations, the solution of the equation can be computed and described by matrices.
Determination of axial error of runout of the rotation platform. The axial error of runout of the rotation platform at sampling positions are determined by computing the distance between the fitted line and the fitted plane.
Composition of axial errors of runout. To composite the axial error caused by the rotation platform, the sampled points of the primary sensor at the corresponding sampling moment should be extracted for process. Assume that the 2D coordinate (x, y) of the points are substituted into the fitted straight line equation as

(5)

z = d x + e y - f

where d, e and f denotes the constant coefficients, z denotes the offset. After z is gotten, the distance of the point (x, y, z) from the fitted plane can be recalculated as(6)

d i s t = a x + b y + c - z

3.2 Compensations of radial offset and measuring angle error

Although radial measurement error is generated by the eccentric center of rotating axis of the workpiece, which could also be resulted from both the random disturbing factor and the manufacturing stage of the rotation platform. All of them would cause a great error in measurement result. For the compensation purpose, the external cylinder surface of the boss on the top of the differential body is captured by secondary sensor. The idea is that the cross section of the external cylinder surface is a circle section in theory, but it would have become into a ellipse section due to the eccentric error. The error composition is to estimate the offsets so as to restore the circle section from the ellipse section. Therefore, the ellipse profile should be constructed from the sampled points so that the eccentric distance can be computed. Then the offset along each sampling orientation of the workpiece can be solved and applied to the sampled position of the primary sensor for composition. On the other hand, the measuring angle inclination of the secondary sensor has a significant impact on the radial offset error. To calculate the inclination angle from the sampled points of the cylinder surface exactly, a laser scanning points-based cylinder fitting algorithm is studied. The cylinder is fitted by using the regression algorithm, which will be discussed in the following sections.

4. Data processing approach

To measure the inner cylinder surfaces of the four holes of the differential body, the light axis of the primary sensor has to be setup unparalleled with the centerline of the holes. In this case, part regions of both cylinder surface and the spherical surface can be covered when the differential body rotates around the centerline of the rotation platform.

4.1 Extraction of object to be measured from sampled points

To recognize the cylinder surface of each hole from the sampled data of primary sensor, in this study, an image-based fast segmentation method is proposed. Noting that the sampled data from laser sensor have only a depth value, and some of them are invalid due to too near or too far away from the surfaces to be measured, for example, the sampled data will be invalid when the light axis just is through the holes at some time period. These invalid sampled data should be removed before data processing. In addition, by analyzing all sampled data of a sensor as an image, we find that the measured values on the regions of the holes and sphere are completely different and their boundaries can be segmented by image gradient. As shown in Figure 3, the vertical axis is the light axis direction of the sensor and the horizontal axis is the rotating direction of the workpiece. When a threshold value is set to the gradient image, the result is shown in Figure 3(b), where the yellow pixels have big gradient. They can be used to distinguish between the areas of the holes and sphere. Although size of the threshold may be varies for different measurement environments, it is easy to be determined for a special case. In this implementation, the gradients are approximately calculated by the differences between adjacent points. The image of Figure 3(b) is created by the threshold 0.05, and it can be clearly seen that the sampled data of the four holes and the partial spherical surface have been completely distinguished out. Known from the geometry of the workpiece, as shown in Figure 3(a), the ellipse-shape regions in Figure 3(b) are belong to both the cylinder surfaces and the spherical surface, and the spherical surface region only appears at the left hand side of the region while the ellipse arc at the right hand side represent the intersection of the cylinder and the sphere. The reason is that due to the geometry structure composed of the cylinder surface and spherical surface the left side regions of them haven't been covered by the primary sensor. Based the symmetry the cylinder surface, however, the region can be easily recognized and localized in the left side region by the ellipse arc at the right side. In addition, noting that the position and orientation of the laser sensor, any vertical straight line of Figure 3(b) represents the intersection of a laser scanning plane and the surface to be measured.

4.2 Surface reconstruction

After the sampled points from the cylinder surfaces and the spherical surface are distinguished the reconstruction can be conducted by using fitting algorithm. Mathematically, the reconstruction is to calculate the intersection of a known surface and the plane consisted of laser scanning lines. The reconstruction procedure can be described as follows.

First, the scanned sampled points from cylinder surfaces and spherical surface are extracted from the image by using contour tracing algorithm. Then, the scanned profile points of the surfaces in each sampling period can be recognized as the pixels that are localized from top to bottom within the white color region. Known in mathematics, the intersection of a plane and a cylinder is ellipse, that is, the scanned profile points should be reconstructed into an ellipse. Considering that the spherical surface region distributes on the enter ends of the holes, the scanned region of the cylinder surface is excluded from the region contained in ellipse arc for reconstruction. Figure 4 shows the reconstructed ellipse and the spherical surface of the workpiece, where the red points and line segment represents the centerline of the hole.

5. Reconstruction of cylinder and ellipse surfaces

In this study, the laser sensor can capture a group of points on a straight line that is called light axis of the sensor. The points are captured based on the triangle measurement theory. Figure 5 shows the sampling process of the primary laser sensor, which is equivalent to find out the intersection of a group of parallel line and the surface to be measured. In the context of this part, only the cylinder surface and spherical surface are concerned, and the captured points at a time by the sensor are the intersections of the parallel lines and the surface.

It should be pointed out that the parallel light beams of the laser sensor are not parallel with the centerline of the hole so that an ellipse arc will be obtained. The parameters of the plane can be determined according to the position and orientation of the sensor, and the intersecting points can be calculated by the position of the hole mathematically. Against the reconstruction is to determine the ellipse arc and center line of hole. The centerline of the hole can be determined by using the center coordinates of several ellipse arcs. Considering that the effect of noise, we have to sample many ellipse arcs to improve measurement accuracy. Based on the same way, the spherical surface can be sampled and reconstructed. Since the intersection of a plane and a sphere is a circle section and the circle can be classified as ellipse that is the same in major and minor axes. Therefore only ellipse fitting method is required in this case.

5.1 Fitting of ellipses

In this study, we have known that the points are sampled by sensor from the geometry with ellipse profile, which can be applied as a constraint for reconstruction. If the constraint did not be considered some non-ellipse curves might be obtained after reconstruction due to noise. Therefore, an improved method is proposed to address this difficulty. The corresponding formula will be derived in detail.

A general conic can be formulated by (Lee et al., 2022; Hu et al., 2022; Li et al., 2022)(7) $f (X) = m_{1} x^{2} + m_{2} x y + m_{3} y^{2} + m_{4} x + m_{5} y + m_{6}$ where $X = {[x^{2}, x y, y^{2}, x, y, 1]}^{T}$ and $m_{1}, m_{2}, m_{3}, m_{4}, m_{5}, m_{6}$ denote constant coefficients, which can be solved by minimizing the following equation:(8) $S (m_{1}, m_{2}, m_{3}, m_{4}, m_{5}, m_{6}) = \min \sum_{i = 1}^{n} f^{2} (X_{i})$

If equation (7) represents an ellipse curve it can be alternatively rewritten as(9) $f (x, y, a, b, A, B, C) = A {(x - a)}^{2} + 2 B (x - a) (y - b) + C {(y - b)}^{2} - 1$ where $x, y, a, b, A, B, C$ denote constant variables.

As known all, if equation (9) describes an ellipse curve, the restraint condition $B^{2} < 4 A C$ should be met. Considering the implementation of the algorithm, a new expression of ${A, B, C}$ is defined by variables ${p, q, r}$ (10) $A = p^{2}, B = 2 p q, a n d C = q^{2} + r^{2}$

Further, equation (10) can be solved by minimizing the following equation(11) $ε = \frac{1}{4} \sum_{i = 1}^{n} \frac{{[p^{2} {(x_{i} - a)}^{2} + 2 p q (x_{i} - a) (y_{i} - b) + (q^{2} + r^{2}) {(y_{i} - b)}^{2} - 1]}^{2}}{{{[p^{2} {(x_{i} - a)}^{2} + p q (y_{i} - b)]}^{2} + [p q (x_{i} - a) + (q^{2} + r^{2}) (y_{i} - b)]}^{2}}$ where $p, q, r$ denote variables, and the initial values for parameters $a, b, A, B, C$ are defined as(12) $a = \frac{1}{N} \sum_{i = 1}^{N} x_{i}, b = \frac{1}{N} \sum_{i = 1}^{N} y_{i}, a n d [\begin{array}{c} A & B \\ B & C \end{array}] = \frac{1}{(u w - v^{2})} [\begin{array}{c} u & v \\ v & w \end{array}]$ where(13) ${\begin{array}{c} u = \frac{1}{N} \sum_{i = 1}^{N} {(x_{i} - a)}^{2} \\ v = \frac{1}{N} \sum_{i = 1}^{N} (x_{i} - a) (y_{i} - b) \\ w = \frac{1}{N} \sum_{i = 1}^{N} {(y_{i} - b)}^{2} \end{array}$

Practically, the ranges of x and y values are dependent on measurable area of the laser sensor, which can be determined in advance.

5.2 Reconstruction of cylinder

Known from previous sections, reconstruction of the external cylinder surface is important for error compensation. The measurement principle of secondary sensor is shown in Figure 6, where the surfaces of the boss will be reconstructed as datum surface for error compensation.

The three-dimensional coordinates of sampled points from secondary sensor can be determined by(14) ${\begin{cases} a = Δ \cos θ, b = Δ \sin θ \\ x = [R + (i - 300) a - d \sin θ] \cos [(j - 1) \cdot 1 . 5^{\circ}] \\ y = [R + (i - 300) a - d \sin θ] \sin [(j - 1) \cdot 1 . 5^{\circ}] \\ z = H - (i - 300) b - d \cos θ \\ d_{l e v e l} = d \sin θ - (i - 300) a \\ d_{v e r t i c a l} = d \cos θ + (i - 300) b \end{cases}$ where θ is the tilt angle of the sensor, ∆ denotes the distance among adjacent rays, i the number of rays arranged in the laser sensor, j the sequence number of sampling process and the sampling period is 1.5 ms, or 240 times per revolution. R denotes the distance from the center of the sensor to the center of the rotation, H height of the sensor that is gotten by calibration, x, y, z is Cartesian coordinates of a sampled point, $d_{l e v e l}$ , $d_{v e r t i c a l}$ are the horizontal and vertical components of the distance from the sensor to the sampled point, respectively.

Since the boss is manufactured with high quality this measurement data contains less noise the external cylinder profile of the boss sampled by the secondary sensor can be reconstructed exactly. The improved algorithm is described as follows.

Step 1. By using the sampled points of the planar surface from the bottom of the boss, a least square algorithm based line fitting method is used to estimate its normal vector. Assume that the straight line is represented as $z = d x + e y - f$ , where constant coefficients d, e and f are determined by minimizing equation. $F (d, e, f) = {\sum_{i = 1}^{n} (z_{i} - d x_{i} - e y_{i} - f)}^{2} .$ The solution can be found with matrices since the system ∂F/∂d = 0, ∂F/∂e = 0 and ∂F/∂f = 0 is a linear equations. Finally coefficients $d, e, f$ are normalized as unit normal vector.
Step 2. Based on the unit normal vector of the cylinder profiles obtained in Step 1, K-mean algorithm is applied to determine the common unit normal vector. First clusters K is set to 3 to define the initial centroids so that some noisy points can be grouped to different clusters to have the data points with high precision in a clusters. After initialization, the centroids are initialized by first shuffling the dataset and then randomly selecting K data points for the centroids without replacement, and iterating until there is no change to the centroids. Further, the sum of the squared distance between data points and all centroids is computed, and each data point to the closest cluster is assigned. Finally, the centroids for the clusters by taking the average of the all data points that belong to each cluster are computed.
Step 3. Different from the traditional methods, only the cluster with most data points is selected, and their average value is computed as the common unit normal vector. As shown in Figure 6, the normal vector belongs to one of the meridian lines of the cylinder. Although the orientation of the vector maybe is not parallel to the centerline of the cylinder the normal vector of the plane on the bottom of Figure 6 can be applied to compute the relative tilt of the vector.

5.3 Determination of the corresponding points of sensors

Since a fixture is designed to setup the work piece in measuring process the error generated due to the tilt of the work piece will be very small, which can be skipped out. However, the errors resulted from the axial and radial run outs of the rotation platform needs to be estimated for compensation. In addition, it should be pointed out that it is almost impossible for the two sensors to be setup with their laser scanning path collinear, but the corresponding relationship between the data points from primary sensor and those of the secondary sensor can be determined. It also is very important for error compensation. After the offset error is found the corresponding points of the sampled data on the primary sensor need to be extracted to perform the radial error compensation. The geometric corresponding relationship between the sampled points of the two sensors is illustrated in Figure 7, where the angle between the scanning vector of the primary sensor and the normal vector of the boss is known, which is determined in subsection 5.2. Let $P_{i}$ represent a point on the surface sampled by the primary sensor. According to the principle of scanning line, its corresponding point $P_{i}^{'}$ is sampled by the secondary. The radial error of ${P_{i}}^{'}$ can be calculated by mapping from ${P_{i}}^{'}$ to $P_{i}$ . Mathematically, the relationship can be represented by using transformation matrices.

6. Experiments

In this experiment, a work piece (differential body) is to be measured. There are four through holes with cylinder surface on the waist of the work piece and a concave spherical surface in the center of the work piece. Both the bosses on the top and the bottom of the work price are designed as mating surface for assembly. In measurement process, the work piece is put on a fixture, as shown in Figure 8.

6.1 Data acquisition

Considering that the manufacturing requirement of the workpiece, the data process is not performed in site so serial communication interface is used to transmit the sampled data from laser sensor to computer. To transmit the sampling data, the serial port with RS 232 of controller of the laser sensor is used. The parameters of serial port and the data transmitting prototype are given as follows:

Serial Port: COM3
Prototype: I2C
Transmitting rate (bits per second): 115,200
Transmitting bits: 9 bits (8 data bit with a stop bit)

Two laser sensors are controlled by single controller, a laser segment with 800 data is captured at a time by a sensor, and the period of the sampling and transmitting is about 16 milliseconds. Figure 9 shows the sampling data of the sensors that is received from a serial port (COM3).

6.2 Errors caused by noise

In order to measure the effect of noise on the roughness of the part surface and light reflection of material on the measurement accuracy, a vernier caliper and a compact disc read-only memory are selected and tested. As shown in Figure 10, the marked rectangle regions are captured by the laser sensor. The results are shown in Figure 10(b) and (c), where the sampled data are illustrated on the right and the fitting curve is shown on the left.

It can be clearly seen from the measured results that the CD compact disc has strongest noises and it generates errors nearly 10 μm. The reason is that the reflection lights from the external environment makes a great influence on the measured results. Also it depends on the material and surface smoothness of the object to be measured. Therefore, a wall maybe should be designed to prevent external lights during measuring.

6.3 Errors caused by vibration

As has been known, the vibration error from the rotation platform is one of error sources, which will cause the radial and axial errors in measurement results. Although the magnitudes of the errors have been controlled within a limit range by manufacturer, it still is necessary to be estimated and compensated for high measurement accuracy.

To estimate the errors caused by the vibration of the rotation platform both the radial and axial errors of the rotation platform have to be respectively measured and calibrated.

As shown in Figure 11, the rotating axis of the rotation platform is set as the origin of the coordinate frame and the ray direction of the laser on the intern of the workpiece is defined as X direction. By calibration, the radial error of rotation platform can be obtained and formulated as(15) $e r r o r = (- 0.8 * Δ x) + (0.6 * Δ y) + (- 1.57 * Δ θ)$

Since the top planar surface of the workpiece is high in manufacturing accuracy it is measured by the secondary sensor to estimate the range of the axial error of the rotation platform.

To estimate the axial error caused by the vibration of the rotation platform, several measurements are conducted, the results are shown in Figure 11. In Figure 11(a), the vertical axis is coincidence with the centerline of top surface of the workpiece and the horizontal axis represents the sampled time. It can be seen that the errors varies for each single measurement. Figure 11(b) shows the corresponding offset of the centerline of holes after reconstruction, where the laser sensor is installed in the inner of the workpiece. It can be concluded that the measurement error of primary sensor caused by the axial vibration of the rotation platform can be measured by calibration for compensation.

6.4 Measurement device and experiment

The objective is to reconstruct the four holes and the concave spherical surface of the differential body for quality evaluation. In this measurement device, as shown in Figure 12, the measurement data are sampled by the primary sensor. In addition, before the measurement begins the primary sensor will be automatically moved down to work position, where the four holes and the concave spherical surface can be “seen” partially. To measure the cylinder surface of the boss and its bottom plane on the top of the work piece for improving measurement accuracy, the secondary sensor is fixed with constant orientation above the workpiece. Two sensors are controlled by a controller, which means their data samplings are performed simultaneously. Before starting to use both sensors are calibrated by using a standard sphere with precision 1 nanometer.

In this experiment, the workpiece is driven by the rotation platform with a constant speed, i.e. 1.0°/second. The sampling period of the two sensors is 1500 millisecond. The rotation platform is approximately rotated 1.5° between adjacent samplings, and 800 points are obtained at a time by sensors. The quality evaluation of the work piece is to check the offsets of the spherical center from the centerlines of holes according to the manufacturing requirement. Based on the part drawing, diameter of the sphere is 180 mm with tolerance (0, +0.10), the diameters of the four holes are the same, i.e. 36 mm with tolerance (−0.022, −0.048), and the maximum allowed offset from centerlines of the holes to center of the sphere is 0.80.

For verifying the measurement accuracy of the proposed measurement instrument and the comparing purpose, the work piece is firstly measured by a contact measurement instrument (coordinate measurement machine, CMM). The results show that diameter of sphere is 183.746 millimeter, the distances from the center of the sphere to centerlines of the four holes are 0.044 millimeter, 0.043 millimeter, 0.042 millimeter and 0.067 millimeter, respectively, and the maximal offset distance among centerlines of the four holes is 0.023 millimeter. According to the design requirements, the manufacturing quality of the differential body is qualified. After that the differential body is measured by proposed measurement instrument. To check the robust of the measuring approach, multiple measurements are conducted repeat on this workpiece, and the reconstruction results are given in Figure 13. It can be seen that the sampling start positions on the work piece are different with small offset but the relative positions among the four holes, i.e. the ellipse regions in Figure 13(a), are constant. To reconstruct the centerline of a hole, the intersection points of the laser scanning lines and the cylinder surface of the hole are first extracted by using boundary trace algorithm. Then, reconstruction is conducted for the hole and the spherical surface region that is hit by laser scanning lines. Quality evaluation can be performed by the distance between the centerline of a hole and the center of the spherical surface.

The error compensation is performed to correct the data sampled by primary sensor by the eccentric distance of workpiece and distance of axial run out of the rotation platform. Both of them are computed by using the sampled data of the secondary sensor.

To demonstrate the efficiency of the proposed compensation method, the spherical diameter is fitted by the sampled spherical data near the holes on two measurements, respectively. Therefore the sphere surface is fitted 8 times in total. Eight reconstruction results of the spherical diameter are obtained and given in Figure 14. As a contrast, the measurement result of the spherical diameter from CMM is illustrated as straight line. The reconstruction results with and without compensations are shown in Figure 14.

It can be seen that the reconstruction results of the spherical diameter using compensated data are closer to that of CMM.

7. Conclusions

The measurement of work piece plays an important role in product quality evaluation. In this paper, a multiply laser sensors-based measurement approach combining measuring with compensating together is studied, and a measurement instrument for differential body measurement is developed. Different from the existing approaches, the characteristics of the workpiece is considered for error compensation purpose. At the same time, multiple features of the workpiece can be measured simultaneously at high speed by laser sensors. To reduce the effect of noisy sampled data on fitting precision, an improved conic fitting algorithm is introduced to make the sampled points fit into ellipse curve. Although a small region of the feature surface is sampled by sensor whole surface can be completely reconstructed. The reason is due to that the surface of the feature is completely continuous for real workpiece. As a result, the valid of proposed instrument and data processing algorithm has been demonstrated by experimental result. The major disadvantage is that the lengths of the holes cannot be determined by the partial sampled data of surface though it's irrelevant to the quality evaluation of the workpiece, and it can be solved by increasing the number of sensors. In addition, on-machine measurement is very important in this case. Since the workpiece is machined in an automatic production system, except for errors caused by the rotation platform, other factors like the vibration of measurement instrument, the oil contamination and residual chip of the work piece will result in errors. In current study, they have not been taken into consideration completely, yet. Further work still need to be conducted.

This project is supported by National Natural Science Foundation of China (Grant No. 51875445).

Figure 1

Proposed measurement scheme with laser sensors

[Figure omitted. See PDF]

Figure 2

Laser sensor and its measurement constraints

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Figure 3

Gradient image generated from the sampled points

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Figure 4

A hole and spherical surface reconstructed from sampled points

[Figure omitted. See PDF]

Figure 5

Laser sampling principle of a hole profile

[Figure omitted. See PDF]

Figure 6

Measurement principle of secondary sensor

[Figure omitted. See PDF]

Figure 7

The mapping relationships among sampled points of the sensors

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Figure 8

Proposed measurement device and the differential body to be measured

[Figure omitted. See PDF]

Figure 9

The sampling data received from a serial port

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Figure 10.

Test parts for the effect of noises on measurement accuracy (a) the measured areas of vernier caliper and a CD compact disc (b) the fitting result (Left) from the sampled data (Right) of the vernier caliper (c) the fitting result (Left) from the sampled data (Right) of the compact disc

[Figure omitted. See PDF]

Figure 11

Offsets of centerline of the four holes on sampling data: (a) profile sampling of holes (b) the obtained centerline of holes from multiply samplings

[Figure omitted. See PDF]

Figure 12

Two sensors are calibrated by using a high precision sphere

[Figure omitted. See PDF]

Figure 13

Experimental results: (a) gradient difference from data sampled by primary sensor; (b) construction of surfaces sampled by secondary sensor; (c) a spherical surface constructed from the data near one of the four hole; (d) construction of surfaces sampled by primary sensor

[Figure omitted. See PDF]

Figure 14

Diameter of the sphere obtained with and without compensation

[Figure omitted. See PDF]

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Geometry error compensation using multiply laser sensors: a case study

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