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A robust and efficient calibration method for spot laser probe on CMM
Journal Pre-proofs A Robust and Efficient Calibration Method for Spot Laser Probe on CMM Yijun Shen, Xu Zhang, Zhenyou Wang, Jianxiang Wang, Limin Zhu PII: DOI: Reference:
S0263-2241(20)30060-9 https://doi.org/10.1016/j.measurement.2020.107523 MEASUR 107523
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
25 October 2019 11 January 2020 15 January 2020
Please cite this article as: Y. Shen, X. Zhang, Z. Wang, J. Wang, L. Zhu, A Robust and Efficient Calibration Method for Spot Laser Probe on CMM, Measurement (2020), doi: https://doi.org/10.1016/j.measurement.2020.107523
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A Robust and Efficient Calibration Method for Spot Laser Probe on CMM Yijun Shen1, Xu Zhang2*, Zhenyou Wang2, Jianxiang Wang2, Limin Zhu1,3 1 School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China 2 Department of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200444, China 3 Shanghai Key Lab of Advanced Manufacturing Environment, Shanghai 200240, China * Author to whom any correspondence should be addressed.
E-mail address: [email protected]
Abstract: Among all the non-contact measurement devices, the spot laser probe achieves a good balance between accuracy and efficiency. When the laser probe is applied to three-dimensional (3-D) measurement，the laser beam direction is essential to transform the probe indication from 1-D distance information into a 3-D vector representing differences in space coordinates. In this article, a calibration method is proposed for the laser probe mounted on the coordinate measuring machine (CMM). During the calibration procedure, the CMM moves along the user-planned path and hundreds of points are scanned in a few seconds. In order to get an accurate laser beam direction estimate, we refine the original inaccurate direction, which is specified by the mechanical structure of the probe fixture, by mapping the scanned points into a sphere. The simulation and experiment manifest that the proposed method is superior to the existing methods and meets the requirement of the micron-level measurement. Keywords: laser probe calibration, non-contact measurement, coordinate measuring machine, sensitivity analysis.
1. Introduction With the rapid development of aerospace and aviation, parts with free-form surface such as turbine blades have been widely used in these industries. The quality control and the reverse engineering during designing and manufacturing require a large quantity of high-quality point cloud data collected from the surface of these parts. Therefore, there is an urgent need of precise and efficient measuring methods for parts with complicated geometry [1]. In recent years, non-contact measurement, especially optical measurement, has been rapidly developing [2]. Compared to the traditional coordinate measuring machine (CMM) with contact probes, non-contact measurement has advantages in its 1
high-speed and non-abrasion. Also, there is no need to compensate the tip radius and to plan a complicated scanning path when using optical probes [3,4]. Both structured light method and laser triangulation method are mature technologies and can scan the object surface efficiently. However, both of them can only reach an accuracy of nearly 10 micrometers which may not meet the requirement of precise measurement [5,6]. The spot laser probe, which reaches sub-micron level accuracy, is designed to be mounted on the CMM to perform accurate and efficient non-contact scanning. Although the spot laser probe cannot scan the surface as quickly as the structured light and laser triangulation probes, the spot laser probe is far more accurate. For those non-contact measurement tasks which require micrometer-level accuracy, the spot laser probe is the best choice. The spot laser probe is an optical displacement sensor which returns the one dimensional (1D) distance information between the probe and the object surface. The direction of the laser beam is necessary to transform the 1D length to 3D coordinates of the measuring point. Considering that the laser length is usually several millimeters, the deviation of a single measuring point can reach several micrometers when the angle error of the calibrated laser beam direction reaches 103 rad. Thus, in the measurement requiring micron-level accuracy, the direction of the laser beam must be precisely calibrated. Many researchers have been working on the calibration of the laser beam direction of the spot laser probe. The existing laser beam direction calibration methods can be divided into two groups. Some researchers solve the laser direction using a geometric constraint such as the plane, sphere and cone constraint. Chen et al. [7] used a plane artefact to calibrate the laser beam direction. The position and pose of this plane should be calculated by fitting point cloud data measured by laser trackers. The operation costs much time and the accuracy of the laser tracker is limited. Smith et al. [8] proposed a method using a polyhedron artifact. Compared to the method using the single plane, this method needs no position information of the plane. Zhou et al. [9] determined the laser beam direction with the help of a cone artifact. The geometric constraint that the cone angle remains unchanged at different height of the cone is used to solve the laser beam direction. Bi et al. [10] used the sphere constraint to solve the equation set. In his method, CMM needs to move the same distance along XYZ axis for several times with high accuracy. All above methods employ geometric constraints to solve the equation set containing the unknown parameters of the laser beam direction. The main disadvantage of these methods is that the CMM positioning error and laser probe indication error may cause an adverse effect on the final results. Other researchers calibrate the laser beam direction by moving the probe along the laser beam direction to several positions and fitting these positions into a line. The line 2
direction is equivalent to the laser beam direction. The key of such methods is to make sure the probe moves exactly along the laser beam direction. Xie et al. [11] proposed the idea of “equivalent probe”, in which the optical probe is regarded equivalent to the contact probe. In the calibration procedure, the user inspects the sphere by the probe with a series of laser beam lengths. However, the accurate length of the laser beam requisite in this method is difficult to obtain in the experiment. Zhu et al. [12] proposed a calibration method to determine the laser sensor mounted on the end of the robot arm. The robot is controlled to reach certain positions to ensure that the laser spot is projected at the same point on the screen. The point is checked by a camera to verify if it maintains unchanged. The accuracy of this method is limited by the accuracy of the camera. The principle of the above calibration methods is the best-fitting algorithm, which is more stable than methods solving the equation sets. However, keeping the probe moving along the laser beam direction is very difficult to realize in the real experiment because of the existence of noise. As discussed above, the previous methods either require precise position information of the artefacts or the precise movement of the measuring machine. Thus, in terms of robustness, the CMM positioning error and the laser probe indication error may cause great deviation in the final results. In addition, in order to ensure the accuracy of sampling, the calibration is usually implemented by point-by-point inspection and the motion of CMM should be set at a low speed in these methods [10]. In other words, the previous calibration methods are lack of efficiency. In this paper, a robust and efficient calibration method for the spot laser probe is proposed. In the first stage, the initial laser beam direction is estimated by the mechanical structure of the probe fixture and the sphere center is estimated by manual inspection. In the second stage, a specially designed scanning path is planned according to the initial laser beam direction and the sphere center. Then hundreds of points are scanned along the path. Finally, the initial laser beam direction is refined by mapping the scanned points into a sphere. Compared to the existing methods, the proposed method has the following advantages: In terms of robustness, an initial inaccurate direction specified by the probe fixture is refined by mapping a large number of sample points into a sphere so that the final result is robust to measuring noise and reaches micron-level accuracy. In terms of efficiency, scanning inspection rather than point-by-point inspection is applied in the calibration procedure so that the proposed method is much faster than the existing methods. 3
Additionally, the input of the proposed algorithm is the measuring points on the sphere surface. Thus the proposed method can be applied not only in the calibration of the spot laser probe but also in the calibration of other similar displacement sensors such as ultrasonic displacement sensors. The rest of this paper is organized as follows: Section 2 describes the laser probe calibration model and algorithm; Section 3 analyzes the robustness of the proposed method using the sensitivity analysis; Section 4 analyzes the important factors in this calibration procedure and compares the proposed method with other existing methods; Section 5 presents the calibration experiments and verifies the result; Section 6 concludes the main opinions in this paper.
2. Laser probe Calibration model 2.1 Coordinate system The non-contact measurement system is a CMM with a laser probe mounted on the end of its Z axis. A reference sphere with known radius is fixed on the CMM operating platform which is used as the calibration artifact. The following coordinate systems are set up for the convenience of explaining the calibration algorithm. 2.1.1
Machine coordinate system The machine coordinate system is denoted as OM X M YM Z M . The origin of the
machine coordinate system is the zero point of all three scales of the CMM. The direction of X, Y and Z axis is parallel to the corresponding axis of the CMM. The machine coordinate system is a standard Cartesian coordinates. 2.1.2
Tool coordinate system The tool coordinate system is denoted as OT X T YT Z T . The origin of the tool
coordinate system is set at the zero point of the laser probe, i.e. the point where the laser probe indication is zero. The X, Y and Z axes are all parallel to the machine coordinate system. 2.1.3
Work piece coordinate system The work piece coordinate system is denoted as OW X W YW Z W . The origin of the
work piece coordinate system is set at the center of the reference sphere. The X, Y and Z axes are all parallel to the machine coordinate system. The set-up of the three coordinate systems is illustrated in Fig. 1. 4
ZM XM
YM
OM
ZT XT
OT
YT XW
ZW OW
YW
Fig. 1. The set-up of the three coordinate systems.
2.2 Term definitions In convenience of introducing the proposed method, some frequently used terms in this article are defined in this section. A general non-contact measurement procedure contains two steps. Firstly, in the planning stage, several sample points are planned on the object surface by the user and the moving trajectory of the CMM is calculated according to these sample points. Secondly, in the measuring stage, the CMM moves along the pre-designed trajectory and the optical probe collects data simultaneously. Fig. 2 shows a specific procedure of the non-contact measurement with a spot laser probe. Firstly, path points on the sphere Ps , which are planned by the user, are regarded as the sample points for measurement. The path points of the CMM Pc can be calculated by offsetting Ps by the planned measuring length l p along the inversed estimated laser beam direction ve . Secondly, CMM is supposed to move to Pc but reaches the actual points of the CMM Pca due to the positioning error. The laser probe emits laser along the actual laser beam direction va and forms a light spot on the object surface which is referred as the measured points Pp . The indication of the laser probe is the actual measuring length la , i.e. the distance between the probe and the object surface. The position of the probe can be obtained by the encoders of the CMM as actual points of the CMM
Pca . These terms will be repeatedly used in the following parts. Thus
corresponding symbols are defined for brief expression as shown in Table 1. It should be emphasized that all the above mentioned variables are expressed in the machine coordinate system OM X M YM Z M . Actually la and va are recorded in the 5
tool coordinate system OT X T YT Z T . However, the conversion relationship between any two coordinate systems is just translation. Therefore, the laser beam direction and the laser length is completely same in these coordinate systems. Besides, Ps is usually planned in the work piece coordinate system OW X W YW Z W as Psw . Then Psw is transformed into the machine coordinate system with a translation vector as Ps =Psw m Tw where
m
Tw is
the translation vector between the work piece coordinate system and the machine coordinate system, i.e. the coordinate of the sphere center in the machine coordinate system in this case. Z axis of CMM
Spot laser probe
Pca Laser beam
va la
Pc ve
Pp
lp
Ps
Fig. 2. A diagram of the measurement with spot laser probe Table 1 Symbols of the terms
Term definition
Symbols
Path points on the sphere
Ps
Path points of the CMM
Pc
Actual points of the CMM
Pca
Measured points
Pp
Planned measuring length
lp
Actual measuring length
la
Estimated laser beam direction
ve
Actual laser beam direction
va 6
2.3 Calibration model The proposed method includes two steps to calibrate the laser beam direction. A standard sphere artefact with known radius R is used in the calibration method. In the first step, an initial inaccurate laser beam direction can be obtained by the mechanical structure of the probe fixture as ve . This direction is not accurate enough for precision measurement but accurate enough as an initial value for the proposed optimization algorithm, which will be proved by simulation in section 4.1.3. N c points denoted by
P pr are inspected manually by users. The positions of these points can be calculated by N c CMM positions Pcar and corresponding laser beam length la as Ppr =Pcar v e la .
The coordinate of the center of the reference sphere Psce can be obtained by fitting P pr into a sphere. The algorithm for sphere fitting is so mature that it won't be discussed in detail here [13,14]. To prevent the influence of the measuring noise, more than 5 points should be measured. The coordinate of the sphere center will be used as the initial value of the optimization. In the second step, the initial inaccurate direction of the laser beam ve will be refined. With the information of estimated sphere center Psce and the laser beam direction
ve , the scanning region on the spherical surface can be determined according to the measurement angle limit of the laser probe. Measurement angle limit refers to the maximum angle between the laser beam of the probe and the normal vector of the surface. Within this region, hundreds of measuring points can be planned according to any rules. For example, in this proposed algorithm, the points are planned on a series of circles on the sphere center as shown in Fig. 3. The radius of the largest circle is R sin where is the measurement angle limit of the laser probe. The path points of the CMM Pc can be calculated by offsetting the path points on the sphere Ps by planned measuring length l p along the inversed estimated laser beam direction ve as Pc =Ps v e l p .
7
Sphere surface
Laser beam
Path points on the sphere
(a) Distribution of the path points on the sphere
(b) A view from the direction of the laser beam
Fig. 3. The path points planned on the sphere surface. (a) Distribution of the path points on the sphere. (b) A view from the direction of the laser beam.
Then the CMM is controlled to scan along the planned path. During this period, hundreds of points on the sphere surface will be obtained. Suppose that N o positions of CMM and N o corresponding lengths of the laser beam are collected in this period. Then i
i
i
N o measuring points on the sphere can be calculated by Pp Pca la v where v is
the actual laser beam direction. Considering the CMM positioning error, the probe indication error and the form error of the reference sphere, the measuring points will not be located exactly on the sphere surface. Thus the direction of the laser beam and the sphere center should be obtained by minimizing the following cost function. No
F v, Psc R Pp(i ) Psc i 1 No
2
2
R P v l Psc i 1
(i ) ca
(i ) a
2
(1)
2
Minimizing (1) is a non-linear least square problem which can be solved as follows. Equation (1) can be rewritten as n
F x R Pca(i ) x1 i 1 n
x3 la(i ) x4 T
x2
fi x 2
i 1
8
x5
x6
T 2
2
(2)
fi x can be approximately linearized by applying first-order Taylor series expansion to fi x at the point x x k as
x x f x x f x x f x
i x f i x k f i x k
T
k
T
k
T
k
i
k
k
i
(3)
i
where x k refers to the estimation of x in the k-th iteration. The original non-linear problem will then be converted to an approximate linear least square problem as n
min x i2 x x
(4)
i 1
The optimal solution of (4) can be used as x k +1 . Let
f 1 Ak f n f 1 bk f n
x
， T
T
k
x x x f x k
T
k
(5)
k
1
x
where f k f1 x k
k
T
k
x k f n x k
=A x k f k k
T
f n x k . One can get
x Ak x bk Ak x bk T
(6)
Thus, the solution of (4) can be obtained as
x k 1 ATk A k ATk b k 1
x
k
A Ak A f
Iterations will be conducted until the
1
T k
ATk f
k
T k
k
(7)
is less than the predefined threshold
. x 0 , i.e. the initial value of v and Psc , is the estimated laser beam direction v e in 9
the first step and the fitting center Psce in the first step respectively. A flow chart of the proposed calibration method is shown in Fig. 4. Estimate the laser beam direction according to the mechanical design
Manually probe 5 points and fit into a sphere to obtain the sphere center
Plan the CMM scanning path and perform the scanning
Optimize the initial laser beam direction
Fig. 4. The flow chart of the proposed algorithm
3. Sensitivity analysis As discussed in Section 2, the proposed method is mainly based on optimization algorithm. The variations of the actual path points of the CMM and actual laser length will result in variations in the final results of laser direction. The form error will not be separately discussed in this section, noticing that the form error of the reference sphere results in the error of the measuring laser beam length, which is same as the result of the probe indication error. In this section, the relationship between the error of the calibrated laser direction and the measuring error of the CMM positions and laser beam lengths is analyzed, which is referred as sensitivity analysis. The sensitivity analysis has already been used in localization and camera calibration problems which essentially are optimization problems [15,16]. The laser direction is calculated by solving the optimization problem in equation (1). The object function in equation (1) can be rewritten as n
F Y , X R Pca(i ) v la(i ) Psc i 1
= R i 1 n
P
(i ) cax
2
2
v x l Pscx P (i ) a
2
(i ) cay
10
v y l Pscy P v z l Pscz (i ) a
2
(i ) caz
(i ) a
2
2
(8)
Where
represents
n
1 n X Pcax Pcax
4n-dimensional Pscx
Pscy
Pscz
Pcay Pcay 1
vector. T
the n
Y v x
total
number
Pcaz Pcaz 1
v z
the
1 la la n
n
vy
of
T
is
a
sample T
points.
is
3-dimensional
a vector.
is not listed in the Y vector because the laser beam direction is what
needs to be analyzed. The gradient vector of the objective function F Y , X is g Y , X
F Y , X Y
(9)
In the optimization problem, g Y , X should, ideally, be zero if Y is the optimal solution. In the practical case, there exists random errors in the measuring data, which results in deviation in the optimization results. Assuming that the variations of the input data and the calibration result are dX and dY respectively, one can get
g Y dY , X dX 0 . Then the first order term of the Taylor series expansion of g Y dY , X dX at Y , X can be obtained as g Y dY , X dX g Y , X
g g dX dY X Y
(10)
dX
(11)
and consequently 1
g g dY Y X 1
g g We define a sensitivity matrix S , which demonstrates the mapping Y X from the variation of the input measuring data to the variation of the estimated laser beam direction. Since the random errors of the measuring points are of spatial independence, the covariance matrix of dX is of diagonal form. In addition, if the number of the measuring points is large enough, we can assume that the variance of the random error at a single point is equal to the variance of the random errors over the whole surface, i.e. the CMM
positioning error M2 . As long as the laser probe is working under the measurement angle limit, the variance of the laser probe indication error should be equal to the probe device 11
accuracy as L2 . Then the covariance matrix of dX can be expressed as
M2 I 3n*3n 0 cov dX L2 I n*n 0
(12)
where I is the identity matrix. The covariance of dY can be obtained by
cov dY S cov dX S T
(13)
According to the multivariate statistics, for a specified probability , the laser direction error vector dY will lie inside a 3D ellipsoid expressed as
dY
T
where
cov dY
1
dY 2
is a size factor associated with . The value of
(14)
can be obtained by
calculating the integral of the probability density function of the chi-square distribution with three degrees of freedom [17]. Since the cov dY is a 3 3 positive definite matrix, one can get the singular values i2 12 22 32 of cov dY through the singular value decomposition. The lengths of the principle axes of this ellipsoid can be expressed as i . Thus the norm of the error vector dY is less than 1 . 300 sampling points are chosen according to the algorithm introduced in section 2. The sphere center is set at
0
0 0
T
and the sphere radius is set as 20mm. The
probability is set as 95% and the corresponding is 2.795. The variance of the measuring points and laser lengths rises from 0 to 0.01mm. The relationship between the variance of the measuring data and the maximum calibration result is shown in Fig. 5.
12
10-3
Maximum error of calibration result (rad)
7
6
5
4
3
2
1
0
0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009
0.01
Variance of the measuring error (mm)
Fig. 5. The relationship between the variance of the measuring data and the maximum calibration result (under 95% probability).
It can be seen in the figure that the maximum calibration error is approximately linear to the variance of the measuring error. In fact, the CMM positioning error and laser indication error is less than 0.004 mm. Thus, the maximum direction error is less than 0.003 rad. Considering that the laser length in actual is less than 4mm, the distance deviation of a single measuring point should be less than 12 m in the worst condition. In this section, the maximum direction calibration error of the proposed method is calculated using the sensitivity analysis. It can be regarded as the theoretical worst condition of the proposed method. In the next section, more simulations will be conducted to verify the accuracy and robustness of the proposed method.
4. Simulations 4.1 Important factors influencing the calibration accuracy In the proposed method, optimization method is used to obtain the precise direction of the laser beam. As is known to all, the initial values, which is the estimated laser beam direction and the sphere center in this case, usually affect the convergence result in optimization problems. In addition, CMM positioning error and laser probe indication error add quite much noise to the calibration method. The form error of the reference sphere also affects the calibration result. However, the form error results in the deviation in the measuring length of the laser beam, which is same as the effect of the probe indication error. Considering the form error of the reference sphere is much heavier than the probe indication error, the laser length measuring error in the simulation is set as 0.004mm, which has synthetically considered the effect of the probe indication error and the form error of the reference sphere. In this section, simulations are carried out to find 13
out the influence of these factors on the calibration results. 4.1.1
Initial sphere center of the optimization
In the proposed method, the estimated sphere center is determined by manual inspection, which will later be used as the initial value of the optimization. The laser beam direction used in this period is estimated by the mechanical structure of the probe fixture. Furthermore, the measured points can only distribute in limited area on the sphere surface due to the maximum measurement angle of the laser probe. Both the factors will introduce error into the measurement of the sphere center. In this simulation, the relationship between the deviation of the laser beam direction and the deviation of the estimated sphere center is analyzed. The theoretical value of the sphere center is set at 0 0 0 . The radius of the sphere is set as 20mm which is same T
as the sphere artefact used in the real experiment. Because of the central symmetry of the sphere, the simulated laser beam direction can be set as
1
0 0
T
without loss of
generality. Different estimated laser beam directions are used to analyze the influence of the direction deviation on the estimation of the sphere center. Without loss of generality, the estimated laser beam direction is set as
cos
sin
0
T
where is the initial angle
error between the estimated laser beam direction and the theoretical value 1 0 0 . 5 T
measured points are used to fit the sphere and to extract the sphere center. Then the distance deviation between the fitting center and the theoretical value
0
0 0
T
is
calculated. The CMM positioning error and the laser probe indication error is set as 0.002mm and 0.004mm respectively, which is accordance with the actual condition. Gaussian noise with 3 =0.002 and 3 =0.004 is added to the path points of the CMM and the actual laser length respectively. The relationship between the deviation of the estimated sphere center and the deviation of the laser beam direction is illustrated in Fig. 6. It can be seen that there is a clear positive correlation between the center distance error and the laser beam direction error. This result will be used in the later simulations.
14
Deviation of the sphere center (mm)
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
Deviation of the laser beam direction (deg)
Fig. 6. The relationship between the deviation of the laser beam direction and the deviation of the sphere center
4.1.2
Planned measuring length for optimization In the planning stage, the planned measuring length l p of each path points on the
sphere Ps should be carefully planned. There are two strategies in choosing l p i.e. each
l p is completely the same or each l p is chosen randomly. These two methods are compared in their performance of calibration with the following simulations. The theoretical value of the laser beam direction is
0
1
0 0
T
and the sphere center is
0 0 . T
In the first simulation, the initial laser beam direction is set as
cos
0
T
sin
where varies from 0 to 10 degrees. The initial value of the sphere center is obtained by virtually probing 5 points on the sphere surface as discussed in section 4.1.1. The CMM positioning error and laser probe indication error is set as 0.002mm and 0.004mm respectively. The calibration results of the two strategies are compared in Fig. 7(a).
In the second simulation, the same initial laser direction cos sin 36 36
T
0
is
chosen for both methods. The CMM positioning error and laser probe indication error varies from 0 to 0.01mm. The calibration results of the two strategies are shown in Fig. 7(b).
15
14
7 Same measuring length Different measuring lengths
10
8
6
4
2
0
Same measuring length Different measuring lengths
6
Final angle error (deg)
Final angle error (deg)
12
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
0
10
0
1
2
3
Initial angle error (deg)
4
5
6
7
8
9
10
Noise of CMM and probe (um)
(a)
(b)
Fig. 7. The influence of different planning strategies on the final results. (a) The influence of initial angle error on the final results under different strategies. (b) The influence of CMM and probe noise on the final results under different strategies.
It can be seen from Fig. 7(a) that the strategy of choosing the same measuring length may converge to the incorrect local optimum when the initial angle error is small. As shown in the Fig. 7, the strategy of choosing different planned measuring lengths shows better robustness when the initial angle error and noise rise. In the following simulations and experiments, the strategy in which planned measuring lengths are chosen randomly will be applied. 4.1.3
Initial laser beam direction of the optimization
The initial values are important in optimization problems. In the proposed method, the initial values are the estimated laser beam direction and the estimated sphere center. It has proved in section 4.1.1 that the estimated sphere center is positive correlated with the laser beam direction. Thus in this section, the influence of the initial laser beam direction on the final calibration result is analyzed in detail in order to verify the robustness of the proposed method. The theoretical value of laser direction is
0
1
0 0
0 0 . The initial laser beam direction is set as T
T
and the sphere center is
cos
sin
0
T
where
varies from 0 to 45 degree. The initial value of the sphere center is obtained by virtually probing 5 points on the sphere surface as shown in section 4.1.1. The CMM positioning error and laser probe indication error is set as 0.002mm and 0.004mm respectively. The relationship between the final result and the initial laser beam direction is shown in Fig. 8.
16
0.14
Final angle error (deg)
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5
10
15
20
25
30
35
40
45
Initial angle error (deg)
Fig. 8. The relationship between the initial angle error and the final angle error
It can be seen in the figure that the final angle error does not increase with the rising of the initial angle error. Under the actual situation, it is impossible that the initial angle error which is mainly caused by mechanical structure will be larger than 45 degrees. It can be concluded that the propose method remains robust when the initial laser beam direction is greatly different from the ground truth value.
4.2 Calibration accuracy of different methods In this section, out proposed method is compared with two existing calibration methods. In Bi’s method [10], the CMM is controlled to moves at an equal step along X, Y and Z axes respectively. Then equation sets are established to calculate the laser beam direction. According to Bi’s method, the following parameters are chosen in the simulation. The step lengths for X, Y and Z axes are chosen as 0.5mm, 0.3mm and 0.45mm respectively. The step numbers for three axes are chosen as 12. In Xie’s method [11], optical probe is regarded equivalent to the contact probe and used to inspect a reference sphere with a series of fixed measuring lengths. According to Xie’s method, the measuring lengths are chosen as 0.6mm, 1.2mm 1.8mm, 2.4mm and 3mm in the
simulation. The initial value of the proposed method is set as cos sin 60 60
T
0 . In
the first simulation, the CMM positioning error is fixed as 0.002mm when the probe indication error rises from 0 to 0.006mm. In the second simulation, the probe indication error is fixed as 0.004mm when the CMM positioning error rises from 0 to 0.005mm. The influence of the measuring error including CMM positioning error and laser probe indication error on the calibration result is analyzed and illustrated in Fig. 9. 17
3.5
3 Bi's method Xie's method Proposed method
2.5
2
1.5
1
2
1.5
1
0.5
0.5
0
Bi's method Xie's method Proposed method
2.5
Final angle error (deg)
Final angle error (deg)
3
0
1
2
3
4
5
0
6
0
1
2
Probe indication error (um)
3
4
5
6
CMM positioning error (um)
(a)
(b)
Fig. 9. The comparison of the proposed method and the existing methods under different noise. (a) The comparison of different methods under different laser indication errors. (b) The comparison of different methods under different CMM positioning errors.
Then the CMM positioning error and the laser probe indication error are set as 0.002mm and 0.004mm respectively which is similar with the actual condition. Other parameters are set as above. Three methods are compared in 20 simulation tests. The result is shown in Fig. 10. 4.5 Bi's method Xie's method Proposed method
4
Final angle error (deg)
3.5 3 2.5 2 1.5 1 0.5 0
0
2
4
6
8
10
12
14
16
18
20
Index
Fig. 10. The comparison of the proposed method and the existing methods under the same noise.
As can be seen in the Fig. 9, the proposed method performs best in the calibration accuracy. In Fig. 10, the mean calibration error of the three calibration methods are 1.5268°, 0.534°and 0.0691°respectively. The proposed method gains 95.5% and 87.1% improvement compared with the existing methods. More specifically, the indication interval of the laser probe is less than 4mm. Thus the angle error of 0.0691 degree will result in a deviation of less than 4.8 m when measuring a single point. 18
In Bi’s method, the laser beam direction is calculated by solving the equation sets. The result is not accurate enough and is easily worsen by noise. In Xie’s method, the laser beam direction is calculated by mapping the point cloud into a series of sphere. The key of this method is to keep the measuring length unchanged, which requires iterative compensation of CMM positions in experiment. This method is time-consuming and difficult to realize because of the existence of noise. Thanks to the scanning and optimization procedure, the noise influence is significantly relieved by a large number of points in the proposed method. Moreover, scanning inspection rather than point-by-point inspection is applied in the proposed method, which reduces the time spent by the air-moves of the CMM. Thus the proposed method is superior in efficiency, accuracy and robustness to the existing methods.
5. Experiments and Verification 5.1 Calibration experiment The laser probe used in the experiment is Precitec CHRocodile C. CHRocodile C is suited for industrial use and is easily integrated into any kind of inspection machine. The technical parameters of this laser sensor are listed in Table 2. The accuracy of the CMM used in the experiment is verified to be
2 4 L /1000 m
where L is the measuring
length. The probe is mounted along two different directions and calibrated with three different algorithms respectively. The reference sphere used in the calibration experiment is a ceramic ball with a diffuse surface so that the laser probe can work stably. The diameter of the reference sphere is 40.040 mm and the form error is 0.0036 mm. The calibration experiment is conducted in a constant-temperature room, so the variations of the environmental conditions during the full calibration run has little influence on the instrument. The calibration configuration is shown in Fig. 11.
19
Fig. 11. Calibration configuration.
According to Bi’s method, the following parameters are chosen. For direction 1, the step lengths for X, Y and Z axes are chosen as 2.4mm, 1.3mm and 0.9mm respectively. For direction 2, the step lengths for X, Y and Z axes are chosen as 0.4mm, 2.4mm and 2.9mm respectively. The step numbers for three axes are chosen as 3. According to Xie’s method, the measuring lengths are chosen as 0.8mm, 1.8mm, 2.8mm for both direction 1 and direction 2. The calibration results are listed in Table 3. The calibration results will be verified by measuring a standard sphere and a step gauge block in the next section. The spent time of these methods is 21s, 135s and 18s respectively. The CMM moving velocity and measuring velocity are totally the same during the entire experiments. Xie’s method costs the longest time because the laser length should be adjusted iteratively to meet the requirement of Xie’s algorithm. The proposed method costs the least time due to using scanning inspection in the calibration procedure. Table 2. Specifications of the laser probe
Descriptions Measuring range Working distance Spot diameter Lateral resolution Axial resolution Accuracy* Measurement angle
Specifications 4 mm 32 mm 8 um 4 um 200 nm 1.2 um
90 20
* Accuracy corresponds to the maximum reading deviation between the laser probe CHRocodile C and a calibrated interferometric reference sensor.
20
Table 3. Calibration results using different algorithms
Bi’s method Xie’s method Proposed method
Direction 1
Direction 2
[0.012004 0.005386 -0.99991] [-0.01085 0.001839 -0.99994] [-0.01394 0.004652 -0.99989]
[-0.99921 0.028263 -0.02798] [-0.99941 -0.00455 0.033976] [-0.99938 0.015429 0.031648]
5.2 Accuracy verification In this section, the accuracy of the laser beam direction calibrated in section 5.1 is verified. A standard sphere and a step gauge block are measured. The standard sphere is another similar ceramic ball whose diameter is 40.038mm. The form error of this sphere is 0.0042mm. The gauge block used is a metallic step gauge block whose step lengths are 20.001mm, 30.001 and 39.996mm. The parallelism between the corresponding pairs is 0.002mm, 0.003mm and 0.002mm. The ground truth value of the sphere radius and the gauge block length is obtained by contact measurement of a high-precision CMM. The measuring scene is shown in Fig. 12. The measuring results of the sphere and gauge block are listed in Table 4 and Table 5 respectively.
(a)
(b)
Fig. 12. The verifying test configuration of a step gauge block. (a) The verifying test in direction 1. (b) The verifying test in direction 2.
Table 4. Measurement results of the standard sphere (mm)
Bi’s method Xie’s method Proposed method
Direction 1 Diameter 40.022 40.027 40.036
Error -0.016 -0.011 -0.002 21
Direction 2 Diameter 40.069 40.046 40.042
Error 0.031 0.008 0.004
Table 5. Measuring results of the step gauge block (mm)
Direction 1 Length Bi’s 20.061 method 30.044 40.018 Xie’s 20.039 30.035 method 40.052 Proposed 19.989 30.005 method 40.002
Error 0.06 0.043 0.022 0.038 0.034 0.056 -0.012 0.004 0.006
Direction 2 Length 20.049 30.074 40.006 20.036 30.042 39.913 19.996 30.006 39.996
Error 0.048 0.073 0.01 0.035 0.041 -0.083 -0.005 0.005 0
It is obvious that the measuring results using the laser direction calibrated by the proposed method is superior to which calibrated by the existing methods. The mean measuring errors of the fitted sphere diameter and the gauge block length are 0.003mm and 0.0054 mm respectively. Unlike the contact measurement, the non-contact measurement by laser probe is also affected by the environment light and material reflectivity. Thus, the actual measurement results may fluctuate according to the actual situation.
6. Conclusions Non-contact measurement has been widely used for those parts with free-form surface due to its capability of efficiently obtaining surface information. Among the optical measurement devices, the spot laser probe achieves a good balance between the accuracy and efficiency. In order to get the information of the 3D surface point from the laser probe indication, the laser beam direction should be precisely calibrated. In this paper, a robust and efficient laser beam direction calibration method is proposed. Compared to the previous researches, the proposed calibration method has the following contributions. In terms of robustness, an initial inaccurate direction specified by the probe fixture is refined by mapping a large number of sample points into a sphere so that the final result is robust to the measuring noise and reaches micron-level accuracy. In terms of efficiency, scanning inspection rather than point-by-point inspection is applied in the calibration procedure so that the proposed method is much faster than the existing methods. With the sensitivity analysis, the proposed method is proved to be insensitive to various measuring noise. Several simulations and experiments have been carried out to 22
verify the effectiveness of the proposed method. Compared to the two existing methods, the proposed method improves the calibration accuracy by 87.1% in simulation. In the actual experiment, the mean measuring errors of the fitted sphere diameter and the gauge block length are 0.003mm and 0.0054 mm which is superior to the existing methods. Additionally, the proposed method can be applied not only in the calibration of the spot laser probe but also in the calibration of other displacement sensors such as ultrasonic sensors.
7. Acknowledgement This research was supported by the National Nature Science Foundation of China (Grant no. 51975344).
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Conflict of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
CReDiT Author Statement Yijun Shen: Conceptualization, Methodology, Writing - Original Draft Xu Zhang: Funding acquisition, Writing - Review & Editing Zhenyou Wang: Formal analysis Jianxiang Wang: Investigation Limin Zhu: Supervision, Project administration
24
S0263-2241(20)30060-9 https://doi.org/10.1016/j.measurement.2020.107523 MEASUR 107523
To appear in:
Measurement
Received Date: Revised Date: Accepted Date:
25 October 2019 11 January 2020 15 January 2020
Please cite this article as: Y. Shen, X. Zhang, Z. Wang, J. Wang, L. Zhu, A Robust and Efficient Calibration Method for Spot Laser Probe on CMM, Measurement (2020), doi: https://doi.org/10.1016/j.measurement.2020.107523
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A Robust and Efficient Calibration Method for Spot Laser Probe on CMM Yijun Shen1, Xu Zhang2*, Zhenyou Wang2, Jianxiang Wang2, Limin Zhu1,3 1 School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China 2 Department of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200444, China 3 Shanghai Key Lab of Advanced Manufacturing Environment, Shanghai 200240, China * Author to whom any correspondence should be addressed.
E-mail address: [email protected]
Abstract: Among all the non-contact measurement devices, the spot laser probe achieves a good balance between accuracy and efficiency. When the laser probe is applied to three-dimensional (3-D) measurement，the laser beam direction is essential to transform the probe indication from 1-D distance information into a 3-D vector representing differences in space coordinates. In this article, a calibration method is proposed for the laser probe mounted on the coordinate measuring machine (CMM). During the calibration procedure, the CMM moves along the user-planned path and hundreds of points are scanned in a few seconds. In order to get an accurate laser beam direction estimate, we refine the original inaccurate direction, which is specified by the mechanical structure of the probe fixture, by mapping the scanned points into a sphere. The simulation and experiment manifest that the proposed method is superior to the existing methods and meets the requirement of the micron-level measurement. Keywords: laser probe calibration, non-contact measurement, coordinate measuring machine, sensitivity analysis.
1. Introduction With the rapid development of aerospace and aviation, parts with free-form surface such as turbine blades have been widely used in these industries. The quality control and the reverse engineering during designing and manufacturing require a large quantity of high-quality point cloud data collected from the surface of these parts. Therefore, there is an urgent need of precise and efficient measuring methods for parts with complicated geometry [1]. In recent years, non-contact measurement, especially optical measurement, has been rapidly developing [2]. Compared to the traditional coordinate measuring machine (CMM) with contact probes, non-contact measurement has advantages in its 1
high-speed and non-abrasion. Also, there is no need to compensate the tip radius and to plan a complicated scanning path when using optical probes [3,4]. Both structured light method and laser triangulation method are mature technologies and can scan the object surface efficiently. However, both of them can only reach an accuracy of nearly 10 micrometers which may not meet the requirement of precise measurement [5,6]. The spot laser probe, which reaches sub-micron level accuracy, is designed to be mounted on the CMM to perform accurate and efficient non-contact scanning. Although the spot laser probe cannot scan the surface as quickly as the structured light and laser triangulation probes, the spot laser probe is far more accurate. For those non-contact measurement tasks which require micrometer-level accuracy, the spot laser probe is the best choice. The spot laser probe is an optical displacement sensor which returns the one dimensional (1D) distance information between the probe and the object surface. The direction of the laser beam is necessary to transform the 1D length to 3D coordinates of the measuring point. Considering that the laser length is usually several millimeters, the deviation of a single measuring point can reach several micrometers when the angle error of the calibrated laser beam direction reaches 103 rad. Thus, in the measurement requiring micron-level accuracy, the direction of the laser beam must be precisely calibrated. Many researchers have been working on the calibration of the laser beam direction of the spot laser probe. The existing laser beam direction calibration methods can be divided into two groups. Some researchers solve the laser direction using a geometric constraint such as the plane, sphere and cone constraint. Chen et al. [7] used a plane artefact to calibrate the laser beam direction. The position and pose of this plane should be calculated by fitting point cloud data measured by laser trackers. The operation costs much time and the accuracy of the laser tracker is limited. Smith et al. [8] proposed a method using a polyhedron artifact. Compared to the method using the single plane, this method needs no position information of the plane. Zhou et al. [9] determined the laser beam direction with the help of a cone artifact. The geometric constraint that the cone angle remains unchanged at different height of the cone is used to solve the laser beam direction. Bi et al. [10] used the sphere constraint to solve the equation set. In his method, CMM needs to move the same distance along XYZ axis for several times with high accuracy. All above methods employ geometric constraints to solve the equation set containing the unknown parameters of the laser beam direction. The main disadvantage of these methods is that the CMM positioning error and laser probe indication error may cause an adverse effect on the final results. Other researchers calibrate the laser beam direction by moving the probe along the laser beam direction to several positions and fitting these positions into a line. The line 2
direction is equivalent to the laser beam direction. The key of such methods is to make sure the probe moves exactly along the laser beam direction. Xie et al. [11] proposed the idea of “equivalent probe”, in which the optical probe is regarded equivalent to the contact probe. In the calibration procedure, the user inspects the sphere by the probe with a series of laser beam lengths. However, the accurate length of the laser beam requisite in this method is difficult to obtain in the experiment. Zhu et al. [12] proposed a calibration method to determine the laser sensor mounted on the end of the robot arm. The robot is controlled to reach certain positions to ensure that the laser spot is projected at the same point on the screen. The point is checked by a camera to verify if it maintains unchanged. The accuracy of this method is limited by the accuracy of the camera. The principle of the above calibration methods is the best-fitting algorithm, which is more stable than methods solving the equation sets. However, keeping the probe moving along the laser beam direction is very difficult to realize in the real experiment because of the existence of noise. As discussed above, the previous methods either require precise position information of the artefacts or the precise movement of the measuring machine. Thus, in terms of robustness, the CMM positioning error and the laser probe indication error may cause great deviation in the final results. In addition, in order to ensure the accuracy of sampling, the calibration is usually implemented by point-by-point inspection and the motion of CMM should be set at a low speed in these methods [10]. In other words, the previous calibration methods are lack of efficiency. In this paper, a robust and efficient calibration method for the spot laser probe is proposed. In the first stage, the initial laser beam direction is estimated by the mechanical structure of the probe fixture and the sphere center is estimated by manual inspection. In the second stage, a specially designed scanning path is planned according to the initial laser beam direction and the sphere center. Then hundreds of points are scanned along the path. Finally, the initial laser beam direction is refined by mapping the scanned points into a sphere. Compared to the existing methods, the proposed method has the following advantages: In terms of robustness, an initial inaccurate direction specified by the probe fixture is refined by mapping a large number of sample points into a sphere so that the final result is robust to measuring noise and reaches micron-level accuracy. In terms of efficiency, scanning inspection rather than point-by-point inspection is applied in the calibration procedure so that the proposed method is much faster than the existing methods. 3
Additionally, the input of the proposed algorithm is the measuring points on the sphere surface. Thus the proposed method can be applied not only in the calibration of the spot laser probe but also in the calibration of other similar displacement sensors such as ultrasonic displacement sensors. The rest of this paper is organized as follows: Section 2 describes the laser probe calibration model and algorithm; Section 3 analyzes the robustness of the proposed method using the sensitivity analysis; Section 4 analyzes the important factors in this calibration procedure and compares the proposed method with other existing methods; Section 5 presents the calibration experiments and verifies the result; Section 6 concludes the main opinions in this paper.
2. Laser probe Calibration model 2.1 Coordinate system The non-contact measurement system is a CMM with a laser probe mounted on the end of its Z axis. A reference sphere with known radius is fixed on the CMM operating platform which is used as the calibration artifact. The following coordinate systems are set up for the convenience of explaining the calibration algorithm. 2.1.1
Machine coordinate system The machine coordinate system is denoted as OM X M YM Z M . The origin of the
machine coordinate system is the zero point of all three scales of the CMM. The direction of X, Y and Z axis is parallel to the corresponding axis of the CMM. The machine coordinate system is a standard Cartesian coordinates. 2.1.2
Tool coordinate system The tool coordinate system is denoted as OT X T YT Z T . The origin of the tool
coordinate system is set at the zero point of the laser probe, i.e. the point where the laser probe indication is zero. The X, Y and Z axes are all parallel to the machine coordinate system. 2.1.3
Work piece coordinate system The work piece coordinate system is denoted as OW X W YW Z W . The origin of the
work piece coordinate system is set at the center of the reference sphere. The X, Y and Z axes are all parallel to the machine coordinate system. The set-up of the three coordinate systems is illustrated in Fig. 1. 4
ZM XM
YM
OM
ZT XT
OT
YT XW
ZW OW
YW
Fig. 1. The set-up of the three coordinate systems.
2.2 Term definitions In convenience of introducing the proposed method, some frequently used terms in this article are defined in this section. A general non-contact measurement procedure contains two steps. Firstly, in the planning stage, several sample points are planned on the object surface by the user and the moving trajectory of the CMM is calculated according to these sample points. Secondly, in the measuring stage, the CMM moves along the pre-designed trajectory and the optical probe collects data simultaneously. Fig. 2 shows a specific procedure of the non-contact measurement with a spot laser probe. Firstly, path points on the sphere Ps , which are planned by the user, are regarded as the sample points for measurement. The path points of the CMM Pc can be calculated by offsetting Ps by the planned measuring length l p along the inversed estimated laser beam direction ve . Secondly, CMM is supposed to move to Pc but reaches the actual points of the CMM Pca due to the positioning error. The laser probe emits laser along the actual laser beam direction va and forms a light spot on the object surface which is referred as the measured points Pp . The indication of the laser probe is the actual measuring length la , i.e. the distance between the probe and the object surface. The position of the probe can be obtained by the encoders of the CMM as actual points of the CMM
Pca . These terms will be repeatedly used in the following parts. Thus
corresponding symbols are defined for brief expression as shown in Table 1. It should be emphasized that all the above mentioned variables are expressed in the machine coordinate system OM X M YM Z M . Actually la and va are recorded in the 5
tool coordinate system OT X T YT Z T . However, the conversion relationship between any two coordinate systems is just translation. Therefore, the laser beam direction and the laser length is completely same in these coordinate systems. Besides, Ps is usually planned in the work piece coordinate system OW X W YW Z W as Psw . Then Psw is transformed into the machine coordinate system with a translation vector as Ps =Psw m Tw where
m
Tw is
the translation vector between the work piece coordinate system and the machine coordinate system, i.e. the coordinate of the sphere center in the machine coordinate system in this case. Z axis of CMM
Spot laser probe
Pca Laser beam
va la
Pc ve
Pp
lp
Ps
Fig. 2. A diagram of the measurement with spot laser probe Table 1 Symbols of the terms
Term definition
Symbols
Path points on the sphere
Ps
Path points of the CMM
Pc
Actual points of the CMM
Pca
Measured points
Pp
Planned measuring length
lp
Actual measuring length
la
Estimated laser beam direction
ve
Actual laser beam direction
va 6
2.3 Calibration model The proposed method includes two steps to calibrate the laser beam direction. A standard sphere artefact with known radius R is used in the calibration method. In the first step, an initial inaccurate laser beam direction can be obtained by the mechanical structure of the probe fixture as ve . This direction is not accurate enough for precision measurement but accurate enough as an initial value for the proposed optimization algorithm, which will be proved by simulation in section 4.1.3. N c points denoted by
P pr are inspected manually by users. The positions of these points can be calculated by N c CMM positions Pcar and corresponding laser beam length la as Ppr =Pcar v e la .
The coordinate of the center of the reference sphere Psce can be obtained by fitting P pr into a sphere. The algorithm for sphere fitting is so mature that it won't be discussed in detail here [13,14]. To prevent the influence of the measuring noise, more than 5 points should be measured. The coordinate of the sphere center will be used as the initial value of the optimization. In the second step, the initial inaccurate direction of the laser beam ve will be refined. With the information of estimated sphere center Psce and the laser beam direction
ve , the scanning region on the spherical surface can be determined according to the measurement angle limit of the laser probe. Measurement angle limit refers to the maximum angle between the laser beam of the probe and the normal vector of the surface. Within this region, hundreds of measuring points can be planned according to any rules. For example, in this proposed algorithm, the points are planned on a series of circles on the sphere center as shown in Fig. 3. The radius of the largest circle is R sin where is the measurement angle limit of the laser probe. The path points of the CMM Pc can be calculated by offsetting the path points on the sphere Ps by planned measuring length l p along the inversed estimated laser beam direction ve as Pc =Ps v e l p .
7
Sphere surface
Laser beam
Path points on the sphere
(a) Distribution of the path points on the sphere
(b) A view from the direction of the laser beam
Fig. 3. The path points planned on the sphere surface. (a) Distribution of the path points on the sphere. (b) A view from the direction of the laser beam.
Then the CMM is controlled to scan along the planned path. During this period, hundreds of points on the sphere surface will be obtained. Suppose that N o positions of CMM and N o corresponding lengths of the laser beam are collected in this period. Then i
i
i
N o measuring points on the sphere can be calculated by Pp Pca la v where v is
the actual laser beam direction. Considering the CMM positioning error, the probe indication error and the form error of the reference sphere, the measuring points will not be located exactly on the sphere surface. Thus the direction of the laser beam and the sphere center should be obtained by minimizing the following cost function. No
F v, Psc R Pp(i ) Psc i 1 No
2
2
R P v l Psc i 1
(i ) ca
(i ) a
2
(1)
2
Minimizing (1) is a non-linear least square problem which can be solved as follows. Equation (1) can be rewritten as n
F x R Pca(i ) x1 i 1 n
x3 la(i ) x4 T
x2
fi x 2
i 1
8
x5
x6
T 2
2
(2)
fi x can be approximately linearized by applying first-order Taylor series expansion to fi x at the point x x k as
x x f x x f x x f x
i x f i x k f i x k
T
k
T
k
T
k
i
k
k
i
(3)
i
where x k refers to the estimation of x in the k-th iteration. The original non-linear problem will then be converted to an approximate linear least square problem as n
min x i2 x x
(4)
i 1
The optimal solution of (4) can be used as x k +1 . Let
f 1 Ak f n f 1 bk f n
x
， T
T
k
x x x f x k
T
k
(5)
k
1
x
where f k f1 x k
k
T
k
x k f n x k
=A x k f k k
T
f n x k . One can get
x Ak x bk Ak x bk T
(6)
Thus, the solution of (4) can be obtained as
x k 1 ATk A k ATk b k 1
x
k
A Ak A f
Iterations will be conducted until the
1
T k
ATk f
k
T k
k
(7)
is less than the predefined threshold
. x 0 , i.e. the initial value of v and Psc , is the estimated laser beam direction v e in 9
the first step and the fitting center Psce in the first step respectively. A flow chart of the proposed calibration method is shown in Fig. 4. Estimate the laser beam direction according to the mechanical design
Manually probe 5 points and fit into a sphere to obtain the sphere center
Plan the CMM scanning path and perform the scanning
Optimize the initial laser beam direction
Fig. 4. The flow chart of the proposed algorithm
3. Sensitivity analysis As discussed in Section 2, the proposed method is mainly based on optimization algorithm. The variations of the actual path points of the CMM and actual laser length will result in variations in the final results of laser direction. The form error will not be separately discussed in this section, noticing that the form error of the reference sphere results in the error of the measuring laser beam length, which is same as the result of the probe indication error. In this section, the relationship between the error of the calibrated laser direction and the measuring error of the CMM positions and laser beam lengths is analyzed, which is referred as sensitivity analysis. The sensitivity analysis has already been used in localization and camera calibration problems which essentially are optimization problems [15,16]. The laser direction is calculated by solving the optimization problem in equation (1). The object function in equation (1) can be rewritten as n
F Y , X R Pca(i ) v la(i ) Psc i 1
= R i 1 n
P
(i ) cax
2
2
v x l Pscx P (i ) a
2
(i ) cay
10
v y l Pscy P v z l Pscz (i ) a
2
(i ) caz
(i ) a
2
2
(8)
Where
represents
n
1 n X Pcax Pcax
4n-dimensional Pscx
Pscy
Pscz
Pcay Pcay 1
vector. T
the n
Y v x
total
number
Pcaz Pcaz 1
v z
the
1 la la n
n
vy
of
T
is
a
sample T
points.
is
3-dimensional
a vector.
is not listed in the Y vector because the laser beam direction is what
needs to be analyzed. The gradient vector of the objective function F Y , X is g Y , X
F Y , X Y
(9)
In the optimization problem, g Y , X should, ideally, be zero if Y is the optimal solution. In the practical case, there exists random errors in the measuring data, which results in deviation in the optimization results. Assuming that the variations of the input data and the calibration result are dX and dY respectively, one can get
g Y dY , X dX 0 . Then the first order term of the Taylor series expansion of g Y dY , X dX at Y , X can be obtained as g Y dY , X dX g Y , X
g g dX dY X Y
(10)
dX
(11)
and consequently 1
g g dY Y X 1
g g We define a sensitivity matrix S , which demonstrates the mapping Y X from the variation of the input measuring data to the variation of the estimated laser beam direction. Since the random errors of the measuring points are of spatial independence, the covariance matrix of dX is of diagonal form. In addition, if the number of the measuring points is large enough, we can assume that the variance of the random error at a single point is equal to the variance of the random errors over the whole surface, i.e. the CMM
positioning error M2 . As long as the laser probe is working under the measurement angle limit, the variance of the laser probe indication error should be equal to the probe device 11
accuracy as L2 . Then the covariance matrix of dX can be expressed as
M2 I 3n*3n 0 cov dX L2 I n*n 0
(12)
where I is the identity matrix. The covariance of dY can be obtained by
cov dY S cov dX S T
(13)
According to the multivariate statistics, for a specified probability , the laser direction error vector dY will lie inside a 3D ellipsoid expressed as
dY
T
where
cov dY
1
dY 2
is a size factor associated with . The value of
(14)
can be obtained by
calculating the integral of the probability density function of the chi-square distribution with three degrees of freedom [17]. Since the cov dY is a 3 3 positive definite matrix, one can get the singular values i2 12 22 32 of cov dY through the singular value decomposition. The lengths of the principle axes of this ellipsoid can be expressed as i . Thus the norm of the error vector dY is less than 1 . 300 sampling points are chosen according to the algorithm introduced in section 2. The sphere center is set at
0
0 0
T
and the sphere radius is set as 20mm. The
probability is set as 95% and the corresponding is 2.795. The variance of the measuring points and laser lengths rises from 0 to 0.01mm. The relationship between the variance of the measuring data and the maximum calibration result is shown in Fig. 5.
12
10-3
Maximum error of calibration result (rad)
7
6
5
4
3
2
1
0
0
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009
0.01
Variance of the measuring error (mm)
Fig. 5. The relationship between the variance of the measuring data and the maximum calibration result (under 95% probability).
It can be seen in the figure that the maximum calibration error is approximately linear to the variance of the measuring error. In fact, the CMM positioning error and laser indication error is less than 0.004 mm. Thus, the maximum direction error is less than 0.003 rad. Considering that the laser length in actual is less than 4mm, the distance deviation of a single measuring point should be less than 12 m in the worst condition. In this section, the maximum direction calibration error of the proposed method is calculated using the sensitivity analysis. It can be regarded as the theoretical worst condition of the proposed method. In the next section, more simulations will be conducted to verify the accuracy and robustness of the proposed method.
4. Simulations 4.1 Important factors influencing the calibration accuracy In the proposed method, optimization method is used to obtain the precise direction of the laser beam. As is known to all, the initial values, which is the estimated laser beam direction and the sphere center in this case, usually affect the convergence result in optimization problems. In addition, CMM positioning error and laser probe indication error add quite much noise to the calibration method. The form error of the reference sphere also affects the calibration result. However, the form error results in the deviation in the measuring length of the laser beam, which is same as the effect of the probe indication error. Considering the form error of the reference sphere is much heavier than the probe indication error, the laser length measuring error in the simulation is set as 0.004mm, which has synthetically considered the effect of the probe indication error and the form error of the reference sphere. In this section, simulations are carried out to find 13
out the influence of these factors on the calibration results. 4.1.1
Initial sphere center of the optimization
In the proposed method, the estimated sphere center is determined by manual inspection, which will later be used as the initial value of the optimization. The laser beam direction used in this period is estimated by the mechanical structure of the probe fixture. Furthermore, the measured points can only distribute in limited area on the sphere surface due to the maximum measurement angle of the laser probe. Both the factors will introduce error into the measurement of the sphere center. In this simulation, the relationship between the deviation of the laser beam direction and the deviation of the estimated sphere center is analyzed. The theoretical value of the sphere center is set at 0 0 0 . The radius of the sphere is set as 20mm which is same T
as the sphere artefact used in the real experiment. Because of the central symmetry of the sphere, the simulated laser beam direction can be set as
1
0 0
T
without loss of
generality. Different estimated laser beam directions are used to analyze the influence of the direction deviation on the estimation of the sphere center. Without loss of generality, the estimated laser beam direction is set as
cos
sin
0
T
where is the initial angle
error between the estimated laser beam direction and the theoretical value 1 0 0 . 5 T
measured points are used to fit the sphere and to extract the sphere center. Then the distance deviation between the fitting center and the theoretical value
0
0 0
T
is
calculated. The CMM positioning error and the laser probe indication error is set as 0.002mm and 0.004mm respectively, which is accordance with the actual condition. Gaussian noise with 3 =0.002 and 3 =0.004 is added to the path points of the CMM and the actual laser length respectively. The relationship between the deviation of the estimated sphere center and the deviation of the laser beam direction is illustrated in Fig. 6. It can be seen that there is a clear positive correlation between the center distance error and the laser beam direction error. This result will be used in the later simulations.
14
Deviation of the sphere center (mm)
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
1
2
3
4
5
6
7
8
9
10
Deviation of the laser beam direction (deg)
Fig. 6. The relationship between the deviation of the laser beam direction and the deviation of the sphere center
4.1.2
Planned measuring length for optimization In the planning stage, the planned measuring length l p of each path points on the
sphere Ps should be carefully planned. There are two strategies in choosing l p i.e. each
l p is completely the same or each l p is chosen randomly. These two methods are compared in their performance of calibration with the following simulations. The theoretical value of the laser beam direction is
0
1
0 0
T
and the sphere center is
0 0 . T
In the first simulation, the initial laser beam direction is set as
cos
0
T
sin
where varies from 0 to 10 degrees. The initial value of the sphere center is obtained by virtually probing 5 points on the sphere surface as discussed in section 4.1.1. The CMM positioning error and laser probe indication error is set as 0.002mm and 0.004mm respectively. The calibration results of the two strategies are compared in Fig. 7(a).
In the second simulation, the same initial laser direction cos sin 36 36
T
0
is
chosen for both methods. The CMM positioning error and laser probe indication error varies from 0 to 0.01mm. The calibration results of the two strategies are shown in Fig. 7(b).
15
14
7 Same measuring length Different measuring lengths
10
8
6
4
2
0
Same measuring length Different measuring lengths
6
Final angle error (deg)
Final angle error (deg)
12
5
4
3
2
1
0
1
2
3
4
5
6
7
8
9
0
10
0
1
2
3
Initial angle error (deg)
4
5
6
7
8
9
10
Noise of CMM and probe (um)
(a)
(b)
Fig. 7. The influence of different planning strategies on the final results. (a) The influence of initial angle error on the final results under different strategies. (b) The influence of CMM and probe noise on the final results under different strategies.
It can be seen from Fig. 7(a) that the strategy of choosing the same measuring length may converge to the incorrect local optimum when the initial angle error is small. As shown in the Fig. 7, the strategy of choosing different planned measuring lengths shows better robustness when the initial angle error and noise rise. In the following simulations and experiments, the strategy in which planned measuring lengths are chosen randomly will be applied. 4.1.3
Initial laser beam direction of the optimization
The initial values are important in optimization problems. In the proposed method, the initial values are the estimated laser beam direction and the estimated sphere center. It has proved in section 4.1.1 that the estimated sphere center is positive correlated with the laser beam direction. Thus in this section, the influence of the initial laser beam direction on the final calibration result is analyzed in detail in order to verify the robustness of the proposed method. The theoretical value of laser direction is
0
1
0 0
0 0 . The initial laser beam direction is set as T
T
and the sphere center is
cos
sin
0
T
where
varies from 0 to 45 degree. The initial value of the sphere center is obtained by virtually probing 5 points on the sphere surface as shown in section 4.1.1. The CMM positioning error and laser probe indication error is set as 0.002mm and 0.004mm respectively. The relationship between the final result and the initial laser beam direction is shown in Fig. 8.
16
0.14
Final angle error (deg)
0.12
0.1
0.08
0.06
0.04
0.02
0
0
5
10
15
20
25
30
35
40
45
Initial angle error (deg)
Fig. 8. The relationship between the initial angle error and the final angle error
It can be seen in the figure that the final angle error does not increase with the rising of the initial angle error. Under the actual situation, it is impossible that the initial angle error which is mainly caused by mechanical structure will be larger than 45 degrees. It can be concluded that the propose method remains robust when the initial laser beam direction is greatly different from the ground truth value.
4.2 Calibration accuracy of different methods In this section, out proposed method is compared with two existing calibration methods. In Bi’s method [10], the CMM is controlled to moves at an equal step along X, Y and Z axes respectively. Then equation sets are established to calculate the laser beam direction. According to Bi’s method, the following parameters are chosen in the simulation. The step lengths for X, Y and Z axes are chosen as 0.5mm, 0.3mm and 0.45mm respectively. The step numbers for three axes are chosen as 12. In Xie’s method [11], optical probe is regarded equivalent to the contact probe and used to inspect a reference sphere with a series of fixed measuring lengths. According to Xie’s method, the measuring lengths are chosen as 0.6mm, 1.2mm 1.8mm, 2.4mm and 3mm in the
simulation. The initial value of the proposed method is set as cos sin 60 60
T
0 . In
the first simulation, the CMM positioning error is fixed as 0.002mm when the probe indication error rises from 0 to 0.006mm. In the second simulation, the probe indication error is fixed as 0.004mm when the CMM positioning error rises from 0 to 0.005mm. The influence of the measuring error including CMM positioning error and laser probe indication error on the calibration result is analyzed and illustrated in Fig. 9. 17
3.5
3 Bi's method Xie's method Proposed method
2.5
2
1.5
1
2
1.5
1
0.5
0.5
0
Bi's method Xie's method Proposed method
2.5
Final angle error (deg)
Final angle error (deg)
3
0
1
2
3
4
5
0
6
0
1
2
Probe indication error (um)
3
4
5
6
CMM positioning error (um)
(a)
(b)
Fig. 9. The comparison of the proposed method and the existing methods under different noise. (a) The comparison of different methods under different laser indication errors. (b) The comparison of different methods under different CMM positioning errors.
Then the CMM positioning error and the laser probe indication error are set as 0.002mm and 0.004mm respectively which is similar with the actual condition. Other parameters are set as above. Three methods are compared in 20 simulation tests. The result is shown in Fig. 10. 4.5 Bi's method Xie's method Proposed method
4
Final angle error (deg)
3.5 3 2.5 2 1.5 1 0.5 0
0
2
4
6
8
10
12
14
16
18
20
Index
Fig. 10. The comparison of the proposed method and the existing methods under the same noise.
As can be seen in the Fig. 9, the proposed method performs best in the calibration accuracy. In Fig. 10, the mean calibration error of the three calibration methods are 1.5268°, 0.534°and 0.0691°respectively. The proposed method gains 95.5% and 87.1% improvement compared with the existing methods. More specifically, the indication interval of the laser probe is less than 4mm. Thus the angle error of 0.0691 degree will result in a deviation of less than 4.8 m when measuring a single point. 18
In Bi’s method, the laser beam direction is calculated by solving the equation sets. The result is not accurate enough and is easily worsen by noise. In Xie’s method, the laser beam direction is calculated by mapping the point cloud into a series of sphere. The key of this method is to keep the measuring length unchanged, which requires iterative compensation of CMM positions in experiment. This method is time-consuming and difficult to realize because of the existence of noise. Thanks to the scanning and optimization procedure, the noise influence is significantly relieved by a large number of points in the proposed method. Moreover, scanning inspection rather than point-by-point inspection is applied in the proposed method, which reduces the time spent by the air-moves of the CMM. Thus the proposed method is superior in efficiency, accuracy and robustness to the existing methods.
5. Experiments and Verification 5.1 Calibration experiment The laser probe used in the experiment is Precitec CHRocodile C. CHRocodile C is suited for industrial use and is easily integrated into any kind of inspection machine. The technical parameters of this laser sensor are listed in Table 2. The accuracy of the CMM used in the experiment is verified to be
2 4 L /1000 m
where L is the measuring
length. The probe is mounted along two different directions and calibrated with three different algorithms respectively. The reference sphere used in the calibration experiment is a ceramic ball with a diffuse surface so that the laser probe can work stably. The diameter of the reference sphere is 40.040 mm and the form error is 0.0036 mm. The calibration experiment is conducted in a constant-temperature room, so the variations of the environmental conditions during the full calibration run has little influence on the instrument. The calibration configuration is shown in Fig. 11.
19
Fig. 11. Calibration configuration.
According to Bi’s method, the following parameters are chosen. For direction 1, the step lengths for X, Y and Z axes are chosen as 2.4mm, 1.3mm and 0.9mm respectively. For direction 2, the step lengths for X, Y and Z axes are chosen as 0.4mm, 2.4mm and 2.9mm respectively. The step numbers for three axes are chosen as 3. According to Xie’s method, the measuring lengths are chosen as 0.8mm, 1.8mm, 2.8mm for both direction 1 and direction 2. The calibration results are listed in Table 3. The calibration results will be verified by measuring a standard sphere and a step gauge block in the next section. The spent time of these methods is 21s, 135s and 18s respectively. The CMM moving velocity and measuring velocity are totally the same during the entire experiments. Xie’s method costs the longest time because the laser length should be adjusted iteratively to meet the requirement of Xie’s algorithm. The proposed method costs the least time due to using scanning inspection in the calibration procedure. Table 2. Specifications of the laser probe
Descriptions Measuring range Working distance Spot diameter Lateral resolution Axial resolution Accuracy* Measurement angle
Specifications 4 mm 32 mm 8 um 4 um 200 nm 1.2 um
90 20
* Accuracy corresponds to the maximum reading deviation between the laser probe CHRocodile C and a calibrated interferometric reference sensor.
20
Table 3. Calibration results using different algorithms
Bi’s method Xie’s method Proposed method
Direction 1
Direction 2
[0.012004 0.005386 -0.99991] [-0.01085 0.001839 -0.99994] [-0.01394 0.004652 -0.99989]
[-0.99921 0.028263 -0.02798] [-0.99941 -0.00455 0.033976] [-0.99938 0.015429 0.031648]
5.2 Accuracy verification In this section, the accuracy of the laser beam direction calibrated in section 5.1 is verified. A standard sphere and a step gauge block are measured. The standard sphere is another similar ceramic ball whose diameter is 40.038mm. The form error of this sphere is 0.0042mm. The gauge block used is a metallic step gauge block whose step lengths are 20.001mm, 30.001 and 39.996mm. The parallelism between the corresponding pairs is 0.002mm, 0.003mm and 0.002mm. The ground truth value of the sphere radius and the gauge block length is obtained by contact measurement of a high-precision CMM. The measuring scene is shown in Fig. 12. The measuring results of the sphere and gauge block are listed in Table 4 and Table 5 respectively.
(a)
(b)
Fig. 12. The verifying test configuration of a step gauge block. (a) The verifying test in direction 1. (b) The verifying test in direction 2.
Table 4. Measurement results of the standard sphere (mm)
Bi’s method Xie’s method Proposed method
Direction 1 Diameter 40.022 40.027 40.036
Error -0.016 -0.011 -0.002 21
Direction 2 Diameter 40.069 40.046 40.042
Error 0.031 0.008 0.004
Table 5. Measuring results of the step gauge block (mm)
Direction 1 Length Bi’s 20.061 method 30.044 40.018 Xie’s 20.039 30.035 method 40.052 Proposed 19.989 30.005 method 40.002
Error 0.06 0.043 0.022 0.038 0.034 0.056 -0.012 0.004 0.006
Direction 2 Length 20.049 30.074 40.006 20.036 30.042 39.913 19.996 30.006 39.996
Error 0.048 0.073 0.01 0.035 0.041 -0.083 -0.005 0.005 0
It is obvious that the measuring results using the laser direction calibrated by the proposed method is superior to which calibrated by the existing methods. The mean measuring errors of the fitted sphere diameter and the gauge block length are 0.003mm and 0.0054 mm respectively. Unlike the contact measurement, the non-contact measurement by laser probe is also affected by the environment light and material reflectivity. Thus, the actual measurement results may fluctuate according to the actual situation.
6. Conclusions Non-contact measurement has been widely used for those parts with free-form surface due to its capability of efficiently obtaining surface information. Among the optical measurement devices, the spot laser probe achieves a good balance between the accuracy and efficiency. In order to get the information of the 3D surface point from the laser probe indication, the laser beam direction should be precisely calibrated. In this paper, a robust and efficient laser beam direction calibration method is proposed. Compared to the previous researches, the proposed calibration method has the following contributions. In terms of robustness, an initial inaccurate direction specified by the probe fixture is refined by mapping a large number of sample points into a sphere so that the final result is robust to the measuring noise and reaches micron-level accuracy. In terms of efficiency, scanning inspection rather than point-by-point inspection is applied in the calibration procedure so that the proposed method is much faster than the existing methods. With the sensitivity analysis, the proposed method is proved to be insensitive to various measuring noise. Several simulations and experiments have been carried out to 22
verify the effectiveness of the proposed method. Compared to the two existing methods, the proposed method improves the calibration accuracy by 87.1% in simulation. In the actual experiment, the mean measuring errors of the fitted sphere diameter and the gauge block length are 0.003mm and 0.0054 mm which is superior to the existing methods. Additionally, the proposed method can be applied not only in the calibration of the spot laser probe but also in the calibration of other displacement sensors such as ultrasonic sensors.
7. Acknowledgement This research was supported by the National Nature Science Foundation of China (Grant no. 51975344).
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Conflict of interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
CReDiT Author Statement Yijun Shen: Conceptualization, Methodology, Writing - Original Draft Xu Zhang: Funding acquisition, Writing - Review & Editing Zhenyou Wang: Formal analysis Jianxiang Wang: Investigation Limin Zhu: Supervision, Project administration
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