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Abstract

As a key part of the articulated arm coordinate measuring machine, the probe can determine the measurement accuracy. Therefore, the error source and influencing factors of the equivalent diameter of the probe are studied. First, the influence of the primary factor of the measuring force on the equivalent diameter of the probe is studied by analyzing the influence degree of its error source. Second, a mathematical model of the relationship between the equivalent diameter and the measuring force of the articulated arm coordinate measuring machine is built to compensate for the equivalent diameter error caused by the measuring force by the simulated annealing method. To illustrate the application advantage of our proposed study, a simple force-measuring device is designed based on this model. The experimental result shows that the maximum error reduction is approximately 43 µm, while the average error reduction ranges from 33 to 4.0 µm, which represents an 87.7% improvement. Overall, our proposed method can effectively compensate for the equivalent diameter error caused by the measuring force. This method can improve the accuracy of the articulated arm coordinate measuring machines on both calibration and measurement.

I. Introduction

The articulated arm coordinate measuring machine (AACMM) is a coordinate measuring system based on a rotating joint and a rotating arm, which replaces the length measurement reference with an angle measurement reference. Compared with the traditional coordinate measuring machine (CMM), the AACMM has the advantages of a simple mechanical structure, small volume, high efficiency, lightweight, large measuring range, flexibility, convenience, low cost, and field measurement capability. It is widely used in automobiles, ships, mechanical processing, and other areas. AACMM is composed of a set of rigid bodies connected by joints and realizing a kinematic chain. Each joint realizes a rotation.¹ The AACMM combines the robot arm with the CMM to have the structural characteristics of the manipulator’s flexible operation and highly accurate coordinate measurement capability, both of which are widely developed and applied characteristics. The probe system consists of a probe and a subsidiary body. The accuracy of the probe, to a large degree, determines the measurement accuracy and repeatability of the AACMM. Many studies have shown that the factors that affect the accuracy of the probe include the geometry of the probe, the direction of the probe contact, and the measuring force.^2–4 Domestic and foreign scholars have performed research on the trigger probe but have primarily considered CMM. Du et al. successfully compensated for the radius of the trigger probe. The method combining dynamic calibration of probe radius and micro plane compensation was put forward to make compensation for the probe radius.⁵ Zheng et al.⁶ improved the measurement efficiency and accuracy of the probe using a simple cone socket component, a calibration tool, and the Levenberg–Marquardt (LM) algorithm, which was edited to solve the probe parameters. Huang et al.⁷ proposed a new three-dimensional (3D) resonant trigger probe based on a quartz tuning fork to achieve true 3D nano-measurement with sub-nanometer resolution and very low touch force through a micro/nano-CMM. With a touch-trigger probe on a micro-CMM, the probe cannot enter into the cavity if the tip-ball diameter is larger than the cavity size. Zhang et al.⁸ presented a vision-assisted noncontact focusing probe to deal with this problem. Xu and Chen⁹ used the CMM to measure the diameter of the device under test (DUT) and described the sources of uncertainty that affect the measurement results. Küng et al. thought that the role of the probe tip was particularly crucial because it was in contact with the sample surface. Understanding how the probe tip wears off would help to narrow the measurement errors.¹⁰ González-Madruga et al.¹¹ analyzed the effect of measuring the force on the diameter by installing a force sensor on the probe.

The above works examine the probe of the measuring machine, and these research insights may also improve the probing accuracy to a certain extent. However, there are certain shortcomings, including the ignorance of thermal deformation,^12–15 installation eccentricity, and measuring force. The stability of AACMM, the choice of probe type, the measurement point acquisition strategy, and the probe processing level all impact the measurement accuracy during the measurement process. Even for the same object, it is impossible to achieve the same results every time. Simultaneously, the measuring force is an important factor affecting the accuracy of AACMM. Local deformation of the probe and the DUT is caused by the presence of the measuring force during the contact measurement. Here, the rod is bent and deformed, which means that the true geometry of the AACMM probe does not match its kinematic equation. We can reduce the probe error by periodically calibrating the AACMM, but the effect of the measuring force is usually overlooked during the calibration and measurement process, which does not compensate for the measuring error.

In this article, first, the composition, working principle, and kinematic equation of the AACMM were introduced. Then, the error of the equivalent diameter of the AACMM was studied by analyzing the eccentricity of the probe and the influence of the thermal deformation and the measuring force on the equivalent diameter, and the maximum influence of measuring force was obtained. Finally, by designing a simple force-measuring device, the measurement coordinates of the AACMM and the corresponding measuring force are built to build the functional relationship between the equivalent diameter and measuring force to realize the measuring force error compensation. The results show that the method is effective.

II. The Working Principle and Kinematic Equation of AACMM

A. Working principle

The AACMM consists of a base, two measuring arms, six angle encoders, and one probe. The utility model is formed by connecting a fixed length arm with an axis perpendicular to each other. Finally, the coordinate measurement is realized by the probe system. In the detection of a fixed point space, AACMM and CMM are completely different. When the position of the probe is determined, the positions of the X, Y and Z axis of the CMM are uniquely determined. However, the spatial positions of each measuring arm of AACMM have infinite combinations, which cannot be uniquely determined. However, the AACMM measuring the probe a fixed space point there is an infinite combination, which means that the angle and position of each arm in space is infinite, not the only. In addition, the distance of the probe from each joint is different. The influence of different levels of the angle error on the measurement results is different. The closer the cornering error of the joint at the base, the greater the influence on the measurement result. A simple AACMM structure is shown in Figure 1 . The physical map is shown in Figure 2 .

Figure 1.

Schematic diagram of AACMM.

Figure 2.

AACMM of Hexagon-Infinite 2.0.

B. Kinematic equation

The motion equation of the AACMM is built using the mathematical model of D-H, which was proposed by Denavit and Hartenberg in 1955 to study the relative position relation of two adjacent parts.¹⁶ The coordinates of the mathematical model of the AACMM are shown in Figure 3 .¹⁷ The calibration value of the main structural parameters of the AACMM is shown in Table 1 .

Figure 3.

Coordinate diagram of the AACMM: (a) front view and (b) side view.

Table 1.

Structural parameters of AACMM (mm).

Structural parameters	Value
l ₁	267.265
l ₃	747.489
l ₅	531.880
d ₂	78.312
d ₄	78.305
d ₆	25.962
l	176.484

According to the D-H mathematical model, the adjacent coordinate systems ${X_{i - 1}, Y_{i - 1}, Z_{i - 1}}$ and ${X_{i}, Y_{i}, Z_{i}}$ can be transformed from the coordinate system ${X_{i - 1}, Y_{i - 1}, Z_{i - 1}}$ to ${X_{i}, Y_{i}, Z_{i}}$ by the translation–rotation of the coordinate system (as shown in Figure 4 ).

Figure 4.

Coordinate transformation.

According to the principle of coordinate transformation, it is concluded that the conversion matrix is as follows¹⁸

\begin{matrix} A_{i - 1, i} = R o t (Z_{i - 1, θ_{i}}) T r a n s (0, 0, d_{i}) T r a n s (l_{i}, 0, 0) R o t (X_{i}, α_{i}) \\ = [\begin{matrix} \cos θ_{i} & - \sin θ_{i} \cos α_{i} & \sin θ_{i} \sin α_{i} & l_{i} \cos θ_{i} \\ \sin θ_{i} & \cos θ_{i} \cos α_{i} & - \cos θ_{i} \sin α_{i} & l_{i} \sin θ_{i} \\ 0 & \sin α_{i} & \cos α_{i} & d_{i} \\ 0 & 0 & 0 & 1 \end{matrix}] \end{matrix}

(1)

where $A_{i - 1, i}$ is the conversion matrix; $l_{1}, l_{3}, and l_{5}$ are the lengths of the arm between adjacent joints; l is the length of the probe; $d_{2}, d_{4}, and d_{6}$ are the lengths of the joint offset; $α_{i}$ is the twist of the joint; and $θ_{i}$ is the corner of the joint. Assuming that the coordinates of the probe in the ${X_{6}, Y_{6}, Z_{6}}$ coordinate system are ( $l_{x}, l_{y}, l_{z}$ ), the position of the probe core is relative to the base coordinate system ${X_{0}, Y_{0}, Z_{0}}$ , that is

[\begin{matrix} x \\ y \\ z \\ 1 \end{matrix}] = A_{0, 1} A_{1, 2} A_{2, 3} A_{3, 4} A_{4, 5} A_{5, 6} [\begin{matrix} l_{x} \\ l_{y} \\ l_{z} \\ 1 \end{matrix}]

(2)

III. Equivalent Diameter

The principle of probe calibration is to measure the true diameter of the probe by measuring a standard part. The calibration of the device has a ring, ring gauge, and standard ball to achieve a full range of probe calibrations, so the standard ball is often chosen. According to the 2013 version of the “Chinese articulated arm coordinate measuring machine calibration standard,” calibrating the probe in the standard ball requires collecting at least five points, that is, one point in the ball pole and the other four on the equator, as shown in Figure 5 . The measurement software will fit a ball according to the collected spherical coordinates and can be used to calculate the diameter of the fit ball. The difference between the calculated spherical diameter and the standard ball diameter is the equivalent diameter of the calibrated probe.

Figure 5.

Probe calibration.

A simple length measurement as an example¹⁹ is shown in Figure 6 . Assume that the outer dimension length and the inner one of the DUT measured by AACMM are L₁ or L₂, respectively. To obtain the actual length L of the DUT, it is necessary to subtract or add a probe diameter on this basis, denoted as $L = L_{1} - d_{0}$ or $L = L_{2} + d_{0}$ , respectively. However, due to temperature changes, probe eccentricity, measuring force, and other factors, the actual lengths of the AACMM measurements are $L_{1}^{'}$ or $L_{2}^{'}$ . Assuming that the probe error due to the above factors is e, the equivalent diameter of the probe should be $d = d_{0} - 2 e$ . The true value of the outer dimension is $L = L_{1}^{'} - d$ , and the true value of the inner dimension is $L = L_{2}^{'} + d$ .

Figure 6.

Diameter error.

IV. Error Analysis of Equivalent Diameter

When the AACMM measures the coordinates, it usually does so at the probe center. We have to get the feature points at the probe’s contact point and the DUT. The difference between these points is half of the equivalent diameter. The compensation measure is to measure the coordinates of the probe center along the measurement direction to obtain a feature point of the DUT. The probe thermal deformation, probe installation eccentricity, contact measuring force, and many physical factors can change the equivalent diameter of the probe. Figure 7 shows the primary error sources of the probe and the influence that each factor has on the probe equivalent diameter.

Figure 7.

Influencing factors of probe equivalent diameter.

Due to experimental equipment and environmental, human, and other factors, the probe equivalent diameter is slightly smaller than the nominal diameter, which is called “equivalent diameter.” During the length measurement, the measured length is subtracted from the nominal diameter of the probe, which results in the inaccuracy of the DUT dimension. In fact, the measured length should be subtracted from the equivalent diameter. Based on the above analysis, this article focuses on the effects that the thermal deformation, eccentricity, and measuring force have the equivalent diameter of the AACMM. The most important factors influencing the equivalent diameter are analyzed theoretically.

A. Thermal deformation error

Temperature affects the performance and precision of mechanical instruments and is always a physical phenomenon. We can take steps to reduce temperature effects, but they cannot be eliminated. The AACMM and other precision instruments are also affected by temperature changes. The primary focus of the following section is to discuss the thermal deformation of the probe diameter. According to the traditional thermal deformation formula^20,21

Δ D_{t} = η D_{0} Δ t + β D_{0} {(Δ t)}^{2} + γ D_{0} {(Δ t)}^{3} + \dots

(3)

where ƞ, β, and γ are the first-order polynomial, quadratic polynomial, and cubic polynomial of the material temperature linear expansion coefficient, respectively; $D_{0}$ is the nominal diameter of the probe; $Δ_{t}$ is the temperature change, and $Δ D_{t}$ is the thermal deformation error. Equation (3) is usually approximated as

Δ D_{t} = η D_{0} Δ t

(4)

Assuming that the temperature change is 1°C, the calculated thermal deformation error of the probe is approximately 0.047 µm. It is concluded that the thermal deformation has little or no effect on the accuracy of the AACMM probe when the temperature change is small. In the experiment, we ensured that the AACMM is measured at constant temperature and humidity. Thus, we could strictly control the experimental environment to reduce the thermal deformation of the AACMM accuracy, so we could ignore the temperature changes on the equivalent diameter of the probe.

B. Eccentricity error

Ideally, the AACMM measures the probe center. However, both the probe error and the installation of the measuring rod lead to a deviation of the actual location of the probe center from the ideal position. This deviation results in the eccentricity of the probe installation error, as shown in Figure 8 . Theoretically, the ideal position of the probe should be point O, but the actual installation position is O₁ due to machining or probe mounting errors, which results in an eccentricity error $Δ {OO}_{1}$ when the feature point P is measured. To avoid the eccentricity of the probe installation, we can improve the probe level to reduce the error or use consistent research methods to achieve the eccentricity of the probe identification and compensation. However, as the AACMM manufacturing process level continues to improve, the probe eccentricity error is only a few micrometers, and we may ignore the probe erection on the equivalent diameter of the impact.

Figure 8.

Schematic diagram of the eccentricity of the probe.

C. Measuring force

The influence of the measuring force on the equivalent diameter is primarily manifested in the deflection of the measuring rod, as shown in Figure 9 . The A state is the theoretical position of the measuring rod when the measuring force F = 0, while the B state is the operative deflect position of the rod when the measuring force F ⩾ 0. The AACMM is conducted in the B state when the coordinate is measured, and the theoretical calculation according to the D-H mathematical model is as in the A state, resulting in measurement errors. If this error is equivalent to the probe diameter, the probe diameter error is formed. According to the manufacturer’s specifications, if the other factors are not considered, the measurement force error is simply analyzed from the mechanical point of view, as shown in Figure 10 .

Figure 9.

The bending deformation of the measuring rod.

Figure 10.

Schematic diagram of bending deformation of the rod.

In Figure 10 , L_P is the rod length; D is the rod diameter; F_Y and F_Z are the projections of F in the XY plane and XZ plane, respectively; and q is the angle between the measuring force F and the Y-axis. Under the action of the measuring force F, the measuring rod is bent and deformed, and the transverse displacement $ω_{y}$ and the axial compression displacement $ω_{z}$ are generated in the Y and Z directions, respectively, which can be expressed as

ω_{y} = F_{Y} \frac{L_{P}^{3}}{3 E J}

(5)

ω_{Z} = F_{Z} \frac{4 L_{P}}{π D^{2} E}

(6)

J = \frac{π D^{4}}{64}

(7)

where E is the elastic modulus of the measuring rod and J is the inertia moment of the cross section of the rod. According to the AACMM shown in Figure 2 , the rod length of the AACMM is 15 mm, the rod diameter is 2.5 mm, and the probe diameter is 6 mm. The rod is made of stainless steel, and its elastic modulus is 190 GN/m². Bringing together the above parameters, the 1 N transverse force to bring the lateral displacement error is approximately 3.1 µm, and the 1 N axial force to bring the axial compression displacement is approximately 0.0161 µm. In the common measurement process, assume that the measurement force applied to the probe is 5 N, resulting in a single point error of approximately 15.5 µm, and a length error of approximately 31 µm. The degree of influence of the measured force on the equivalent diameter is greatest compared with the thermal deformation error and eccentricity.

V. Equivalent Diameter Measuring Force Experiment

Through the above analysis, the measuring force has the greatest influence on the equivalent diameter of the probe. To quantitatively analyze the influence of the measuring force on the equivalent diameter, the external dimension measurement is selected as the research object. The equivalent diameter is identified by the indirect measurement method. The research method is as follows: Use the AACMM to measure the calibrator length. The measured length minus the calibrator length is the equivalent diameter of the AACMM probe, which is expressed as $d = L_{1}^{'} - L$ . Change the measuring force to repeat the measurement and obtain the relationship between the equivalent diameter and the measuring force.

A. Calibration experiment by CMM

We design a simple force-measuring device, as shown in Figure 11 . The two high-precision pressure sensors were fixed to the two ends of the metal base, and the metal base was fixed to the test bench and equipped with a display module to obtain the measuring force. The distance between two planes was measured by two sensors. The experimental data were collected on the sensor measurement plane. To avoid introducing unnecessary influence factors, we chose to measure the experimental data within the circle of a radius 3 mm from the sensor-measuring surface. With this force-measuring device, the AACMM could be achieved in the coordinate measurement while obtaining the corresponding measurement force. Then, we obtained the trend of the equivalent diameter with the measured force.

Figure 11.

A simple force-measuring device.

Before the experiment, it was necessary to test and calibrate the force-measuring device to check whether it met the experimental requirements. Considering that the detection object is the shape and error of the two pressure sensor measurement surfaces, it primarily included the flatness of the sensor-measuring surface and the parallelism between the sensor measurement planes. When the flatness error and the parallelism error met the experimental requirements, the distance between the measurement planes could be calibrated. A more accurate CMM of the Global Class 9158 for verification and calibration is used, as shown in Figure 12 . The steps for verification and calibration are as follows: First, 50 coordinate points on the left $P_{N} (N = 1, 2, \dots, 50)$ and 50 points on the right $P_{M} (M = 1, 2, \dots, 50)$ of the sensor measurement plane are randomly measured (the measured points are within the limited range). Then, on the left side of the sensor, 5 of the measured 50 coordinates are taken for plane fitting. A total of 10 flat planes $V_{i} (i = 1, 2, \dots, 10)$ were fitted. On the right side of the same sensor, a total of 10 planes $V_{j} (j = 1, 2, \dots, 10)$ were fitted. Finally, the flatness $V_{i}$ and $V_{j}$ of the two pressure sensors and the parallelism $V_{i, j}$ of the two planes were verified, and the calibration distance L between the planes was calculated. The measured form and position error and calibration distance are shown in Figures 13 and 14 .

Figure 12.

Distance calibration by CMM.

Figure 13.

Form and position error of the two sensors.

Figure 14.

Standard distance between the two planes of the two sensors.

The test results are shown in Figure 13 , and the calibration results in Figure 14 indicate that the two measurement surfaces of the maximum flatness error of 0.004 mm have an average error of 0.0017 and 0.0021 mm, respectively. The maximum parallelism error between the measurement planes is 0.007 mm, and the average error is 0.0050 mm. The flatness error and parallelism error are relatively small, so we can ignore their impact on the experimental results. According to the experiment, the average of the 10 calibration results is considered the calibration distance between the measurement planes, denoted as L ≈ 186.201mm.

B. Effect of measuring force on equivalent diameter of AACMM

The influence of measuring the force on the equivalent diameter by the AACMM of the Hexagon-Infinite 2.0 is analyzed. The nominal diameter is 6 mm, as given by the manufacturer. The experimental scenario is shown in Figure 15 .

Figure 15.

Equivalent diameter experiment by AACMM.

To analyze the effect of the measuring force on the equivalent diameter of the AACMM, the experimental method is as follows:

First, $N (N ⩾ 3)$ coordinate points $P_{i} (i = 1, 2, \dots, N)$ are measured at approximately the same measuring force on the measurement plane of the left sensor, and the measured points are fitted into the plane V by the least squares fitting method. Next, a coordinate point $P_{k} (x_{k}, y_{k}, z_{k})$ is measured on the measurement plane of the right sensor with a measuring force of approximately the same size as the left side. Finally, according to the formula of a point to the surface, we calculate that the distance from $P_{k} (x_{k}, y_{k}, z_{k})$ to the least squares plane V is the AACMM measured value $L^{'}$ . To continue the iterations, we change the measurement force to repeat the experiment. The experimental part of the measurement data is shown in Table 2 .

Table 2.

Measuring force and the corresponding distance.

	Left force (N)	Right force (N)	Distance (mm)		Left force (N)	Right force (N)	Distance (mm)		Left force (N)	Right force (N)	Distance (mm)
1	0.40	0.39	192.186	34	3.82	3.80	192.179	67	7.46	7.46	192.161
2	0.55	0.54	192.188	35	3.90	3.92	192.184	68	7.52	7.52	192.159
3	0.63	0.63	192.195	36	4.10	4.12	192.177	69	7.65	7.64	192.160
4	0.75	0.75	192.190	37	4.22	4.17	192.177	70	7.73	7.75	192.162
5	0.81	0.85	192.184	38	4.33	4.32	192.181	71	7.83	7.86	192.157
6	0.99	0.93	192.176	39	4.47	4.37	192.179	72	7.97	7.94	192.152
7	1.14	1.13	192.186	40	4.53	4.51	192.170	73	8.14	8.16	192.157
8	1.24	1.26	192.194	41	4.61	4.59	192.174	74	8.23	8.22	192.151
9	1.30	1.30	192.187	42	4.69	4.71	192.176	75	8.32	8.37	192.166
10	1.45	1.45	192.186	43	4.82	4.82	192.175	76	8.42	8.42	192.161
11	1.55	1.51	192.192	44	4.87	4.94	192.170	77	8.53	8.52	192.158
12	1.65	1.65	192.185	45	5.04	5.01	192.164	78	8.63	8.60	192.159
13	1.76	1.75	192.193	46	5.11	5.10	192.171	79	8.78	8.74	192.162
14	1.85	1.84	192.190	47	5.25	5.25	192.164	80	8.88	8.79	192.161
15	1.92	1.92	192.189	48	5.32	5.30	192.167	81	8.91	8.93	192.156
16	2.05	2.03	192.183	49	5.45	5.45	192.172	82	9.13	9.14	192.151
17	2.16	2.16	192.188	50	5.56	5.58	192.175	83	9.25	9.25	192.153
18	2.25	2.25	192.184	51	5.67	5.60	192.172	84	9.34	9.32	192.153
19	2.32	2.31	192.189	52	5.70	5.72	192.181	85	9.41	9.38	192.148
20	2.46	2.47	192.194	53	5.87	5.82	192.177	86	9.59	9.56	192.152
21	2.54	2.52	192.188	54	6.02	6.07	192.174	87	9.66	9.62	192.154
22	2.65	2.67	192.192	55	6.18	6.14	192.173	88	9.76	9.75	192.152
23	2.71	2.75	192.197	56	6.25	6.23	192.170	89	9.85	9.82	192.154
24	2.82	2.85	192.186	57	6.33	6.32	192.169	90	9.95	9.92	192.152
25	2.93	2.96	192.180	58	6.41	6.45	192.163	91	10.13	10.14	192.145
26	3.05	3.07	192.177	59	6.53	6.52	192.160	92	10.28	10.23	192.143
27	3.15	3.16	192.181	60	6.64	6.62	192.156	93	10.35	10.36	192.143
28	3.26	3.25	192.171	61	6.70	6.74	192.157	94	10.47	10.41	192.145
39	3.35	3.33	192.180	62	6.86	6.85	192.161	95	10.53	10.48	192.145
30	3.43	3.44	192.184	63	6.91	6.95	192.166	96	10.59	10.62	192.144
31	3.53	3.51	192.182	64	7.11	7.09	192.163	97	10.72	10.74	192.149
32	3.63	3.60	192.180	65	7.26	7.23	192.164	98	10.83	10.82	192.144
33	3.77	3.72	192.181	66	7.33	7.31	192.165	99	10.90	10.93	192.146

According to the data in Table 2 and the formula d = L′ − L, L = 186.201 mm, the corresponding relationship between the equivalent diameter d and the measured force F is obtained. The experimental results of the equivalent diameter and the nominal diameter are shown in Figure 16 .

Figure 16.

Effects of measuring force on equivalent diameter of the AACMM.

Figure 16 shows that the equivalent diameter decreases with an approximately linear trend as the measured force increases. Additionally, the test results are smaller than the nominal diameter. This discrepancy explains why the equivalent diameter of the probe is less than the nominal diameter during the calibration process. The results also show that the maximum error caused by the measuring force on the equivalent diameter of the probe is approximately 65 µm. This force has greater influence on the equivalent diameter of the probe and thus cannot be ignored.

VI. Error Compensation

To reduce the influence of the measuring force on the equivalent diameter, we must compensate for the equivalent diameter error caused by the measuring force. According to the theoretical analysis of the equivalent diameter error and the experimental results in Figure 16 , the equivalent diameter is linearly related to the measuring force. This linear relationship can be obtained by establishing the equivalent diameter and the measuring force database. The simulated annealing method is used to find the slope and intercept of the best fitting line, and the data are fitted.

The simulated annealing was first proposed by Metropolis et al. in 1953. This approach is a stochastic optimization algorithm based on the Monte-Carlo iterative solution strategy. The basic idea of simulated annealing is to find the global optimal solution of the objective function in all the solution space with the initial temperature increase and the temperature decrease or to jump out of the local optimal solution with a certain probability to achieve the global optimum.

The function $f (F; k, b) = k F + b$ containing the parameter to be determined is constructed according to the above experimental data $(F_{i}, d_{i}), i = 1, 2, \dots$ , where k is the slope of the fitted straight line and b is the intercept of the straight line. The set objective function H is the degree of deviation from $(F_{i}, d_{i}), i = 1, 2, \dots, n$ , to the fitting function $f (F; k, b)$

H = \sum_{i = 1}^{n} {[d_{i} - f (F_{i}; k, b)]}^{2}

(8)

According to equation (8), the constants k and b are the parameters that determine the relationship between the equivalent diameter and the measuring force, that is, the optimized variable. Its optimization core is as follows: (1) provide the initial solution of the optimization variable; (2) iteratively optimize the variables along the appropriate optimization path, forcing the objective function to proceed in the optimal direction; and (3) when the objective function infinitely approaches zero, the optimal solution is obtained by setting the optimal termination condition. The primary flow is shown in Figure 17 .

Figure 17.

Line fitting compensation model based on simulated annealing.

According to the experimental data collected, set the initial temperature t = 900K, the attenuation factor w = 0.95, the total number of iterations N = 1000, and the initial values are $k_{0} = - 0.0055$ and b₀ = 5.99. Simulated annealing linearly fits the simulated data, and the deviation degree of each data point to the fitted straight line is compared. The optimal slope and intercept of the fitting straight line are −0.00498 and 5.99519, respectively. Finally, the equivalent diameter and the measuring force of the fitting line are shown in Figure 18 , and the function of the relationship is

d = 5.99519 - 0.00498 F

(9)

Figure 18.

Equivalent diameter fitting.

Define the difference between the equivalent diameter and the nominal diameter of the experimental acquisition as the equivalent diameter error. Equation (9) is used as the compensation model of the equivalent diameter of the AACMM probe, which compensates for the equivalent diameter error caused by the measuring force. The error of the equivalent diameter caused by the measuring force is compensated.

Figure 19 shows that the compensation method can compensate the equivalent diameter error to a certain extent. To compare the effect of error compensation, select the compensation effect in González-Madruga et al.¹¹ In González-Madruga et al.,¹¹ the authors proposed a probe deflection model based on the finite element method (FEM) in order to obtain the AACMM probe deflection caused by measuring force. From the point of view of theoretical analysis, the measurement error of probe diameter was compensated. Through his research, the precision of the AACMM is improved to a certain extent. However, this error compensation model is based on theoretical analysis, and the effect of compensation is not very satisfactory. The error compensation model in this article is based on the experimental data, the compensation model is more in line with the currently used AACMM, and the compensation effect is better. A comparative analysis of this article and González-Madruga et al.,¹¹ the compensation effect, is shown in Table 3 .

Figure 19.

Equivalent diameter error compensation before and after.

Table 3.

Comparative analysis of this article and González-Madruga et al.,¹¹ equivalent diameter error compensation before and after (mm).

	Comparison object	Before error compensation	After error compensation
This article	Maximum	0.063	0.020
	Minimum	0.004	0.000
	Average	0.033	0.004
González-Madruga et al.¹¹	Maximum	0.054	0.049
	Minimum	0.029	0.039
	Average	0.024	0.011

Table 3 shows that the error of the equivalent diameter is compensated such that the maximum error of the equivalent diameter is reduced by approximately 43 µm, and the average error is reduced by 29 µm. On average, the probing accuracy increased by approximately 87.7%. Comparing with the González-Madruga et al.,¹¹ we can see that the error compensation method proposed in this article is slightly better. In terms of the average value of compensation, the average error after compensation in this article is 0.004 mm, which is better than the compensation error of 0.011 mm in González-Madruga et al.¹¹

VII. Conclusion and Future Work

The influencing factors of the equivalent diameter during the calibration period and measurement of the AACMM are analyzed. This analysis shows that the measuring force is the primary source of the equivalent diameter of the probe. The innovation of this study is to design a force-measuring device to realize the combination of force and coordinate measurements. The relationship between the equivalent diameter and the measuring force is defined. The equivalent diameter error caused by the measuring force is also compensated. The compensation method is verified by theoretical and experimental analyses. Our suggested method can compensate for the equivalent diameter error to a large extent and has laid a foundation for both calibration and measurement precision improvement of the AACMMs.

This research is not only for the specific AACMM but also for any AACMM when calibrating the probe. This method can be used to obtain the functional relationship between the equivalent diameter of the probe and the measuring force. For different physical materials and dimensions of the probe, the functional relationship may be different, but the method is generally applicable. At present, the research method can be realized on the basis of theory and experiment, but the actual engineering application still needs further study. In the follow-up study, we plan to apply this method to the probe calibration of the AACMM. The measuring force calibration module will be added to the probe calibration software system to further improve the calibration accuracy of the probe and through the calibration of the measuring force to improve the AACMM’s measurement accuracy.

Footnotes

Funding

This work was supported by the National Natural Science Foundation of China (project 51675499 and project 51405463) and Natural Science Foundation of Zhejiang Province (project LY15E050013).

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Analysis and Compensation of Equivalent Diameter Error of Articulated Arm Coordinate Measuring Machine

Abstract

I. Introduction

II. The Working Principle and Kinematic Equation of AACMM

A. Working principle

B. Kinematic equation

III. Equivalent Diameter

IV. Error Analysis of Equivalent Diameter

A. Thermal deformation error

B. Eccentricity error

C. Measuring force

V. Equivalent Diameter Measuring Force Experiment

A. Calibration experiment by CMM

B. Effect of measuring force on equivalent diameter of AACMM

VI. Error Compensation

VII. Conclusion and Future Work

Footnotes

Funding

References