# A Novel Kinematic Parameters Identification Method for Articulated Arm Coordinate Measuring Machines Using Repeatability and Scaling Factor

Authors: Lin Cheng; Wen Wang; Yixin Weng; Guang Shi; He Yang; Keqing Lu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Kinematic parameters identification and compensation are effective ways to improve the accuracy of articulated arm coordinate measuring machines (AACMMs) and robotic arms without increasing the cost of hardware. Generally, kinematic parameters identification methods based on standard references are relatively high in accuracy but time-consuming and not suitable for industrial sites, while kinematic parameters identification methods based on repeatability are flexible and easy to implement but lack reliability in accuracy. A novel kinematic parameters identification method for AACMMs using repeatability and scaling factor is proposed in this paper, which combines the advantages of methods based on both standard references and repeatability. Through theoretical analysis and numerical simulations, we found that the commonly used single-point-repeatability-based identification method has problems in identifying the length parameters, which is due to that high repeatability cannot guarantee the accuracy of the kinematic parameters and the measurement accuracy of the AACMM. Further analysis showed that the error of the length parameters is determined by a scaling factor which can be used to remove the error of length parameters. Therefore, a two-step novel kinematic parameters identification method for the AACMMs using repeatability and scaling factor was proposed to get accurate parameters with convenient operation. Experimental studies showed the effectiveness of the proposed identification method, which indicated that 93% more error in spatial length can be decreased comparing to the traditional method of repeatability-based identification.


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## Body

## 1. Introduction

The Articulated Arm Coordinate Measuring Machine (AACMM) is a portable coordinate measurement instrument, which transforms the measurement of geometry into the measurement of coordinates [1, 2]. Due to its flexibility and portability, there is a growing demand for applications at industrial sites [3]. However, as a multi-joint serial structure like robotic arm [4, 5] AACMMs have lower accuracy comparing with the normal coordinate measuring machines. Limited accuracy of AACMMs restricts the applications and development [6, 7].Some studies have shown that 95% errors of the AACMM are due to the errors of the kinematic parameters [8–10]. As the systematic errors, the errors of kinematic parameters can be identified by the identification calculation to improve the accuracy of the AACMM.According to the kinematic chain characteristics of the AACMM, we can obtain multiple sets of kinematic equations under simple physical constraints without using the external high-precision measurement instrument. The kinematic parameters of the AACMM can be identified by optimization algorithms, and the identified parameters will be input to the controller to compensate the errors. According to different evaluation references of the identification methods, the parameters identification methods for AACMMs can be classified into two types. One is based on repeatability, whose objective functions are evaluated by repeatability error with training data acquired from physical constraints. The other is based on standard references, whose objective functions are evaluated by residual error with training data acquired from standard gauges.Single-point-repeatability-based identification method is one of the most representative methods of repeatability-based identification methods. Wang et al. [11] and Gao et al. [12] used a cone hole, which can restrict the center of the probe in the same position, to collect multiple groups of joint angles by changing the poses AACMMs. Under the constraint of single-point cone hole, the coordinates calculated by different group of joint angles will be the same if the kinematic parameters have no errors. However, this is not the fact for all the kinematic parameters which have errors so far. Generally, the differences of the coordinates calculated are used as the objective function of the optimization algorithm to search the actual kinematic parameters. Single-point-repeatability-based identification method only uses a simple calibration tool providing a physical constraint of single-point coordinates, which has high data-sampling efficiency and is suitable for parameters identification at industrial sites. However, this kind of identification method has a limited accuracy because the standard geometric values are not introduced in the identification.Distance-reference-based identification method is a commonly used standard-reference-based identification method. Santolaria et al. estimated the parameters’ error of the AACMM by a spherical gauge with 14 standard balls and a self-centering probe [13, 14]. Cheng et al. [15] took the homemade standard cone hole rod as the distance reference which was calibrated by high-precision measuring instrument. Then the residual error between the experimental distance and the standard distance, measured from the cone hole rod, was applied as the objective function along with the improved least square method used as the optimization algorithm. And the kinematic parameters of the AACMM were identified. As a standard-reference-based identification method, distance-reference-based identification method introduces a standard spatial geometric value; thus the kinematic parameters of the AACMM can be identified definitely theoretically. However, distance-reference-based identification method is more complicated than the single-point cone hole identification method during the sampling process, and it is hard to guarantee the stabilization of calibration tool, which will introduce much more the sampling error. Meanwhile the standard distance of the cone hole rod needs to be measured before identification, which increases the manufacturing costs of the calibration tool and the calibration costs.Comparing the two kinds of identification methods presented above, standard-reference-based identification method can achieve a higher accuracy for AACMMs, but the identification procedure is complex and not suitable for industrial site. Repeatability-based identification method is more convenient and more efficient, but the identification accuracy is lower.Based on the working principle of the repeatability-based identification method, we studied its working mechanism and found that the commonly used single-point-repeatability-based identification method has defects in identifying the length parameters, that is, high repeatability cannot guarantee the accuracy of the kinematic parameters and the measurement accuracy of the AACMM. Further analysis showed that the error of the length parameters is determined by a scaling factor which can be used to remove the error of length parameters. To improve the accuracy of AACMMs, a novel two-step identification method was proposed in this paper. The first step is to identify the angular parameters and initial length parameters based on single-point-repeatability-based. The second step is to achieve the scaling factor for modifying the length parameters based on standard references. Numerical simulations and practical experiments were carried on to verify the method proposed in this paper.The paper is organized as follows. Section2 presents the analysis of repeatability-based identification method and the two-step identification method; the numerical simulations of the scaling factor law and the parameters identification process are introduced in Section 3; and experimental results are given in Section 4. Section 5 provides conclusions.


## 2. Scaling Factor of Repeatability-Based Identification

### 2.1. Scaling Factor

As shown in Figure1, theoretically, when the probe of the AACMM is poised in the same single-point cone hole, a group of coordinates of the probe can be obtained with a serial of poses. And the groups of coordinates are all the same as the probe never moves. When all the length parameters are scaled in a determined factor, another group of coordinates of the probe with the same serial of poses can be obtained. The second group of coordinates are all the same, too. And they are proportional to the first group of coordinates. Moreover, the proportional factor is equal to the scaling factor. This scaling factor law can be described as when all angle parameters of the AACMM remain unchanged, if all length parameters are scaled by a factor k, the coordinates of the probe will be scaled by the same factor k. The scaling factor law can be proved as follows.Figure 1
Scaling with the same factor.
(a)
Before scaling (b)
After scalingThe kinematic model of AACMM is established according to D-H method [16]. The coordinates of the probe are obtained by homogeneous matrix transformations which consist of six joints’ kinematic parameters (each joint contains joint angle θi, linkage length di, joint length ai, and torsion angle αi) and the probe offset parameter l, as follows:(1)
x
y
z
1
=
∏
i
=
1
6
T
i
0
0
l
1
=
∏
i
=
1
6
cos
⁡
θ
i
-
sin
⁡
θ
i
cos
⁡
α
i
sin
⁡
θ
i
sin
⁡
α
i
a
i
cos
⁡
θ
i
sin
⁡
θ
i
cos
⁡
θ
i
cos
⁡
α
i
-
cos
⁡
θ
i
sin
⁡
α
i
a
i
sin
⁡
θ
i
0
s
i
n
α
i
cos
⁡
α
i
d
i
0
0
0
1
·
0
0
l
1The transformation matrixTi for adjacent joints can be divided into 2 × 2 matrices and all the length parameters are in the divided matrix in the upper right corner. The probe offset matrix can be divided into upper and lower matrices, and the length parameter is in the upper matrix, if we set Ti=AiBi01, 00l1=C1,xyz1=X1, then X1=∏i=16AiBi01·C1.If the length parameters are scaled in the same factor ofk, we will obtain the following equation:(2)Ti·Ti+1=AikBi01·Ai+1kBi+101=AiAi+1kBi+AiBi+101In the process of transformation,k always exists in the upper right corner of Ti and Ti+1, and it also exists in the upper right corner of the product of Ti·Ti+1. To multiply the result by the probe offset matrix scaled with same factor, we can get:(3)Ti·Ti+1·kC1=kAiAi+1C+Bi+AiBi+11The scaling factor still exists in the upper partitioned matrix which is linear. It can be deduced that the scaled coordinates of the probe are(4)x′y′z′1=∏i=16AikBi01·kC1=kX1=kxkykz1According to (4) we can get the conclusion: when all angle parameters of the AACMM remain unchanged, if all length parameters are scaled by a factork, the coordinates of the probe will be scaled by the same factork. In other words, if the size of the AACMM is scaledk, the coordinates of the probe will be scaledk times accordingly.From the scaling factor law it can be deduced that the distance calculated from the scaled coordinates are also scaled, as shown in (6). Hence, if the length parameters of the AACMM are scaled, there will be a systematic error in measuring the distance.(5)L=x2-x12+y2-y12+z2-z12(6)L′=kx2-kx12+ky2-ky12+kz2-kz12=kL


### 2.2. First Step of Identification Method Using Repeatability

The kinematic parameters that determine the coordinates of the probe of the AACMM consist of the length parametersD including linkage lengthdi, joint lengthai, the probe offsetl, and the angle parametersR including torsion angle αi and joint angle θi. The joint angle θi is composed of the sensor data θsi (experimentally obtained with high accuracy encoders) and the joint zero angle deviation θ0i. Here we take coordinate x, for example,(7)xj=fxθj,si,D,Rwhere j is the order of repeated measurements of the same point, j=1,2,...,n.If the length parametersD are equal to the actual length parametersD0 and the angle parametersR are equal to the actual angle parametersR0, it can be deduced that the coordinatexj calculated by anyjth measurement is equal to the real coordinatex0. Hence, all the coordinatesxj calculated are equal. The standard deviation σ that evaluates repeatability is zero:(8)
D
=
D
0
R
=
R
0
→
x
j
=
x
0
→
x
j
=
x
j
+
1
→
σ
=
0According to the law of scaling factor of length parameters, there is another situation in which standard deviationσ is also zero. When the length parameter D is equal to kD0 (i.e., k times the actual length parameters D0), the angle parameters R are equal to the actual angle parameters R0. Furthermore, it can be deduced that the coordinate xj calculated by any jth measurement is equal to kx0 (k times the real coordinate). Also the values of all the calculated coordinates xj are still equal. So the standard deviation σ is still zero, i.e., (9)
D
=
k
D
0
R
=
R
0
→
x
j
=
k
x
0
→
x
j
=
x
j
+
1
→
σ
=
0It can be inferred thatσ = 0 is a necessary but insufficient condition for the identifiability of the kinematic parameters. And the objective function here with the standard deviation can be extended to any other function that evaluates repeatability error.(10)
D
=
D
0
R
=
R
0
→
σ
=
0Therefore, when all angle parameters are unchanged and all length parameters are scaled byk times, the repeatability error is zero. So the repeatability-based identification method cannot accurately identify the kinematic parameters of each joint.


### 2.3. Second Step of Identification Method Using Scaling Factor Law

According to (4) and (6), if the length parameters of AACMM are scaled with the same factor, the measuring results will contain systematic error. This paper presents a novel kinematic parameters identification method for the AACMMs using repeatability and the law of scaling factor. The novel method includes the repeatability-based identification, and the absolute measuring to get the scaling factor of systematic errors with the distance reference. Firstly, the accurate angle parameters and preliminary length parameters are identified with the repeatability-based identification method. Then, after compensating with the identified parameters, the AACMM are used to measure a gauge to get the scaling factor k=L′/L, where L is the nominal length of the gauge, and L′ is the measured length of the gauge. Finally, the accurate length parameters will be obtained by D′=D/k.


## 2.1. Scaling Factor

As shown in Figure1, theoretically, when the probe of the AACMM is poised in the same single-point cone hole, a group of coordinates of the probe can be obtained with a serial of poses. And the groups of coordinates are all the same as the probe never moves. When all the length parameters are scaled in a determined factor, another group of coordinates of the probe with the same serial of poses can be obtained. The second group of coordinates are all the same, too. And they are proportional to the first group of coordinates. Moreover, the proportional factor is equal to the scaling factor. This scaling factor law can be described as when all angle parameters of the AACMM remain unchanged, if all length parameters are scaled by a factor k, the coordinates of the probe will be scaled by the same factor k. The scaling factor law can be proved as follows.Figure 1
Scaling with the same factor.
(a)
Before scaling (b)
After scalingThe kinematic model of AACMM is established according to D-H method [16]. The coordinates of the probe are obtained by homogeneous matrix transformations which consist of six joints’ kinematic parameters (each joint contains joint angle θi, linkage length di, joint length ai, and torsion angle αi) and the probe offset parameter l, as follows:(1)
x
y
z
1
=
∏
i
=
1
6
T
i
0
0
l
1
=
∏
i
=
1
6
cos
⁡
θ
i
-
sin
⁡
θ
i
cos
⁡
α
i
sin
⁡
θ
i
sin
⁡
α
i
a
i
cos
⁡
θ
i
sin
⁡
θ
i
cos
⁡
θ
i
cos
⁡
α
i
-
cos
⁡
θ
i
sin
⁡
α
i
a
i
sin
⁡
θ
i
0
s
i
n
α
i
cos
⁡
α
i
d
i
0
0
0
1
·
0
0
l
1The transformation matrixTi for adjacent joints can be divided into 2 × 2 matrices and all the length parameters are in the divided matrix in the upper right corner. The probe offset matrix can be divided into upper and lower matrices, and the length parameter is in the upper matrix, if we set Ti=AiBi01, 00l1=C1,xyz1=X1, then X1=∏i=16AiBi01·C1.If the length parameters are scaled in the same factor ofk, we will obtain the following equation:(2)Ti·Ti+1=AikBi01·Ai+1kBi+101=AiAi+1kBi+AiBi+101In the process of transformation,k always exists in the upper right corner of Ti and Ti+1, and it also exists in the upper right corner of the product of Ti·Ti+1. To multiply the result by the probe offset matrix scaled with same factor, we can get:(3)Ti·Ti+1·kC1=kAiAi+1C+Bi+AiBi+11The scaling factor still exists in the upper partitioned matrix which is linear. It can be deduced that the scaled coordinates of the probe are(4)x′y′z′1=∏i=16AikBi01·kC1=kX1=kxkykz1According to (4) we can get the conclusion: when all angle parameters of the AACMM remain unchanged, if all length parameters are scaled by a factork, the coordinates of the probe will be scaled by the same factork. In other words, if the size of the AACMM is scaledk, the coordinates of the probe will be scaledk times accordingly.From the scaling factor law it can be deduced that the distance calculated from the scaled coordinates are also scaled, as shown in (6). Hence, if the length parameters of the AACMM are scaled, there will be a systematic error in measuring the distance.(5)L=x2-x12+y2-y12+z2-z12(6)L′=kx2-kx12+ky2-ky12+kz2-kz12=kL


## 2.2. First Step of Identification Method Using Repeatability

The kinematic parameters that determine the coordinates of the probe of the AACMM consist of the length parametersD including linkage lengthdi, joint lengthai, the probe offsetl, and the angle parametersR including torsion angle αi and joint angle θi. The joint angle θi is composed of the sensor data θsi (experimentally obtained with high accuracy encoders) and the joint zero angle deviation θ0i. Here we take coordinate x, for example,(7)xj=fxθj,si,D,Rwhere j is the order of repeated measurements of the same point, j=1,2,...,n.If the length parametersD are equal to the actual length parametersD0 and the angle parametersR are equal to the actual angle parametersR0, it can be deduced that the coordinatexj calculated by anyjth measurement is equal to the real coordinatex0. Hence, all the coordinatesxj calculated are equal. The standard deviation σ that evaluates repeatability is zero:(8)
D
=
D
0
R
=
R
0
→
x
j
=
x
0
→
x
j
=
x
j
+
1
→
σ
=
0According to the law of scaling factor of length parameters, there is another situation in which standard deviationσ is also zero. When the length parameter D is equal to kD0 (i.e., k times the actual length parameters D0), the angle parameters R are equal to the actual angle parameters R0. Furthermore, it can be deduced that the coordinate xj calculated by any jth measurement is equal to kx0 (k times the real coordinate). Also the values of all the calculated coordinates xj are still equal. So the standard deviation σ is still zero, i.e., (9)
D
=
k
D
0
R
=
R
0
→
x
j
=
k
x
0
→
x
j
=
x
j
+
1
→
σ
=
0It can be inferred thatσ = 0 is a necessary but insufficient condition for the identifiability of the kinematic parameters. And the objective function here with the standard deviation can be extended to any other function that evaluates repeatability error.(10)
D
=
D
0
R
=
R
0
→
σ
=
0Therefore, when all angle parameters are unchanged and all length parameters are scaled byk times, the repeatability error is zero. So the repeatability-based identification method cannot accurately identify the kinematic parameters of each joint.


## 2.3. Second Step of Identification Method Using Scaling Factor Law

According to (4) and (6), if the length parameters of AACMM are scaled with the same factor, the measuring results will contain systematic error. This paper presents a novel kinematic parameters identification method for the AACMMs using repeatability and the law of scaling factor. The novel method includes the repeatability-based identification, and the absolute measuring to get the scaling factor of systematic errors with the distance reference. Firstly, the accurate angle parameters and preliminary length parameters are identified with the repeatability-based identification method. Then, after compensating with the identified parameters, the AACMM are used to measure a gauge to get the scaling factor k=L′/L, where L is the nominal length of the gauge, and L′ is the measured length of the gauge. Finally, the accurate length parameters will be obtained by D′=D/k.


## 3. Numerical Simulation

To verify the method proposed in Section2, numerical simulations were conducted.We consider the theoretical kinematic parameters of the AACMM asF=[α1α2α3α4α5a1a2a3a4a5d2d3d4d5d6θ02θ03θ04θ05θ06l] = [−1.5708 −1.5708 −1.5708 −1.5708 −1.5708 0 62 0 62 0 0 750 0 500 11.5 0 0 0 0 0 102].According to the kinematic model and the theoretical kinematic parameters of the AACMM, 70 groups of joint anglesθsi (i.e., different poses) are obtained from the assumed coordinates of single-point position.According to definitions in Certification Standard JJF 1408-2013 [17] and ASME B.89.4.22 standard (hereinafter referred to asthe certification standard), the errors in x, y, and z directions are(11)δx=maxxj-minxj2δy=maxyj-minyj2δz=maxzj-minzj2The single-point repeatability accuracy is designed as(12)Deltam=max⁡δx,δy,δzThe single-point repeatability accuracy of the AACMM calculated by (11) and (12) with the 70 groups of simulation data is 3×10−6 mm.


### 3.1. The Effects of Scaling Factor on Repeatability Accuracy

We can scale the length parameters with a scaling factor of 1/2, i.e.,F = [−1.5708 −1.5708 −1.5708 −1.5708 −1.5708 0 31 0 31 0 0 375 0 250 5.75 0 0 0 0 0 51].The single-point repeatability accuracy of the AACMM calculated by (11) and (12) with the 70 groups of simulation data is 2×10−6 mm. The deviation between the repeatability accuracies before and after scaling length parameters can be neglected with regard to the normal accuracy (5×10−2 mm) of AACMMs. This numerical simulation suggests that the scaling factor of the length parameters has little effect on the single-point repeatability accuracy of the AACMM.


### 3.2. Simulation of Single-Point-Repeatability-Based Identification

The coordinates (xj, yj, zj) of all measuring points are calculated according to homogeneous matrix transformation formula (1), wherej = 1,2,...,N. The average coordinates of the measuring points inx,y, andz directions are as follows:(13)x¯=1N∑j=1Nxj,y¯=1N∑j=1Nyj,z¯=1N∑j=1NzjThe error distance of each measuring point from the average coordinate point can be derived:(14)Ej=xj-x-2+yj-y-2+zj-z-2The mean error distance of all measuring points can be calculated, which can represent the repeatability accuracy of the AACMM:(15)DE=1N∑j=1NEjWhen the kinematic parameters are unknown, we carry out the identification with D-H model and the joint angles. Here we use 70 groups of joint angles as the input and use the mean error distanceDEas the objective function according to (15) with the optimization algorithm of the interior point.(1) The lower and upper boundaries of the variables (kinematic parameters) to be searched are set to be:lb = [−1.5883 −1.5883 −1.5883 −1.5883 −1.5883 −10 52 −10 52 −10 −10 740 −10 490 1.5 −0.0175 −0.0175−0.0175 −0.0175 −0.0175 92] and ub = [−1.5533 −1.5533 −1.5533 −1.5533 −1.5533 10 72 10 7210 10 760 10 510 21.5 0.0175 0.0175 0.0175 0.01750.0175 112].The kinematic parameters obtained through identification are as follows:(16)
F
=
-
1.570801
-
1.570798
-
1.570804
-
1.570798
-
1.570797
0.000103
61.966195
0.001734
61.966583
0.000112
-
0.000039
749.597077
0.000363
499.730124
11.493096
-
0.000001
0.000000
0.000004
0.000002
0.000000
101.945075
.The results of the identification are shown in Figure2. The mean error distanceDE is 7×10−5 mm, and single-point repeatability accuracyDeltam is 3.7×10−4 mm according tothe certification standard.Figure 2
Simulation identification results.(2) Keeping the searching range of the angle parameters unchanged, the searching center of the length parameters deviate is set to +5 mm:l b = [−1.5883 −1.5883 −1.5883 −1.5883 −1.5883 −5 57−5 57 −5 −5 745 −5 495 6.5 −0.0175 −0.0175−0.0175 −0.0175 −0.0175 97] and ub = [−1.5533 −1.5533 −1.5533 −1.5533 −1.5533 15 77 15 7715 15 765 15 515 26.5 0.0175 0.0175 0.0175 0.01750.0175 117].The identified kinematic parameters are as follows:(17)
F
=
-
1.570799
-
1.570802
-
1.570800
-
1.570797
-
1.570801
-
0.001005
62.547732
-
0.001035
62.547997
-
0.000070
-
0.000025
756.618307
-
0.000419
504.412566
11.601591
-
0.000001
-
0.000002
-
0.000001
-
0.000001
0.000000
102.900101
.As identification results shown in Figure3, the mean error distanceDE is 8 × 10−5 mm, and single-point repeatability accuracyDeltam can be calculated as 1.5 × 10−4 mm according tothe certification standard.Figure 3
Simulation identification results after changing the searching range.Two simulations of single-point-repeatability-based identification are compared in Table1.Table 1
Comparison of identifications in simulation.
No. of simulations Length searching range[mm] Angle searching Range[°] DE[mm] Deltam[mm] Length Max Error[mm] Angle Max Error 1 [-10,+10] [-1,+1] 7×10−5 3.7×10−4 0.4 4×10−6 2 [-5,+15] [-1,+1] 8×10−5 1.5×10−4 6.6 3×10−6Although the searching boundaries of the two groups of variables are different, the single-point repeatability accuracies of the two identification results are both in the range of 10−4 mm. Hence, the difference between the two accuracies can be neglected. However, by comparing each kinematic parameter of the results of simulation identifications with the theoretical kinematic parameters, we know the maximum error of the length parameters is 0.4 mm in the first simulation and 6.6 mm in the second simulation. It should be noted that the errors of the angle parameters may be neglected. Therefore, it suggests that while the objective function of repeatability error (accuracy) converges, the obtained length parameters will have different factor scaling with different settings of searching boundaries.In addition, after a serial of numerical simulations, it is found that the scaling of the length parameters will decrease when the ranges of parameters are small. But in practical identification experiments, it is impractical to set the ranges of parameters small for the actual parameters that are unknown. Once the small ranges are set incorrectly, the identification algorithms cannot achieve the actual parameters.The above numerical simulations illustrate that the identification using the repeatability error as the objective function cannot assure the accuracy of the obtained length parameters. In general, the length parameters obtained by repeatability-based identification should be scaled with same factor compared with the nominal values.


## 3.1. The Effects of Scaling Factor on Repeatability Accuracy

We can scale the length parameters with a scaling factor of 1/2, i.e.,F = [−1.5708 −1.5708 −1.5708 −1.5708 −1.5708 0 31 0 31 0 0 375 0 250 5.75 0 0 0 0 0 51].The single-point repeatability accuracy of the AACMM calculated by (11) and (12) with the 70 groups of simulation data is 2×10−6 mm. The deviation between the repeatability accuracies before and after scaling length parameters can be neglected with regard to the normal accuracy (5×10−2 mm) of AACMMs. This numerical simulation suggests that the scaling factor of the length parameters has little effect on the single-point repeatability accuracy of the AACMM.


## 3.2. Simulation of Single-Point-Repeatability-Based Identification

The coordinates (xj, yj, zj) of all measuring points are calculated according to homogeneous matrix transformation formula (1), wherej = 1,2,...,N. The average coordinates of the measuring points inx,y, andz directions are as follows:(13)x¯=1N∑j=1Nxj,y¯=1N∑j=1Nyj,z¯=1N∑j=1NzjThe error distance of each measuring point from the average coordinate point can be derived:(14)Ej=xj-x-2+yj-y-2+zj-z-2The mean error distance of all measuring points can be calculated, which can represent the repeatability accuracy of the AACMM:(15)DE=1N∑j=1NEjWhen the kinematic parameters are unknown, we carry out the identification with D-H model and the joint angles. Here we use 70 groups of joint angles as the input and use the mean error distanceDEas the objective function according to (15) with the optimization algorithm of the interior point.(1) The lower and upper boundaries of the variables (kinematic parameters) to be searched are set to be:lb = [−1.5883 −1.5883 −1.5883 −1.5883 −1.5883 −10 52 −10 52 −10 −10 740 −10 490 1.5 −0.0175 −0.0175−0.0175 −0.0175 −0.0175 92] and ub = [−1.5533 −1.5533 −1.5533 −1.5533 −1.5533 10 72 10 7210 10 760 10 510 21.5 0.0175 0.0175 0.0175 0.01750.0175 112].The kinematic parameters obtained through identification are as follows:(16)
F
=
-
1.570801
-
1.570798
-
1.570804
-
1.570798
-
1.570797
0.000103
61.966195
0.001734
61.966583
0.000112
-
0.000039
749.597077
0.000363
499.730124
11.493096
-
0.000001
0.000000
0.000004
0.000002
0.000000
101.945075
.The results of the identification are shown in Figure2. The mean error distanceDE is 7×10−5 mm, and single-point repeatability accuracyDeltam is 3.7×10−4 mm according tothe certification standard.Figure 2
Simulation identification results.(2) Keeping the searching range of the angle parameters unchanged, the searching center of the length parameters deviate is set to +5 mm:l b = [−1.5883 −1.5883 −1.5883 −1.5883 −1.5883 −5 57−5 57 −5 −5 745 −5 495 6.5 −0.0175 −0.0175−0.0175 −0.0175 −0.0175 97] and ub = [−1.5533 −1.5533 −1.5533 −1.5533 −1.5533 15 77 15 7715 15 765 15 515 26.5 0.0175 0.0175 0.0175 0.01750.0175 117].The identified kinematic parameters are as follows:(17)
F
=
-
1.570799
-
1.570802
-
1.570800
-
1.570797
-
1.570801
-
0.001005
62.547732
-
0.001035
62.547997
-
0.000070
-
0.000025
756.618307
-
0.000419
504.412566
11.601591
-
0.000001
-
0.000002
-
0.000001
-
0.000001
0.000000
102.900101
.As identification results shown in Figure3, the mean error distanceDE is 8 × 10−5 mm, and single-point repeatability accuracyDeltam can be calculated as 1.5 × 10−4 mm according tothe certification standard.Figure 3
Simulation identification results after changing the searching range.Two simulations of single-point-repeatability-based identification are compared in Table1.Table 1
Comparison of identifications in simulation.
No. of simulations Length searching range[mm] Angle searching Range[°] DE[mm] Deltam[mm] Length Max Error[mm] Angle Max Error 1 [-10,+10] [-1,+1] 7×10−5 3.7×10−4 0.4 4×10−6 2 [-5,+15] [-1,+1] 8×10−5 1.5×10−4 6.6 3×10−6Although the searching boundaries of the two groups of variables are different, the single-point repeatability accuracies of the two identification results are both in the range of 10−4 mm. Hence, the difference between the two accuracies can be neglected. However, by comparing each kinematic parameter of the results of simulation identifications with the theoretical kinematic parameters, we know the maximum error of the length parameters is 0.4 mm in the first simulation and 6.6 mm in the second simulation. It should be noted that the errors of the angle parameters may be neglected. Therefore, it suggests that while the objective function of repeatability error (accuracy) converges, the obtained length parameters will have different factor scaling with different settings of searching boundaries.In addition, after a serial of numerical simulations, it is found that the scaling of the length parameters will decrease when the ranges of parameters are small. But in practical identification experiments, it is impractical to set the ranges of parameters small for the actual parameters that are unknown. Once the small ranges are set incorrectly, the identification algorithms cannot achieve the actual parameters.The above numerical simulations illustrate that the identification using the repeatability error as the objective function cannot assure the accuracy of the obtained length parameters. In general, the length parameters obtained by repeatability-based identification should be scaled with same factor compared with the nominal values.


## 4. Experiments

To verify the novel kinematic parameters identification method and scaling factor law, experiments were conducted. We carried out the repeatability-based identification, following with obtaining scaling factor of systematic error by measuring the distance reference, then compensating the length parameters by using the scaling factor.


### 4.1. Single-Point-Repeatability-Based Identification Experiment

The AACMM of Hexagon RA7125 was used in the experiment at the environmental temperature of 20°C. 70 groups of joint angles (“Data1”) were collected at the same single-point position in different poses with the constraint of the three-ball cone hole, as shown in Figures4 and 5.Figure 4
Hexagon RA7125.Figure 5
Single-point data acquisition.Like the parameters identification in the simulation, we used 70 groups of joint angles as the input with the mean error distanceDEas the objective function according to (15).The lower searching boundary and the upper searching boundary of the kinematic parameters were set as follows:l b = [−1.5883 −1.5883 1.5533 −1.5883 1.5533−10 52 −10 52 −10 −10 740 −10 490 5 −0.0175−0.0175 −0.0175 −0.0175 −0.0175 87] and ub = [−1.5533 −1.5533 1.5883 −1.5533 1.5883 1072 10 72 10 10 760 10 510 25 0.0175 0.01750.0175 0.0175 0.0175 107].The identified kinematic parameters are as follows:(18)
F
=
-
1.570742
-
1.570856
1.570870
-
1.570865
1.571311
-
0.022551
61.883904
-
0.006059
61.870128
0.004900
-
0.052339
747.933132
0.059462
498.683691
15.013116
-
0.000160
0.000009
0.000104
0.000172
-
0.000069
97.445020
.As shown in Figure6, after single-point-repeatability-based identification, the mean error distanceDE is 0.0095mm, and single-point repeatability accuracyDeltam is 0.0186mm according tothe certification standard.Figure 6
The result of single-point-repeatability-based identification.


### 4.2. System Error Compensation

A gauge was measured four times with the AACMM after single-point-repeatability-based identification, whose nominal length is of 201.513mm. The measuring results were as follows: 200.761mm, 200.759mm, 200.755mm, and 200.759mm, so the average is 200.759mm with the deviation of 0.006mm between maximum and minimum, and maximum measurement error is 0.758mm. According to the analysis in Section2, the systematic errors are caused by the scaling factor of the length parameters in the repeatability-based identification.By (6), we know the scaling factork = 200.759 / 201.513. Then, the accurate length parameters D′=D/k and the final kinematic parameters can be calculated which is shown in Table 2.Table 2
The kinematic parameters identified of the AACMM.
Joint numberi α i [°] a i [mm] θ 0 i [°] d i[mm] 1 -89.9683 -0.0226 0 376.5 2 -90.0312 62.1163 0.0040 -0.0525 3 90.0071 -0.0061 0.0466 750.7422 4 -90.0338 62.1025 -0.0374 0.0597 5 90.0373 0.0049 0.0127 500.5566 6 -90 0 -0.0060 15.0695 Probe offsetl= 97.8110mm


### 4.3. Single-Point Repeatability Accuracy Verification

Another two groups of single-point data were used to verify the single-point repeatability accuracy of the identified kinematic parameters. As shown in Table3, the average is 0.026mm.Table 3
Single-point repeatability accuracy verification by different datasets.
Datasets (70 groups) “Data1” “Data2” “Data3” Single-point repeatability accuracy/mm 0.0187 0.0326 0.0274


### 4.4. Spatial Length Accuracy Verification

According tothe certification standard, the spatial length deviation is(19)Dev=Lj-LcalwhereLj is the result of each length measurement, Lcal is actual length value.The maximum spatial length deviation is designed as(20)MaxDev=max⁡DevAs shown in Figure7, we used the AACMM after compensation with the parameters of Table 2 to measure a quartzes plate (the nominal length of 300.0005 mm) 30 times, which was placed in different positions and orientations. The results of measurements were shown in Figure 8. The maximum value is 300.036mm, the minimum measured value is 299.950mm, the maximum deviation between the measured value and the standard value is 0.050mm, the range of the deviations is 0.086mm, and the two times the root-mean-square of the deviations is 0.061mm.Figure 7
Length accuracy verification experiment.Figure 8
Length accuracy verification results.According tothe certification standard, the single-point repeatability of the AACMM finally calibrated in this research is 0.026mm and the spatial length accuracy is ± 0.050mm, which has 93% improvement in length accuracy compared with traditional method of repeatability-based identification with no length scaling compensation.


## 4.1. Single-Point-Repeatability-Based Identification Experiment

The AACMM of Hexagon RA7125 was used in the experiment at the environmental temperature of 20°C. 70 groups of joint angles (“Data1”) were collected at the same single-point position in different poses with the constraint of the three-ball cone hole, as shown in Figures4 and 5.Figure 4
Hexagon RA7125.Figure 5
Single-point data acquisition.Like the parameters identification in the simulation, we used 70 groups of joint angles as the input with the mean error distanceDEas the objective function according to (15).The lower searching boundary and the upper searching boundary of the kinematic parameters were set as follows:l b = [−1.5883 −1.5883 1.5533 −1.5883 1.5533−10 52 −10 52 −10 −10 740 −10 490 5 −0.0175−0.0175 −0.0175 −0.0175 −0.0175 87] and ub = [−1.5533 −1.5533 1.5883 −1.5533 1.5883 1072 10 72 10 10 760 10 510 25 0.0175 0.01750.0175 0.0175 0.0175 107].The identified kinematic parameters are as follows:(18)
F
=
-
1.570742
-
1.570856
1.570870
-
1.570865
1.571311
-
0.022551
61.883904
-
0.006059
61.870128
0.004900
-
0.052339
747.933132
0.059462
498.683691
15.013116
-
0.000160
0.000009
0.000104
0.000172
-
0.000069
97.445020
.As shown in Figure6, after single-point-repeatability-based identification, the mean error distanceDE is 0.0095mm, and single-point repeatability accuracyDeltam is 0.0186mm according tothe certification standard.Figure 6
The result of single-point-repeatability-based identification.


## 4.2. System Error Compensation

A gauge was measured four times with the AACMM after single-point-repeatability-based identification, whose nominal length is of 201.513mm. The measuring results were as follows: 200.761mm, 200.759mm, 200.755mm, and 200.759mm, so the average is 200.759mm with the deviation of 0.006mm between maximum and minimum, and maximum measurement error is 0.758mm. According to the analysis in Section2, the systematic errors are caused by the scaling factor of the length parameters in the repeatability-based identification.By (6), we know the scaling factork = 200.759 / 201.513. Then, the accurate length parameters D′=D/k and the final kinematic parameters can be calculated which is shown in Table 2.Table 2
The kinematic parameters identified of the AACMM.
Joint numberi α i [°] a i [mm] θ 0 i [°] d i[mm] 1 -89.9683 -0.0226 0 376.5 2 -90.0312 62.1163 0.0040 -0.0525 3 90.0071 -0.0061 0.0466 750.7422 4 -90.0338 62.1025 -0.0374 0.0597 5 90.0373 0.0049 0.0127 500.5566 6 -90 0 -0.0060 15.0695 Probe offsetl= 97.8110mm


## 4.3. Single-Point Repeatability Accuracy Verification

Another two groups of single-point data were used to verify the single-point repeatability accuracy of the identified kinematic parameters. As shown in Table3, the average is 0.026mm.Table 3
Single-point repeatability accuracy verification by different datasets.
Datasets (70 groups) “Data1” “Data2” “Data3” Single-point repeatability accuracy/mm 0.0187 0.0326 0.0274


## 4.4. Spatial Length Accuracy Verification

According tothe certification standard, the spatial length deviation is(19)Dev=Lj-LcalwhereLj is the result of each length measurement, Lcal is actual length value.The maximum spatial length deviation is designed as(20)MaxDev=max⁡DevAs shown in Figure7, we used the AACMM after compensation with the parameters of Table 2 to measure a quartzes plate (the nominal length of 300.0005 mm) 30 times, which was placed in different positions and orientations. The results of measurements were shown in Figure 8. The maximum value is 300.036mm, the minimum measured value is 299.950mm, the maximum deviation between the measured value and the standard value is 0.050mm, the range of the deviations is 0.086mm, and the two times the root-mean-square of the deviations is 0.061mm.Figure 7
Length accuracy verification experiment.Figure 8
Length accuracy verification results.According tothe certification standard, the single-point repeatability of the AACMM finally calibrated in this research is 0.026mm and the spatial length accuracy is ± 0.050mm, which has 93% improvement in length accuracy compared with traditional method of repeatability-based identification with no length scaling compensation.


## 5. Conclusions

Repeatability-based identification methods are convenient and efficient in acquiring data and identification calculation, but they are low in accuracy. This limits their applications in the field of AACMMs greatly. Through theoretical analysis and numerical simulations, we found that the length parameters of AACMMs cannot be identified accurately by repeatability-based identification method for there being a scaling factor law in the repeatability-based identification. And accurate kinematic parameters cannot be identified without introducing any standard reference. Therefore, a two-step novel kinematic parameters identification method for the AACMMs using repeatability and scaling factor is proposed in this paper to get accurate parameters by convenient operation. Experimental studies have verified the feasibility of the proposed identification method, which indicates that 93% more error in spatial length can be decreased comparing to the traditional method of repeatability-based identification. Therefore, the accuracy of the AACMM is improved greatly.The scaling factor problem also exists in the kinematic identifications of similar linkage mechanisms, e.g., industrial robots. The novel parameters identification method proposed in this paper also can be applied in the parameters identification for industrial robots to improve the accuracy of them.



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*Source: 1760190-2018-07-15.xml*