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

Kinematic parameters identification and compensation are effective ways to improve the accuracy of articulated arm coordinate measuringmachines (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 identificationmethods 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.


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][9][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-repeatabilitybased 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, distancereference-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 repeatabilitybased identification method, we studied its working mechanism and found that the commonly used single-pointrepeatability-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 singlepoint-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.Section 2 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.

Scaling Factor of Repeatability-Based Identification
2.1.Scaling Factor.As shown in Figure 1, theoretically, when the probe of the AACMM is poised in the same singlepoint 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 , the coordinates of the probe will be scaled by the same factor .The scaling factor law can be proved as follows.
The 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   , linkage length   , joint length   , and torsion angle   ) and the probe offset parameter , as follows: The transformation matrix   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 If the length parameters are scaled in the same factor of , we will obtain the following equation: In the process of transformation,  always exists in the upper right corner of   and T i+1 , and it also exists in the upper right corner of the product of T i ⋅ T i+1 .To multiply the result by the probe offset matrix scaled with same factor, we can get: The scaling factor still exists in the upper partitioned matrix which is linear.It can be deduced that the scaled coordinates of the probe are According to (4) we can get the conclusion: 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. In other words, if the size of the AACMM is scaled k, the coordinates of the probe will be scaled k 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.

First Step of Identification Method Using Repeatability.
The kinematic parameters that determine the coordinates of the probe of the AACMM consist of the length parameters D including linkage length d  , joint length a  , the probe offset l, and the angle parameters R including torsion angle   and joint angle   .The joint angle   is composed of the sensor data   (experimentally obtained with high accuracy encoders) and the joint zero angle deviation  0 .Here we take coordinate , for example, where  is the order of repeated measurements of the same point,  = 1, 2, ..., .
If the length parameters D are equal to the actual length parameters D 0 and the angle parameters R are equal to the actual angle parameters R 0 , it can be deduced that the coordinate x  calculated by any jth measurement is equal to the real coordinate x 0 .Hence, all the coordinates x  calculated are equal.The standard deviation  that evaluates repeatability is zero: According to the law of scaling factor of length parameters, there is another situation in which standard deviation  is also zero.When the length parameter  is equal to  0 (i.e.,  times the actual length parameters  0 ), the angle parameters  are equal to the actual angle parameters  0 .Furthermore, it can be deduced that the coordinate   calculated by any th measurement is equal to  0 ( times the real coordinate).Also the values of all the calculated coordinates   are still equal.So the standard deviation  is still zero, i.e., It can be inferred that  = 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.
Therefore, when all angle parameters are unchanged and all length parameters are scaled by k times, the repeatability error is zero.So the repeatability-based identification method cannot accurately identify the kinematic parameters of each joint.

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  =   /, where  is the nominal length of the gauge, and   is the measured length of the gauge.Finally, the accurate length parameters will be obtained by   = /.

Numerical Simulation
To verify the method proposed in Section 2, numerical simulations were conducted.
We consider the theoretical kinematic parameters of the AACMM as According to the kinematic model and the theoretical kinematic parameters of the AACMM, 70 groups of joint angles   (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 as the certification standard), the errors in , , and  directions are The single-point repeatability accuracy is designed as The single-point repeatability accuracy of the AACMM calculated by ( 11) and ( 12) with the 70 groups of simulation data is 3×10 −6 mm.
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.

Simulation of Single-Point-Repeatability-Based Identification.
The coordinates (x  , y  , z  ) of all measuring points are calculated according to homogeneous matrix transformation formula (1), where j = 1,2,..., N. The average coordinates of the measuring points in x, y, and z directions are as follows: The error distance of each measuring point from the average coordinate point can be derived: The mean error distance of all measuring points can be calculated, which can represent the repeatability accuracy of the AACMM: When 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 distance DE as 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: As identification results shown in Figure 3, the mean error distance DE is 8 × 10 −5 mm, and single-point repeatability accuracy Deltam can be calculated as 1.5 × 10 −4 mm according to the certification standard.
Two simulations of single-point-repeatability-based identification are compared in Table 1.
Although 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 repeatabilitybased identification should be scaled with same factor compared with the nominal values.

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 and  5.
Like the parameters identification in the simulation, we used 70 groups of joint angles as the input with the mean error distance DE as the objective function according to (15).
As shown in Figure 6, after single-point-repeatabilitybased identification, the mean error distance DE is 0.0095mm, and single-point repeatability accuracy Deltam is 0.0186mm according to the certification standard.

System Error Compensation.
A gauge was measured four times with the AACMM after single-point-repeatabilitybased 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 Section 2, the systematic errors are caused by the scaling factor of the length parameters in the repeatability-based identification.
By (6), we know the scaling factor k = 200.759/ 201.513.Then, the accurate length parameters   = / and the final kinematic parameters can be calculated which is shown in Table 2.

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 Table 3, the average is 0.026mm.

Spatial Length Accuracy Verification.
According to the certification standard, the spatial length deviation is where L  is the result of each length measurement,   is actual length value.The maximum spatial length deviation is designed as As shown in Figure 7, 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 rootmean-square of the deviations is 0.061mm.
According to the 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.

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 twostep 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.

Figure 1 :
Figure 1: Scaling with the same factor.

Figure 3 :
Figure 3: Simulation identification results after changing the searching range.

Table 1 :
Comparison of identifications in simulation.

Table 2 :
The kinematic parameters identified of the AACMM.

Table 3 :
Single-point repeatability accuracy verification by different datasets.