A SELF-CALIBRATION METHOD FOR ARTICULATED ARM COORDINATEMEASURING MACHINES

Guanbin Gao, Jing Na, Xing Wu and Yu GuoFaculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming, China

E-mail: gbgao@163.com; najing25@163.com; xingwu@aliyun.com; kmgary@163.com

IMETI 2015 SM5011_SCINo. 16-CSME-20, E.I.C. Accession Number 3906

ABSTRACTTo improve the accuracy of articulated arm coordinate measuring machines (AACMM) and simplify the cal-ibration process, an improved self-calibration method was proposed. Unlike the traditional calibration meth-ods, which need external expensive precision instruments and elaborate setups, the proposed self-calibrationmethod only requires a gauge to assist the data acquisition operation. By designing a movement trajectoryof the AACMM, a series of joint angles can be obtained to form overdetermined equations based on thekinematic model of the AACMM. Therefore, the structural parameters of the AACMM can be obtained bysolving the equations. Consequently, the calibration can be achieved by solving these equations. The coeffi-cient matrix of the equations was further analyzed to simplify the equations, and a constructive method waspresented to identify the structural parameters by solving the simplified equations with a modified simulatedannealing algorithm, in which an optimized search strategy was applied to improve the robustness and ef-ficiency. Experimental studies on an AACMM validate the convenience and effectiveness of the proposedAACMM self-calibration approach.

Keywords: articulated arm; self-calibration; simulated annealing algorithm; kinematic model.

UNE MÉTHODE D’AUTO-ÉTALONNAGE POUR UNE MACHINE À BRAS ARTICULÉ POURMESURER LES COORDONNÉES

RÉSUMÉAfin d’améliorer la précision des machines à mesurer tridimensionnelles à bras articulé (AACMM) et desimplifier le processus d’étalonnage, une méthode d’auto-étalonnage améliorée a été proposée. Contraire-ment aux méthodes traditionnelles d’étalonnage qui requiert un dispositif de calibrage externe coûteux etdes configurations complexes, le procédé d’auto-étalonnage proposé ne nécessite qu’un gabarit pour aiderl’opération d’acquisition de données. En concevant une trajectoire de déplacement de l’AACMM, une séried’angles des articulations peut être obtenue pour former des équations surdéterminées basées sur le modèlecinématique de l’AACMM de sorte que les paramètres structurels de l’AACMM peuvent être obtenus enrésolvant les équations. Par conséquent, l’étalonnage peut être effectué par la résolution des équations. Lamatrice de coefficients des équations a été en outre analysée pour simplifier les équations, et un procédéconstructif a été présenté pour identifier les paramètres structuraux en résolvant les équations simplifiées,et en utilisant un algorithme de recuit simulé modifié, dans lequel une stratégie de recherche optimisé a étéappliquée pour améliorer la robustesse et l’efficacité. Des études expérimentales sur une AACMM validentla commodité et l’efficacité de l’approche d’auto-étalonnage de l’AACMM proposée.

Mots-clés : bras articulé; auto-étalonnage; algorithme de recuit simulé; modèle cinématique.

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1. INTRODUCTION

The articulated arm coordinate measuring machine (AACMM) is a multiple degrees of freedom (DOF) andnon-orthogonal coordinate (Cartesian coordinate) measuring machine which simulates the structural style ofhuman arm joints [1]. The AACMM is made of a series of linkages connected by joints in series. It possessesa lot of special characters, such as simple mechanical structure, small size, light weight, large measurementrange, enabling flexible measurement in field, etc. [2]. Moreover, tremendous research works had beenconducted since AACMM invention in 1980s. Soon after, relevant products were quickly applied not onlyin the mold design fields including cars, motorcycles, plastic products, toys and others, but also in the fieldsof online product quality inspection, equipment maintenance, aircraft assembly, medical application andothers [3, 4].

Currently, the measuring accuracy of the AACMM is much belower than that of the orthogonal coordinatemeasuring machine [5]. The problem of measurement accuracy has limited its applications greatly. Toimprove the measuring accuracy of the AACMM, the strategies of choosing high precision components,enhancing manufacturing and assembly precision are usually suggested. However, the increasing precisionof individual components cannot guarantee an overall high-precision measuring accuracy, at the same timethe cost of the measuring machine increases dramatically [6]. The reason is that small errors in somekinematic parameters may accumulate and influence the measuring accuracy of the AACMM greatly.

Referring to robots and parallel machines theories, kinematic calibration is one of the most importantmethods to eliminate errors of structural parameters and improve the accuracy of AACMM [7]. The kine-matic calibration is a process where advanced measurement means or appropriate methods of identificationof the kinematic model parameters are used to improve the accuracy [8, 9]. Generally, there are two kindof kinematic calibration methods, external calibration and self-calibration [10]. Self-calibration process isrelatively simple, convenient, efficient, and easy-to-use in industrial field, but its precision is affected by thecalibration method greatly due to its indirect measurement on structural parameters. In this sense, if theself-calibration method is improper, reliability of the results cannot be guaranteed [11].

Kovac et al. [12] developed a new high precision device for the AACMM testing and calibration based onlaser interferometer measurements along a line gauge beam. Santolaria et al. [11] used four laser trackers toconstruct a geometric multilateration system, and obtained data from the laser trackers and an AACMM toidentify the structural parameters of the AACMM. Hamana et al. [13] presented a method that the kinematicparameters of ACMM were calibrated by using spherical center coordinates as the artifact. Howerever, onlypart of the measuring volume can be calibrated, so the AACMM was not sufficiently calibrated.

The above calibration methods need external expensive calibration instruments and elaborate setups;therefore, it is difficult for them to be applied to industrial fields. This paper proposed a simple self-calibration method which only needs a fixed conical hole in acquisition data to improve AACMM accuracy.Then, the identification equations are deduced from the kinematic model of the AACMM. And a method ofenhancing the numerical stability of the identification equations is proposed. The self-calibration methodcan be applied in industrial fields without using any precision instruments.

2. KINEMATIC MODELING AND ANALYSIS

2.1. Kinematic ModelingThe structure of the AACMM shown in Fig. 1a is similar to an articulated robot. Therefore, AACMM modelcan be established by using modeling method for robotics. In the robotics kinematic modeling field, Denavitand Hartenberg proposed a four-parameter model [14], i.e., the Denavit–Hartenberg model (DH model),which has been widely used due to its clear physical meaning; it uses a 4×4 homogeneous transform matrixto represent the spatial relations of adjacent joints coordinate systems. Because all the adjacent joints of theAACMM are perpendicular, there is no nominally-parallel problem, we can use DH method to establish the

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Fig. 1. (a) Structure of the AACMM; (b) coordinate systems of the AACMM.

kinematic model. For a 6-DOF AACMM, the coordinate values of the probe in the reference coordinatesystem can be derived through seven successive transformations.

According to the DH method, the AACMM has four basic structural parameters, e.g. linkage length di,joint length ai, torsion angle αi, and joint angle θi. These parameters are defined as follows: di is the distancebetween Xi−1 axis and Xi axis, ai is the distance between Zi−1 axis and Zi axis, al phai is the angle betweenZi−1 axis and Zi axis, and θi is the angle between Xi−1 axis and Xi axis. As shown in Fig. 1b, the values ofθ1,θ3,θ5 are within [0◦,360◦], while θ2,θ4,θ6 range from [−180◦,0◦] due to the hardware limitations.

The transform from the coordinate system {Oi Xi Yi Zi} to {Oi−1 Xi−1 Yi−1 Zi−1} is equivalent to aprocess in which {Oi−1 Xi−1 Yi−1 Zi−1} finishes rotation, translation and then completely coincide with{Oi Xi Yi Zi}. The whole transform procedure can be expressed mathematically as

Ti−1,i = Rot(zi−1,θi)Trans(0,0,di)Trans(ai,0,0)Rot(xi,αi) (1)

Due to the transform principle of coordinate system, the homogeneous transform matrix between the adja-cent coordinate systems {Oi Xi Yi Zi} and {Oi−1 Xi−1 Yi−1 Zi−1} can be expressed as

Ti−1,i =

cosθi −sinθi cosαi sinθi sinαi ai cosθi

sinθi cosθi cosαi −cosθi sinαi ai sinθi

0 sinαi cosαi di

0 0 0 1

(2)

While calculating the probe coordinates in the reference coordinate system except for four basic parame-ters of the above DH model, the probe coordinates (lx, ly, lz) in the last joint coordinate system need to becalculated namely probe bias, the fifth set of parameters of AACMM. After the homogeneous coordinatetransform, the probe coordinates in the reference coordinate system {O0 X0 Y0 Z0} which can be expressed

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as follows: xyz1

= T0,1 ·T1,2 ·T2,3 ·T3,4 ·T4,5 ·T5,6 ·

lxlylz1

= Π6i=1

cosθi −sinθi cosαi sinθi sinαi ai cosθi

sinθi cosθi cosαi −cosθi sinαi ai sinθi

0 sinαi cosαi di

0 0 0 1

·

lxlylz1

. (3)

Among the five parameters of AACMM, only the joint angle θi measured by the angle sensor is a variable.The other four parameters are constant. However, the assembly conditions and linkage deformation willaffect the values of these parameters. Therefore, it is necessary to re-assemble the measuring machine orre-calibrate the parameters after a certain working period so as to recover the accuracy of the AACMM.

2.2. Kinematic AnalysisThe probe coordinates can be expressed as a function of structural parameters of joints

x = fx(θ1,θ2,θ3,θ4,θ5,θ6,α1,α2,α3,α4,α5,α6,d1,d2,d3,d4,d5,d6,a1,a2,a3,a4,a5,a6, lx, ly, lz)y = fy(θ1,θ2,θ3,θ4,θ5,θ6,α1,α2,α3,α4,α5,α6,d1,d2,d3,d4,d5,d6,a1,a2,a3,a4,a5,a6, lx, ly, lz)z = fz(θ1,θ2,θ3,θ4,θ5,θ6,α1,α2,α3,α4,α5,α6,d1,d2,d3,d4,d5,d6,a1,a2,a3,a4,a5,a6, lx, ly, lz)

(4)

The total differential of Eq. (4) can be written in a compact form as

dxdydz

=

∂ fx∂θ1

∂ fx∂θ2

∂ fx∂θ3

∂ fx∂θ4

∂ fx∂θ5

∂ fx∂θ6

∂ fx∂α1

∂ fx∂α2

∂ fx∂α3

∂ fx∂α4

∂ fx∂α5

∂ fx∂α6

∂ fy∂θ1

∂ fy∂θ2

∂ fy∂θ3

∂ fy∂θ4

∂ fy∂θ5

∂ fy∂θ6

∂ fy∂α1

∂ fy∂α2

∂ fy∂α3

∂ fy∂α4

∂ fy∂α5

∂ fy∂α6

∂ fz∂θ1

∂ fz∂θ2

∂ fz∂θ3

∂ fz∂θ4

∂ fz∂θ5

∂ fz∂θ6

∂ fz∂α1

∂ fz∂α2

∂ fz∂α3

∂ fz∂α4

∂ fz∂α5

∂ fz∂α6

∂ fx∂d1

∂ fx∂d2

∂ fx∂d3

∂ fx∂d4

∂ fx∂d5

∂ fx∂d6

∂ fx∂a1

∂ fx∂a2

∂ fx∂a3

∂ fx∂a4

∂ fx∂a5

∂ fx∂a6

∂ fx∂ lx

∂ fx∂ ly

∂ fx∂ lz

∂ fy∂d1

∂ fy∂d2

∂ fy∂d3

∂ fy∂d4

∂ fy∂d5

∂ fy∂d6

∂ fy∂a1

∂ fy∂a2

∂ fy∂a3

∂ fy∂a4

∂ fy∂a5

∂ fy∂a6

∂ fy∂ lx

∂ fy∂ ly

∂ fy∂ lz

∂ fz∂d1

∂ fz∂d2

∂ fz∂d3

∂ fz∂d4

∂ fz∂d5

∂ fz∂d6

∂ fz∂a1

∂ fz∂a2

∂ fz∂a3

∂ fz∂a4

∂ fz∂a5

∂ fz∂a6

∂ fz∂ lx

∂ fz∂ ly

∂ fz∂ lz

×[dθ1,dθ2,dθ3,dθ4,dθ5,dθ6,dα1,dα2,dα3,dα4,dα5,dα6,dd1,dd2,dd3,dd4,dd5,dd6,

da1,da2,da3,da4,da5,da6,dlx,dly,dlz]T (5)

Define this 3×27 dimension matrix as J, which denotes the error coefficient matrix of the 6-DOF AACMM.Further analysis shows that elements of the 13th column of the matrix J are all zero, i.e.

∂ fx

∂d1=

∂ fy

∂d1=

∂ fz

∂d1= 0

This means that the deviation of the structural parameters d1 has no effect on the measurement accuracy.Consequently, it is not necessary to conduct an accurate calibration on it, i.e., we can keep its value as aconstant in the calibration process.

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3. SELF CALIBRATION

3.1. Self-Calibration PrincipleThe measurement results of an AACMM are derived through the kinematic model which is based on thejoint angles. The joint angle θi consists of the angle sensor reading θs,i and the joint angle deviation θ0,iat the zero position, that is, θi = θS,i +θ0,i. θ0,i shows the deviation between the physical zero position ofangle sensor and the joint zero position, whose value is related to the installation of the angle sensor. Thecoordinate transform model of an AACMM can be expressed as follows:

xyz1

=6

∏i=1

cos(θS,i +θ0,i) −sin(θS,i +θ0,i)cosαi sin(θS,i +θ0,i)sinαi ai cos(θS,i +θo,i)

sin(θS,i +θ0,i) cos(θS,i +θ0,i)cosαi −cos(θS,i +θo,i)sinαi ai sin(θS,i +θ0,i)

0 sinαi cosαi di

0 0 0 1

lxlylz1

(6)

In addition to the angle sensor readings θS,i in Eq. (6), the remaining 27 parameters correspond to thestructural parameters of the AACMM. Without considering the deformation in the work process, the valuesof these 27 parameters can be determined when the measuring machine assembly is completed.

Set p=(θ0,1,θ0,2,θ0,3,θ0,4θ0,5,θ0,6,α1,α2,α3,α4,α5,α6,d1,d2,d3,d4,d5,d6,a1,a2,a3,a4,a5,a6, lx, ly, lz)T ,the probe coordinates (x,y,z)T can then be written as a function of the angle sensor readings(θs,1,θs,2,θs,3,θs,4,θs,5,θs,6)

T and the structural parameters p.x = fx(θS,1,θS,2,θS,3,θS,4,θS,5,θS,6, p)y = fy(θS,1,θS,2,θS,3,θS,4,θS,5,θS,6, p)z = fz(θS,1,θS,2,θS,3,θS,4,θS,5,θS,6, p)

(7)

In the ideal case, 27 equations can be established by measuring nine points Q j ( j = 1,2, . . . ,9) fromEq. (7). Then the structural parameters in p can be obtained by solving

x1 = fx(θ1S,1,θ

1S,2,θ

1S,3,θ

1S,4,θ

1S,5,θ

1S,6,p)

...

x9 = fx(θ9S,1,θ

9S,2,θ

9S,3,θ

9S,4,θ

9S,5,θ

9S,6,p)

y1 = fy(θ1S,1,θ

1S,2,θ

1S,3,θ

1S,4,θ

1S,5,θ

1S,6,p)

...

y9 = fy(θ9S,1,θ

9S,2,θ

9S,3,θ

9S,4,θ

9S,5,θ

9S,6,p)

z1 = fz(θ1S,1,θ

1S,2,θ

1S,3,θ

1S,4,θ

1S,5,θ

1S,6,p)

...

z9 = fz(θ9S,1,θ

9S,2,θ

9S,3,θ

9S,4,θ

9S,5,θ

9S,6,p)

(8)

where xi is the X axis coordinate value of point j ( j = 1,2, . . . ,), yi is the Y axis coordinate value of pointj, and zi is the Z axis coordinate value of point j. θ

jS,i(i = 1,2, . . . ,6, and j = 1,2, . . . ,9) is the jth value of

angle sensor readings θs,i of point Q j ( j = 1,2, . . . ,9).On the left side of Eq. (8), there are the measured coordinates x j,y j,z j ( j = 1,2, . . . ,9) of the 9 points.

These coordinates should be obtained by highly-precise measurement equipments or special measuring de-vices which make this calibration method difficult to be used in industrial field. One solution to eliminate

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the need for high-precision measuring devices and to simplify the calibration process is the single-pointcalibration, where a single point with unknown coordinate (x,y,z) is measured for 10 times with each mea-surement being done in different encoder reading positions. Theoretically, 30 formulae can be establishedto solve the 30 unknowns (27 AACMM parameters and 3 unknown coordinates of the point). This methodgreatly reduces the difficulty of conventional calibration operation and expands the applications of AACMMdramatically.

In the single-point calibration process, the measurement error and sensor reading error may affect the so-lution accuracy of the above equations. In order to reduce the impact of these random errors and increase therobustness of the solution, more points should be measured to construct more equations, so that an overde-termined equation group is created. The overdetermined equation group is then solved by the least squaremethod (LSM) or other optimization methods. Since there are many drawbacks using the LSM for solvingmulti-variable equations (One of them is that the least square method tends to generate the local minimumsolution), this paper adopted a modified simulated annealing algorithm to solve the overdetermined equa-tion group and obtain a global optimal solution, which will also serve as the optimal structural parametersof AACMM.

3.2. Objective FunctionA 6-DOF AACMM has multiple redundant degrees of freedom. Any point within the measured space canbe detected theoretically from innumerable different directions (i.e. different position and orientation). Nomatter from which direction the coordinates of the measured point should be the same. However, thereexists deviation between the actual structural parameters of the measuring machine and the theoretical pa-rameters which is used in the coordinate transform model. Therefore, each measured coordinate values ofthe measuring machine are not actually the same. The larger deviation between the theoretical parametersand actual parameters, the greater difference between the measured coordinates. Conversely, if every timethe measured coordinates are the same or the difference is small, it can also be concluded that the structuralparameters of the measuring machines is identical or close to the actual parameters. Based on this princi-ple, an optimal searching method is used to determine the optimum structural parameters of the measuringmachines with the goal to make the coordinate values of the measured point equal.

To reduce the random error in the point measuring process, a fixed conical hole is used to keepthe probe center at the same location (measure this point throughout). A series of joint angle dataθ i (θ i

1,θi2,θ

i3,θ

i4,θ

i5,θ

i6) are obtained by changing the angle of each joint. The corresponding coordinates

(xi,yi,zi) can be calculated through the joint angle data, so that the deviation between the measured coordi-nates and the actual coordinates (x,y,z) can be expressed as a distance between the two points

E i =√(xi− x)2 +(yi− y)2 +(zi− z)2 (9)

The average deviation of coordinates of each group is

E =1N

N

∑i=1

E i. (10)

As the average error cannot explain the repeatability or the dispersion of measurements, the standard devi-ation σ is introduced to represent the repeatability of measurements. The standard deviation of measurementresults is calculated by the Bessel Equation as

σ =

√1

N−1

N

∑i=1

(E i− E)2 (11)

650 Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 4, 2016

Fig. 2. The data acquisition based on a conical hole.

Generally, it is not easy to get accurate coordinates of point Q in the reference coordinate system, partic-ularly when it is measured in the industrial field. Then, in Eq. (9), x,y and z are substituted by x, y and z,which are the average value of xi,yi and zi, respectively. It is not necessary to know the exact coordinatesof point Q because the location of probe can always remain the same conical hole during the measurementprocess.

Equation (11) is used as the objective function to search for the optimal structural parameters of AACMM,which uses the measuring machine kinematics model as the theoretical parameters to improve the repeata-bility of the measuring machines.

3.3. Simulated Annealing Algorithm

Simulated annealing algorithm (SAA) has ability to escape from the local minimum solution. However,it is difficult for SAA to achieve global convergence conditions. Sometimes the probability of acceptancemakes the current state worse than some intermediate states in the search process [15]. Thus, in practicalimplementation of the algorithm, an approximate optimal solution is usually obtained which is possiblyworse than the best solution in the search process. Another drawback of normal SAA is the low searchingefficiency. During the iteration process, the size of search range delta (delta as a vector, including the errorlimit range of the structural parameters) does not change, which results in poor searching efficiency and lowquality of the final solution.

In order to get the best state in the whole search procedure and improve the efficiency and quality of thefinal solution, a modified simulated annealing algorithm (MSAA) is proposed. In the proposed searchingprocess, we firstly retain the intermediate optimal solution p∗ and update it in time. When it is close tothe optimal solution, we reduce the search range to deltaS. This can improve the searching efficiency andaccuracy in the final solution. With the following two conditions, we can determine whether or not thecurrent state is closer to the optimal solution.

Condition 1: current repeatability of f (p) is smaller than a constant of ε . Here ε is small enough to be ableto indicate the internal energy of the solid system, which has been reduced to critical state.

Condition 2: after m times of Markov chain calculation the repeatability of f (p) has not been significantlyimproved.

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Table 1. The initial values of the structure parameters in the self-calibration.Joint number i di (mm) ai (mm) αi (

◦) θ0,i (◦)

1 401 42 –90 02 0 42 –90 –111.09503 335 37 –90 –133.20774 0 37 –90 –142.95395 335 37 –90 –116.42046 0 37 –90 –84.0994Probe bias (mm) lx = 0 ly = 0 lz = 210

4. EXPERIMENTS

4.1. Parameters CalibrationThe repetitive data of single point used in the AACMM self-calibration were measured at the same pointfor many times. In the measurement process, the probe center is requested to remain at the same location,and a conical hole was designed whose bottom is smaller than the probe in diameter but the top is larger. Aslong as the probe is placed in the conical hole and contacts tightly with the cone, the center of the probe willnot change as shown in Fig. 2. According to this principle, the probe placed in the conical hole is measuredby changing the angle of each joint, and then a series of joint angle data can be obtained for the followingself-calibration calculation. In the data acquisition process, the external surface of the probe ball shouldcontact tightly with the conical hole’s inner surface, otherwise, the collected data may not be at the samelocation of the point. Besides, the joint angle should change adequately in order to achieve good robustnessof calibration results.

A self-calibration method based on the MSAA is used to calibrate the structural parameters of theAACMM. The initial parameters are shown in Table 1.

The MSAA takes a first-global-later-local search strategy. The setting of the initial searching range ofstructural parameters is different. Since the initial iterative value of deviation θ0,i from joint zero positionis an estimated value, the error will be large. Therefore a search range of ±5◦ is set. The accuracy of jointtorsion angle αi is relatively high, so that a search range of±1◦ is set. The search range of the linkage lengthdi, joint length ai and probe bias (lx, ly, lz) are set to ±3 mm. Refining the search range based on the initialsearch results will be narrowed. As the accuracy of the structural parameters become higher, the refinedsearch range is accordingly smaller. The search range of deviation θ0,i from joint zero position and jointtorsion angle αi are ±0.5◦ and the search range of linkage length di, joint length ai and probe bias (lx, ly, lz)are±0.5 mm. According to the kinematic analysis results, the linkage length d1 does not affect the accuracyof measuring machine. Therefore, the value of d1 remains the same throughout the iterative search.

The control parameters of MSAA have an important impact on the algorithm process. In this study, theyare set as follows

1. Termination conditions are e = 0.01 and Tf = 1E−6.

2. Markov algorithm equilibrium is q = 10.

3. The switch refined search terms are ε = 0.5, m = 20 and ζ = 0.01.

When the parameters are set, the calibration data will be imported into the calibration software. Then,the iterative search starts. After about 10,000 times iterations, the algorithm ends. The search results of thestructural parameters are shown in Table 2.

652 Transactions of the Canadian Society for Mechanical Engineering, Vol. 40, No. 4, 2016

Table 2. The results of the structure parameters after the self-calibration.Joint number i di (mm) ai (mm) αi(

◦) θ0,i(◦)

1 401.000 41.448 –90.0982 0.00002 0.001 41.604 –90.0037 –110.19643 334.293 37.306 –90.0327 –133.37964 0.542 37.062 –89.8327 –143.52335 334.000 36.901 –90.0802 –115.01176 0.001 36.865 –90.0618 –85.4598Probe bias (mm) lx =−2.230 ly =−0.307 lz = 209.201

Fig. 3. Accuracy test of length measurement after the self-calibration (a) Length measurement accuracy test with agauge block of 52.1 mm; (b) Length measurement accuracy test with a gauge block of 121.8 mm.

4.2. Verification of Calibration ResultsBecause the structural parameters of the AACMM cannot be measured directly and accurately, we verify thecalibration accuracy by measuring the length of the standard gauge blocks and conducting a repeatabilitytest of the three dimensional coordinates values of a single-point.

(1) Accuracy experiment of length measurement: Standard gauge blocks were used as a reference inthe length measurement experiment. The gauge blocks were placed at four different locations respectivelyand they were measured for 5 times at each location. The sizes of the standard gauge blocks used in theexperiment are 51.200 and 121.800 mm, with the accuracy of ±1 µm under temperatures of 20∼ 24◦.

Measurement results of structural parameters after calibration are shown in Fig. 3. In Fig. 3a the maximummeasuring value is 51.253 mm while the minimum measuring value is 51.141 mm with a maximum errorrange of 0.112 mm; in Fig. 3b the maximum value is 121.860 mm while the minimum value is 121.744 mmwith a maximum error range of 0.116 mm.

(2) Repeatability experiment of a single-point: The repetitive data of a single-point used in the AACMMself-calibration were measured at the same point for many times. Measurement results are shown inFig. 4. In Fig. 4, the x,y and z coordinates values ranges of a single point are x ∈ [19.268,19.179] mm,y ∈ [−463.825,−463.918] mm and z ∈ [−18.557,−18.649] mm, respectively. The data distributes within abelt area of ±0.05 mm. That means the repeatability of the single-point is ±0.05 mm.

5. CONCLUSIONS

The kinematic model of a 6-DOF AACMM was established based on DH Methods. The number of struc-tural parameters requiring accurate calibration is reduced through the kinematic analysis. The principle of

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Fig. 4. Repeatability test of single-point after the self-calibration. (a) Value of coordinates x; (b) value of coordinatesy; (c) value of coordinates z.

self-calibration of AACMM was studied, and a self-calibration method based on an overdetermined equa-tion group was proposed. Global optimization of the calibrated structural parameters was achieved by usinga modified simulated annealing algorithm. With this integrated method, a 6-DOF AACMM was calibrated.Experimental results show that the accuracy of the measuring machine and repeatability in length measure-ment have been greatly improved after using the suggested self-calibration.

ACKNOWLEDGEMENT

The authors gratefully acknowledge the support from the National Natural Science Foundation of China(Grant No. 51465027).

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