Influence of measurement noise and laser arrangement on measurement uncertainty of laser tracker multilateration in machine tool volumetric verification

Precision Engineering 37 (2013) 929–943

Contents lists available at SciVerse ScienceDirect

Precision Engineering journal homepage: www.elsevier.com/locate/precision

Review

Influence of measurement noise and laser arrangement on measurement uncertainty of laser tracker multilateration in machine tool volumetric verification S. Aguado ∗ , J. Santolaria, D. Samper, J.J. Aguilar Design and Manufacturing Engineering Department, Universidad de Zaragoza María de Luna, 3, 50018 Zaragoza, Spain

a r t i c l e

i n f o

Article history: Received 19 March 2012 Received in revised form 8 March 2013 Accepted 13 March 2013 Available online 1 April 2013 Keywords: Volumetric verification Laser tracker self-calibration Multilateration Laser tracker positioning

a b s t r a c t This paper aims to present different techniques and factors that affect the measurement accuracy of a commercial laser tracker responsible for capturing checkpoints used in machine tool volumetric verification. This study was conducted to uncover various sources of error affecting the measurement uncertainty of the laser tracker, additional sources of error that further contributed to the uncertainty, and the factors influencing these techniques. We also define several noise reduction techniques for the measurements. The improvement in the accuracy of captured points focuses on a multilateration technique and its various resolution methods both analytically and geometrically. Similarly, we present trilateration and least squares techniques that can be used for laser tracker self-calibration, which is an essential parameter in multilateration. This paper presents the influence of the spatial distribution of laser trackers (LTs) in measurement noise reduction by multilateration, which produces an improvement in volumetric error machine tool reduction. A study of the spatial angle between LTs, the distance and the visibility of the point to be measured are presented using a synthetic test. All of these factors limit the scope of multilateration. Similarly, a comparison of self-calibration techniques using the least squares and trilateration methods with which to determine the relative position of the laser tracker employees is presented. We also present the influence of the relationship between the radial and angular measurement noise self-calibration processes as it relates to the volumetric error reduction achieved by the machine tool with multilateration. All studies were performed using synthetic tests generated using a synthetic data parametric generator. © 2013 Elsevier Inc. All rights reserved.

Contents 1. 2. 3.

4.

5. 6.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of a commercial laser tracker for machine tool volumetric verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multilateration with laser tracker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Factors and number of laser tracker to use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Multilateration by intersection of spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Transformation of a quadratic to a linear system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Intersection of the three spheres: the geometric resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Self-calibration techniques of laser trackers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Self-calibration by least squares (HST) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Obtaining the roto-translation matrix using homogenous spatial transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Self-calibration by trilateration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The influence of laser tracker positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Multilateration tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

∗ Corresponding author. Tel.: +34 686346726. E-mail addresses: [email protected] (S. Aguado), [email protected] (J. Santolaria), [email protected] (D. Samper), [email protected] (J.J. Aguilar). 0141-6359/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.precisioneng.2013.03.006

930 930 931 931 931 932 932 932 933 933 934 934 934 935

930

7.

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

6.2. Self-calibration tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Three spheres intersect – Geometric resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction A laser tracker is a portable measurement system that provides the position of a point in spherical coordinates. The point coordinates are determined by comparing a measurement beam with a reference beam from the laser interferometer and the combination of azimuth and polar angle encoders of its head, which provide two rotational degrees of freedom of the LT. Since its first application in tracking missiles and aircraft in the early 60 s, the tracking process has undergone major changes. In 1980, Allen Greenleaf proposed a surface measurement system with four interferometers and a tracking system [1]. In 1986, Lau and Hocken [2] patented a new optical tracking system with a laser beam, angular encoders and photo sensors that track reflector robots with three or five degrees of freedom to create a more manageable system with the capability to maintain accuracy and perform both static and dynamic measurements. In 1994, Vincze [3] managed to obtain six degrees of freedom of a robot. Despite the recent developments in laser trackers, they are still affected by various sources of random or systematic errors. Gallagher [9] divided the sources of error as follows: angular encoders, tracking system, perpendicularity, distance measurement and alignment of the beam. Teoh [10] divided the sources of error as follows: errors in the environmental factors, errors in the data capture and errors due to approximations and simplifications. The variation in the environmental conditions alters the wavelength of the measuring beam; therefore, the distance measure [8] and the variation in temperature, humidity and pressure produce the greatest condition [3,9]. The mechanism of the orientation of the laser [10,11], the reflector [11] and the sensor calibration and resolution PSD [12,13] also influence the measurement uncertainty. The most significant random source of error is the measurement noise formed by the uncertainty of the interferometer and the uncertainty of the angular encoders, which provide a greater contribution to the measurement uncertainty. By capturing a single point, the laser tracker provides the spherical coordinates with respect to the laser tracker home, the radial distance affected by the uncertainty of the interferometer, the azimuth angle affected by the uncertainty of the azimuth angular encoder and the elevation angle affected by the uncertainty of the polar encoder. By obtaining the coordinates of at least one point with three LTs, the multilateration technique can be utilized [5,6] in which we are only given the distance of each LT to the reflector, and it is possible to achieve a reduction in the influence of the measurement noise and consequently, reduce the overall uncertainty. This technique was first used in aviation in the 60 s as a method to determine the position of the aircraft in real time. It was then extended to other fields that required greater precision. In 1994, Vincze [3] generated the first trilateration tests for the positioning of robots with three trackers that differed from those of Takatsuji in 1998 [4], which extends the use of laser trackers to CMM with the concept of an independent LTs trilateration that aims to improve the accuracy of the data captured. In 2000, Hughes [7] applied this technique to design an CMM that reduces uncertainty and improves accuracy in all fields with a single LT. Tkatsuji [15] conducted a study on the relationship between the measurement error of one LT and the applied positioning of trilateration. This development was discovered by designing

939 941 942 943

and manufacturing individual LTs to implement testing with cat eye reflectors. This paper aims to present the influence, the techniques and the scope of each of the factors that affect the measurement accuracy of a commercial laser tracker used to capture a point with which the volumetric verification of a machine tool is realized [5,14]. Consequently, the multilateration technique along with various analytical and geometrical methods of resolution and their limitations are presented. To identify the relative positions of the laser trackers, various calibration techniques, such as self-calibration using least squares, homogenous spatial transformation (HST) [29], and calibration by trilateration, are presented. Similarly, there has been a particular effort in studying the influence of the spatial distribution of the LTs among themselves with the multilateration limiting factor. In addition to the influence of the relationship between the radial measurement noise and the angular measurement noise in the self-calibration process is the limiting reduction in the uncertainty of the measurement using multilateration. To improve the accuracy of the points captured by a laser tracker with multilateration, several factors must be considered prior to the measurement: (I) Restrictions that incorporate the use of the laser tracker as a system of measurement in volumetric verification. (II) Self-calibration of the LTs as a prerequisite to the measurement. (III) The scope of multilateration as a result of the spatial distribution of the LTs’ respective points to be measured. 2. Use of a commercial laser tracker for machine tool volumetric verification Volumetric verification, which is understood to be an intensive process of identifying errors through a nonlinear model, aims to reduce the volumetric error of a machine tool [5,14]. The volumetric error is determined by the positioning error of the machine within its workspace due to its geometric errors. Minimization with indirect measurements of the geometric errors [16–21,30–33] through the points captured by the LT and the theoretical points obtained through a kinematic model machine will reduce the volumetric error of machine tool (MT) to improve its accuracy. The kinematic model of the machine depends on the type of machine, the number of axles, the sequence of motion and the errors that affect the machine [5,14,22–28,31] (Fig. 1). In this way, we obtain a mathematical model based on the configuration of the MT in which the tool position with respect to the measurement system (LT) is determined as a function of the movement of the machine. The addition of the LT to the kinematic model of the MT [14,33] will result from the MT structure in which the laser tracker replaces the part and the reflector replaces the tool. The LT will be placed over the bench or over a mobile structure depending on the machine tool that will be verified (Fig. 2). An incorrect laser tracker self-calibration directly influences the difference between the theoretical and actual points, altering the volumetric error of the machine tool, which affects the characterization of the geometric error. When the laser tracker is removed and the compensation is realized, the geometric errors attempt

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

931

Fig. 1. Machine tool configuration.

Fig. 2. Kinematic model of the machine tool used in tests.

to compensate with a self-calibration error of the measurement system. Therefore, the MT accuracy that is obtained is lower than it would have been if a more accurate self-calibration had been realized. Similarly, the measurement uncertainty of the measurement system also affects the difference between the real and theoretical points. The measurement uncertainty will attempt to be compensated by the geometric error obtained in the volumetric verification. By performing the measurement of a point from three different positions using multilateration techniques, the measurement uncertainty can be reduced. Subsequently, the characterization of the geometric errors will improve as well as the accuracy of the machine tool after compensation.

3. Multilateration with laser tracker The measurement uncertainty is composed of the design and manufacturing errors of the measurement system, the test setup, the data capture and the environmental conditions. Of these errors, the greatest contribution is made by the measurement noise originating mainly from the angular resolution of the encoders used. The multilateration technique reduces the measurement uncertainty and eliminates the contribution of the angular noise measurement. To apply this technique, it is required to obtain information of the measured point from at least three different positions.

3.1. Factors and number of laser tracker to use The use of one, three or four laser trackers to obtain the information required will depend on the environmental conditions in which the measurement is performed, the repeatability of the measuring point and the relationship between the cost and the improvement obtained. If the capture points are performed in a controlled environment and without the influence of external factors when positioning the reflector at the same point, the capture points can be created by moving a single LT to three different positions capturing the same points in a multipath. In this case, the relationship between the improvement obtained by using three LTs and the cost would render it feasible to apply a multipath with a single LT. However, for the application of volumetric verification multilateration for a machine tool in a specific work environment [14], using a multipath with a single LT may not be allowed. The result of the temperature changes that unevenly affect the

Fig. 3. Variation of temperature with time at different positions.

different elements that form the kinematic chain of the machine reduces the repeatability of the measuring point (Fig. 3). 3.2. Multilateration by intersection of spheres When a point with known CSRef coordinates is measured with three LTsi = 1, 2, 3 , the point will be measured in three coordinate systems CSLTii = 1, 2, 3 . If the measured point is changed from CSLTii = 1, 2, 3 to CSRef, three close but different points will result due to the measurement noise. The measurement noise of each LT consists of two angle components, the azimuth and the polar, and a radial component (Fig. 4). Multilateration aims to obtain a point with less involvement of the measurement uncertainty. Therefore, it only uses the radial

Fig. 4. Influence of measurement noise.

932

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

Fig. 5. Laser tracker positioning in multilateralized coordinate system.

component of each LT to determine the new multilateration point. To do this, the new point is defined by the intersection of the three spheres that define the radial component of each of the LTs. With (xi yi zi ) defining the center of sphere LTi(i = 1, 2, 3) .

3.2.2. Intersection of the three spheres: the geometric resolution The geometric resolution of the intersection of the spheres requires two steps to obtain the resolution (see Appendix A):

D12 = (x − x1 )2 + (y − y1 )2 + (z − z1 )2

(1)

D22 = (x − x2 )2 + (y − y2 )2 + (z − z2 )2

(2)

• Calculate the intersection between sphere 1 and sphere 2 to obtain circle C1 . • Calculate the intersection between sphere 3 and circle C1 .

D32 = (x − x3 )2 + (y − y3 )2 + (z − z3 )2

(3)

The resolution of the system of equations can be obtained by the transformation of a quadratic equation system to a linear system accounting for the restrictions and the limitations of the method. The resolution can also be obtained by the geometric intersection resolution of the three spheres. In both cases, it is essential to know the position and orientation of the LTs by self-calibration. Otherwise, the new point that is calculated differs from the real point.

In contrast to the previous method, this method does not require the creation of a new coordinate system depending on the laser tracker position. This method presents the coordinates of the intersection point in a reference coordinate system CSR. This coordinate system may be the machine coordinate system CSMT. Similarly, if there is no intersection of the spheres, there will not be an approximation point and the measured point is discarded in the subsequent calculations.

4. Self-calibration techniques of laser trackers 3.2.1. Transformation of a quadratic to a linear system A process that is most widely used involves transforming the quadratic system of Eqs. (1)–(3) into a linear system of equations. For these equations, it is necessary to generate a new reference plane that meets the following specifications (Fig. 5). • The origin of the new coordinate system CSN is the origin of CSLTi, i.e., CSLT1. • The X-axis results from the intersected origin of CSLT1-CSLT2. • The Y-axis is defined so that the origin of CSLT3 is included in the XY plane.

The realization of a measurement aims to identify the position of the measured points in a coordinate system different from the measurement system (Fig. 6) and/or the distance between the points. In both cases, the points are affected by the uncertainty of the measurement. The reduction in the measurement uncertainty by multilateration requires knowing the precise spatial arrangement of each LT relative to the others. This result is obtained either by a least squares fit (HST) between the measured points for each of the three or four laser tracker employees or by trilateration or quadrilateration of the points measured by the LTs.

By solving the system of equations, the coordinates of the point are obtained. x=

y=

D12 − D22 + x22

(4)

2x1 D12 − D32 + x22 + y22 − 2xx2 2y2

z = (D12 − x2 − y2 )

1/2

(5)

(6)

The sign of the z component is obtained with a quarter LT or by the appropriate positioning of the three LT employees through the study of the z component of each of the LTs. Using information from the fourth laser tracker, trilateration is transformed into quadrilateration. This method provides a unique value for the z component of the measured point. The system of equations expressed above returns a solution even when there is no intersection of the three spheres. In this case, the resolution of the system provides an approximation point, which is located at an equal distance from all three of the spheres.

Fig. 6. Obtaining transformation matrix.

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

933

Fig. 7. Least squares (HST) self-calibration.

⎡

4.1. Self-calibration by least squares (HST) Self-calibration using a least squares fit provides the rototranslation matrix between the laser tracker reference coordinate system CSLTj and a coordinate laser tracker system LTi =/ j . The greater the number of common points that are measured with both of the LTs, the better the fit between both coordinate systems will be. It is necessary to have a minimum of five points in common. This method is not a method of multilateration. This method uses XYZ coordinates instead of using only the distances from the outputs of the LTs in which the influence of angular measurement noise is incorporated to obtain the positioning of the LTs. When the position of the LTs is known, the coordinates of the multilateralized point are obtained considering only the radial distance of each laser tracker (Fig. 7).

−a1

0

−a2

⎢ ⎢ a1 0 Qa,i = ⎢ ⎢ a −a 3 ⎣ 2

⎥ ⎥ ⎥ a1 ⎦

0

a2

−a1

0

0

−b1

−b2

−b3

0

−b3

b2

b3

0

−b2

b1

⎢ ⎢ b1 Qb,i = ⎢ ⎢b ⎣ 2 b3 n 

⎤

−a2 ⎥

a3

a3

⎡

M=

−a3

(11)

⎤

⎥ ⎥ ⎥ ⎥ −b1 ⎦

(12)

0

Q  a,i Qb,i

(13)

i

4.1.1. Obtaining the roto-translation matrix using homogenous spatial transformation A is a 3 × N matrix that contains the coordinates of the points measured in the LT system. N represents the number of measured points, and B is a 3 × N matrix containing the coordinates of the reference system. The center of each set of points must be calculated.

n  i=1

Ca =

xa,i

ya,i

za,i

n

Cb =

xb,i

yb,i

zb,i



a1

b1

b2

a3

b3

 



= xa,i



= xb,i

⎡

yb,i

 

za,i − Cax

 

zb,i − Cbx

Cay

Cby

Caz

Cbz



(9)

(10)

E1,4

E2,4

E1,4

−E2,4

−E3,4

−E4,4

E1,4

−E4,4

E3,4

E4,4

E1,4

−E3,4

E2,4

W = M  1 ∗ M2

⎤ ⎥ ⎥ ⎥ ⎦

−E3,4 ⎥

E1,4

⎣ E3,4

(14)

⎤

⎥ ⎥ ⎥ ⎥ −E2,4 ⎦

(15)

E1,4 (16)

The rotation matrix between the two coordinate systems originates from matrix R, while the translation vector is given by vector T.

⎡

⎤

W2,2

W2,3

W2,4

= ⎣ W3,2

W3,3

W3,4 ⎦

W4,2

W4,3

W4,4

⎢



E4,4

−E4,4

−E2,4

⎢ ⎢ E2,4

(8)

−E3,4

E3,4

E4,4

ya,i

−E2,4

E4,4

M2 = ⎢ ⎢



n

a2

E1,4

⎢ ⎢ E2,4 E1,4 M1 = ⎢ ⎢E ⎣ 3,4 −E4,4

(7)

This value is subtracted from each of the points of matrices A and B. By summing the product of the least squares (HST) of each pair of points, the rotation matrix between the different coordinate systems is obtained.



⎡



n

i=1 

The eigen values and the eigenvectors of matrices M, E and D are obtained. The rotation matrix is determined by:

⎥

(17)

934

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

Fig. 8. Trilateration self-calibration.

T = Cb − R · Ca

(18)

It should be noted that not every 3 × 3 matrix is a rotation matrix. 4.2. Self-calibration by trilateration Self-calibration by trilateration allows us to obtain the position of the LTs with respect to the optimization points measured previously by each of the LTs rather than the position of the measured points directly (Fig. 8). Therefore, the relative position and the multilateralized points [15,34] are obtained simultaneously. (Xi Yi Zi ) are the coordinates of laser ith and the distance di between the reflector and the laser, which fix the origin and orientation of the coordinate system using theses as initial values of optimization. The distance kij between the measured point jth and the laser ith can be expressed using the value measured lij by the laser ith. kij = di + lij

(19)

Using two different laser trackers to measure the point (xij yij zij ) and based on the principle of trilateration by transforming the intersection of a linear system of equations, the distance kij is expressed as: kij = [(xij − Xi )2 + (yij − Yi )2 + (zij − Zi )2 ]

1/2

(20)

The distance should be the same when calculated both ways. If that is not the case, the assumptions for the positioning parameters are incorrect. Positional parameters of the different systems are calculated by minimizing Rest, and squaring the sum of the difference kij for all points and all LTs, with n being the number of measured points. Rest 2 =

n n  

{[(xij − Xi )2 + (yij − Yi )2 + (zij − Zi )2 ]

1/2

− (di + lij )}

2

j=1 j=1

(21)

The minimization of the difference is obtained by the variation in the distance between the LT and the position of the origin di=1, ..., 3 , the relative positioning of the LTs x1 , x2 , y2 , x3 , y3 , z3 and the coordinates of the points xi yi zi measured with the method of Levenber Marquardt. 5. The influence of laser tracker positioning Each point measured for each laser tracker is surrounded by a characteristic error ellipse, which indicates the region of uncertainty for each point. Multilateration reduces the area of uncertainty by the intersection of the spheres, creating a zone of lower uncertainty. The size of this zone is determined by the relative spatial position between the laser trackers. It is especially important that the spatial angle that forms the measurement beam of each LT with the point is measured for a different error reduction in each case [15,34]. Similarly, the positioning of the LTs determines the volume of space where points must be measured. This position is determined by the tracking capability of the reflector for all of the LTs. The viewing angle for a standard reflector is ±30◦ . Therefore, the spatial angle between the LT and any point to be measured with respect to the line of the initial alignment between the laser and the reflector must be within this range. Another parameter to consider is the ability to turn the head of each LT to avoid a cut in the measuring beam that may exceed this limit (Fig. 9). 6. Test and results When a volumetric verification of a real machine tool is realized, the operator only can modify the relative position of each laser tracker and it is directly related to the reduction of the measurement error reduction by multilateration.

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

935

Table 1 Measurement noise. LTs noise Azimuth Polar Radial

3.00E−05 ␮rad 3.00E−05 ␮rad 4 ␮m 0.8 ␮m/m

From Figs. 10 and 11, several conclusions can be made:

Fig. 9. Angles of interest in a LT.

By performing the synthetic test presented below, the influence of the spacial distribution of LTs in measurement noise reduction was studied. Thus, the theoretical reduction in measurement noise reduction using multilateration was obtained. Similarly, the accuracy in the calculation of laser tracker location is related to the self-calibration technique employed, which results in different values depending on the laser tracker radial noise. In order to know the influence of radial noise in laser tracker self-calibration techniques, several synthetic tests are presented.

• Any influence of the angular error is eliminated by applying multilateration. • The maximum reduction achievable with this technique is based on two parameters: the distance of the LTs to the measurement points and the relative positioning between them. To verify the influence of the positioning of the LTs in the application of the multilateration, a study on the influence of the spatial angle between the LTs was conducted. From a workspace defined by 0 ≤ X ≤ 1400, 0 ≤ Y ≤ 500 and 0 ≤ Z ≤ 600, 10.000 measurements of the midpoint of the workspace were obtained with each of the LTs modifying the spatial position of the laser trackers.

6.1. Multilateration tests

• Initially, the three LTs are at the same point at a radial distance of 1200 mm from the workspace midpoint. • By maintaining a constant radial distance to the measured point, o the spatial angle ˛i,j between LTi and LTj varies from ˛i,j = 0 to ◦ ˛i,j = 120 , which is the case for all of the LTs.

First, we tested the principle of multilateration for a random distribution of three laser trackers. For this test, a random mesh of points in a workspace defined by 0 ≤ X ≤ 1400, 0 ≤ Y ≤ 500 and 0 ≤ Z ≤ 600. The workspace was not created arbitrarily, it was obtained through a real milling machine studied in [5,14] with three laser trackers located arbitrarily to generate the measurement results (simulation). In the first test, the synthetic points were generated and affected by the radial and angular components of the measurement noise (Fig. 10), which are modeled using a uniform distribution similar to that of each of the LTs (Table 1) [12,13]. The system of equations constituted by Eqs. (1)–(3) was solved by a transformation of this linear system of Eqs. (4)–(6) with the geometric resolution of Eqs. (A.1)–(A.20). In a second experiment (Fig. 11), these same points were affected only by the radial component of the measurement noise and the systems of equations were solved for each point with the techniques used in the first test.

Depending on the type of technique that was used when performing multilateration, we observed a different behavior of the residual mean error in the module for the same spatial angle of the laser trackers. If multilateration is performed by transforming a quadratic to a linear system of Eqs. (4)–(6), this method will provide a solution even if there is no intersection between the three spheres. The influence of the approximate points depends on the module variation and the sense of direction in the radial error for each of the spheres represented by Eqs. (1)–(3). The measured noise reduction is also influenced by the distance of the measured point with respect to the plane formed by the LTs in Section 3. For this reason, a series of peaks are observed when the spatial angles of the LT are far from 0◦ and 120◦ producing an unexpected behavior. This mathematical error can be corrected with the geometric error method of calculating the intersection of spheres by Eqs. (A.1)–(A.29). The graphs of Fig. 12 show the results of using various multilateration spatial angles. By using a spatial angle of 90◦ between the laser tracker measurement beams, the reduction in the

Fig. 10. Measurement error of the positioning of random laser trackers.

936

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

Fig. 11. Measurement error with only radial component on a random laser trackers positioning.

Fig. 12. Influence of spatial angle in error measurement and error reduction.

measurement error reaches the level of the radial component of the noise of a single LT. However, if the LTs angle away from this value to the left toward smaller angles of 90◦ , reducing the scope is more progressive than if it moved to the right with angles greater than 90◦ . o From a spatial angle of ˛1,2 = ˛1,3 = ˛2,3 = 90 , the behavior of the error while driving the LTs and maintaining the same angle space was studied by realizing the measurement of ten thousand

points present in the workspace. First, the points were only affected by the radial measurement error (Fig. 13). Subsequently, the points were affected by the radial and angular measurement noise (Fig. 14). The spatial distribution of the LTs in which a solid angle of 90◦ is formed between them is not always feasible. This phenomenon will be limited by the configuration of the machine tool to be verified and the availability of space. In most trials, the laser tracker

Fig. 13. Measurement error with only radial component with a spatial angle of 90◦ between LTs.

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

937

Fig. 14. Measurement error with a spatial angle of 90◦ between LTs.

Table 2 Color scale of Fig. 16. Residual Error

Min (␮m)

Max (␮m)

Red Yellow Green

5.65 12.4 Higher

12.4 1504

Fig. 15. Laser tracker positioning.

employees form a horizontal plane. Extenders or extensions can be used to modify the height and, therefore, the spatial angle between the rays. Of the many possible tests to perform, we needed to consider only the tests that were representative regardless of the structure of the machine to verify. In these new studies, three LTs directly measured the midpoint of the workspace generated synthetically from the same position. Subsequently, LT2 and LT3 moved by the same angle in opposite directions about the midpoint of the workspace of the MT to verify while LT1 was fixed. After visiting all of the positions of LT2 and LT3, the height of LT1 was modified until a maximum, which was determined by the length of the extender, was reached and the first operation resumed (Fig. 19). The variation in the height of LT1 modified the spatial angle ˛1,2 y1,3 . By turning LT2 and LT3 with respect to the midpoint by an angle ␤ in the opposite direction, the spatial angle ˛2,3 was modified. When we place the LTs, it is difficult to determine the spatial angle between them. However, a color map (Fig. 15) provides the best LT positioning in terms of three parameters: the height of LT1,

the angle ˇ between LT2 and LT3 and the height variation between LT2-3 and the midpoint of the workspace. The graph on the left of Fig. 16 shows the case where the height of LT2 and LT3 coincides with the height of the midpoint measured. In that case, the worst possible arrangement is one in which the LT2 and LT3 are facing each other. The left side and the right side are two symmetrical areas of behavior. In the event that the height of the LT2-3 is 250 mm above the midpoint of the workspace, the symmetry of the positioning is lost, which indicates a better performance with ˇ > 180◦ (Table 2). Similarly, the maximum error does not occur for ˇ = 180 but has a trend line approaching that value. In all cases, the best position o closely resembles ˛1,2 = ˛1,3 = ˛2,3 = 90 . By maintaining the LTs’ spatial angles of Fig. 17, the angle of vision of the LTs must be verified to comply with restrictions on the vision angle and the tracking angle reflector. Of all possible positions of the LTs, the best position allows the visibility of the angles, and is as close as possible to the workspace. The improvement is the result of a reduction in the variable radial component of the measurement noise, which reduces the global error by 2–4% depending on the distance to measure.

Fig. 16. Measurement error reduction color map (real LT positioning).

938

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

Fig. 17. Visibility and influence of distance in reducing the measurement uncertainty.

Table 3 Work space.

Initial Final ˜ Intervan

X (mm)

Y (mm)

Z (mm)

0 1400 100

0 500 100

0 600 100

The tests that were performed show the importance of the laser tracker positioning on the measured point. In our case, the measured point is the workspace midpoint of the machine to verify. This point was not chosen arbitrarily. If the spatial angle formed by the laser tracker measurement beams is as close as possible to the ideal angle, the noise reduction variation at points far from the midpoint will be less than if the ideal spatial angle is formed at a point different from the midpoint. However, to determine if the measurement noise reduction affects the volumetric verification, new tests are required. Using a parametric synthetic data generator [14], points captured from each of the three laser trackers were generated. The measured points were affected by the geometric error of the machine depending on the position of the target reflector point within the workspace (Table 3) and measurement noise according

to the LTs’ positioning (Tables 4 and 5). The measurement noise was characterized using data from Table 1. The position of the LTs, which limits the scope of multilateration, was extracted from the color map in Fig. 17. The positioning of the LTs results in an error of 5.27 ␮m in measuring the midpoint of the workspace. To determine the influence of the geometric errors introduced into the volumetric errors, the combined influence of all of the errors is presented. Table 6 presents the mean and the maximum of the geometric errors before and after optimization. Two different tests were conducted, one with an LT positioning close to the workspace and the other with an LT positioning far from the workspace. The volumetric error reduction and the compensation of the geometric errors were found using the method of parameter identification [14] with the addition of an improvements package [5]. The results show how multilateration improves the volumetric verification results by improving the machine tool accuracy. If the LTs are placed far from the workspace to be verified, the influence of the measurement error on the volumetric error is greater. This finding is a result of the angular measurement noise and the variable component of the variable radial noise, which depend on the distance to measure. In this case, the improvement experienced in the verification of multilateralitation is greater than for the previous case.

Table 4 Laser tracker positioning near work space. N◦ LT

Coordenate X (mm)

Coordenate Y (mm)

Coordenate Z (mm)

Turn about X (◦ )

Turn about Y (◦ )

Turn about Z (◦ )

LT 1 LT 2 LT 3

700 −330 1730

1445 1478 1478

1369 300 300

15 60 5

35 5 60

20 35 40

Table 5 Laser tracker positioning away work space. N◦ LT

Coordenate X (mm)

Coordenate Y (mm)

Coordenate Z (mm)

Turn about X (◦ )

Turn about Y (◦ )

Turn about Z (◦ )

LT 1 LT 2 LT 3

700 −3276 3276

3237 3320 2826

2972 300 300

15 60 5

35 5 60

20 35 40

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

939

Table 6 Volumetric verification results.

LT1 (near) LT multi (near) LT1 (away) LT multi (away)

A. initial E.M (␮m)

Max. initial error (␮m)

466.12 463.20 471.09 463.57

1371.05 1352.09 1472.62 1350.37

Noise % error 9.0% 18.3%

A. opt. E.M (␮m)

Max. opt. error (␮m)

Residual error %

58.09 12.27 175.33 20.09

390.07 43.17 468.20 52.22

12.5% 2.6% 37.2% 4.3%

A.E.M, average error in module; Max, maximum.

However, the overall volumetric error reduction is greater for the case of in which the positioning of the LTs is near the points to be measured. Variations in the environmental conditions, the temperature, the relative humidity and the pressure directly affect the wavelength of the measurement beam and the measurement noise in the radial direction. The influence of the uncertainty in the radial direction on the variation not only is not remedied by the use of the technique of multilateralitation, but it is amplified by the spatial arrangement of the LTs. The influence of this factor on the volumetric error reduction was studied by changing the characterization of the radial noise for each test. By maintaining the laser tracker positions (far from and near the work space), the work space in which the verification was carried out, the points to be measured, the geometric error that affects the points to be measured and the angular noise characterization, a study on the influence of the variation of

the constant term of the radial noise from 0 ␮m to 8 ␮m (equivalent to variations of 0–200% in the nominal noise (Table 1)) on the volumetric error reduction was realized. Figs. 18 and 19 show how applying this technique results in a greater error reduction if the LTs are far from the target point. The result of the angular measurement noise increases with the distance to the measured point. However, the smallest error occurs when the laser trackers are as close as possible to the measuring points. 6.2. Self-calibration tests As demonstrated by the tests presented above, multilateralization reduces the measurement uncertainty of the measured points through the adequate positioning of the LT employees by providing a greater number of accuracy points. Therefore, laser

Fig. 18. Influence of radial error in the residual error of the volumetric verification.

Fig. 19. Influence of radial error in the residual error of the volumetric verification.

940

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

Table 7 Laser tracker positioning for self-calibration. N◦ LT

Coordenate X (mm)

Coordenate Y (mm)

Coordenate Z (mm)

Turn about X (◦ )

Turn about Y (◦ )

Turn about Z (◦ )

LT 1 LT 2 LT 3 LT 4

700 −1228 2628 700

2769 2548 2548 250

1950 550 550 1300

15 60 5 40

35 5 60 35

20 35 40 5

tracker positioning should be as accurate as possible. An error in the LT positioning will influence the coordinates of the measured points. This error will affect the multilateration results as well as the approximation functions obtained in the verification. Self-calibration techniques allow us to obtain the relative position between the laser trackers using a series of common points measured using all of the LTs. The points are affected by the uncertainty in the measurement of each LT as well as the position of each LT with respect to the measured point. To verify the adequacy of each of the self-calibration techniques discussed in Section 4 of this article, a study of their suitability was performed by conducting a test generated from a synthetic data parameter generator. The test is based on the measurement of 60 points distributed in the workspace of Table 1, which are affected only by the measurement noise of each of the LTs (Table 7). A comparison of both of the methods of self-calibration using a least squares fit (HST) Eqs. (8)–(19) and trilateration Eqs. (20)–(22) was performed by comparing the multilateralized mesh obtained with the same method of resolution (Section 3) to obtain the LTs’ spatial distributions by both methods of self-calibration. To compare both techniques, it is necessary to obtain the following information: • The mean of the module between the multilateralized and the nominal points. • The average module distance of each mesh point to all other points without the affects from the measurement noise. The self-calibration method that provides the smallest value of the first parameter of control results in the final best relation between the theoretical and multilateration points but not necessarily the best self-calibration. The second parameter is the difference between the average of all of the possible distances between all of the points of the mesh. The smallest difference between the average module distance of all of the points and the average of the same points without noise provides a better overall positioning of the measured points (Table 8). In view of the results presented in Table 8, self-calibration by trilateration improves self-calibration using the least squares method.

Table 8 Self-calibration techniques.

Initial error without multilateration (␮m) Multilateration with least squares (␮m) Multilateration with trilateration (␮m)

First parameter

Second parameter

65.18 13.06 11.82

45.02 11.88 8.74

Table 9 Self-calibration real test.

LT position 1 (␮m) LT position 2 (␮m) LT position 3 (␮m) Multilateration with Trilateration (␮m)

First parameter

Second parameter

30.99 25.04 163.01 15.92

18.67 18.06 79.59 21.03

However, obtaining the parameters of the positioning of the LTs and the coordinates of the LTs by trilateration strongly depends on the initial parameters of the optimization. Similarly, the computation cost of this method is greater than that of using HST. Similar to multilateralitation, the scope of self-calibration is influenced by the measurement noise in the radial direction of each LT. By maintaining the laser tracker positions, the measuring points and the influence of the angular noise, several tests were conducted by varying the modeled radial noise from 0% to 200% of its nominal value (Table 1). Table 9 The greater the measurement noise in the radial direction, the greater the global positioning error measured by LT1 will be (Fig. 20). For a linear variation in the radial noise, the global positioning error reacts in the same way. This error is equal the error that occurs when least squares self-calibration is used. However, a small improvement is observed when quadrilateration self-calibration is used. However, when performing multilateration, there is a significant error reduction from the theoretical to the actual points. The use of different self-calibration techniques does not indicate a clear

Fig. 20. Influence of radial error in self-calibration techniques.

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

941

Fig. 21. Influence of radial error in self-calibration techniques.

Fig. 22. Real self-calibration test.

differentiation between least squares and trilateration in the second parameter of control (Fig. 21). In order to prove the simulated results of LT self-calibration several tests were realized. In these tests are used 8 SMRs uniformly distributed in the workspace of a CMM, a CMM and a LT (Fig. 22). Once the SMRs have been placed, these were not touch at any time during the test. The geometric centers of the SMRs are measured using the CMM. If it is assumed that the geometric center of the SMR is the same as the optical one, the CMM measurements provides a reference nominal mesh to calculate the parameters of control. 7. Conclusions Test results determined that the multilateration technique presents different results depending on the calculation method used. If the measured points are far from the multilateration coordinate system created by the plane of the three laser trackers, the transformation of the quadratic system of equations to a linear system is an appropriate process. Otherwise, the measurement error is greater than that of the theoretical results as a consequence of the calculation process used. Similarly, the trilateration equation systems present a solution even if there is no intersection of the spheres. The error that is introduced by this method is related to

the module and the direction of change in the radius of each of the spheres. For this reason, the best calculation process for obtaining multilateralized points is the geometric resolution of the spheres’ intersection. This process eliminates the need for a new coordinate system and indicates if the intersection of the spheres is found. If an intersection does not exist, this method eliminates the point measured by the LTs to subsequent operations. The results of our synthetic test show that a reduction in the measurement errors is directly related to the spatial angle formed by the measurement laser beams with the measuring point. If there are no restrictions on the LTs’ positioning, they should be placed o forming a spatial angle of ˛ = 90 between them. In this case, the residual error of the measurement noise will be similar to the radial error of the LTs. Depending on the height of the laser tracker and the workspace used to measure, the position of the LTs must be modified. If the laser tracker height matches the height of the workspace midpoint to be measured, the performance of the measurement is symmetric. If the LTs are located above the midpoint to be measured, the largest reduction occurs when the LTs are located in different quadrants. This is in contrast to what occurs when the height is lower and an improvement of approximately 2% is obtained for the same spatial angle. An additional error reduction between 2% and 4% occurs when the LTs approach the measured points and the spatial angle between the LTs is fixed. In this case,

942

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

the retroreflector visibility limits are ±30◦ for all of the measured points, and the maximum rotation angle of the LT is reached. Using a suitable spatial distribution of LTs, which is limited by the configuration of the machine tool used to verify and the available space, an improvement between 75% and 90% is realized using multilateration rather than a single LT to perform volumetric verification. Similarly, there is a linear decrease in the accuracy obtained with multilateration with the incising measurement noise in the radial direction. The self-calibration method was used to obtain the relative positioning of the laser tracker that directly influenced multilateration by modifying the scope of the multilateration. Regarding the test results, there is a slight improvement in the noise reduction when the self-calibration method was used to obtain the LT positioning with the trilateration self-calibration technique rather than the least squares technique. However, this reduction is only possible through a suitable initial parameter optimization, which can be obtained using a least squares fit. By modifying the model of the noise in the radial direction, a lower overall position error of the cloud results if quadrilateration self-calibration is used to obtain the positioning of the LTs. As a result of simplifications used, real tests present a smaller noise reduction than synthetic ones. However this method provides an important reduction of 36.4% compared to the LT with less measurement noise and 90.23% compared to the LT with the most measurement noise. This paper presents the sources of error that affect laser tracker measurements and techniques, the scope and the main parameters that affect measurement error reduction. A software module has been generated to study the positioning of LTs in real tests. Appendix A. Three spheres intersect – Geometric resolution The geometric resolution of the intersection of three spheres requires two steps for resolution: • Calculate the circle intersection of sphere 1 and sphere 2, C1 . • Calculate the intersection between the sphere 3 and the circle C1 . The spheres 1 and 2 are defined by Eqs. (1) and (2). If we subtract Eqs. (1) and (2), it provides the equation of the plane where the intersection of both spheres is. D12 − D22 − x12 − y12 − z12 + x22 + y22 + z22 = 2(x2 − x1 )x + 2(y2 − y1 )y + 2(z2 − z1 )z

d = D12 − D22 + x12 + y12 + z12 − x22 − y22 − z22

D12 − D22 − x12 − y12 − z12 + x22 + y22 + z22

t=

4(x2 − x1 )2 + 4(y2 − y1 )2 + 4(z2 − z1 )2

(A.9)

The radius of the intersection circle is the result of applying Pythagorean Theorem. Where O1 is the origin of the circumference of the first sphere, P1 a point of the circumference and C1 the center of it. C1 P12 = O1 P12 − O1 C12 .

(A.10)

The intersection of the circle intersects with the third sphere is determined by the intersection of the circle formed between the intersection of the plane created previously and the third sphere. The center of the circle intersection plane – third sphere is determined by: xr2 = x3 + 2ka

(A.11)

yr2 = y3 + 2kb

(A.12)

zr2 = z3 + 2kc

(A.13)

d − 2ax3 − 2by3 − 2cz3 4a2 + 4b2 + 4c 2

k=

(A.14)

The radius is determined by the Pythagorean Theorem. Where O2 is the origin of the second circle, P2 a point of the circumference and C2 is the center of it. C2 P22 = O2 P22 − O2 C22 .

(A.15)

Once the center and radius of each of the spheres is known, the roto-translation matrix between the circumferences plane a reference system is obtained. It will be determined by the direction cosines of three orthogonal vectors that form the background. [a1 a2 a3 ] =

[xr yr zr ] − [xr2 y2 z2 ] norm([xr yr zr ] − [xr2 y2 z2 ])

(A.16)

[b1 b2 b3 ] =

[2a2b2c] norm([2a2b2c])

(A.17)

(A.1)

[c1 c2 c3 ] = [a1 a2 a3 ] × [b1 b2 b3 ]

(A.2)

The center of the spheres in the new coordinate system is determined by:

a = (x2 − x1 )

(A.3)

b = (y2 − y1 )

(A.4)

c = (z2 − z1 )

(A.5)

The intersection of two spheres is either a circle that can have a zero radius in the event that the two spheres are tangent, or an empty set. The radius of the sphere perpendicular to this plane passes through the center of the circle of intersection. The vector normal to the plane passes through [2(x2 − x1 ), 2(y2 − y1 ), 2(z2 − z1 )] and the parametric equations of the radio are: xr = 2(x2 − x1 )t

(A.6)

yr = 2(y2 − y1 )t

(A.7)

zr = 2(z2 − z1 )t

The normal radius intersects with the plane in the center of the circle of intersection of the spheres is the result of replacing Eqs. (A.6)–(A.8) in Eq. (A.9). The center point of the circle is to replace the Eq. (16) in Eqs. (13)–(15).

(A.8)

⎡

xrn

⎤

⎡

a1

b1

c1

xr

⎤−1 ⎡

⎢ ⎥ ⎢ ⎥ ⎢ yrn ⎥ ⎢ a2 b2 c2 yr ⎥ ⎢ ⎥=⎢ ⎥ ⎢z ⎥ ⎢a b c z ⎥ r ⎦ ⎣ rn ⎦ ⎣ 3 3 1

⎡

0

xr2n

⎤

⎡

0

a1

b1

0 c1

xr

0

0

0

1

xr

⎤

⎢ ⎥ ⎢ yr ⎥ ⎢ ⎥ ⎢z ⎥ ⎣ r⎦

(A.19)

1

1

⎤−1 ⎡

⎢ ⎥ ⎢ ⎥ ⎢ yr2n ⎥ ⎢ a2 b2 c2 yr ⎥ ⎢ ⎥=⎢ ⎥ ⎢z ⎥ ⎢a b c z ⎥ r ⎦ ⎣ r2n ⎦ ⎣ 3 3 1

(A.18)

xr2

⎤

⎢ ⎥ ⎢ yr2 ⎥ ⎢ ⎥ ⎢z ⎥ ⎣ r2 ⎦

(A.20)

1

Note that the center of the first circle is at the origin of the new coordinate system. The intersection points of the two areas are: xi =

2 C1 P1 2 − C2 P2 2 + xr2n

2xr2n

(A.21)

S. Aguado et al. / Precision Engineering 37 (2013) 929–943

yi = (C1 P1 2 − xi2 )

1/2

(A.22)

Then the previous intersection points must be changed to previously reference system. References [1] Watson JT. A new approach to large coordinate measurement using four tracking interferometers. In: Proceedings of the third international precision engineering seminar. 1985. p. 68–71. [2] Lau KC, Hocken RJ. Three and five axis laser tracking systems. Patent 4714339; 1987. [3] Vincze M, Prenninger J, Gander H. A laser tracking system to measureposition and orientation of robot end effectors under motion. The International Journal of Robotics Research 1994;13(4):305–14. [4] Takatsuji T, Goto M, Kurosawa T, Tanimura Y, Koseki Y. The first measurement of a three-dimensional coordinate by use of a laser trackering interferometer system based on trilateration. Measurement Science and Technology 1998;9:38–41. [5] Aguado S, Samper D, Santolaria J, Aguilar JJ. Towards an effective strategy in volumetric error compensation of machine tools. Measurement Science and Technology 2012;23:12. [6] Kim S-W, Rhee H-G, Chu J-Y. Volumetric phase-measuring interferometer for three-dimensional. Precision Engineering 2003;27:205–15. [7] Hughes E, Wilson A, Peggs G. Design of a high-accuracy CMM based on multi-lateration techniques. CIRP Annals – Manufacturing Technology 2000;49:391–4. [8] Montgomery Smith L, Dobson Chris C. Absolute displacement measurements using modulation of the spectrum of white light in a Michelson interferometer. University of Tennessee Space Institute, Center for Laser Applications; 1988. [9] Gallagher BB. Optical shop applications for laser tracking metrology systems. Master’s thesis. Department of Optical Sciences, University of Arizona; 2003. [10] Teoh P, Shirinzadeh B, Foong C, Alici G. The measurement uncertainties in the laser interferometry based sensing and tracking technique. Measurement 2002;32:130–50. [11] Zhuang H, LI B, Roth Z, Xie X. Self-calibration and mirror center offset elimination of a multi-beam laser tracking system. Robotics and Autonomous Systems 1992;9:255–69. [12] Zhuan H, Motaghedi SH, Roth ZS, Bai Y. Calibration of multi-beam laser tracking systems. Robotics and Computer Integrated Manufacturing 2003;19:301–14. [13] Faro Laser Tracker Faro Ion specificactions. www.faro.com. [03.07.13]. [14] Aguado S, Samper D, Santolaria J, Aguilar JJ. Identification strategy of error parameter in volumetric error compensation of machine tool based on laser tracker measurements. International Journal of Machine Tools and Manufacture 2012;53:160–9. [15] Tkatsuji T, Goto M, kirita A, Kurosawa T, Tanimura Y. The relationship between the measurement error and the arrangement of laser trackers in laser trilateration. Measurement Science and Technology 2000;11:477–83.

943

[16] Tsutsumi M, Identification Saito A. Compensation of particular deviations of 5-axis machining centers. International Journal of Machine Tools and Manufacture 2003;43:771–80. [17] Schellekens P, Spaan H, Soons J, Trapet E, Loock V, Dooms J, et al. Development of methods for numerical correction of machine tools. In: Proceedings of the 7th international precision engineering seminar. 1993. p. 213–23. [18] Donmez MA, Blomnquist DS, Hocken RJ, Liu CR. A general methodology for machine tool accuracy enhancement by error compensation. Precision Engineering 1986;4:187–96. [19] Lin D, Ehmman KF. Direct volumetric error evaluation for multiaxis machines. International Journal of Machine Tools and Manufacture 1993;33(5):675–93. [20] Zhan G, Veale R, Charlton T, Borchardt B, Hocken R. Error compensation of coordinate measuring machines. Annals of CIRP 1985;34(1):445–8. [21] Teeuwsen J, Soons JA, Schellekens P. A general method for error description of CMMs using polynomial fitting procedures. Annals of CIRP 1989;38(1):505–10. [22] Mou J, Richard Liu C. A method for enhancing the accuracy of CNC machining tools for on-machining inspection. Journal of Manufacturing System 1992:229–37. [23] Brecher C, Hoffmann F. Multi-criteria comparison of standardised kinematic structures for machine tools. In: Proceedings of the 5th chemnitz parallel kinematics seminar. 2006. [24] Bringmann B. 3D error compensation for parallel kinematics. In: The 5th chemnitz parallel kinematics seminar. 2006. p. 531–46. [25] Lin Y, Shen Y. Modelling of Five-Axis Machine tool metrology models using the matrix summation approach. Advanced Manufacturing Technology 2003;21: 243–8. [26] Alexander H. Slocum precision machine design. Society of Manufacturing Engineers 1992. ISBN:0-13-690918-3.A. [27] Duffie NA, Bollinger JG. Generation of parametric kinematic error correction functions from volumetric error measurements. Manufacturing Technology 1985;34(1):435–8. [28] Lin Y, Shen Y. Modeling of five-axis machine tool metrology models using the matrix summation approach. Advance Manufacturing Technology 2003:243–8. [29] Horn Berthold KP. Closed-form solution of absolute orientation using unit quaternions. Journal of Optical Society of America 1987;4:629–43. [30] Schwenke H, Schmitt R, Jutzkowski P, Warmann C. On-the fly calibration of linear and rotary axes of machine tools and CMMs using tracking interferometer. CIRP Annals-Manufacturing Technology 2009;58:477–80. [31] Schwenke H, Haitjema H, Knapp W, Weckenmann A, Schmitt R, Delbressine F. Geometric error measurement and compensation of machines – an update. CIRP Annals-Manufacturing Technology 2008;57:660–75. [32] Ibaraki S, Sawada M, Matsubara A, Matsushita T. Machining test to identify kinematic errors on five-axis machine tools. Precision Engineering 2010;34:387–98. [33] Schwenke H, Franke M, Hannaford J. Error mapping of CMMs and machine tools by a single tracking interferometer. CIRP Annals 2005;54:475–9. [34] Takatsuji T, Goto M, Kirita A, Kurosawa T, Tanimura Y. The relationship between the measurement error and the arrangement of laser tracker in laser trilateration. Measurement Science and Technology 2000;11:477–83.