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Precision Engineering 39 (2015) 173–178

Contents lists available at ScienceDirect

Precision Engineering

jo ur nal ho me p age: www.elsev ier .com/ locate /prec is ion

traightness evaluation using inclinometers with a pair of offset bars

. Kumea,∗, M. Satohb, T. Suwadab, K. Furukawab, E. Okuyamac

Mechanical Engineering Center, High Energy Accelerator Research Organization (KEK), Oho 1-1, Tsukuba, Ibaraki 305-0801, JapanAccelerator Laboratory, High Energy Accelerator Research Organization (KEK), Oho 1-1, Tsukuba, Ibaraki 305-0801, JapanFaculty of Engineering Science, Akita University, 1-1 Tegata Gakuen-machi, Akita 010-8502, Japan

r t i c l e i n f o

rticle history:eceived 30 April 2014eceived in revised form 28 July 2014ccepted 18 August 2014vailable online 16 September 2014

eywords:traightness evaluationnclinometer

a b s t r a c t

To evaluate the straightness of large objects, the use of an inclinometer is advantageous because it requiresneither straight shape references nor transferring mechanisms. Herein, we consider adopting it for precise(with greater accuracy than 1 mm) evaluation of the straightness of linear particle accelerators (linacs)that are several hundred meters long or longer. In this study, the straightness evaluation of a 206-m-longpart of the KEK injector linac was demonstrated using inclinometers with a pair of cantilevers called offsetbars. The offset bars were adopted to extend the evaluation length by avoiding obstacles that block theevaluation path. Errors caused by the offset bars can be eliminated by reversal measurement consideringthe slope angles of the offset bars. The derived straightness corresponded with those derived by an

eversal measurementarticle acceleratorslignment

alignment telescope and a laser-based alignment system within several millimeters and partly withinseveral hundred micrometers. The reproducibility of slope angles for an arbitrary measurement pointwas 15 �rad at standard deviation. This corresponds to a standard deviation of 0.47 mm for straightness,with a total evaluation length of 500 m and measurement intervals of 2 m. The results indicate that ournewly devised method is applicable for evaluating the straightness.

. Introduction

Straightness evaluation is usually performed by scanning aetector along an object. Because errors in the scanning locusirectly affect this evaluation, several kinds of error separatingechniques have been proposed and developed for eliminatinghese errors [1–3]. In principle, they require no straight shape ref-rences, which become difficult to accurately define and maintainor increased evaluation lengths. It follows that they are inherentlydvantageous for evaluating relatively large objects. Straightnessvaluation detecting tangential angles or differential straightness4–6] is a simple and classical technique. Above all, the methodsing an inclinometer is considered to be advantageous for largebjects because it requires no transferring mechanism that scanshe detector with sufficient accuracy that the error separatingechniques can be adopted in practice. The transferring mecha-ism often requires unacceptably excessive space and costs for

arge objects (over 10 m). The method using an autocollimator

oes not require such mechanisms; however, the evaluation length

s restricted by the effective range of the measurement beam,

∗ Corresponding author. Tel.: +81 29 879 6013; fax: +81 29 864 4051.E-mail address: tatsuya.kume@kek.jp (T. Kume).

ttp://dx.doi.org/10.1016/j.precisioneng.2014.08.006141-6359/© 2014 Elsevier Inc. All rights reserved.

© 2014 Elsevier Inc. All rights reserved.

whereas the method using an inclinometer does not have suchlimitations of evaluation length.

We have considered adopting the method using an inclinome-ter for accurately (with greater accuracy than 1 mm) evaluatinglarge (with lengths of several hundred meters or longer) linearparticle accelerators (linacs). We evaluated the straightness of a71-m-long part of the KEK injector linac by using a precise incli-nometer and demonstrated that the reproducibility of the derivedstraightness was less than 49 �m at standard deviation and corre-sponded with those by two other methods, within submillimeters[7]. This showed that the evaluation was fairly reliable; however,the evaluation length was limited by obstacles that blocked theevaluation path, and the length could not be extended.

Here, we added a pair of cantilevers called offset bars at eachmeasurement point for avoiding the obstacles and extended theevaluation length by shifting the evaluation path. Errors arisingfrom the offset bars were eliminated by newly devised reversalmeasurement, which considers the slope angle of each offset bar.

2. Principle

2.1. Straightness evaluation using an inclinometer

Fig. 1(a) and (b) shows the principles of straightness evaluationusing an inclinometer. The inclinometer sequentially detects the

174 T. Kume et al. / Precision Engineering 39 (2015) 173–178

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from measurements, �ma, �mb, �mc, �na, �nb, and �nc, made by thethree inclinometers before and after reversal.

Before reversal (Fig. 3(a)), each root, ar and br, of offset bars, aand b, is located on points, p and q; and two ends, c1 and c2, of the

ig. 1. (a) Straightness evaluation based on tangential angles, �(xi), of an object, f(x),sing an inclinometer. (b) Straightness derived by integrating the detected angles.

angential angle, �(xi), of the profile shape, f(x), for an object at eacheasurement point, xi (i = 0–n), where the inclinometer is moved

n the x-direction along which the measurement points are alignedFig. 1(a)). Straightness is evaluated by the shape, f(xn), which iserived by integrating the detected angles as

(xn) = h0 +n∑

i=1

si · �(xi), (1)

here h0 stands for the arbitrarily defined shape of the first mea-uring point and si stands for the distance between neighboringeasuring points (Fig. 1(b)).Here, the detected angle, �(xi), and consequently, the derived

hape, f(xn), are not affected by the transferring locus, e(x). The locuss predominantly used for reference in conventional straightnessvaluations that directly detect the shape; however, it is difficult toaintain a sufficiently straight or accurate locus as the evaluation

ength increases. It follows that straightness evaluation using annclinometer is advantageous for evaluating relatively large objectsecause it is not affected by the transferring locus.

.2. Obstacle avoidance using a pair of offset bars

We previously evaluated the alignment of base plates in the KEKnjector linac by using an inclinometer with a straight bar, as shownn Fig. 2(a) [7]. However, the evaluation length was limited by obsta-les that blocked the evaluation path, as shown in Fig. 2(b). Here, wextended the evaluation length by using a pair of cantilevers calledffset bars on each base plate to avoid obstacles by shifting the eval-ation path. Slope angles for lines connecting each end of the offsetars were determined in place of those connecting each of theiroots, which were the original targets of evaluation. Errors arisingrom the offset bars can be eliminated by reversal measurement,hich considers the slope angles of the offset bars.

.3. Reversal measurement considering slope angles of the offset

ars

Fig. 3(a) and (b) shows reversal measurement [8] consideringhe slope angles of the offset bars before and after reversal. The

Fig. 2. (a) Straightness evaluation for the alignment of base plates using a straightbar and an inclinometer. (b) Pair of offset bars for avoiding obstacles that block theevaluation path.

measurement setup consists of a pair of offset bars, a and b; astraight bar, c; and three inclinometers, A, B, and C, which forma U-shape. Here, we aim to determine an angle, �t, which is theslope angle of a line connecting two measurement points, p and q,

Fig. 3. Reversal measurement for angle measurement (a) before and (b) after rever-sal using a pair of offset bars, a and b; a straight bar, c; and three inclinometers, A,B, and C.

ngineering 39 (2015) 173–178 175

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traight bar, c, are located on each end, ae and be, of the offset bars.fter reversal (Fig. 3(b)), the measurement setup is turned, main-

aining relative positions between the components, in which pairsf the offset bar and the inclinometer, (a, A) and (b, B), exchangeheir positions so that their roots, ar and br, are located on points, qnd p, whereas the set of the straight bar and the inclinometer, (c,), is turned so that its ends, c1 and c2, are located on ends, ae ande.

Here, two angle components, (�, �), are defined to express theirection of each offset bar. They are orthogonal, where � is on aertical plane, on which the axis of the offset bar is aligned, whereas

is on a horizontal plane, on which each measurement point islaced. Before reversal, the angle components of offset bar, a, atoint, p, are expressed as (�ap, �ap), whereas those of b at q arexpressed as (�bq, �bq). After reversal, those of b at p and those of

at q are expressed as (�bp, �bp) and (�aq, �aq), respectively.Next, loads applied to the offset bars at their roots and ends are

efined as war and wae for a, and wbr and wbe for b, whereas loadspplied to c at its two ends are defined as wc1 and wc2. The loadpplied to each gravity center of the bar and inclinometer pairs, (a,), (b, B), and (c, C), is defined as wag, wbg, and wcg. The momentspplied to the roots, ar and br, of bars, a and b, are defined as Mar

nd Mbr, respectively. All are stable between reversals because theelative positions between the pairs of the bar and the inclinometer,a, A), (b, B), and (c, C), and the relative positions between the pairso not change.

Angles, �ma and �mb, detected by inclinometers, A and B, beforeeversal are expressed as

ma = �ap + �a + �A, (2)

mb = �bq + �b + �B, (3)

hereas those after reversal, �nb and �na, are expressed as

nb = �bp + �b + �B, (4)

na = �aq + �a + �A, (5)

here �a and �b stand for offset errors caused by straightness errorsnd deformations of the offset bars, and �A and �B stand for offsetngles caused by the inclinometers. Here, we assume that �a andb are stable between reversals because they depend on the initialhapes of the bars and the loads applied to the bars, which do nothange between reversals. We also assume that �A and �B are stableetween reversals because they only depend on the initial states ofhe inclinometers.

Angles, �mc and �nc, detected by inclinometer, C, before and aftereversal are expressed as

mc = �u + �c + �C, (6)

nc = −�v + �c + �C, (7)

here �c stands for an offset error caused by c and �C stands forn offset error caused by C. We assume that �c is stable betweeneversals because �c depends on the initial shape of the bar and theoads applied to the bar, which do not change between reversals.

e also assume that �C is stable between reversals because �C onlyepends on the initial state of the inclinometer.

�u and �v stand for the slope angles of lines connecting eachnd, ae and be, of the offset bars before and after reversal, respec-ively. If the angles, �t, �u, and �v, are small, they have the followingelations:

t · s + hmb = �u · dc + hma, (8)

t · s + hna = �v · dc + hnb, (9)

here s stands for the distance between measurement points, p and; dc stands for the length of c; �t stands for the slope angle that

Fig. 4. Schematic view of a typical accelerator unit.

we aim to determine; hma and hmb stand for heights of ends, ae

and be, against roots, ar and br, for offset bars, a and b, respectively,before reversal; and hna, and hnb stand for heights of ends, ae andbe, against roots, ar and br, for offset bars, a and b, respectively, afterreversal. If the angles, �ap, �bq, �bp, and �aq, are small, they have thefollowing relations:

hma = da · �ap + ha, (10)

hmb = db · �bq + hb, (11)

hnb = db · �bp + hb, (12)

hna = da · �aq + ha, (13)

where da and db stand for the lengths of offset bars, a and b, respec-tively, and ha and hb stand for displacements of ends, ae and be,respectively. ha and hb depend on the initial shapes of the bars andthe loads applied to the bars, and we assume that they are stablebetween reversals because the loads applied to them do not change.

Then, Eqs. (2)–(13) lead to the following relation:

�t = dc

2s(�mc − �nc) − 1

2s{da · (�na − �ma) + db · (�nb − �mb)}. (14)

This suggests that slope angle, �t, which we aim to determine,can be derived from measurements, �ma, �mb, �mc, �na, �nb, and�nc, by the three inclinometers before and after reversal, unaffectedby offset errors, �a, �b, and �c, caused by the offset bars and thestraight bar; displacements, ha and hb, caused by the offset bars;offset errors, �A, �B, and �C, of the inclinometers; and directions,(�ap, �ap), (�bq, �bq), (�bp, �bp), and (�aq, �aq), of the offset bars.

In the right-hand side of Eq. (14), the first term,(dc/2s)(�mc − �nc), corresponds to slope angles obtained byprevious reversal measurements [7,8] (slope angles withoutconsidering the effects of the offset bars), whereas the secondterm, (1/2s) {da · (�na − �ma) + db · (�nb − �mb)}, expresses the effectsof the offset bars.

3. Straightness evaluation

3.1. Setup for straightness evaluation

We applied our newly devised method for evaluating thestraightness of the KEK injector linac. The linac has two straightsections with lengths of 125 m and 476 m connected by an arc sec-tion, which form a J-shaped 600-m-long beam line [9]. The linac is

typically composed of 9.6-m-long accelerator units using an 8.44-m-long steel pipe girder with a pair of legs on both ends, as shownin Fig. 4. Components of the linac, such as accelerator structures,magnets, and beam monitors, are typically mounted on the girder

176 T. Kume et al. / Precision Engine

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“Inclinometers,” compared with those by a laser-based alignment

ig. 5. Setup for the straightness evaluation of the vertical alignment of base platesn the KEK injector linac.

ia 15-mm-thick base plates. The base plates are mounted on theirder at intervals of approximately 2 m, and each girder typicallyas five base plates. Surfaces of the base plates are machined toe flat (±0.05 mm) and used as vertical position references. Eachase plate has a pair of rails, whose side faces are also machined toe flat (±0.05 mm) and used as horizontal position references. Rel-tive positions between the units are aligned using a laser-basedlignment system, which was installed in a vacuum pipe set into theirders throughout each straight section [9,10]. This system detectseviations from its reference laser beam in the vacuum pipe withptical position detectors (quadrant photodiodes) installed in bothnds of the girders [10]. Relative positions between the compo-ents, base plates, pairs of rails, and detectors were aligned with aolerance of ±0.05 mm when they were assembled.

The straightness was evaluated for the vertical alignment of 113ase plates of an approximately 206-m-long downstream end partf the linac. Fig. 5 shows our setup.

An electric inclinometer, Talyvel 4 (range, ±3 mrad (±600 s);esolution, 0.5 �rad (0.1 s); Taylor-Hobson [11]), was adopted.ecause the inclinometer has only two angle detectors, a dummyeight, which has the same weight and the same footprint as theetectors, was alternately used with the detectors so that the loadspplied to the bars would not be changed.

Three kinds of straight bars with lengths, dc, of 640 mm,344 mm, and 2304 mm were adopted to adjust to the variations inhe intervals of 563–2266 mm between the plates. Each straight baras made of an extruded aluminum pipe with a thickness of 2 mm

r 3 mm and a rectangular cross section with height of 25 mm andidth of 50 mm.

Each offset bar was also made of a pipe with thickness of 2 mmnd the same cross section as the straight bars. These pipes were00 mm long and their effective lengths, da and db, were 375 mm.efore reversal (Fig. 3(a)), their directions, which are �ap and �bq,ere aligned in a horizontal plane and equated by placing their end

aces perpendicular to their axes on the side faces of the rails. Aftereversal (Fig. 3(b)), their �bp and �aq directions were also alignednd equated in the same way. In this way, the length, dc, of straightar, c, and interval, s, between each measurement point, p and q,ere equated, in case their vertical direction angles, �ap, �bq, �bp,

nd �aq, were small. This is also effective to ensure the direction ofeasurement.Flat square glass plates, optical parallels (contact area,

0 mm × 50 mm; thickness, 2 mm; flatness, better than 633 nm),ere used as contact feet between the base plates and the off-

et bars, and between the offset bars and the straight bar. These

lates acted as spatial low pass filters to prevent local minor shapes,uch as surface roughness and waviness of the bars, base plates,nd glass plates, affecting measurements. They were also expected,

ering 39 (2015) 173–178

together with the offset and straight bars, to reduce the influenceof positioning errors through their averaging effect.

3.2. Results

Fig. 6 shows the results of the measurements. White trianglesexpress slope angles without considering the effects of the off-set bars (in other words, values to be corrected), whereas blackcircles express the effects of the offset bars (in other words, thecorrection values). They correspond to the first and second termson the right-hand side of Eq. (14). Their root-mean-square valuesfor the whole measurements are 0.63 mrad and 0.26 mrad, respec-tively. This shows that the correction is fairly valuable because thecorrection value is comparable to the value to be corrected.

Reproducibility of the slope angles for an arbitrary measure-ment point, which were derived considering the effect of the offsetbars, displayed a standard deviation of 15 �rad for four repeat mea-surements. The standard deviation of straightness, �d, derived fromslope angles with a standard deviation of �a, is estimated as

�d =√

l · s · �a, (15)

where l and s stand for total evaluation length and average mea-surement interval, respectively, assuming that s has no error [7].The standard deviation for the derived straightness, �d, is estimatedto be 0.47 mm with l = 500 m, s = 2 m, and �a = 15 �rad. This indi-cates that our newly devised method is applicable for evaluatingthe alignment of large (several hundred meters long) linacs accu-rately (with accuracy greater than 1 mm), despite the complexitycaused by the added offset bars.

Fig. 7 shows the straightness derived by integrating the slopeangles shown in Fig. 6. White triangles express the straight-ness without the correction, whereas black circles express thestraightness with the correction. Here, the derived straightnessesare subtracted by their regression lines because straightnessevaluations usually focus on deviations from an ideal straightline and a straight shape component in the derived straight-ness is not important. Moreover, straightness for a distancelonger than 100 m is affected by globe roundness when consid-ering an accuracy of submillimeters or greater, because theproposed method uses the direction of gravity as a refer-ence. Globe roundness compensates for the straightness valuesexpressed in Fig. 7, assuming that the globe radius is 6371 km.The compensation value is also expressed by “Globe round-ness” in Fig. 7. The differences between the straightness in thecases considering and not considering the effect of the offsetbars are 0.78 mm (maximum) and 0.27 mm (root mean square),respectively. These differences are not negligible for evaluatingstraightness with an accuracy of submillimeters.

Finally, with respect to the derived straightness, some featurescan be observed. Sags with a period of approximately 10 m overthe whole straightness are considered to be caused by the bendingof the girders supported by their two legs on both ends. The peakshown at approximately 138 m is considered to be caused by defor-mation of the tunnel in which the linac was installed, because theexpansion joint of the tunnel is located near this peak.

4. Discussions

We evaluate the derived straightness by comparing it to thosederived for an identical object using two other independent meth-ods. Fig. 8 shows the straightness derived by the proposed method,

system, “Laser” [10], and an alignment telescope, “Telescope”(Taylor-Hobson [11]). The straightnesses are subtracted by theirregression lines for the overlapping range (from 0 m to 206 m).

T. Kume et al. / Precision Engineering 39 (2015) 173–178 177

Fig. 6. Angles without considering the effect of the offset bars and correction values for the effect of the offset bars derived from the measurements.

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ig. 7. Straightness derived by integrating the slope angles for cases considering aoundness”.

These values are for different alignment targets; namely, centersf the optical position detectors installed in the vacuum pipe at bothnds of each girder for the alignment telescope and the laser-basedlignment system, and centers of the upper surfaces of the baselates for the proposed method. However, the comparison is con-idered to be valid because they were aligned within ±0.05 mm

n their primary assembly stage and their relative positions arexpected be aligned with an accuracy better than submillimeters.

Actually, the three straightnesses correspond within severalm and display similar trends, even though they were performed

ig. 8. Comparison between the straightness determined by inclinometers, “Inclinometersates express when each measurement was performed.

t considering the effect of the offset bars, with the estimated globe curve, “Globe

with intervals of several months, during which the linac may havedeformed as a result of aging. Here, the straightnesses derived bythe proposed method and the laser-based alignment system par-tially agree within several hundred micrometers. The straightnessderived by the alignment telescope differs from the other two, par-ticularly for the long (longer than several 10 m) wavelength regions,

but displays similar trends to the other methods in the straightnessfor the short (shorter than several 10 m) wavelength region.

The straightness derived by the alignment telescope wasobtained by stitching together each 30-m-long measurement with

”; a laser-based alignment system, “Laser”; and an alignment telescope, “Telescope.”

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78 T. Kume et al. / Precision E

n overlap of approximately 10 m. This may cause straightnessrrors for longer (>30 m) wavelength regions, but the straightnesss considered to be fairly reliable for each 30-m-long measurement.t follows that the straightness values derived by the three meth-ds agree fairly well in the shorter (<30 m) wavelength region,hereas the straightnesses derived by the proposed method and

he laser-based alignment system agree well for the whole region.his indicates that the straightness values derived by the proposedethod and the laser-based alignment system are more reliable for

he whole region.

. Conclusion

We used a pair of cantilevers called offset bars to avoid obstacleshat block evaluation paths and extended the evaluation length for

straightness evaluation using an inclinometer.Error arising from the offset bars can be eliminated,

ogether with the offset errors of the inclinometers, by rever-al measurements considering the slope angles of the offsetars.

The reproducibility, expressed by the standard deviation oflope angles for an arbitrarily sampled measurement point, was5 �rad. This corresponds to a standard deviation of 0.47 mm in thetraightness, with a total evaluation length of 500 m and averageeasurement intervals of 2 m.A straightness evaluation for a 206-m-long part of the KEK injec-

or linac was demonstrated. The derived straightness correspondedell within several millimeters and partly within several hundredicrometers with those derived by an alignment telescope and a

aser-based alignment system.

The results indicate that our newly devised method is applica-

le for evaluating the straightness of large (several hundred meterong) linacs accurately (with greater accuracy than 1 mm), despitehe complexity caused by the offset bars.

[

[

ering 39 (2015) 173–178

Acknowledgements

The authors thank Prof. Y. Kobayashi and the members of Accel-erator Division 7 of the Accelerator Laboratory at KEK and Prof. M.Katoh of the Institute of Molecular Science for providing us withTalyvel 4; Y. Iino of Toyama Co., Ltd. and Y. Mizukawa of MitsubishiElectric System & Service Co., Ltd. for assisting with our experimen-tal studies; and the engineers and researchers at the MechanicalEngineering Center at KEK for encouraging and supporting ourstudies. This work was supported by the Collaborative Develop-ment Research Program of KEK (2010-ARL-03 and 2011-ARL-03)and JSPS KAKENHI Grant Number 22560130.

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