ALIGNMENTS OF THE TRISTAN MR MAGNETS
Atsushi KABE and Kuninori ENDO
National Laboratory for High Energy Physics
Oho-machi, Tsukuba-gun, Ibaraki-ken, 305, Japanor in the matrix form.

ABSTRACTAt the beginning of the alignment of the TRISTAN
MR magnets, the provisional positions which are thebases to align the magnets, are marked on the floor ofthe tunnel. The relative distances between these provisional monuments are surveyed. Using the computer,the relative displacement is transformed to the radialand angular displacements of each monuments.
Using
these radial and angular correction values, the alignment is achieved. The procedures of the calculations
to obtain the correction values and an example of thesimulation are presented.

--ai-1ai ai+1 ___ bi-1 bi bi+1 __ _I

I
I
I

I
I
I

I

I

I
I

I

I

I

I

I
I

I

I

(5)

II
I
I

I
I

------c. ci+1 _______ di d 1+1 --~INTRODUCTION
where the coefficients a, b, c and d are calculated
from Eqs. (1) and' (2). Solving Eq. (5), the correction
values ri and e. are obtained for all monuments.

For the TRISTAN MR magnets, the reference points(monuments) for the magnet alignment, which are determined from the geodetic survey, are marked on the floorof the tunnel. In order to correct these monuments tothe desirable positions horizontally, about 400 monuments are ·selected out of 670 monuments in the TRISTAN
MR, and at representative monuments two distances aremeasured.
One is the distance between the adjacentmonuments, and the other is the perpendicular to thestraight line between the next nearest neighbor monuments. Making use of the difference between the theoretical value and measurement, the radial and angulardisplacements (correction values) in the horizontal
plane are calculated for each monument.
Thus, eachmonument can be corrected, and the magnets are aligned
based on the corrected monuments.
As a target isattached precisely· on each magnet, the final magnet
positions are determined by the alignment using thetarget, applying the same method as the monument.
The calculation of the displacements treats 800 x
800 matrix and requires the large computer memories,
because the t,otal number of monuments or targets isabout 400. Therefore, some expedients are required tosave the memory of the computer in the actual calculation.

~M;II

M; monument fori-th magnet
perpendicularP;
S; short chord

II

I
I:R;_,II

I

I
I ""1-11

I

Ia,I

9;

I
1I

R;

OMi

®;

~MiOMi+l: / .,,"' ®;.,~~':.---0------R;.2

Fig. 1

Perp.endicular and shortchord of each monument.

EXPRESSION OF THE CORRECTION VALUESPROCEDURE OF THE ACTUAL COMPUTATIONAs seen from Fig. 1, at the monument M.,the shortchord Si (distance between the nearest neithbor monuments) and perpendicular P. to the straight line between the next nearest n€ighbor monuments ri-1 andMi+ 1 ' are expressed with Ri and ei as followsl :·Eq. (5) is simplified as follows:... ...yAx(6)

where A is the 2N x 2N coefficients matrix, and+rlr2

X

(1)

II
I

(2)

i=1,2,----,N (N is the total number of monuments).

To obtainforms to...

H xyPlP2
III.

rNele2

PNsls2

IIIIeNsN

IIn Eqs. (1) and (2), if perpendicular Pi and shortchord S. have errors p. and s. respectively, theseerrors lre transformed t~ the riaial and angular displacements r. and e. of the i-th monument. Therefore,
the error p ~ and s~ are described by the linearized
observational equation as follows:+Ithe least squares solution ~. Eq. (6) trans+z(7)ywhere H AT . • A (2N x 2N matrix) and ; = AT
(2Nvector).
In the TRISTAN MR, the total number of the representative monuments is 396 points, so in Eq. (6) or(7) A or H is 792 x 792 square matrix, and has 627264

(3)

(4)
215

elements. Therefore, in the actual computation it requires some techniques to save the memory of the computer. Namely, as the matrix A has only 5 nonzero elements per column, it is reduced to 5 x 792 matrix. The
other side, the matrix H is symmetric, because of H =AT x A, therefore the upper triangular elements of H
are sufficient in the form of one dimensional array,and total elements of matrix H is 314028 in calculation. By the above mentioned technique,the number ofelements stored in the computer is less than 26 % of
the total elements of A and H.
In Eq. (7), since the matrix H has 3 degrees offreedom, the order of H must be reduced by 3. As thematrix H is symmetric, the modified Cholesky' s method
can be used. Consequently Eq. (7) transforms to theequation containing the coefficients of the upper
triangular matrix as follows:(~(radial)
The relation of the radial direction for each monument
does not change by rotation, therefore(13)r'.'L = r'i(angular)
The angular displacement is reduced by 6~, thus

(14)

1,2,-- N).

Now the correction values are ri and ei (i(8))(:) = ;.

Thus from the backward substitution of the Gauss's
method, the solution ri and e. are obtained as theradial and angular displacementsLo f the i-th monument.
To remove the unnecessary components in the correction values (r. and e }, by the parallel and rotational transformaf"ions of the coordinates and the reduction of the harmonic components, the final correction values are obtained from the following processes.
Parallel transformation
The amounts of the parallel transformation (6X forX-direction and 6Y for Y-direction) are given by theaverage of the displacement for all monuments. Thatis,

theoretical monument before transformstion
theoretical monument after transformation
measured monument
coordinates before transformation
X' ,Y' coordinates after transformationi-1
N
6X = - E {(Ri+ri}sin( E 0j+6i}-Xi}
N i=1
j=1
i-1
N
6Y = ..!:. E { (R.+r. )cos ( E 0j+6i}-Yi}
j= 1
L L
N i=1(9)Fig. 2where (X., Y.) is the theoretical coordinates of i-thmonument; anl the angle is measured from Y-axis. Afterthe parallel transformation of the coordinates, thecorrection values of each monument are obtained using
the relation shown in Fig. 2 as follows:

Reduction of harmonic components
There are introduced the irrational harmonic components in the correction values due to the measureTo reduce these components the Fourier
ment errors.
analysis is applied to both correction values r. and ei
independently. The harmonic components are obtlined asfollows:(radial}i-1r!L({CR.+r.)sin( E 0.+6.}-6X}Z
j= 1 J LL L

the k-th component of the radial correction values isi-1+ {(R.+ri)cos( E 0.+6.)-6Y}z] 1 12_R.
L
j= 1 J L
L

i-1

i-1

(10)

akcos(2nkE S./c) + bksin(2nk E S./c)

j=1

j=1

J(15)

J

where(angular)e•iCorrection values after
parallel transformation.

i-1
E 0!
j=1 J-j=1•

M =-NEN i=1,.

..

ri (Si_ 1+S.
2L)j(R!)l+(R! )Z-(S!)ZL }L+1L.
2R' R'i(11)i+1andRotational transformation
The rotation angle 6~ is given by the averageof the angular correction values of all monuments asfollows:
1Nc i=10E0 ,., = cos -1{

whereao = -1i-1ei

After the rotation of the coordinates,
values are described as follows.

with the circumference c of the TRISTAN main ring, andk = 1,2,3,----.
The low order harmonics usually have the large
amplitude and a little effect to the particle motion in
the accelerator even if the low order harmonic deviations are really exist, so they are eliminated mathe-(12)the correction

216

The results of the simulation under the following
conditions are shown in Fig. 3 and 4.

matically without gLvLng serious effects to the machine
performance. An example of this kind is given below.

1) standard deviation of dummy data

EXAMPLE OF SIMULATIONa

In order to estimate the accuracy of the horizontal alignment, an example of simulation is presented.
Random data are given to the correction values r.
and e .. Using these dummy data r. and e., p. and s. ofeach fuonument are calculated, whtch arl corfsidereJ- tobe deviations from the theoretical value. Further toreproduce the actual alignment, the measurement errorsof Pi and Si are reflected in the dummy data pi and si
as follows:rfor

3 mmr.
Lae

6 X 10- 3 m rada

3.97 mmforpi

4.16 mm

fors.
L

pas

for e.
L2) standard deviation of measurement error

p~LFig. 3 shows measurement errors ~p. and ~s., andFig. 4 shows the radial and azimuthal ~ror dis~lace­ments of the horizontal alignment. The result of thesimulation shows that the error of the horizontalalignment is 1.1 mm rms.

where ~p. and ~s. are the measurement errors and p~ andUsings~ are Ehe fin~l dummy data for computation.
p~ and s~, the computation to obtain the correction
v~lues r.L and e., is executed, and the accuracy isgiven by Lthe diff~rence between the computation results and dummy values of ri and ei.

"'

0.3

LIPj

(rms 0.1 mm)"'

REFERENCE1)"'

K. Endo and M. Kihara, KEK 74-3, 1974.

"' "'

"'

0"'

"'
.....

0.....

"'"'(')

LISi(rms(')

.....

w
0(')

(')

(')
(')

(')

(')

(')

(')

-0.
199Monument number

100

Fig. 3396

298

Measurement errors.

liE--- radial

32C)---azimuthal

EE~-3100

199
Monument
Fig. 4. 298
numberError displacements.

217396