Parameters to define the mapper geometry. Version 5
Here is a new set of parameters to define the geometry of the mapping machine used in
the analysis of real data. It is based on the survey so it should be quite close to reality. As
the moving NMR probe was not working, it has been removed from the analysis.
However it is still a somewhat idealised geometry. Corrections of up to a few mm will
have to be applied to the position of each Hall sensor to get its true position.
We number the Hall probes from 1 to 48. The surveyors used a labelling system based on
A or C for the end of the machine, E or I for external or internal, and numbers 1-12. So
the relation between the systems is given in the table below.
Survey label range
Our label range
start
AE1
AI1
CI1
CE1
start
end
AE12
AI12
CI12
CE12
end
1
13
25
37
12
24
36
48
Zenc0, the Z position in Atlas coordinates of the mapper carriage centre when its Z
encoder reads zero. Not surveyed, treated as 0 m.
Fenc0, the phi position in Atlas coordinates of the mapper axle when the phi encoder
reads zero. Not surveyed, treated as 0 degrees.
ZarmA, the distance in Z from the carriage centre to the mid-line of the arm at the A end.
Utilised 0.2225 m based on survey.
dZarmA, the distance in Z from the mid-line of the arm A to the Hall sensors. Utilised
0.0275 m based on survey.
FarmAE, FarmAI, FarmCI, FarmCE the difference in phi from the mapper axle to the
arms. Utilised 0,180,270,90 degrees. The survey measured the arms to be perpendicular
to within 0.04 degrees so we simulate them perfectly perpendicular.
ZarmC, dZarmC. Same as arm A. Based on survey, utilised 0.2225 m and 0.0275 m.
Rhall(12), the radial positions of the hall probes on the arm, utilised 118, 228, 338, 438,
538, 638, 718, 798, 878, 938, 998, 1058 mm.
dznmr. No longer used.
Rnmr, the radius of the fixed NMR probes in atlas coordinates. Utilised 1.133 m. The
probes are simulated at 45, 135, 225 and 315 degrees and at Z=0.
The Z encoder step size and the phi encoder were not used as the mapping machine
output actual positions + angles rather than encoder values.
Mapper machine coordinate system offset
The coordinate system of the mapper machine was found to be offset from the IWV
coordinate system. When the machine z-encoder was 0, the machine was found to be at
zIWV = -1.8 mm.
Arm tilts
The final machine survey found that the two arms were tilted at different angles relative
to the machine central axis. The survey data for arms AE and CI have been analysed to
provide tilt information, and arms AI and CE are assumed to have the same tilts as AE
and CI respectively. The plane in which an arm moves is defined by the normal unit
vector axˆ  byˆ  czˆ , where a = -0.000438, b = -0.001267 for arm A, and a = -0.000422, b
= +0.000482 for arm C, and c  1  a 2  b 2 for all cases.
The positions and field components are corrected in Cartesian coordinates, and the results
are then transformed back to cylindrical coordinates. These transformations assume that a
and b are small:
x   x1  a  , y   y 1  b  , z   z  ax  by
c
ab

Bx 
Bx 
B y  aB z
1 b2
1 b2

B y  1  b 2 B y  bB z

B z  cB z 
a
1 b2
Bx 
bc
1 b2
By
John Hart also derived the rotation of field components in cylindrical coordinates. Tests
showed that these were numerically equivalent to the transformation in Cartesian
coordinates. However, the Cartesian form was retained because of its convenience for
subsequent transformations.
Rotation centre offset
The centre of rotation of the arms is also offset from the IWV axis. This leads to a
correction of the positions but not the field components in Cartesian coordinates.
However, once transformed back to cylindrical coordinates, field components also
change because the r and phi directions have changed. The centre of rotation is shifted Δx
= -0.0001, Δy = +0.00105 for arm A, and Δx = +0.0001, Δy = +0.00125 for arm C (all
values in metres). These shifts were for when the mapper machine was at a z-encoder
value of -2.49843 (near end C).
Machine skew
Additionally, the machine chariot may be skewed because the wheels on the rails do not
travel exactly the same amount in z. This skew is accounted for by correcting the
positions and field components of the Hall probes. The chariot was assumed to be a rigid
body which rotated about its centre. The centre was defined to lie on the z-axis (i.e., x=0,
y=0) with zmap = average of the two rail positions z0 and z1.
z 0  zmap  , z  z  zmap
zmap  12  z 0  z1 , skew 
RIW V
sin   skew , cos   1  skew
x   x cos   z sin  , y   y , z   zmap  z cos   x sin 

B x  B x cos   B z sin 

By  By

B z  B z cos   B x sin 
Rail tilt
A survey of the rails found that they had a tilt of +0.0775 mm/m. The correction for the
rotation centre of the arms becomes
y  rail tilt  zmap  2.49843  y zmap 2.49843