# **SIMULATION OF THE FULL EMU ALIGNMENT SCHEME**

The successful reconstruction of sensor locations in the prototype ISR
system strongly suggests that the design of the EMU Alignment System is
promising. However, the ISR prototype system does not provide sufficient
enough grounds to make larger generalizations about the entire EMU
Alignment scheme nor characterize the interactions between many of the
components. A general simulation of the EMU alignment scheme is needed
to evaluate the operational viability of the system as a whole, estimate
the precision with which the system can reconstruct CSC chamber
positions, and relate the contribution of uncertainties in the
construction of individual components to the performance of the system
as a whole. Simulations of an idealized EMU system were performed with
COCOA v2.0.0. The simulations included all major system components and
appropriate CSC chamber geometries.

## Construction and Extraction of Simulation Parameters and Objects

To optimize the development of Endcap simulations in COCOA, it was
decided to compose the simulated Endcap solely from objects arranged in
a 'parent-child' relationship. This means that relatively simple
collections of objects already defined in COCOA can be placed together
to compose more complex objects. In this scheme, two DCOPS sensors can
be arranged in a local coordinate system to define a CSC chamber (Figure
A.5). Four CSC chambers, in turn, are arranged to form a SLM line, three
SLM lines are arranged to form an Muon Endcap (ME) layer, and six ME
layers are assembled to form the entire CMS Endcap Muon system. The
modeling of components in this manner means that the location and
orientation of all the individual components (dowel pins, chamber active
reference centers, lasers, etc) does not need to be known in the general
CMS coordinate system beforehand. Rather, the spatial relationships
between parent and child objects can be specified directly from CMS
production drawings. This method works very well for six of the Muon
Endcaps (ME±2, ±3, and ±4 Endcap), however the unique construction of
the inner most Endcaps (ME ±1) requires a somewhat less symmetric
approach, as SLM lines do not traverse the ME discs and a Secondary
Laser Line must be introduced.

The extraction of the information necessary to define these
relationships from CMS production drawings involves several steps and is
not a simple exercise of comparing two points in a single drawing. For
this reason, the exact manner in which parent-child objects were
composed and assigned in the simulation is specified in Appendix A.
Components included in the simulation include inclinometers, DCOPS
sensors, LINK 2D transparent sensors, crosshair and beam lasers, and
distance measuring devices. The objects created from these components
include all the CSC chambers, transfer plates, and MABs used in the EMU
alignment scheme.

The most important object in the simulation is the basic CSC chamber
object. Though the details of CSC construction in the simulation are
found in Appendix A, the approach taken to describe the basic ME chamber
object is shown in Figure 3.1. Of principle importance is the manner in
which the chamber 'center' is defined. Since tolerances between the
separation of strip layers is not controlled in CSC chamber production,
it has been decided to define a 'Reference Center' (or 'Active
Centerpoint') for the chambers as the average strip position projected
onto the upper strip plane. For a perfectly constructed CSC chamber (no
uncertainties in construction), the centerline of the CSC chambers will
fall along the Active Reference Center of the simulated chambers. Note:
the 'Reference Center' and 'Active Centerpoint' will be simply referred
to as the 'chamber center' for the remainder of this discussion.

> ![](media/image1.jpeg){width="5.995138888888889in"
> height="4.559027777777778in"}

Figure 3.1: The Simulated CSC Chamber Geometry and Local Coordinate
System in the Idealized COCOA Model. The 'Average Active Center' of the
CSC chambers is taken as the Reference Center for COCOA EMU simulations.

## Final Simulation Script Geometry Compared to Theoretical Placement

A complete model of the CMS EMU Alignment System constructed with the
components detailed in Appendix A show layout and placement errors in
the COCOA EMU simulation components of \<5 μm along the Transfer Line
(average deviation of .05 μm) and 10 μm along the SLM Line (average
deviation of .81 μm). The discrepancies on SLM on the order of 10 μm
occur only on the placement the inner ME ±2/1 DCOPS sensors relative to
the ME ±2/1 CSC chamber centers. It has been decided that this error
lies in the corresponding ME ±2/1 production drawings. The decision has
been made to match components in the COCOA simulation to the final
production drawings rather than a perfect CMS geometry. It is expected
that the tolerance on the construction of the components comprising the
system will greatly exceed any of these small discrepancies in the
production drawings.

A snapshot of a VRML model of the final simulation as generated by
COCOA, showing only active components (lasers and all measurement
devices), is shown in Figure 3.2.

![](media/image2.jpeg){width="5.992361111111111in"
height="4.956944444444445in"}

Figure 3.2: The Simulated EMU Alignment System. This is a COCOA
generated VRML representation of the simulated geometry used for the
idealized simulations of the EMU Alignment System.

## Estimation of Uncertainties Used in EMU Simulations

The primary goal of the COCOA simulation of the EMU alignment scheme is
to obtain an estimate of the uncertainties associated with the
reconstruction of CSC chamber positions along the SLM laser lines and an
understanding of the relationships and correlations between the various
components of the alignment system. For simplicity, the most studies and
simulations were done with the idealized EMU. The idealized EMU system
contains all the EMU system components arranged in the manner detailed
in Appendix A with uncertainties on the location and placement of the
components set to the estimations detailed below. Since the system is
ideal, it is assumed that all measurement devices are able to make a
successful measurement and the full redundancy of the alignment scheme
(multiple measurements of opposing lasers) is exploited. The effects of
component failure (missing CCDs measurements from DCOPS, faulty laser
modules, inadequate resolutions) should be examined separately.

### Digital CCD Optical Position Sensor (DCOPS) Sensors

DCOPS sensors are viewed as a collection of four independent vectors,
corresponding to the four CCD pixel arrays, referenced to a common
calibration slot on the DCOPS hardware assembly (where the CSC
calibration pin is to be inserted). Systematic uncertainties in the
location of the origin of these vectors, i.e. the first active pixel
position, are generally independent of the uncertainties associated with
the pixel array orientation. Uncertainties in the location of any given
pixel in a CCD to the calibration slot can be separated into
uncertainties directly associated with the calibration of the first
pixel position and those associated with the pixel array orientation and
length.

#### Uncertainties in Direct Calibration of First Pixel Position

Calibration of first active pixel positions to the reference pins for
large numbers of sensors is done on a specially designed test bench at
Northeastern University (NEU). The calibration bench is constructed of a
laser diode assembly, a single directional DCOPS sensor that has been
previously calibrated under a microscope, and a mount for the DCOPS
sensor that is to be calibrated. In this arrangement, the calibrated
DCOPS sensor acts as a mask to calibrate the uncalibrated DCOPS sensor.
NEU indicates typical uncertainties in the first pixel location using
this calibration technique will not exceed 40 μm \[3.1\].

#### Uncertainties in Pixel Array Orientation and Length

For simulation and reconstruction purposes, it is assumed that the four
CCD vectors lie in the local DCOPS XY reference plane and run parallel
to the X or Y axis. The error introduced by this assumption manifests
itself as a correction to the Sony specified pixel-to-displacement
conversion of 14μm per pixel along the array. The uncertainty in the
determination of the position of the charge distribution will scale
across the pixel array as the cosine of the angle by which the CCD array
deviates from its optimal orientation.

The uncertainty introduced by the error in pixel array length and the
misorientation of the pixel arrays can be estimated as the quadrature of
the maximal error due to the misalignment of the pixel arrays within the
CCD packages, the uncertainty in the length of the pixel array, and the
maximal error due to misalignment of the CCD package within the DCOPS
window frame.

#### Uncertainty of CCD Array and Packaging Positioning

Direct measurement of the pixel array length was performed on a
representative sample of ILX-551 CCDs. In addition to the 2048 active
pixels in each pixel array, one end of the array contains an additional
33 dummy pixels while the other end contains an additional 6 dummy
pixels. The total length of the pixel strip is calculated to be 29.218
mm based on the Sony pixel specification of 14 x 14 µm^2^ with no
manufacturer quote on the uncertainty in pixel size.

Sony has provided estimates of uncertainties for the placement of the
first pixel in the array in the CCD package, but does not offer any
estimate as to the uncertainty in the pixel array length or orientation.
These uncertainties where estimated as the maximal deviation from Sony
specifications found in the small set of CCDs studied. Systematic errors
embedded in the ILX551A CCD packages have been studied on a small sample
of unmounted CCDs. Direct measurements of the pixel array lengths on
these samples revealed a maximum discrepancy of (50 ± 10) µm with the
Sony specification \[3.1\]. All measurements of the array lengths
yielded a result which was always greater than the Sony specified value.

Uncertainty occurring in the final active pixel position as the result
of a misalignment of the pixel array within the CCD package has been
determined to be less than 4 μm based on a measured 15 mrad deviation of
the array with the package edge (maximum misorientation found in the
small sample of measured CCDs) \[3.1\].

![](media/image3.jpeg){width="3.202777777777778in"
height="3.0993055555555555in"}

Figure 3.3: SONY ILX-551 CCD Specification and Direct Measurement (mm).
The dimensional specifications and tolerances for the ILX-551 (A) quoted
from Sony and the dimensions of a CCD taken from the small sample of
studied ILX-551s (B) which exhibit the greatest deviation of the pixel
array (in red) placement from the optimal location.

#### Uncertainty of CCD Package Orientation inside DCOPS Window Frame

Uncertainty occurring in the final active pixel position as the result
of the misalignment of the CCD package, and thus the encased pixel
array, scales as the cosine of misalignment. The present DCOPS window
frame design incorporates a specially designed polycarbonate mount for
the CCDs. Tolerances for the positioning of each end of the 53.71 mm
long CCD mounts in the window frame is ±100 µm, contributing an error of
less than 1 µm in the determination of the distance along the pixel
array. Tolerances for the placement (orientation) of the CCDs within the
polycarbonate mounts are estimated at \>20 mrad, contributing a maximal
uncertainty of less than 10 µm in the determination of the last active
pixel in the array.

#### Final Estimation of Uncertainty in CCD Pixel - DCOPS Reference Pin Calibration

The final estimation of the uncertainty associated with the
determination of the location of any given pixel in the DCOPS sensor
relative to the CSC reference slot is taken to be the quadrature of all
known errors in the determination of the first and last active pixel
positions in the pixel array. In addition to the systematic errors
associated with the placement of CCD arrays relative to the primary
reference pin, the manner and stability in which the centroid of charge
distributions are determined must be considered. It has been determined
from the 2000 ISR Tests that centroids from successive measurements on
the CCD are repeatable to ±14 µm. This uncertainty is characterized as
the uncertainty associated with a particular measurement and considered
separately in the COCOA simulation. The contributions and final estimate
of the systematic uncertainty associated with the placement of pixels is
given in Table 3.1.

Table 3.1: Contributions to Final Error in Determination of Pixel-to-CSC
Reference Pin Calibration. Uncertainties due to misalignment and pixel
array length are estimated as the worst case errors in the determination
of the final active pixel position.

  --------------------------------------------------- -------------------
  **Uncertainty Origin**                              **Magnitude (µm)**

                                                      

  Direct Calibration of First Pixel (NEU Estimate)    40

  Array Misalignment (inside package)                 4

  Array Length                                        50

  Package Misalignment                                10

                                                      
  --------------------------------------------------- -------------------

### CSC Active Center - DCOPS Reference Pin Calibration

As previously discussed, Cathode Strip Chambers are described in terms
of their 'active center', which corresponds to the average of the
individual panel active areas projected onto the plane of the first
panel (See Appendix A for more detail on chamber definition).
Determining the certainty with which this active center can be
externally referenced is crucial to the successful simulation and
reconstruction of the Endcap Muon System. Uncertainties in the
determination of the relationship between CSC active cathode strips and
DCOPS reference pins can be separated into uncertainties associated with
individual panel definitions and manufacture, assembly of multiple
panels to form the complete CSC chamber, the mounting assemblies which
affix the DCOPS sensors to the surface of the assembled chamber, and
deformations in the chamber after installation in CMS.

All estimations of uncertainties addressed here regarding CSC chamber
and DCOPS mounting hardware tolerances are typically gathered from the
specified tolerances placed on the fabrication of components. In most
cases, particularly CSC panel definitions, adherence to these tolerances
has been confirmed by direct measurement on an appropriate number of
preproduction samples. However, there are many components which have not
yet been produced on a large scale. Uncertainties for these components
have been estimated from production drawings. A far more meaningful
estimation of uncertainties for such components should be taken from the
rms value of deviations found from a sufficient sample of the finished
products.

In keeping with the convention established in Appendix A, the local
chamber coordinate system is taken as right handed with the local Z axis
running across the chamber centerline from the narrow end of the chamber
to the wider end and the local Y axis running from the bottom layer to
the top layer of cathode strips.

#### CSC Panel Definition

All cathode strip chambers are constructed of a polycarbonate honeycomb
panels with 1.5 mm G-10 epoxy fiberglass skins coated with a 34 μm layer
of copper. Individual panels are first drilled with two CSC Alignment
Holes (where the CSC Alignment Pin will ultimately be inserted) along
the centerline (See Figure 3.4). These two holes (±25 μm tolerance on
the diameters) establish the reference system from which all other
machining on the panel is established. A high precision router is then
used to mill the cathode strips and associated artwork directly into the
copper surface of the panel. Accuracy of the router has been confirmed
by direct measurement. Errors in absolute strip position exhibit
accumulative systematics over chamber width due to the manner in which
they are milled. Quality control measurements on more than 300 panels
have shown a σ(average active area) of 25 μm and rotational uncertainty
of 7 μrad. However, measurements of the total active area across the
widest end of these panels show a σ(width) of 40 μm whereas the same
measurement across the narrow end of the panel reveals a σ(width) of 113
μm. Since the straightness strips have been measured as having a σ(rms)
\< 22 μm, the effect is likely due to the fact that the strips are
milled into the panel in only one direction (from the wide end to narrow
end).

![](media/image4.jpeg){width="5.998611111111111in"
height="2.859722222222222in"}

Figure 3.4**:** Placement of Pins, Holes, and Etchings on CSC Chamber
\[3.2\].

Additionally, a series of ten slot cuts are placed on each panel for
future calibration and quality control as well. The slot cuts are
referenced directly to the center line defined by the two CSC Alignment
Holes as well as a particular cathode strip number \-- but not
necessarily the first strip nor the same strip in all chambers. Each
mark is referenced within ±25 μm to the center point of the CSC
Alignment Holes and within ±25 μm to the particular strip being
referenced (the "Reference Strip").

A series of secondary dowel pins and slots used to mount the DCOPS
sensors to the chamber surface are also drilled at this time. On the
wider end of the chamber, these holes position the precision pins and
bolts used by the DCOPS mounting brackets to reference the CCDs directly
to the cathode strips. On the narrow end, a series of holes for
alignment pins are established which locate the plates for DCOPS
mounting hardware to be glued to the panel surface. The locations of all
holes are placed with a certainty of ±25 μm on the surface of each panel
and have their diameters drilled to a certainty of ±25 μm.

#### Assembly of Final Chamber from Multiple Panels

Once all six panels are milled and prepared, they are layered to form
the composite six layer CSC chamber. The precision with which each panel
in the chamber is positioned with respect to neighboring panels can be
estimated from the uncertainty with which the central CSC Alignment
Holes are placed (±25 μm) and the tolerance placed on the diameter of
the hole itself. Since the placement of all six layers are independent
of each other, the maximal misalignment of any two given planes (two
sets of holes, each with 25μm uncertainty in the diameter and relative
location of their center points) is estimated to be ±50 μm. The
uncertainty in the average center point across all six chamber planes is
±87 μm.

After the panels are stacked together, a frame is mounted to the edges
of the chamber to add rigidity to the structure, protect connections,
and form a Faraday shield around the electronics. Though the DCOPS
mounting tower assemblies will mount on top of the frame structure,
placement of the tower mounting hardware is completely determined by an
alignment pin that sits in the previously drilled alignment holes. Since
the CMS triggering requirements will accommodate relatively large
uncertainties in the final determination of the chambers along the CMS Z
coordinate, the placement of elements along the local Y axis of the
chamber (CMS Z), including the frame, is not tightly controlled during
manufacturing. The certainty with which the placement of the frame above
the first strip plane will be known is estimated to be ±127 μm. The
frame itself is estimated as being flat to ±25 μm whereas chamber
flatness is estimated as no worse than ±200 μm across any 60 cm span.
Tolerances on the panel thickness are asymmetric, +508 μm to -254 μm,
and determined by the manner in which the panels are manufactured and
cut.

#### Alignment Mounting Hardware

Mounting hardware to attach the DCOPS sensors to the chamber frames is
similar across all chamber varieties. The mounting hardware consists of
a series of spacers, shims, and mounting plates which secure the primary
DCOPS mounting bar (where CCD-dowel calibration is done). CSC chambers
in both inner and outer rings of ME ±2, ±3, and ±4 disks are placed into
two layers. The chamber configuration for Trigger towers makes it
necessary to mount hardware on the back layer for one outer CSC.
Chambers on back layers will have mounting hardware called towers to
give the attached DCOPS sensor an elevation in the SLM line which
matches the front layer.

As previously noted, sensors affixed to the larger end of the chambers
(+Z local end) are mounted directly to the chamber frame while sensors
on the smaller end of the chamber (-Z local end) are attached to a
special plate which is glued into position with the aid of a special pin
and slot set during the fabrication of the chamber. The precision with
which the inner plate can be glued to the surface of the chamber is
determined from the tolerance with which the pin and slot are placed on
the chamber surface (± 25 μm each) as well as the precision with which
the receiving pin and slot are placed on the plate to be glued (± 25 μm
each). Therefore, it is estimated that the DCOPS plates on the narrow
end of the chamber can be glued in position within ±50 μm of the
specified location - approximately the same tolerance with which they
can be attached on the +Z local end to the chamber frame.

Although the heights of the towers on the back layers differ by 290 mm
from the standard brackets, the plates and assemblies used are
essentially the same. The mounting brackets are attached across two
standoffs, each with an asymmetric tolerance of +.100 -0.000 mm. The
standoffs locate a set of dowel pins on a stock Aluminum jig plate (±
125 μm thickness). The DCOPS mounting bar connects directly to this
plate. Holes and pins used to secure the DCOPS are positioned and
drilled to ±25μm, contributing an additional ±50 μm uncertainty in the
referencing of the sensors to the strip positions.

The potential uncertainty in the orientation of the DCOPS sensors about
the primary DCOPS dowel pin due to the uncertainty of the standoffs
(separation of 88mm) is less than ±1μrad. Likewise, the uncertainties of
the chamber frame positioning above the first panel on the chamber
contribute an angular uncertainty about the same axis of less than
±1μrad as well. Rotational uncertainties about other axes are found to
be only slightly larger, but can be safely neglected by COCOA as each
CCD is one dimensional.

#### Deformation of Chambers and Chamber Components

One final consideration in defining the precision with which individual
layers can be externally referenced is the deformation of the chambers
when mounted in the detector. Since the CSC chambers are mounted
vertically, the weight of the chambers creates a shearing effect across
the six layers of the chamber. This effect is will be most prevalent
when the shearing occurs perpendicular to the direction of the cathode
strips. Thus, the effect is largest for chambers lying along the CMS X
axis. The displacement of the individual layers due to the shearing
force increases from the bottom layer (closest to the RF iron mounts) to
the top layer. The maximal displacement of the uppermost layer has been
directly measured as less than 25 μm.

#### Angular/Rotational Uncertainties

Components in COCOA are specified by their location and orientation
making it necessary to estimate the rotational uncertainties with which
the DCOPS sensors are placed on the chamber. Since the CCDs used in the
DCOPS sensors are one dimensional pixel arrays, rotational uncertainties
about the axis normal to the plane of the CCDs (DCOPS local Z axis) are
the most important. Estimations of uncertainties about this axis, where
it might be expected the slope and flatness of surfaces will contribute
most, can also be estimated from the tolerance and separation on the
mounting bracket standoffs, CSC panel/frame uncertainty (127μm), length
(\>\>100cm), and flatness (±200 μm across 60 cm), and other tolerances
placed on the components in the mounting bracket (Al plate, DCOPS window
mounting bar - both ∼25-50μm across ∼90mm). The uncertainties about the
DCOPS Z axis are summarized in Table 3.3.

Uncertainties about the DCOPS X and Y axis are determined from the
uncertainties of the two precision pins or holes used to position
components on the chamber and mounting brackets and the separation
between them. In all cases, the uncertainties associated with the
placement of pins and holes (∼25-50μm) and the relatively large span of
between them (∼90mm) mean individual contributions to the rotational
uncertainty in the orientation of the DCOPS sensors is on the order of 1
μrad. The uncertainties about the DCOPS X and Y axes are summarized in
Table 3.2.

#### Final Estimation of DCOPS - CSC Active Center Uncertainties

The final uncertainty associated with the DCOPS reference pin - CSC
active centers is determined from the quadrature of all estimated
uncertainties in the plane of the strips (Table 3.3) and perpendicular
to the plane of the strips (Table 3.2). The estimated rotational
uncertainties about the DCOPS X, Y, and Z axes are summarized in Table
3.4 (DCOPS Z axis) and Table 3.5 (DCOPS X and Y axes). The rotational
uncertainties have been estimated from the uncertainty and separation of
the pins and/or holes which join components.

Table 3.2: Estimation of Error of DCOPS Positioning Above First Strip
Layer (local Y axis). This table shows the uncertainties associated with
the determination of the displacement between the DCOPS mounting plate
and the first plane of cathode strips.

  ------------------------------------------------------ ----------------
  **Uncertainty Origin**                                 **Magnitude
                                                         (µm)**

                                                         

  Panel Thickness (Maximal deviation)                    508

  Frame to Panel Placement                               127

  Mounting Bracket Chamber-Shim Standoff                 100

  Mounting Bracket Al. Plate                             125

                                                         

  **Final Estimation of Uncertainty in Y Plane of        **548 µm**
  Chamber**                                              
  ------------------------------------------------------ ----------------

Table 3.3: Estimation of Error Transverse to CSC Chamber Centerline.
This table shows the uncertainties associated with the determination of
the displacement between the DCOPS alignment pin and the cathode strips
transverse to the chamber centerline (local chamber X axis).

  ------------------------------------------------------ ----------------
  **Uncertainty Origin**                                 **Magnitude
                                                         (µm)**

                                                         

  Central Alignment Pin - Notched Alignment Marks        25

  Notched Alignment Mark - Numbered Reference Strip      25

  Intrinsic Strip Positioning (from milling)             30

  Averaged Centerline Across 6 Assembled Planes          87

  Positioning of Primary DCOPS Alignment Pins/Holes      25

  Diameter of Primary DCOPS Alignment Pins/Holes         25

  Placement of Mounting Plate On Chamber                 50

  Placement of DCOPS Mounting Plate                      50

  Maximal Shearing Effect                                25

                                                         

  **Final Estimation of Uncertainty Along X Axis of      **129 µm**
  Chamber**                                              
  ------------------------------------------------------ ----------------

Table 3.4: Estimation of Error of DCOPS Orientation About DCOPS CCD
Plane Normal (local DCOPS Z axis). This table shows the uncertainties
associated with the determination of the orientation between the DCOPS
CCDs and the first plane of cathode strips. Most uncertainties were less
than 1 µrad and have been rounded up.

  ----------------------------------------------------- -----------------
  **Uncertainty Origin**                                **Magnitude
                                                        (µrad)**

                                                        

  Upper Cathode Panel - Frame Relationship              1

  Frame (or Fwd Glue Plate) - DCOPS Mounting Bracket    1
  Base                                                  

  DCOPS Mounting Bracket Base - Mounting Bracket Shim   1
  Plate                                                 

  Mounting Bracket Shim Plate - DCOPS Mount Bar         1

  Straightness of DCOPS Mount Bar - DCOPS Window Frame  1

                                                        

  **Total Uncertainty in DCOPS Orientation on Chamber** **2.2 µrad**
  ----------------------------------------------------- -----------------

Table 3.5: Estimation of Error of DCOPS Orientation of DCOPS CCD Plane
(local DCOPS X/Y axis). This table shows the uncertainties associated
with the determination of the orientation between the DCOPS CCDs and the
first plane of cathode strips. Most uncertainties were less than 1 µrad
and have been rounded up.

  ----------------------------------------------------- -----------------
  **Uncertainty Origin**                                **Magnitude
                                                        (µrad)**

                                                        

  Upper Cathode Panel Pins/Holes                        1

  DCOPS Mounting Bracket Base                           1

  Mounting Bracket Shim Plate                           1

  DCOPS Mount Bar                                       1

                                                        

  **Total Uncertainty in DCOPS Orientation on Chamber** **2 µrad**
  ----------------------------------------------------- -----------------

### Estimation of Other Simulation Parameters and Uncertainties

The complete simulation of the EMU alignment scheme includes several
components in the Link Alignments system. Link components found in the
COCOA EMU simulations include the MABs, Secondary Link lines, and the ME
±1/2 'ALMY' sensors. The uncertainties in the construction and/or
performance of these components have been taken from a COCOA simulation
of a quarter Link plane \[3.3\] and discussions \[3.4\] with the Link
Alignment group. A summary of the uncertainties assigned to these
components in the simulation is given in Table 3.6.

Table 3.6: Estimation of LINK System Uncertainties. This table shows the
estimated uncertainties associated with components in the Link Alignment
System included in the EMU Idealized COCOA Simulation

  ----------------------------------------------------- -----------------
  **Uncertainty Origin**                                **Magnitude**

                                                        

  LINK Reconstructed MAB Position                       135 µm

  LINK Reconstructed MAB Orientation                    10 µrad

  DCOPS Location within MAB                             50 µm

  DCOPS Orientation within MAB                          2 µrad

  Definition of Secondary Link Line                     10 µm, 2 µrad

  ME ±1/2 ALMY Sensor Resolution                        5 µm

                                                        
  ----------------------------------------------------- -----------------

## Simulation Results (Idealized EMU System)

The primary goal of the COCOA simulation of the EMU alignment scheme is
to obtain an estimate of the uncertainties associated with the
reconstruction of CSC chamber positions along the SLM laser lines and an
understanding of the relationships and correlations between the various
components of the alignment system. For simplicity, most studies and
simulations were done with the idealized EMU. The idealized EMU system
contains all the EMU system components arranged in the manner detailed
in Appendix A with uncertainties on the location and placement of the
components set to the estimations previously discussed. Since the system
is ideal, it is assumed that all measurement devices are able to make a
successful measurement and the full redundancy of the alignment scheme
(multiple measurements of opposing lasers) is exploited. The effects of
component failure (missing CCDs measurements from DCOPS, faulty laser
modules, inadequate resolutions) are examined separately.

The layout of CSC chambers, transfer plates, and SLM lines exhibit a
high degree of symmetry between ME layers on YE Iron discs. The symmetry
of most components is reflective between discs, which is to say chambers
on ME+2 appear mirror symmetric to chambers on ME-2. This is not
generally true for DCOPS sensors located on the transfer plates. The
placement of DCOPS sensors into the transfer plate design for each ME
disc was based on the ensuring that the pointing (i.e. the normal to the
CCD plane) of the DCOPS sensors was consistent for all layers, the
orientations of otherwise symmetric DCOPS sensors were rotated 180º
about the local Y axis of the calibrated dowel pin. This effectively
changes the distance between several of the transfer plate SLM DCOPS
dowel pins and transfer plate Transfer Line DCOPS dowel pins.
Additionally, the transfer plates located at Point 5 on the ME layers
must be elongated to avoid possible conflict with the YE Iron supports.
The extension of the distance between the calibrated dowel pins results
in a leveraging of the uncertainties in the definition of the SLM laser
and, ultimately, slight discrepancies in the spatial uncertainties
between otherwise symmetric CSC chambers. This effect is seen across all
eight ME discs. For this reason, simulation results presented here will
deal with the average of all SLM chamber uncertainties for a particular
coordinate in a chamber ring, or the specification of a small range of
chamber uncertainties for a set of objects or chambers in a single ME
disc.

The software was also found to put an enormous drain on the available
computer memory. The full EMU simulation of ME ±1, ±2, ±3 and ±4 Endcap
Alignment System contained approximately 6200 separate objects to fit
with at least 4700 non-zero correlations and was simply too large for
the software to compute without exhausting its memory allocation. Though
a memory leak was identified (but could not be traced) when running the
code, the sheer number of parameters to compute requires a large amount
of memory. Since each element in the matrix requires eight bytes of
memory (double precision) and there are at least 9,000,000 elements in
each matrix (≈ 6000 x 1500) for the full simulation, at least 72 MB of
memory are required for each matrix constructed. As Section 2.1
indicates, at least seven unique matrices are required for a COCOA fit
(\> .5 GB). However since the fitting process requires an iterative
update of matrices, the actual amount of memory required for a complete
COCOA simulation of the EMU System can quickly exceed 1 GB of memory
after several iterations. It was found that the largest simulation which
could be successfully fit with the available computing resources (1 GB
of memory) contained the ME ±2, ±3, and ±4 layer alignment systems.

### Comparison of Large Simulations vs. Small Simulations

Using the full COCOA simulation model, several attempts were made to
obtain a fit of the complete 8 ME disc system. These attempts failed
with the computers reporting abnormal utilization and allocation of
memory. Indeed, when the memory usage of the system was examined, it was
found that a serious memory leak was present. The memory leak was not so
serious as to preclude the fitting of smaller subsets of the full EMU
simulation model. The largest simulation successfully fit consisted of
the ME ±2, ±3, ±4 discs and transfer line system. Fifty-three smaller
subsets of the ME ±2, ±3, ±4 discs and transfer line simulation were
also completed and compared amongst themselves and the larger 6 ME disc
simulations. These subsets included all possible permutations of systems
composed of at least two ME discs with full transfer line systems (i.e.
all simulations had the same number of transfer plates in the system).

Comparisons made between systems within the 53 subsets of two ME disc
systems showed very little variation (\< 10%) between the certainties
with which identical chambers could be reconstructed. Comparisons made
between these two ME disc systems and the larger six ME disc system also
yielded very little variation (\< 5%) between the certainty with which
identical chambers could be reconstructed. This seems to confirm that
there is very little coupling of components across different ME discs.

### Simulation of the Idealized System

The uncertainty in location for CSC chambers located along a particular
SLM line may vary from the uncertainty of similar chambers located in
other SLM lines. The discrepancies between chamber uncertainties in ME
discs with a similar SLM arrangement of components is expected to be
small (\<10µm). For chambers in the same Endcap layer, this is primarily
due to variations in the separation (i.e. lever arm) between transfer
line and reference sensors placed on transfer plates located at the SLM
endpoints. Chambers located on different ME discs are affected by the
manner in which the SLM laser line crosses the chamber and/or the
compounding of errors to inner ME chambers. Since SLM lines on ME ±1
discs are constructed in a very different manner from those on SLM ME
±2, ±3, and ±4 discs, more substantial variations are expected between
chamber uncertainties when comparisons are made to ME ±1 chambers. The
mean uncertainty with which chamber locations and orientations along the
SLM lines can be reconstructed in the EMU alignment scheme is summarized
in Table 3.7 and Table 3.8. The average deviations presented alongside
the estimates represent the average deviation from the mean uncertainty
determined for the all chambers located along a particular ME ring.

Table 3.7: Uncertainty in CSC locations along the SLM lines for the
Idealized EMU System. The uncertainty estimates for chambers in ME ±1/2
layer have been done using two sets of resolutions for the inclinometers
placed on the ME ±1/2 CSC chamber frames

+--------------------+------------+----------+------------+----------+
| **CSC Chamber**    | **Mean     |          | **Average  |          |
|                    | U          |          | Deviation  |          |
|                    | ncertainty |          | of         |          |
|                    | in Chamber |          | Unc        |          |
|                    | L          |          | ertainty** |          |
|                    | ocations** |          |            |          |
+--------------------+------------+----------+------------+----------+
|                    | **CMS RΦ   | **CMS Z  | **CMS RΦ   | **CMS Z  |
|                    | (µm)**     | (µm)**   | (µm)**     | (µm)**   |
+--------------------+------------+----------+------------+----------+
|                    |            |          |            |          |
+--------------------+------------+----------+------------+----------+
| ME±1/2             | 515        | 717      | 17         | 9        |
|                    |            |          |            |          |
| (no inclinometer)  |            |          |            |          |
+--------------------+------------+----------+------------+----------+
| ME±1/2             | 90         | 385      | 2          | 5        |
|                    |            |          |            |          |
| (σ inclinometer =  |            |          |            |          |
| short term ISR σ)  |            |          |            |          |
+--------------------+------------+----------+------------+----------+
| ME±1/2             | 187        | 415      | 2          | 1        |
|                    |            |          |            |          |
| (σ inclinometer =  |            |          |            |          |
| long term ISR σ)   |            |          |            |          |
+--------------------+------------+----------+------------+----------+
| ME±1/3             | 216        | 878      | 3          | 22       |
+--------------------+------------+----------+------------+----------+
| ME±2/1             | 205        | 467      | 10         | 12       |
+--------------------+------------+----------+------------+----------+
| ME±2/2             | 221        | 509      | 7          | 24       |
+--------------------+------------+----------+------------+----------+
| ME±3/1             | 230        | 491      | 14         | 15       |
+--------------------+------------+----------+------------+----------+
| ME±3/2             | 248        | 520      | 20         | 22       |
+--------------------+------------+----------+------------+----------+
| ME±4/1             | 241        | 525      | 14         | 17       |
+--------------------+------------+----------+------------+----------+
| ME±4/2             | 259        | 524      | 20         | 17       |
+--------------------+------------+----------+------------+----------+

Table 3.8: Uncertainty in CSC Orientations About Axes Parallel to CMS
Coordinate System and Through CSC Chamber Center Idealized EMU System.
The uncertainty estimates for chambers in ME ±1/2 layer have been done
using two sets of resolutions for the inclinometers placed on the ME
±1/2 CSC chamber frames.

+--------------------+------------+----------+------------+----------+
| **CSC Chamber**    | **Mean     |          | **Average  |          |
|                    | U          |          | Deviation  |          |
|                    | ncertainty |          | of         |          |
|                    | in Chamber |          | Unc        |          |
|                    | Ori        |          | ertainty** |          |
|                    | entation** |          |            |          |
+--------------------+------------+----------+------------+----------+
|                    | **CMS RΦ   | **CMS Z  | **CMS RΦ   | **CMS Z  |
|                    | (µrad)**   | (µrad)** | (µrad)**   | (µrad)** |
+--------------------+------------+----------+------------+----------+
|                    |            |          |            |          |
+--------------------+------------+----------+------------+----------+
| ME±1/2             | 95         | 95       | N/A        | N/A      |
|                    |            |          |            |          |
| (inclinometer      |            |          |            |          |
| short term ISR σ)  |            |          |            |          |
+--------------------+------------+----------+------------+----------+
| ME±1/2             | 698        | 698      | N/A        | N/A      |
|                    |            |          |            |          |
| (inclinometer long |            |          |            |          |
| term ISR σ)        |            |          |            |          |
+--------------------+------------+----------+------------+----------+
| ME±1/3             | 138        | 1168     | 3          | 360      |
+--------------------+------------+----------+------------+----------+
| ME±2/1             | 113        | 812      | 0          | 252      |
+--------------------+------------+----------+------------+----------+
| ME±2/2             | 74         | 771      | 0          | 297      |
+--------------------+------------+----------+------------+----------+
| ME±3/1             | 834        | 888      | 244        | 261      |
+--------------------+------------+----------+------------+----------+
| ME±3/2             | 742        | 777      | 303        | 323      |
+--------------------+------------+----------+------------+----------+
| ME±4/1             | 725        | 909      | 383        | 241      |
+--------------------+------------+----------+------------+----------+
| ME±4/2             | 1105       | 839      | 126        | 315      |
+--------------------+------------+----------+------------+----------+

In addition to variations in chamber reconstruction uncertainties
between different SLM lines and ME discs, chamber reconstruction
uncertainties within a particular SLM line may vary. This is expected
since the spacing of DCOPS sensors is not the same for chambers on the
inner and outer rings of the discs. Chambers of the same type and on the
same SLM line are expected to have similar uncertainties.

The original ME ±1 alignment scheme did not incorporate inclinometers on
ME ±1/2 chambers. It had been assumed that the ME ±1/3 SLM lines, having
been offset from the Secondary Link laser lines, would provide
sufficient angular definition about the local chamber Z axis for the ME
±1/2 chambers (sensors on these chambers cannot discern rotations about
the laser lines). Simulations of the ME ±1 layer (results shown in Table
3.9 and Table 3.10without inclinometers revealed poor spatial and
rotational resolution on all ME ±1 chambers and did not meet the
estimated system performance. For this reason, inclinometers similar to
those employed on the transfer plates were incorporated into the ME ±1/2
chamber frames. Simulations of the idealized system are performed with
the resolution of these inclinometers set to be equivalent to the long
term resolution of the inclinometers studied during the 2000 ISR tests.

Table 3.9: Uncertainty in ME ±1 CSC locations along the SLM lines for
the Idealized EMU System without ME ±1/2 Inclinometers. The uncertainty
estimates where prepared in an Idealized EMU ME±1 simulation.

+-----------------+-------------+-----------+-------------+----------+
| **CSC Chamber** | **Mean      |           | **Average   |          |
|                 | Uncertainty |           | Deviation   |          |
|                 | in Chamber  |           | of          |          |
|                 | Locations** |           | Un          |          |
|                 |             |           | certainty** |          |
+-----------------+-------------+-----------+-------------+----------+
|                 | **CMS RΦ    | **CMS Z   | **CMS RΦ    | **CMS Z  |
|                 | (µm)**      | (µm)**    | (µm)**      | (µm)**   |
+-----------------+-------------+-----------+-------------+----------+
|                 |             |           |             |          |
+-----------------+-------------+-----------+-------------+----------+
| ME±1/2          | 515         | 717       | 17          | 9        |
|                 |             |           |             |          |
| (no             |             |           |             |          |
| inclinometer)   |             |           |             |          |
+-----------------+-------------+-----------+-------------+----------+
| ME±1/3          | 291         | 965       | 5           | 111      |
|                 |             |           |             |          |
| (no             |             |           |             |          |
| inclinometer)   |             |           |             |          |
+-----------------+-------------+-----------+-------------+----------+

Table 3.10: Uncertainty of ME ±1 CSC Orientations About Axes Parallel to
CMS Coordinate System and Through CSC Chamber Center for Idealized EMU
System without ME ±1/2 Inclinometers. The uncertainty estimates where
prepared in an Idealized EMU ME±1 simulation.

+--------------+-------------+------------+-------------+------------+
| **CSC        | **Mean      |            | **Average   |            |
| Chamber**    | Uncertainty |            | Deviation   |            |
|              | in Chamber  |            | of          |            |
|              | Or          |            | Un          |            |
|              | ientation** |            | certainty** |            |
+--------------+-------------+------------+-------------+------------+
|              | **CMS RΦ    | **CMS Z    | **CMS RΦ    | **CMS Z    |
|              | (µrad)**    | (µrad)**   | (µrad)**    | (µrad)**   |
+--------------+-------------+------------+-------------+------------+
|              |             |            |             |            |
+--------------+-------------+------------+-------------+------------+
| ME±1/2       | 1517        | 1521       | 553         | 538        |
|              |             |            |             |            |
| (no          |             |            |             |            |
| i            |             |            |             |            |
| nclinometer) |             |            |             |            |
+--------------+-------------+------------+-------------+------------+
| ME±1/3       | 1331        | 1333       | 425         | 460        |
|              |             |            |             |            |
| (no          |             |            |             |            |
| i            |             |            |             |            |
| nclinometer) |             |            |             |            |
+--------------+-------------+------------+-------------+------------+

### Relationships and Correlations Between EMU Components

Since COCOA calculations are based on variance-covariance matrix
operations, the final matrix returned from the fitting process contains
additional information concerning the relationship between reconstructed
quantities in the simulated system. However, relationships between most
components must be inferred by a series of simulations, with small
variations in each simulation.

A large number of such simulations were conducted to explore the
relationship between the largest sources of uncertainties within the EMU
alignment scheme and several key components of the system. In general
(details to follow), the uncertainties ascribed to system components in
Section 3.3.2.6 were varied in a systematic manner to gauge their impact
on the determination of CSC chamber positions. This information is
particularly important during the design and pre-production stages of
the EMU project, as the cost of manufacturing and materials is highly
dependent upon the tolerances placed on the components and equipment
being employed. It is also instrumental in validating the operational
viability of the EMU alignment scheme. Since the EMU Alignment scheme
will be staged with an initial deployment of ME ±1 and ME ±2 layer
systems, the focus of these studies has been directed toward these ME
layers.

#### Correlations Between 'Unknown' Quantities

The correlations between unknown entries in the simulation are given by
the off-diagonal elements of the variance-covariance matrix returned by
the COCOA fit. CSC chambers in the simulation typically showed hundreds
of correlations with other components, however only a few correlations
exceed .01 (1%). A typical sample of the most significant correlations
between the five unknown chamber parameters (2 spatial, 3 rotational)
and other unknown simulation parameters is shown in Table 3.11. The
correlations given in Table 3.11 are representative of most CSC chambers
in the simulation: the most significant correlations between unknown
parameters were generally confined within SLM lines and connecting
transfer lines.

Looking at a sample of the correlations between various transfer plates
in Table 3.12, it is evident most correlations are less than 15%. The
largest correlations exist between transfer plates located along the
same Transfer laser line (see Appendix A, Section 5.2.6, for labeling
scheme). The coupling of individual Transfer Laser Lines across SLM
lines (i.e. opposing Transfer Lines) was expected to be weak since the
transfer line MAB reference points are relatively well known and the six
indirect interactions between the three pairs of opposing Transfer Laser
Lines are connected only by initially undefined SLM laser lines. Since
correlations between transfer plates are weak, and correlations between
transfer plates and chamber along the same SLM lines are always less
than 1, correlations between CSC chambers which do not fall along the
same SLM laser lines are weak (\< 5%).

Table 3.11: Sample Correlations between Reconstructed Entries for CSC
Chamber ME--22_15 (SLM-23). The correlation between entries shown in the
table is taken directly from the off-diagonal matrix elements of the
variance-covariance matrix used in the COCOA fit. This sample (it is not
complete) of entries represents the largest correlations found for this
chamber and was prepared from an ME ±1 and ME ±2 simulation. Entries are
taken to be in the local coordinate system of the objects they describe
(Appendix A).

  ---------------------- --------------------------------- -----------------
  **ME-22_15 Parameter** **Dependent Object**              **Correlation**

                                                           

  Centre_Y               slm-23/me-22_32_angles_Z          0.143

  Centre_Y               slm-23/me-21_8_centre_Y           0.136

  Centre_Y               transfer1_angles_X                0.081

  Centre_Y               slm-23/me-21_8_centre_Y           0.059

  Centre_Z               slm-23/me-22_15_angles_X          0.769

  Centre_Z               transfer1_angles_Z                0.326

  Centre_Z               slm-23/me-21_8_angles_X           0.326

  Centre_Z               slm-23/me-22_32_angles_Y          0.294

  Centre_Z               slm-23/me-22_15_angles_Z          0.273

  Centre_Z               transfer-1_angles_X               0.017

  Angles_X               transfer-1_angles_X               0.457

  Angles_X               slm-23/me-22_32_angles_Y          0.279

  Angles_X               transfer1_angles_Z                0.265

  Angles_X               slm-23/me-21_8_angles_X           0.265

  Angles_X               slm-23/me-22_15_angles_Z          0.187

  Angles_Y               transfer-1_centre_Y               0.651

  Angles_Y               transfer1_angles_Z                0.183

  Angles_Y               transfer1_angles_X                0.183

  Angles_Y               transfer-1_angles_Y               0.040

  Angles_Z               slm-23/me-21_8_centre_Y           0.618

  Angles_Z               slm-23/me-22_32_angles_Y          0.177

  Angles_Z               slm-23/me-21_8_angles_X           0.111

  Angles_Z               transfer1_angles_Z                0.111

  Angles_Z               transfer-1_angles_X               0.017
  ---------------------- --------------------------------- -----------------

Table 3.12: Sample Correlations between Reconstructed Entries for
Transfer Plate. The correlation between entries shown in the table is
taken directly from the off-diagonal matrix elements of the
variance-covariance matrix used in the COCOA fit. This sample (it is not
complete) of entries represents the several non-zero correlations found
between transfer plates in an ME ±2, ±3, ±4 simulation. Transfer plates
labeling has all objects designated as 'transfer_plateAB', where two
plates with the same value of 'B' lay on the same Transfer laser line
(see Appendix A, Section 5.2.6)

  ---------------------------- ------------------------------ -----------------
  **Object 1**                 **Object 2**                   **Correlation**

                                                              

  transfer_plate21_centre_X    transfer_plate24_centre_Z      0.03

  transfer_plate-21_centre_X   transfer_plate-31_centre_X     0.10

  transfer_plate-21_centre_Z   transfer_plate-31_centre_Z     0.10

  transfer_plate24_centre_Z    transfer_plate26_centre_Y      0.02

  transfer_plate24_centre_Z    transfer_plate34_centre_Z      0.14

  transfer_plate24_centre_Z    transfer_plate36_centre_Z      0.03

  transfer_plate22_centre_Z    transfer_plate32_centre_Z      0.14

  transfer_plate26_centre_Y    transfer_plate36_centre_Z      0.09

  transfer_plate34_centre_Z    transfer_plate36_centre_Z      0.08

  transfer_plate-32_centre_X   transfer_plate-42_centre_X     0.02

  transfer_plate44_centre_X    transfer_plate-31_centre_Z     0.01

  transfer_plate44_centre_X    transfer_plate-21_centre_Z     0.01
  ---------------------------- ------------------------------ -----------------

#### Correlations Between 'Known' and 'Unknown' Quantities

Correlations between the uncertainties in the construction of components
within the simulation and the certainty with which chambers can be
reconstructed provides valuable information for the design of these
components being studied. The goal of exploring the relationships
between the design of EMU components and the performance of the system
is to maximize system performance while minimizing costs. Particular
attention was paid to chambers in the ME ±1 and ±2 layers, as muons
traversing these layers experience a much greater magnetic field, and
thus a larger sagita, than in the ME ±3 and ±4 layers.

In general, these studies examined the dependence of chamber spatial CMS
RPhi uncertainties versus the resolution or tolerances of a particular
component. Simulations were done with two opposing ME layers (i.e. ME ±1
or ±2), a full Transfer Line system, inclusion of inclinometers on ME
±1/2 (resolutions set at long term ISR values), and in a manner
otherwise consistent with the simulations used to produce Table 3.7 and
Table 3.8 unless specified.

#### LINK Interface to EMU Alignment System

The interface of the EMU alignment scheme to the Link alignment scheme
provides Tracker coordinate information to the MABs and Secondary Link
lines. The certainty with which MAB and Secondary Link lines are defined
directly impacts the precision with which chambers centers can be
reconstructed. Uncertainties in the Link interface fall into three
categories: uncertainties in the construction of the MABs, uncertainty
in the definition of the Secondary Link lines, and uncertainties in the
rhomboid prism which split the Primary and Secondary Link Lines.
Uncertainties in the rhomboid prism can also be regarded as any other
potential uncertainty which propagate to both the MABs and Secondary
Link lines.

The estimated uncertainty in MABs position and orientation is given in
Section 3.3.3. This estimated includes allowances for distortions in the
MAB as well as the placement of a DCOPS sensors within the structure.
Figure 3.5 shows how the uncertainty in reconstructed CMS RPhi chamber
positions in ME ±1 and ±2 varies as a function of the uncertainty on the
MABs.

Since ME ±1 chambers are reconstructed in part by the Secondary Link
line, the dependence on Transfer Laser lines (and thus the MABs) is
relatively weak. This is particularly true for the ME ±1/2 chambers,
where there is no dependence on the MABs for spatial reconstruction in
CMS RPhi. The short lever arm between the ME ±1 (outer) Transfer Plates
and placement of the ME ±1/3 SLM reference sensor on the ME ±1/2 chamber
frame dampens the interaction between the MABs and ME ±1/3 chamber
reconstruction as well.

ME ±2 chambers (as well as ME ±3 and ME ±4 chambers) must be
reconstructed solely from the MABs. As a result, a strong coupling
between the uncertainties on the MABs and reconstructed chamber
locations is seen.

![](media/image5.wmf)

Figure 3.5: Reconstructed CSC RPhi Chamber Uncertainty vs. Tolerance on
MAB Position. The plot shows the average uncertainty in reconstructed
chamber locations as a function of the uncertainty estimate on the
determination of the placement of the MABs within CMS.

The correlation between ME ±1 chambers and the Secondary Link lines is
likely to be strong for ME ±1/2 chambers, but weaker for ME ±1/3
chambers, as the ME ±1/3 chambers reconstruction relies strongly on
MABs. Figure 3.6 shows the uncertainty of reconstructed RPhi chamber
location in ME ±1 as a function of Secondary Link line definition. As
expected ME ±1/3 chambers show very little change as the Secondary Link
line resolution is varied. Surprisingly, the coupling of the Secondary
Link line resolution and the ME ±1/2 chambers is non-linear. This is
most likely due to either the dominance of other errors in the
reconstruction of ME ±1/2 chambers when the Secondary Link line is well
defined, or an indication that ME ±1 SLM reconstruction is more
complicated than previously thought.

![](media/image6.wmf)

Figure 3.6: Reconstructed CSC RPhi Chamber Uncertainty vs. Secondary
Link Line Resolution. The plot shows the average uncertainty in
reconstructed chamber locations as a function of the uncertainty
estimate on the placement of the Secondary Link Line.

#### DCOPS-Reference Center Tolerance

Figure 3.7 shows how the uncertainty in the CMS RPhi position of the CSC
chambers vary with the uncertainty in the chamber construction (along
the chamber's local X axis). For all chambers, this relationship is
linear. The slope indicates the correlation between uncertainties in
chamber construction and reconstructed chamber RPhi positions is roughly
1:3, except on the ME ±1/2 chambers where it is closer to 1:4. The
significantly higher correlation on the ME ±1/2 chambers is due the much
lower uncertainties associated with the definition of the Secondary Link
lines and resolution of the ME ±1/2 LINK CMOS sensors. Correlations
decrease slightly for chambers located in ME layers further away from
the MABs as the net uncertainty is generally larger (See Figure 3.5).

![](media/image7.wmf)

Figure 3.7: Reconstructed CSC RPhi Chamber Uncertainty vs. DCOPS-CSC
Reference Center Tolerance. The plot shows the average uncertainty in
reconstructed chamber locations as a function of the uncertainty
estimate on the relative placement of the primary DCOPS calibration pin
with the chamber reference centerpoint.

####  {#section .unnumbered}

#### DCOPS Pixel Resolution

DCOPS sensors make measurements by fitting the charge distribution
across the CCD pixel arrays. Each pixel in the array is approximately 14
µm wide. The resolution of the DCOPS sensors is in large part determined
by the quality of the fit and the shape of the distribution to be fit.
Typical short term fluctuations in the fitted centroids during the 2001
ISR tests suggested that the short term resolution of the DCOPS sensors
in the ISR tunnel to be on the order of 1 pixel, however it is certainly
possible that the conditions in the CMS detector will decrease the
stability of these measurements. Figure 3.8 shows the uncertainty in the
reconstructed chamber RPhi locations as a function of DCOPS pixel
resolution. As the figure indicates, an uncertainty of up to three
pixels in the location of the centroids still provides a resolution of
better than 250 μm on the reconstruction (in CMS RPhi) of chamber
centers. The independence of ME ±1/2 chambers in Figure 3.8 is expected
since the measurement devices are not DCOPS sensors (thus no CCD pixel
resolutions to degrade). The uncertainties in ME ±1/2 chamber locations
are plotted to illustrate the independence of these chambers to degraded
measurements in ME ±1/3 chambers positions.

![](media/image8.wmf)

Figure 3.8: Reconstructed CSC RPhi Chamber Uncertainty vs. DCOPS Pixel
Resolution. The plot shows the average uncertainty in reconstructed
chamber locations as a function of DCOPS pixel resolution. One pixel is
14 µm wide. ME ±1/2 chamber sensors are not DCOPS sensors and have not
been varied.

#### ME ±1/2 Inclinometer Resolution

Although it was quickly established that the inclusion of inclinometers
were required on ME ±1/2 chamber frames, it was unclear if the use of
the same inclinometers on the transfer plates would provide sufficient
resolution (as well as a sufficient margin of safety) to successfully
reconstruct chamber locations. A series of simulations was performed to
investigate the dependence of ME±1 chamber reconstruction on
inclinometer resolution. Figure 3.9 shows the dependence of the
uncertainty in the determination of reconstructed CMS RPhi ME ±1chamber
positions as a function of ME ±1/2 inclinometer resolution. The
uncertainty in the ME ±1/2 chamber RPhi location is seen to
approximately double in a linear as the fashion inclinometer resolution
is varied from its nominal short term resolution to the long term
resolution. ME ±1/3 RPhi chamber resolution remains roughly independent
of the inclinometer resolution, though, as Table 3.7-Table 3.10
indicate, there are other chamber parameters which depend upon the
resolution of the inclinometers.

![](media/image9.wmf){width="5.997222222222222in" height="3.625in"}

Figure 3.9 ME±1 CSC Chamber Resolution vs ME±1/2 Inclinometer
Resolution.

The determination of the ISR inclinometer resolutions used in the
simulation is based upon the results of the extended, uncorrelated, and
uncorrected measurements over the approximately three months of testing
during the 2000 ISR Tests. It is likely that further work with these
inclinometers will increase the precision of these measurements.

## Conclusions

The results garnered from COCOA simulations of the Idealized EMU System
indicate that the design of the EMU Alignment scheme is viable and can
reconstruct chamber locations along SLM lines with an uncertainty of
≈200μm in CMS RФ and ≈500μm in CMS Z. A detailed error analysis and
subsequent examination of key sources of uncertainties across several
simulations show that the tolerances on component construction and DCOPS
calibration are well understood and meet the necessary requirements to
ensure the successful reconstruction of chamber positions.

Although the simulation assumes the entire system operates perfectly
(unlikely for any complex system), in many cases the estimations of
uncertainties and tolerances in components modeled in the simulation
have been very conservative (symmetrizing asymmetric tolerances,
assuming little or no additional calibration on CSC chambers, taking
long term uncorrelated resolutions of ISR sensors). It is also likely
that several sets of measurements could be simultaneously fit in the
COCOA simulation to compensate for failed or damaged components. More
importantly, the additional knowledge about components in the system
acquired over the next 3-5 years prior to the commissioning of the CMS
detector is likely to further constrain the fits made by COCOA and
further minimize the uncertainties associated with the reconstruction of
SLM sensors and CSC chambers.