Alignment
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Alignment. Which way is up?. w. Local Plane Coordinate System. q. v. r. u. z. r i. Plane i. r 0,i. Tower Origin. z’. y’. r 0. x’. y. x. Coordinate Systems.

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Alignment

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Alignment

Alignment

Which way is up?


Coordinate systems

w

Local Plane Coordinate System

q

v

r

u

z

ri

Plane i

r0,i

Tower Origin

z’

y’

r0

x’

y

x

Coordinate Systems

R = Rotation matrix transforming from local to global systemr0 = Position of Tower originr0,i = Translation vector from global to local originri = Position, relative to tower origin, of plane origin

Local to Global:

r = Rq + r0,i

Global to Local:

q = R-1(r – r0,i)

Global Coordinate System


Coordinate systems another view

w

Local Plane Coordinate System

q

v

r

u

z

ri

Plane i

r0,i

Tower Origin

z’

y’

r0

x’

y

x

Coordinate Systems(another view)

R = Rotation matrix transforming from local to global systemr0 = Position of Tower originr0,i = Translation vector from global to local originri = Position, relative to tower origin, of plane origin

Local to Global:

r = Rq + r0,i

Global to Local:

q = R-1(r – r0,i)

Global Coordinate System


Transformations

Transformations

  • Rotation R and translation r0,i are for a perfectly aligned detector

    • For Glast, we can take R = I (the identity matrix)

  • Vector to origin of the ith plane:

    • r0,i = r0 + ri

    • For Glast we have ri = (0,0,zi)

  • Corrections to perfect alignment will be small, above are modified by and incremental rotation R and translation r:

    • R→RR

    • r0→ r0 + r0

  • These corrections give:

    • r0,I = r0 + r0 + Rri

    • r = RRq + r0 + r0+ Rri = R(Rq + ri) + r0 + r0

    • q = (RR)-1(r - r0 - r0 - Rri)


Incremental rotation matrix

cos β 0 sin β

0 1 0

-sin β 0 cos β

Ry(β)=

  • 0 0

  • 0 cos α -sin α

  • 0 sin α cos α

Cos γ -sin γ 0

Sin γ cos γ 0

0 0 1

Rx(α) =

Rz(γ ) =

Incremental Rotation Matrix

  • Express the incremental rotation matrix as:R = Rx(α)Ry(β)Rz(γ )where Rx(α), Ry(β)and Rz(γ )are small rotations by α, β, γ about the x-axis, y-axis and z-axis, respectively

  • In General


Incremental rotation matrix continued

cos β cos γ cos γ sin αsin β- cos α sin γ cos α cos γ sin β + sin α sin γ

cos β sin γ cos αcos γ + sin α sin β sin γ -cos γ sin α + cos α sin β sin γ

- sin β cos β sin α cos  α cos β

R =

1 -γβ

γ 1 -α

-βα 1

R =

Incremental Rotation Matrix(continued)

  • Multiplying it out, we get:

  • Taking α, β and γ to be small (and ignoring terms above 1st order) gives:


Local to global transformation

1 -γβ

γ 1 -α

-βα 1

x0 + x0

y0 + y0

z0 + z0

ui

vi

zi

+

r=

Local to Global Transformation

  • Start with: r = R(Rq + ri) + r0 + r0

  • For Glast, R = I, the identity matrix

  • R as given on the previous page

  • ri = (0, 0, zi) since, for Glast, the silicon planes are parallel to x-y plane

  • q = (ui, vi, 0) since the measurement is in the sense plane (no z coordinate)

  • This gives:

x = ui - γvi + βzi + x0 + x0y = vi + γui - αzi + y0 + y0z = zi - βui + αvi + z0 + z0


Global to local transformation

1 γ -β

-γ 1 α

β -α 1

x - x0 - x0 - βzi

y - y0 + y0 + αzi

z - z0 + z0 - zi

q=

Global to Local Transformation

  • Start with: q = (RR)-1(r - r0 - r0 - Rri)

  • For Glast, R = I, the identity matrix

  • R as given on the previous page, to 1st order R-1 = RT

  • Rri = (βzi, - αzi, zi)

  • This gives (keeping terms to 1st order only):

ui = x – x0 –x0 + γ (y – y0 – y0) – β (z – z0 – z0)

vi = y – y0 – y0 – γ (x – x0 – x0) + α (z – z0 –z0)

wi = z – z0 –z0 – β (x – x0 – x0) + α (y – y0 – y0) – zi


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