26/10/20051 BARREL ALIGNMENT: LIST OF PARAMETERS REQUIRED FOR THE POSITIONING AND THE CALIBRATION OF THE VARIOUS ELEMENTS Praxial system: - Platforms - Praxial sensors - Axial sensors Projective system: - Platforms - Extension plates - RASNIK systems Reference system/Chamber to Chamber Connection: - Plates on toroid ribs - Platforms on chambers - LED sources on chambers - Cameras/LED on plates and chambers BIM-BIR connections: - Platforms - RASNIK systems B sensors platforms Appendix: Coordinate system defined by a 3 balls triplet C.Guyot (Saclay)

26/10/20052 Praxial system: Platforms on BIS, BML, BMS, BOL, BOS x ch y ch x z x z ch x ch z x z z x x y x y Local reference frame built out of the 6 balls mounted on the praxial sensor mechanics. Balls define (x’P z) plane Balls 5-6 define (x’P y) plane Ball 4 defines the z axis + translation  x,  z (fixed numbers) to bring the origin P close to the centre of the platform Platform parameters: - Identification number (chamber id, corner location) - Nominal coordinates (x,y,z) of the platform origin P in the local frame of the chamber - Angular corrections (  x,  y  z ) given by the control towers at the end of the chamber assembly X’ Z X P Z’ 52 67

26/10/20053 Praxial system: Platforms on BIL x ch y ch x z x z ch x ch z x z z x x y x y Local reference frame built out of the 6 balls mounted on the praxial sensor mechanics. Balls define (x’P z) plane Balls 5-6 define (x’P y) plane Ball 4 defines the z axis + translation  x,  z (fixed numbers) to bring the origin P close to the centre of the platform Platform parameters: - Identification number (chamber id, corner location) - Nominal coordinates (x,y,z) of the platform origin P in the local frame of the chamber - Angular corrections (  x,  y  z ) given by the control towers at the end of the chamber assembly X’ Z X P Z’ 52 67

26/10/20054 Praxial system: Praxial sensor Local frame identical to the platform frame Two orientations (mask at x 0) Mask Image sensor lens x z z x Parameter list: Identification number - Proximity part: Type (orientation, gap size…) (x,y,z) of mask center  mask line /x axis (x,y,z) of image sensor center  pixel line/x axis (x,y,z) of lens optical axis RASNIK calibration x z - Axial part: RASNIK element type: 1=mask, 2=lens, 3=image sensor Type (x,y,z)  line /x axis Type (x,y,z)  line/x axis Proximity sensor Axial sensor RASNIK element at x<0 RASNIK element at x>0 Caliprax calibration

26/10/20055 Projective system: Platforms on BIL x ch y ch ypyp x y The local reference frame of the platform is defined by the centres of the 3 balls mounted on the extension plate mechanics. Platform parameters: - Identification number (chamber id, side location) - Nominal coordinates (x,y,z) of the platform origin P in the local frame of the chamber - Angular corrections (  x,  y  z ) (provisional) xpxp zpzp Elongated cone flat P xpxp z ch x ch xpxp zpzp zpzp xpxp Template square The z axis is defined by the cone ball (origin P) and the elongated cone ball. Then the flat ball defines the (z,x) plane. The x axis is perpendicular to the z axis. xpxp ypyp HV RO

26/10/20056 Projective system: Platforms on BML,BOL x ch y ch ypyp x y The local reference frame of the platform is defined by the centres of the 3 balls mounted on the extension plate mechanics. Platform parameters: - Identification number (chamber id, side location) - Nominal coordinates (x,y,z) of the platform origin P in the local frame of the chamber - Angular corrections (  x,  y  z ) (provisional) xpxp z ch x ch Template square The z axis is defined by the cone ball (origin P) and the elongated cone ball. Then the flat ball defines the (z,x) plane. The x axis is perpendicular to the z axis. See appendix with d f = 0. HV RO xpxp zpzp Elongated cone flat P xpxp ypyp xpxp zpzp zpzp xpxp

26/10/20057 Projective system: Extension plate, rays# 2,3,4 The local reference frame of the extension plate (P,x p,y p,z p ) is identical to the platform local frame. It is defined by the centre of the 3 balls. The local reference frame of the RASNIK element support (R,x r,y r,z r ) is defined by the surface defined by 3 steel balls (plane x r,z r ), a large ball and a pin. For RASNIK support zrzr yryr zrzr ypyp xpxp xrxr z r axis in the direction of the projective ray light R R Extension plate parameters : - Identification number (chamber id, side location) - Nominal coordinates (x,y,z) R of the RASNIK origin R in the local frame of the extension plate - Three angles:  y and  z define x r,  xr defines y r w.r.t. z p P CMM pin Large precise ball Steel balls Ball on platform flat Ball on platform cone zpzp xpxp P Ball on platform elongated cone BIL extension plate

26/10/20058 Projective system: Extension plate, ray# 1 (  =0) The local reference frame of the extension plate (P,x p,y p,z p ) is identical to the platform local frame. It is defined by the centre of the 3 balls. The local reference frame of the RASNIK element support (R,x r,y r,z r ) is defined by the surface defined by 3 steel balls (plane x r,z r ), a large ball and a pin. For RASNIK support zrzr yryr zrzr ypyp zpzp z r axis parallel to the projective ray trajectory R Extension plate parameters : - Identification number (chamber id, side location) - Nominal coordinates (x,y,z) R of the RASNIK origin R in the local frame of the extension plate - Three angles:  y and  z define x r,  xr defines y r w.r.t. x p P CMM pin Large precise ball Steel balls Ball on platform flat Ball on platform cone xpxp zpzp P Ball on elongated cone xrxr BML extension plate

26/10/20059 Projective system: RASNIK element yryr xrxr R xMxM yMyM M Extension plate RASNIK mask The local reference frame of the RASNIK element support (R,x r,y r,z r ) is identical to its counterpart on the extension plate. The local reference frame of the RASNIK element (mask as an example) (M,x M,y M,z M ) is ideally identical to the support frame, up to a translation. RASNIK element support RASNIK element parameters: - Identification number and type (mask/lens/image sensor ) - Nominal coordinates (x,y,z) M of the RASNIK element M in the local frame of the support - One angle:  z = angle of mask or pixel lines w.r.t. x r axis RASNIK calibration Holes

26/10/ Reference system: Toroid plates Reference plates on toroid ribs : The local reference frame of the reference plate (P,x P,y P,z P ) is defined by 3 balls glued on the plate. The local reference frame of each camera (C i,x ci,y ci,z ci ) is defined by the ball triplets (See appendix with d f > 0). Ball triplet for camera support yPyP xPxP P Toroid rib Ball to define the local frame of the plate Reference plate parameters for each ball triplet: - Identification number (plate id, camera location id = i) - Nominal coordinates (x,y,z) R of the camera triplet origin C i in the local frame of the reference plate - Three angles:  x and  y define y c,  yc defines z c w.r.t. x P CMM + Measured distance between the 2 plates (e.g. the 2 upper balls) on the same coil on 2 neighbouring ribs ( parameter per coil) ycyc zczc xcxc CiCi zcizci xcixci Ball on elongated cone Ball on cone Ball on flat Camera axis ycyc CiCi zcizci CiCi Ball on flat

26/10/ Platform parameters: - Identification number (chamber id, side) - Nominal coordinates (x,y,z) P of the platform origin P in the local frame of the chamber - Three angles:  x and  z define y P,  y defines x P w.r.t. z ch zpzp xpxp P flat Elongated cone The local reference frame of the platform (P,x P,y P,z P ) is defined by the 3 balls triplet (See appendix with d f > 0). The V underneath the platform is always put on the tube at large |z|. By Construction or CMM z ch x ch xpxp zpzp cone Reference system: Platforms for Sacled (BIL) zpzp ypyp xpxp ypyp xpxp zpzp

26/10/ Platform parameters: - Identification number (chamber id, side) - Nominal coordinates (x,y,z) P of the platform origin P in the local frame of the chamber - Three angles:  x and  z define y P,  y defines x P w.r.t. z ch zpzp xpxp P flat Elongated cone The local reference frame of the platform (P,x P,y P,z P ) is defined by the 3 balls triplet (See appendix with d f > 0). By Construction or CMM z ch x ch xpxp zpzp zpzp xpxp cone Reference system : Platforms for Sacled (BIL H8 only) zpzp ypyp xpxp ypyp

26/10/ Platform parameters: - Identification number (chamber id, side) - Nominal coordinates (x,y,z) P of the platform origin P in the local frame of the chamber - Three angles:  x and  z define y P,  y defines x P w.r.t. z ch zpzp xpxp P flat Elongated cone The local reference frame of the platform (P,x P,y P,z P ) is defined by the 3 balls triplet (See appendix with d f > 0). The V underneath the platform is always put on the tube at large |z|. By Construction or CMM z ch x ch xpxp zpzp cone zpzp ypyp xpxp ypyp xpxp zpzp CCC: Platforms for Sacled (BIL)

26/10/ Platform parameters: - Identification number (chamber id, side) - Nominal coordinates (x,y,z) P of the platform origin P in the local frame of the chamber - Three angles:  x and  z define y P,  y defines x P w.r.t. z ch zpzp xpxp P flat Elongated cone The local reference frame of the platform (P,x P,y P,z P ) is defined by the 3 balls triplet (like for the BIL CCC platform). (See appendix with df > 0). The V underneath the platform is always put on the tube at large |z|. By Construction or CMM z ch x ch xpxp zpzp zpzp ypyp xpxp ypyp cone CCC: Platforms for camera extension plates (BIS) xpxp zpzp

26/10/ zpzp Elongated cone flat P xpxp Reference system: Platforms for Sacled extension plates (BML) CCC: Platforms for camera extension plates (BMS,BOS) The local reference frame of the platform (P,x P,y P,z P ) is defined by the 3 balls triplet (See appendix with d f =0). The V underneath the platform is always put on the tube at large |z|. Platform parameters: - Identification number (chamber id, side) - Nominal coordinates (x,y,z) P of the platform origin P in the local frame of the chamber - Three angles:  x and  z define y P,  y defines x P w.r.t. z ch By Construction or CMM z ch x ch xpxp zpzp zpzp ypyp xpxp ypyp xpxp zpzp

26/10/ zpzp Elongated cone cone flat P xpxp Reference system: Platforms for Sacled extension plates (BML H8 only) The local reference frame of the platform (P,x P,y P,z P ) is defined by the 3 balls triplet (See appendix with d f =0). Platform parameters: - Identification number (chamber id, side) - Nominal coordinates (x,y,z) P of the platform origin P in the local frame of the chamber - Three angles:  x and  z define y P,  y defines x P w.r.t. x ch By Construction or CMM z ch x ch xpxp zpzp zpzp xpxp xpxp ypyp zpzp ypyp

26/10/ CCC: Platforms for Sacled extension plates (BML/BOL) The V underneath the platform is always put on the edge tube. The local reference frame of the platform (P,x P,y P,z P ) is defined by the 3 balls triplet (See appendix with d f > 0). Platform parameters: - Identification number (chamber id, side) - Nominal coordinates (x,y,z) P of the platform origin P in the local frame of the chamber - Three angles:  x and  z define y P,  y defines x P w.r.t. z ch By Construction or CMM z ch x ch zpzp xpxp P flat Elongated cone cone zpzp xpxp zpzp xpxp zpzp xpxp zpzp xpxp zpzp ypyp xpxp ypyp

26/10/ xpxp The local reference frame of the transfer plate (P,x p,y p,z p ) is identical to the platform local frame. It is defined by the 3 ball triplets. The local reference frame of the Sacled (S,x s,y s,z s ) is also defined by 3 balls. Extension plate parameters: - Identification number (chamber id, side location) - Nominal coordinates (x,y,z) S of the origin S in the local frame of the extension plate - Three angles:  x and  x define y s,  y defines x s w.r.t. x p CMM The z axis is defined by the cone ball (origin S) and the elongated cone ball. Then the flat ball defines the (x,z) plane (see appendix with df>0). The x axis is perpendicular to the z axis. Reference system/CCC: Transfer plate for Sacled supports (BIL) xpxp ypyp xsxs ysys S P Bottom face P zpzp zszs cone ball flat S xsxs xpxp P zpzp cone ball S xsxs zszs Ref : Type R (x ch >0) CCC: Type L (x ch <0) Ref : Type L (x ch <0) CCC: Type R (x ch >0) Top views: There are 2 types of extension plates which differ only by the orientation of the C frame w.r.t. the P frame.

26/10/ Extension plate parameters: - Identification number (chamber id, side location) - Nominal coordinates (x,y,z) S of the origin S in the local frame of the extension plate - Three angles:  x and  y define y s,  z defines x s w.r.t. x p The local reference frame of the extension plate (P,x p,y p,z p ) is identical to the platform local frame. It is defined by the 3 balls triplet. The local reference frame of the Sacled (S,x s,y s,z s ) is also defined by 3 balls. CMM The z axis is defined by the cone ball (origin S) and the elongated cone ball. Then the flat ball defines the (x,z) plane (d f >0). The x axis is perpendicular to the z axis. Reference system: Extension plate for Sacled supports (BML) zszs xsxs S Ball on platform cone zpzp xpxp P Type L (x ch <0) zszs xsxs S Cone ball Elongated cone ball Flat ball Ball on platform flat Ball on platform cone zpzp xpxp P Type R (x ch >0) ypyp zpzp P zszs ysys Type L There are 2 types of extension plates which differ only by the orientation of the S frame w.r.t. the P frame.

26/10/ Reference system: Extension plate for Sacled supports (BML H8 only) The local reference frame of the extension plate (P,x p,y p,z p ) is identical to the platform local frame. It is defined by the 3 balls triplet. The local reference frame of the Sacled (S,x s,y s,z s ) is also defined by 3 balls. Extension plate parameters: - Identification number (chamber id, side location) - Nominal coordinates (x,y,z) S of the origin S in the local frame of the extension plate - Three angles:  y and  z define y s,  x defines x s w.r.t. z p CMM The z axis is defined by the cone ball (origin S) and the elongated cone ball. Then the flat ball defines the (x,z) plane (d f >0). The x axis is perpendicular to the z axis. ypyp xpxp P zszs ysys zszs xsxs S Cone ball Elongated cone ball Flat ball Ball on platform flat Ball on platform cone zpzp xpxp P Elongated cone ball Bottom view

26/10/ CCC: Extension plate for camera supports (BIS, BMS, BOS) The local reference frame of the extension plate (P,x p,y p,z p ) is identical to the platform local frame. It is defined by the 3 balls triplet. The local reference frame of the camera (C,x c,y c,z c ) is also defined by 3 balls. Extension plate parameters: - Identification number (chamber id, side location) - Nominal coordinates (x,y,z) C of the origin C in the local frame of the extension plate - Three angles:  x and  y define y c,  z defines x c w.r.t. x p CMM The z axis is defined by the cone ball (origin C) and the elongated cone ball. Then the flat ball defines the (x,z) plane (d f >0). The x axis is perpendicular to the z axis. zczc xcxc C Cone ball Flat ball Ball on platform flat Ball on platform cone zpzp xpxp P Type L (x ch <0) zczc xcxc C zpzp xpxp P Type R (x ch >0) Ball on platform cone ypyp zpzp P zczc ycyc Type L (x ch <0) There are 2 types of extension plates which differ only by the orientation of the C frame w.r.t. the P frame.

26/10/ CCC: BML, BOL extension plate for Sacled support The local reference frame of the extension plate (P,x p,y p,z p ) is identical to the platform local frame. It is defined by the 3 ball triplets. The local reference frame of the Sacled (S,x s,y s,z s ) is also defined by 3 balls. Extension plate parameters: - Identification number (chamber id, side location) - Nominal coordinates (x,y,z) S of the origin S in the local frame of the extension plate - Three angles:  x and  x define y s,  y defines x s w.r.t. x p CMM zszs xsxs S Cone ball Elongated cone ball Flat The z axis is defined by the cone ball (origin S) and the elongated cone ball. Then the flat ball defines the (x,z) plane (see appendix with df>0). The x axis is perpendicular to the z axis. Ball on platform flat Ball on platform cone zpzp xpxp P xpxp ypyp RPC xsxs ysys S zszs xsxs S Type R (x ch >0) Type L (x ch <0) Type R (x ch >0) zpzp xpxp P There are 2 types of extension plates which differ only by the orientation of the S frame w.r.t. the P frame.

26/10/ Reference system: Cameras (except for ref-BML connection) zczc ycyc Camera parameters: - Identification number - Lens optical center coordinates (x L,y L,z L ) in the reference frame of the plate - Image sensor center coordinates (x s,y s,z s ) in the reference frame of the plate - Angle  s of the pixel lines w.r.t. the x axis - Coordinates (x 1,y 1,z 1 ) and (x 2,y 2,z 2 ) of the LED holes in the local frame of the platform Camera calibration Plate Cameras on toroid rib plates: The local reference frame of the camera is given by the ball triplet frame on the toroid plate (C,x c,y c,z c ). C 4 LED Holes Diffusor zczc xcxc cone Elongated cone Flat Image sensor By construction or from calibration Bottom view C

26/10/ CCC + Ref-BML connections: Cameras zczc ycyc Camera parameters: - Identification number - Lens optical center coordinates (x L,y L,z L ) in the reference frame of the plate - Image sensor center coordinates (x s,y s,z s ) in the reference frame of the plate - Angle  s of the pixel lines w.r.t. the x axis - Coordinates (x 1,y 1,z 1 ) and (x 2,y 2,z 2 ) of the LED holes in the local frame of the platform Camera calibration Plate Cameras on the CCC extension plates or on the reference plates: The local reference frame of the camera is given by the extension plate reference frame (C,x c,y c,z c ). C zczc xcxc cone Elongated cone Flat By construction or from calibration Bottom view C Image sensor

26/10/ Reference system: Sacled support (BIL H8 only) The local reference frame of the support is given by the 3 balls and hence by the platform reference frame (P,x P,y P,z P ). zpzp ypyp 4 Holes Diffusor SACLED support parameters: - Identification number - Coordinates (x 1,y 1,z 1 ) to (x 4,y 4,z 4 ) of the 4 LED holes in the local frame of the platform Platform tube By construction or from calibration Ball on platform cone zpzp xpxp P Ball on platform flat Bottom view LED

26/10/ Reference system (BIL,BML), CCC(BIL,BML,BOL): Sacled supports The local reference frame of the support is given by the extension plate reference frame (C,x c,y c,z c ) or (S,x s,y s,z s ). 4 Holes Diffusor SACLED support parameters: - Identification number, type L/R - Coordinates (x 1,y 1,z 1 ) to (x 4,y 4,z 4 ) of the 4 LED holes in the local frame of the extension plate Extension plate LED By construction or from calibration S cone zszs xsxs S flat Bottom view ysys zszs

26/10/ BIM-BIR connection: z ch x ch xpxp zpzp zpzp xpxp Mask Lens-support Image sensor BIM BIR The BIM-BIR connection platform is identical to the projective platform. The local reference frame is defined via the 3 balls on the platform. BIM-BIR connection platform parameters: - Identification number (chamber id, side location) - Nominal coordinates (x,y,z) of the platform origin P in the local frame of the chamber - Angular corrections (  x,  y  z ) (provisional) Template square (= proj. square) BIM-BIR RASNIK parameters: - Identification number - Zero of the RASNIK: (x c,y c,  c,mag) - Angle of the mask w.r.t platform (  m ) BIM-BIR calibration

26/10/ Platform parameters: - Identification number (chamber id, side) - Nominal coordinates (x,y,z) P of the platform origin P in the local frame of the chamber - Coordinates of the unit vectors x p and z p in the local coordinate frame of the chamber (-1,0,0) and (0,0,1) in the above example The local reference frame of the platform (P,x P,y P,z P ) is defined by the platform edge and the V-shape at the connector side. By Construction z ch x ch B field sensors : Platforms zpzp ypyp xpxp ypyp xpxp zpzp P xpxp xpxp P xpxp xpxp P P P

26/10/ Appendix: Coordinate system defined by a 3 balls triplet Ball on platform flat Ball on platform cone x Ball on platform elongated cone z F E C Unit vectors (u x,u z ) coordinates as a function of the ball center coordinates measured by a CMM (all expressed in the CMM coordinate frame): u z = (E-C)/||E-C|| f = (F-C)/||F-C|| u x = [ f – u z.f *u z ]/sqrt[1.-(u z.f) 2 ] u y = u z x u x If d f > 0 then iterate: F’ = F – d f u y f = (F’-C)/||F’-C|| u x = [ f – u z.f *u z ]/sqrt[1.-(u z.f) 2 ] u y = u z x u x z E C y F d f = nominal flat ball offset (from the drawings)