1 Reconstruction and statistical modelling of geometric measurements from the LiCAS project Patrick Brockill LiCAS Group Oxford, 6 February, 2008 Talk prepared for the Industrial and Interdisciplinary Workshop Mathematical Institute at the University of Oxford

2 Outline  Introduction  ILC, alignment requirements  Prototype survey robot built by LiCAS  Problems  Global survey  “Toy model”  Calibration of the robot  Constraints

3  International Linear Collider  About 30-50km long  Late 2010’s  Cost: About US$6.65 billion  500 GeV (later upgrade to 1000 GeV)  Difficult requirements, incl. alignment  Goal: Build Straight Line ILC

4 Why Not a Circle? Circular Accelerator (CERN) Pros  Reuse uncollided particles  “Kick” each time round  (LHC 14 TeV) Cons  Synchrotron radiation, E loss worse with smaller mass  Only “heavy” particles (protons) so messy collisions, quantum #’s initial state unknown Linear Accelerator (ILC) Pros  Light particles, more fundamental  Cleaner collisions Cons  Only get one shot at accelerating (500 GeV)  Cannot reuse uncollided particles

5 Source: “Status of ILC Accelerator and Detectors”, Nobu Toge, ICHEP08, 5 August 2008 (FF Test Beamline,ATF2) Example of What a Test Beam Line Looks Like Alignment of Accelerator Components Being Performed by Laser Tracker Slow: ≈ 10 m/h Essentially, we will want to replace the laser trackers with something faster and hopefully at least as accurate  Req. Accuracy: 200µm each 600m!

6 Traditional Methods  Vertical alignment: Hydrostatic levelling systems  Follow geoids  Long wires  The wire will sag under gravity: Only good for horizontal alignment  Laser line  In open-air, it will be refracted by temperature gradients: A 600m line of sight can be bent by 4.5mm for 0.1°C/m temperature gradient  Laser trackers  John showed traditional methods won’t work  Also: time and ground movement Measured Vertical Height

7 Key Principles Retroreflector (Corner Cube)/Laser Combination:  Two properties:  (1) Direction  (2) Length  Use with FSI  Freq. Scanning Interferometry LiCAS Principle :  Overlapping measurements of retroreflector array for a global survey  Moving survey robot, Rapid Tunnel Reference Surveyor (RTRS)  Determines positions of RR's  After coordinates of RR's determined, local surveying methods

8 Prototype at DESY

9 RTRS in Operation

10 Global RTRS Operation Accelerator Component to Align Tunnel wall Retroreflector Wall Markers RTRS Moves Into Position External FSI Laser Measures Distances to Wallmarkers RTRS Moves Into New Position ExtFSI Measures Distances to Old and New Wallmarkers LSM/FSI See Rotations and Translations Wallmarker Coordinates Are Noted in this Frame Wallmarker Coordinate Sections Overlap, Allow For Global Coordinate System (Global) Coordinates of Wallmarkers Serve as Basis For Component Alignment Our Goal: Determine the coordinates for the wall markers which best describe our measurements (“Reconstruction”), 200µm over 600m

11 Local RTRS Operation Multilateration to Determine Wall Marker “External FSI” Need to Relate Positions Of Cars: Need To Introduce a Common Object Between Them Laser line passed through all cars… Then reflected back along itself… Then split off and observed. Able to determine some rotations and translations this way. “Laser Straightness Monitor” (LSM) Of course, we use gravity, i.e. tilt sensors. But this is not enough. But none of these systems determines distance between cars so this must also be measured: “Internal FSI” Basic Elements (“Subsystems”): 1.Distance between two points 2.Intersection of a line and a plane 3.Tilt sensor Our Goal: Determine the best positions for the wall markers which agree with our data …But first we have to find the positions/orientations of the internal elements (“Calibration”). Caveat: we can’t just measure this...

12 LSM Principle z y Translation: Spots move same direction Rotation: Spots move opposite directions CCD Camera Used to measure carriage transverse translations and rotations Source: Armin Reichold

13 Approach Two subproblems:  Global Reconstruction  Local Calibration Both require finding parameters X which best describe measurements L=F(X), where we think we understand F  (nonlinear) Least Squares. Both subproblems have issues which make their least square processes non-trivial.

14 Local Calibration: The Rub  Local calibration also determines success of global reconstruction since any mistakes “exponentiated”  Any hidden symmetries must be identified and resolved by using constraints  Where my efforts, and questions, are currently  Currently only middle car, but eventually train-wide

15 Global Reconstruction: The Rub Least squares point of view  Relatively large-scale problem: ≈100,000×100,000  Only some general results here We need a simpler (“toy”) model for qualitative analysis  Systematic errors and random walk model  Interesting problem, being studied by John Dale

16 Random Walk Model (Brief) We determine local coordinates, but these are slightly wrong Say a systematic error, pushing us to the right each time Begin with a network of points to be determined. Say they’re exactly on a straight line. From our point of view, these are the proper coordinates Simplified problem: ignore the horizontal offsets. Define the “residuals” as the vertical differences. These match up very well with data from models using linear algebra. Residuals Linear algebra model for entire reconstruction is complicated. We would like to simulate this with a “toy model” which provides both insight and speed. Random walk has been shown to provide this. (We’ll rotate for comparison) Source: John Dale Some results from Least Squares analysis Simplified problem: ignore the horizontal offsets. Define the “residuals” as the vertical differences. These match up very well with data from models using linear algebra.

17 Random Walk Model (Brief) Begin with a network of points to be determined. Say they’re exactly on a straight line. Question: how can we modify the random-walk model in such a way as to reproduce the linear algebra results? What if we added more measurements, say GPS on the first and last points? i.e. we overlay a new network onto our problem. How do the residuals change? Some results from least squares analysis. GPS points have been “pinned down” better. Source: John Dale

18 Local Calibration  Least Squares  Given L=F(X), if we make enough measurements L, we can determine X (hopefully)  Subsystems  Distance between two points: FSI  Intersection of line (laser) and plane (CCD): LSM  “Virtual LSM”: Beam splitters function just to allow CCD’s out of the way, can be removed altogether  Angle: Tilt Sensors/(In)Clinometres  Distance between points and angles: Laser Tracker

19 Laser Tracker Aided Calib. Additional observations of the car movements to be used if necessary

20 Subsystem Overview LSM: Laser Straightness Monitor Rotation about x, y axes Translations in x,y External FSI: Freq. Scanning Interferometer Measures distances from quills to wall marker Internal FSI Translations in z Tilt sensor/(In)clinometre Measure tilt about z, x axes Laser Tracker Platform Laser Tracker

21 LSM External FSI Internal FSI Internal FSI (Next Car) Laser Tracker Platform Laser Tracker Wall Marker Slight rotation Slight translation LSM Beam Intersects CCD’s External FSI Sees Wall Marker Internal FSI Between Cars Laser Tracker Observations Angles also measured In its own frame Subsystem Operation Car 2 Assuming cars 1, 3 stationary

22 “Auto-/Self-” Calibration  Idea: Least Squares and Repeated Measurements  Idea: if make enough measurements, the positions and orientations will be determined for subsystems independently.  Try to implement this idea for various subsystems independently  LSM  External FSI  Internal FSI: seems too underdetermined  But we ran into some surprises

23 “Auto-/Self- Calibration”  Nasty surprises: “symmetries”  Assoc. with zero eigenvalues  E.g. Hoberman sphere LSM External FSI: All of these symmetries are overcome if we calibrate all subsystems together with the laser tracker included. 

24 Calibration Demo (w/o IntFSI) Constraints: Wallmarker y,z Lasertracker Platform LT Platform Roll Sum of yaw, pitch Sum of translations TOTAL Laser Tracker Wall Marker LSM Beam Laser Tracker Platform External FSI Quills LSM CCD’s Z-tilt Sensor X-tilt Sensor Gravity How/where converges Depends on the Constraints

25 Reproduce Measurements CCD0x RR1 Pitch Z-Tilt Sensor External FSI  No: we still have to consider the constraints and errors... So we are finished, right?

26 Cheated: First Results Different Actually, we’ve lied a bit. The first results did not look so nice: What happened? All of the coordinates/orientations were given with respect to CCD0, whose own position is not particularly well-determined with respect to the other components.

27 Constraints  Single car problem: 14 zero eigenvalues in matrix of first derivatives ∂F(X)/ ∂X  14 constraints  Choice of constraints:  Orientations and positions of internal elements  Orientations and positions of unit  Positions of the wall marker  Orientation and position of the Laser Tracker  Orientation of gravity  Orientations of tilt sensors  All together, about 350 elements in X, many possible combinations for constraints

28 Choosing Constraints  Different sets of constraints seem to be as motivated as others  Classification system?  Mysteries: some “logical” sets do not seem to minimise (F-L)∙P∙(F-L)  Errors  Self-referential system: basing coordinates on a particular weakly determined element gives large errors  Which choice of constraints gives the smallest errors?  Need equivalent of a “centre of mass”  Basis for classification system?  Effects of local constraints on global reconstruction  Need to be studied and understood

29 Constraints in Physics  We need a “guiding principle”, look to physics?  Constraints in (classical) FT  Global, gauge symmetries, gauge fixing  These were difficult there, too (e.g. higher-spin theories), but had guiding principles (global symmetries and conserved quantities, group representations, etc.)  Dirac’s formalism, first-order, primary constraints, etc.  Is there some analogue to the condition of “smallest errors”?

30 Conclusions  Global reconstruction  We need a “toy model” which reproduces residuals somewhat faithfully  Local calibration  We need to choose one set of constraints amongst many possibilities  which has associated small errors  understand the impact on global reconstruction

31 Local Operation Subsystems/Building Blocks (1)Distance between two points: FSI Distances to about 1 micron Problem: measured from where? Some laser lines encased encased in vacuum (“internal”), others not (“external”) (2)Intersection of line (laser) and plane (CCD): LSM Errors under 1 micron, but ghosts Problem: CCD's small: only very small movements “Virtual LSM”: Beam splitters function just to allow CCD’s out of the way, can be removed altogether (3)Angle: Tilt Sensors/(In)Clinometres Problem: orientation, gravity, zero offset (4)Distance between points and angles: Laser Tracker Problem: errors

32 Laser Straightness Monitor(LSM) Virtual LSM Rotated positions Original positions “Virtual” positions

33 CCD0 Local Coordinate System Choose virtual position/orientation of CCD0 as the (self-referential) coordinate system Problem: If CCD0’s position and orientation not well-determined, errors of positions/orientations of all elements are horrible Question: What should we take as the coordinate system? Remember that the entire global functioning of the train may depend on this. Really a question of how to choose constraints. What kind of constraints are we talking about?

34 Constraints Let’s show two constraints. We’ll begin as before… And now we will simply rotate about The line which represents the LSM laser… But our measurements remain the same  A symmetry which must be constrained. We find a similar constraint for movement along the LSM laser (i.e. along z) Similarly: 6 constraints to define “original” coordinate system. 6 constraints to define laser tracker platform. A total of 6+6+2=14 things to constrain.

35 collider component Tunnel Wall Reconstructed tunnel shapes (relative co- ordinates) wall markersinternal FSI external FSISM beam Survey Implementation