Coordinate Systems

Catherine LeCocqSLAC

USPAS, Cornell UniversityLarge Scale Metrology of Accelerators

June 27 - July 1, 2005

Coordinate Systems 1

Coordinate Systems

• Purpose:– To express the size and aspect of an object.– To locate other objects with respect to the first

one.

• Requirements in 3D:– an origin– an orientation– a scale



June 27 - July 1, 2005


Examples of Coordinate Systems

• Magnet

• Accelerator

• Earth



June 27 - July 1, 2005


Fiducialization Procedures for the ALS Ring Magnets and the Booster Synchrotron Girders

Jack Tanabe, Roderich Keller and Ted LauritzenLawrence Berkeley Laboratory, Berkeley, CA 94720, U. S. A.

presented at IWAA90

The mechanical coordinate system of each magnet is defined with respect to the mechanical features of the core. The cores for each magnet are made from precision stamped laminations and the upper surfaces of assembled magnets and parting planes of two and three piece magnets are precisely parallel to the central axis of the magnet. Moreover, great care is taken in assembling the core segments so that the axes of each core segment are precisely normal to the planes of the laminations. Thus, the u/w plane (defining u’ and w’, the pitch and roll) of the magnet is determined from the upper plane of the assembled core or the parting plane of a core segment.



June 27 - July 1, 2005


PEPII Coordinate Systems

There are five coordinate systems commonly used in the PEP-II Interaction Region.– PEP-II Coordinate System

Origin: Center of the PEP-II rings. Elevation above mean sea level = 65.986 m

+X: Direction from the PEP-II ring center through the center of IR-12 (nominally north).

+Y: Up, parallel to gravity vector through the ring center. +Z: Direction from the PEP-II ring center through the symmetry point

(mid-arc) in Arc 3 (nominally east).

– PEP Control Line – IR Reference Frame – Collision Axis Coordinate System – Detector Coordinate System



June 27 - July 1, 2005


PEPII Coordinate Systems



June 27 - July 1, 2005


Earth Coordinate Systems

• Origin:– Center of Mass– Point at the surface

• Orientation:– Axis of rotation– Vertical



June 27 - July 1, 2005


Vertical and Axis of Rotation

ΛΦ

g

x

y

z plumb line

level surfacesW = const.

P



June 27 - July 1, 2005


Geocentric Systems

y

sinhe-1N

sincoshN

coscoshN

z

y

x

2

2

322

2

sine1

e1 aM

2122sine1

aN

The principal radii of curvature:

in the plane of the meridian: M

in the plane of the prime vertical: Nx

zP

0

h

φλ

p



June 27 - July 1, 2005


How to realize such CS?

• Today the center of mass and the axis of rotation of the Earth are well observed. The IERS was established as the International Earth Rotation Service in 1987 by the International Astronomical Union and the International Union of Geodesy and Geophysics and it began operation on 1 January 1988. In 2003 it was renamed to International Earth Rotation and Reference Systems Service. The primary objectives of the IERS are to serve the astronomical, geodetic and geophysical communities by providing the following:

– The International Celestial Reference System (ICRS) and its realization, the International Celestial Reference Frame (ICRF).

– The International Terrestrial Reference System (ITRS) and its realization, the International Terrestrial Reference Frame (ITRF).

– Earth orientation parameters required to study earth orientation variations and to transform between the ICRF and the ITRF.

– Geophysical data to interpret time/space variations in the ICRF, ITRF or earth orientation parameters, and model such variations.

– Standards, constants and models (i.e., conventions) encouraging international adherence.

• Before only local astronomical observations were possible and the common method was to decide that the geodetic and the astronomical latitude and longitude were set at one point. For the US, this was Meades Ranch in Kansas. So there were a wide variety of origin, orientation and ellipsoidal parameters.



June 27 - July 1, 2005


Ellipsoids

• Before satellite geodesy– In the USA: Clarke 1866

• a = 6378206.4 m b = 6356584 m• f-1 = 294.9786982

– In France: Clarke 1880 • a = 6378249 m b = 6356515 m

– World: Hayford 1909/1924 • a = 6378388 m f-1 = 297.0 b = 6356912 m

• Now: GRS80 (Geodetic Reference System of 1980)– a = 6378137 m– b = 6356752.3141 m f-1 = 298.25722101



June 27 - July 1, 2005


International Terrestrial Reference System http://www.iers.org/iers/earth/itrs/

• The ITRS definition fulfills the following conditions: – 1. It is geocentric, the center of mass being defined for the whole earth, including

oceans and atmosphere. – 2. The unit of length is the metre (SI). This scale is consistent with the TCG time

coordinate for a geocentric local frame, in agreement with IAU and IUGG (1991) resolutions. This is obtained by appropriate relativistic modelling.

– 3. Its orientation was initially given by the BIH orientation at 1984.0. – 4. The time evolution of the orientation is ensured by using a no-net-rotation

condition with regards to horizontal tectonic motions over the whole earth.

• The ITRS is realized by estimates of the coordinates and velocities of a set of stations observed by VLBI, LLR, GPS, SLR, and DORIS. Its name is International Terrestrial Reference Frame.

• The ITRS can be connected to the International Celestial Reference System (ICRS) by use of the IERS Earth Orientation Parameters (EOP).

• Reference: http://tai.bipm.org/iers/conv2003/conv2003.html

http://www.iers.org/iers/earth/itrs/



http://tai.bipm.org/iers/conv2003/conv2003.html



June 27 - July 1, 2005


Transformation between CRF and TRFH-W and all, pages 25, 30-34

CRFPNSM

TRF xRRRRx

precessionfor matrix rotation R

nutationfor matrix rotation R

timesiderealfor matrix rotation R

motionpolar for matrix rotation R

P

N

S

M

where

X01

X03 = X3

X1

eart

h’s

rota

tion

axi

s

pole

Greenw

ich

geocenterec

liptic

equator

vernal equinox

Θ0



June 27 - July 1, 2005


PrecessionH-W and all, page 31-32

ζR R zRR 323P

cossinsincossin

sinsincoscossincossinsincoscoscossin

sincoscossinsincoscossinsincoscoscos

RP zzzzz

zzzzz

X01(to)

mean equator (to)

mean equator (t)

X01(t)

X02(t0)

X03(t0)X0

3(t)

Eo

E90º + z

90º - ζ



June 27 - July 1, 2005


NutationH-W and all, pages 32-33

Δψ

mean equator

true equatorΔε

ε

E

Et

eclip

tic

εR ΔψR ΔεεRR 131N

1sin

1cos

sincos1

R N



June 27 - July 1, 2005


Rotation and Polar MotionH-W and all, pages 33-34

CEP

mean G

reenwich

meridian

y

yp

xp

CIO

1yx

y10

x01

yR xRR

pp

p

p

p1p2M

timesideralGreenwich Θ0



June 27 - July 1, 2005


Other Geocentric Systems

• NAD83 is the North American Datum of 1983. It is the horizontal control datum for the United States, Canada, Mexico, and Central America, based on a geocentric origin and the Geodetic Reference System 1980. It is based on the adjustment of 250,000 points including 600 satellite Doppler stations which constrain the system to a geocentric origin.

– http://www.ngs.noaa.gov/faq.shtml#WhatDatum

• WGS84 is the World Geodetic System of 1984. It is the reference frame used by the U.S. Department of Defense (DoD) and is defined by the National Imagery and Mapping Agency (NIMA formerly the Defense Mapping Agency).

– WGS 84 was defined in January 1987 using Doppler satellite surveying techniques. It was used as the reference frame for broadcast GPS ephemerides beginning January 23, 1987.

– At 0000 GMT January 2, 1994, WGS 84 was upgraded in accuracy using GPS measurements. The formal name then became WGS 84 (G730) since the upgrade date coincided with the start of GPS Week 730. It became the reference frame for broadcast orbits on June 28, 1994.

– At 0000 GMT September 30, 1996 (the start of GPS Week 873), WGS 84 was redefined again and was more closely aligned with International Earth Rotation Service (IERS) Terrestrial Reference Frame (ITRF) 94. It is now formally called WGS 84 (G873). WGS 84 (G873) was adopted as the reference frame for broadcast orbits on January 29, 1997.

http://www.ngs.noaa.gov/faq.shtml#WhatDatum



June 27 - July 1, 2005


Plane Coordinateshttp://gge.unb.ca/Research/GeodesyGroup/tutorial/tutorial.htm

http://gge.unb.ca/Research/GeodesyGroup/tutorial/tutorial.htm

Documents

Coordinate Systems