Transcript No Slide Title

SEDRIS Spatial Reference Model (SRM)
It’s not just coordinate systems any more!
Presented at the
SEDRIS Technology Conference
September 28-30, 1999
Arlington, VA
by
Dr. Paul A. Birkel, MITRE Corporation
&
Dr. Ralph Toms, SRI International
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Tutorial Organization
I.
Simulation Interoperability
from a Physical Environment Interface Perspective
II. Introduction
to the SEDRIS Spatial Reference Model (SRM)
III. Earth Reference Models (ERMs)
and Coordinate Frameworks
IV. Map Projections
V.
Augmented Map Projections
VI. Selection of a Coordinate Framework
for Models and Simulations
VII. ERM Geometry
VIII. Computational Considerations
IX. Interface Specification
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Section I
Simulation Interoperability from a
Physical Environment Interface
Perspective
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The Interfaces Between Each Class of
Simulations Makes Interoperability Difficult
• The traditional hierarchy of models does not work very well.
Complex, labor intensive interface processing is required. Although
there are software based linkages for connecting dissimilar models this
does not guarantee that it is meaningful to do so.
• A major problem is the lack of commonality of interfaces due to the
use of different earth reference models and coordinate systems. This
leads to inconsistent positions and environmental representations.
• The lack of commonality is intensified by traditional aggregation
policies.
• It is much easier, but certainly not pro forma, to interface between
entity level simulation classes.
• The interface between aggregated constructive simulations and entity
level simulations has been referred to as the “Grand Canyon”.*
*Blumenthal, Bridging the Grand Canyon, 1997
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A Spectrum of Constructive, Live and Virtual
Capabilities Support Training, Planning & Analysis
Aggregated
Constructive
Large
S
c
o
p
e
Low
• Closed for
Analysis
• Interactive for
Training
• Units normally
are Bns or higher
• Environmental
effects aggregated
• Usually
deterministic
not stochastic
• Attrition often
based on Lanchester
differential equations
• Entity states not
maintained
Entity Level
Constructive
Tactical
Simulators
Ranges and
Live Exercises
• Closed
for analysis
• Interactive for
training &
planning
• Player inputs
tactics or uses
SAF
• Generally
“stochastic”
• More detailed
environment
• Acquisition
is modeled
• 2-D Graphics
with 3-D Disp.
• Virtual, always
interactive
• Principally for
training
• 3-D graphics
with 2-D
display
• Acquisition by
humans
• Principally for
distributed play
• OPFOR is
primarily
constructive
• Protocol stds.
enforced
• Real Platforms
• Emulated
wpns. delivery
• Simulated BDA
• Acquisition by
HITL in real
environment
• Training and
procurement
support
Low
Level of Detail
War
• Simulations
used for fire
control solutions,
sensor pointing,
guidance and
control
High
Lots of detail does not necessarily imply accuracy, fidelity or functional completeness.
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A Spectrum of Constructive, Live and Virtual
Capabilities Support Training, Planning & Analysis
Aggregated Closed
& Interactive
Constructive
JTF/Theater
S
c
o
p
e
Constructive
Seminar CEM,THUNDER,
RSAS, METRIC
War
ENWGS, JTLS
Games
TACWAR, ITEM
Corps/
CVBG/ARV
CBS
TACSIM
Div
VIC
EAGLE
Bde/
CVW
Entity
Level
SPECTRUM ELAN
(MOOTW)
BBS
MTWAS
War
Live Exercises
and
Ranges
Simulators
NTC
CASTFOREM
JCM
Janus
JTS
JTCTS
JCATS
CMTC/
JRTC
EADSIM
Operational
Planners
Bn/Wng
MODSAF/
CCTT SAF
Co/Sqdn
SIMNET
Plt
CCTT
DFIRST
SOFNET
Sqd
Low
Engineering Simulations
Embedded
Fire Control,
Guidance, etc.
High
Level of Detail
Lots of detail does not necessarily imply accuracy, fidelity or functional completeness.
R. Toms 2/18/97
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The Interface Canyons
• This is a notional view of the scope of the problem.
Aggregated Constructive
Entity Level
Constructive
Low
Simulators
Level of Detail
Ranges & Live
Exercises
War
High
A consistent SRM is critical to addressing the interface problem.
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Section II
Introduction to the SEDRIS
Spatial Reference Model
(SRM)
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The Synthetic Natural Environment
(SNE) begins with a location ...
Systems
The void ...
Systems, where?
The SNE starts with locating
your forces; sometimes that’s
about all you could afford in
legacy simulations.
Systems, and what else?
The SNE continues with defining the
context within which forces engage;
and that context can advantage, or
disadvantage, forces ...
• Defining and using a consistent spatial reference framework is
critical for M&S interoperability
– Military models (men, material, …)
– Environmental data, models, phenomena
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Why is a Spatial Reference Model
(SRM) Needed?
• Traditionally the M&S community has not been consistent in the
treatment of models of the earth and related coordinate systems.
• Consistency is required for joint distributed simulation in order to:
– achieve a reasonably level playing field,
– to support meaningful VV&A.
• A number of different earth reference models (ERMs) are currently
employed and this affects:
– representation of the environment in simulations & authoritative data
bases.
– dynamics formulations, both kinematics and kinetics (movement).
– acquisition modeling and processing (inter-visibility).
• Approximations in coordinate transformation algorithms made to
reduce processing time may introduce additional inconsistencies.
• An SRM is needed to promote lossless and accurate transformations.
• A nomenclature inconsistency evolves when there is no SRM
– For example, how do these variables relate?
Altitude, elevation, height, geodetic height, ellipsoidal height, orthometric
height, height above sea level, height above mean sea level, terrain height,
pressure altitude, temperature altitude, nap of the earth, ...
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SRM Requirements
• Completeness
– Must include coordinate frameworks in common usage.
– Must tie those systems together into a common framework.
– Must educate the system developer.
• E.g., What’s a horizontal datum? A vertical datum?
• Accuracy
– Generally higher than required for C4ISR systems.
– Typically better than 1 cm. up past geosynchronous orbit.
• Performance
– Never fast enough!
– Many environmental data sets dominated by location data
• Therefore efficient interconversion key to meeting 72 hour
“ready-to-run” mandate.
– Federate costs for distributed simulation using heterogeneous
coordinate systems can be substantial (e.g., 20% or more).
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Suggested References
1. “Handbook for Transformation of Datums, Projections, Grids and
Common Coordinate Systems”, U.S. Army Corps of Engineers,
Topographic Engineering Center, TEC-SR-7, 1998.
2. “Department of Defense World Geodetic System 1984”, National Imagery
and Mapping Agency, Third Edition, TR8350.2, 1997.
3. “Geodesy for the Layman”, National Imagery and Mapping Agency,
on-line at http://164.214.59/geospatial/products/GandG/geolay/toc.htm.
4. Richard H. Rapp, “Geometric Geodesy Part I & II”, The Ohio State
University, Dept. of Geodetic Science & Surveying, 1993.
5. John P. Snyder, “Map Projections -- A Working Manual”, U.S.
Geological Survey Professional Paper 1395, 1987.
6. Paul D. Thomas, “Conformal Projections in Geodesy and Cartography”,
U.S. Department of Commerce, Coast and Geodetic Survey, Special
Publication 251.
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Section III
Earth Reference Models (ERMs)
and Coordinate Frameworks
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Earth Reference Models (ERMs)
• Earth’s shape
–
–
–
–
Sphere: Used by meteorology community (see JMCDM).
Ellipsoid: 21 (as per NIMA TR 8350.2).
Mathematical approximations are not the earth.
ERMs do not include the natural environment (smooth surfaces).
• Horizontal Datum
– 200+ (as per NIMA TR 8350.2)
– Common interoperability problem in C4ISR community
• Vertical Datum
– Many ...
•
•
•
•
Earth Reference Model (i.e., sphere/ellipsoid)
WGS-84 Geoid
MSL (local)
Others (e.g., NAVD-88, EGM-96)
– Interoperability problem in littoral regions.
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SEDRIS Reference Ellipsoids
Reference Ellipsoid
Code
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Name
Semi-major Axis
Inverse Flattening
a (meters)
f--1 (unitless)
AA
Airy 1830
6377563.396
299.3249646
AM
Modified Airy
6377340.189
299.3249646
AN
Australian National
6378160
298.25
BN
Bessel 1841 (Namibia)
6377483.865
299.1528128
BR
Bessel 1841 (Ethiopia Indonesia Japan Korea)
6377397.155
299.1528128
CC
Clarke 1866
6378206.4
294.9786982
CD
Clarke 1880
6378249.145
293.465
EA
Everest (India 1830)
6377276.345
300.8017
EB
Everest (Sabah & Sarawak)
6377298.556
300.8017
EC
Everest (India 1956)
6377301.243
300.8017
ED
Everest (W. Malaysia 1969)
6377295.664
300.8017
EE
Everest (W. Malaysia & Singapore 1948)
6377304.063
300.8017
FA
Modified Fischer 1960
6378155
298.3
HE
Helmert 1906
6378200
298.3
HO
Hough 1960
6378270
297
IN
International 1924
6378388
297
KA
Krassovsky 1940
6378245
298.3
RF
Geodetic Reference System 1980 (GRS 80)
6378137
298.257222101
SA
South American 1969
6378160
298.25
WD
WGS 72
6378135
298.26
WE
WGS 84
6378137
298.257223563
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SEDRIS Reference Spheres
Reference Sphere
Code
NOG
NOGAPS
6371000.00
COA
COAMPS
6371229.00
MMR
MMR (AFWA)
6370000.00
ACM
ACMES
6371221.30
MFE
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Name
Radius
(meters)
MultiGen
Flat Earth
6366707.02
MOD_T
Tropical
6378390.00
MOD_M
Midlatitude
6371230.00
MOD_S
Subartic
6356910.00
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Horizontal Datums
Geodetic Datums
Code
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Reference
Ellipsoid
Name
Transformation
Parameters
Applicable Extent
of Datum
delta X
delta Y
delta Z
(meters
)
(meters)
(meters)
ADI-A Adindan
CD
-165
-11
206 Ethiopia
ADI-B Adindan
CD
-161
-14
205 Sudan
ADI-C Adindan
CD
-123
-20
220 Mali
ADI-D Adindan
CD
-128
-18
224 Senegal
ADI-E
Adindan
CD
-118
-14
218 Burkina Faso
ADI-F
Adindan
CD
-134
-2
ADI-M Adindan
CD
-166
-15
AFG
Afgooye
KA
-43
-163
AIA
Antigua Island Astro
1943
CD
-270
13
AIN-A Ainel Abd 1970
IN
-150
-250
...
...
..
...
...
... ...
W72
WGS 1972
WD
0
0
0 Global Definition
W84
WGS 1984
WE
0
0
0 Global Definition
WAK
Wake Island Astro 1952
IN
276
-57
YAC
Yacare
IN
-155
171
37 Uruguay
ZAN
Zanderij
IN
-265
120
-358 Suriname
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210 Cameroon
204 MEAN FOR Ethiopia,
Sudan
45 Somalia
62 Antigua (Leeward
Islands)
-1 Bahrain
149 Wake Atoll
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Standardizing Coordinate Frameworks
• A coordinate framework is a combination of an
ERM and a coordinate system.
• Not much hope in getting everyone to use one
coordinate framework.
• Some coordinate systems, combinations of systems
and ERMs are natural to a specific application.
• Trend is towards standard ERMs, but not there yet.
• Some hope of reducing the number of coordinate
systems required to a manageable set.
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SRM Reference Frames
Arbitrary
LSR
Local Space Rectangular Coordinate System
Earth-Surface, Global
GDC
Geodetic Coordinate System
Earth-Centered, Earth-Fixed
GCC
Geocentric Coordinate System
M
Mercator Projected Coordinate System (PCS)
OM
Oblique Mercator PCS
PS
Polar Stereographic PCS
UPS
Universal Polar Stereographic PCS
LCC
EC
Lambert Conformal Conic PCS
Equidistant Cylindrical PCS
TM
Transverse Mercator PCS
UTM
Universal Transverse Mercator PCS
GCS
LTP
Global Coordinate System
Local Tangent Plane Coordinate System
GM
GEI
Geomagnetic Coordinate System
GSE
Geocentric Solar Ecliptic Coordinate System
GSM
Geocentric Solar Magnetospheric Coordinate System
SM
Solar Magnetic Coordinate System
Earth-Surface, Projected
Earth-Surface, Local
(Topocentric)
Earth-Centered, Rotating
(Inertial & Quasi-Inertial)
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Geocentric Equatorial Inertial Coordinate System
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Ellipsoidal Earth Reference Model (ERM)
Geometry & Notation
• P(X, Y, Z) or P(Ø, , h)
Z
• P(W,Z)
Z
h
Pe
Ze
Pe
ø

Y
ø
We
W
Where W2 = X2 + Y2
X
Generic ERM & notation
Meridian plane geometry
• Ellipsoids are standard in current geodesy practice.
• For SNE data modeling, spheres are often used to simplify
dynamics equations.
• Geocentric coordinates (GCC) are defined by the point P(X, Y, Z).
• Geodetic coordinates (GDC) are defined by the point P(Ø, , h).
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Latitude, Longitude and Height
for Ellipsoids & Spheres
• P(X, Y, Z) or P(Ø, , h)
Z
• P(X, Y, Z) or P( , , h)
Z
h
hos
Pe
Pe
ø

Y
X
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
Y
X
For ellipsoids
For spheres
Latitude, longitude and geodetic height
are defined as per this diagram. The line
through P is perpendicular to the ellipsoid.
Longitude  is generally referenced to the
Prime Meridian.
Longitude is the same as for the ellipsoidal case,
is the geocentric latitude & hos is height above
the sphere. The line through P is perpendicular to
the sphere. In mapping, charting and geodesy
spherical ERMs are almost never used.
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North/South Cross Section of the
Geoid, Ellipsoid and the Earth’s Surface
Earth's Physical Surface
Geoid
H
Geoid
Ellipsoid
Geoid Separation: + N
Ellipsoid
Geoid Separation: - N
The geoid is a gravity
equipotential surface selected to
match mean sea level as well as
possible.
• h is the geodetic height
• H is the orthometric height
• N is the separation of the geoid
For more on this see NIMA’s “Geodesy for the Layman” at
http://164.214.2.59/geospatial/products/GandG/geolay/toc.htm
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Gravitational Field and the
Geoid, Ellipsoid and the Earth’s Surface
Gravity vector depends on:
latitude, longitude,
and H (or h)
Earth's Physical Surface
Geoid
Gravity potential
results in a
gravity field
•P
H
h
Geoid
Geoid Separation: + N
Ellipsoid
• h is the geodetic height
• H is the orthometric height
• N is the separation of the geoid
Geoid Separation: - N
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Section IV
Map Projections
Map projections were invented to support paper
map development ---- a long time ago.
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Development of Surfaces to Generate Maps
Developable Surfaces
A cone or cylinder can
be cut and laid out flat.
Non-developable Surfaces
QuickTime™ and a
Phot o - JPEG decompressor
are needed to see t his picture.
Qui ckTime™ and a
Photo - JPEG decompressor
are needed to see this pictur e.
The surface of an ellipsoid cannot be cut so it will lie flat
without tearing or stretching.
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Map Projections associate points on the
surface of an ERM with points on an X-Y plane
• A map projection is a mathematical transformation from a
three dimensional ellipsoidal or spherical ERM surface onto
a two dimensional plane.
Y
X
• Since spheres and ellipsoids are not developable, distortions must occur.
• Note that the transformation is from three to two dimensions and
there is no vertical axis in the plane.
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A Simple Projection
Projection from the point N of all points on the circle onto a line.
N
Note that the red points do not map!
s
s
s
• Note the stretching of the length of the arc s after the projection.
• The concept of a projection can be extended to projecting the points
on the surface of an ERM onto a plane.
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Cylindrical Projections
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Planar Projections
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Stereographic Projection
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Conic Projections
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Simple Conic Projection*
A simple conic projection.
A simple conic map of the northern hemisphere.
* From: N. Bowditch, American Practical Navigator, U.S. Navy Hydrographic Office, 1966 Ed.
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Cylindrical Projection*
* From: N. Bowditch, American Practical Navigator, U.S. Navy Hydrographic Office, 1966 Ed.
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Mercator Projection
A Mercator projection is
a cylindrical projection.
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Oblique Mercator Projection*
A Transverse Mercator (TM) Projection is defined when
the cylinder is parallel to the equator.
* From: N. Bowditch, American Practical Navigator, U.S. Navy Hydrographic Office, 1966 Ed.
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Transverse Mercator Map of the
Western Hemisphere*
• In geodetic coordinates
the origin is at (0, -π/2, 0)
• The longitude of the origin
is shown as 90º W.
* From: N. Bowditch, American Practical Navigator,
U.S. Navy Hydrographic Office, 1966 Ed.
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Transverse Mercator Map - the Meridians
are curved and the Spacing between them is stretched*
Y
• In geodetic coordinates
the origin is at (0, -π/2, 0)
• The longitude of the origin
is shown as 90º W.
X
* From: N. Bowditch, American Practical Navigator,
U.S. Navy Hydrographic Office, 1966 Ed.
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Universal Transverse Mercator
• Widely used for paper maps by the U. S. Army.
• Defined on six degree wide regions with 60 origins on the equator.
• A grid numbering scheme is used to define the Military Grid Reference System.
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Lines of Constant Heading on a Mercator Map*
• This line is called a rhumb line or loxodrome.
• Mercator projection is used for maritime navigation.
* Adapted from: N. Bowditch, American Practical Navigator, U.S. Navy Hydrographic Office, 1966 Ed.
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Rhumb Line or Loxodrome on a Globe*
* From: N. Bowditch, American Practical Navigator, U.S. Navy Hydrographic Office, 1966 Ed.
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Great Circle Arc between Moscow and Washington D.C.*
This is a Mercator map.
This is an oblique Mercator map for a sphere with the
central meridian on the great circle arc between cities.
* From: N. Bowditch, American Practical Navigator, U.S. Navy Hydrographic Office, 1966 Ed.
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Many Map Projections Are Conformal
• Conformal means that the mathematical transformation preserves angles.
• What does this mean?
• The curves between A & B and B & C are on the surface of an ERM.
• The projected curves (not necessarily the same shape) are on the plane.
• For a conformal transformation the angles ABC and abc are the same.
A
Y
B
a
C
b
c
X
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Transformations may or may not
change Geometrical Relationships
• The taxonomy for classifying mathematical
transformations is complex and there are a lot of types:
– isometric, linear, bi-linear, conformal, orthogonal, affine, isomorphic, ...
• For SEDRIS, two classifications are sufficient for
transformations associated with earth referenced
coordinate frameworks.
– Geometry Invariant (GI):
that class of transformations
between coordinate reference frames that
do not distort geometrical relationships.
– Non-Geometry Invariant (NGI):
that class of transformations
between coordinate reference frames that
distort some geometrical relationships.
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SRM Coordinate System Relationships
for an Ellipsoidal ERM
Transformations between
Transformations
map projections
among earth referenced
and these are NGI.
3D systems are GI.
Global
Coordinate
System
Local Tangent Plane
Map projections (2D)
Universal Transverse
Mercator
Transverse Mercator
Mercator
Geocentric
Geodetic
Coordinate
System
Oblique Mercator
Lambert Conformal
Conic
Geomagnetic
Geocentric Equatorial Inertial
Geocentric Solar Ecliptic
Geocentric Solar Magnetospheric
Solar Magnetic
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Polar Stereographic
Universal Polar
Stereographic
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Section V
Augmented Map Projections
These are used, used and used
but,
they are distorted, distorted and distorted.
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Augmented Map Projections
• Models & simulations usually require three dimensions.
• Some frameworks are three dimensional by definition.
• Map projections are commonly augmented with a
vertical axis to create a three dimensional system.
• Various vertical measures are used, such as mean sea
level height, orthometric height, geodetic height, pressure
altitude and others.
• This practice adds additional geometric distortions.
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Projection from 2-D to 1-D and then
Augmentation with a Vertical Axis causes distortions
N
Note that the red points do not map
ss
s
s
Distance is distorted by the projection
Z
X
An augmented projection produces
another 2D system. Note that there
are now two distortions with respect
to the original rectangular system.
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This process can be extended to the
3D case but even if the projection is
conformal, elevation angles are not
preserved.
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Distance Distortion
can be mitigated, somewhat
N
Note that the red points do not map
s
s
s
s
O
Scale here = 1
Scale here < 1
On the green line the average distortion is reduced
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Augmented UTM
effectively “flattens the ERM”
• Augmented UTM (AUTM) was often used in legacy ground combat
•
simulations because, under simplifying assumptions, unit dynamics
equations take on a simple form that minimizes processing time.
For AUTM several distortions are introduced, especially at the higher
latitudes.
plane of the projection
90o East
•
•
•
UTM plane inset to
reduce average distortion
(scale factor .9996 at the
central meridian)
•
Central meridian
The results of such distortions may not be so apparent when all simulations
involved use UTM. However, in a federation involving real world coordinate
systems the distortions may become evident. Use of AUTM increases visibility,
causes the battle to prosecute too fast, leads to an uneven playing field and is not
recommended for use in joint simulations.
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Transverse Mercator Map - Revisited*
Y
• In geodetic coordinates
the origin is at (0, -π/2, 0)
• The longitude of the origin
is shown as 90º W.
X
* From: N. Bowditch, American Practical Navigator,
U.S. Navy Hydrographic Office, 1966 Ed.
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Augmented 3D coordinate systems
are often used by modelers
These are all augmented map projection based coordinate systems.
Spherical Universal Transverse
Mercator, add vertical axis
Universal Transverse
Mercator, add vertical axis
Spherical Transverse
Mercator,
add vertical axis
Transverse Mercator,
add vertical axis
Oblique Mercator,
add vertical axis
Mercator,
add vertical axis
Geodetic
Coordinate
System
Spherical
Coordinate
System
Lambert Conformal
Conic 1 & 2,
add vertical axis
Spherical Mercator,
add vertical axis
Spherical Lambert
Conformal Conic 1 & 2,
add vertical axis
Polar Stereographic,
add vertical axis
Spherical Polar
Stereographic, add
vertical axis
Equidistant Cylindrical,
add vertical axis
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Spherical Oblique
Mercator,
add vertical axis
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Spherical Equidistant
Cylindrical, add
vertical axis
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Section VI
Selection of a Coordinate Framework
for Models and Simulations
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Modelers often prefer
Cartesian Coordinate Frameworks
• Dynamics equations can be “simplified”so that they are cheaper
computationally. Velocity and acceleration components generally
do not contain trigonometric functions.
• In Cartesian real world systems straight lines are linear functions.
– Shortest distance paths are straight lines.
– The Euclidean metric requires only a square root operation.
• In other coordinate systems minimum distance paths may be nontrivial to compute.
• Segments of ellipses lead to elliptic integrals.
– Shortest path on the surface of an ellipsoid is a geodesic (not an arc
segment of an ellipse).
• The Earth and its natural environment are modeled with an ERM
and a SNE.
– In this model shortest distance (or time) paths are not unique, and
– Almost always are not geodesics or straight lines.
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Non-real World 3D Systems are
Often Used. Why ?
Newton’s second law of motion for a fixed or inertial reference frame is
d aVa
M
dt
For a rotating system Va can be written
Va  V    r
In which case the relative acceleration becomes
dV
  p  2  V  g a    (  r)  F
dt
Transforming these to any system of earth related coordinate systems
will result in very complex equations. However, under certain
assumptions they can take on a relatively simple form, particularly in
rectangular coordinates for a spherical ERM. An example would be
using Augmented Lambert Conformal Conic projected off of a sphere
as a framework for a numerical weather prediction model.
The assumptions involved must be understood when using such data to
create an SNE data set in a simulation coordinate system.
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Relationships Between
Coordinate Systems and Simulations
Rectangular Inertial Coordinates
Rotating
Inertial
RECTANGULAR
Mapping, Charting,
Geodesy & Imagery
Quasi-Inertial
CURVILINEAR
geocentric (ECEF)
GCS, topocentric
geodetic,
spherical
Dynamics
(No direct force
modeling)
F = m •a
Earth Reference Models
archaic
2-dimensional
distorted
3-dimensional
more distorted
spherical
ellipsoidal
Kinetics
current practice
Map
Projections
Earth Reference
Models & Datums
TM, UTM, LCC,
PS, UPS & others
geocentric
or geodetic
coordinates
Augmented
Map Projections
Add vertical
axis to projections
Earth gravity
modeling
(Geoid)
Models in
augmented
projection
coordinates
position, velocity
geometry distorted
( ModSAF,
many SNE
models, ...)
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Kinematics
simplified gravity
model to specify
“down”
Correct coordinate
systems, simplifications
may be made to dynamic
models and ERMs to
increase performance
Models in
augmented
projection
coordinates
Correct coordinate
systems, simplifications
may be made to dynamic
models and ERMs to
increase performance
(JointSAF, ModSAF,
CCTT, SNE models...)
(Janus, JCATS
VIC, EAGLE,
CBS, JTLS, ...)
(JointSAF, ModSAF,
CCTT ...)
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MCG&I and Dynamics Modeling
Rectangular Inertial Coordinates
Rotating
Mapping,
Charting,
Geodesy &
Imagery
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Inertial
RECTANGULAR
Geocentric (ECEF)
GCS, Topocentric
Quasi-Inertial
CURVILINEAR
Geodetic
Spherical
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Dynamics
56
MCG&I
Mapping,
Charting,
Geodesy &
Imagery
Earth Reference Models
archaic
2-dimensional
distorted
3-dimensional,
more distorted
Spherical
Ellipsoidal
current practice
Map Projections
Earth Reference
Models & Datums
TM, UTM, LCC,
PS, UPS & others
Geocentric or Geodetic
Coordinates
Augmented
Map Projections
Add vertical
axis to projections
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Dynamics Modeling
Dynamics
F = m•a
No direct force
Kinematics modeling
Kinetics
Earth Gravity
Modeling
(Geoid)
position,
velocity,
geometry
distorted
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Models in
augmented
projection
coordinates
(ModSAF,
many SNE
models, ...)
simplified gravity
model to specify
“down”
Correct coordinate
systems, simplifications may be made
to dynamic models &
ERMs to increase
performance
(JointSAF, ModSAF,
CCTT, SNE models...)
Models in
augmented
projection
coordinates
(Janus, JCATS
VIC, EAGLE,
CBS, JTLS, ...)
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Correct coordinate
systems, simplifications may be made
to dynamic models &
ERMs to increase
performance
(JointSAF, ModSAF,
CCTT ...)
58
Section VII
ERM Geometry
• Distance measures may have several meanings.
• Angular measures and direction may also have
several meanings.
• True north is referenced to the north pole of the ERM
– which is different than the north pole in the real world,
– and different from the magnetic north pole.
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Euclidean Distance in a Rectangular Framework
z
b•
y
a•
x
• The Euclidean distance in a rectangular space is given by
D = (xa - xb)2 + (ya - yb)2 +(za - zb)2 .
• A straight line is the minimum distance path between a and b.
• The variables are linearly related by the parametric equations
of a line.
• The equation of the line segment from a to b is given by
x = xa + (x - xb) µ
y = ya + (y - yb) µ
z = za + (z - zb) µ where the parameter µ is in [0, 1].
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Distance on a Spherical ERM
z
B
A
O
y
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• The blue curve is a small circle arc
generated by the intersection of a plane,
that does not contain O, with the sphere.
• The red curve is a great circle arc
generated by the intersection of a plane,
containing O, with the sphere.
• The great circle arc is the minimum
distance path on the surface of a sphere.
• The length of the great circle arc can be
x
computed using spherical trigonometry
which involves trigonometric functions.
• The normals at A & B are coplanar.
• When an environmental model is added,
e. g., terrain, minimum distance paths on a
sphere are almost never great circle arcs.
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Normal Section of an Ellipsoid
NA
NB
A
B
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• The gray plane is tangent to the
ellipsoid at A.
• The normal NA at A is orthogonal to
the tangent plane.
• The red plane, containing NA is the
normal plane at A.
• The curve from A to B, the intersection
of the ellipsoidal surface and the the
normal plane, is called the normal
section.
• The normals at A & B are generally
not coplanar.
• Construction of a normal plane at B
which passes through A will generate a
different normal section.
• Normal planes may not contain the
origin.
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Distance on an Ellipsoidal ERM
z
B
A
O
y
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• The red curve A to B is the normal section
at A.
• The red curve B to A is the normal section
at B.
• The green curve, the minimum distance
path on the surface of an ellipsoid, is
called a geodesic.
• There is no plane that contains the geodesic.
• The length of a geodesic is an incomplete
x
elliptic integral.
• The curvature in the figure is exaggerated
for the purpose of exposition. For short
distances all three curves have nearly the
same length.
• When an environmental model is added,
e. g. terrain, minimum distance paths on an
ellipsoid are almost never geodesics.
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Bearing Angles (Ellipsoidal ERM)
• From the previous slide there are three possible definitions for
defining the bearing angle: ß1, ß2 and ß3.
N
Top view
ß1
A
ß2
B
ß3
• The angle ß2 is preferred because of its unique definition but it is
computationally complex to compute.
• For distances under a hundred kilometers all three angles are
nearly the same.
• The normal section from A to B is generally used for bearing
computations in practice.
• Note that A and B are on the surface of the ERM.
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Bearing when a point P is above
the ERM Surface (Ellipsoid)
• This is an edge view of
the tangent plane
(purple) and the normal
NA
plane (red) at A.
• A and B are on the ERM A
• NB
surface (B is below the
B•
horizon).
• The green and blue
arrows represent the
normals NA and NB.
• This is defining geometry
for bearing angle.
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• In this case the normal plane
is rotated to the point P.
• A and B are on the ERM
surface (B is below the
horizon).
• The green and blue arrows
represent the normals NA
and NB.
• This does not define the
same bearing angle.
• Mathematical adjustments
need to be made to compute
bearing. This process is
called “reduction”.
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NA
•P
NB
B
•
•
A
65
Once the Environment is included,
there are many Feasible Paths
•A
•B
• Minimum distance (or time) paths are much harder to
determine and are not unique.
• This is true for land, maritime and airborne assets.
• Sometimes paths are constrained by roads, trafficability,
political boundaries, hostile sites, underwater structure
and many others.
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Siting a Solid Cube on an ERM
• Excavation is not allowed because
there is no environmental model.
The ERM is a mathematical
concept.
• The cube can only contact the
ERM at one common point.
• Once the terrain is modeled there
are many ways to site a cube by
using excavation.
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Transforming a Cube from an
Augmented Conformal Map Projection to GDC
• Every point on the base (red) is on
the plane.
• Interior angles of the base are 90°.
• All other interior angles are 90°.
• All sides are of the same length.
• The vertical sides are parallel planes.
• The cube is a convex hull.
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• Only points in the red region are
transformed by the map projection.
• Every point on the base (red) is on the
ERM.
• Since the projection is conformal interior
angles of the base are 90°.
• All other angles are generally not 90°.
• In general none of the sides are equal.
• The vertical lines are not parallel and are
not even coplanar.
• The 3D volume is no longer convex.
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Siting a cube when the terrain model is present
• These might not be very acceptable.
• These might be more acceptable, but only the user can decide
how to site objects.
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Transformation of Long Linear Structures
from a Projection-based system to GDC
Z
X
Y
• In map projection coordinates.
• Side view in geodetic coordinates.
Y
• Top view.
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X
• Top view in geodetic coordinates.
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Representation of a Region
is Likely to be Discrete
•
•
•
•
• •
•
becomes
•
•
•
•
• •
•
• •
•
•
•
•
•
•
• • •
• •
•
• Regular boundaries are generally curved after a transformation.
• Will likely require intermediate points to determine boundaries to
support accurate interpolation.
• Each point will be transformed by a computationally expensive
coordinate transformation.
• This can happen in either direction: source system to target system
and conversely.
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Section VIII
Computational Considerations
Accuracy
Errors
Efficiency
Testing
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Why is Accuracy Needed
for Coordinate Transformations?
• In real world, it has been difficult to measure positions on the
surface of the earth to better than 1 meter
– This situation is changing due to new technology developments.
– GPS now can achieve absolute accuracy of about 21 cm (SEP 90%)
over large regions.
• Real-world weapons applications mostly use relative coordinate
systems
– Dynamically correct location errors by using on-board sensors.
• In the simulation environment, relative coordinate systems must be
accurately portrayed.
• Mixing of live & synthetic environments has special accuracy
requirements.*
• Mission planning, rehearsal, & conduct of real operations have
situation-dependent accuracy requirements.
* Lucha, G. V., On the Consequences of Neglecting Measurement
Accuracy Issues in Live and Virtual interactions”, SIW Spring 1997
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Shooting at a Target in the
Real World (Relative Coordinate System)
• Set up paper target with aim-point approximately 1000 meters
away
• Shoot N rounds
• Measure miss distances using bullet holes on paper target &
compute CEP or some other measure of accuracy
Never used any precise location or environmental data
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Simulation of Shooting at a Target
(Simulation of a Relative Coordinate System)
• Select rectangular (topocentric) coordinate system origin at shooter.
• Define position location of target & shooter.
– Both with target plane oriented perpendicular to LOS
• Develop an aiming model with random inputs.
• Define shoot time T.
• Integrate bullet trajectory in time from T until it pierces the plane
of the target (need air temperature, density, speed of sound, wind,
etc.).
– Will have to access geodetic system for each of these.
– Will need an iterative scheme to get the impact point.
• Compute radial miss at target plane impact.
Any errors made in any of the position-location computations,
including those needed to compute the correct environmental
parameters, can & will dilute the accuracy of the result.
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Error Sources in
Coordinate Transformation Software
•
There are many possible error sources in development of
software for coordinate transformations.
- Did we mention distortions yet?
- Truncation errors are due to the use of a finite number of terms in an
infinite series.
- Approximation error is due to approximating one function with another
(simpler to compute) function.
- Iteration error is the due to the use of a finite number of iterations in an
iterative process.
- Formulation errors are due to the analyst developing the incorrect
equations or logic. This includes improper formulations near singular
points, improper treatment of signs, incorrect treatment of units and others.
- Implementation errors are are due to improper coding of the correct
formulation.
- Roundoff errors are those caused by finite word length computers.
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Mathematical Definition Of Error
• Position error
– If (X,Y,Z) is the true value of a point and (XA, YA, ZA) the approximate value.
– Use the Euclidean metric E2 = [(X- XA)2 + (Y- YA)2 + (Z- ZA)2] to determine an error
ball of radius E. For two dimensional systems, set the Zs to 0.
• Angular error
– There two types of geodetic points: (lat, lon, h) or for the map projections (lat, lon, 0).
– Except for UTM, the forward transformations are exact.
• General approach
– Generate a known set of points {(lat, lon, h)}.
– When the exact transformation is available, generate the corresponding exact set of
points {(X,Y,Z)}.
– E in terms of position errors can always be calculated in two or three dimensions.
• UTM is a special case because there is no exact transformation in either direction
– Angular measures can be converted to distance measures using s = r•ø.
– Again, start with a known set of exact points {(lat, lon)}.
– Given the approximate point (latA, lonA) compute e2 = [(lat - latA)RM]2 + [( lon lonA)RN]2
– Where RN is the radius of curvature in the prime vertical and RM is the radius of
curvature in the meridian.
– e is the (approximate) radius of the positional error ball.
– When the angular errors are small, the error measure e is nearly E.
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Computational Methods
• Analytic (closed form) solutions
• Taylor’s series
• Iteration
• Approximation methods (curve fitting the inverse)
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Efficient evaluation of special functions
common to Coordinate Transformations
• Developed a generic machine independent timing capability for SEDRIS.
• Transcendental functions are frequently occurring and expensive to compute.
• Eliminate them using identities or in-line approximations.
• Relative cost of evaluating common functions on modern workstation below.
20
15
10
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pow
log
exp
sin
cos
tan
asin
acos
atan
atan2
f0
f1
f2
sqrt
0
int*
float*
int/
float/
double/
5
Int =
float =
double=
int+
float+
double+
Time w/ respect to double multiply
25
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Normalized floating
multiply = 1
79
Analytic (closed form) solutions
• Often can not be found.
• When available, always used in mapping, charting
and geodesy applications.
• Advantages:
- They provide exact reference values,
- Useful for derivations,
- ERM parameters embedded as variables.
• Disadvantages
- Usually involve many transcendental functions,
- Generally least efficient,
- Usually too accurate (wasted computation time).
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Taylor/Maclaurin Series Methods
• Taylor series always exists for the type
of transformations considered.
• Advantages
- Very useful for derivations,
- Can be used to get theoretical error bounds,
- ERM parameters embedded as variables,
- Power series can be inverted to yield the inverse function.
• Disadvantages
- Successive terms get very complex and hard to derive,
- Truncation error tends to grow rapidly away from expansion point,
- Almost always not as efficient as curve fitting or direct approximation.
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Iterative Methods
• Coordinate transformation calculations can
almost always be formulated using an iterative method.
• Advantages
- ERM parameters embedded as variables,
- If properly formulated they are almost always more efficient for the
same accuracy as a power series,
- Usually the expressions involved are compact.
• Disadvantages
- Convergence rate depends on formulation and quality of the initial value,
- Initial guess must be efficient to compute,
- Efficient to compute stopping criterion needed,
- Even when this approach works, the direct approximation is almost
always more efficient.
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Direct Approximation
of a Function or its Inverse
• Advantages:
- By far the most efficient approach for a fixed accuracy criterion,
- Maximum flexibility in efficiency vs. accuracy tradeoffs,
- Can use piecewise approximation for increased efficiency.
• Disadvantages:
- More difficult to include ERM parameters as variables,
- Requires non-linear approximation tools to get coefficients,
- Requires considerable analyst experience and intuition.
For s = sin(ø)

 1  s in  2
f  ,s in   

 1  s in 
becomes
2
2 2
32
4 2
a1  a2 s  a3 s  a4 s  a5 s 

f s,   
1 b2 2 s  b32 s2  b43 s2  b5 4 s2 
and once the ERM is fixed, becomes f s  C1 
(C2  C3s)
(C4  s(C5  s))
where the Ci are constants.
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Error in Power Series Expansions
• Typical plot of the truncation error in one dimension.
error for non-alternating series
error for alternating series
Error
D1
Distance from expansion point
• Checking at one point, say near D1, can give you unwarranted warm feelings.
• When the series expansions three dimensional (the usual case for coordinate
system applications) there may be many more zeroes of the error function.
The error must be evaluated at all or nearly all points
where the series is expected to be used.
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First three terms of the TM series for x
1
For     0 , t  tan( ) , N  (1   sin ( ) ,    2 cos2 ( ),
E is the truncation error, and where  is a small parameter of the ERM,
the series expansion for X is given by,
2
2
X
3 cos3 ( )
  cos( ) 
(1 t 2   2 ) 
N
6
 cos ( )
2
4
2
2 2
4
6
4 2
6 2
(5  18t  t  14  58t   13  4  64 t  24 t ) 
120
7
 cos7 ( )
(61 479t 2  179t 4  t 6 ) + E
5040
(from P. D. Thomas)
5
5
This is an alternating series which depends on latitude, longitude,
longitude of the origin, and the eccentricity of the ERM.
Note that E is small only if delta is small
and the latitude does not get too near 90º.
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Authoritative Sources sometimes
appear to Disagree
• These are truncated series expansions for Transverse Mercator
taken from three different sources are shown below.
X
3 cos3 ( )
  cos( ) 
(1 t 2   2 ) 
N
6
5 cos5 ( )
2
4
2
2
(5  18t  t  72 - 58 )
120
X
3 cos3 ( )
  cos( ) 
(1  t 2   2 ) 
N
6
5 cos5 ( )
(5  18t 2  t 4  14 2 - 58 2t 2 )
120
(from Snyder)
(from TEC SR-7)
X
3 cos3 ( )
  cos( ) 
(1  t 2   2 ) 
N
6
5 cos5 ( )
(5  18t 2  t 4  14 2  58 2t 2  13 4  4 6  64 4t 2  24 6t 2 )
120
(from P.D. Thomas)
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Error Analysis
and Resolution of Disagreements
• For power series expansion, it is possible to compute an exact upper
bound on the error and this should be done.
• Testing should be done over the entire region of application using a
very large and dense set of test points. This will insure sampling away
from zeros of the error function. This will also validate the analytical
error analysis and help find possible coding errors.
• The three representations shown on the previous slide, while appearing
to be different, may be equivalent in the following sense:
- Suppose that the test region is bounded and closed,
- That is R = [min. lon. , max. lon.] x [0 , max. lon.] x [min. e , max. e]
where e is the eccentricity of the set of meridian ellipses being considered,
- Then generate a dense grid on the three dimensional region R,
- Then compare the three alternatives at each grid point,
- If the maximum absolute difference between them is less than some
acceptable value (say one millimeter) then they have equivalent accuracy.
Proper error analysis requires a very dense set of test points in R.
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Effect of Small Errors
“For small regions all map projections are the same.”*
• Take such statements with a grain of salt!
- The original Bowditch book is very old,
- In navigation - meters of error are considered small,
- In geodesy a meter is small (to some),
- In GPS applications a meter is big,
- To a gunner who aims at the turret ring of an enemy tank
a meter is really big,
- It only takes a small curvature distortion to hide or uncover
a target and in some terrains this is a frequent occurrence,
- In some real time embedded systems small errors may accumulate.
• The application domain determines what is small, in M&S very
accurate representations of spatial reference frameworks are
required for VV&A support and to promote a level playing field.
* From: N. Bowditch, American Practical Navigator, U.S. Navy Hydrographic Office, 1966 Ed.
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Coordinate Transformations Project Goals
• Provide accurate, robust, transformation services
software and supporting documentation
– All software to be in ANSI Standard C and algorithm designs to be
as portable as is practical
– Design to all 21 ERMs in the SEDRIS SRM
– Accuracy goal of one mm position error in transformation
algorithms
– NIMA DTCC 4.1 results were the gold standard for transformations
in the 2.5 release
– Perform timing and accuracy comparisons of new S/W against
DTCC 4.1 procedures
• Develop user interface consistent with SEDRIS API
Project goals were all met for 2.5 release
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Transformation Performance
• Substantive testing indicates that the accuracy requirement
was met for all baseline transformations (error ball less than
1 millimeter).
• The DTCC4.1 and SEDRIS inner loop procedures were timed and
compared in terms of the ratio DTCC4.1/SEDRIS. This was done
on a Dual Pentium 90 in ANSI C, optimizing compiler on, in-line
function option on. Results are summarized for the -10 to +50
kilometer region.
Transformation
GCC to GDC
GDC to GCC
GDC to UTM
UTM to GDC
GDC to PS
PS to GDC
GDC to LCC2
LCC2 to GDC
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Time Ratio
(SEDRIS divided
by DTCC4.1)
0.520
0.938
0.524
0.551
0.650
0.650
0.711
0.800
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Section IX
Interface Specification
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Interface Specification
• Manipulating the retrieval coordinate system
– SE_GetCoordinateSystemParameters
– SE_UseDefaultWorldCoordinateSystemParameters
– SE_SetCoordinateSystemParameters
For use with the SEDRIS Read API
• Initialization of coordinate system parameters
– SE_CreateCoordConversionConstants
• Convert a coordinate
– SE_ConvertCoordToGivenSystem
– SE_ConvertCoordToGivenSystemWithoutBoundaryChecking
• Free coordinate system parameters
– SE_FreeCoordConversionConstants
For use independent of the SEDRIS Read API
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Coordinate System Parameters
• GCC: <none>
• GDC: horizontal_datum, vertical_datum, elevation_units
• PS: horizontal_datum, vertical_datum, latitude (pole), longitude (y axis),
•
•
•
•
•
•
•
•
•
•
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central_scale_factor, z_units
LCC: horizontal_datum, vertical_datum, north_parallel,
south_parallel, latitude (origin), longitude (origin), z_units
TM: horizontal_datum, vertical_datum, latitude (origin),
longitude (origin), x_offset, y_offset, central_scale_factor, z_units
UTM: horizontal_datum, vertical_datum, x_offset, y_offset, z_units
EC: horizontal_datum, vertical_datum, latitude (origin),
longitude (origin), x_offset, y_offset, scale_factor, z_units
LTP: horizontal_datum, vertical_datum, latitude (origin),
longitude (origin), azimuth, x_offset, y_offset, z_units
GCS: x_offset, y_offset
GM: epoch, earth_radius
GEI: epoch
GSE, GSM, SM: <none>
LSR: up_direction, forward_direction
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Coordinate Specifications
•
•
•
•
•
•
•
•
•
•
•
•
•
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GCC: { x, y, z }
GDC: { latitude, longitude, elevation }
PS: { x, y, z }
LCC: { x, y, z }
TM: { x, y, z }
UTM: { zone, hemisphere, x, y, z }
EC: { x, y, z }
LTP: { x, y, z }
GCS: {cell_id, x, y, z }
GM: { gm_latitude, gm_longitude, radius }
GEI: { right_ascension, declination, radius }
GSE, GSM, SM: { latitude, longitude, radius }
LSR: { x, y, z }
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Controlling the Retrieval Coordinate System
• Obtain retrieval coordinate system in current use
 Default: transmittal coordinate system of <Environment Root>
extern SE_STATUS_CODE_ENUM SE_GetCoordinateSystemParameters
(
SE_OBJECT
synth_env,
SE_COORD_SYSTEM_PARAMETERS *coord_system_params_out_ptr
/* out parameter */
);
• (Re)Set to transmittal coordinate system of <Environment Root>
extern void SE_UseDefaultWorldCoordinateSystemParameters(void);
• Set to specific retrieval coordinate system
extern SE_STATUS_CODE_ENUM SE_SetCoordinateSystemParameters
(
SE_COORD_SYSTEM_PARAMETERS new_coordinate_system_parameters
);
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Coordinate System Parameter Initialization
• Initialize once, use many times, de-initialize
• Setup includes calculation of many derived parameters
– Derivations depend on both in- and out- coordinate system
• Multiple conversion-pairs can be setup at the same time
• Provides handle to conversion-specific data structure
– Context for all subsequent conversions
extern SE_COORD_STATUS_CODE_ENUM SE_CreateCoordConversionConstants
(
const SE_COORD_SYSTEM_PARAMETERS *source_system_parameters_ptr,
const SE_COORD_SYSTEM_PARAMETERS *destination_system_parameters_ptr,
SE_CONVERT_COORD_SYSTEM_PAIR *convert_coord_system_pair_ptr
/* out parameter */
);
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Convert a Coordinate
extern SE_COORD_STATUS_CODE_ENUM SE_ConvertCoordToGivenSystem
(
const SE_CONVERT_COORD_SYSTEM_PAIR convert_coord_system_pair,
const SE_COORD_3D
*original_coord_ptr,
SE_COORD_3D
*new_coord_ptr
/* out parameter */
);
extern SE_COORD_STATUS_CODE_ENUM
SE_ConvertCoordToGivenSystemWithoutBoundaryChecking
(
SE_CONVERT_COORD_SYSTEM_PAIR convert_coord_system_pair,
const SE_COORD_3D
*original_coord_ptr,
SE_COORD_3D
*new_coord_ptr
/* out parameter */
);
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Free Coordinate System Parameters
extern SE_COORD_STATUS_CODE_ENUM SE_FreeCoordConversionConstants
(
SE_CONVERT_COORD_SYSTEM_PAIR
to_free
);
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Conversion Support Matrix
L
S
R
G
D
C
G
C
C
M
O
M
T
M
U
T
M
P
S
L
C
C
E
C
Local Space
Rectangular (LSR)
X
2.5
2.5
2.6
2.6
2.5
2.5
2.5
2.5
2.6
Geodetic (GDC)
--
X
Geocentric (GCC)
--
2.5
X
M
--
2.6
2.6
X
OM
--
2.6
2.6
2.6
X
TM
--
2.5
2.5
2.6
2.6
X
UTM
--
2.5
2.5
2.6
2.6
2.5
X
PS
--
2.5
2.5
2.6
2.6
2.5
2.5
X
LCC
--
2.5
2.5
2.6
2.6
2.5
2.5
2.5
X
EC
--
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
X
GCS
--
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
X
LTP
--
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
2.6
X
GM
--
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
GEI
--
GSE
--
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
X
GSM
--
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
SM
--
From \ To
Projected
Topo
centric
Inertial &
Quasi
Inertial
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Projected
Topo
centric
G L
C
T
S
P
Inertial & QuasiInertial
G
G
G
G
S
M E
S
S
M
I
E
M
2.7
2.7
2.6
3.0
3.0
X
X
X
X
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Local Space
Rectangular
3D
SRM Conversion Support Status
Earth Referenced 3D Systems
Local Space
Rectangular
2D
2D Systems (Map Projections)
Universal Transverse
Mercator
Global
Coordinate
System
Local Tangent Plane
Transverse Mercator
Oblique Mercator
Geocentric
(ECEF)
Geodetic
Coordinate
System
Mercator
Lambert Conformal Conic
Geomagnetic
Geocentric Equatorial Inertial
Geocentric Solar Ecliptic
Geocentric Solar Magnetospheric
Solar Magnetic
2.5: Defined, but no conversion support
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2.5: Undefined (in API)
Polar Stereographic
Equidistant Cylindrical
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Augmented Map-Projection Based
Coordinate Systems Support Status
Spherical Universal Transverse
Mercator, add vertical axis
Universal Transverse
Mercator, add vertical axis
Spherical Transverse
Mercator,
add vertical axis
Transverse Mercator,
add vertical axis
Oblique Mercator,
add vertical axis
Mercator,
add vertical axis
Geodetic
Coordinate
System
Spherical
Coordinate
System
Lambert Conformal
Conic 1 & 2,
add vertical axis
Spherical Mercator,
add vertical axis
Spherical Lambert
Conformal Conic 1 & 2,
add vertical axis
Polar Stereographic,
add vertical axis
Spherical Polar
Stereographic, add
vertical axis
Equidistant Cylindrical,
add vertical axis
Spherical Equidistant
Cylindrical, add
vertical axis
2.5: Defined, but no conversion support
2.5: Undefined (in API)
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Spherical Oblique
Mercator,
add vertical axis
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SRM “without” SEDRIS
• Preparation and publication of SISO product nomination
–
–
–
–
Released 2-5-99
Completed 30 day review period for public comment
Nomination approval by SAC on hold
Standards Development Group (SDG) will then be formed
• A self-contained copy of the conversions API is available
– Contains source and header files and a README.
• README lists the files and gives a brief description of each.
• No build environment
– Intended for those who are interested in the coordinate conversions
capability of the SEDRIS API as a separate capability.
– Contact: [email protected]
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