3451288469a4f7450c899a98181f6171.ppt

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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 9/29/99 SEDRIS Technology Conference 1

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 9/29/99 SEDRIS Technology Conference 2

Section I Simulation Interoperability from a Physical Environment Interface Perspective 9/29/99 SEDRIS Technology Conference 3

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 9/29/99 SEDRIS Technology Conference 4

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 Tactical Simulators Constructive • 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 Low Level of Detail Ranges and Live Exercises • Real Platforms • Emulated wpns. delivery • Simulated BDA • Acquisition by HITL in real environment • Training and procurement support 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. 9/29/99 SEDRIS Technology Conference 5

A Spectrum of Constructive, Live and Virtual Capabilities Support Training, Planning & Analysis Aggregated Closed & Interactive Constructive JTF/Theater S c o p e Entity Level Constructive Seminar CEM, THUNDER, RSAS, METRIC War Games ENWGS, JTLS TACWAR, ITEM Corps/ CVBG/ARV VIC EAGLE Live Exercises and Ranges Simulators CBS TACSIM Div War Bde/ CVW SPECTRUM ELAN (MOOTW) BBS MTWAS 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 9/29/99 SEDRIS Technology Conference 6

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. 9/29/99 SEDRIS Technology Conference 7

Section II Introduction to the SEDRIS Spatial Reference Model (SRM) 9/29/99 SEDRIS Technology Conference 8

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 9/29/99 SEDRIS Technology Conference 9

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, . . . 9/29/99 SEDRIS Technology Conference 10

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 C 4 ISR 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). 9/29/99 SEDRIS Technology Conference 11

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, TR 8350. 2, 1997. 3. “Geodesy for the Layman”, National Imagery and Mapping Agency, on-line at http: //164. 214. 59/geospatial/products/Gand. G/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. 9/29/99 SEDRIS Technology Conference 12

Section III Earth Reference Models (ERMs) and Coordinate Frameworks 9/29/99 SEDRIS Technology Conference 13

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 C 4 ISR 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. 9/29/99 SEDRIS Technology Conference 14

SEDRIS Reference Ellipsoids 9/29/99 SEDRIS Technology Conference 15

SEDRIS Reference Spheres Reference Sphere Code Name Radius (meters) NOGAPS 6371000. 00 COAMPS 6371229. 00 MMR (AFWA) 6370000. 00 ACMES 6371221. 30 MFE Multi. Gen Flat Earth 6366707. 02 MOD_T 6378390. 00 MOD_M Midlatitude 6371230. 00 MOD_S 9/29/99 Tropical Subartic 6356910. 00 SEDRIS Technology Conference 16

Horizontal Datums 9/29/99 SEDRIS Technology Conference 17

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. 9/29/99 SEDRIS Technology Conference 18

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) 9/29/99 Geocentric Equatorial Inertial Coordinate System SEDRIS Technology Conference 19

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 W 2 = X 2 + Y 2 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). 9/29/99 SEDRIS Technology Conference 20

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 Y X X For ellipsoids 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. 9/29/99 For spheres 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. SEDRIS Technology Conference 21

North/South Cross Section of the Geoid, Ellipsoid and the Earth’s Surface Earth's Physical Surface Geoid H 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/Gand. G/geolay/toc. htm 9/29/99 SEDRIS Technology Conference 22

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 Separation: + N Ellipsoid • h is the geodetic height • H is the orthometric height • N is the separation of the geoid Geoid Separation: - N 9/29/99 SEDRIS Technology Conference 23

Section IV Map Projections Map projections were invented to support paper map development ---- a long time ago. 9/29/99 SEDRIS Technology Conference 24

Development of Surfaces to Generate Maps Developable Surfaces A cone or cylinder can be cut and laid out flat. Non-developable Surfaces The surface of an ellipsoid cannot be cut so it will lie flat without tearing or stretching. 9/29/99 SEDRIS Technology Conference 25

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. 9/29/99 SEDRIS Technology Conference 26

A Simple 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. 9/29/99 SEDRIS Technology Conference 27

Cylindrical Projections 9/29/99 SEDRIS Technology Conference 28

Planar Projections 9/29/99 SEDRIS Technology Conference 29

Stereographic Projection 9/29/99 SEDRIS Technology Conference 30

Conic Projections 9/29/99 SEDRIS Technology Conference 31

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. 9/29/99 SEDRIS Technology Conference 32

Cylindrical Projection* * From: N. Bowditch, American Practical Navigator, U. S. Navy Hydrographic Office, 1966 Ed. 9/29/99 SEDRIS Technology Conference 33

Mercator Projection A Mercator projection is a cylindrical projection. 9/29/99 SEDRIS Technology Conference 34

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. 9/29/99 SEDRIS Technology Conference 35

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. 9/29/99 SEDRIS Technology Conference 36

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. 9/29/99 SEDRIS Technology Conference 37

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. 9/29/99 SEDRIS Technology Conference 38

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. 9/29/99 SEDRIS Technology Conference 39

Rhumb Line or Loxodrome on a Globe* * From: N. Bowditch, American Practical Navigator, U. S. Navy Hydrographic Office, 1966 Ed. 9/29/99 SEDRIS Technology Conference 40

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. 9/29/99 SEDRIS Technology Conference 41

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 9/29/99 SEDRIS Technology Conference 42

Transformations may or may not change Geometrical Relationships • The taxonomy for classifying mathematical transformations is complex and there 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. 9/29/99 SEDRIS Technology Conference 43

SRM Coordinate System Relationships for an Ellipsoidal ERM Transformations between Transformations map projections among earth referenced and these are NGI. 3 D systems are GI. Global Coordinate System Local Tangent Plane Map projections (2 D) Universal Transverse Mercator Geocentric Geodetic Coordinate System Oblique Mercator Lambert Conformal Conic Geomagnetic Geocentric Equatorial Inertial Geocentric Solar Ecliptic Geocentric Solar Magnetospheric Solar Magnetic 9/29/99 Polar Stereographic Universal Polar Stereographic SEDRIS Technology Conference 44

Section V Augmented Map Projections These are used, used and used but, they are distorted, distorted and distorted. 9/29/99 SEDRIS Technology Conference 45

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. 9/29/99 SEDRIS Technology Conference 46

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 s s Distance is distorted by the projection Z X An augmented projection produces another 2 D system. Note that there are now two distortions with respect to the original rectangular system. 9/29/99 This process can be extended to the 3 D case but even if the projection is conformal, elevation angles are not preserved. SEDRIS Technology Conference 47

Distance Distortion can be mitigated, somewhat N Note that the red points do not map s s O Scale here = 1 Scale here < 1 On the green line the average distortion is reduced 9/29/99 SEDRIS Technology Conference 48

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 90 o 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. 9/29/99 SEDRIS Technology Conference 49

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. 9/29/99 SEDRIS Technology Conference 50

Augmented 3 D coordinate systems are often used by modelers These are all augmented map projection based coordinate systems. Spherical Universal Transverse Mercator, add vertical axis Spherical Transverse Mercator, add vertical axis Oblique 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 9/29/99 Spherical Oblique Mercator, add vertical axis SEDRIS Technology Conference Spherical Equidistant Cylindrical, add vertical axis 51

Section VI Selection of a Coordinate Framework for Models and Simulations 9/29/99 SEDRIS Technology Conference 52

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. 9/29/99 SEDRIS Technology Conference 53

Non-real World 3 D Systems are Often Used. Why ? Newton’s second law of motion for a fixed or inertial reference frame is For a rotating system Va can be written In which case the relative acceleration becomes 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. 9/29/99 SEDRIS Technology Conference 54

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 ( Mod. SAF, many SNE models, . . . ) 9/29/99 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 (Joint. SAF, Mod. SAF, CCTT, SNE models. . . ) (Janus, JCATS VIC, EAGLE, CBS, JTLS, . . . ) (Joint. SAF, Mod. SAF, CCTT. . . ) SEDRIS Technology Conference 55

MCG&I and Dynamics Modeling Rectangular Inertial Coordinates Rotating Mapping, Charting, Geodesy & Imagery 9/29/99 Inertial RECTANGULAR Geocentric (ECEF) GCS, Topocentric Quasi-Inertial CURVILINEAR Geodetic Spherical SEDRIS Technology Conference Dynamics 56

MCG&I Mapping, Charting, Geodesy & Imagery Earth Reference Models archaic Spherical Ellipsoidal current practice Map Projections 2 -dimensional distorted 3 -dimensional, more distorted Earth Reference Models & Datums TM, UTM, LCC, PS, UPS & others Geocentric or Geodetic Coordinates Augmented Map Projections Add vertical axis to projections 9/29/99 SEDRIS Technology Conference 57

Dynamics Modeling Dynamics F = m • a No direct force Kinematics modeling Kinetics Earth Gravity Modeling (Geoid) position, velocity, geometry distorted 9/29/99 Models in augmented projection coordinates (Mod. SAF, many SNE models, . . . ) simplified gravity model to specify “down” Correct coordinate systems, simplifications may be made to dynamic models & ERMs to increase performance (Joint. SAF, Mod. SAF, CCTT, SNE models. . . ) Models in augmented projection coordinates (Janus, JCATS VIC, EAGLE, CBS, JTLS, . . . ) SEDRIS Technology Conference Correct coordinate systems, simplifications may be made to dynamic models & ERMs to increase performance (Joint. SAF, Mod. SAF, 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. 9/29/99 SEDRIS Technology Conference 59

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]. 9/29/99 SEDRIS Technology Conference 60

Distance on a Spherical ERM z B A O y 9/29/99 • 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 almost never great circle arcs. SEDRIS Technology Conference 61

Normal Section of an Ellipsoid NA NB A B 9/29/99 • 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 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. SEDRIS Technology Conference 62

Distance on an Ellipsoidal ERM z B A O y 9/29/99 • 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. SEDRIS Technology Conference 63

Bearing Angles (Ellipsoidal ERM) • From the previous slide there are three possible definitions for defining the bearing angle: ß 1, ß 2 and ß 3. N Top view ß 1 A ß 2 B ß 3 • The angle ß 2 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. 9/29/99 SEDRIS Technology Conference 64

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. 9/29/99 • In this case the normal plane is rotated to the point P. NA • A and B are on the ERM • P surface (B is below the NB horizon). A • • The green and blue arrows • represent the normals NA B and NB. • This does not define the same bearing angle. • Mathematical adjustments need to be made to compute bearing. This process is called “reduction”. SEDRIS Technology Conference 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. 9/29/99 SEDRIS Technology Conference 66

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. 9/29/99 SEDRIS Technology Conference 67

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. 9/29/99 • 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 3 D volume is no longer convex. SEDRIS Technology Conference 68

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. 9/29/99 SEDRIS Technology Conference 69

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. 9/29/99 X • Top view in geodetic coordinates. SEDRIS Technology Conference 70

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 • • 9/29/99 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. SEDRIS Technology Conference 71

Section VIII Computational Considerations Accuracy Errors Efficiency Testing 9/29/99 SEDRIS Technology Conference 72

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 9/29/99 SEDRIS Technology Conference 73

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 9/29/99 SEDRIS Technology Conference 74

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. 9/29/99 SEDRIS Technology Conference 75

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 due to improper coding of the correct formulation. - Roundoff errors are those caused by finite word length computers. 9/29/99 SEDRIS Technology Conference 76

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 E 2 = [(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 (lat. A, lon. A) compute e 2 = [(lat - lat. A)RM]2 + [( lon. A)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. 9/29/99 SEDRIS Technology Conference 77

Computational Methods • Analytic (closed form) solutions • Taylor’s series • Iteration • Approximation methods (curve fitting the inverse) 9/29/99 SEDRIS Technology Conference 78

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 9/29/99 pow log exp sin cos tan asin acos atan 2 0 f 1 f 2 sqrt 5 Int = float = double= int+ float+ double+ int* float* int/ float/ double/ Time w/ respect to double multiply 25 SEDRIS Technology Conference 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). 9/29/99 SEDRIS Technology Conference 80

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. 9/29/99 SEDRIS Technology Conference 81

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. 9/29/99 SEDRIS Technology Conference 82

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) becomes and once the ERM is fixed, becomes where the Ci are constants. 9/29/99 SEDRIS Technology Conference 83

Error in Power Series Expansions • Typical plot of the truncation error in one dimension. error for non-alternating series error for alternating series Error D 1 Distance from expansion point • Checking at one point, say near D 1, 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. 9/29/99 SEDRIS Technology Conference 84

First three terms of the TM series for x For , , E is the truncation error, and where is a small parameter of the ERM, the series expansion for X is given by, +E (from P. D. Thomas) This is an alternating series which depends on latitude, 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. 9/29/99 SEDRIS Technology Conference 85

Authoritative Sources sometimes appear to Disagree • These are truncated series expansions for Transverse Mercator taken from three different sources are shown below. (from Snyder) (from TEC SR-7) (from P. D. Thomas) 9/29/99 SEDRIS Technology Conference 86

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 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. 9/29/99 SEDRIS Technology Conference 87

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. 9/29/99 SEDRIS Technology Conference 88

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 9/29/99 SEDRIS Technology Conference 89

Transformation Performance • Substantive testing indicates that the accuracy requirement • 9/29/99 was met for all baseline transformations (error ball less than 1 millimeter). The DTCC 4. 1 and SEDRIS inner loop procedures were timed and compared in terms of the ratio DTCC 4. 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. SEDRIS Technology Conference 90

Section IX Interface Specification 9/29/99 SEDRIS Technology Conference 91

Interface Specification • Manipulating the retrieval coordinate system – SE_Get. Coordinate. System. Parameters – SE_Use. Default. World. Coordinate. System. Parameters – SE_Set. Coordinate. System. Parameters For use with the SEDRIS Read API • Initialization of coordinate system parameters – SE_Create. Coord. Conversion. Constants • Convert a coordinate – SE_Convert. Coord. To. Given. System. Without. Boundary. Checking • Free coordinate system parameters – SE_Free. Coord. Conversion. Constants For use independent of the SEDRIS Read API 9/29/99 SEDRIS Technology Conference 92

Coordinate System Parameters • GCC:

Coordinate Specifications • • • • 9/29/99 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 } SEDRIS Technology Conference 94

Controlling the Retrieval Coordinate System • Obtain retrieval coordinate system in current use - Default: transmittal coordinate system of

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 9/29/99 SEDRIS Technology Conference 96

Convert a Coordinate 9/29/99 SEDRIS Technology Conference 97

Free Coordinate System Parameters 9/29/99 SEDRIS Technology Conference 98

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. 6 2. 5 2. 6 Geodetic (GDC) -- X Geocentric (GCC) -- 2. 5 X M -- 2. 6 X OM -- 2. 6 X TM -- 2. 5 2. 6 X UTM -- 2. 5 2. 6 2. 5 X PS -- 2. 5 2. 6 2. 5 X LCC -- 2. 5 2. 6 2. 5 X EC -- 2. 6 2. 6 X GCS -- 2. 7 2. 7 X LTP -- 2. 6 2. 6 X GM -- 2. 7 2. 7 GEI -- GSE -- 3. 0 3. 0 X GSM -- 3. 0 3. 0 SM -- From To Projected Topo centric Inertial & Quasi Inertial 9/29/99 Projected Topo centric G L C T S P Inertial & Quasi. Inertial G G S M E S S M I E M 2. 7 2. 6 3. 0 X X SEDRIS Technology Conference 99

Local Space Rectangular 3 D SRM Conversion Support Status Earth Referenced 3 D Systems Local Space Rectangular 2 D 2 D 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 9/29/99 2. 5: Undefined (in API) Polar Stereographic Equidistant Cylindrical SEDRIS Technology Conference 100

Augmented Map-Projection Based Coordinate Systems Support Status Spherical Universal Transverse Mercator, add vertical axis Spherical Transverse Mercator, add vertical axis Oblique 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) 9/29/99 Spherical Oblique Mercator, add vertical axis SEDRIS Technology Conference 101

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] org 9/29/99 SEDRIS Technology Conference 102