Navigation coordinate systems

Transcription

1 Lecture 3 Navigation coordinate systems Topic items: 1. Basic Coordinate Systems. 2. Plane Cartesian Coordinate Systems. 3. Polar Coordinate Systems. 4. Earth-Based Locational Reference Systems. 5. Reference Ellipsoids. 6. Geodetic Datums. 7. Global systems 8. Rhumb and Great Circle Line.

2 Basic Coordinate Systems There are many basic coordinate systems. These systems can represent points in twodimensional or three-dimensional space. Ren. Decartes ( ) introduced systems of coordinates based on orthogonal (right angle) axes. These two and three-dimensional systems used in analytic geometry are often referred to as Cartesian systems. Similar systems based on angles from baselines are often referred to as polar systems. 2

3 Plane Coordinate Systems Two-dimensional coordinate systems are defined with respect to a single plane (Cartesian Coordinates in a Plane) 3

4 Polar Coordinates in a Plane and Conversion from Polar to Cartesian Coordinates 4

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6 Tree-Dimensional Cartesian Coordinates X, Y, Z 6

7 Three-Dimensional Polar Coordinates (φ, θ, r) 7

8 Conversion of Three Dimensional Polar Coordinates (φ, θ, r) to Cartesian Coordinates (X, Y, Z) 8

9 Earth-Based Locational Reference Systems Reference systems and map projections extend the ideas of Cartesian and polar coordinate systems over all or part of the earth. Map projections portray the nearly spherical Earth in a two-dimensional representation. Earth-based reference systems are based on various models for the size and shape of the Earth. Earth shapes are represented in many systems by a sphere However, precise positioning reference systems are based on an ellipsoidal earth and complex gravity models. 9

10 Reference Ellipsoids Ellipsoidal earth models are required for precise distance and direction measurement over long distances. Ellipsoidal models account for the slight flattening of the Earth at the poles. This flattening of the earth's surface results at the poles in about a twenty kilometer difference between an average spherical radius and the measured polar radius of the earth. The best ellipsoidal models can represent the shape of the earth over the smoothed, averaged sea-surface to within about one-hundred meters. Reference ellipsoids are defined by either: semi-major (equatorial radius) and semi-minor (polar radius) axes, or the relationship between the semi-major axis and the flattening of the ellipsoid (expressed as its eccentricity). 10

11 Ellipsoidal Parameters WGS-84 World Geodetic System 84 is an earth-fixed global reference frame, including an earth model Primary parameters: Angular velocity ω = x 10-8 rad s -1 Secondary parameters: Geocentric gravitation constant GM= km 3 s -2 Coefficients of an earth gravity field model (EGM) of degree and order n=m=180 11

12 Selected Reference Ellipsoids Ellipse Semi-Major Axis Flattening Airy Bessel Clarke Clarke Everest Fischer 1960 (Mercury) Fischer G R S G R S G R S Hough International Krassovsky South American WGS WGS WGS WGS PZ (IFRS-2000)

13 What is a Geodetic Datum? Definitions 1 (adopted by international community) Geodetic reference system (GRS): Conceptual idea of an earth-fixed Cartesian system (X, Y, Z) Geodetic reference frame: Practical realization of a geodetic reference system by observations and mesyrments Global GRS: Origin Earth s centre of mass Z-axis: Coincides with mean rotational axis of Earth X-axis: Mean meridian plane of Greenwich and to Z-axis Y-axis: Orthogonal 13

14 What is a Geodetic Datum? Definitions 2 Local GRS: Origin and orientation of axes is arbitrary Geodetic datum: Minimum set of parameters required to define location and orientation of local system with respect to the global reference system/frame Note: The term datum is often used when one actually means reference frame 14

15 What is a Geodetic Datum? Z WGS 84 Cartesian datum: Z D Y Ellipsoidal datum: Z b a X, Y X X Set of shift parameters: X, Y, Z Set of rotation angles α, β, Ɣ Scale factor parameter: ϻ Y Additionally the shape of the meridian ellipse of mean earth ellipsoid is defined Memo rule: Ellipsoidal Datum=Cartesian Datum + Shape of Earth Ellipsoid 15

16 Geodetic Datums Precise positioning must also account for irregularities in the earth's surface due to factors in addition to polar flattening. Topographic and sea-level models attempt to model the physical variations of the surface: The topographic surface of the earth is the actual surface of the land and sea at some moment in time. Aircraft navigators have a special interest in maintaining a positive height vector above this surface. Sea level can be thought of as the average surface of the oceans, though its true definition is far more complex. Specific methods for determining sea level and the temporal spans used in these calculations vary considerably. Tidal forces and gravity differences from location to location cause even this smoothed surface to vary over the globe by hundreds of meters. 16

17 Gravity models and geoids are used to represent local variations in gravity that change the local definition of a level surface Gravity models attempt to describe in detail the variations in the gravity field. The importance of this effort is related to the idea of leveling. Plane and geodetic surveying uses the idea of a plane perpendicular to the gravity surface of the earth which is the direction perpendicular to a plumb bob pointing toward the center of mass of the earth. Local variations in gravity, caused by variations in the earth's core and surface materials, cause this gravity surface to be irregular. Geoid models attempt to represent the surface of the entire earth over both land and ocean as though the surface resulted from gravity alone. Geodetic datums define reference systems that describe the size and shape of the earth based on these various models. While cartography, surveying, navigation, and astronomy all make use of geodetic datums, they are the central concern of the science of geodesy. 17

18 Reference system can be divided into two groups: Global systems can refer to positions over much of the Earth (WGS-84, PZ-90.02). Regional systems have been defined for many specific areas, often covering national, state, or provincial areas (Russia СК-42 and СК-95, the North American Datum of 1927 (NAD27, Ellipsoid Clarke 1866 ) and the North American Datum of 1983 (NAD83, Ellipsoid GRS 80 ) are the datum now used to define the geodetic network in North America). 18

19 Latitude, Longitude, Height The Prime Meridian and the Equator are the reference planes used to define latitude and longitude. There are several ways to define these terms precisely. 19

20 From the geodetic perspective these are: The geodetic latitude of a point is the angle between the equatorial plane and a line normal to the reference ellipsoid. The geodetic longitude of a point is the angle between a reference plane and a plane passing through the point, both planes being perpendicular to the equatorial plane. The geodetic height at a point is the distance from the reference 20 ellipsoid to the point in a direction normal to the ellipsoid.

21 Earth Centered, Earth Fixed (ECEF) Cartesian coordinates can also be used to define three dimensional positions 21

22 ECEF X, Y, and Z Cartesian coordinates define three dimensional positions with respect to the center of mass of the reference ellipsoid. The Z-axis points from the center toward the North Pole. The X-axis is the line at the intersection of the plane defined by the prime meridian and the equatorial plane. The Y-axis is defined by the intersection of a plane rotated 90 east of the prime meridian and the equatorial plane. 22

23 Spherical and geodetic (geodesic) coordinates ϕspherical =ϕgeodetic sin2ϕgeo і λsph =λgeo 23

24 World Geographical Reference System (GEOREF) 24

25 Universal Transverse Mercator (UTM) coordinates define two dimensional, horizontal, positions UTM zone numbers designate individual 6 wide longitudinal strips extending from 80 South latitude to 84 North latitude 25

26 Rhumb Line A rhumb line is a line on the surface of Earth which cross every meridian (line of longitude) at the same angle. On a sphere, where the meridians are converging at the poles, a rhumb line will form a spiraling curve that eventually ends at either of the Earth's poles. The spiral that is created by a rhumb line is a Loxodromic Curve or a Loxodrome. Since a loxodrome is not a great circle, it follows that by tracking a loxodrome a longer distance must be traveled compared to a great circle line. 26

27 Orthodrome A great circles arc that is casted between two points on a surface of a sphere. Is the shortest geodetic connecting line between two points on a sphere. 27

28 Great Circle Line A great circle is a circle on a sphere's surface whose plane is passing exactly through the center of the sphere. An arc on a great circle represents the shortest distance between two points on a sphere. In long range navigation, great circle routes are desired. Since the great circle is not a straight line and therefore difficult to follow, it is divided into a sequence of shorter rhumb lines segments. 28

29 Distance and direction Distance S in angular value (1 minute = km) coss = sinϕ sinϕ + cosϕ cosϕ cos( λ λ ) Direction cos C = sin ϕ 1 sin sin ϕ D NM cos cos ϕ D 1 NM 60 29

30 Track line in orthodromic net TP 2 Conditional Equator Y(θ) IP y+ FP x+ TP 1 X(Z) Conditional Meridian 30

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32 Position lines Aircraft equal bearing line Radiostation equal bearing line Equal distance line Equal difference distances line Constant azimuth line and ets. 32