Learning from the Past, Looking to the Future James R. (Jim) Beaty, PhD - NASA Langley Research Center Vehicle Analysis Branch, Systems Analysis & Concepts Directorate Bldg 1209, Room 128B, M/S North Dryden Street Hampton, VA Page: 1 Fundamentals of Geodetic Kinematics Part 1 of 2 (“Introduction and Definitions”)

Learning from the Past, Looking to the Future Overview Part 1, “Introduction and Definitions” – Mathematical relationships for ellipses – Coordinate frames and definition of terms – Equations used for geodetic kinematics – Spherical to rectangular position coordinates and vice-versa Part 2, “Implementation” – Sidereal “Time” – Summary of WGS-84 standard constants used for Earth and moon – Coordinate frames commonly used for modeling of geodetic kinematics – Equations for navigation over an oblate spheroid – Great circle distance calculations – Calculation of position and velocity of an object relative to planet fixed frames Page: 2

Learning from the Past, Looking to the Future What is Geodetic Kinematics? According to Wickipedia ®, Geodetic is defined as: “Geodesy also called geodetics, a branch of earth sciences, is the scientific discipline that deals with the measurement and representation of the Earth, including its gravitational field, in a three-dimensional time-varying space. Geodesists also study geodynamical phenomena such as crustal motion, tides, and polar motion For this they design global and national control networks, using space and terrestrial techniques while relying on datums and coordinate systems.” For the current topic, we will limit our discussion primarily to the geometry and kinematics necessary to define position and velocity of an object relative to an oblate spheroid representation of a planet – oblate spheroid refers to the three-dimensional shape created by rotating an ellipse about its minor axis – oblate spheroids can be thought of as spheres that are flattened at their poles (two points on the spheroid which the axis of rotation intersects) Page: 3

Learning from the Past, Looking to the Future Mathematical relationships for ellipses Because an oblate spheroid is created by rotating an ellipse about its minor axis, we begin with a review of some preliminary mathematical relationships for ellipses – Points f and f ‘ are referred to as the focii of the ellipse f is primary focus f ‘ is secondary focus – For any point, P, on the ellipse, the sum of the lengths from each focus to P is constant: r + r ‘ = constant – The equation of the ellipse in rectangular coordinates, x and y, is where a and b are the semi-major and semi-minor axis lengths respectively The equation of the ellipse in polar coordinates, r and , is where K is the semilatus rectum, and e is the eccentricity of the ellipse Page: 4 f f '  r r ' P x y 2 a 2 b cc

Learning from the Past, Looking to the Future Definition of Eccentricity Using the polar form of the equation of the ellipse, we can solve for K, the semilatus rectum: At the positive end of the semi-major axis (  = 0): At the negative end of the semi-major axis (  =  ): Solving these two equations gives K and c K = a ( 1 – e 2 )( Eq. 5 ) c = a e( Eq. 6 ) Or, from the latter relationship, we get the definition of the eccentricity: e = ( c / a )( Eq. 7 ) Page: 5 f f '  r r ' P x y 2 a 2 b cc Eccentricity is the ratio of the distance between the two focii and the major axis length

Learning from the Past, Looking to the Future Relationship between a and b Using the value just computed for for K, the semilatus rectum, the polar form of the equation of the ellipse, can be written as: Recall that the sum of the distances from each focus, f and f ‘, is constant for any point on the ellipse r + r ‘ = constant( Eq. 9 ) At the positive end of the semi-major axis: r = a – c = a – ae = a ( 1 – e ) r ‘ = a + c = a + ae = a ( 1 + e ) So, the constant in Eq. 9 is equal to 2a or, for any point P on the ellipse, r + r ‘ = 2 a( Eq. 10 ) Now, at the positive end of the semi-minor axis (point B). r = r ‘ = a so that Page: 6 f f ' r = a r ‘ = a B y 2 a 2 b a e b

Learning from the Past, Looking to the Future Eccentricity and Flattening Ratio From the last result on the previous page, the semi-minor axis length, b, of the ellipse can be expressed as a function of its semi-major axis length, a, and its eccentricity: or ( 1 – e 2 ) = ( b / a) 2 ( Eq. 13 ) The flattening ratio of an ellipse, f, is defined as: or, in terms of the eccentricity, ( 1 – f ) 2 = ( 1 – e 2 )( Eq. 15 ) Page: 7

Learning from the Past, Looking to the Future Radius of Curvature of Ellipse The radius of curvature of an ellipse at any point, P, is: But, from Eq. (10), r + r ‘ = 2a, so Also, at the ends of the semi-major axis, the equitorial radius of curvature is  = b 2 / a = a ( 1 – e 2 ) (Eq. 18) At the ends of the semi-minor axis, the polar radius of curvature is   = a 2 / b = a / √ 1 – e 2 ( Eq. 19 ) Page: 8 x Equal angles (radius of curvature , bisects angle between r and r ‘ ) f r r ' P y  Normal to ellipse at point P

Learning from the Past, Looking to the Future Position of Objects Relative to Oblate Spheroids Positions of object relative to the oblate spheroid are commonly defined using either rectangular or spherical coordinates: – rectangular coordinates (X, Y, Z) relative to a coordinate frame with its origin at the centroid of the oblate spheroid, which may be fixed to the oblate spheroid (and rotates with it), or which may be fixed relative to inertial space – spherical coordinates (latitude, , longitude,, altitude, h) relative to a prime meridian plane and equatorial plane fixed to the oblate spheroid latitude can be specified as geodetic latitude, or geocentric latitude longitude can be specified as geodetic longitude or inertial longitude geodetic altitude is the distance of the object above the surface of the oblate spheroid along a direction normal to the surface of the oblate spheroid directly below the object – actual planet shape is more complex than an oblate spheroid (irregularities, tidal motion on Earth, etc), but oblate spheroid represents the “mean” planet shape Page: 9

Learning from the Past, Looking to the Future Geodetic Coordinate Frames From figure at right, point P is some arbitrary point on surface of oblate spheroid Position of P is defined relative to two reference frames, both with their origins at centroid of oblate spheroid: – Earth centered, inertial (ECI) frame: – Earth centered, earth fixed (ECEF) frame (also called earth centered, rotating (ECR) ECI frame has its x- and y-axes in equatorial plane with x-axis pointing toward Vernal Equinox (first point of Ares), z-axis pointing toward mean axis of rotation, y-axis completes right handed frame. ECI frame is inertial ECEF frame has its x- and y-axes in equatorial plane with x-axis pointing toward prime meridian (Greenwich Meridian on earth), z-axis pointing toward mean axis of rotation, y-axis completes right handed frame. ECEF frame rotates with earth at a constant rate Page: 10 I Meridian Equinox X E P   Z I, Z E Y E Y Equatorial Plane Greenwich Vernal X I  g   meridian line containing P

Learning from the Past, Looking to the Future Spherical Geodetic Coordinates From figure at upper right, point P is some arbitrary point on surface of oblate spheroid – The geodetic latitude (  E ) defines the angle along a meridian line from the equatorial plane to point P – The geodetic longitude ( E ) defines the angle in the equitorial plane from the prime meridian (Greenwich meridian on Earth) to the meridian line containing point P (see lower figure at right) – The inertial longitude ( I ) defines the angle in the equatorial plane from the vernal equinox to the meridian line containing point P – The sidereal “time” ( g ) is the angle between the vernal equinox and the prime meridian – The inertial longitude ( i ) is the sum of the geodetic longitude ( E ) and its sidereal “time” ( g ) ( I = g + E ) In the figure at lower right, the geocentric radius (r P ) to a point on the surface of the oblate spheroid, and the geocentric latitude (  C ) are defined, along with the geodetic latitude (  E ) Page: 11 Normal to ellipse at point P P EE CC rPrP I Meridian Equinox X E P   Z I, Z E Y E Y Equatorial Plane Greenwich Vernal X I  g   meridian line containing P

Learning from the Past, Looking to the Future Spherical Geodetic Coordinates From figure at upper right, point P is some arbitrary point on surface of oblate spheroid directly “below” the current position of some object, O – “below” refers to coordinate in the direction of the local vertical toward the surface of the spheroid The figure at lower right, represents a view perpendicular to the meridian plane containing the object. Shown in this figure are: – Radius of curvature in prime meridian, (  ) – Radius of curvature in prime vertical, (  V ) – Part of radius of curvature in prime vertical above equitorial plane, (   ) – Geodetic altitude of object, ( h ) – Geodetic latitude, (  E ) – Geocentric latitude, (  C ) – Geocentric radius to a point on the surface, ( r P ) Page: 12 O EE CC rPrP h P  VV ZZ I Meridian Equinox X E P   Z I, Z E Y E Y Equatorial Plane Greenwich Vernal X I  g   meridian line containing P

Learning from the Past, Looking to the Future Spherical Geodetic Coordinates From equation (17) on page 8, the radius of curvature at point P (also called the radius of curvature in the prime meridian at P), , is computed from It can be shown that  can be written in terms of the semi- major axis length ( a ), eccentricity ( e ), and geodetic latitude (  E ) as Likewise, the radius of curvature in the prime vertical (  V ) can be written as Finally, the part of  V that is above the equatorial plane (  Z ) can be written as Page: 13 O EE CC rCrC h P  VV ZZ

Learning from the Past, Looking to the Future Geocentric Radius and Latitude The figure at the upper right shows the geocentric latitude (  C ) and geocentric radius vector (r P ) for point P on the surface of the oblate spheroid, the declination (  O ) and geocentric radius vector (r C ) for point O above the surface of the oblate spheroid, along with the geodetic latitude (  E ) The figure on the lower right also shows the relationships of the positions and angular measurements for the triangle represented by the vectors r P, r C, and h, bounded by points E, P, and O It is easy to show that the geocentric latitude can be computed from the geodetic latitude as follows: It is also easy to show that the geocentric radius of point P can be computed from the geocentric latitude (  C ) as: Page: 14 EE CC rPrP P O h rCrC E h O P E CC OO rCrC rPrP

Learning from the Past, Looking to the Future Declination of Point Above Geoid If we define O, P, and E as the interior angles of the triangle bounded by r C, r p, and h at those points on the triangle, and applying the law of cosines to the triangle at the lower right Also, it is easy to show that the angle P can be written as By applying the law of sines to the triangle shown on the lower right, So, it can be shown that the angle O is: Finally, it can be shown that the declination angle,  0, of point O relative to the equatorial plane is Page: 15 EE CC rPrP P O h rCrC E h O P E CC OO rCrC rPrP

Learning from the Past, Looking to the Future Declination of Point Above Geoid Summarizing the equations shown in this section, the equations for calculating the geocentric radius (r C ), geocentric latitude (  C ), declination angle (  O ), and geocentric radius (r P ) to point on the surface of the geoid below an object (r P ) are Page: 16 EE CC rPrP P O h rCrC E h O P E CC OO rCrC rPrP

Learning from the Past, Looking to the Future Spherical to Rectangular Coordinates The figures at right show a meridian view and top view – Meridian view is looking normal to the meridian plane containing the object, O – Top view is looking down the Z E axis The vertical component of the object position ( z E ) can be computed as: The horizontal components of the object position ( x E and y E ) can be computed as: Page: 17 Normal to ellipse at point P O EE h P VV ZZ zEzE  xy xExE yEyE E YEYE XEXE ZEZE Top view Meridian view

Learning from the Past, Looking to the Future Spherical to Rectangular Coordinates The figures at right show a meridian view and zoomed in view of that meridian geometry – Meridian view (upper figure) is looking normal to the meridian plane containing the object, O – Lower figure zooms in to show the geocentric parameters The ECEF rectangular coordinates can be computed from their geocentric counterparts as: where the geocentric radius (r C ) and declination angle (  O ) were presented on pages 15 and 16 Page: 18 EE CC rPrP P O h rCrC E h O P E CC OO rCrC rPrP

Learning from the Past, Looking to the Future Rectangular to Spherical Coordinates The calculation of the geodetic longitude is exact, and simple: Next, the geodetic latitude must be computed, but it is not as simple – Closed form solutions are available (e.g. – Sofair’s method), though very complicated, and prone to truncation errors, even for double precision calculations – A preferred method is derived from a single pass Newton-Raphson iterative solution of the non-linear, coupled equations – It can be shown that some closed form exact solutions (such as Sofair’s) are not as accurate as the approximate solution, due to numerical precision / roundoff errors – After the geodetic latitude is computed (see next page), the exact equation for computing the geodetic altitude can then be computed as: Page: 19

Learning from the Past, Looking to the Future Rectangular to Spherical Coordinates The approximate solution for the geodetic latitude: Principal value arc tangent calculations are adequate for both equations Page: 20

Learning from the Past, Looking to the Future Fundamentals of Geodetic Kinematics, Part 1 of 2 Summary In Part 1, “Introductions and Definitions” we covered the following: – Mathematical relationships for ellipses – Coordinate frames and definition of terms – Equations used for geodetic kinematics – Spherical to rectangular position coordinates and vice-versa In Part 2, “Implementation”, we will cover the following: – Sidereal “Time” – Summary of WGS-84 standard constants used for Earth and moon – Coordinate frames commonly used for modeling of geodetic kinematics – Equations for navigation over an oblate spheroid – Great circle distance calculations – Calculation of position and velocity of an object relative to planet fixed frames Page: 21