Global Navigation Satellite Systems, Inertial Navigation, and Integration. Mohinder S. Grewal

target="_blank" rel="nofollow" href="#ulink_5909012e-778e-5732-8f39-4fa6c6f9b1d0">Figure 3.9 WGS84 geoid heights.

Figure 3.10 Ellipse and osculating circles.

      The formula for meridional radius of curvature as a function of geodetic latitude ( is

      where is the semimajor axis of the ellipse, is the semiminor axis, and = ( – / is the eccentricity squared.

       Geodetic Latitude Rate

The rate of change of geodetic latitude as a function of north velocity is then

      (3.9)

      and geodetic latitude can be maintained as the integral

      (3.10)

      where is height above (+) or below () the ellipsoid surface and will be in radians if is in meters per second and and are in meters.

       Transverse Radius of Curvature

The radius of curvature of the reference ellipsoid surface in the east–west direction (i.e. orthogonal to the direction in which the meridional radius of curvature is measured) is called the transverse radius of curvature. It is the radius of the osculating circle in the local east–up plane, as illustrated in Figure 3.11, where the arrows at the point of tangency of the transverse osculating circle are in the local ENU coordinate directions. As this figure illustrates, on an oblate Earth, the plane of a transverse osculating circle does not pass through the center of the Earth, except when the point of osculation is at the equator. (All osculating circles at the poles are in meridional planes.) Also, unlike meridional osculating circles, transverse osculating circles generally lie outside the ellipsoidal surface, except at the point of tangency and at the equator, where the transverse osculating circle is the equator.

Figure 3.11 Transverse osculating circle.

The formula for the transverse radius of curvature on an ellipsoid of revolution is

      (3.11)

      where is the semimajor axis of the generating ellipse and is its eccentricity.

       Longitude Rate

      The rate of change of longitude as a function of east velocity is then

      (3.12)

      and longitude can be maintained by the integral

      (3.13)

      where is height above () or below () the ellipsoid surface and will be in radians if is in meters per second and and are in meters. Note that this formula has a singularity at the poles, where cos(, a consequence of using latitude and longitude as location variables.

       WGS84 Reference Surface Curvatures

The apparent variations in meridional radius of curvature in Figure 3.10 are rather large because the ellipse used in generating Figure 3.10 has an eccentricity of about 0.75. The WGS84 ellipse has an eccentricity of about 0.08, with geocentric, meridional, and transverse radius of curvature as plotted in Figure 3.12 versus geodetic latitude. For the WGS84 model,

      • Mean geocentric radius is about 6371 km, from which it varies by –14.3 km (–0.22%) to +7.1 km (+0.11%).

      • Mean meridional radius of curvature is about 6357 km, from which it varies by –21.3 km (–0.33%) to 42.8 km (+0.67%).

      • Mean transverse radius of curvature is about 6385 km, from which it varies by –7.1 km (–0.11%) to +14.3 km (+0.22%).

Because these vary by several parts per thousand, one must take radius of curvature into account when integrating horizontal velocity increments to obtain longitude and latitude.

      3.4.5 Attitude Models

      Attitude models for inertial navigation represent