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

Gravitational acceleration is the negative gradient of gravitational potential. Potential is a scalar function, and its gradient is a vector. Because gravitational potential increases with altitude, its gradient points upward and the negative gradient points downward.

An equipotential surface is a surface of constant gravitational potential. If the ocean and atmosphere were not moving, then the surface of the ocean at static equilibrium would be an equipotential surface. Mean sea level is a theoretical equipotential surface obtained by time‐averaging the dynamic effects. Orthometric altitude is measured along the (curved) plumbline.

The WGS84 Earth model approximates mean sea level (an equipotential surface) by an ellipsoid of revolution with its rotation axis coincident with the rotation axis of the Earth, its center at the center of mass of the Earth, and its prime meridian through Greenwich. Its semimajor axis (equatorial radius) is defined to be 6 378 137 m, and its semiminor axis (polar radius) is defined to be 6 356 752.3142 m.

Geoids are approximations of mean sea‐level orthometric height with respect to a reference ellipsoid. Geoids are defined by additional higher‐order shapes, commonly modeled by spherical harmonics of height deviations from an ellipsoid, as illustrated in Figure 3.8. There are many geoid models based on different data, but the more recent, most accurate models depend heavily on GPS data. Geoid heights deviate from reference ellipsoids by tens of meters, typically.

The WGS84 geoid heights vary about 100 m from the reference ellipsoid. As a rule, oceans tend to have lower geoid heights and continents tend to have higher geoid heights. Coarse 20‐m contour intervals are plotted versus longitude and latitude in Figure 3.9, with geoid regions above the ellipsoid shaded gray.

The second integral of acceleration in locally level coordinates should result in the estimated vehicle position. This integral is somewhat less than straightforward when longitude and latitude are the preferred horizontal location variables.

The rate of change of vehicle altitude equals its vertical velocity, which is the first integral of net (i.e. including gravity) vertical acceleration. The rates of change of vehicle longitude and latitude depend on the horizontal components of vehicle velocity, but in a less direct manner. The relationship between longitude and latitude rates and east and north velocities is further complicated by the oblate shape of the Earth.

Figure 3.8Equipotential surface models for Earth.

The rates at which these angular coordinates change as the vehicle moves tangent to the surface will depend upon the radius of curvature of the reference surface model. Radius of curvature can depend on the direction of travel, and for an ellipsoidal model there is one radius of curvature for north–south motion and another radius of curvature for east–west motion.

The radius of curvature for north–south motion is called the “meridional” radius of curvature, because north–south travel is along a meridian (i.e. line of constant longitude). For an ellipsoid of revolution, all meridians have the same shape, which is that of the ellipse that was rotated to produce the ellipsoidal surface model. The tangent circle with the same radius of curvature as the ellipse is called the “ osculating circle ” (osculating means “kissing”). As illustrated in Figure 3.10for an oblate Earth model, the radius of the meridional osculating circle is smallest where the geocentric radius is largest (at the equator), and the radius of the osculating circle is largest where the geocentric radius is smallest (at the poles). The osculating circle lies inside or on the ellipsoid at the equator and outside or on the ellipsoid at the poles and passes through the ellipsoid surface for latitudes in between.

Figure 3.9WGS84 geoid heights.

Figure 3.10Ellipse 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.

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.