9.4.3 Earth Models

Inertial navigation and satellite navigation require models for the shape, gravity, and rotation of the earth.

9.4.3.1 Navigation Coordinates Descriptions of the major coordinates used in inertial navigation and GNSS/INS integration are described in Appendix C (on the CD). These include coordinate systems used for representing the trajectories of GNSS satellites and user vehicles in the near-earth environment and for representing the attitudes of host vehicles relative to locally level coordinates, including the following:

1. Inertial coordinates:
   (a) Earth-centered inertial (ECI), with origin at the center of mass of the earth and principal axes in the directions of the vernal equinox (defined in Section C.2.1) and the rotation axis of the earth.
   (b) Satellite orbital coordinates, as illustrated in Fig. C.4 and used in GPS ephemerides.
2. Earth-fixed coordinates:
   (a) Earth-centered, earth-fixed (ECEF), with origin at the center of mass of the earth and principal axes in the directions of the prime meridian (defined in Section C.3.5) at the equator and the rotation axis of the earth.
   (b) Geodetic coordinates, based on an ellipsoid model for the shape of the earth. Longitude in geodetic coordinates is the same as in ECEF coordinates, and geodetic latitude as defined as the angle between the equatorial plane and the normal to the reference ellipsoid surface. Geodetic latitude can differ from geocentric latitude by as much as 12 arc minutes, equivalent to about 20 km of northing distance.
   (c) Local tangent plane (LTP) coordinates, also called "locally level coordinates," essentially representing the earth as being locally flat. These coordinates are particularly useful from a human factors standpoint for representing the attitude of the host vehicle and for representing local directions. They include
       i. East-north-up (ENU), shown in Fig. C.7
       ii. North-east-down (NED), which can be simpler to relate to vehicle coordinates and
       iii. Alpha wander, rotated from ENU coordinates through an angle a about the local vertical (see Fig. C.8)
3. Vehicle-fixed coordinates:
   (a) Roll-pitch-yaw (RPY) (axes shown in Fig. C.9).

Transformations between these different coordinate systems are important for representing vehicle attitudes, for resolving inertial sensor outputs into inertial navigation coordinates, and for GNSS/INS integration. Methods used for representing and implementing coordinate transformations are also presented in Appendix C, Section C.4.

9.4.3.2 Earth Rotation Our earth is the mother of all clocks. It has given us the time units of days, hours, minutes, and seconds we use to manage our lives. Not until the discovery of atomic clocks based on hyperfine transitions were we able to observe the imperfections in our earth clock. Despite these, we continue to use earth rotation as our primary time reference, adding or subtracting leap seconds to atomic clocks to keep them synchronized to the rotation of the earth. These time variations are significant for GNSS navigation, but not for inertial navigation.

WGS84 Earthrate Model The value of earthrate in the World Geodetic System 1984 (WGS84) earth model used by GPS is 7, 292, 115, 167 10^-14 radians per second, or about 15.04109°/h. This is its sidereal rotation rate with respect to distant stars. Its mean rotation rate with respect to the nearest star (our sun), as viewed from the rotating earth, is 15°/h, averaged over one year.

9.4.3.3 GPS Gravity Models Accurate gravity modeling is important for maintaining ephemerides for GPS satellites, and models developed for GPS have been a boon to inertial navigation as well. However, spatial resolution of the earth gravitational field required for GPS operation may be a bit coarse compared to that for precision inertial navigation, because the GPS satellites are not near the surface and the mass concentration anomalies that create surface gravity anomalies. GPS orbits have very little sensitivity to surface-level undulations of the gravitational field on the order of 100 km or less, but these can be important for high-precision inertial systems.

9.4.3.4 INS Gravity Models Because an INS operates in a world with gravitational accelerations it is unable to sense and unable to ignore, it must use a reasonably faithful model of gravity.

Gravity models for the earth include centrifugal acceleration due to rotation of the earth as well as true gravitational accelerations due to the mass distribution of the earth, but they do not generally include oscillatory effects such as tidal variations.

Gravitational Potential Gravitational potential is defined to be zero at a point infinitely distant from all massive bodies and to decrease toward massive bodies such as the earth. That is, a point at infinity is the reference point for gravitational potential.

In effect, the gravitational potential at a point in or near the earth is defined by the potential energy lost by a unit of mass falling to that point from infinite altitude. In falling from infinity, potential energy is converted to kinetic energy, mν²_escape/2, where ν_escape is the escape velocity. Escape velocity at the surface of the earth is about 11 km/s.

Gravitational Acceleration 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.

Equipotential Surfaces 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.

Ellipsoid Models for Earth Geodesy is the process of determining the shape of the earth, often using ellipsoids as approximations of an equipotential surface (e.g., mean sea level), as illustrated in Fig. 9.17. The most common ones are ellipsoids of revolution, but there are many reference ellipsoids based on different survey data. Some are global approximations and some are local approximations. The global approximations deviate from a spherical surface by about ±10 km, and locations on the earth referenced to different ellipsoidal approximations can differ from one another by 10² - 10³ m.

Geodetic latitude on a reference ellipsoid is measured in terms of the angle between the equator and the normal to the ellipsoid surface, as illustrated in Fig. 9.17.

Orthometric height is measured along the (curved) plumbline.

WGS84 Ellipsoid The World Geodetic System (WGS) is an international standard for navigation coordinates. WGS84 is a reference earth model released in 1984. It approximates mean sea level 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.

Geoid Models Geoids are approximations of mean sea-level orthometric height with respect to a reference ellipsoid. Geoids are defined by additional higher order shapes, such as spherical harmonics of height deviations from an ellipsoid, as illustrated in Fig. 9.17. 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 Fig. 9.18, with geoid regions above the ellipsoid shaded gray.

9.4.3.5 Longitude and Latitude Rates 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.

The rates at which these angular coordinates change as the vehicle moves tangent to the surface will depend on the radius of curvature of the reference

surface model, which is an ellipsoid of revolution for the WGS84 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.

Meridional Radius of Curvature 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 (the WGS84 model), 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 Fig. 9.19 for 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.

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

where a is the semimajor axis of the ellipse, b is the semiminor axis, and e² = (a² -b²)/a² is the eccentricity squared.

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

and geodetic latitude can be maintained as the integral

where h(t) is height above (+) or below (-) the ellipsoid surface and φ_geodetic(t) will be in radians if ν_N(t) is in meters per second and r_M(t) and h(t) 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 Fig. 9.20, 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.

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

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

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

and longitude can be maintained by the integral

where h(t) is height above (+) or below (-) the ellipsoid surface and θ will be in radians if ν_E(t) is in meters per second and r_T (t) and h(t) are in meters. Note that this formula has a singularity at the poles, where cos(φ_geodetic) = 0, a consequence of using latitude and longitude as location variables.

WGS84 Reference Surface Curvatures The apparent variations in meridional radius of curvature in Fig. 9.19 are rather large because the ellipse used in generating Fig. 9.19 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 Fig. 9.21 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.