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
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
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
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
Figure 3.12 Radii of WGS84 reference ellipsoid.
1 The relative rotational orientations of two coordinate systems, usually represented by coordinate transformation matrices but also represented in terms of rotation vectors.
2 Attitude dynamics, usually represented in terms of three‐dimensional rotation rate vectors but also represented by four‐dimensional quaternion.
3.4.5.1 Coordinate Transformation Matrices and Rotation Vectors
Appendix B on www.wiley.com/go/grewal/gnss is all about the coordinates, coordinate transformation matrices, and rotation vectors used in inertial navigation.
3.4.5.2 Attitude Dynamics
Rate gyroscopes used in inertial navigation measure components of a rotation rate vector
