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

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 3.11Transverse 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.

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.

The apparent variations in meridional radius of curvature in Figure 3.10are rather large because the ellipse used in generating Figure 3.10has 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 3.12versus 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.

Attitude models for inertial navigation represent

Figure 3.12Radii 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.

Appendix B on www.wiley.com/go/grewal/gnssis all about the coordinates, coordinate transformation matrices, and rotation vectors used in inertial navigation.

Rate gyroscopes used in inertial navigation measure components of a rotation rate vector in ISA‐fixed coordinates, which had historically been used to express its effect on a coordinate transformation matrix from ISA coordinates to the coordinates for integration in terms of a linear differential equation of the sort

(3.14)

which is not particularly well‐conditioned for numerical integration.

The alternative representation in terms of quaternions is described in Section 3.6.1.2and (in greater detail) in Appendix B.

This is only possible with gimbaled systems and it adds to the start‐up time, but it can improve performance. Navigation accuracy is very sensitive to accelerometer biases, which can shift due to thermal transients in turn‐on/turn‐off cycles, and can also drift randomly over time for some accelerometer design. Fortunately, gimbals can be used to calibrate accelerometer biases in a stationary 1g environment. Bias and scale factors can both be determined by using the gimbals to point each accelerometer input axes straight up and then straight down (by nulling the horizontal accelerometer outputs). Then each accelerometer's bias is the average of the up and down outputs and scale factor is half the difference divided by the local gravitational acceleration.