Phase Correction for GPS Antenna With Nonunique Phase Center

Lyndon B. Johnson Space Center, Houston, Texas

A method of determining the position and attitude of a body equipped with a Global Positioning System (GPS) receiver includes an accounting for the location of the nonunique phase center of a distributed or wraparound GPS antenna. The method applies, more specifically, to the case in which (1) the GPS receiver utilizes measurements of the phases of GPS carrier signals in its position and attitude computations and (2) the body is axisymmetric (e.g., spherical or round cylindrical) and wrapped at its equator with a single- or multiple-element antenna, the radiation pattern of which is also axisymmetric with the same axis of symmetry as that of the body.

An Axisymmetric Body With a Wraparound GPS Antenna at its equator contains a GPS receiver that measures the phase of the signal from the ith GPS satellite at a nonunique phase center at position vector rpi relative to the center of the body. For simplicity, rpi is depicted as equal to the radius of the wraparound antenna, but it could differ.

The figure depicts the geometric relationships among the GPS-equipped object centered at position r_B, the ith GPS satellite at position r_si, and the phase center at position r_pi relative to the center of the body during observation of the ith satellite. The main GPS observable calculated from the phase measurement for the ith satellite is the pseudorange ||v_i||, which is nominally the distance from the phase center to the ith satellite. However, what is needed to determine the position of the center of the body is another pseudorange — that which one would obtain if the phase center were at the center of the body. That pseudorange would nominally equal ||r_si – r_B||. In order to determine ||r_si – r_B|| from phase measurements, it is necessary to account for the phase difference attributable to r_pi . A straightforward mathematical derivation that starts with the law of cosines for this geometry and that incorporates simplifying assumptions based on the axisymmetry and on the smallness of ||r_pi|| relative to ||r_si – r_B|| leads to the following equations:

||r_si – r_B|| = ||v_i|| + ||r_pi|| cos(ß_i) (1) and

cos(ß_i) = [ 1 - (r^_sib • z^_B)² ] ^1/2 (2) where

ß_iis the angle between r_si – r_B and r_pi as shown in the figure,
zˆ_B is the unit vector along the axis of symmetry as shown in the figure, and
r^_sibis the unit vector along r_si – r_B.

The computation of the desired pseudorange ||r_si – r_B|| begins with a coarse estimate of r_B — for example, a previously computed value or a value computed anew without the phase correction. The coarse estimate of r_B is used to obtain an estimate of rˆ_sib, which is used in iterations of equation 1 to obtain successively refined estimates of r_B. Optionally, one can also obtain successively refined estimates of rˆ_sib from the iterations, though in most GPS applications, the error in the initial estimate of rˆ_sib should be negligible.

The iterations follow one of two courses, depending on whether or not ||r_pi|| and the attitude of the body are known a priori. If the attitude is known, then zˆ_B is known and can be inserted in equation 2, which yields cos(ß_i) for use in equation 1. Then ||r_pi|| and cos(ß_i) can be used in equation 1 without further ado. If ||r_pi|| and zˆ_B are not known a priori, then it is necessary to determine ||r_pi||,the attitude, and the phase-correction term ||r_pi|| cos(ß_i) from a least-squares or other fit of (a) an approximate geometric model of the amount by which the phase at r_pi leads the phase at r_B to (b) phase measurements for all of the GPS signals detected by the receiver.

This work was done by Patrick W. Fink and Justin Dobbins of Johnson Space Center.

This invention is owned by NASA, and a patent application has been filed. Inquiries concerning nonexclusive or exclusive license for its commercial development should be addressed to the Patent Counsel, Johnson Space Center, (281) 483-0837. Refer to MSC-23228.