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.

The figure depicts the geometric relationships among the GPS-equipped object centered at position**r**, the ith GPS satellite at position

_{B}**r**, and the phase center at position

_{si}**r**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 ||

_{pi}**v**||, 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 ||

_{i}**r**–

_{si}**r**||. In order to determine ||

_{B}**r**–

_{si}**r**|| from phase measurements, it is necessary to account for the phase difference attributable to

_{B}**r**. 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 ||

_{pi}**r**|| relative to ||

_{pi}**r**–

_{si}**r**|| leads to the following equations:

_{B}||**r _{si}** –

**r**|| = ||

_{B}**v**|| + ||

_{i}**r**||

_{pi}**cos**(

**ß**) (1) and

_{i}**cos**(**ß _{i}**) =

**[ 1**- (

**r^**•

_{sib}**z^**)

_{B}

^{2}**]**(2) where

^{1/2}**ß**is the angle between_{i }**r**–_{si}**r**and_{B}**r**as shown in the figure,_{pi}**zˆ**is the unit vector along the axis of symmetry as shown in the figure, and_{B}**r^**is the unit vector along_{sib }**r**–_{si}**r**._{B}

The computation of the desired pseudorange ||**r _{si}** –

**r**|| begins with a coarse estimate of

_{B}**r**— for example, a previously computed value or a value computed anew without the phase correction. The coarse estimate of

_{B}**r**is used to obtain an estimate of

_{B}**rˆ**, which is used in iterations of equation 1 to obtain successively refined estimates of

_{sib}**r**. Optionally, one can also obtain successively refined estimates of

_{B}**rˆ**from the iterations, though in most GPS applications, the error in the initial estimate of

_{sib}**rˆ**should be negligible.

_{sib}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**is known and can be inserted in equation 2, which yields

_{B}**cos**(

**ß**) for use in equation 1. Then ||

_{i}**r**|| and

_{pi}**cos**(

**ß**) can be used in equation 1 without further ado. If ||

_{i}**r**|| and

_{pi}**zˆ**are not known a priori, then it is necessary to determine ||

_{B}**r**||,the attitude, and the phase-correction term ||

_{pi}**r**||

_{pi}**cos**(

**ß**) from a least-squares or other fit of (a) an approximate geometric model of the amount by which the phase at

_{i}**r**leads the phase at

_{pi}**r**to (b) phase measurements for all of the GPS signals detected by the receiver.

_{B}*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.*

##### NASA Tech Briefs Magazine

This article first appeared in the September, 2005 issue of *NASA Tech Briefs* Magazine.

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