A method has been proposed for extracting information on the rate of rotation of an aircraft, spacecraft, or other body from differential Doppler shifts of Global Positioning System (GPS) signals received by antennas mounted on the body. In principle, the method should be capable of yielding low-noise estimates of rates of rotation. The method could eliminate the need for gyroscopes to measure rates of rotation.

The method is based on the fact that for a given signal of frequency ft transmitted by a given GPS satellite, the differential Doppler shift is attributable to the difference between those components of the instantaneous translational velocities of the antennas that lie along the line of sight from the antennas to the GPS satellite. On the basis of straightforward geometric considerations (see figure), it can be readily shown that the differential Doppler shift is related to the angular velocity (?) of the rotating body by ƒr1 − ƒr2 = 2ƒt(ω×r) • a/c , where ƒr1 and ƒr2 are the instantaneous Doppler-shifted frequencies of the replicas of the ft signal received by the two antennas, r is half of the baseline vector between the two antennas, a is a unit vector along the line of sight from the antennas to the GPS satellite, and c is the speed of light.

must be noted that the equation above can be solved to obtain only partial information about ?. However, if there are three or more antennas and if signals can be received from two or more GPS satellites, then one can form simultaneous independent equations for different pairs of antennas and different unit vectors that can be solved to obtain all of the components of ?.

It is assumed that the received ƒr1 and ƒr2 signals would be subjected to the usual GPS processing, including phaseshifting and cross-correlation with the applicable GPS pseudorandom-noise code for acquisition and tracking. To obtain the differential Doppler frequency ƒr1 − ƒr2 for a given antenna pair and a given GPS satellite, the ƒr1 and ƒr2 signals would be fed to a multiplier. By virtue of the trigonometric identity for the product of sines of different arguments, the low-frequency multiplier output would be a sinusoidal waveform of frequency ƒr1 − ƒr2. For high accuracy, the multiplier output could be fed to a subsystem containing a zero-crossing detector coupled with a counter driven by a quartz-crystal clock circuit. Such a subsystem could accumulate counts over times long enough to enable estimation of periods of rotation to within microseconds.

This work was done by Charles E. Campbell, Jr., of Goddard Space Flight Center. For further information, access the Technical Support Package (TSP) free online at www.techbriefs.com/tsp under the Electronics/Computers category.

This invention has been patented by NASA (U.S. Patent No. 6,593,879). Inquiries concerning nonexclusive or exclusive license for its commercial development should be addressed to

###### the Patent CounselGoddard Space Flight Center(301) 286-7351.

Refer to GSC-14087-1.