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Claims  |
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What is claimed is:
1. An apparatus for determining a direction or heading in a reception environment that includes radio reflections, reception noise, antenna phase center motion,
temperature-sensitive drifts and detected signal phase corruption, comprising:
a pair of GPS antennas separated by a known first physical distance that exceeds one wavelength of a radio carrier signal emitted by a GPS satellite for simultaneously receiving radio carrier signals emitted by radio transmitter in at least four
corresponding GPS satellite each having a known position;
a pair of first and second phase coherent GPS correlators connected to the pair of GPS antennas and having respective first and second correlator outputs derived from said GPS satellite radio carrier signal as independently received by each of
the pair of GPS antennas wherein said first and second correlator outputs include a phase difference representative of a phase difference that exists in the pair of GPS antennas for said GPS satellite radio carrier signal;
first computer means connected to the pair of first and second phase coherent GPS correlators for determining a first plurality of possible headings to a first of said GPS satellites based on a single first phase difference between said first and
second correlator outputs for a first radio carrier signal from said first GPS satellite, wherein said first plurality of possible headings includes more than one possible heading due to whole cycle carrier phase ambiguities;
second computer means connected to the pair of first and second phase coherent GPS correlators for determining a second plurality of possible headings to a second of said GPS satellites based on a single second phase difference between said first
and second correlator outputs for a second radio carrier signal from said second GPS satellite, wherein said second plurality of possible headings includes more than one possible heading due to whole cycle carrier phase ambiguities;
third computer means connected to the pair of first and second phase coherent GPS correlators for determining a third plurality of possible headings to a third of said GPS satellites based on a single third phase difference between said first and
second correlator outputs for a third radio carrier signal from said third GPS satellite, wherein said third plurality of possible headings includes more than one possible heading due to whole cycle carrier phase ambiguities;
fourth computer means connected to the pair of first and second phase coherent GPS correlators for determining a fourth plurality of possible headings to a fourth of said GPS satellites based on a single fourth phase difference between said first
and second correlator outputs for a fourth radio carrier signal from said fourth GPS satellite, wherein said fourth plurality of possible headings includes more than one possible heading due to whole cycle carrier phase ambiguities; and
simultaneous heading solution computer means connected to the first through fourth computer means for matching said first through said fourth pluralities of possible headings and for identifying at least one geometry for the pair of antennas that
satisfies at least three of said first through fourth plurality of possible headings and eliminating any additional geometries for the pair of antennas that satisfies at least three of said first through fourth plurality of possible headings that have a
less perfect fit to said respective geometries before identifying any geometry for the pair of antennas that satisfies four of said first through fourth plurality of possible headings.
2. The apparatus of claim 1, further comprising:
a single local oscillator connected to phase coherently drive the pair of first and second GPS correlators.
3. The apparatus of claim 1, further comprising:
fifth computer means connected to the pair of first and second phase coherent GPS correlators for determining a fifth plurality of possible headings to a fifth of said GPS satellites based on a single fifth phase difference between said first and
second correlator outputs for a fifth radio carrier signal from said fifth GPS satellite, wherein said fifth plurality of possible headings includes more than one possible heading due to whole cycle carrier phase ambiguities; and
wherein the simultaneous heading solution computer means matches said first through said fifth pluralities of possible headings and identifies at least one geometry for the pair of antennas that satisfies at least three of said first through
fifth plurality of possible headings and eliminates any additional geometries for the pair of antennas that satisfy at least three of said first through fifth plurality of possible headings that have a less perfect fit to said respective geometries
before identifying any geometry for the pair of antennas that satisfies four of said first through fifth plurality of possible headings.
4. An apparatus for determining a direction or heading in a reception environment that includes radio reflections, reception noise, non-common antenna phase center motion, temperature-sensitive drifts and other corruption of detected signal
phase, comprising:
a pair of GPS antennas for simultaneously receiving a plurality of radio carrier signals emitted by radio transmitters aboard a plurality of GPS satellites wherein the pair of antennas are separated by a known physical distance that exceeds one
wavelength of said radio carrier signals;
a pair of first and second correlators having respective first and second correlator outputs derived from said GPS satellite radio carrier signals as received by each of the pair of GPS antennas;
means for coherently driving the pair of first and second correlators and for obtaining a phase difference that exists between said first and second correlator outputs responsive to said GPS satellite radio carrier signals and said separation of
the pair of GPS antennas;
first computer means connected to the pair of first and second phase coherent GPS correlators for determining a plurality of possible unambiguous carrier phase differences between said first and second correlator outputs for each respectively
received GPS satellite radio carrier signal, wherein all the possible unambiguous carrier phase differences of any given received signal include a single common whole cycle ambiguity integer;
second computer means connected to the pair of first and second phase coherent GPS correlators for determining a unique combination of unambiguous carrier phase differences, from among all possible combinations inclusive of all said received GPS
satellite radio carrier signals, having a superior heading solution fit with said unambiguous carrier phase differences compared to all other possible combinations likewise inclusive of all said received GPS satellite radio carrier signals; and
third computer means connected to the pair of first and second phase coherent GPS correlators for determining direction or heading based on said unambiguous carrier phase differences of a best-fit combination inclusive of all said received GPS
satellite radio carrier signals.
5. The apparatus of claim 4, wherein:
the first, second and third computer means each include means to determine combinations of unambiguous carrier phase differences that exclude one or more of said received GPS satellite radio carrier signals, and in cases where said exclusive
combination exhibits a weighted-fit error that exceeds another combination error inclusive of all said received GPS satellite radio carrier signals, the first, second and third computer means avoid evaluating of all other combinations, inclusive or
exclusive, that contain said present exclusive combination as a smaller part, wherein the detection of an exclusive combination with weighted-fit error that exceeds a predetermined maximum value does not further process larger combinations that contain
said present exclusive combination.
6. A computer-implemented method for determining a navigational heading in a reception environment that includes radio reflections, reception noise, antenna phase center motion, temperature-sensitive drifts and detected signal phase corruption,
the method comprising the steps of:
inputting phase measurement data from a set of GPS correlators connected to a trio of GPS antennas mounted to a vehicle and separated from one another by more than one GPS carrier frequency wavelength and having a phase output measurement subject
to whole-cycle carrier phase ambiguity;
calculating in a computer connected to said GPS correlators a maximum likelihood estimation (MLE) optimization procedure expressible in computer program pseudocode as:
where .theta..sub.i are first-difference carrier phase measurements derived from said phase coherent GPS correlators, U.sub.i is a weighted-fit error that appears at a stage "i" of a tree, W.sub.best is a best weighted-fit error,
from.sub.1...to.sub.1, from.sub.2...to.sub.2 through from.sub.m...to.sub.m are the outer limits of whole cycle carrier ambiguities, H.sub.i =[LOS.sub.i .vertline.1] is an observation matrix containing a line-of-sight unit vector that points into a
direction of a GPS signal arrival, and where v.sub.i and K.sub.i are pre-computed, wherein a final integer combination is selected on the basis of a best fit with the measured carrier phase, as indicated by a weighted-fit error, and X-.sub.best,
n.sub.best, and W.sub.best, represent a desired optimum antenna position, an integer combination and an associated weighted-fit error;
determining a range of integers that are possible for a first baseline drawn between two of said GPS antennas, given only the known constraints on vehicle roll and pitch, wherein only a small range of integers is possible if said vehicle cannot
roll or pitch significantly relative to any GPS satellites that are substantially vertically overhead;
processing all phase measurements from a shortest baseline between said GPS antennas to determine a first-antenna position, searching a range of integers determined in the previous step using a four-state filter that solves in geodetic
coordinates for a single antenna position and a single cable bias;
verifying that antenna vertical coordinates are within a limit set by a maximum vehicle roll and pitch for integer combinations that survive the previous step;
verifying that the baselength computed from said antenna coordinates matches a known baselength to within a noise tolerance;
using said known baselength as a reference measurement and normalizing an antenna position vector to unit length to serve as Kalman observation matrix H, and assigning a measurement variance R to cover a non-linearity that results from H being
slightly perturbed by noise in the phase measurements from said GPS correlators;
if said integer combination and antenna coordinate survive to this step, calculating an arc of coordinates that are possible for a second antenna, assuming that said first antenna coordinate is true, and determining a range of integers that are
possible for said second baseline, while adding a margin to allow for a phase measurement noise;
processing all phase measurements of said second baseline to determine a second antenna position;
verifying that a surviving one of said antenna vertical coordinates is within a limit set by a maximum vehicle roll and pitch;
verifying that a baselength computed from a second antenna coordinates matches a known baselength to within a noise tolerance;
verifying that an included angle relative to a first antenna agrees with a known angle q to within a noise tolerance;
using a known baselength of said second baseline as a very accurate measurement, a measurement variance R is assigned to cover a non-linearity, accumulating U with this measurement and abandoning a search at any step where U exceeds W.sub.best ;
computing a Y-axis of a reference antenna coordinate system by normalizing a second antenna position vector, which yields a vector of unit length, in geodetic coordinates, that points along the roll axis of an antenna coordinate system, and
determines two degrees of freedom of this coordinate frame, wherein the other antenna position determines a plane containing both antenna baselines and a Z-axis of said antenna coordinate system is defined to be normal to this plane, with a polarity
chosen in such a way that the Z-axis always points somewhere toward the sky;
determining whether any GPS satellite line-of-sight vectors lie below said X-Y plane, wherein valid phase measurements are not possible for signal direction of arrival that is below the antenna ground plane and where a cutoff angle of -5.degree.
is used to allow for noise in a frame orientation, and to allow a margin for low-elevation gain;
mechanizing a five-state filter for small roll, pitch and yaw angles that rotate this frame into an antenna orientation that yields a closest agreement with measured phase, wherein closest agreement is equivalent to minimum weighted-fit error W
using eight phase measurements;
comparing W from with a best W, if the new W is better, then updating W.sub.best and saving this set of integers and associated attitude as a MLE optimum;
continuing to search until all integer combinations have been exhausted; and
outputting a value representing the relative attitude of said baselines drawn among said GPS antennas.
7. The method of claim 6, wherein:
said v.sub.i and K.sub.i are pre-computed in said computer in a computer-implemented method expressible in computer program pseudocode as: |
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Claims |
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Description |
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A portion of the disclosure of this patent
document contains material which is subject to copyright protection. The copyright owner has no objection to facsimile reproduction by anyone of the patent document or the patent disclosure, as it appears in the Patent and Trademark Office patent file
or records, but otherwise reserves all rights whatsoever.
BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention relates generally to attitude determination with global positioning system (GPS) satellite signals and more specifically to the resolution of carrier cycle integer ambiguities in the carrier phase difference measurement between
separate GPS antennas attached to a rigid structure.
2. Description of the Prior Art
The United States Department of Defense has placed in orbit a group of satellites as part of a global positioning system (GPS) that can be used by civilians and the military alike to get automated and highly-accurate earth position coordinates on
easy-to-read digital displays. Determining where you are has been a particular problem for seafarers for thousands of years. Now, GPS enables small sailboat owners and even combat soldiers to get their positions to within several meters using hand-held
portable equipment.
GPS-based attitude determination offers significant cost savings in applications where inertial guidance has traditionally been the standard approach. Attitude is measured by differential measurements of GPS carrier phase between two or more
antennas. Performance may be characterized in terms of accuracy and bandwidth, both being dependent on application specific parameters, such as the antenna spacing and the carrier-to-noise ratio.
Factors which limit performance are multipath, carrier-to-noise ratio, and integer resolution. Techniques are available for working around multipath and increasing the bandwidth of differential carrier phase tracking.
The rapid resolution of integer ambiguities in measured GPS carrier phase data is a principal obstacle in high performance systems. Integer ambiguity makes it difficult to determine the integer number of carrier cycles that occur between the
antennas and the cable paths. For example, as shown in FIG. 1, a single carrier intersects two antennas. The phase angle at the first antenna is zero degrees, and the phase angle at the second antenna is 72 degrees. There may, however, be an
additional full cycle between the antennas. As the signals travel their respective paths from the antennas to a pair of correlators, additional phase shifts appear. If the first antenna cable path is 3.6 wavelengths, and the second is 0.8 wavelengths,
the correlator connected to the first antenna sees a signal sin (.omega.t-1296 degrees), while the correlator connected to the second antenna sees a signal sin (.omega.t+72+360-288 degrees). The signal correlators measure the first-difference carrier
phase, the phasor difference between the signals seen at a channel one and a channel two correlator input. For this example, the basic output is a phase measurement of 288 degrees, or 0.8 wavelengths.
Integer Ambiguity Resolution
Since GPS carrier frequencies are so high, the wavelengths are short enough (e.g., 19.02 cm) that phase differences used in attitude determination can easily exceed 360.degree.. Resolving whether a phase difference is within
0.degree.-360.degree., or 360.degree.-720.degree., or 720.degree.-1080.degree., or some other whole cycle, requires more than just a simple phase measurement. Integer resolution has developed into a significant performance issue for attitude
determination. It is clear that for attitude determination with GPS to be viable, the integers must be resolvable quickly and reliably under all conditions.
An important element of attitude determination is the separation between translational and rotational degrees of freedom. The choice of a reference point on a platform can be completely arbitrary, if kinematic considerations alone govern the
separation of translation and attitude. Platform translation can be effectively removed from a differential measurement. For example, a platform with a single baseline can be constructed using two antennas, a master and a slave. Without any loss of
generality, the master antenna location may be defined as a fixed reference point. The possible slave antenna locations are constrained to lie on the surface of a virtual sphere surroundings the master antenna location and having a radius equal to the
length of the baseline between the antennas.
There are numerous prior art methods that are used for resolving integer ambiguities. These include integer searches, multiple antennas, multiple GPS observables, motion-based approaches, and external aiding.
Search techniques work well for smaller baselines (on the order of a couple of carrier wavelengths), but they often require significant computation time and are vulnerable to erroneous solutions when longer baselines are used or when only a few
satellites are visible. More antennas can improve reliability considerably. If carried to an extreme, a phased array results whereby the integers are completely unambiguous and searching is unnecessary. But for economy, the minimum number of antennas
required to quickly and unambiguously resolve the integers, even in the presence of noise, is preferred.
One method for integer resolution is to make use of the other observables that modulate a GPS timer. The pseudo-random code can be used as a coarse indicator of differential range, although it is very susceptible to multipath problems.
Differentiating the L1 and L2 carriers provides a longer effective wavelength, and reduces the search space. However, dual frequency receivers are expensive because they are more complicated. Motion-based integer resolution methods make use of
additional information provided by platform or satellite motion. But such motion may not always be present when it is needed.
Another prior art method and apparatus for precision attitude determination and kinematic positioning is described by Hatch, in U.S. Pat. No. 4,963,889, comprises the steps of:
determining the approximate initial relative position of a secondary antenna that is freely movable with respect to a reference antenna;
making carrier phase measurements based on the reception of "N" number of satellites, where N is the minimum number of satellites needed to compute the relative position of the secondary antenna;
deriving from the carrier phase measurements an initial set of potential solutions for the relative position, wherein the initial set of potential solutions all fall within a region of uncertainty defined by a sphere having a radius equal to the
maximum distance between the two antennas, and wherein multiple potential solutions arise because of whole-cycle ambiguity of the carrier signal;
making redundant carrier phase measurements based on the reception of a carrier signal from an additional satellite (N+1); and
eliminating false solutions from the initial set of potential solutions, based on a comparison of the redundant carrier phase measurements with the initial set of potential solutions, to reduce number of potential solutions to close to one,
whereby the number of potential solutions is not increased by use of the redundant carrier phase measurements.
"Deriving from the carrier phase measurements an initial set of potential solutions" means deriving the initial set from just two satellites. The rest of the Hatch specification explains why N is exactly two in the case of attitude
determination. Planar intersections of wave fronts are formed from the two satellites, thus obtaining a collection of parallel lines. The intersection points of these lines and a baseline sphere are determined, producing the initial set of potential
solutions. For example, at column 12, line 35 of Hatch, there are 188 points or potential solutions in the initial set. The phrase "eliminating false solutions from the initial set of potential solutions," means eliminating 187 of those 188 points.
The idea of potential solutions refers to the initial set of 188 points.
The Hatch method and the present invention are fundamentally different. The Hatch method is to form an initial collection of around 188 potential solutions using just two satellites, and then to use phase measurements of the remaining satellites
to whittle away at that small initial collection, leaving only one candidate solution if phase measurements are accurate enough. The Hatch method avoids having to deal with large numbers of integer combinations. The present invention considers all of
the thousands or millions of integer combinations as potential solutions, and solves for the one integer combination that produces the maximum likelihood best fit with the observed phase measurements. The present invention makes a frontal attack on the
problem of large numbers of combinations, and finds a practical solution. Hatch teaches against the present invention right in his patent application. The Hatch method has the advantage that it is workable and fast provided that phase measurement
errors are on the order of one or two degrees. The present invention has the advantage that it is much more robust in the presence of phase measurement errors which may easily be on the order of ten to twenty degrees in a practical embodiment.
The Hatch method, as well as the other prior art methods, require more computation power or time than is typically available and sometimes do not produce unique solutions. A system of rapidly determining unique solutions by rapidly and simply
resolving integer ambiguities is therefore needed.
SUMMARY OF THE PRESENT INVENTION
It is therefore an object of the present invention to provide a method for comparing the relative phase of carrier signals received from GPS satellites to determine the roll, pitch and azimuth of an antenna platform in real-time.
It is a further object of the present invention to provide a system for the rapid resolution of integer ambiguities in measured GPS carrier phase data.
Briefly, a preferred embodiment of the present invention is a computer-implemented method that substantially eliminates integer ambiguity in GPS carrier phase measurements. The method comprises receiving GPS carrier signals and measuring phase
differences containing integer ambiguities and then finding the one integer combination and associated antenna platform attitude that gives the best fit with the measured phase differences. If antenna baselengths are one meter, then there are ten
carrier phase integer possibilities associated with each satellite-antenna combination. With two antenna baselines and four satellites, there are ten integer possibilities associated with eight such combinations, producing
0.times.10.times......times.10=10.sup.8, or one hundred million integer combinations. The method effectively searches the entire collection of 10.sup.8 integer combinations and finds the one-and-only combination that gives the best fit with the measured
data. The method accomplishes a maximum likelihood optimization over the full range of integer and attitude possibilities. It does not search out a small subset of possibilities obtained by a rule-of-thumb. The method includes a practical means to
search 10.sup.8 combinations efficiently, so that a typical embodiment can search the entire range of combinations in less than half a second. A decision tree is formed expressing the possible combinations of integers. A weighted fit error is computed
at one or more stages of the tree whereby a determination may be made that none of the integer combinations connected to the current branch of the tree can be optimum. The current branch accordingly may be cut, thus reducing computation time. A final
integer combination is selected on the basis of best fit with the measured phase data, as determined by the weighted fit error.
An advantage of the present invention is that it provides a system in which roll, pitch and azimuth outputs are available at two Hz, even in the complete absence of integer continuity (i.e., continuos receiver lock on the GPS signals) between
successive phase measurements.
Another advantage of the present invention is that it provides a system that will produce usable outputs with phase measurement errors of up to 40.degree. RMS.
Another advantage of the present invention is that it provides a system that arrives at a maximum likelihood estimation (MLE) optimum solution over the full range of integers and vehicle attitudes, and does so efficiently. With antenna
separations on the order of a meter, with two antenna baselines and three to six satellites, there can be thousands of possible integer combinations. The present method finds the one and only optimum integer combination and associated vehicle attitude
out of all possible integers and attitudes.
Another advantage of the present invention is that it provides a system that can operate with large phase error levels (e.g., by at least a factor of ten compared to the prior art). At a three degree RMS phase error, the present method will
support antenna baselengths of eight feet, yielding a solution under dynamics on 95.6% of the two Hz cycles, with RMS absolute azimuth accuracy of 0.43 mils unfiltered, for example.
These and other objects and advantages of the present invention will no doubt become obvious to those of ordinary skill in the art after having read the following detailed description of the preferred embodiments which are illustrated in the
various drawing figures.
IN THE DRAWINGS
FIG. 1 is a block diagram of a differential phase measurement unit for a GPS positioning system, according to an embodiment of the present invention, with a phase difference of 432.degree. between the two antennas;
FIG. 2 is a decision tree illustrating the integer search method of an embodiment of the present invention;
FIG. 3 is a diagram showing an antenna configuration of an embodiment of the present invention;
FIG. 4 illustrates a two-dimensional example problem which is used as a simple analog to describe a method embodiment of the present invention;
FIG. 5 geometrically illustrates the source of a non-linearity problem in baselength measurement;
FIG. 6 geometrically illustrates the difference between linearized estimates and nonlinear estimates;
FIG. 7 geometrically illustrates non-linearity in an included angle measurement;
FIGS. 8(a)-8(n) each are probability distribution diagrams for W-U found in Monte Carlo simulations of the method of the present invention; and
FIG. 9 diagrams a computer-implemented method of the present invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
FIG. 1 shows an embodiment, referred to by the general reference number 10, of the present invention for comparing the relative phase of carrier signals received from GPS satellites to determine the roll, pitch and azimuth attitude of ships,
aircraft, land vehicles and survey instruments carrying an antenna array. A primary technical obstacle overcome with the present invention is the rapid resolution of integer ambiguities in the measured carrier phase data. The cause of the integer
ambiguity is the inability of the equipment to determine the integer number of carrier cycles that have occurred between the antennas and the cable paths. FIG. 1 shows the basic system 10 measurement generation. A GPS carrier signal 11, intersects a
pair of antennas 12 and 14, which are separated in space. System 10 further comprises a channel-one RF stage 16, a computer 17, a channel-one in-phase (I) correlator 18, a 386-type personal computer (PC) 19, a channel-one quadrature (Q) correlator 20, a
387-type math coprocessor 21 for 64-bit precision arithmetic, a channel-two in-phase correlator 22, a computer memory 23, a channel-two quadrature correlator 24, a computer program 25, a voltage controlled oscillator (VCO) 26, and a channel-two RF stage
28. The phase angle at antenna 12 is 0.degree. (for reference) and the phase angle at antenna 14 is measured at 72.degree.. There is, however, an additional full cycle between the antennas, so the actual phase at antenna 14 is 360.degree.+72.degree.,
or 432.degree.. Additional phase shifts occur as the signals travel along their respective cable paths from antennas 12 and 14 to correlators 18, 20, 22, and 24. In the example shown, the channel-one antenna cable path is 3.6 wavelengths, and the
channel-two is 0.8 wavelengths. Thus, correlators 22 and 24 see a signal sin (.omega.t-1296.degree.) when correlators 18 and 20 see a signal sin (.omega.t+72+360-288.degree.). The signal correlators 18, 20, 22, and 24 are arranged so as to measure the
channel-one difference carrier phase, the phasor difference between the signals seen at channel-one and channel-two correlator inputs. For this example, the basic system 10 hardware output is a phase measurement of 288.degree., or 0.8 wavelengths.
There are typically eight such measurements at two Hz, corresponding to four satellites and two antenna pairs.
System 10 is applicable to a military tank for aiming a cannon while moving at top speed through rough terrain and slamming into firing position. The method herein is operable in such environments because it computes a completely new solution
each cycle. Outputs of roll, pitch and azimuth are available at two Hz, even in the complete absence of integer continuity between successive phase measurements and prior information about vehicle attitude. It is anticipated that frequent interruption
of signals may arise due to obstructions such as freeway overpasses, shading from the vehicle itself, and antenna sky coverage problems when the vehicle is rolled or pitched. High levels of multipath interference are also anticipated due to reflections
off the vehicle, the ground and surrounding objects. Also, cable electrical path lengths can vary with differential heating and rough handling.
Method of Resolving Integer Ambiguities
An attitude determination problem is formulated as a maximum likelihood estimation (MLE) optimization, where vehicle attitude and the integers are regarded as unknown parameters to be adjusted in order to maximize the probability of the
first-difference carrier phase measurements that are actually generated in the hardware of system 10. That formulation results in weighted-fit error "W" as the objective criterion to minimize. A Kalman filter is introduced, having the same objective
criterion. Avoiding unnecessary computation in the Kalman filter leads to a decision tree for the integers. A nested loop is devised that implements the tree, produces all the integer combinations, and merges the optimization over vehicle attitude with
the optimization over integers. Two ways are then introduced to prune the tree, cutting off entire branches at points where it can be guaranteed that all integer combinations further down the same branch are not optimum. The first way is to exclude
impossible combinations, for example, those that produce an antenna upside down. The second is to generate a lower bound for W at each branch of the tree using: ##EQU1## where m is the number of measurements, y.sub.i is the measurement residual, and
v.sub.i is the innovations variance, all produced as byproducts of the Kalman filter method. A running sum U.sub.i of y.sub.i.sup.2 /v.sub.i is kept at each stage i moving down the tree. When that sum exceeds a reasonableness bound, or the current best
W found elsewhere in the search, it is guaranteed that all subsequent integer combinations further down the current branch will produce an even larger W and are not optimum. The remainder of the current branch can then be cut off, speeding up the
search. Equation (1) is derived in one dimension and can be demonstrated to eleven-digit accuracy in four dimensions.
Once past the above basic approach, the application of the approach to a three antenna system is developed. Next, the performance of the method is described, as implemented with three antennas. Satisfactory performance has been obtained with
antenna baselengths of a meter, and in the presence of phase measurement errors of nine degrees, RMS. There is a performance tradeoff between antenna baselength and phase measurement accuracy. The method has been demonstrated with phase measurement
errors of up to forty degrees RMS when the antenna baselengths have been shortened. Alternatively, if antenna location and other factors permit phase measurement accuracies better than nine degrees RMS, the antenna baselines may be longer.
Problem Formulation and Basic Solution
The basic system problem formulation and solution as an MLE optimization, including development of an efficient method to locate the MLE best estimate, is treated next. Performance of the estimate is treated herein later, as is the application
of the basic solution to a three-antenna system. This explanation is limited to a single antenna pair or baseline, and assumes the system to be linear, aside from the integer ambiguity of the measurements.
If system 10 could measure whole-value phase r without the integer ambiguity:
where
i=satellite 1..6;
j=antenna baseline 1..2, or more;
LOS.sub.i =line-of-sight unit vector that points from receiving antenna toward satellite i, expressed in the local East/North/Up (ENU) coordinate system;
a.sub.j =antenna baseline vector, in units of wavelengths, connecting the two antennas, the displacement vector from antenna 12 to 14 in FIG. 1, expressed in ENU coordinates;
b.sub.j =line variation; the residual error remaining after calibration of cable electrical path length, in units of wavelengths; and
.epsilon..sub.ij =phase measurement error; the accumulation of all errors in the phase measurement process, excluding line variation.
All units of distance and phase are in wavelengths, .lambda.=19.02 cm, to simplify the notation. The local East/North/Up (ENU) coordinate system is used exclusively. It is assumed that cable bias, 3.6.lambda.-0.8.lambda. in FIG. 1, has been
removed from the measured phase data by an earlier calibration procedure, but there remains a residual line variation b.sub.j .about.N(0,R.sub.b), denoting a Gaussian random error with mean zero and variance R.sub.b. Phase measurement error,
.epsilon..sub.ij -N(0,R.sub.ij) is the Sum of multipath, differential antenna phase center motion, thermal noise, crosstalk, chip bias, smear, transient response of the correlators, and anything else that affects phase measurement accuracy, unless it
affects all r.sub.ij equally, in which case it is lumped together with line variation: ##EQU2##
The receiver is unable to detect whole-value phase r.sub.ij and can only detect phase modulo 360.degree. (one wavelength),
The unknown integers are,
and equation (2) becomes,
An optimal estimate is sought for a.sub.j given the available phase measurements .theta..sub.ij. The n.sub.ij and b.sub.j are nuisance parameters.
Estimation problems with a non-linearity such as (r.sub.ij m | | |
