NavCom SF-3050 A User Manual

A COMPUTATIONALLY EFFICIENT AMBIGUITY
RESOLUTION TECHNIQUE
Ron Hatch, Tenny Sharpe, NavCom Technology, Inc.
BIOGRAPHY
Ron Hatch is the Director of Navigation Systems at NavCom Technology, a company of which he was a co­founder. He has developed a number of innovative techniques for processing GPS measurements and has obtained more than a dozen patents related to GPS. Ron is currently the president of the Institute of Navigation, is a Fellow of the ION and has received the Kepler and Thurlow awards from the ION.
Mr. Tenny Sharpe is Director of Advanced Development at NavCom Technology Inc. Mr. Sharpe received a B.S. in Physics from Case Institute of Technology and a M.S. in Computer Science from the University of California, Los Angeles. Mr. Sharpe has over 30 years experience in the development of aerospace and industrial electronics. His specializations are software and systems design for GPS navigation systems.
ABSTRACT
A new method of carrier-phase ambiguity resolution is described. The new technique is a variation of the least­squares residual search technique in the ambiguity domain. It uses a very efficient algorithm to compute the residuals associated with each potential combination of ambiguities to be tested. Several other techniques are employed to simplify the calculations and to enhance the probability of identifying the correct ambiguity vector. The intent is to minimize the number of data epochs required to correctly identify the integer ambiguity values.
The capability of the technique to rapidly identify the correct ambiguity vector is illustrated by plotting the results of scoring runs which exercise the search algorithms using recorded field data taken over different rover-to-base separation distances. To maximize the number of searches exercised in these runs, as soon as the ambiguity vector is identified, the results are cleared and a new search is initiated with all navigation parameters re­initialized. Some navigation results are also shown which are typical of RTK carrier-phase navigation results.
INTRODUCTION
The RTK ambiguity resolution technique implemented within the NavCom dual-frequency receiver has several unique features designed to minimize the computational task while ensuring that a minimal data collection interval is required. The fundamental approach is similar to several least-squares residual search techniques which have been employed by others. [1-4]
The first unique feature is that the base station transmits corrections rather than the raw data, which most RTK implementations transmit. This has several advantages:
1) it offloads part of the computation from the user receiver to the base station receiver; 2) it allows code smoothing of the base station data to occur even before the user receiver is turned on, which ensures a more accurate initial code solution; 3) it simplifies the processing algorithms because no differencing across receivers is required.
The second unique feature is a simplified computational technique to generate the residuals resulting from candidate ambiguity vectors. This technique is the main subject of the paper and will be explored in some detail. The search process is a two-stage process which depends upon the availability of dual-frequency measurements at both the base-station receiver and the user receiver. First a wide-lane search is performed and up to 10 ambiguity vectors, which meet specific requirements (e.g. residuals less than a threshold value) are saved for further processing. In the second stage, each of the ambiguity vectors determined in the first stage are tested by searching each satellite (except the first to avoid redundant clock solutions) across the two narrow-lane ambiguity values which result in a narrow-lane measured range closest to the corresponding wide-lane measured range. Those narrow-lane combinations, which meet a number of specific criteria, are scored and if more than one are acceptable their relative score is used to determine whether or not one of them can be declared the correct ambiguity vector.
λφφ
INITIAL CODE SOLUTION
The primary motivation in the initial code solution is to obtain the most accurate code solution possible with as few measurement epochs as possible. Since the baseline distances over which the ambiguities can be easily resolved are typically less than 15 kilometers, it is assumed that the ionospheric refraction effects can be modeled at both base station and user with sufficient accuracy that the residual ionospheric errors are almost negligible. This means that the largest errors, which will affect the differential code solution, will be the multipath effects upon the dual-frequency code measurements. The combination of code measurements which has the minimum multipath error (or at least very close to the minimum) is the frequency-weighted sum of the code measurements. This combination also has the advantage that it is exactly matched in ionospheric refraction effects by the wide-lane difference carrier-phase measurement. This means that it can be smoothed by the wide-lane carrier-phase measurement with an increasing time constant, i.e. the multipath error will become smaller and smaller as the satellite is continuously tracked.
Because the base-station receiver is typically put in operation long before the user receiver, our choice of generating the code and carrier phase corrections at the base-station receiver allows the base-station code measurements to be smoothed such that most of the multipath error is removed before the code corrections are generated. This means that the multipath noise contributed by the base station receiver is largely eliminated. This represents a substantial advantage over the normal practice of sending raw base-station measurements to the user.
The frequency weighted combined measurement is:
+
PfPf
where f represents the frequency and P the pseudorange code measurement
The carrier-phase measurement which is affected by the same amount of ionospheric refraction error is:
where N is the integer ambiguity, measurement and
By setting equation (1) equal to equation (2) and solving for the wide-lane ambiguity, N floating ambiguity can be computed each epoch.
N
=
P
.
NC
λ
the wavelength.
=
w
+
+
)(
ff
21
2211
+
PfPf
2211
λ
w
(1)
ff
21
)(
+= (2)
ww
21
φ
is the carrier-phase
, a new value for the
w
)(
φφ
(3)
21
This value for the wide-lane floating ambiguity value can be improved each epoch by computing the recursive average over all epochs up to the present time. This smoothed value of N equation (2) to give a smoothed carrier-phase pseudorange:
One can use the process represented by equations (1) through (4) at both the base-station receiver and at the user receiver. At the base station the smoothed carrier­phase pseudorange is differenced with the true range to yield a pseudorange correction which, after removing the common clock effect, is sent to the user receiver. At the user receiver the correction is applied to the combined frequency-weighted code measurement as represented by equation (1). A wide-lane carrier-phase correction is also generated at the base station by arbitrarily selecting a whole-lane integer in equation (2) and differencing it with the true range. The whole-lane error, which is made by arbitrarily selecting the ambiguity value at the base station is absorbed into the whole-lane ambiguity value at the user receiver. After clock removal, this carrier-phase correction is sent to the user receiver and applied to the wide-lane carrier-phase measurement as represented by equation (2). The significant advantage to this process, as indicated above, is that, due to the separate smoothing, the multipath corruption of the base station measurements is usually largely removed even before the user receiver is turned on. This means that the initial smoothed code measurements at the user receiver are more accurate and a more accurate code solution will be obtained.
At the user receiver, the corrected and smoothed value of equation (4) for each satellite is now used in a least­squares process to obtain an initial code solution.
WIDE-LANE SEARCH
While the process described above, i.e. equation (4), gave us a set of floating (non-integer) ambiguity values for each satellite, a better set (fewer degrees of freedom) can be obtained from the initial code solution. Also we note that given a set of integer ambiguities, a second set, differing from the first by a constant integer value for each satellite, will result in the same identical spatial position but with a different clock value. Because our clock solution is of no interest and can be selected arbitrarily, this means that we can constrain one of the ambiguity values arbitrarily. In our process, we choose to use the highest elevation satellite as the reference satellite and force its ambiguity value to zero. Using the initial code position solution, we can compute a range, R, to each satellite and set it equal to the measurement represented by equation (2). Solving the result for the ambiguity value and subtracting the ambiguity value of the highest elevation satellite gives:
can then be substituted back into
w
NC
λφφ
)(
+= (4)
ww
21
2
)()(/)(
ZHX
IHX
λφφλφ
=
+
w
rii
RRN
w
rrii
φφφφλ
+= (5)
2121
where the superscript, i, represents each satellite in turn and the superscript, r, represents the reference satellite— in this case the highest elevation satellite.
Our initial wide-lane carrier-phase RTK solution involves rounding the floating ambiguities so obtained to their integer values, substituting these values into equation (4) and computing a new least-squares position solution. The floating ambiguity given by equation (5) is used in the search process to sequentially select the closest integer values over the selected search range.
We have found that searching over the four closest values is almost always sufficient to include the true ambiguity value. Given n satellites this means that there will be 4
n-1
permutations of the ambiguity vector. Thus, there will be 1024 permutations when six satellites meet the appropriate elevation criterion and 4096 when seven satellites are available. With each ambiguity permutation a set of wide-lane carrier-phase measurement residuals can be computed. It is these residuals which play a prominent role in determining the correct set of integer ambiguity values. However, obtaining the residuals by re­computing a least-squares solution for each permutation is computationally prohibitive. A much faster way to compute the residuals is to use a residual sensitivity matrix, S.
RESIDUAL SENSITIVITY MATRIX
Given the linearized measurement equation:
where H is the sensitivity matrix (i.e. the direction cosines) of the state vector (position and clock corrections), x, to the innovations (difference between the measurements and their expected value), z.
zHx = (6)
Equation (6) can be expanded to include a set of position and clock corrections corresponding to a set of innovations:
= (7)
Now if we want to see the separate effect on the position of a single whole-cycle change in the ambiguity value of each satellite, Z will become the identity matrix (or depending on units, the identity matrix times the scalar wavelength).
= (8)
The least-squares solution for X is then:
TT
1
= (9)
HHHX
)(
For a weighted least-squares solution this becomes:
111
= RHHRHX
TT
)(
(10)
where R is the measurement covariance matrix.
Note that the left hand side of equation (9) or (10) can be pre-computed and when multiplied by the appropriate column of the identity matrix (or scaled identity matrix) gives the appropriate column of X, which is the associated change in the position and clock caused by the integer ambiguity change.
Multiplying equation (9) by H tells us how much that change in position will affect the innovations.
TT
1
= (11)
HHHHHX
)(
Now if we subtract this change in the innovations from the input value of the innovations (I) we get the effect on the residuals of a whole cycle change in the ambiguity value for each satellite. This is called the residual sensitivity matrix, S, and is given by:
TT
1
= (12)
HHHHIS
)(
The residual sensitivity matrix, S, for a weighted least squares solution is:
111
= RHHRHHIS
The S matrix has a number of interesting properties. It is symmetric. It is idempotent, i.e. S=S
TT
)(
2=S3
=… The sum of
(13)
any row or column equals zero, i.e. residuals are zero mean. The length of any row or column is equal to the square root of the associated diagonal element. Since the solution vector, x, has four elements, the rank of S is n-4 where n is the number of satellites.
The residuals of the initial RTK solution, described in the prior section are updated by adding the product of the S matrix and the matrix formed by scaling the identity matrix diagonal elements by the specific permutation of changes in the integer ambiguity values to be tested. The ten permutations with the smallest root-sum-square (rss) of residuals are saved for further narrow-lane processing if their rss residuals is less than an acceptable threshold value.
STEPPING TO THE NARROW-LANE
Assuming the differential ionospheric refraction is negligible allows us to equate the narrow-lane range to the wide-lane range. Thus:
Solving equation (14) for the L
9
4
17
NN
ambiguity value gives:
1
9
NN (15)
3(
w
17
)()(
+
+=
21111
9
4
17
(14)
ww
)
φφ
211
It is interesting to note that the deviation from an integer value obtained for N
in equation (15) is identical in
1
3
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