CN107966718B - Improved integer ambiguity searching method - Google Patents

Improved integer ambiguity searching method Download PDF

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CN107966718B
CN107966718B CN201711182460.0A CN201711182460A CN107966718B CN 107966718 B CN107966718 B CN 107966718B CN 201711182460 A CN201711182460 A CN 201711182460A CN 107966718 B CN107966718 B CN 107966718B
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卢立果
刘万科
马立烨
吴汤婷
鲁铁定
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East China Institute of Technology
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Abstract

The invention discloses an improved integer ambiguity searching method, and relates to the technical field of satellite navigation positioning. Aiming at the problem that the search time is long when the floating ambiguity resolution precision of the conventional SEVB algorithm is poor, the improved algorithm can effectively reduce the number of ambiguity search candidate points and unnecessary redundant calculation by limiting the size of an initial search space and optimizing the calculation process, so that the search time is reduced, and the ambiguity resolution efficiency is improved; compared with an SEVB algorithm, the improved algorithm has more obvious efficiency advantage in the aspects of low-precision and high-dimensional ambiguity searching, and can ensure that the ambiguity resolving efficiency is relatively more stable and reliable on the whole.

Description

Improved integer ambiguity searching method
Technical Field
The invention relates to the technical field of satellite navigation and positioning, in particular to an improved integer ambiguity searching method.
Background
The fast and accurate resolving of the phase integer ambiguity is a key problem of GNSS real-time high-precision positioning and a hotspot problem in the GNSS research field. Among many Integer ambiguity resolution methods, a search method based on the Integer Least Squares (ILS) is widely adopted because of having the highest success rate. ILS search refers to finding out a group of integer solutions meeting the minimum quadratic form of ambiguity residual errors in a given space, and in order to quickly obtain the group of integer solutions, the SEVB algorithm based on an oscillatory shrinkage strategy is the most popular at present. Wherein, the oscillation type search strategy means that enumeration is in a Z shape from Bootstrap estimation points to two sides; the contraction strategy means that each time a set of solutions with smaller quadratic form is obtained, the search space is updated. Through the two strategies, the SEVB algorithm can quickly reduce the search space, so that the number of search nodes is reduced, and the ambiguity is quickly estimated, therefore, the algorithm is also adopted by the LAMBDA method.
When the SEVB algorithm is adopted for processing, firstly, an initial search space is set to be infinite so as to ensure that the obtained first group of integer candidate solutions are Bootstrap estimation solutions, and then, the contraction and oscillation enumeration processes of the search space are carried out based on the group of integer solutions. However, when the ambiguity resolution accuracy is poor, the success rate of the Bootstrap is low, the obtained Bootstrap estimation solution deviates from the ILS estimation solution more, the obtained search space is relatively large, and the search radius can be updated only by performing enumeration for multiple times, so that the search process is slow and the time consumption is obviously increased.
In summary, in the prior art, there is a problem that the ambiguity search space is large, which results in low search efficiency.
Disclosure of Invention
The embodiment of the invention provides an improved integer ambiguity searching method, which is used for solving the problem of low searching efficiency caused by large ambiguity searching space in the prior art.
The embodiment of the invention provides an improved integer ambiguity searching method, which comprises the following steps:
step 1: determining an original ambiguity float solution and an original variance covariance matrix from data received by a terrestrial receiver, assuming
Figure BDA00014794566800000215
And
Figure BDA00014794566800000216
respectively carrying out integer transformation on an original ambiguity floating solution and an original variance covariance matrix to obtain an ambiguity floating solution and a variance covariance matrix, wherein according to an integer least square estimation criterion, the objective function is as follows:
Figure BDA0001479456680000021
wherein z is an integer solution with optimal ambiguity; and is aligned with
Figure BDA0001479456680000022
Cholesky decomposition of the matrix yields:
Figure BDA0001479456680000023
wherein the content of the first and second substances,
Figure BDA0001479456680000024
is a lower triangular matrix of the unit,
Figure BDA0001479456680000025
is a diagonal matrix;
step 2: assuming that the size of the search space corresponding to the objective function in step 1 is χ2Then, there are:
Figure BDA0001479456680000026
defining sequential condition estimates
Figure BDA0001479456680000027
Substituting the formula to obtain:
Figure BDA0001479456680000028
wherein the content of the first and second substances,
Figure BDA0001479456680000029
is a matrix
Figure BDA00014794566800000210
The main diagonal elements of (a) further include:
Figure BDA00014794566800000211
Figure BDA00014794566800000212
wherein the content of the first and second substances,
Figure BDA00014794566800000213
is ziThe condition estimate of (1);
and step 3: firstly, a proper initial search space is calculated before the first search
Figure BDA00014794566800000214
Deriving an improved integer ambiguity search formula from the first two formulas in the step 2, and performing integer ambiguity search by using the improved integer ambiguity search formula to obtain a positioning result; the improved integer ambiguity search formula is as follows:
Figure BDA0001479456680000031
starting from a position where i is equal to n, the improved integer ambiguity searching algorithm searches layer by adopting a depth-first method, and the size of a searching interval in each layer depends on a searching space chi2And the conditional variance d corresponding to the layeri。
Preferably, the calculation process of the sequential condition estimation in step 2 is optimized:
Figure BDA0001479456680000032
Figure BDA0001479456680000033
where kold is the number of layers in which the node was last enumerated.
Preferably, the initial search space in step 3
Figure BDA0001479456680000034
The calculation method specifically includes:
if ncands is not more than n +1, n is the ambiguity dimension, and ncands is the number of candidate groups to be output, constructing n +1 quadratic values; the first quadratic value is obtained by sequentially performing integer calculation on the ambiguity floating point solution; then, sequentially measuring a sub-component in the n-dimensional ambiguity to be a sub-integer according to a sequential integer method to obtain the rest n quadratic values; the n +1 quadratic values are sorted from small to large, and the ncands quadratic values are taken as the initial space size
Figure BDA0001479456680000035
If ncands > n +1, the ellipsoid volume formula is used:
Figure BDA0001479456680000036
in the formula, EnFor searching for an ellipsoid volume, andrelationships betweenFormula intEnNcands; | is a determinant value; vnAs a volume function, the formula is:
Figure BDA0001479456680000037
obtaining an initial space size:
Figure BDA0001479456680000038
in the embodiment of the invention, an improved integer ambiguity searching method is provided, and compared with the prior art, the improved integer ambiguity searching method has the following beneficial effects: the invention provides an effective integer ambiguity searching algorithm, which can effectively reduce the number of ambiguity searching candidate points and unnecessary redundant calculation, thereby improving the searching efficiency. Compared with the existing SEVB algorithm, the improved algorithm has more obvious efficiency advantage in the aspects of low-precision and high-dimensional ambiguity searching, and can ensure that the ambiguity resolving efficiency is relatively more reliable and stable on the whole.
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FIG. 1 is a flow chart of an improved integer ambiguity search method according to an embodiment of the present invention;
fig. 2 is a depth-first search tree according to an embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Fig. 1 is a flowchart of an improved integer ambiguity searching method according to an embodiment of the present invention. Referring to fig. 1, the method includes:
step 1: determining an original ambiguity float solution and an original variance covariance matrix from data received by a terrestrial receiver, assuming
Figure BDA0001479456680000041
And
Figure BDA0001479456680000042
respectively, for performing integer on original ambiguity floating solution and original variance covariance matrixThe ambiguity floating solution and the variance covariance matrix after number transformation can obtain a target function according to an integer least square estimation criterion, wherein the target function is as follows:
Figure BDA0001479456680000043
wherein z is the ambiguity-optimized integer solution.
To pair
Figure BDA0001479456680000044
The matrix was subjected to Cholesky decomposition:
Figure BDA0001479456680000045
wherein the content of the first and second substances,
Figure BDA0001479456680000046
is a lower triangular matrix of the unit,
Figure BDA0001479456680000047
is a diagonal matrix.
It should be noted that the integer transformation refers to multiplying the original ambiguity floating solution and variance covariance matrix by an integer transformation matrix Z to improve the characteristics, and in practical implementation, the decorrelation algorithm in LAMBDA is directly adopted.
Step 2: assuming that the size of the search space corresponding to the objective function in step 1 is χ2Then, there are:
Figure BDA0001479456680000051
defining sequential condition estimates
Figure BDA0001479456680000052
Substituting the formula to obtain:
Figure BDA0001479456680000053
wherein the content of the first and second substances,
Figure BDA0001479456680000054
is a matrix
Figure BDA0001479456680000055
The main diagonal elements of (a) further include:
Figure BDA0001479456680000056
Figure BDA0001479456680000057
wherein the content of the first and second substances,
Figure BDA0001479456680000058
is ziThe condition estimation of (1).
It should be noted that the calculation process of the condition estimation in this step is improved on the basis of the SEVB. The reason for this is that more dead nodes will be generated when the search space is larger (as shown in fig. 2, where 2, 4 and 5 of the third row areIs composed of Dead node) Since the integer search node is not included after the dead node, if the rest layer is continuously updated based on the dead node
Figure RE-GDA0001550490160000059
Will result in unnecessary values of
Figure RE-GDA00015504901600000510
And (4) calculating.
The existing SEVB algorithm calculates sequential condition estimates using the following formula:
Figure BDA00014794566800000511
the above equation indicates that S (k, j), (j ═ 1, 2.., k) needs to be recalculated each time the condition estimation is calculated, and when the search encounters a dead node (especially when the search space is large, more dead nodes are included), the calculation information of the 1 st layer k-1 is actually unnecessary, i.e., calculation redundancy is generated.
The improved calculation formula is as follows:
Figure BDA00014794566800000512
Figure BDA00014794566800000513
wherein, kold is the number of layers in which the node is positioned in the last node enumeration; an improved algorithm converts S (k, j), (j ═ 1, 2., k) into a calculation S (k, k), which can reduce the amount of unnecessary calculation, thereby reducing the overall search time.
And step 3: firstly, a proper initial search space is calculated before the first search
Figure BDA0001479456680000061
The reason is that when the precision of the floating ambiguity is poor, the difference between the Bootstrap estimation solution and the ILS estimation solution is large, which causes the first search space χ′2The search space can be reduced to a smaller range only by repeating unnecessary node enumeration for many times, so that the search time is greatly increased. The improved algorithm calculates an initial value which satisfies both the existence of a solution in the space and is relatively small by constraining the initial space size
Figure BDA0001479456680000062
Unnecessary node enumeration processes can be effectively reduced, and therefore searching efficiency is improved. Firstly, a proper initial search space is calculated before the first search
Figure BDA0001479456680000063
Deriving an improved integer ambiguity search formula (starting from the position where i is equal to n) from the first two formulas in the step 2, and performing integer ambiguity search by using the improved integer ambiguity search formula to obtain a positioning result; improved integer ambiguity search formulaAs follows:
Figure BDA0001479456680000064
in this step, an improved SEVB search algorithm (MSEVB) adopts a depth-first method to search layer by layer, and the size of a search interval in each layer depends on a search space χ2And the conditional variance d corresponding to the layeriBefore the first search, a more appropriate initial search space needs to be calculated
Figure RE-GDA0001550490160000065
Firstly, sequentially rounding from the nth layer to the 1 st layer to obtain a first ambiguity candidate set (called Bootstrap estimation solution), and taking an objective function value F (z) of the set of solution as a new search space size χ′2(ii) a Then updated χ′2And continuing searching, if the layer 1 still has a solution, obtaining a candidate group again, and if the layer 1 does not have a solution, returning to the next integer point of the layer 2. And so on, when the target function value of the new candidate group is less than the current space size x′2Then, update χ′2Value, enabling the continual narrowing of the search space.
To make the search space shrink faster, condition estimation from ambiguity is needed
Figure BDA0001479456680000066
Begin a "zigzag" oscillatory search:
Figure BDA0001479456680000067
where i is the current number of layers and [ ] is the rounded symbol.
The initial search space can be calculated in the following two cases:
the first case: if ncands is less than or equal to n +1(n is the ambiguity dimension, and ncands is the number of candidate groups to be output), n +1 quadratic values are constructed. The first quadratic value is obtained by sequentially rounding the ambiguity floating-point solutionCalculating to obtain; and then, sequentially measuring a sub-component in the n-dimensional ambiguity to be close to an integer according to a sequential integer method to obtain the rest n quadratic values. The n +1 quadratic values are sorted from small to large, and the ncands quadratic values are taken as the initial space size
Figure BDA0001479456680000071
The second case: if ncands > n +1, the ellipsoid volume formula is used:
Figure BDA0001479456680000072
in the formula, EnFor searching for an ellipsoid volume and having an approximate relationship intEnNcands; | is a determinant value; vnAs a volume function, the formula is:
Figure BDA0001479456680000073
the initial space size can be found:
Figure BDA0001479456680000074
in summary, the present invention provides an effective integer ambiguity search algorithm, which can effectively reduce the number of ambiguity search candidate points and unnecessary redundant computation, thereby improving the search efficiency. Compared with the existing SEVB algorithm, the improved algorithm has more obvious efficiency advantage in the aspects of low-precision and high-dimensional ambiguity searching, and can ensure that the ambiguity resolving efficiency is relatively more reliable and stable on the whole.
The above disclosure is only a few specific embodiments of the present invention, and those skilled in the art can make various modifications and variations of the present invention without departing from the spirit and scope of the present invention, and it is intended that the present invention encompass these modifications and variations as well as others within the scope of the appended claims and their equivalents.

Claims (1)

1. An improved integer ambiguity search method, comprising:
step 1: determining an original ambiguity float solution and an original variance covariance matrix from data received by a terrestrial receiver, assuming
Figure FDA0002844604470000011
And
Figure FDA0002844604470000012
respectively carrying out integer transformation on an original ambiguity floating solution and an original variance covariance matrix to obtain an ambiguity floating solution and a variance covariance matrix, wherein according to an integer least square estimation criterion, the objective function is as follows:
Figure FDA0002844604470000013
wherein z is an integer solution with optimal ambiguity; and is aligned with
Figure FDA0002844604470000014
Cholesky decomposition of the matrix yields:
Figure FDA0002844604470000015
wherein the content of the first and second substances,
Figure FDA0002844604470000016
is a lower triangular matrix of the unit,
Figure FDA0002844604470000017
is a diagonal matrix;
step 2: assuming that the size of the search space corresponding to the objective function in step 1 is χ2Then, there are:
Figure FDA0002844604470000018
defining sequential condition estimates
Figure FDA0002844604470000019
Substituting the formula to obtain:
Figure FDA00028446044700000110
wherein the content of the first and second substances,
Figure FDA00028446044700000111
is a matrix
Figure FDA00028446044700000112
The main diagonal elements of (a) further include:
Figure FDA00028446044700000113
Figure FDA00028446044700000114
wherein the content of the first and second substances,
Figure FDA00028446044700000115
is ziThe condition estimate of (1);
and step 3: firstly, a proper initial search space is calculated before the first search
Figure FDA00028446044700000116
Deriving an improved integer ambiguity search formula from the first two formulas in the step 2, and performing integer ambiguity search by using the improved integer ambiguity search formula to obtain a positioning result; the improvement beingThe integer ambiguity search formula of (a) is as follows:
Figure FDA00028446044700000117
starting from a position where i is equal to n, the improved integer ambiguity searching algorithm searches layer by adopting a depth-first method, and the size of a searching interval in each layer depends on a searching space chi2And the conditional variance d corresponding to the layeri；
And (2) optimizing the calculation process of the sequential condition estimation:
Figure FDA0002844604470000021
Figure FDA0002844604470000022
wherein, kold is the number of layers in which the node is positioned in the last node enumeration;
initial search space in step 3
Figure FDA0002844604470000023
The calculation method specifically includes:
if ncands is not more than n +1, n is the ambiguity dimension, and ncands is the number of candidate groups to be output, constructing n +1 quadratic values; the first quadratic value is obtained by sequentially performing integer calculation on the ambiguity floating point solution; then, sequentially measuring a sub-component in the n-dimensional ambiguity to be a sub-integer according to a sequential integer method to obtain the rest n quadratic values; the n +1 quadratic values are sorted from small to large, and the ncands quadratic values are taken as the initial space size
Figure FDA0002844604470000024
If ncands > n +1, the ellipsoid volume formula is used:
Figure FDA0002844604470000025
in the formula, EnFor searching for an ellipsoid volume, andrelationships betweenFormula intEnNcands; | is a determinant value; vnAs a volume function, the formula is:
Figure FDA0002844604470000026
obtaining an initial space size:
Figure FDA0002844604470000027
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