CN111650615B - Ambiguity lattice reduction quality evaluation method - Google Patents

Ambiguity lattice reduction quality evaluation method Download PDF

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CN111650615B
CN111650615B CN202010038297.6A CN202010038297A CN111650615B CN 111650615 B CN111650615 B CN 111650615B CN 202010038297 A CN202010038297 A CN 202010038297A CN 111650615 B CN111650615 B CN 111650615B
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CN111650615A (en
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吴汤婷
李大军
鲁铁定
王胜平
王建强
卢立果
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East China Institute of Technology
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Abstract

The invention discloses a fuzzy degree lattice reduction quality evaluation method, which comprises the following steps: firstly, establishing an observation equation by using GNSS observation values, and estimating an ambiguity floating solution by adopting a adjustment method
Figure DDA0002366806510000011
Variance matrix of the system
Figure DDA0002366806510000012
Second, to
Figure DDA0002366806510000013
Performing George decomposition to obtain an upper triangular matrix B, and converting the integer least square problem of the ambiguity into a nearest vector problem on a lattice; again, the B Schmitt is orthogonalized to obtain an orthogonalized matrix B * And a triangular matrix U on a unit; thereafter, the lattice reduction algorithm is adopted for U and B * Sequentially carrying out scale specification and orthogonal base vector length exchange; compared with the existing ambiguity lattice reduction quality evaluation method, the index defined by the invention can consider the whole length of the orthogonal basis vectors and reflect the ordering trend of the orthogonal basis vectors, so that the reduction quality of the reduction algorithm can be accurately and quantitatively evaluated, reference can be provided for the selection of the ambiguity reduction algorithm, and the method has good application value in the aspect of GNSS rapid high-precision positioning.

Description

Ambiguity lattice reduction quality evaluation method
Technical Field
The invention relates to the technical field of satellite navigation positioning, in particular to a ambiguity lattice reduction quality evaluation method.
Background
The key of GNSS high-precision positioning is the resolution of carrier phase ambiguity, and only if the ambiguity is fixed correctly, the carrier phase observation value can be converted into a millimeter-level precision distance observation value, so that high-precision navigation positioning is realized. Therefore, ambiguity resolution is critical for GNSS high precision data processing. To quickly and accurately resolve the integer ambiguity, an integer least squares estimation criterion is typically used to resolve the integer ambiguity. Because integer least square is equivalent to the problem of nearest vector on lattice, in order to realize quick estimation of ambiguity, lattice reduction (reduction for short) is needed to be carried out on the base vector first, so that the search space of the ambiguity is reduced, and the subsequent estimation of integer ambiguity is conveniently carried out by adopting a search algorithm, so that the resolution efficiency of the ambiguity depends on the quality of the reduction.
The evaluation indexes commonly adopted for measuring the protocol quality at present comprise: the reduction of correlation coefficients (Teunissen, 1994), the degree of orthogonality defects (Wang and Feng, 2013) and the condition number (Liu et al, 1999; xu, 2012). Jazaeri et al (2014) and Lu Liguo (2015) indicate that none of these indicators accurately measure protocol quality. Therefore, a new evaluation method is urgently needed to be searched for accurately evaluating the protocol quality of the protocol algorithm, and meanwhile, the method can be used for selecting a more efficient protocol algorithm, so that the quick estimation of the ambiguity is realized, and the requirements of GNSS high-precision real-time quick positioning are met.
Disclosure of Invention
The invention aims to provide a fuzzy degree lattice reduction quality evaluation method for solving the problems in the background technology.
In order to achieve the purpose, the invention provides the following technical scheme:
a fuzzy degree lattice reduction quality evaluation method comprises the following steps:
A. establishing an observation equation by using GNSS observation values, and estimating an ambiguity floating solution by adopting a adjustment method
Figure GDA0004162960690000011
And its variance matrix->
Figure GDA0004162960690000021
B. For a pair of
Figure GDA0004162960690000022
Performing George decomposition to obtain an upper triangular matrix B, and converting the integer least square problem of the ambiguity into a nearest vector problem on a lattice;
C. orthogonalizing B Schmidt to obtain an orthogonalization matrix B * And a triangular matrix U on a unit;
D. the lattice reduction algorithm is adopted for U and B * Sequentially carrying out rulerDegree specification and orthogonal basis vector length exchange;
E. defining the length stability rho of the orthogonal base as an index for evaluating the quality of the specification, and calculating the length stability rho of the orthogonal base before and after the specification Front part ，ρ Rear part (S) Judging ρ Rear part (S) ≤ρ Front part If so, the success of the protocol is indicated, the ambiguity is estimated by adopting a search algorithm, and otherwise, the protocol fails and the protocol needs to be re-regulated.
As a further scheme of the invention: in the step A, an adjustment method is used for estimating an ambiguity floating point solution
Figure GDA0004162960690000023
And variance matrix thereof>
Figure GDA0004162960690000024
The specific implementation steps are as follows:
general formula of GNSS observation equation:
E(y)＝Aa+Bb，P yy (1)
wherein E (& gt) and D (& gt) represent the expected and variance symbols, respectively; y represents an observed value; a and b represent ambiguity and baseline components, respectively; a and B are corresponding coefficient matrixes; p (P) yy Is the weight of the observed value.
The floating solution can be obtained by adopting a classical least square adjustment method
Figure GDA0004162960690000025
Sum of variances matrix->
Figure GDA0004162960690000026
Figure GDA0004162960690000027
wherein ,
Figure GDA0004162960690000028
P B ＝B(B T P yy B) -1 B T ) -1 B T P yy 。
as still further aspects of the invention: the pair in the step B
Figure GDA0004162960690000029
And (3) performing the Cholesky decomposition to convert the integer least squares problem of the ambiguity into the nearest vector problem on the lattice.
As still further aspects of the invention: for a pair of
Figure GDA0004162960690000031
The process of cholesky decomposition was as follows:
Figure GDA0004162960690000032
in the formula, the element B in the upper triangular matrix B ij Solving the formula:
Figure GDA0004162960690000033
wherein ,aij Is that
Figure GDA0004162960690000034
An element; n is->
Figure GDA0004162960690000035
Dimension number.
As still further aspects of the invention: the integer least squares problem is converted into the nearest vector problem on the lattice, and the specific process is as follows:
obtaining ambiguity floating point solution according to step A
Figure GDA0004162960690000036
Sum of variances matrix->
Figure GDA0004162960690000037
Obtaining an integer value of the ambiguity by adopting integer least square estimation, wherein the estimation criterion is as follows:
Figure GDA0004162960690000038
according to
Figure GDA0004162960690000039
Substitution estimation criteria may be:
Figure GDA00041629606900000310
in the formula ,
Figure GDA00041629606900000311
is a constant.
As still further aspects of the invention: in the step C, the Schmidt orthogonalization is adopted, and then the orthogonal matrix B is obtained by decomposing B * And a triangular matrix U on a unit, which comprises the following processes:
Figure GDA0004162960690000041
/>
wherein ,
Figure GDA0004162960690000042
is an orthogonal basis vector, and->
Figure GDA0004162960690000043
u ji Is an orthogonalization coefficient, and->
Figure GDA0004162960690000044
As still further aspects of the invention: the step D adopts a lattice reduction algorithm to pair U and B * The length exchange of the scale protocol and the orthogonal base vector is carried out, and the scale protocol and the orthogonal base vector can be directly realized by adopting an LLL type protocol algorithm in polynomial time such as a classical LLL algorithm, a greedy algorithm, a block algorithm, a base vector deep insertion and a minimum column rotation algorithmExchange process.
As still further aspects of the invention: in the step E, the length stability ρ of the orthogonal base is adopted as an index for evaluating the quality of the specification, and the specific definition is as follows:
Figure GDA0004162960690000045
in the formula ,
Figure GDA0004162960690000046
since the product of the successive orthogonal basis lengths is equal to the determinant of the matrix, i.e
Figure GDA0004162960690000047
The determinant of the variance matrix of the ambiguity is equal to the square of the B matrix after the George decomposition of the variance matrix, i.e.>
Figure GDA0004162960690000048
The range of values of the smoothness ρ is therefore:
ρ∈[1，+∞) (9)
as can be seen from the range of values of ρ, the value of ρ depends on the degree of smoothness of the length of the orthogonal basis, when the orthogonal basis
Figure GDA0004162960690000049
ρ=1 when the length sizes are equal. The quality of the reduction algorithm depends on the smoothness of the orthogonal base vectors, and the smaller the length fluctuation of the base vectors, the better the reduction quality is. Therefore, the closer the ρ value is to the 1-basis vector reduction effect is, the better the ambiguity search is facilitated.
Compared with the prior art, the invention has the beneficial effects that: the invention adopts the adjustment method to estimate the ambiguity floating solution
Figure GDA0004162960690000051
And its variance matrix->
Figure GDA0004162960690000052
For->
Figure GDA0004162960690000053
Performing George decomposition to obtain an upper triangular matrix B, and converting the integer least square problem of the ambiguity into a nearest vector problem on a lattice; orthogonalizing B Schmidt to obtain an orthogonalization matrix B * And a triangular matrix U on a unit; the protocol algorithm is adopted for U and B * Performing scale reduction and orthogonal basis vector length exchange; defining the length stability rho of the orthogonal base as an index for evaluating the quality of the specification, and respectively calculating the length stability rho of the orthogonal base before and after the specification Front part ，ρ Rear part (S) Judging ρ Rear part (S) ≤ρ Front part If so, the success of the protocol is indicated, a search algorithm can be adopted to estimate the ambiguity, otherwise, the protocol failure needs to be re-regulated. Compared with the existing ambiguity lattice reduction quality evaluation method, the index defined by the invention can consider the overall size of the length of the orthogonal basis vector and reflect the ordering trend of the orthogonal basis vector, so that the reduction quality of a reduction algorithm can be accurately and quantitatively evaluated.
Drawings
FIG. 1 is a flow chart of the ambiguity lattice reduction quality assessment method of embodiment 1 of the present invention.
Fig. 2 is a graph of orthogonal reduction base length trends for three reduction methods.
FIG. 3 is a graph of two orthogonal reduction base length trends for three reduction methods.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Example 1: referring to fig. 1-3, to achieve the above object, the present invention provides the following technical solutions:
the ambiguity lattice reduction quality evaluation method specifically shown in fig. 1 comprises the following steps:
step 1: establishing an observation equation by using GNSS observation values, and estimating an ambiguity floating solution by adopting a adjustment method
Figure GDA0004162960690000054
And its variance matrix->
Figure GDA0004162960690000061
In particular, floating point solution
Figure GDA0004162960690000062
And its variance matrix->
Figure GDA0004162960690000063
The calculation process is as follows:
general formula of GNSS observation equation:
E(y)＝Aa+Bb，P yy (1)
wherein E (& gt) and D (& gt) represent the expected and variance symbols, respectively; y represents an observed value; a and b represent ambiguity and baseline components, respectively; a and B are corresponding coefficient matrixes; p (P) yy Is the weight of the observed value.
The floating solution can be obtained by adopting a classical least square adjustment method
Figure GDA0004162960690000064
Sum of variances matrix->
Figure GDA0004162960690000065
Figure GDA0004162960690000066
wherein ,
Figure GDA0004162960690000067
P B ＝B(B T P yy B) -1 B T P yy 。
step 2: for a pair of
Figure GDA0004162960690000068
Performing George decomposition to obtain an upper triangular matrix B, and converting the integer least square problem of the ambiguity into a nearest vector problem on a lattice;
specifically, to
Figure GDA0004162960690000069
The process of cholesky decomposition was as follows:
Figure GDA00041629606900000610
in the upper triangular matrix B element B ij Solving the formula:
Figure GDA00041629606900000611
wherein ,aij Is that
Figure GDA0004162960690000071
An element; n is->
Figure GDA0004162960690000072
Dimension number.
Specifically, the process of converting the integer least squares problem to the nearest vector on grid problem is as follows:
obtaining ambiguity resolution according to step 1
Figure GDA0004162960690000073
Sum of variances matrix->
Figure GDA0004162960690000074
Obtaining an integer value of the ambiguity by adopting integer least square estimation, wherein the estimation criterion is as follows:
Figure GDA0004162960690000075
according to
Figure GDA0004162960690000076
Substitution estimation criteria may be: />
Figure GDA0004162960690000077
in the formula ,
Figure GDA0004162960690000078
is a constant.
The integer least squares ambiguity problem is converted into a nearest vector problem, and the integer ambiguity can be solved directly by adopting a lattice reduction algorithm.
Step 3: orthogonalizing B Schmidt to obtain an orthogonalization matrix B * And a triangular matrix U on a unit;
specifically, the schmitt orthogonalization process is as follows:
Figure GDA0004162960690000079
wherein ,
Figure GDA00041629606900000710
is an orthogonal basis vector, and->
Figure GDA00041629606900000711
u ji Is an orthogonalization coefficient, and->
Figure GDA00041629606900000712
Step 4: the lattice reduction algorithm is adopted for U and B * Sequentially carrying out scale specification and orthogonal base vector length exchange;
specifically, a classical LLL algorithm, a greedy algorithm, a blocking algorithm, a base vector deep insertion algorithm, a minimum column rotation algorithm and other polynomials can be adoptedLLL class reduction algorithm pair U and B in time * And performing scale reduction and orthogonal basis vector exchange.
Step 5: defining the length stability rho of the orthogonal base as an index for evaluating the quality of the specification, and calculating the length stability rho of the orthogonal base before and after the specification Front part ，ρ Rear part (S) Judging ρ Rear part (S) ≤ρ Front part If so, the success of the protocol is indicated, a search algorithm can be adopted to estimate the ambiguity, otherwise, the protocol failure needs to be re-regulated.
Specifically, the orthogonal base length smoothness ρ is defined as follows:
Figure GDA0004162960690000081
in the formula ,
Figure GDA0004162960690000082
the value range rho E [1, + ] of the stationarity rho is better as the rho value is closer to the 1-base vector reduction effect, and the ambiguity searching is facilitated.
Specifically, the ambiguity estimation can directly adopt a depth-first search algorithm such as FP/VB/SEVB and the like to realize quick search of the ambiguity.
The ambiguity lattice reduction quality evaluation method provided by the embodiment adopts a adjustment method to estimate an ambiguity floating solution
Figure GDA0004162960690000083
And its variance matrix->
Figure GDA0004162960690000084
For->
Figure GDA0004162960690000085
Performing George decomposition to obtain an upper triangular matrix B, and converting the integer least square problem of the ambiguity into a nearest vector problem on a lattice; orthogonalizing B Schmidt to obtain an orthogonalization matrix B * And a triangular matrix U on a unit; using lattice reduction algorithm pairsU and B * Performing scale reduction and orthogonal basis vector length exchange; defining the length stability rho of the orthogonal base as an index for evaluating the quality of the specification, and respectively calculating the length stability rho of the orthogonal base before and after the specification Front part ，ρ Rear part (S) Judging ρ Rear part (S) ≤ρ Front part If so, the success of the protocol is indicated, a search algorithm can be adopted to estimate the ambiguity, otherwise, the protocol failure needs to be re-regulated. Compared with the existing lattice reduction quality evaluation method, the method can accurately and quantitatively evaluate the performance of the reduction algorithm, and can be used for selecting a better reduction algorithm for quick estimation of the ambiguity. Because the key of GNSS high-precision data processing is quick and accurate resolving of ambiguity, the method has good application value in the aspect of GNSS quick and high-precision positioning.
In example 2, to verify whether the lattice reduction quality evaluation method of this example can accurately reflect the reduction quality of different algorithms based on example 1, two sets of ambiguity data are adopted to perform experimental verification, the stationarity of the orthogonal base lengths of the three methods of the original variance matrix (Origin), the LLL and the Deep are respectively counted, and an orthogonal reduction base length trend graph is adopted as a basis for judging whether the evaluation index is reasonable. The specific experimental results are shown in table 1 and fig. 2 and 3.
TABLE 1 orthogonal base Length smoothness for three protocol methods
Figure GDA0004162960690000091
The experimental results in table 1 and fig. 2 show that the length stability of the orthogonal base can accurately measure the protocol quality of different algorithms, and therefore can be used as an evaluation index of the protocol quality.
It will be evident to those skilled in the art that the invention is not limited to the details of the foregoing illustrative embodiments, and that the present invention may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The present embodiments are, therefore, to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. Any reference sign in a claim should not be construed as limiting the claim concerned.
Furthermore, it should be understood that although the present disclosure describes embodiments, not every embodiment is provided with a separate embodiment, and that this description is provided for clarity only, and that the disclosure is not limited to the embodiments described in detail below, and that the embodiments described in the examples may be combined as appropriate to form other embodiments that will be apparent to those skilled in the art.

Claims (7)

1. The ambiguity lattice reduction quality evaluation method is characterized by comprising the following steps of:
A. establishing an observation equation by using GNSS observation values, and estimating an ambiguity floating solution by adopting a adjustment method
Figure FDA00041629606800000110
And its variance matrix->
Figure FDA0004162960680000011
B. For a pair of
Figure FDA0004162960680000012
Performing George decomposition to obtain an upper triangular matrix B, and converting the integer least square problem of the ambiguity into a nearest vector problem on a lattice;
C. orthogonalizing B Schmidt to obtain an orthogonalization matrix B * And a triangular matrix U on a unit;
D. the lattice reduction algorithm is adopted for U and B * Sequentially carrying out scale specification and orthogonal base vector length exchange;
E. defining the length stability rho of the orthogonal base as an index for evaluating the quality of the specification, and calculating the length stability rho of the orthogonal base before and after the specification Front part ,ρ Rear part (S) Judging ρ Rear part (S) ≤ρ Front part Whether or not it isIf yes, the protocol is successful, the ambiguity is estimated by adopting a search algorithm, otherwise, the protocol fails and the protocol needs to be re-regulated;
in the step E, the length stability ρ of the orthogonal base is adopted as an index for evaluating the quality of the specification, and the specific definition is as follows:
Figure FDA0004162960680000013
in the formula ,
Figure FDA0004162960680000014
since the product of the successive orthogonal basis lengths is equal to the determinant of the matrix, i.e
Figure FDA0004162960680000015
The determinant of the variance matrix of the ambiguity is equal to the square of the B matrix after the George decomposition of the variance matrix, i.e.>
Figure FDA0004162960680000016
The range of values of the smoothness ρ is therefore:
ρ∈[1，+∞)
as can be seen from the range of values of ρ, the value of ρ depends on the degree of smoothness of the length of the orthogonal basis, when the orthogonal basis
Figure FDA0004162960680000017
ρ=1 when the length sizes are equal; the quality of the reduction algorithm depends on the stability degree of the orthogonal base vector, and the smaller the base vector length fluctuation, the better the reduction quality is indicated, so that the closer the rho value is to the 1 base vector, the better the reduction effect is, and the more the ambiguity is searched.
2. The method for evaluating quality of ambiguity lattice reduction as claimed in claim 1, wherein said step a uses a variance method to estimate an ambiguity floating solution
Figure FDA0004162960680000018
And variance matrix thereof>
Figure FDA0004162960680000019
The specific implementation steps are as follows:
general formula of GNSS observation equation:
E(y)＝Aa+Bb,P yy
wherein E (& gt) and D (& gt) represent the expected and variance symbols, respectively; y represents an observed value; a and b represent ambiguity and baseline components, respectively; a and B are corresponding coefficient matrixes; p (P) yy Weights for observations;
the floating solution can be obtained by adopting a classical least square adjustment method
Figure FDA0004162960680000021
Sum of variances matrix->
Figure FDA0004162960680000022
Figure FDA0004162960680000023
wherein ,
Figure FDA0004162960680000024
P B ＝B(B T P yy B) -1 B T P yy 。
3. the method for evaluating the quality of an ambiguity lattice reduction in accordance with claim 1, wherein in said step B
Figure FDA0004162960680000025
And (3) performing the Cholesky decomposition to convert the integer least squares problem of the ambiguity into the nearest vector problem on the lattice.
4. A method for evaluating the quality of an ambiguity lattice reduction in accordance with claim 3, wherein for
Figure FDA0004162960680000026
The process of cholesky decomposition was as follows: />
Figure FDA0004162960680000027
In the formula, the element B in the upper triangular matrix B ij Solving the formula:
Figure FDA0004162960680000028
wherein ,aij Is that
Figure FDA0004162960680000029
An element; n is->
Figure FDA00041629606800000210
Dimension number.
5. The method for evaluating the quality of an ambiguity lattice reduction according to claim 4, wherein the integer least squares problem is converted into a lattice nearest vector problem, comprising the steps of:
obtaining ambiguity floating point solution according to step A
Figure FDA00041629606800000211
Sum of variances matrix->
Figure FDA00041629606800000212
Obtaining an integer value of the ambiguity by adopting integer least square estimation, wherein the estimation criterion is as follows:
Figure FDA00041629606800000213
according to
Figure FDA0004162960680000031
Substitution estimation criteria may be:
Figure FDA0004162960680000032
in the formula ,
Figure FDA0004162960680000033
is a constant.
6. The method for evaluating quality of ambiguity lattice reduction as claimed in claim 1, wherein in said step C, said step B is performed by using Schmitt orthogonalization, and said step B is decomposed to obtain an orthogonalization matrix B * And a triangular matrix U on a unit, which comprises the following processes:
Figure FDA0004162960680000034
wherein ,
Figure FDA0004162960680000035
is an orthogonal basis vector, and->
Figure FDA0004162960680000036
u ji Is an orthogonalization coefficient, and->
Figure FDA0004162960680000037
7. The method for evaluating quality of fuzzy degree lattice reduction according to claim 1, wherein said step D adopts lattice reduction algorithm to pair U and B * Performing scale reduction and orthogonal basis vector length exchangeThe scale reduction and orthogonal basis vector exchange process can be directly realized on the matrix by adopting a LLL type reduction algorithm in polynomial time such as a classical LLL algorithm, a greedy algorithm, a partitioning algorithm, a basis vector deep insertion and a minimum column rotation algorithm.
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