CN118226375A - Wireless positioning method based on iterative least square weighting and position residual calculation - Google Patents
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Abstract
The invention discloses a wireless positioning method based on iterative least square weighting and position residual calculation. The method aims to solve the problems of larger error and lower positioning precision of the existing wireless positioning mode; the invention obtains the estimated value of the arrival time of each base station, thereby establishing an original LLOP equation set between the static node SS and the mobile node MS, and obtaining the initial position estimation of the mobile node MS by means of least square method. Defining a position residual error and calculating weights, and obtaining a weight matrix synthesized by the weighting coefficients of all LLOP equations based on the position residual error. And reconstructing a positioning equation set aiming at the original LLOP equation set and the weight matrix, and estimating the position of the mobile node MS based on a weighted least square principle. And returning to the residual calculation part to continue iteration until the termination condition is met, and outputting a final mobile node MS position estimation result. And limiting NLOS influence in the next WLS estimation by using the weight matrix, and generating a more accurate positioning result.
Description
Technical Field
The invention relates to the technical field of wireless positioning, in particular to a wireless positioning method based on iterative least square weighting and position residual calculation.
Background
Wireless communication and internet of things technology develop rapidly, and location-based services, such as emergency road assistance, route navigation, and location-based internet of things services, have become popular research directions, and are important components of the internet of things.
The positioning technology based on the node geometric relationship has better positioning performance and is widely concerned. For example, a method for geometrically locating a node in a wireless sensor network disclosed in chinese patent literature, publication No. CN110351657a, is improved on the basis of a conventional DV-Hop locating algorithm. Firstly, starting from a global network, considering all beacon nodes participating in the positioning of an unknown node, and calculating the average hop distance of the unknown node by setting a weighting factor wi. And simultaneously, calculating the position coordinates of the unknown node by adopting a centroid positioning algorithm according to the geometric position distribution from the unknown node to the nearest beacon node. This type of method typically requires acquisition of certain positioning parameters such as time of arrival (TOA), time difference of arrival (TDOA, TIME DIFFERENCE of arrival) and angle of arrival (AOA).
The measurement of the positioning parameters is affected by two types of non-idealities: gaussian measurement noise and non-line-of-sight (NLOS) errors. In general, when the direct path between the Motion Sensors (MS) and the Static Sensors (SS) is blocked by an obstacle, non-line-of-sight measurement errors occur, resulting in significant deviations in the time-of-arrival TOA and angle-of-arrival AOA measurements. Therefore, suppressing and eliminating the effects of non-line-of-sight errors is critical to improving the accuracy of wireless network positioning algorithms. In-field measurements have found that non-line-of-sight errors tend to rise with increasing linear distance between the mobile sensor MS and the stationary sensor SS, which has a greater impact on the accuracy of conventional positioning algorithms.
Disclosure of Invention
The invention mainly solves the problems of larger error and lower positioning precision of the existing wireless positioning mode; the wireless positioning method based on the iterative least square weighting and the position residual calculation is provided, weight vectors of a positioning equation set are calculated based on the position residual iteration, and then the position of the mobile sensor is estimated based on the weighted least square principle.
The technical problems of the invention are mainly solved by the following technical proposal:
the wireless positioning method based on iterative least square weighting and position residual calculation comprises the following steps:
S1: obtaining a positioning parameter measurement value, and constructing a distance equation between the static node SS and the mobile node MS according to the static node SS coordinates and the arrival time TOA;
s2: linearly transforming the distance equation to obtain a linear position line LLOP equation set; obtaining an initial position of the mobile node MS through least square solution;
S3: respectively selecting the intersection points of two static node SS ranging circles according to the position of the mobile node MS, and calculating the position residual error of the mobile node MS and the intersection points;
S4: calculating corresponding weighting coefficients according to the position residual errors, and synthesizing a weighting matrix when the current iteration is performed by all the weighting coefficients;
s5: reconstructing a positioning equation set by using the linear position line LLOP and the weight matrix in the current iteration, and solving to obtain the position of the mobile node MS;
s6: and outputting the position of the mobile node MS when the iterative positioning is finished, otherwise, returning to the step S3.
Starting from the relation between the residual error and the importance degree of a positioning equation, the scheme provides a position residual error definition based on double SS combination calculation and a reasonable residual error-weight calculation mode, and finally provides an iterative weighted least square positioning algorithm. The positioning equation with larger influence of NLOS is restrained through the weighting matrix in each iteration process so as to reduce positioning errors.
Preferably, in step S1, the measured distance of each static node SS to the mobile node MS is estimated from the time of arrival TOA of that static node SS.
Preferably, the equation of the distance between each stationary node SS and the mobile node MS is expressed as the square of the measured distance is equal to the square of the difference between the stationary node SS abscissa and the mobile node MS abscissa plus the square of the stationary node SS ordinate and the mobile node MS ordinate.
Preferably, in step S3, the intersection point acquisition process of the ranging circle is:
two static nodes SS are included for each linear position line LLOP equation in the set of linear position line LLOP equations;
And taking the measured distance of the two static nodes SS as a radius, correspondingly obtaining two ranging circles, and obtaining an intersection point by intersecting the two ranging circles.
Preferably, in step S2, the matrix form of the linear position line LLOP equation set obtained after the linear transformation is:
Wherein, the expression of each matrix is:
Wherein, The measurement distance from the ith static node SS to the mobile node MS;
N is the total number of static nodes SS;
position coordinates of the i-th static node SS;
(x, y) is the true position coordinates of the dynamic node MS.
Preferably, the mobile node MS initial position estimation based on least square principleThe method comprises the following steps:
。
matrix can be found Including ranging, i.e., it is subject to non-line-of-sight NLOS errors; and matrix/>Contains no items that would be affected by non line of sight NLOS.
It is then apparent that non-line-of-sight NLOS errors can only be passed through the matrixTo cause a change in the estimated position. If matrix/>The influence of components influenced by non-line-of-sight NLOS errors on the whole least square solution is reduced through proper weighting, and the positioning performance in a non-line-of-sight NLOS environment can be improved.
Preferably, the intersection point of the ranging circle closer to the calculated mobile node MS position estimate is selected:
in the case of only the initial position estimation of the mobile node MS, the intersection point closer to the initial position estimation of the mobile node MS is selected and recorded as ；
In the kth iteration, the mobile node MS position estimation obtained by the previous iteration estimation is taken as the standard position estimation, and the intersection point which is closer to the standard position estimation is selected and is recorded as。
Preferably, the expression of the position residual error of the ith static node SS combination is the distance between the position estimation coordinate of the mobile node MS and the selected ranging circle intersection point coordinate in the kth iteration.
Preferably, in the kth iteration, the linear position line LLOP is a weighting matrix of the system of equationsCan be expressed as:
In the method, in the process of the invention, Position residual errors combined for the ith static node SS in the kth-1 cycle;
diag { } represents a diagonal matrix with bracketed elements as diagonal elements.
This equation represents the mobile node MS position estimate that needs to be used for the k-1 iteration to calculate the residual and thus the weighting matrix for the current k iteration.
Preferably, the weighted least squares solution of the linear position line LLOP equation set in the kth iteration is expressed as:
。
By means of iterative least square weighting and position residual calculation, the method can gradually optimize position estimation of the mobile node, effectively reduce positioning errors and accordingly improve positioning accuracy.
The beneficial effects of the invention are as follows:
On the basis of the current WLS positioning result, a weight matrix reflecting the NLOS severity is obtained by calculating a position residual error, NLOS influence is limited in the next WLS estimation by using the weight matrix, and a more accurate positioning result is generated.
Drawings
Fig. 1 is a flowchart of a wireless sensor network positioning method of the present invention.
Fig. 2 is a schematic diagram of the position residual of the present invention.
FIG. 3 is a graph showing the relationship between the position residual mean and the iteration number.
FIG. 4 is a comparative graph of RMSE for different ranging noise standard deviations for the 2NLOS-7SS of the present invention.
FIG. 5 is a comparative graph of RMSE for different ranging noise standard deviations for the 3NLOS-7SS of the present invention.
FIG. 6 is a comparative graph of RMSE for different ranging noise standard deviations for the 5NLOS-7SS of the present invention.
Detailed Description
The technical scheme of the invention is further specifically described below through examples and with reference to the accompanying drawings.
Examples:
According to the wireless positioning method based on iterative least square weighting and position residual calculation, as shown in fig. 1, an arrival time TOA estimated value of each base station in positioning base stations is obtained, an original ranging equation set between a static node SS and a mobile node MS is established accordingly, and initial position estimation of the mobile node MS is obtained by means of least square solution. The position residuals are defined and the inverse of the position residuals is selected to calculate weights, based on which a weight matrix for the weighted coefficient synthesis of all LLOP equations is obtained. And reconstructing a positioning equation set aiming at the original LLOP equation set and the weight matrix, and estimating the position of the mobile node MS based on a weighted least square principle. And judging whether the iterative positioning process is finished according to the termination condition, if the termination condition is not met, returning to the residual calculation part to continue iteration, and if the termination condition is met, outputting a final mobile node MS position estimation result.
Specifically, the scheme of the embodiment includes the following steps:
S1: and obtaining a positioning parameter measurement value, and constructing a distance equation between the static node SS and the mobile node MS according to the static node SS coordinates and the arrival time TOA.
The positioning parameter measurement values include the coordinate position of the stationary node SS and the corresponding time of arrival TOA in the signal transmitted by the mobile node MS.
In the present embodiment, the N stationary nodes SS receive the transmission signals of the mobile node MS, and it is assumed that the arrival time TOA in the signals has been estimated, and these estimated information and SS coordinates involved in positioning are available in the positioning master stationary node SS.
The coordinates of the ith static node SS are expressed as，/>。
And calculating the measurement distance between the static node SS and the dynamic node MS according to the TOA estimated value of the arrival time of each static node SS, and establishing a distance equation between the static node SS and the dynamic node MS based on the measurement distance.
The TOA estimate directly reflects the time of transmission of the signal from the stationary node to the mobile node, so that the distance between the two can be directly calculated. This substantivity makes the calculation process simple, intuitive, and easy to understand and implement.
Specifically, the distance equation between the i-th static node SS and the dynamic node MSIs expressed as:
Where (x, y) is the true position coordinates of the dynamic node MS.
The expression of the distance equation for all N static nodes SS is:
S2: linearly transforming the distance equation to obtain a linear position line LLOP (LINEAR LINE of position) equation set; the initial position of the mobile node MS is obtained by solving the linear position line LLOP by least squares.
The linear position line LLOP is obtained by linearly transforming the distance equation, and the initial position of the mobile node is solved by using a least square method. Meanwhile, as only partial data is calculated in the iterative process, the calculation complexity is further reduced.
The distance equation is linearly transformed to obtain LLOP (LINEAR LINE of position) equation set, and the matrix expression is:
Wherein, the expression of each matrix is:
specifically, in the formula:
that is, the expression that can give the linear position line LLOP equation set is:
And carrying out least square solution on the original linear position line LLOP equation set to obtain an initial estimated position of the mobile node MS. Mobile node MS initial position estimation based on least square principle The method comprises the following steps:
by observing the above equation, a matrix can be found Including ranging, i.e., it is subject to non-line-of-sight NLOS errors; and matrix/>Contains no items that would be affected by non line of sight NLOS.
It is then apparent that non-line-of-sight NLOS errors can only be passed through the matrixTo cause a change in the estimated position. If matrix/>The influence of components influenced by non-line-of-sight NLOS errors on the whole least square solution is reduced through proper weighting, and the positioning performance in a non-line-of-sight NLOS environment can be improved.
S3: based on the position of the mobile node MS, the intersection point of the two static node SS ranging circles is selected respectively, and the position residual error of the mobile node MS and the intersection point is calculated.
Two static nodes SS are included for each linear position line LLOP equation in the set of linear position line LLOP equations. As shown in fig. 2, the ranging values of the two stationary nodes SS correspond to two ranging positioning circles. The intersection point of the two ranging positioning circles is calculated, and two intersection point coordinates, namely an intersection point A and an intersection point B in fig. 2, can be obtained.
Any one of the linear position lines LLOP equations contains two ranging circle equations as shown by the distance equation. In order not to lose generality, in the present embodiment, it is assumed that the corresponding is the i-th static node SS and the 1-th static node SS. The essence of the equation is to obtain a ranging positioning circle as shown in fig. 2 by taking the ith static node SS or the 1 st static node SS as a circle center and taking the corresponding ranging as a radius.
According to the binary quadratic equation system solving method, the two ranging positioning circles have two intersection points, namely an intersection point A and an intersection point B in fig. 2, and the distances AC and BC between the estimated position C of the mobile node MS and the two intersection points in the line-of-sight LOS environment are close to 0. In a non line of sight NLOS environment, however, the estimated position is at least a large distance from one of the intersections, so that the estimated position of the mobile node MS can be used to define a residual, called position residual. The specific solving process is not described in detail.
And selecting an intersection point A or B which is closer to the mobile node MS according to the calculated position estimation of the mobile node MS.
In the initial position estimation of only mobile node MS, the coordinates of the mobile node MS are measuredThe closer intersection is selected and denoted/>。
At the kth iteration, the mobile node MS position estimation obtained by the previous iteration estimationAs a criterion, the intersection point nearer to it is selected and denoted/>。
For each linear position line LLOP equation, the position residual of the linear position line LLOP equation is calculated using the mobile node MS position estimate and the selected range location circle intersection.
In the line-of-sight LOS environment, the distance between the estimated position of the mobile node MS and two intersection points of two ranging positioning circles should approach 0; in a non line-of-sight NLOS environment, the mobile node MS estimates a location that is at least a large distance from one of the intersections. The position residual expression of the ith static node SS combination is thus defined as:
Wherein, Is the positioning result of the mobile node MS in the kth iteration;
Is the selected circle intersection;
Representing the vector norm, essentially representing the distance between the two coordinates.
Further explaining the iterative weighted least squares positioning algorithm based on the position residuals, assuming an estimation result with a smaller positioning error is already present, substituting this result into the ranging equation, respectively. For the line of sight LOS measurement equation, the deviations on both sides of the equation are caused by small order of magnitude positioning errors and measurement noise; for the static node SS of the non-line-of-sight NLOS, a larger value of the non-line-of-sight NLOS error causes a larger imbalance in both sides of the equation. The residual can reflect how important this equation is to the position estimate, i.e. the larger the residual the more likely it is that it is not. Comparing with least square estimation without weight (which can be regarded as weighting vector 1), calculating weight based on residual error and carrying out WLS positioning estimation, the estimation result is expected to be improved, and an iterative process can be further designed to gradually improve WLS result.
S4: and calculating corresponding weighting coefficients according to the position residual errors, and combining all the weighting coefficients into a weighting matrix at the current iteration.
For each linear position line LLOP equation in the set of linear position line LLOP equations, a corresponding weighting coefficient is calculated using its position residual. In this embodiment, the inverse of the position residual is selected to calculate the weighting coefficients. The inverse of the position residual error is used for calculating the weight, so that the reliability and importance of ranging information of different base stations can be reflected more accurately, and the positioning result is further optimized.
On the basis of the current weighted least square WLS positioning result, a weight matrix reflecting the severity of the non-line-of-sight NLOS is obtained by calculating a position residual error, and the influence of the non-line-of-sight NLOS is limited in the next weighted least square WLS estimation by using the weight matrix, so that a more accurate positioning result is generated. By means of iterative least square weighting and position residual calculation, the method can gradually optimize position estimation of the mobile node, effectively reduce positioning errors and accordingly improve positioning accuracy.
The weighting coefficients of all linear position line LLOP equations are synthesized into a weight matrixK indicates what number of iterations is currently.
To not lose generality, the inverse of the position residual is chosen in this embodiment to calculate the weights, so in the kth iteration, the weighting matrix of the linear position line LLOP system of equationsCan be expressed as:
In the formula, diag { } represents a diagonal matrix having elements in brackets as diagonal elements. This equation represents the mobile node MS position estimate that needs to be used for the k-1 iteration to calculate the residual and thus the weighting matrix for the current k iteration.
S5: reconstructing a positioning equation set by using the linear position line LLOP and the weight matrix in the current iteration, and solving to obtain the position of the mobile node MS;
a set of positioning equations is reconstructed for the original LLOP set of equations and the weight matrix and the mobile node MS position is estimated based on a weighted least squares (WLS, WEIGHTED LEAST squares) principle.
According to the weighted least squares principle, the weighted least squares solution for the linear position line LLOP equation set in the kth iteration can be expressed as:
s6: and outputting the position of the mobile node MS when the iterative positioning is finished, otherwise, returning to the step S3.
Judging whether the iterative positioning process is finished, if not, jumping to the step S3 for execution; if yes, the final mobile node MS position estimation value is output.
Steps S3 through S5 may be performed iteratively to improve the final positioning result, and a series of termination conditions may be used to determine whether the iteration may be terminated. Such as the mobile node MS position estimate difference for two adjacent iterations being small enough, or the number of iterations reaching an upper limit, etc.
Because an iterative mode is adopted, the method can adapt to positioning requirements under different environments and noise conditions, and has stronger robustness. Relatively accurate positioning results can be obtained even in the presence of poor signal quality or interference.
The linear position line LLOP is obtained by linearly transforming the distance equation, and the initial position of the mobile node is solved by using a least square method. Meanwhile, as only partial data is calculated in the iterative process, the calculation complexity is further reduced.
In practical application, the iteration times and the termination conditions can be adjusted as required to realize the balance between faster positioning speed and higher positioning accuracy. Without loss of generality, in the present embodiment, iteration convergence is described on the condition of the number of iterations. As can be seen from the observation of the simulation results, as the iteration number increases, the position residual mean value gradually decreases, and convergence is achieved after about 15 iterations. When the iteration termination condition is reached, the whole positioning process is terminated, a final positioning result is output, and otherwise, the step S3 is skipped.
As shown in fig. 3. The abscissa on the graph is the iteration number, and the ordinate is the position residual error average value. The standard deviation of the ranging noise is 1m, and the non-line-of-sight NLOS errors are uniformly distributed in [10,30] m. As can be seen from the figures: as the number of iterations increases, the position residual mean gradually decreases, reaching convergence after about 15 iterations.
As shown in fig. 4-6, RMSE comparisons of different ranging noise standard deviations are shown at a maximum possible non-line-of-sight NLOS error (NLOSmax) of 40 m. Wherein, FIG. 4 is a 2NLOS-7SS, i.e. 2 static nodes SS among 7 static nodes SS are in NLOS transmission environment; FIG. 5 shows a 3NLOS-7SS, i.e., 3 of the 7 static nodes SS are in NLOS transmission environment; fig. 6 shows 5NLOS-7SS, i.e., 7 static nodes SS, and 5 static nodes SS are in NLOS transmission environment. The abscissa on the figure shows the ranging noise level, and the ordinate shows RMSE (in meters).
The simulation environment is set as a classical honeycomb type static sensor group, and the SS coordinate positions of static nodes are respectively as follows: (0, 0),、/>、/>、/>、/>、/>The topology radius R is set to 100m. The positioning algorithm provided by the embodiment and the traditional wireless positioning algorithm are comprehensively compared, wherein the curve of the traditional positioning algorithm is a dotted line, and the algorithm provided by the embodiment is a solid line. The MS location is randomly generated.
The comparison algorithm used in the simulation graph comprises the following steps: the residual weighting algorithm (RWGH), the constrained least squares method (CLS), the known TDOA measurement, the two-step WLS algorithm (TS-WLS), the semi-forward programming algorithm (SDP), the optimization solution based LLOP algorithm (OptLLOP), and the algorithm proposed in this example (position res-2 SS) are used.
The positioning algorithm provided by the embodiment is comprehensively compared with the traditional wireless positioning algorithm, and the simulation result can be observed: for all positioning algorithms, the performance of the positioning algorithm is reduced along with the increase of the standard deviation of the ranging noise, but the positioning algorithm provided by the application always keeps the lowest RMSE and has the best positioning performance.
The scheme of the embodiment defines a position residual error based on the combination of the dual-static nodes SS, and accordingly provides a wireless positioning algorithm applying the iterative weighted least square principle. The key point is that the algorithm introduces an iterative updating mechanism of the weight matrix, so that the estimation error of the weighted least square WLS in each iteration process tends to be reduced. Specifically, the algorithm obtains a weight matrix reflecting the severity of the non-line-of-sight NLOS through calculating a position residual on the basis of the current weighted least square WLS positioning result, and limits the influence of the non-line-of-sight NLOS in the next weighted least square WLS estimation by using the weight matrix to generate a more accurate positioning result.
The scheme of the embodiment starts from the relation between the residual error and the importance degree of the positioning equation, provides position residual error definition based on double static node SS combination calculation and a reasonable residual error-weight calculation mode, and finally provides an iterative weighted least square positioning algorithm. The positioning equation with larger influence on the non-line-of-sight NLOS is suppressed through the weighting matrix in each iteration process so as to reduce positioning errors. Simulation results show that the performance of the proposed positioning algorithm of the embodiment is superior to that of the traditional algorithm in a non-line-of-sight NLOS transmission environment. The specific expression is as follows: when the number of static nodes of the non-line-of-sight NLOS is small, the performance of the positioning algorithm provided by the embodiment is superior to that of other traditional positioning algorithms. As the number of non line-of-sight NLOS static nodes increases, the algorithm performance of this embodiment still has advantages, and the error rises more slowly.
It should be understood that the examples are only for illustrating the present application and are not intended to limit the scope of the present application. Furthermore, it should be understood that various changes and modifications can be made by one skilled in the art after reading the teachings of the present application, and such equivalents are intended to fall within the scope of the application as defined in the appended claims.
Claims (10)
1. The wireless positioning method based on iterative least square weighting and position residual calculation is characterized by comprising the following steps:
S1: obtaining a positioning parameter measurement value, and constructing a distance equation between the static node SS and the mobile node MS according to the static node SS coordinates and the arrival time TOA;
s2: linearly transforming the distance equation to obtain a linear position line LLOP equation set; obtaining an initial position of the mobile node MS through least square solution;
S3: respectively selecting the intersection points of two static node SS ranging circles according to the position of the mobile node MS, and calculating the position residual error of the mobile node MS and the intersection points;
S4: calculating corresponding weighting coefficients according to the position residual errors, and synthesizing a weighting matrix when the current iteration is performed by all the weighting coefficients;
s5: reconstructing a positioning equation set by using the linear position line LLOP and the weight matrix in the current iteration, and solving to obtain the position of the mobile node MS;
s6: and outputting the position of the mobile node MS when the iterative positioning is finished, otherwise, returning to the step S3.
2. The method according to claim 1, wherein in step S1, the measured distance from each static node SS to the mobile node MS is estimated based on the time of arrival TOA of the static node SS.
3. The method of claim 2, wherein the equation of the distance between each stationary node SS and the mobile node MS is expressed as a square of the measured distance being equal to a square of the difference between the stationary node SS abscissa and the mobile node MS abscissa plus a square of the stationary node SS ordinate and the mobile node MS ordinate.
4. The wireless positioning method based on iterative least squares weighting and position residual calculation according to claim 2, wherein in step S3, the intersection point obtaining process of the ranging circle is:
two static nodes SS are included for each linear position line LLOP equation in the set of linear position line LLOP equations;
And taking the measured distance of the two static nodes SS as a radius, correspondingly obtaining two ranging circles, and obtaining an intersection point by intersecting the two ranging circles.
5. The wireless positioning method based on iterative least squares weighting and position residual calculation according to claim 1 or 2 or 3 or 4, wherein in step S2, the matrix form of the linear position line LLOP equation set obtained after the linear transformation is:
；
Wherein, the expression of each matrix is:
；
；
；
；
；
Wherein, The measurement distance from the ith static node SS to the mobile node MS;
N is the total number of static nodes SS;
position coordinates of the i-th static node SS;
(x, y) is the true position coordinates of the dynamic node MS.
6. The method for wireless location based on iterative least squares weighting and position residual calculation of claim 5 wherein the mobile node MS initial position estimate based on least squares principlesThe method comprises the following steps:
。
7. the method of claim 1,4 or 6, wherein the intersection of the ranging circle is selected to be closer to the calculated mobile node MS position estimate:
in the case of only the initial position estimation of the mobile node MS, the intersection point closer to the initial position estimation of the mobile node MS is selected and recorded as ；
In the kth iteration, the mobile node MS position estimation obtained by the previous iteration estimation is taken as the standard position estimation, and the intersection point which is closer to the standard position estimation is selected and is recorded as。
8. The method of claim 7, wherein the expression of the position residual of the i-th stationary node SS is the distance between the position estimation coordinates of the mobile node MS and the selected ranging circle intersection coordinates in the k-th iteration.
9. The method of claim 5, wherein in the kth iteration, the weighting matrix of the linear system of position line LLOP equationsCan be expressed as:
；
In the method, in the process of the invention, Position residual errors combined for the ith static node SS in the kth-1 cycle;
diag { } represents a diagonal matrix with bracketed elements as diagonal elements.
10. The method of claim 9, wherein the weighted least squares solution for the linear position line LLOP equation set in the kth iteration is expressed as:
。
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