Method for quickly acquiring highly reliable integer solution for satellite positioning

11686860 · 2023-06-27

Assignee

Southeast University (Jiangsu, CN)

Inventors

Xianwen Yu (Jiangsu, CN)

Cpc classification

International classification

Abstract

The present disclosure discloses a method for quickly acquiring a highly reliable integer solution for satellite positioning. The method includes: acquiring observation data by a data computing platform from a GNSS receiver; establishing a GNSS carrier observation equation; solving a real solution for ambiguity and the corresponding variance matrix, a real solution for other unknown parameters including positioning parameters and the corresponding variance matrix, and a covariance matrix of the and by using the least squares method; determining integer vectors with the same dimension as the ambiguity according to a given reliability probability; computing a posterior weighted probability with the integer vectors being the true value of the ambiguity; computing an integer solution for other unknown parameters including positioning parameters by using the posterior weighted probability; computing a variance matrix of the integer solution for other unknown parameters; and outputting a computed result by the data computing platform.

Claims

1. A method for quickly acquiring a highly reliable integer solution for satellite positioning, wherein estimates and a variance matrix of satellite positioning parameters under a constraint of an integer property of an ambiguity are obtained by synthesizing a plurality of integer vectors with a same dimension as the ambiguity according to a probability, wherein the method comprises following steps: step 1. acquiring observation data by an input unit of a data computing platform from a GNSS receiver, and sending the observation data to a processor of the data computing platform for solving, and executing following steps by the processor: (1) establishing a GNSS carrier observation equation, solving a real solution â for ambiguity a and a variance matrix D.sub.ââ, a real solution {circumflex over (b)} for other unknown parameters b including positioning parameters and a corresponding variance matrix D.sub.{circumflex over (b)}{circumflex over (b)}, and a covariance matrix D.sub.â{circumflex over (b)} of the real solutions â and {circumflex over (b)} by using a least squares method; (2) determining t integer vectors with the same dimension as the ambiguity according to a given reliability probability p; (3) computing a posterior weighted probability P.sub.i(â) with the t integer vectors being a true value of the ambiguity; (4) computing an integer solution b.sub.p for the other unknown parameters b by using the posterior weighted probability P.sub.i(â); and (5) computing a variance matrix D.sub.b.sub.p.sub.b.sub.p of the integer solution b.sub.p; and step 2. outputting a computed result of the processor by an output unit of the data computing platform.

2. The method for quickly acquiring the highly reliable integer solution for satellite positioning according to claim 1, wherein the given reliability probability p in the step 1 is given based on reliability requirements for positioning results, wherein 0.95≤p<1.

3. The method for quickly acquiring the highly reliable integer solution for satellite positioning according to claim 1, wherein the method for determining t integer vectors with the same dimension as the ambiguity in the step 1 comprises following steps: (1) computing a chi-square value χ.sub.1-p.sup.2(n) according to an ambiguity dimension n and the reliability probability p; (2) computing a hyperellipsoid radius comprising at least two integer vectors by a formula (1) according to the ambiguity dimension n $\begin{matrix} χ_{(2)}^{2} (n) = {\frac{1}{π \sqrt[n]{| D_{\hat{a} \hat{a}} |}} [3 Γ (\frac{n}{2} + 1)]}^{\frac{2}{n}} & (1) \end{matrix}$ in the formula (1), ∥ is a determinant compute sign and Γ(g) is a gamma function; (3) determining a hyperellipsoid size for search by a formula (2):
χ.sup.2(n)=max(χ.sub.1-p.sup.2(n),χ.sub.(2).sup.2(n)) (2); and (4) acquiring t n-dimensional integer vectors z.sub.t by searching and comparing within a restriction of ∥â−z.sub.i∥D.sub.ââ.sup.2≤χ.sup.2(n), i=1, 2, L, t.

4. The method for quickly acquiring the highly reliable integer solution for satellite positioning according to claim 1, wherein the computing the posterior weighted probability P.sub.i(â) with the t integer vectors being the true value of the ambiguity in the step 1 is obtained by formulas (3) and (4): $\begin{matrix} \begin{matrix} T_{i} = \exp (- \frac{1}{2} {.Math. \hat{a} - z_{i} .Math.}_{D_{\hat{a} \hat{a}}}^{2}) & i = 1, 2, L, t \end{matrix} & (3) \end{matrix}$ $\begin{matrix} P_{i} (\hat{a}) = \frac{T_{i}}{{.Math.}_{j = 1}^{t} T_{j}} . & (4) \end{matrix}$

5. The method for quickly acquiring the highly reliable integer solution for satellite positioning according to claim 1, wherein the computing the variance matrix D.sub.b.sub.p.sub.b.sub.p of the integer solution b.sub.p in the step 1 is obtained by formulas (5), (6) and (7): $\begin{matrix} \begin{matrix} \frac{\partial P_{i} (\hat{a})}{\partial {\hat{a}}^{T}} = \frac{z_{i}^{T} \overset{t}{\underset{j = 1}{.Math.}} T_{j} - \overset{t}{\underset{j = 1}{.Math.}} T_{j} z_{j}^{T}}{{[\overset{t}{\underset{j = 1}{.Math.}} T_{j}]}^{2}} T_{i} D_{\hat{a} \hat{a}}^{- 1} & j = 1, 2, L, t \end{matrix} & (5) \end{matrix}$ $\begin{matrix} K = - D_{\hat{b} \hat{a}} D_{\hat{a} \hat{a}}^{- 1} (E - {.Math.}_{i = 1}^{t} z_{i} \frac{\partial P_{i} (\hat{a})}{\partial {\hat{a}}^{T}}) & (6) \end{matrix}$ $\begin{matrix} D_{{\bar{b}}_{p} {\bar{b}}_{p}} = K D_{\hat{a} \hat{a}} K^{T} + D_{\hat{b} \hat{a}} K^{T} + K D_{\hat{a} \hat{b}} + D_{\hat{b} \hat{b}} & (7) \end{matrix}$ wherein, D.sub.{circumflex over (b)}â=D.sub.â{circumflex over (b)}.sup.T.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a system block diagram of a method for quickly acquiring a highly reliable integer solution for satellite positioning; and

(2) FIG. 2 is a technical flow chart for implementing the solving method in step 1 of the present disclosure.

DESCRIPTION OF THE EMBODIMENTS

(3) The present disclosure will be further described with reference to the accompanying drawings.

(4) Data in the embodiment are taken from two consecutive observation stations approximately 18 km apart in Nanjing. Observation instruments in the observation stations include Leica GPS receivers and choke ring antennae. The observation epoch interval is set to 1 s and the elevation mask angle is set to 15°.

(5) Step 1. acquiring observation data by a computer input unit from a Leica GPS receiver through a data transmission line, and sending the observation data to a computer processor for solving, and executing the following steps by the processor:

(6) (1) establishing a carrier observation equation, and solving real solutions for ambiguity a and other unknown parameters b and the corresponding variance matrix and covariance matrix by using the least squares method:
â=(113.774002783 106.941104245 122.642171765 111.095951045 123.947487435 112.885194093).sup.T
{circumflex over (b)}=(−496.8076 10167.1846−14650.3774).sup.T

(7) $\begin{matrix} D_{\hat{a} \hat{a}} = (\begin{matrix} 5.84 5 1 8 7 0 7 9 7 & 4.7615884152 & 3.8785096512 & 2.9268425342 & 4.261534911 & 0.7405175034 \\ 4.7615884152 & 7.8501924873 & 5.8494655809 & 6.9656592755 & 6.551971314 & 3.2529734117 \\ 3.8785096512 & 5.8494655809 & 6.944092374 & 3.5575665839 & 6.4386031314 & 3.2511323776 \\ 2.9268425342 & 6.9656592755 & 3.5575665839 & 8.055727467 & 4.3524838521 & 2.582849755 \\ 4.261534911 & 6.5519771314 & 6.438631314 & 4.3524838521 & 6.864703327 & 3.4612275634 \\ 0.7405175034 & 3.2529734117 & 3.2511323776 & 2.582849755 & 3.4612275634 & 2.4107938586 \end{matrix}) \\ D_{\dot{b} \dot{b}} = (\begin{matrix} 0.2 2 6 2 4 8 7 8 0 3 & - 0.2081789014 & - 0.1446332556 \\ - 0.2081789014 & 0 4 9 2 4 673105 & 0.2130609355 \\ - 0.1446332556 & 02130609355 & 0.2 6 0 2 3 06315 \end{matrix}) \\ D_{\hat{a} \hat{b}} = (\begin{matrix} 0.1570727438 & - 1.3865400525 & - 0.11323269598 \\ - 0.2724232268 & - 1.1429777232 & - 0.5911368275 \\ 0.3883469099 & - 1.4557706845 & - 0.9955747836 \\ - 0.8698467808 & - 0.2483236004 & - 0.1154227406 \\ 0.3023076141 & - 1.4939503925 & - 0.9922523355 \\ 0.0681649499 & - 0.4836315569 & - 0.6689350383 \end{matrix}) \end{matrix}$

(8) (2) giving the reliability probability value p=0.997 and dimension n=6 of the ambiguity, acquiring χ.sub.1-p.sup.2(6)=19806;

(9) computing

(10) $χ_{(2)}^{2} (n) = {\frac{1}{π \sqrt[n]{| D_{\hat{a} \hat{a}} |}} [3 Γ (\frac{n}{2} + 1)]}^{\frac{2}{n}} = 3.92$

(11) then acquiring the hyperellipsoid size for search:
χ.sup.2(n)=max(χ.sub.1-p.sup.2(n),χ.sub.(2).sup.2(n))=19.80465

(12) and acquiring 9 6-dimensional integer vectors z by searching within the restriction of ∥â−z.sub.i∥D.sub.ââ.sup.2≤χ.sup.2(n), as shown in Table 1. Here, t=9.

(13) TABLE-US-00001 TABLE 1 Sequence z.sub.i T.sub.j P.sub.i(â) 1 112 104 121 108 122 112 0.314167380677157 0.841680320106421 2 116 112 124 118 126 114 0.027007829467721 0.072356202298198 3 109 103 119 109 120 112 0.026409479339390 0.070753172962484 4 119 113 126 117 128 114 0.003873696459373 0.010377952252372 5 115 105 123 107 124 112 0.001437667389506 0.003851629491253 6 118 117 132 119 134 119 0.000122530693910 0.000328269833267 7 114 109 129 109 130 117 0.000119811706495 0.000320985441767 8 113 111 122 119 124 114 0.000072435232337 0.000194059960677 9 110 99 113 107 114 107 0.000051289072077 0.000137407653559

(14) (3) computing a posterior weighted probability with the 9 integer vectors being the true value of the ambiguity;

(15) $\begin{matrix} \begin{matrix} T_{i} = \exp (- \frac{1}{2} {.Math. \hat{a} - z_{i} .Math.}_{D_{\hat{a} \hat{a}}}^{2}) & i = 1, 2, L, t \end{matrix} \\ P_{i} (\hat{a}) = \frac{T_{i}}{{.Math.}_{j = 1}^{i} T_{j}} \end{matrix}$

(16) and acquiring the computed results shown in Table 1.

(17) (4) computing an integer solution for other unknown parameters b by using the posterior weighted probability; and

(18) $\begin{matrix} {\bar{b}}_{p} = \hat{b} - D_{\hat{b} \hat{a}} D_{\hat{a} \hat{a}}^{- 1} (\hat{a} - {.Math.}_{i = 1}^{t} P_{i} (\hat{a}) z_{i}) \\ = {(\begin{matrix} - 496.663 & 10167.492 & - 14650. 3 0 6 \end{matrix})}^{T} \end{matrix}$

(19) (5) computing a variance matrix D.sub.b.sub.p.sub.b.sub.p of the integer solution b.sub.p:

(20) using

(21) $\frac{\partial P_{i} (\hat{a})}{\partial {\hat{a}}^{T}} = \frac{z_{i}^{T} \overset{t}{\underset{j = 1}{.Math.}} T_{j} - \overset{t}{\underset{j = 1}{.Math.}} T_{j} z_{j}^{T}}{{[\overset{t}{\underset{j = 1}{.Math.}} T_{j}]}^{2}} T_{i} D_{\hat{a} \hat{a}}^{- 1}$

(22) to acquire the corresponding computed result:

(23) 0 $\frac{\partial P_{i} (\hat{a})}{\partial {\hat{a}}^{T}} = \begin{matrix} \overset{.Math. i = 1}{(1.7755439015} & - 2.2695977158 & - 8.0577409889 & 0.0517902046 & 6.8116373134 & 3.6003646778) \end{matrix}$ $\frac{\partial P_{i} (\hat{a})}{\partial {\hat{a}}^{T}} = \begin{matrix} \overset{.Math. i = 2}{(- 1.201059133} & - 0.38411168 & 3.9745420053 & 0.9731903367 & - 1.5924312068 & - 3.1741454932) \end{matrix}$ $\frac{\partial P_{i} (\hat{a})}{\partial {\hat{a}}^{T}} = \begin{matrix} \overset{.Math. i = 3}{(- 0.5387715195} & 3.4149888812 & 4.5655136157 & - 1.3140054108 & - 6.3741063979 & - 0.0451463375) \end{matrix}$ $\frac{\partial P_{i} (\hat{a})}{\partial {\hat{a}}^{T}} = \begin{matrix} \overset{.Math. i = 4}{(- 0.0713476908} & - 0.5839814749 & - 0.1989506758 & 0.3316808251 & 0.790530339 & - 0.4042487682) \end{matrix}$ $\frac{\partial P_{i} (\hat{a})}{\partial {\hat{a}}^{T}} = \begin{matrix} \overset{.Math. i = 5}{(0.0455796034} & - 0.2066755322 & - 0.322281749 & 0.0710572408 & 0.4093324737 & 0.035409052) \end{matrix}$ $\frac{\partial P_{i} (\hat{a})}{\partial {\hat{a}}^{T}} = \begin{matrix} \overset{.Math. i = 6}{(- 0.0056765603} & 0.0133267213 & 0.018182327 & - 0.004267167 & - 0.018847988 & - 0.0081964766) \end{matrix}$ $\frac{\partial P_{i} (\hat{a})}{\partial {\hat{a}}^{T}} = \begin{matrix} \overset{.Math. i = 7}{(0.0004546497} & 0.0138694484 & - 0.0029258654 & - 0.0085094799 & - 0.0087677242 & 0.007439544) \end{matrix}$ $\frac{\partial P_{i} (\hat{a})}{\partial {\hat{a}}^{T}} = \begin{matrix} \overset{.Math. i = 8}{(- 0.0051083534} & 0.008859636 & 0.0250397412 & - 0.000981974 & - 0.0233241558 & - 0.0094670211) \end{matrix}$ $\frac{\partial P_{i} (\hat{a})}{\partial {\hat{a}}^{T}} = \begin{matrix} \overset{.Math. i = 9}{(0.0003851024} & - 0.006678284 & - 0.00137841 & 0.0036258337 & 0.0059773466 & - 0.002009177) \end{matrix}$

(24) and computing

(25) $K = - D_{\hat{b} \hat{a}} D_{\hat{a} \hat{a}}^{- 1} (E - {.Math.}_{i = 1}^{t} z_{i} \frac{\partial P_{i} (\hat{a})}{\partial {\hat{a}}^{T}}) = (\begin{matrix} 1.934746 & - 0.937101 & - 7.922566 & - 0.642889 & 5.15104 & 4.544514 \\ - 0.328698 & 4.303042 & 5.00038 & - 1.932506 & - 7.412875 & 0.44421 \\ - 0.64987 & 0.614295 & 2.516321 & - 0.033225 & - 1.840195 & - 1.053978 \end{matrix})$

(26) then obtaining

(27) $D_{{\bar{b}}_{p} {\bar{b}}_{p}} = K D_{\hat{a} \hat{a}} K^{T} + D_{\hat{b} \hat{a}} K^{T} + K D_{\hat{a} \hat{b}} + D_{\hat{b} \hat{b}} = (\begin{matrix} 0.157508 & - 0.010105 & - 0.039551 \\ - 0.010105 & 0.051018 & 0.0 0 7 5 6 0 \\ - 0.039551 & 0.0 0 7 5 6 0 & 0.010525 \end{matrix})$

(28) Step 2. outputting a computed result of the processor by an output unit of the data computing platform.

(29) The basic principles, main features and advantages of the present disclosure have been shown and described above. It should be understood by a person skilled in the art that the present disclosure is not limited to the embodiments set forth herein. The embodiments and specification herein are only for purposes of illustrating the principle of the present disclosure, various changes and improvements can be made to the present disclosure without departing from the spirit and scope of the present disclosure, and the changes and improvements will fall into the protection scope of the present disclosure. The protection scope set forth in the present disclosure is defined by the appended claims and equivalents thereof.