GDOP minimum in multi-GNSS positioning

Accepted Manuscript GDOP minimum with six satellites in the single-point positioning Bi Liu, Yunlong Teng, Qi Huang PII: DOI: Reference:

S0273-1177(17)30475-1 http://dx.doi.org/10.1016/j.asr.2017.06.049 JASR 13303

To appear in:

Advances in Space Research

Received Date: Revised Date: Accepted Date:

18 April 2017 23 June 2017 26 June 2017

Please cite this article as: Liu, B., Teng, Y., Huang, Q., GDOP minimum with six satellites in the single-point positioning, Advances in Space Research (2017), doi: http://dx.doi.org/10.1016/j.asr.2017.06.049

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GDOP minimum with six satellites in the single-point positioning Bi Liu, Yunlong Teng*, Qi Huang School of Energy Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan Province, 611731, PR China Abstract: In positioning, navigation and timing (PNT) applications with the Global Navigation Satellite System (GNSS), the geometric dilution of precision (GDOP) offers an important index for selecting satellites from all tracked satellites for positioning calculation. In general, the lower the GDOP values are, the more accurate the PNT solution is. Therefore, the GDOP minimum should be pursued. In this paper, we mainly focused on the GDOP minimum when the single-point positioning is based on the integration of three GNSSs. The GDOP minimum for any number of tracked satellites is theoretically derived in this paper. In addition, when the number of the satellites is equal to that of the unknown parameters, the correctness of GDOP minimum obtained has also been validated from two different perspectives. Keywords: Global Navigation Satellite System (GNSS); Single-point positioning; Geometric Dilution of Precision (GDOP); Minimum

1. Introduction Providing positioning, navigation and timing (PNT) information for global users, the Global Navigation Satellite System (GNSS) has played very important roles in many scientific fields and daily life. With the PNT applications using GNSS, the GDOP is an important parameter for selecting satellites from all tracked satellites and for evaluating the accuracy of positioning (Hofmann-Wellenhof et al., 2008; Leick et al., 2015). In general, the lower the GDOP values are, the more accurate the PNT solution is. Thus, how to calculate the GDOP minimum is worthy of further discussion. If the single-point positioning is based on a single GNSS (such as the GPS), at least four satellites are required. In the four-satellite case, many algorithms for calculating the minimum of GDOP have been presented (Parkinson et al., 1996; Yarlagadda et al., 2000; Sairo et al., 2003; Zheng et al., 2004; Blanco-Delgado et al., 2017; Xue and Yang, 2017). These methods derived the GDOP minimum from different points. On the other hand, when discussing the single-point positioning using the integration of two or more GNSSs (i.e., the GPS/GLONASS positioning, the GPS/GLONASS/Galileo positioning, etc), the number of unknown parameters is equal to three (relative to the receiver position in three-dimensions) plus the number of the individual GNSSs. *Corresponding author. Email address: [email protected] (B. Liu), [email protected] (Y. Teng), [email protected] (Q. Huang)

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For instance, as there are five unknowns to be estimated in the single-point positioning with dual-GNSS (i.e., the GPS/GLONASS positioning), at least five satellites are required. In the five-satellite case with dual-GNSS, how to calculate the minimum of GDOP has been discussed in our former studies (Teng et al., 2016). In addition, the optimal geometric distribution of the five satellites with regard to the minimum of GDOP is also given. However, if the single-point positioning is based on the integration of three GNSSs (such as the GPS/GLONASS/Galileo positioning), it is harder to calculate the GDOP minimum since one more unknown are involved in comparison to that with dual-GNSS. In this paper, we mainly focus on calculating the GDOP minimum suitable for the integration of three GNSSs. The remainder of this paper is organized as follows. In Section 2, the GDOP concept in the single-point positioning is briefly reviewed. The detailed procedures for calculating the GDOP minimum for any number of satellites are theoretically derived in Section 3. Moreover, aiming at the special case that the number of satellites is equal to that of the unknown parameters (i.e., the six-satellite case in this work), two different methods for calculating the minimum are also discussed in Section 4. Finally, conclusions are drawn and future research directions are highlighted in Section 5. 2. The GDOP concept When discussing the single-point positioning applications using GNSS, the GDOP is defined as 1 GDOP  tr  H T H    

(1)

with H being the design matrix. And it is given by

HA H   H B  H C

1A 0B 0C

0A 1B 0C

0A  0B  1C 

(2)

with   A, B, C being the different GNSSs. In addition, H  and N  denote the corresponding geometric matrix and the number of the tracked satellites in  , respectively. It should be noted that, the ones vectors ( 1 ) and the zeros vectors ( 0 ) are located in different columns in order to get each receiver clock bias, because the biases for different constellations are different (Choi et al., 2011; Torre and Caporali, 2015; Teng and Wang, 2016). Supposing that hi denotes the direction cosine vector between the receiver and the i-th satellite, and then the sub-matrix H  in Equation (2) can be expressed as

2

 hN A 1   hN A  N B 1   h1        H A    , HB    , HC     hN  N   hN   hN   A  A B  

(3)

where the vector hi  hix , Niy , Niz  can be calculated by the approximate position of the receiver and the approximate position of the i-th satellite, and it is a unit one. That is, the tips of these vectors lie on the surface of a unit sphere. 3. GDOP minimum in a generic case The GDOP minimum in a generic case, namely, for any number of satellites, will be discussed in this section. According to the design matrix in Equation (2), we have  A U HT H   T D  U

(4)

where N  NA A   hiT hi , U   hiT i 1  i 1

 hiT  i  N A  N B 1 

(5)

   1  D  U T A1U  

(6)

N A  NB



N

i  N A 1

hiT



and D  diag  N A , N B , NC  . Thus, the inverse of Equation (4) can be expressed as  A  UD 1U T 1 1  1   Φ  HT H    1  Ψ    

where the non-diagonal elements are not given. By combining Equation (6) with the GDOP definition in the single-point positioning, we have 1 GDOP 2  tr  H T H    tr  Φ1   tr  Ψ 1   

(7)

Supposing that i  Φ  and i  Ψ  denote the eigenvalues of Φ and Ψ , respectively. On the basis of the relationship between the eigenvalues and the trace of a matrix, we can obtain 3  1 1  (8) GDOP 2  tr  Φ1   tr  Ψ 1       i  Ψ   i 1  i  Φ  Herewith we begin with the bound of tr  Φ1  . The sum of the three diagonal elements of

the sub-matrix Φ is given by

1 ii    h  h  h    NA i 1 i 1 3

N

2 ix

2 iy

2 iz

 N A  2  N A  2  N A  2    hix     hiy     hiz    i 1   i 1   i 1  

 N A +N B  2  N A +N B  2  N A +N B  2    hix     hiy     hiz    i  N A 1   i  N A 1   i  N A 1   2 2 2   N   N   1  N   hix     hiy     hiz    N C  i  N A +N B 1   i  N A +N B 1   i  N A +N B 1     1  NB

(9)

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As hix2  hiy2  hiz2  hi

2 2

 1 for any i  1,

, N , then we have

    h 3

i 1

N

ii

i 1

2 ix

 hiy2  hiz2   N

(10)

Therefore, 1  Φ  2  Φ  3  Φ  11  22  33  N holds. By means of the characteristics of inequality, we can obtain

tr  Φ1    i1  Φ   3

i 1

9 N

(11)

If and only if 1  Φ   2  Φ   3  Φ  , the equality in Equation (11) holds.

On the other hand, after obtaining the inequality for tr  Φ1  , the bound on tr  Ψ 1  will be

derived below. To find the bound of tr  Ψ 1  , we firstly consider a theorem (Yang et al., 2014): Let

A and B being two Hermitian matrices, if

B  A is positive semi-definite, then

i  A  i  B  holds, where i  A and i  B  , arranged in an decreasing order, are the eigenvalues of A and B , respectively. If B  A is a positive definite matrix, the inequalities are strict. In this paper, Ψ  D  U T A1U is positive semi-definite, which means Ψ  D . In this case, the corresponding eigenvalues of the two matrices satisfy i  Ψ   i  D  . Thus, the inequality for tr  Ψ 1  is given by

tr  Ψ 1    i1  Ψ   3

i 1

1 1 1   N A N B NC

(12)

Substituting Equation (11) and Equation (12) into Equation (8), then the minimum of GDOP in a generic case is given by

GDOP 

9  1 1 1      N  N A N B NC 

(13)

which gives the detailed expression of the minimum GDOP for any number of satellites. In addition, for the given expression in Equation (13), if we set N A  N B  NC  2 (i.e., each single constellation has two tracked satellites), the GDOP minimum is

3 . This minimum in

the six-satellite case will be validated from different perspectives below. 4. The GDOP in the special case In the single-point positioning, it is required that the number of the tracked satellites is no less than that of the unknown parameters. Actually, when the number of the satellites equals that of the unknown parameters to be estimated (i.e., the special case in this paper), the minimum of GDOP can also be obtained in different ways. 4.1 The mathematical minimum of GDOP As there are six unknowns to be estimated when the single-point positioning is based on the integration of three GNSSs, at least six satellites are required. In this paper, we focus on the 4

special case of six satellites. Under six satellites, there are three different situations about the combination of the satellite number, namely,

 N A , NB , NC    4,1,1 ,  3, 2,1

or

 2, 2, 2  .

In the first situation, the single-point positioning based on the combination of three GNSSs can then be achieved using the four satellites from one single GNSS. In other words, in such a situation, the calculation of the GDOP minimum can be regarded as that under four satellites in a single GNSS, which has been discussed widely. So this situation will not be considered in this work. In the second situation, how to calculate the GDOP minimum is equivalent to that of five satellites with dual-GNSS, which has also been discussed. Therefore, this paper mainly targets the third situation, which has not been solved currently. Aiming at the special case of

 N A , N B , NC    2, 2, 2 , if the design matrix in Equation (2) is

non-singular, then we can obtain 1 1 tr  H T H    tr  HH T    tr  M 1     

(14)

with M  HH T being a symmetric positive definite matrix. Supposing that N  M 1 , then the GDOP can also be calculated by

GDOP  tr  M 1   tr  N 

(15)

Applying the theorem from Graybill (1983) and Yarlagadda et al. (2000), then we have mii nii  1 with mii and nii being the diagonal elements of

(16)

M and N , respectively.

As mii  hi hiT  1  2 for all 1  i  6 , then the inequality nii  1 2 satisfies. Combining this inequality with the GDOP in Equation (15) leads to

GDOP 

6

n i 1

ii

 3

which means that the GDOP minimum in the six-satellite case is

(17)

3 . And it is consistent with

the results obtained in Section 3. 4.2 More discussions about the GDOP minimum Using the characteristics of diagonal elements of the inverse matrix, the GDOP minimum

in

the special case has been derived in Section 4.1. Herewith we will validate the correctness of the minimum from a different view. According to the GDOP concept in the single-point positioning, we partition the design matrix as H   H1

H2 

(18)

with the two sub-matrices H1 and H 2 being given by

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 h1  1 0 0  h  1 0 0   2    h3  0 1 0  H1    , H 2     h4  0 1 0   h5  0 0 1       h6  0 0 1 

(15)

Actually, H1 and H 2 are relative to the positioning information in three dimensions and the timing information corresponding to the three individual constellations, respectively. This is consistent with the GDOP concept. In general, the GDOP is a quality measure to specify the additional multiplicative affect of measurement error on positioning accuracy and the timing accuracy. Actually, the GDOP are composed of two parts, PDOP (position dilution of precision) and TDOP (time dilution of precision). They are related to the positioning accuracy and the timing accuracy, respectively. Referring to the GDOP concept, if we assume that the positioning information is known, there are only three unknown parameters (three receiver clock biases) to be calculated. In this situation, the design matrix H degenerates the sub-matrix H 2 . As a consequent, the GDOP is simplified as the TDOP, which is given by 1 TDOP  tr  H 2 H 2T    3 2  

(19)

In the same way, if the timing information has not been taken into consideration, we can reduce the GDOP to the PDOP. In this case, we have 1 PDOP  tr  H1T H1    

(20)

On the other hand, the PDOP in Equation (20) is also called as the first-kind GDOP in Levanon (2000), and Xue and Yang (2015). On the basis of the corresponding results in Xue and Yang (2015), if and only if H1T H1  2 I with I being an identify matrix, the PDOP (i.e., the first-kind GDOP) gets the minimum 3

6.

Therefore, when the TDOP and the PDOP achieve their minimums, respectively, the GDOP gets its minimum. And the GDOP minimum can be calculated by 2

2 GDOPmin  PDOPmin  TDOP 2 

3  3    3 2  6 

(21)

5. Conclusions Aiming at the single-point positioning using the integration of three GNSSs, how to calculate the GDOP minimum for any number of tracked satellites has been theoretically derived in this paper. In addition, for the special case that the number of tracked satellites is equal to that of the unknown parameters, the correctness of the GDOP minimum has also been validated from 6

two different views. The GDOP minimum for any number of satellites in the integration of three GNSSs, together with that for the five-satellite case with dual-GNSS in our former studies, can further enrich the knowledge on the GDOP minimum in the single-point positioning. In this work, we have focused on the GDOP minimum in the integration of three GNSSs. Actually, this minimum does not take the practical constraints (such as the elevation of the tracked satellite) into consideration, and it is theoretical. How to calculate the GDOP minimum under practical constraints is an important issue to be investigated in future studies.

Acknowledgments This work is supported by the National Natural Science Foundation of China under grant Nos. 51277022 and 61603075. References Blanco-Delgado, N., Nunes, F., Seco-Granados, G., 2017. On the relation between GDOP and the volume described by user-to-satellite unit vectors for GNSS positioning. GPS Solutions, http://dx.doi.org10.1007/ s10291-016-0592-3, Published online: 11 January, 2017. Choi, M., Blanch, J., Akos, D., Heng, L., Gao, G., Walter, T., Enge, P. 2011. Demonstrations of multi-constellation advanced RAIM for vertical guidance using GPS and GLONASS signals. Proceedings of the 24th international technical meeting of the satellite division of the institute of navigation, 3227-3234. Graybill, F. A., 1983. Matrices with applications in statistics. Wadsworth, Belmont. Han, S., Chen, P., Li, G., Du, Y., 2014. Minimum of PDOP and its application in inter-satellite-link (ISL) of Walker-δ constellation. Adv. Space Res., 54(4), 726-733. Hofmann-Wellenhof, B., Lichtenegger, H., Walse, E., 2008. GNSS-Global navigation satellite systems GPS, GLONASS, Galileo and more, SpringerWien, New York. Leick, A., Rapoport, L., Tatarnikov, D., 2015. GPS satellite surveying, 4th edn. Wiley, New York. Levanon, N., 2000. Lowest GDOP in 2-D scenarios. IEEE Proc. Radar Sonar Navig., 147(1): 149-155. Parkinson, B., Spilker, J. J., Axelrad, P., Enge, P., 1996. Global positioning system theory and applications. American Institute of Aeronautics and Astronautics. Sairo, H., Akopian, D., Takala, J., 2003. Weighted dilution of precision as quality measure in satellite positioning. IEE Proc. Radar Sonar Navig., 150(6), 430-436. Teng, Y., Wang, J., 2016. A closed-form formula to calculate geometric dilution of precision (GDOP) for multi-GNSS constellations. GPS Solutions, 20(3), 331-339. Teng, Y., Wang, J., Huang, Q., 2016. Mathematical minimum of Geometric Dilution of Precision (GDOP) for dual-GNSS constellations. Adv. Space Res., 57(1), 183-188. Torre, A. D., Caporali, A., 2015. An analysis of intersystem biases for multi-GNSS positioning. GPS Solutions, 19(2), 297-307. Xue, S., Yang, Y., 2015. Positioning configurations with the lowest GDOP and their classification. J. Geod., 89(1): 49-71. Xue, S., Yang, Y., 2017. Understanding GDOP minimization in GNSS positioning: infinite solutions, finite 7

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