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GDOP’s influence on observable degree is studied by analyzing a multi-antenna GPS/SINS attitude measuring system. Firstly, the mathematical model of the attitude measuring system, whose observations are single-differences, is provided; Secondly, it is proved that the model can be studied by PWCS theory, and GDOP’s influence on observable degree is explained intuitively. Then, the variance of observable degree is studied by simulation while GDOP is different, and the result is analysed. Simulation results show that, the observable degree becomes worse with the increasing GDOP value. Therefore, while designing such kind of Kalman filter, the influence to observable degree which made by GDOP should be considered adequately.

A multi-antenna GPS/SINS attitude measuring system，whose observe vectors are single-differences, is established. Those observe including baseline vector, velocity and position. By such system, the error which cause by signal delay is decreased. While designing such a Kalman Filter, the observability of states should be considered. Usually, it can be analysed by PWCS theory [

Reference [

How about it while the number of visible stars is the same but geometric dilution of precision (GDOP) is not? Generally speaking, GDOP affects the precision of GPS receiver and still the integrated system. But no reference shows how it affects the observable degree of GPS/SINS integrated system. By analyzing a GPS/SINS integrated attitude measuring system, which is another type of CIS, the GDOP’s influence to the observable degree is studied.

Error equation of Mathematics Platform,

Error equation of velocity,

Error equation of position,

The GPS/SINS integrated system state space equation is established by error of SINS parameters.

In Equation (7),

State-transition matrix is

In Equation (8),

The attitude measuring part of system is realized by GPS multi-antenna layout and carrier phase relative positioning technique.

Calculate the double difference of GPS measurements, attitude measurement equations can be obtained as,

While,

Pseudo-range observations based on single-difference equation:

Doppler velocity measurement based on single-difference equation:

where,

Make the system measurements to be

Therefore, the observation equation is

where,

The model established before is a time-varying one. It should be analyzed by PWCS, before which, the model can be studied by PCWS needs to be proven.

It is proved that the model multi-antenna GPS/SINS integrated system accord with the PWCS theorem.

According to the definition of observability analysis matrix,

Elementary row transformation to

Set

In Equation (30),

Therefore,

Analyzes the observability by PWCS method, and results show that the rank n = 18 while the N Asset Coverage is 4, which means all the state are observable. It is obvious that the observability of states will be different while the GDOP is different. AS baseline vector B.

Similar, the velocity and position part have the same conclusion. While the transition matrix F is the same, GDOP affect the observation equation H, and better GDOP makes better H.

PWCS theory does not show the difference when GDOP is different. Therefore, quantitative analysis is needed to study the observability degree. SVD is a common method to analyze the observable degree. But SVD is still defective because of the different dimension of states. So analyzes the observable degree changes of system-level only when GDOP changes.

Simulation based on a 1000 sec long track, analyze the observable degree while GDOP changes. Set the N Asset Coverage 4, the length of baseline in Y axis 10 m, in X axis 5 m. Angle heading

Choose satellites number as 4, 13, 16, 19, and the averaged GDOP during simulation is 3.4; Choose satellites with number 4, 18, 19, 21, and the averaged GDOP during simulation is 12.3;

A GPS/SINS integrated attitude measuring system is established, and proof of it can be studied by PWCS theory is given. By using SVD and condition number method, how GDOP affects the observable degree of integrated is studied. Conclusions are obtained:

1) The integrated system this paper introduced can be studied by PWCS theory, and all its states are observable while the N Asset Coverage is more 4.

2) The relationship between GDOP and measurement matrix is studied. The less the GDOP is, the more significant the measurement matrix is. And still the precision become better.

3) The relationship between GDOP and system-level observable degree is studied via SVD and condition. The result shows that observable degree becomes worse with the increase of GDOP.

SV | GDOP = 3.4 | GDOP = 12.3 | SV | GDOP = 3.4 | GDOP = 12.3 | SV | GDOP = 3.4 | GDOP = 12.3 |
---|---|---|---|---|---|---|---|---|

5.3218e7 | 2.4836e7 | 68.8517 | 59.4036 | 3.1070 | 2.2073 | |||

1.2806e7 | 1.2737e7 | 48.7350 | 43.3406 | 2.6805 | 1.6535 | |||

233.6498 | 186.3406 | 11.9525 | 8.7298 | 2.4561 | 0.9920 | |||

169.0660 | 138.5101 | 6.9809 | 5.6404 | 2.2782 | 0.2777 | |||

142.0541 | 110.2614 | 5.6903 | 4.01241 | 1.9021 | 0.2446 | |||

108.0004 | 84.5738 | 3.9480 | 2.9204 | 1.1478 | 0.2443 |

Meanwhile, the conclusion comes from a certain model of GPS/SINS, of which the measurements are single- differences and double-differences. Further research is needed, for whether the conclusion is also correct for other models.

Hao He,Yuhang Zheng,Dongfang Yang,Jinsheng Zhang,Shicheng Wang, (2015) GDOP’s Influence on Observable Degree of Multi-Antenna GPS/SINS Integrated Attitude Measuring System. World Journal of Engineering and Technology,03,354-359. doi: 10.4236/wjet.2015.33C054