BDS / GPS Integrated Positioning Method Research Based on Nonlinear Kalman Filtering

In order to realize fast and accurate BDS/GPS integrated positioning, it is necessary to overcome the adverse effects of signal attenuation, multipath effect and echo interference to ensure the result of continuous and accurate navigation and positioning. In this paper, pseudo-range positioning is used as the mathematical model. In the stage of data preprocessing, using precise and smooth carrier phase measurement value to promote the rough pseudo-range measurement value without ambiguity. At last, the Extended Kalman Filter(EKF), the Unscented Kalman Filter(UKF) and the Particle Filter(PF) algorithm are applied in the integrated positioning method for higher positioning accuracy. The experimental results show that the positioning accuracy of PF is the highest, and UKF is better than EKF. * Corresponding author


INTRODUCTION
With the growing of navigational information resources, the use of each system in combination in order to improve the positioning accuracy and reliability will be the inevitable development of satellite positioning technology.GPS is the most widely used navigation and positioning system currently, which is able to provide high-precision navigation and positioning services for users in worldwide.BeiDou Navigation Satellite System (BDS) is being implemented in China as an independent research and development system.BeiDou global satellite navigation system with 35 satellites will be built around 2020, which will together with GPS, GLONASS and Galileo provide high accuracy as well as reliable navigation and positioning service for the global users.Therefore, BDS/GPS combination navigation and positioning will also be an important development direction of future high-precision positioning.The paper carried out BDS/GPS integrated positioning method based on summary of the current integrated positioning technology research status and analysis the problems and trends.Based on Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) and Particle Filter (PF), BDS/GPS integrated positioning solution is proposed and constructed as well as the corresponding observation equation and measurement equation.

INTEGRATED POSITIONING PRINCIPLE
Using the pseudo-range positioning as the mathematical model, the parameters in the formula are shown as following.
The time of signal transmitted from satellite to the receiver is as following, In the pseudo-range positioning model, the pseudo-range observation is as below, ( ) ( ) ( ) As the signal transmits with the speed of light except the ionosphere and troposphere, the geometric distance between the satellite and observation station is calculated to consider the ionospheric delay ( , ) t is the geometric distance between the satellite position of time j t and the receiver position of time i t . (5)

State Equation of Integrated Positioning Method
The state vector X of the system is transferred form tk-1 to tk according to the following dynamic model.
is the state vector of the system, including three-dimensional position, three-dimensional velocity, as well as threedimensional accelerated velocity.
As the mobile devices are low dynamic in this research, the Singer model is chosen as the motion model.The system state variables are as following, The moving speed of the mobile device is considered as a first-order Gauss Markov process.
The system state equation is as following, where,

Measurement Equation of Integrated Positioning System
The single positioning system equation could be written as following represents the geometric observation matrix,  represents the measurement error vector, x is the state vector of the user.Thus, the observation equations of BDS and GPS are respectively as following: Combining the formula of ( 11) and ( 12), we can conclude that the pseudo-range observation model of the integrated positioning system.
The observation equation of BDS/GPS integrated positioning system is obtained as following: The system observation equation is nonlinear.
In the equation, 1  and 2  represent the white Gaussian noise.

Test Environment
The positioning module is used in the test with a sports car in the outskirts of Beijing.The test road map is showed in Figure 1.The test distance is 1349m, and the time is 568s as well as the output of positioning result is 1Hz.

Result Analysis
Figure3 shows the detection probability's change with the satellite carrier to noise ratio and non-coherent integration times.The false alarm probability Pfa is 0.001, the coherent integration time is 10ms, and the non-coherent integration frequency N is indefinite.When the signal carrier to noise ratio is fixed, if the non-coherent integration times are many enough, the detection probability is higher.The detection probability could be optimized with the increase of the non-coherent integration times.
Figure 3: Change of detection probability with carrier to noise ratio (Pfa=0.001, the coherent integration time is 10ms) As Figure 4 shows that we could capture satellite signal of -130dbm through 10ms coherent integration using the capture algorithms in this paper.In view of the weak signal environment, we are able to capture weak signal of -140dbm through 10 times of non-coherent integration based on the coherent integration.The result analysis is based on 1000 epochs of positioning results.Figure 5, Figure 6 and Figure 7 are show that the positioning accuracy of PF is the best and the UKF is better than EKF.
For the X direction error statistics under three kinds of filter, Figure 8, Figure 9

CONCLUSITION
In this paper, the research of BDS/GPS integrated positioning method is carried out.The feasibility and stability of the method have also been verified by EKF, UKF and PF.The results of the test show that the positioning result accuracy of PF is the best, and the positioning result accuracy of UKF is better than EKF.As the algorithm used pseudo-range positioning model without carrier value observation, the accuracy of positioning result could be improved in high precision demand area with carrier value observation and the pseudo-range smoothing.
j  --the real time of signal transmitted of Satellite named basic model of the pseudo-range positioning is as following.

Figure1.
Figure1.Test road map The paper uses BDS/GPS dual-mode compatible thumb receiver (MXTOS2-200) as the development module.The development module is shown in Figure2.The basic technical indicators of the receiver are shown in Table.

Figure
Figure 2: BDS/GPS dual-mode compatible thumb receiver

Figure4:
Figure4 : Signal acquisition correlation diagram EKF, UKF and PF are used to deals with the test data respectively, and the following results are obtained.Figure5 represents the X direction positioning error comparison under three kinds of filter.Figure 6 represents the Y direction positioning error comparison and Figure 7 represents the Z direction positioning error comparison.

Figure5
Figure5.X direction positioning error comparison under three kinds of filter Figure8.horizontal direction of error statistics about the integrated positioning under EKF

Table 2 .
Table 1 is the error analysis's statistical table.It shows that the result of PF is better than the other two.The reason may be that EKF and UKF are improved based on the linear Kalman filter algorithm for the nonlinear system.Error Analysis Statistical Table of Filter Therefore the filter is constrained by the linear Kalman filter algorithm, the system state should satisfy the Gaussian distribution.For the state model with non-Gaussian distribution, the filtering performance is not better enough.Instead, PF algorithm does not require the probability density of state variables with too many constraints.