基于改进SR-UKF算法的BDS / GPS双系统定位

Article

BDS/GPS Dual Systems Positioning Based on the

Modified SR-UKF Algorithm

JaeHyok Kong, Xuchu Mao * and Shaoyuan Li

School of Electronic Information and Electric Engineering, Shanghai JiaoTong University, 800 Dongchuan Street,

Minhang District, Shanghai 200240, China; jhkong2014@163.com (J.K.); syli@sjtu.edu.cn (S.L.)

* Correspondence: maoxc@sjtu.edu.cn; Tel.: +86-133-9100-7759

measurement model and the state model are described. Furthermore, the modified square-root

Unscented Kalman Filter (SR-UKF) algorithm is employed in BDS and GPS conditions, and analysis

of single system/multi-system positioning has been carried out, respectively. The experimental results

are compared with the traditional estimation results, which show that the proposed method can

perform highly-precise positioning. Especially when the number of satellites is not adequate enough,

the proposed method combine BDS and GPS systems to achieve a higher positioning precision.

Keywords:

Global Navigation Satellite System (GNSS); positioning algorithm; modified square-root

Unscented Kalman filter (modified SR-UKF); BeiDou navigation System (BDS)

1. Introduction

With the development of space information technology, some countries are constructing Global

of the single system, and shows better effects on system performance [2].

The most conventional positioning estimation method is the iterative least square method (ILS).

Furthermore, extended Kalman filter (EKF) and unscented Kalman filter (UKF) are also used to

estimate the positioning data. ILS can solve three-dimensional positioning only when it receives the

signals from at least four satellites. This method is simple, and its computing speed is fast, but it has

a large linearization error and a low positioning estimation precision. The EKF can only be accurate

for a first order Taylor series. There may be a larger nonlinear error, and it needs to compute the

Currently, unscented Kalman filter (UKF) and square root UKF (SR-UKF) are the widely used

nonlinear filtering strategies, their applications are proposed. In the process of filtering, the calculation

error exists. The accumulation of the calculation error reduces the filtering precision, and even it

can make the error covariance matrices gradually lose their positive semidefiniteness [

to improve the numerical performance, the SR-UKF was proposed [

covariance matrices are directly used to calculate the state estimate, the QR decomposition, as well

as Cholesky factor updates. Thus, by this way, the numerical stability can be improved and also the

positive semi-definiteness of covariance matrices can usually be guaranteed [8].

This paper proposes a BDS-GPS system model, using the modified SR-UKF algorithm to perform

position estimation, and the experimental results illustrate the effectiveness of our proposed method.

Section 2 lists the general concept of the GNSS positioning. Section 3 introduces the modified

SR-UKF algorithm which will be used in our proposed models for GNSS position estimation. Section 4

describes the unification of the reference systems and the time systems of GPS and BDS. Section 5

addresses the developed nonlinear model and the filter implementation . Section 6 includes recent

experimental results and provides the comparison of these results from GPS and BDS with ILS, UKF,

SR-UKF and the proposed method. Finally, Section 7 summarizes this paper, and puts forward the

future developments.

2. GNSS Positioning Overview

coordinates, to determine the position of the corresponding user receiver antenna. According to

the different positions of the user receiver antennas, GNSS positioning can be divided into dynamic

positioning and static positioning. Currently, GNSS positioning has a variety of modes, such as precise

and satellite clock bias data provided from the International GNSS Service (IGS) and calculates the

precise user coordinates from the corrected carrier phase and pseudorange. The relative positioning

uses single or double difference of the carrier phase, and calculates the relative coordinates comparing

to one or several base stations. These technologies can perform high precise positioning, but they

need some accessories besides the user GNSS receiver, such as radio, network equipment, and the

(1)

cδt

is the receiver clock offset (scaled by speed of light

c

), and

contains the residual errors after

satellite-based and atmospheric error corrections [10].

with deterministic samplings. These points evolve according to the dynamics of the true nonlinear

precision, but also is easier to be implemented. Moreover, different from EKF, the evaluations of the

new EKF and UKF algorithm [12–15], and were used in GPS positioning [16–20]. The fuzzy adaptive

UKF algorithm was applied in spacecraft celestial navigation [21]. When no more than four satellites

can be received, a precision of data processing can be obtained by considering the UKF algorithm’s

small linearization error.

The UKF is mainly used for an arbitrary nonlinear system, and numerical instability often causes

P

design, the demanding operation is the evaluation of the square root of the covariance matrices at each

time instant for the updated set of sigma points. To solve this problem, the SR-UKF was proposed [

Meanwhile, in the application of positioning, the exact knowledge of the noise matrix which

is required in the framework of the Kalman filter is usually unknown and time-varying in practice.

The inappropriate prior statistics in the Kalman filter cause large estimation errors or even errors

possibly diverging. Because of the uncertain process noise, the standard UKF yields poor performance

in robustness and tracking accuracy.

with time, thus the effects of observations for correction of the state estimation are more and more

weakened. As the recent observations contain more information on the changed dynamic system

z

(2)

where

x

is state vector,

z

is zero-mean independent Gaussian white

noise, of which the covariance matrix is

is zero-mean independent Gaussian white noise of the

measurement, of which the covariance matrix is R.