BDS/GPS Multi-System Positioning Based on Nonlinear Filter Algorithm

Home , Kalman filter

Global Journal of Computer Science and Technology: G Interdisciplinary Volume 16 Issue 1 Version 1.0 Year 2016 Type: Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 0975-4172 & Print ISSN: 0975-4350

BDS/GPS Multi-System Positioning based on Nonlinear Filter Algorithm By JaeHyok Kong, Xuchu Mao & Shaoyuan Li Shanghai Jiao Tong University Abstract- The Global Navigation Satellite System can provide all-day three-dimensional position and speed information. Currently, only using the single navigation system cannot satisfy the requirements of the system's reliability and integrity. In order to improve the reliability and stability of the satellite navigation system, the positioning method by BDS and GPS navigation system is presented, the measurement model and the state model are described. Furthermore, Unscented Kalman Filter (UKF) is employed in GPS and BDS conditions, and analysis of single system/multi-systems’ positioning has been carried out respectively. The experimental results are compared with the estimation results, which are obtained by the iterative least square method and the extended Kalman filtering (EFK) method. It shows that the proposed method performed high-precise positioning. Especially when the number of satellites is not adequate enough, the proposed method can combine BDS and GPS systems to carry out a higher positioning precision. Keywords: global navigation satellite system (GNSS), positioning algorithm, unscented kalman filter (UKF), beidou navigation system (BDS). GJCST-G Classification : G.1.5, G.1.6, G.2.1

BDSGPSMultiSystemPositioningbasedonNonlinearFilterAlgorithm

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BDS/GPS Multi-System Positioning based on Nonlinear Filter Algorithm

JaeHyok Kong α, Xuchu Mao σ & Shaoyuan Li ρ

Abstract- The Global Navigation Satellite System can positioning only when it receives the signals from at provide all-day three-dimensional position and speed least four satellites. This method is simple, and its information. Currently, only using the single navigation system computing speed is fast, but it has a large linearization cannot satisfy the requirements of the system's reliability and

error and a low positioning estimation precision. The 201 integrity. In order to improve the reliability and stability of the EKF can only be accurate for a first order Taylor series. satellite navigation system, the positioning method by BDS and GPS navigation system is presented, the measurement There may be a larger nonlinear error, and it needs to Year model and the state model are described. Furthermore, compute the Jacobian matrix, in addition to the Unscented Kalman Filter (UKF) is employed in GPS and BDS calculation being difficult and one of the main sources of 9 conditions, and analysis of single system/multi-systems’ error [2]. The UKF represents statistical properties of the positioning has been carried out respectively. The system by deterministic sampling and avoids the experimental results are compared with the estimation results, disadvantage that the EKF must compute the Jacobian which are obtained by the iterative least square method and matrix. Theory shows that the EKF predicts the means the extended Kalman filtering (EFK) method. It shows that the correctly up to the second order of Taylor series and proposed method performed high-precise positioning. covariances up to fourth order. In contrast, the UKF Especially when the number of satellites is not adequate enough, the proposed method can combine BDS and GPS predicts the means and covariances correctly up to the systems to carry out a higher positioning precision. fourth order [3]. Keywords: global navigation satellite system (GNSS), The positioning accuracy of the GNSS, positioning algorithm, unscented kalman filter (UKF), especially the low-price GNSS receiver, cannot always beidou navigation system (BDS). be satisfied. This is because during the positioning process many errors, especially the measurement and I. INTRODUCTION satellite position errors, cannot be deleted. Traditional G

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methods can hardly minimize their influence. Filtering ith the development of space information algorithm provides an important method to reduce these technology, some countries are constructing errors and improve the GNSS positioning precision. Global Navigation Satellite System (GNSS). W This paper proposes a BDS-GPS system Now, in addition to USA’s GPS and Russia's GLONASS, model, using UKF to perform position estimation, and the Europe's Galileo and China's Bei Dou navigation the experimental results illustrate the effectiveness of our satellite System (BDS) are being built. Japan, India and proposed method. other countries are also planning to build their own SectionⅡlists the general concept of the GNSS regional navigation satellite systems. Even though it is positioning. SectionⅢ summarizes the general used GPS that is the most developed navigation satellite algorithm for GNSS receiver positioning and introduces system and it has many advantages, it also has some the UKF algorithm which will be used in our proposed disadvantages of system reliability that cannot satisfy models for GNSS position estimation. SectionⅣ the requirements of a single navigation system in the describes the unity of space coordinate system and certain situations [1]. Recently, the idea of multi- time system of GPS and BDS. SectionⅤ addresses the navigation positioning that consists of GPS, GLONASS, developed nonlinear model and the filter Galileo and regional satellite positioning system is implementation. SectionⅥ includes recent experimental gradually getting more interest in the field of satellite results and provides the comparison of these results navigation. Especially, the combination of GPS and BDS from GPS and BDS with ILS, EKF and UKF. Finally, can overcome the deficiency of the single system, and SectionⅥconcludes the paper and describes the future shows better effects on system performance. developments. Global Journal of Computer Science and Technology Volume XVI Issue I Version The most conventional positioning estimation methods are the iterative least square method (ILS), II. GNSS Positioning Overview extended Kalman filter (EKF) and unscented Kalman filter (UKF), etc. ILS can solve three-dimensional GNSS is a worldwide, all-weather navigation systems able to provide tridimensional position, velocity, and time synchronization to UTC scale. GNSS considers Author α σ ρ: School of Electronic Information and Electric Engineering, Shanghai Jiao Tong University, Shanghai 200240, China. the earth’s center as the reference point, to determine e-mails: [email protected], [email protected] the position of the receiver antenna in the reference

coordinate system. Since the positioning operation (GRV), but it is now represented using a minimal set of requires only one receiver, it is called the standalone carefully chosen points. Those points are propagated positioning. The basic principle of the standalone GNSS through the true nonlinear system. Besides the positioning is that taking the observed distance between possibility of improvement in precision, the UKF is much GNSS satellite and the user receiver antenna as the easier to implement than EKF. Unlike EKF, the UKF do benchmark, which is based on the known instantaneous not require Jacobian evaluations or any partial derivative satellites’ coordinates, to determine the position of the calculation. Some papers were proposed the new UKF corresponding user receiver antenna. According to the algorithm [6], and used in GPS positioning [7,8,9].When different positions of the user receiver antennas, GNSS no more than 4 satellites can be received, a precision of positioning can be divided into dynamic positioning and data processing can be obtained by considering UKF static positioning. algorithm’s small linearization error. Based on this fact, GNSS positioning is based on the one-way this paper proposes UKF as the position estimation

201 ranging technique: the propagation time to transmit algorithm for composing GNSS based navigation from satellite to user receiver is measured and multiplied systems. Year by the signal propagation velocity to obtain satellite-to- First, assume that state and measurement user range. The offset of the receiver clock relative to the equations of the system are discrete time nonlinear

10 system time scale should be estimated to position. The systems: measured range between receiver and satellite is x+ = fxw(, ) referred to as pseudorange, and can be represented as  t1 tt (2) z= hx() + v follows:  t tt ρ=++R ct δε (1) where is state vector, is measurement vector, and i i ui xt zt

wt is zero -mean independent gaussian white noise of where ρi is the i th satellite’s pseudorange which the covariance matrix is Q . v is zero -mean measurement, R is the geometric distance receiver- t i independent gaussian white noise of the measurement satellite, ctδ is the receiver clock offset (scaled by u of which the covariance matrix is R. speed of light c ) and εi contains the residual errors after Details about the UKF algorithm are described satellite-based and atmospheric error corrections. as follows: Equation (1) holds for both single GNSS (i.e.

G Initialize with

() BDS or GPS only) and ctδ u is referred to the time scale

of the considered system. In multi-constellation case, a xˆ00= Ex[] further unknown, representing the inter-system time =−−T PExxxx0() 0000ˆˆ () offset, must be estimated. (3) xˆˆa= Ex aT = [ x 0 0] T III. GNSS Positioning Algorithm 00 Pa= E() x aaaaT − xˆˆ ()x −=x diag( P, Q, R) a) Iterative Least Square Method 0 0000 0

ILS based position estimation is conventionally Calculate the sigma point used by GNSS receiver because of its simplicity and

rapidity. The algorithm is based on the pseudorange a aa a χλt−1=xxˆˆ tt −− 11±+ ()LPt−1 (4) equation. Since the pseudorange equation contains four 

unknowns: 3-D coordinates and one clock bias, at least Calculate weight coefficient four satellites are received to take 3-D positioning. Since these equations are nonlinear, they need to be W()mc=(λλ LW+ ) () =( λλ L+ )(1 +− ξη2 + ) 00 (5) linearized using the first order Taylor series WW(mc )= () =1/[2(L+λ)] i= 1, … , 2L approximation. Then a least squares solution is used ii and the receiver position can be calculated out [4]. This where λξ=2 (Lk+ ) − L， 0 ≤≤ξ 12, in order to control algorithm is simple and easy to implement, but the the distribution of the sigma points; η=2 , in order to position estimation accuracy is low and the stability is

Global Journal of Computer Science and Technology Volume XVI Issue I Version introduce the high order information. not strong. Time update: b) Unscented Kalman Filtering UKF algorithm is the minimum variance estimation based on UT (Unscented Transform).It was first proposed by Julier et al. [5] in 1995. In the UKF with the deterministic samplings, the state distribution is again approximated by Gaussian Random Variables

x xw χtt\1−−=f( χχ t 1 , t ) V. The Positioning of bds and gps using 2L the ukf ()mx xWˆtt\1−−= ∑ iχ itt,\1 i=0 Nonlinear Kalman filter can be well applied in 2L the GNSS positioning estimation, because of the =()cxχχ −−−−x T Ptt\1−∑Wi[ itt,\1−− xˆˆ tt \1][ itt ,\1 −− x tt \1] (6) characteristics of the Kalman filter in which the current = i 0 state parameter is updated according to the observed = χχxv zhtt\1−−(t 1 ,) t value using the predictive value. The system model 2L consists of the process model and the measurement zˆ− = Wz()m tt\1−−∑ i itt,\1 model. i=0 Measurement update equations: a) Process model 2L The state includes the receiver position and 201 P= Wzzzz()cT[ −−ˆˆ−− ][ ] velocity coordinates in CGCS2000 ECEF coordinate zztt ∑ i itt,\1−− tt \1 itt ,\1 −− tt \1 i=0

system, and the receiver clock offset which is related to Year 2L −− states and the clock drift caused by the Doppler P= W()cT[χ −− xzˆ ][ zˆ ] xztt ∑ i itt,\1−− tt \1 i ,\1tt −− tt \1 i=0 deviation. It also includes the non-white error in each 11 − satellite channel. So the overall system state has 10 K= PP1 (7) t xzt t zz tt fundamental states plus one shaping state for each −− xˆtˆ =+− x tt\1−− Kz t() tzˆ tt\1 observable channel: T sys T Pt=P tt]1− − KP t zz K t   tt xt= (,,,,,,,,,,, X ttttttttt X YYZ Z ctctctδδδ δεε R 12tt,..., ε nt ) where, where, is the receiver’s position,   is (XYZtt ,, t ) (XYZtt ,, t ) TT xa= xw T T v T  ; χχ a = ( xT ) ( χ wT ) ( χ vT )  the receiver velocity, ctctδδ,  are the receiver clock   . tt offset and the clock drift bias, respectively, ctδ sys is the t IV. THE UNITY OF TIME AND SPACE SYSTEM OF BDS offset between the two systems, δ R is the offset AND GPS between the Doppler shift and pseudorange’s rate, εε ε is the non-white error in each satellite a) The unity of the two coordinate systems ( 12t, t,..., nt )

channel. Because the error in each satellite is G The definition of coordinate system of BDS’s () CGCS2000 is essentially equal to GPS’s WGS84. Their independent, it can be modeled as a first-order reference ellipsoids are very similar, and there mainly Gaussian-Markov process. exists an extremely small flat rate difference. Flat rate Since there is a deviation in the two systems' difference between two coordinate systems causes a times, in order to eliminate positioning error caused by deviation of 0.105mm in latitude and altitude, and the time deviation between BDS and GPS, a variable δ sys ctt largest deviation of gravity is 0.016× 10−82ms . In the should be added. current level of the accuracy of measurement, such Considering the above state model, and a deviation can be neglected. So we can directly use the generic kinematics model for the receiver coordinates, we obtain the associated system model. position data GPS and BDS without coordinate transformation. x=⋅+⋅ Fx Cw t+1 tt b) The unity of the two time systems Since BDS and GPS use atomic time, there is The state transition matrix F is given by no leap second in both systems. The difference between A 0 the two systems is 1356 weeks in whole week and 14 F =  seconds in second. The navigation message of BDS 0 B contains the synchronization parameters between BDS where A is a 10× 10 time invariant matrix and B is a and GPS, but its implementation content in the message nn× diagonal matrix, which are given respectively by

is unpublished. So we cannot directly realize the full Global Journal of Computer Science and Technology Volume XVI Issue I Version synchronization between the two systems. Instead this problem, we simply add 14 seconds to BDT’s time for the rough calculation, and add a variable to represent the different system’s time deviation.

with 222 1T 00000000 Qδ XYZ  = diag(,,)σXYZ σσ    = σσ22 σ 2 0100000000 Qct diag(c12 , c ,,cn ) 001T 000000 2 sys 2 where σδ sys is the variance associated with theδt ,σδ R 0001000000 2 2 2 is the variance assoc iated with theδ R ,σ  ,σ  and σ  00001T 0000 X Y Z A = are the process noise variances associated with δ X , 0000010000 t   22 2 δYt and δ Zt ,σσc12,, c , σcn are the variances 0000001T 00 associated with the shaping filters driving noises and, 0000000100 Qδ t is defined by following equations. 0000000010  QQ 0000000001 11 12 201 = = Qδδtt Eww{}δt  = αα α QQ21 22  B diag(,, ,) Year wt is the system driving noise, which is given by h0 22 23

Q11 =++ T2 hT−−1 π hT2 12 T 23    22 wt=δ XYZwwwwww t ,,,,,,,,,, δδ t tt12 tsysδ Rct1c 2t w cnt with = +π  Q122 hT−− 1 hT2 h where, δ X , δY and δ Z are noises due to models’ 0 8 2 t t t Q22 = T ++2 h−−1 π hT2 receiver acceleration and other system disturbances , 23T w is the drivin g noise of the i th shaping filters of the where ci channel, and are the clock offset model driving w1t w2t ττ 2 -1，， 2 noises, is noise of the δt sys , and w is noise of the α= βτ= =100 wsys δ R 2ττ +1 2 +1 δ R . These noises are assumed to be zero – mean and −−20 19 −21 uncorrelated white random sequences. hh0=×=×=×9.4 10 ,−−12 1.8 10 , h 3.8 10 The noise matrix C is given by b) Measurement model D 0 The pseudoranges and Doppler shifts form the C =  G measurements set, the measurement equations of the

() 0 E BDS’s pseudorange ρ b and GPS’s pseudorange ρ g are where, D is a 10× 7 time invariant matrix and E is a as follows. nn× diagonal matrix which are given respectively by ρbb=−+r r ct δε ++ v T 2 2 0 0 0000 it t it b it it  bb22 b 2 T 0 0 0000 =(Xt − X it )( + Y t − Y it )( + Zt − Z it ) + ctδεb ++ it v it 0T 2 2 0 0000 ρgg=−+r r ct δε ++ v 0T 0 0000 jt t jt g jt jt 0 0 T 2 20000 =(X − Xgg )(22 +− Y Y )( + Z − Z g ) 2 + ctδε ++ v D = t jt t jt t jt g jt jt 0 0T 0000 where, XYZ,, are the receiver position coordinates, 0 0 0c 000 tt t XYZbbb,, are the i th BDS satellite’s coordinates, 000000c it it it XYZggg,, are the j th GPS satellite’s coordinates, ε is 0000010 jt jt jt it the non-white error in th satellite channel, and is the 0 0 0 0001 i vit

E= diag(,,ββ ,) β measurement noise of i th channel. Doppler shifts give information related to the The correspondent process noise covariance receiver velocity. Doppler is also used in our formulation, Global Journal of Computer Science and Technology Volume XVI Issue I Version matrix Qt is given by modeling it as:

Q   0 0 00 δ XYZ (X− X )( X − X  ) +− ( YYYY )( − ) = t it t it t it t it 0Q 0 00 Dit δ t (X− X )(22 +− YY )( + Z − Z ) 2 T 2 t it t it t it Qt= E{} ww tt = 0 0σδ sys 00  −− 2 ()()ZZZZt it t it  0 00σδ R 0 + ++ctδδ Rt 22 2  (Xt−X it )( +−YY t it )( + Z t − Z it ) 0 000Qct

  where XYZtt,, tare velocity coordinate of receiver at Position-X UKF EKF ILS 10   time t ; XYZit,, it it are velocity coordinate of i th satellite 0 X/m at time t . -10 -20 0 500 1000 1500 2000 2500 3000 So, the system measurement zt for n satellites is given by: Position-Y 20 T z=ρρρρρρbbbggg , ,, , , ,, , DD , ,,  D Y/m 0 tttntttntttn12 1212 1 2 t -20 0 500 1000 1500 2000 2500 3000 where nn=12 + n. n1 and n2 are the number of Position-Z 40 measured BDS and GPS satellites. 20 Z/m

0 VI. XPERIMENT AND NALYSIS E A 0 500 1000 1500 2000 2500 3000 201 t/s In the experiment, a set of BDS/GPS data

Year collected by the NovAtel OEM-615 receiver is used. The Figure 3 : Position Estimation Error using BDS/GPS’s position estimation is performed using the matlab in the

Data 13 PC environment. The raw data are processed using three algorithms: ILS, EKF and UKF. The number of visible satellites during the The position estimation errors (,xyz ,)are experiment is shown in figure 4. Since the observation shown in Figures 1-3. environment was good in the outdoor with a clear view of the sky, the number of BDS satellites was stable at Position-X UKF EKF BDS-ILS 4 10, GPS satellite numbers changed between 6 and 7. 2 X/m 0 The current BDS satellites are distributed in -2 0 500 1000 1500 2000 2500 3000 geostationary orbit (GEO) and inclined geosyn-chronous

Position-Y 15 satellite orbit (IGSO) over China. That’s why the BDS

10 satellite number that is received is stable at present Y/m 5

0 stage. 0 500 1000 1500 2000 2500 3000

Position-Z 10 Visible Satellites Number 14

5 BDS Satellites Number Z/m 13 GPS Satellites Number

0 G

0 500 1000 1500 2000 2500 3000 12 () t/s 11

Figure 1 : Position Estimation Error using BDS’s Data 10

9 Position-X UKF EKF GPS-ILS 8 10 0 X/m 7 -10 -20 0 500 1000 1500 2000 2500 3000 6

Position-Y 5 40 0 500 1000 1500 2000 2500 3000 20 Y/m 0

-20 0 500 1000 1500 2000 2500 3000 Figure 4 : The number of visible satellites

Position-Z 40 The correspondent root mean square 20 Z/m errors(RMSE) are compared in table 1-3. It's not difficult 0 0 500 1000 1500 2000 2500 3000 to find whether BDS, GPS or BDS/GPS positioning, the t/s result of the UKF has a higher accuracy than that of ILS Figure 2 : Position Estimation Error using GPS’s Data and EKF. Table 1 : Position Estimation Error using BDS’s data RMSE/m Global Journal of Computer Science and Technology Volume XVI Issue I Version X Y Z 3D ILS 3.429 4.487 4.879 7.463

EKF 3.437 4.065 4.770 7.148 UKF 3.438 3.136 3.655 5.917

Visible Satellites Number Table 2 : Position Estimation Error using GPS’s data 4 BDS Satellites Number RMSE/m GPS Satellites Number 3.5 X Y Z 3D

3 ILS 3.464 9.086 12.948 16.193

2.5 EKF 3.013 8.396 12.743 15.555

2 UKF 3.031 7.055 11.330 13.687 1.5 Table 3 : Position Estimation Error using BDS/GPS’s

1 data 0 500 1000 1500 2000 2500 3000

201 RMSE/m

X Y Z 3D Figure 6 : The number of visible satellites after remove Year ILS 2.979 6.077 10.437 12.439 When the number of satellites is less than four, the single system cannot be calculated; the advantage 14 EKF 2.880 4.681 9.521 10.994 of multi-system positioning can now be represented. As long as the sum of BDS and GPS satellites is not less UKF 2.573 4.345 9.308 10.589 than five, the multi-system can be calculated effectively.

When the number of the visible satellites is In figure 7, we can see that the number of large, this method cannot fully represent its superiority. satellites is going down, ILS algorithm's positioning We have eliminated some of the original data so as to precision is getting worse, and the rapid change in the simulate the condition of less visible satellites, the case number of satellites causes unsatisfactory results. when the numbers of BDS and GPS were in the 2-3 Kalman filtering overcomes these difficulties well. Reduction in the number of satellites does not have a range. significant impact on the positioning result.

Position-X UKF EKF ILS 10

0 X/m -10

G -20

() 0 500 1000 1500 2000 2500 3000

Position-Y

Y/m 0

-20 0 500 1000 1500 2000 2500 3000

Position-Z 40

m / 20 Z

0 0 500 1000 1500 2000 2500 3000 t/s

Figurer 7 : Estimation Error using BDS/GPS’s Data after Figure 5 : Positioning with GPS and BDS under bad removal conditions

The positioning with BDS and GPS in bad Table 4 : Position Estimation Error using BDS/GPS’s data after removal environments is shown as figure 5.

In a single system, where there are less than RMSE/m four visible satellites, the positioning cannot be obtained X Y Z 3D by the conventional method, but a multi-system ILS 4.491 8.981 13.240 16.488 positioning model is a good way to deal with this

Global Journal of Computer Science and Technology Volume XVI Issue I Version situation. The number of visible satellites after part of EKF 4.425 6.456 13.078 15.241 them have been removed is shown in figure 6. BDS UKF 3.492 6.371 12.519 14.475 remained at three, but GPS changed in 2-3. The correspondent root mean square errors of the various methods are compared in table 4. It is not difficult to find that the result of the modified UKF has the highest accuracy. When the number of BDS or GPS satellites is too few to calculate in each single

©20161 Global Journals Inc. (US) BDS/GPS Multi-System Positioning based on Nonlinear Filter Algorithm positioning system, we can adopt the multi-system 6. Psiaki, Mark L, “Kalman Filtering and Smoothing to positioning method. We can get the results by using the Estimate Real-Valued States and Integer ILS algorithm, but the error is too large and it does not Constants, ” Journal of Guidance, Control, and meet the requirements of practical application. The Dynamics, Vol. 23, No. 5, 2000, pp. 1404–1417. Kalman filtering algorithm of the multi-system can still 7. Cooper, S. and Durrant, W., “A Kalman filter Model maintain high precision, and it has a huge advantage. for GPS navigation of Land Vehicles,” IEEE Conference on Intelligent Robots and Systems VII. CONCLUSION (IR0S’94), Munich, Germany, 1994. 8. Psiaki, M. L., Batch Algorithm for Global-Positioning- In this paper, the multi-system position System Attitude Determination and Integer estimation algorithm which uses the UKF algorithm was Ambiguity Resolution, Journal of Guidance, Control, proposed, which was verified by the real BDS/GPS data. and Dynamics, Vol. 29, No. 5, 2006, pp. 1070–1079. We improved new nonlinear positioning models for the 9. Perea, L., Elosegui, P., New state update equation 201 two navigation systems. After analysing the for the unscented Kalman filter, Journal of characteristics of BDS and GPS, the unity of time and Guidance, Control, and Dynamics, Vol. 31, No. 5, Year space of the two systems was considered. We selected 2008, pp. 1500–1504.

15 the BDT as the time standard, and the CGCS2000 as 10. Antonio, A., Salvatore, G. and Ciro, G., the coordinate standard. The proposed models can well “Performance assessment of aided Global address the frequent change of the GNSS satellites. The Navigation Satellite System for land navigation,” IET experimental result shows that the performance of the Radar, Sonar and Navigation, Vol. 7, No. 6, 2013, algorithm is better than ILS and EKF algorithms. On the pp. 671-680. other hand, UKF and EKF algorithms reach higher 11. Wei, Z., “Chinese geodetic coordinate system 2000 stability than ILS algorithm. Thus the proposed method and its comparison with WGS84,” Journal of also can be used in multi-system position estimation Geodesy and Geodynamics, Vol. 28, No.5, 2008, under the bad positioning conditions. In a single system pp. 1-5. when there are less than four visible satellites, the 12. Liu, J. and Lu, M., “An Adaptive UKF Filtering positioning cannot be obtained by the conventional Algorithm for GPS Position Estimation,” Wireless method. However, using the proposed nonlinear Communications, Networking and Mobile positioning models, the high precision positioning can Computing, 2009. WiCom '09. 5th International be solved as long as the sum of the satellites is no less Conference. Beijing China. 2009, pp. 1-4. G () than five in the dual system. In addition, the positioning accuracy is much higher than the result obtained by the iterative least square algorithm. Future work will include the implement of the other nonlinear filters with the proposed GNSS positioning models.

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