Performance Enhancement for a GPS Vector-Tracking Loop Utilizing an Adaptive Iterated Extended Kalman Filter

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1 Sensors 24, 4, ; doi:.339/s Article OPEN ACCESS sensors ISSN Performance Enhancement for a GPS Vector-racing Loop Utilizing an Adaptive Iterated Extended Kalman Filter Xiyuan Chen,2, *, Xiying Wang,2 and Yuan Xu 3 School of Instrument Science and Engineering, Southeast University, Nanjing 296, China; wangxiying@outloo.com 2 Key Laboratory of Micro-Inertial Instrument and Advanced Navigation echnology Ministry of Education, Southeast University, Nanjing 296, China 3 School of Automation and Electrical Engineering, University of Jinan, Jinan 2522, China; xy_abric@26.com * Author to whom correspondence should be addressed; chxiyuan@seu.edu.cn; el./fax: External Editor: Assefa M. Melesse Received: 6 August 24; in revised form: 27 October 24 / Accepted: 3 December 24 / Published: 9 December 24 Abstract: his paper deals with the problem of state estimation for the vector-tracing loop of a software-defined Global Positioning System (GPS) receiver. For a nonlinear system that has the model error and white Gaussian noise, a noise statistics estimator is used to estimate the model error, and based on this, a modified iterated extended Kalman filter (IEKF) named adaptive iterated Kalman filter (AIEKF) is proposed. A vector-tracing GPS receiver utilizing AIEKF is implemented to evaluate the performance of the proposed method. hrough road tests, it is shown that the proposed method has an obvious accuracy advantage over the IEKF and Adaptive Extended Kalman filter (AEKF) in position determination. he results show that the proposed method is effective to reduce the root-mean-square error (RMSE) of position (including longitude, latitude and altitude). Comparing with EKF, the position RMSE values of AIEKF are reduced by about 45.%, 4.9% and 54.6% in the east, north and up directions, respectively. Comparing with IEKF, the position RMSE values of AIEKF are reduced by about 25.7%, 9.3% and 35.7% in the east, north and up directions, respectively. Compared with AEKF, the position RMSE values of AIEKF are reduced by about 2.6%, 5.5% and 3.7% in the east, north and up directions, respectively.

2 Sensors 24, Keywords: Global Positioning System (GPS); iterated extended Kalman filter (IEKF); model error; nonlinear filtering; vector-tracing. Introduction raditional Global Positioning System (GPS) receivers utilize scalar tracing loops (SL) to trac the received GPS signals. All scalar tracing loops are independent of each other and ignore the internal relationship of each satellite []. When the received GPS signals degrade, the tracing loop inside the receiver may fail, therefore, reliable tracing loop operation is required to improve the tracing performance of GPS receivers. Recently, the vector-tracing loop (VL) consisting of a vector delay loc loop (VDLL) and a vector frequency loc loop (VFLL) was proposed for GPS receivers at low C/N ratios condition. For instance, Kim et al., proposed an adaptive VL for low-quality GPS signals []; Zhao et al., proposed implementation and performance assessment of a vector-tracing method based on a software GPS receiver in [2]; Jafarnia-Jahromi proposed a detection and mitigation of spoofing attacs on a vector-based tracing GPS receiver [3]. he VL was first proposed by Spiler [4]; it is an appealing and advanced structure which is able to provide an improved performance over traditional scalar loc loops. Depending on the correlation of each satellite signal, the VL technique processes received satellite signals and user dynamics together rather than separately. hus, a VL GPS receiver can get sufficient total signal power to trac the signal even if the signal quality from individual satellites is low [5 8]. In a VL GPS receiver, the VL navigation filter used to control the numerically controlled oscillators (NCO) in each satellite channel can aid all the tracing loops to trac wea received signals with other satellite channels. hus, as the core of the VL navigation filter, the nonlinear Kalman filter should be carefully designed. In the field of the estimation for the nonlinear system, extended Kalman filter is one of the most common examples [3,5]. However, although the EKF has the advantage in real-time estimation, the linearization of a nonlinear system by aylor series expansion, neglecting of the truncated high-order terms will introduce a truncated error, it is a biased estimator [9 ]. In order to overcome this problem, unscented Kalman filter (UKF) [2] and iterated EKF (IEKF) [3] are proposed. he UKF employs a deterministic sampling technique nown as the unscented transform to pic a minimal set of sample points (so called sigma points) around the mean. hese sigma points are then propagated through the non-linear functions, from which the mean and covariance of the estimate are then recovered [4]. However, the UKF needs to compute large numbers of samples. In order to overcome this problem, the IEKF is proposed to reduce the bias and the estimation error by increasing only a few simple iterative operations [9 ]. It should be point out that, the model error statistics in the UKF and IEKF are still prior estimates [9], while the noise is unnown in practice. Hence, motivated by the problems mentioned above, new methods need to be studied. In this wor, the AIEKF which combines the advantages of the Adaptive EKF (AEKF) [9] and the IEKF is proposed for the VL GPS receiver. he remainder of this paper is organized in four sections: Section 2 introduces the IEKF, AEKF and AIEKF. Section 3 presents the AIEKF method applied to a GPS

3 Sensors 24, vector-tracing loop in detail. In Section 4, the performance of the proposed filter is illustrated by a road test and compared with that of the AEKF and IEKF in estimation accuracy. Finally, the conclusions are given. 2. Adaptive Iterated Extended Kalman Filter In this section, a brief introduction to the IEKF is given. Furthermore, a modified IEKF named AIEKF is proposed. 2.. Iterated Extended Kalman Filter For a discrete-time nonlinear system, its model can be given by the following equations: X = f ( X ) + Γ W () Z = h( X) + V (2) where X is the state vector at step, f ( X ) is the nonlinear system function, Γ is the process noise driving matrix, Z is the observation vector, h( X ) Is the nonlinear observation model, W is the process noise which is assumed to be drawn from zero mean Gaussian white noise with covariance Q, and V is the observation noise which is assumed to be zero mean Gaussian white noise with covariance R. he IEKF used in this paper involves the following recursive equations [9,]: where X ˆ / is the priori state estimate at step, Xˆ =Φ X ˆ (3) / / P / =Φ P Φ + Q (4) ( ˆ / ) f X Φ = Xˆ / is the Jacobian matrix of f ( X ˆ / ), P / is the priori error covariance matrix. In addition to the standard EKF functions, a few iterative operations are involved in IEKF to reduce the bias and estimation errors of X ˆ / and P / given from Equations (3) and (4). he recursive steps are: ˆ n n Step : Initialize X / and P / where n =, 2, 3..., N is the number of the iteration. Step 2: Iterative processing Xˆ = X ˆ (5) / / P / = P / (6) ( ˆ ) ( ˆ ) ( ˆ ) h ( ) ( ) ( ) K = P H X H X P H X + R (7) n n n n n n n / / / / / ˆ ˆ ˆ ˆ ˆ ˆ X n+ n n n n n n n / = X / + K Z X / H X / X / X / (8)

4 Sensors 24, where n n K is the filter gain, ( ˆ n H X / ) iteration steps. ( ˆ ) P n n n n n / = I K H X P / / (9) + ( ˆ n / ) h X = Xˆ n / is the Jacobian matrix, n is the number of Step 3: Iteration finished Xˆ X () ˆ N / = / N P / = P / () 2.2. Adaptive Iterated Extended Kalman Filter In the EKF and IEKF algorithms, the nonlinear system function and nonlinear measurement function are linearized by aylor series expansion. Hence, the ignoring of higher order terms may introduce a truncation error. It is evident that both the Q and R for EKF and those for IEKF are prior estimates. here are uncertainties in the noise description, and the assumptions on the statistics of disturbances are violated since the availability of a precisely nown model is unrealistic in practical situations. In order to overcome these problems, the noise statistics estimator is employed in the IEKF. For a discrete-time nonlinear system described in Equations (3) and (4), the adaptive iterated extended Kalman filter (AIEKF) algorithm utilizes a set of equations as follows: ˆ X = X ˆ (2) where R ˆ n is the estimate of following equations: Rˆ where ( ) ( ) / / P / = P / (3) ( ) ( ) ( ) K n n ˆ n n ˆ n n ˆ n ˆ n = P H X / / / / / + H X P H X R (4) ( ) ( ) ( ) ˆ ˆ ˆ ˆ ˆ ˆ X n+ n n n n n n n / = X / + K Z h X / H X / X / X / (5) ( ˆ ) P n n n n n / = I K H X P / / (6) + ˆ N / = / Xˆ X (7) N P / = P / (8) ˆ N R = R (9) n R, it is estimated by the time-varying noise statistics estimators with the n( ˆ n n / ) ( ˆ n / ) ( ˆ / ) / ( ˆ / ) ( d ) ˆ I H X V V I H X + = R + d n n n n n n n H X P H X d = b b, < b <. (2)

5 Sensors 24, Adaptive Iterated Extend Kalman Filter to Vector-racing GPS Receiver 3.. he Architecture of a Vector-racing Loop he term Vector-racing Loop was first proposed by Spiler [5]. He presented a vector delay-loc loop (VDLL) algorithm, combining all the tracing channels and navigation functions. A performance analysis about VL was well described by Benson [6] in 27. His study showed that the EKF-based VL has potential advantages to improve the noise performance and reacquisition ability of a tracing loop. As software receivers developed, many studies on VL were conducted [7]. Several VL implementation methods in software receivers and their field test results showing improved tracing performance were reported in [8,9]. Figure shows the VL architecture in the Global Navigation Satellite System (GNSS) receiver. Comparing with a conventional scalar tracing loop, a VL tracing loop based on discriminator consists of a correlator, a discriminator and a code/carrier generator. he loop filter is removed in each channel. he discriminator outputs of each channel are directly connected to the navigation filter [2]. hese are used as the measurement of the EKF in navigation filter. Figure. he architecture of the vector tracing loop. Correlator Carrier and Code Discriminator Carrier and Code NCO GPS IF Signal Correlator Channel Carrier and Code Discriminator Navigation Filter (EKF is included) Carrier and Code NCO Channel N he Doppler frequency and the pseudo range are calculated from the estimated user position and velocity of the navigation filter. hus, there is one big loop including tracing channels and navigation module. Because the navigation result is derived from all channel tracing results, all the channels and navigation function are combined. his structure can trac temporarily attenuated or bloced satellite signals because the navigation result can be derived from other visible satellites. In general, it is nown that the vector-tracing loop based on the discriminator gives users a more accurate position and Doppler frequency than the scalar tracing loop. here have been many studies of VL implementation. Recently, So [2] described VDLL, VFLL and VDFLL with EKF. As mentioned above, it can be seen that the filtering output accuracy of VL is dependent on the EKF; however, the EKF will generate truncation errors due to aylor s series expansion to linearize the nonlinear system. he VL implementation in this paper is different from the previous studies. We implemented VL with AIEKF in a VL GPS software receiver.

6 Sensors 24, Design of the AIEKF In a conventional GPS receiver, all channels process incoming signals independently. his architecture is easy to implement and channels do not affect each other if one of them loses loc. However, this independency also prevents one channel from helping another because information obtained from one is not utilized by others. Since all channels share the same receiver position and velocity, and feedbacs of the position and velocity from the navigation filter should be exploited by all tracing channels so that they can comprehensively process signals from different satellites. A GPS receiver s position error and velocity error are determined by the code phase errors and pseudo range error through a line-of-sight (LOS) projection, as shown in Figure 2. Ignoring all other non-gaussian error sources such as satellite cloc, multipath, hardware bias, etc., the relationship between position error and code phase error can be written as the equation below [2]: Δ τ ˆ j, = τ j, τ j, + wj, ˆ code = tb, + ( X X) a j, + wj, (2) code = t +ΔX a + w b, j, j, where the subscript j is the satellite number, the subscript refers to measurement epoch, Δ τ j, is the code phase error in meters, τ j, is the code phase measurement in meters, the symbol ^ represents the estimation of a variable, t b, is the receiver cloc bias in meters, X is the receiver position vector, code a j, is the unit LOS vector from the receiver to the jth satellite, and w j, is the white Gaussian noise. Similarly, the carrier frequency error impacts the receiver velocity measurement error: Δ f = f fˆ + w carrier j, j, j, j, carrier = td, + a j, + wj, ( V Vˆ ) = t +ΔV a + w carrier d, j, j, where Δ f j, is the carrier frequency error, f j, is the carrier frequency measurement, the symbol ^ represents the estimation of a variable, t d, is the receiver cloc drift in meters per second, V is the carrier GPS receiver velocity vector, and w j, is the white Gaussian noise. In a typical GPS receiver, the code phase and carrier frequency measurements in Equations (2) and (22) can be obtained from the tracing loops. he following sections show how to use these measurements so that a vector-tracing loop can be formed. o obtain the dynamic equation for the local filter, the signal dynamic models are derived as follows. he user dynamics can be modeled using the shaping filter driven by white noise. When the user is stationary or moving with nearly constant velocity, an adequate model for the LOS range dynamic would be the integrated random wal. he discrete time state model for this integrated random wal model is: (22) X = X + + V (23) where is the discrete time interval. he acceleration of the receiver is modeled as a Gaussian distribution white noise, and thus the velocity at epoch + is expressed as below:

7 Sensors 24, V = V + v + (24) he drift of the receiver cloc is assumed a constant plus a small white noise. So the cloc bias and drift are modeled by Equations (25) and (26): t = b, t + b, + td, (25) t = + (26) d, + td, + η where t b, is the GPS receiver cloc bias at step, t d, is the GPS receiver cloc drift at step, η is the GPS receiver cloc drift noise. Figure 2. Relationship between GPS receiver position error and code phase error. GPS satellite Real GPS receiver position Code phase error Position error Line of sight direction Estimated GPS receiver position Process Equation of Vector-racing Loop he process equation can be established based on Equations (23) (26). he receiver position error, velocity error, cloc bias, and cloc drift are the state variables. Equation (27) shows the discrete process equation: ΔX+ ΔX ΔV + ΔV = Φ, + + W tb, + tb, (27) td, + td, [ x xˆ y yˆ z zˆ ] Δ X = (28) Δ ˆ ˆ ˆ = vx, vx, vy, vy, vz, vz, (29) Φ +, = (3)

8 Sensors 24, where ΔX is the GPS receiver position error at step, ΔV is the GPS receiver velocity error at step, Φ +, is the state transition matrix Measurement Equation of Vector-racing Loop he code phase discriminator and carrier frequency discriminator provide noisy code phase error and carrier frequency error when woring within their linear range [7]. As shown in Equations (23) and (24), the receiver position and velocity are directly affected by the code phases and carrier frequencies observables. herefore, the outputs of the discriminators are used as the measurements for the AIEKF as shown in Equation (32): where Z is the measurements, code, j, ( δ ) Z = h X + V (3) Z = E code,, Ecode, J, E carrier,, Ecarrier, J, (32) δ X = Δ X Δ X tb, td, (33) E is the code phase discriminator output of channel j ( j J), E carrier, j, is the carrier frequency discriminator output of channel j ( j J), δx is the state vector, h() is the nonlinear observation equation, V is Gaussian distribution white noise. When Equation (3) is used in AIEKF, it can be linearized by aylor series expansion and expressed as in Equation (34). ( δx ) Z = HδX + V (34) ax, ay, az, axj, ayj, azj, H = ax, ay, az, (35) axj, ayj, azj, h where H = is the Jacobian matrix, a x, j, a y, j and a z, j are the components of the LOS vector δx pointing from the receiver to the number j satellite Implementation of AIEKF to Vector-racing Loop As mentioned in the previous section, the GPS receiver position error, velocity error, receiver cloc bias and cloc drift terms consist of the AIEKF state vector, while the code phase and carrier frequency discriminator outputs are the measurement of AIEKF. he bloc diagram for GPS software defined VL receiver using AIEKF is shown in Figure 3. Figure 4 illustrates the flow chart for implementing the proposed AIEKF.

9 Sensors 24, Figure 3. Flowchart of vector-tracing loop based on AIEKF. Channel ~Channel N baseband Navigation filter LOS vector calculation GPS Antenna Code phase and Carrier frequency correction parameters Receiver position and velocity prediction GPS satellites position and velocity calculation GPS IF signal digital sampler Code and Carrier Generator GPS signal Correlator Code phase and Carrier frequency discriminator - Receiver position and velocity calculation Receiver position and velocity errors Code phase and carrier frequency discriminator output AIEKF Figure 4. Flowchart for implementing AIEKF. ˆ δ X P n ˆR δxˆ =Φ δxˆ / / P =Φ P Φ + Q / δxˆ = δxˆ P / / = P / / ( ) ( ) ( ) n n ˆ n n ˆ n n ˆ n ˆ n K = P/ H δx / H δx/ P/ H δx/ + R ˆ n+ ˆ n n n( ˆ n n ) ( ˆ n ) ( ˆ ˆ n δx / = δx / + K Z h X / H δx / δx / δx ) / P n+ n n / ( ˆ n n = δ / ) I K H X P / n( ˆ n n / ) ( ˆ n δ δ / ) + ˆ n ( ) ˆ n I H X V V I H X R = d R + d n( ˆ n n n / ) / ( ˆ n H δx P H δx/ ) δxˆ P ˆ N / = δx / = P N / / ˆ N = R R

10 Sensors 24, Effectiveness of IEKF in VL Shojaie et al. [2] uncovered the fact that the iteration of the observation updating step in a Kalman filter family will reduce the linearization error and improve estimation accuracy. he detailed ideas are explained with IEKF. he observation function h( X ) is expanded with aylor series at x ˆ,, which is x () in Figure 5. he linear expansion of the observation function can be described as: ( ˆ ) ( ˆ xh ) zlin = h x, + X x, (36) If the real observation is z, the state estimate x () at time can be obtained with Equation (36) and the Equation z lin = z. Figure 5. Graphic fundamental of IEKF with two iterations. z z = h ( x) x () x(2) x() x It can be seen from Figure 5 that there is an obvious distance between x () and x. his distance comes from the truncation error brought by the linearization of h( X ), which is expanded with aylor series. Shojajie found that the re-linearization of the observation equation can reduce the error between estimated value and observed value. he iterated extended Kalman filter repeatedly calculates the Kalman gain and an intermediate posterior state estimate x ˆ i, where i is the iteration number. It should be noted that the IEKF achieves good results if the measurement model is close to linear between the true state x and the calculated posterior intermediate state estimate after one linearization. Otherwise, the linearization error increases due to re-linearization and the IEKF fails to improve the state estimate. hus, it s necessary to mae sure that the measurement model of VL is close to linear around the true state x. In VL GPS receiver implementation, the observation Equation h () is used to calculate the code phase errors and carrier frequency errors of each satellite channel. For easily to analyses, we suppose there is only one satellite channel, and then h () is shown in Equation (37): ( ) h X ( ) ( ) ( z ) xr + x xs yr + y ys r + z zs x + y + z ρ ρ ρ = ( xr x) xs ( yr y) ys ( zr z) z s vx, + vy, + v z, ρ ρ ρ (37)

11 Sensors 24, where ρ = ( ) 2 ( ) 2 ( ) 2 xr + x xs + yr + y ys + zr + z zs and satellite, ( x, y, z ) is the GPS receiver position, (,, ) r r r is the distance between GPS receiver x y z is the satellite position, s s s X = x y z vx, vy, vz, is the state vector, x, y and z are the GPS receiver position error, v x,, v y, and v z, are the GPS receiver velocity error. It is nown that the GPS satellites wor on medium Earth orbit (above an altitude of 5 m). Also, x, y and z are usually less than 2 m. It is obvious that ρ, x s, y s and z s are much bigger than x, y and z. hus, Equation (37) can be approximated to Equation (38): ( ) h X x xr x s yr y s zr z s y ρ ρ ρ z x-x r s y-y v r s zr z s x, ρ ρ ρ v y, v z, (38) ρ r s r s r s where ( x x ) ( y y ) ( z z ) = + +. It can be seen that Equation (38) is a linear equation. Hence, the measurement model is close to linear function around x, and IEKF can achieve good results. 4. Positioning est Results and Discussion In this wor, road tests were done to assess the performance of the proposed method. he test platform consists of a GPS IF signal digital sampler and a real-time inematic (RK) GPS system (Figure 6). he sampler used in this wor is shown in Figure 7, and RK system is shown in Figure 8. he characteristics of the digital sampler used in this wor are listed in able. he digital sampler is a digital down converter which can receive GPS signals through the GPS antenna and then convert the high frequency GPS signals down to lower frequency signals. he lower frequency signals are called intermediate frequency (IF) signals. IF signals are digitalized by analog to digital converter which is included in GPS IF signal digital sampler. Figure 6. est platform. (a) RK base station; (b) RK mobile unit; (c) GPS IF signal digital sampler.

12 Sensors 24, Figure 7. GPS IF signal digital sampler. Figure 8. RK system. RK Base Station RK Mobile Unit Radio ransmitter Antenna Battery RK Base Station GPS Antenna Radio Receiver Antenna RK Mobile Unit GPS Antenna RK Base Station GPS Receiver RK Base Station Radio ransmitter Battery RK Mobile Unit Radio Receiver RK Mobile Unit GPS Receiver able. Position errors of the EKF, IEKF and AIEKF method. Method RMSE(m) East North Up EKF IEKF AEKF AIEKF he road test was carried out in the sports ground of Southeast University, Nanjing, China. he GPS IF signal digital sampler, GPS antenna, portable computer, and mobile unit of RK are carried on a cart. he cart runs along the sports round one circle. he GPS IF signals are recorded by computer. In the same time, RK mobile unit recorded the precise position of the cart (Figure 9).

13 Sensors 24, Figure 9. Road test. he DELA-G2 RK system is manufactured by Javad GNSS Company (San Jose, CA, USA). RK data rate is Hz. he position accuracy of RK is mm + ppm. In this test, the RK base station was placed at a nown location on the sports ground (Figure ), and the maximal distance between the GPS mobile unit and the RK base station is less than 4 m. In this case, the RK can provide a theoretical position accuracy of less than cm. he RK data were served as the reference in the evaluation of the VL performance. he AIEKF implements a discretized set of differential equations described in Section 3. Differences between the estimated velocity/position and the measured ones are processed in the AIEKF, to estimate the code phase errors and carrier frequency errors. he estimated code phase errors and carrier frequency errors are used to control the numerically controlled oscillators (NCO) of code/carrier generators. Figure shows the trajectory of the cart obtained from RK. he road test data were processed by EKF, IEKF, AEKF and AIEKF methods respectively and the trajectories of EKF, IEKF, AEKF and AIEKF are shown in Figure. he position errors, code phase errors and carrier frequency errors are important performance indexes of the VL GPS receiver. Figure 2 is the position error curves about EKF, IEKF, AEKF and AIEKF method. In Figure 2, green square line is EKF position error curve, blue circle line is IEKF position error curve, red point line is AEKF position error curve, and yellow line is AIEKF position error curve. It can be seen from Figure 2 that IEKF, AEKF and AIEKF can obtain higher accuracy on position compared with the EKF. Comparing with IEKF and AEKF method, the AIEKF method proposed in this paper can more effectively to reduce position error of VL GPS receiver.

14 Sensors 24, Figure. rajectory of the cart obtained the RK. Figure. rajectory of the road test obtained from RK, EKF, IEKF, AEKF and AIEKF. (a) Longitude and latitude; (b) Altitude EKF IEKF AEKF AIEKF RK 4 3 EKF IEKF AEKF AIEKF RK 2 2 North[m] 8 Up[m] Sampling Number East[m] (a) (b)

15 Sensors 24, Figure 2. Position errors of VLL GPS receiver. (a) Longitude error; (b) Latitude error; (c) Altitude error. East Error[m] EKF IEKF AEKF AIEKF Sampling Number North Error[m] 5 5 (a) EKF IEKF AEKF AIEKF Sampling Number (b)

16 Sensors 24, Figure 2. Cont. 4 3 EKF IEKF AEKF AIEKF 2 Up Error[m] Sampling Number (c) able shows the position RMSE values of the EKF, IEKF, AEKF, and AIEKF during the road test. From able, it can be seen that, compared with EKF, IEKF and AEKF, the RMSE values of AIEKF are reduced by about 45.%, 25.7% and 2.6% in the east direction, respectively. In able, similar results can also be seen in the north and up directions. Similarly, Figure 3 shows the code phase error and carrier frequency error curves. From able 2 we can see that, the RMSE value of code phase errors and carrier frequency errors based on AIEKF are lower compared with EKF, IEKF and AEKF. It should be noted that the code phase accuracy of tracing-loop is very important for GPS receiver. Higher code phase accuracy means higher positioning accuracy. hus, the VL GPS receiver based on AIEKF has better performance. In addition to the road test, a static positioning test ( s) was carried out at the Baima Par square, Nanjing, China. he static positioning GPS IF signal was processed by GPS software receiver based on SL method and VL method (based on EKF, IEKF, AEKF and AIEKF) respectively. he GPS software receiver generates a positioning result per 2 milliseconds. Figure 4 shows the static positioning results based on scalar, EKF, IEKF, AEKF and AIEKF method respectively. It can be seen from able 3 that, compared with EKF, IEKF and AEKF, the RMSE values of AIEKF are reduced by about 4.7%, 8.% and 4.% in the east direction, respectively. Meanwhile, the similar results can be seen in the north and up directions.

17 Sensors 24, Figure 3. Code phase error and carrier frequency error of PRN 2 satellite. (a) Code phase error; (b) Carrier frequency error..3.2 EKF IEKF AEKF AIEKF Code Phase Error[Chip] ims(ms) 8 6 (a) EKF IEKF AEKF AIEKF Carrier Frequency Error[Hz] ims(ms) (b) able 2. Code phase and carrier frequency errors of the EKF, IEKF and AIEKF method. Method RMSE(Hz) RMSE(Chip) Carrier Frequency Error Code Phase Error EKF IEKF.3.63 AEKF..56 AIEKF.84.5

18 Sensors 24, Figure 4. Static positioning result. 5 EKF IEKF AEKF AIEKF 5 North(m) -5 - able 3. RMSE of static positioning test depending on scalar, EKF, IEKF and AIEKF method. 5. Conclusions Method RMSE(m) East North Up EKF IEKF AEKF AIEKF his wor proposed the AIEKF method for a VL GPS software defined receiver. In this model, AIEKF is employed instead of the EKF in VL. In AIEKF, IEKF and AEKF are used together. IEKF can reduce the truncation error of EKF by a simple iterative procedure. Furthermore, the noise statistics estimator is employed in the IEKF to combine the advantages of the AEKF and the IEKF. he experimental results show that the proposed AIEKF outperforms the EKF. Acnowledgments East(m) his wor was supported in part by National Natural Science Foundation of China (No , 42425, ), Ocean Special Funds for Scientific Research on Public Causes (No ), Specialized Research Fund for the Doctoral Program of Higher Education (No ), the 52th China Postdoctoral Science Foundation (No. 22M52967).

19 Sensors 24, Author Contributions Xiyuan Chen and Xiying Wang designed the mathematical model of the proposed navigation method; Xiying Wang and Yuan Xu performed the experiments; Xiying Wang and Yuan Xu analyzed the data; Xiyuan Chen and Xiying Wang wrote the paper. Conflicts of Interest he authors declare no conflict of interest. References. Kim, K.; Jee, G.-I.; Im, S.-H. Adaptive Vector-racing Loop for Low-Quality GPS Signals. Int. J. Control Autom. Syst. 2, 4, Zhao, S.-H.; Lu, M.-Q.; Feng, Z.-M. Implementation and Performance Assessment of a Vector racing Method Based on a Software GPS Receiver. J. Navig. 2, 64, Jafarnia-Jahromi, A.; Lin,.; Broumandan A.; Nielsen, J.; Lachapelle, G. Detection and Mitigation of Spoofing Attacs on a Vector-Based racing GPS Receiver. In Proceedings of the 22 International echnical Meeting of the Institute of Navigation, Newport Beach, CA, USA, 3 January February 22; pp Parinson, B.W.; Spiler, J.J. heory and Applications Volume I. In Global Positioning System: heory and Applications; Aiaa: Washington, DC, USA, Lashley, M.; Bevly, D.M. Vector Delay/Frequency Loc Loop Implementation and Analysis. In Proceedings of the 29 International echnical Meeting of the Institute of Navigation (IM 29), Anaheim, CA, USA, January 29; pp Petovello, M.G.; Lachapelle, G. Comparison of vector-based software receiver implementations with application to ultra-tight GPS/INS integration. In Proceedings of the Institute of Navigation (ION GNSS 26), Fort Worth, X, USA, September 26; pp Won, J.-H.; Eissfeller, B. Effectiveness analysis of vector-tracing-loop in signal fading environment. In Proceedings of the 2 5th ESA Worshop on Satellite Navigation echnologies and European Worshop on GNSS Signals and Signal Processing (NAVIEC), Noordwij, Netherlands, 8 December 2; pp Lashley, M.; Bevly, D.M.; Hung, J.Y. Performance analysis of vector tracing algorithms for wea GPS signals in high dynamics. Sel. op. Signal Process. IEEE J. 29, 4, Fang, J.-C.; Gong X.-L. Predictive iterated Kalman filter for INS/GPS integration and its application to SAR motion compensation. IEEE rans. Instrum. Meas. 2, 4, Xu, Y.; Chen, X.-Y.; Li, Q.H. Autonomous Integrated Navigation for Indoor Robots Utilizing On-Line Iterated Extended Rauch-ung-Striebel Smoothing. Sensors 23, 2, Xu, Y.; Chen, X.-Y.; Li, Q.H. Adaptive Iterated Extended Kalman Filter and Its Application to Autonomous Integrated Navigation for Indoor Robot. Sci. World J. 24, 24, doi: org/.55/24/38548.

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