The third-generation BeiDou Navigation Satellite System (BDS-3), which transmits new frequencies, has been completed on June 23rd, 2020. It means that BDS can provide Positioning, Velocimetry, and Timing (PVT) services for global users. However, its performance would be degraded under challenging users' environments. In this paper, we provide the loose and tight integration model between BDS-3 B1I/B2b Precise Point Positioning (PPP) and Inertial Navigation System (INS) to enhance the performance of BDS. Meanwhile, different precise satellite orbit and clock products are used in data processing. Experiment results show that the tight integration model can provide more accurate positioning solutions than that of PPP and PPP/INS loose integration. The impact of orbit/clock products' accuracy on the positioning accuracy is visible. Wherein, the positioning accuracy based on final orbit/clock products presents the best performance with about 3%, 19%, and 19% improvements in the north, east, and vertical components, respectively, compared to that calculated by rapid and ultra-rapid orbit/clock products.
usually, the Inertial Navigation System (INS) is utilized to form the PPP/INS integration system. Because INS can autonomically provide continuous position, velocity, and attitude by only processing measurements of the carrier output from Inertial Measurement Units (IMU) without external observations. Such character makes it possible to restrain the drawback of PPP in a poor environment effectively. However, IMU errors will accumulate over time. The integration of PPP and INS can compensate IMU errors online and restrain the divergence [9-11].
The definition of GPS and INS was proposed in 1978 [12]. Last decades, researchers did many works on the integration models of PPP and INS. In [13], a PPP/INS Loosely Coupled Integration (LCI) model is presented, and the results show that INS has a positive impact on PPP accuracy improvement. Own to the fact that LCI mode only can be worked when there are PPP solutions, hence, LCI will stop work under the satellite signal blocked areas where no PPP solutions are obtained. Therefore, a Tightly Coupled Integration (TCI) model is mentioned in [14] to get ideal solutions under satellite-denied environments. According to its conclusions, the horizontal positioning accuracy can be better than 15cm even when the satellite number is less than 4. In [15], PPP/INS TCI system is realized based on a lowcost IMU.
Currently, real-time PPP-related algorithms are becoming the research hotspot. Besides the final products, International GNSS Service (IGS) officially provided ultra-rapid products in November 2011 and established the Real-Time Service (RTS) centers in 2013. Recently, IGS centers such as Wuhan University (WHU) and Centre National d’Etudes Spatiales (CNES) also provide real-time satellite corrections products [16].
Therefore, we provide a tight integration model based on BDS-3 B1I/B2b PPP and low-cost IMU in this paper. Meanwhile, to evaluate the performance of such a model in real-time, a set of vehicle-borne data and the final/rapid/ultra-rapid orbit/clock products are utilized.
The mathematical models of B1I/B2b PPP, PPP/INS loose integration, and PPP/INS tight integration are described. 2.1.
Ionosphere-free pseudo-range ( PIF ) and carrier-phase ( LIF ) based on B1I and B2b can be described as [9]:
PIF f12 f12 f22 P1 f12 f22 f22 P2 c tr ts Trs Rers Errs Pcors,PIF Pcvrs,PIF Tidrs Grars P,PIF (1) LIF f12 f12 f22 L1 f12 f22 f22 L2 c tr t s Trs IF NIF Rers Errs Pcors,LIF Pcvrs,LIF Tidrs Grars Phrs,LIF L,LIF (2) where r and s represent receiver and satellite; is the geometric range between satellite and receiver; c is the speed of light; tr and t s represent the receiver clock offset and satellite clock offset; Trs is the tropospheric delay; IF is the IF wavelength; NIF is the IF float ambiguity in cycles; Pcvrs,LIF and Pcvrs,PIF are the phase center variation project to carrier phase and pseudo-range observation; Pcors,LIF and Pcors,PIF represent the phase center offset project carrier phase and pseudo-range observation; Tidrs is tidal loading; Grars is gravity error; Phrs,LIF is phase windup; Errs represents the earth rotation delay; Rers is the relativistic delay; L,LIF and P,PIF are the measurement noise and unmodeled errors of carrier phase and pseudorange observation.
The measurement function and state function of loosely coupled integration can be described respectively as [13,16]
ZLCI ,k H LCI ,k X LCI ,k LCI ,k , LCI ,k ~ N 0, RLCI (3)
X LCI ,k LCI ,k,k 1 X LCI ,k1 LCI ,k1, LCI ,k1 ~ N 0, QLCI ,k (4) where X LCI ,k represents the parameter vector; H LCI ,k is coefficient matrix; ZLCI ,k is the innovation vector; LCI ,k represents the vector of observation noise with the prior covariance of RLCI ; LCI ,k,k 1 is the state transition matrix which can be obtained by using the PSI angle model and first-order Gauss-Markov model [13]; LCI ,k 1 represents the state noise with the prior covariance of QLCI ,k .
The innovation vector ZLCI ,k can be expressed as
ZLCI ,k n Cne pPePP pIeNS C1 (5) vPPP (vInNS enn ine C1 C1ibb
C1 Cbnlb (6) where n , e , b , i are the navigation frame (n), the Earth-Centered Fixed reference frame (e), the body frame (b), and the inertial frame (i); Cn ( Cbn ) represents the rotation matrix from the n-frame (b-frame) e to the e-frame (n-frame); pINS and vInNS are the position and velocity of INS; pPPP and vPPP represent n n n the position and velocity results of PPP; l b represents the lever-arm; iee is the angular rotation rate of e-frame related to i-frame project to e-frame; ibb represents gyro’s angular measurements in b-frame; enn is the angular rotation rate of n-frame related to e-frame project to n-frame.
The parameter vector X LCI ,k can be described as
X LCI ,k pInNS vInNS (7) where pInNS and vInNS represent the position and velocity corrections under n frame; is attitude correction; Sg and Bg represent the scale factor and bias of gyroscope; Sa and Ba represent the scale factor and bias of accelerometers. The coefficient matrix H LCI ,k can be expressed as HLCI ,k 0I 0I CHbnvl,b 00 0H1 00 H10ibb (8)
Hv, inn lb lb ibb (9)
H1 Cbn lb (10) Based on the models above, the extended Kalman filter can be used for parameter estimation. Ba Bg Sa Sg T X LCI ,k,k 1 LCI ,k,k 1 X LCI ,k 1 PLCI ,k,k 1 LCI ,k,k 1 X LCI ,k 1LTCI ,k,k 1 QLCI.k 1 X LCI ,k X LCI ,k,k 1 Kk ZLCI ,k HLCI ,k X LCI ,k ,k 1 PLCI ,k I Kk H LCI ,k PLCI ,k ,k 1 I Kk HLCI ,k T Kk RLCI KkT where Kk is the Kalman gain matrix. (11) (12)
Different from loose integration, PPP/INS tight integration model uses the raw observation of BDS-3. Similarly, the observation function can be expressed as [10,17]
ZTCI,k HTCI,k XTCI,k TCI,k ,TCI,k ~ N (0, RTCI ) (13) with (14) ZPPC P1 P2 pre pse CneCbnlb PPC Ppc
ZLLC L1 L2 pre pse CneCbnlb LLC LLC where and are coefficient of Ionosphere-free combination; ZPPC , ZLLC , are the innovation vector of pseudo-range, carrier-phase, and Doppler that are calculated by making a difference operation between BDS-3 measurements ( PGNSS,PC , LGNSS,LC , and ) and the corresponding INS predicted values ( PINS,PC , LINS,LC , and ); represents modular operation; is vector cross product operation, pse and vse are satellite’s position and velocity in e-frame; pre and vre represent receiver’s position and velocity; PPC , LLC and are sum of pseudo-range errors, carrier-phase errors, and Doppler errors; Ppc , LLC , and are observing noise; TCI ,k represents the vector of observation noise with the prior covariance of RTCI ; other symbols have the same meanings as above.
The parameter vector can be expressed as (18) where tr and are receiver clock offset and receiver clock drift, dwet is wet component of tropospheric zenith delay and NIF represents carrier ambiguity.
The coefficient matrix HTCI ,k can be obtained by making the differential operation on Eqs. (15), (16), and (17)
H2 AC1 Cbnlb
H3 ACne H1diag(ibb ) Hv, ,TCI ACne (enn ine)H1 Cbn (lb ibb ) AD1C2 (Cbnlb)
1 / (RM h) 0 0 D1 0 1 / (RN h) cos(B) 0 (23)
0 0 1 where A represents the direction cosine matrix of satellite-receiver; Htr and are the coefficient of receiver clock offset and receiver clock drift; C1 is the transition matrix to transform position corrections from the e-frame to n-frame.
The state equation of TCI can be expressed as
XTCI ,k TCI ,k,k1 XTCI ,k 1 TCI ,k 1,TCI ,k 1 ~ (0, QTCI ,k ) (24) where TCI ,k,k1 is the system transition matrix from epoch k 1 to epoch k ; TCI ,k 1 represent the state noise with the covariance of QTCI ,k .
The algorithm structure of TCI and LCI can be shown in Fig. 1.
To evaluate the performance of those positioning methods using BDS-3 final/rapid/ultra-rapid orbit/clock products, a vehicle-borne experiment was arranged in Beijing on December 23, 2021. The equipments are a NovAtel GNSS receiver and a low-cost IMU INS616. The data sampling rate of BDS3 and IMU are 1HZ and 125HZ. The solutions calculated by Inertial Explorer (IE) software’s RTK/INS tight integration are used as reference values. The position differences by making a difference operation between the reference values and the solutions from PPP, PPP/INS LCI, and PPP/INS TCI are transformed into North-East-Up coordinate system. The trajectory of the experiment is shown in Fig. 2. The average number of satellites is 9.24, and the corresponding PDOP value is 2.07 (as shown in Fig. 3). According to Fig. 3, the data before 1300 s are collected almost in open sky conditions, and the observational condition becomes unexpected after 1300 s.
Fig. 4 shows the position differences of PPP, PPP/INS LCI, and PPP/INS TCI in the north, east, and vertical directions by using the ultra-rapid precise BDS-3 orbit/clock products. Similar to the trend of the number of satellites, position accuracy before 1300 s performs much better than those after 1300 s. In contrast, PPP/INS TCI mode provides the highest accuracy solutions compared to those of PPP and PPP/INS LCI, especially during the satellites partially blocked environments. The corresponding statistics in terms of RMS are listed in Table 1. Accordingly, the position RMSs of PPP are upgraded from 58.64 cm, 45.15 cm, and 99.47 cm to 38.40 cm, 26.84 cm, and 52.37 cm by PPP/INS TCI with the improvements of 34.52%, 40.55%, and 47.35% in the north, east, and vertical components. Compared to the solutions of TCI with that of LCI, visible improvements in the east and vertical directions (35.79% and 43.55%) can also be found. It may be due to the fact that BDS-3 PPP/INS TCI mode can work even while there are not enough available satellite for PPP calculation. In order to furtherly evaluate the high-accuracy positioning capability of the PPP/INS TCI, the distribution of these position differences calculated by the schemes above are shown in Fig.5. The results show that there are about 0.05%, 0.38%, and 24.06% horizontal position differences within 0.1 m for the PPP, PPP/INS LCI, and PPP/INS TCI, respectively. Such percentages of the horizontal position differences larger than 1.0 m are about 13.22%, 10.16%, and 3.38%. For the vertical component, the percentages of position differences larger than 1.0 m are 17.78% , 17.55%, and 6.78% for PPP, PPP/INS LCI, and PPP/INS TCI, respectively.
Fig.6 shows the position differences of PPP/INS TCI by using the final/rapid/ultra-rapid BDS-3 satellite orbit/clock products. The corresponding statistics are given in Table 2. The position RMS of PPP/INS TCI using final product are 37.20 cm, 21.71 cm, and 42.02 cm with the improvements of 3.1%, 19.11%, and 19.76% in the north, east, and up directions compared with those calculated by using ultrarapid products. However, the RMS differences between the rapid products-based solutions and those based on ultra-rapid products are invisible. Fig.7 shows the distribution of the position differences of PPP/INS TCI using different orbit/clock products. The results indicate that the percentages of the horizontal position differences within 0.1 m are 4.8%, 15.8% and 24.06% while using the final, rapid and ultra-rapid products in the PPP/INS TCI. Such percentages of the horizontal position differences larger than 1.0 m are 3.33%,3.36%, and 3.38%. For the vertical component, the percentages of position differences larger than 1.0 m are 3.69% , 5.98%, and 5.88% of the three type products.
This paper evaluates the impacts of different orbit/clock products on the positioning accuracy of the BDS-3 B1I and B2b signal-based PPP/INS tight integration model. The results carried out from vehicle-borne experiment data demonstrate that PPP/INS tight integration can provide more reliable and continuous positioning solutions than PPP and IPPP/INS loose integration, especially while suffering poor BDS observing conditions. Meanwhile, the positioning accuracy of BDS-3 PPP/INS tight integration can provide decimeter-level positioning accuracy by using ultra-rapid products.
6. Reference