*2.2. GPS/BDS-2/BDS-3 SF-PPP/INS Tight Integration Model*

In the GPS/BDS-2/BDS-3 SF-PPP/INS tight integration model, the theoretical geometrical distance in SF-PPP can be replaced by the INS-predicted values. Meanwhile, the doppler observation was proved to be effective to estimate the INS sensor errors [21]. Hence, the corresponding functions can be expressed as:

$$\begin{bmatrix} \mathbf{P}\_{\text{GNNS},f\_{1}} - \mathbf{P}\_{\text{INS},f\_{1}} - l\_{p}^{s} \\ \mathbf{L}\_{\text{GNNS},f\_{1}} - \mathbf{L}\_{\text{INS},f\_{1}} - l\_{p}^{s} \\ \mathbf{D}\_{\text{GNNS},f\_{1}} - \mathbf{D}\_{\text{INS},f\_{1}} - l\_{v}^{s} \\ \mathbf{F}\_{r,f\_{1}}^{\text{F}} - \mathbf{f}\_{\text{INS},r,f\_{1}}^{\text{F}} \\ d^{s}\_{r,f\_{1}} - d\_{\text{INS},r,f\_{1}}^{\text{s}} \end{bmatrix} = \begin{bmatrix} \mathbf{u}\_{r}^{s} \cdot \mathbf{x} + \delta t\_{r} + \mathbf{M} \mathbf{w}\_{r}^{s} \cdot \delta \mathbf{Z} \mathbf{W} \mathbf{D}\_{r} + \delta I\_{r,f\_{1}}^{s} + \delta d\_{r,f\_{1}}^{s} \\ \mathbf{u}\_{r}^{s} \cdot \mathbf{x} + \delta t\_{r} + \mathbf{M} \mathbf{w}\_{r}^{s} \cdot \delta \mathbf{Z} \mathbf{W} \mathbf{D}\_{r} - \delta I\_{r,f\_{1}}^{s} - \lambda\_{f\_{1}}^{s} \delta N\_{r,f\_{1}}^{s} \\ \mathbf{u}\_{r}^{s} \delta \nu\_{r} + \delta t\_{r} \\ \delta I\_{r,f\_{1}}^{s} \\ \delta d\_{r,f\_{1}}^{s} \end{bmatrix} \tag{14}$$

where *l s <sup>p</sup>* and *l s <sup>ν</sup>* are the lever arms on position and velocity; *<sup>D</sup>* stands for the doppler observations with the unit of m/s.

In general, the mathematical model of INS mechanization can be written as [21]

$$
\begin{bmatrix}
\boldsymbol{\nu}\_{\text{INS},t\_k}^{\boldsymbol{n}} \\
\boldsymbol{\nu}\_{\text{INS},t\_k}^{\boldsymbol{n}} \\
\boldsymbol{\mathcal{C}}\_{b,t\_k}^{\boldsymbol{n}}
\end{bmatrix} = \int\_{\begin{subarray}{c}t\_{k-1} \\ \end{subarray}}^{t\_k} \begin{bmatrix}
\boldsymbol{f}^{\text{n}} - \left(\boldsymbol{\omega}\_{\text{i}\varepsilon}^{\boldsymbol{n}} + \boldsymbol{\omega}\_{\text{in}}^{\boldsymbol{n}}\right) \times \boldsymbol{\mathcal{V}}\_{\text{INS},t\_{k-1}}^{\boldsymbol{n}} + \boldsymbol{\mathcal{g}}^{\text{n}} \\
& \boldsymbol{\mathcal{V}}\_{\text{INS},t\_{k-1}}^{\boldsymbol{n}} \\
& \left(\boldsymbol{\omega}\_{\text{i}\boldsymbol{b}}^{\boldsymbol{n}} \times \right) - \left(\boldsymbol{\omega}\_{\text{in}}^{\boldsymbol{n}} \times \right) \boldsymbol{\mathcal{C}}\_{\text{b},t\_{k-1}}^{\boldsymbol{n}}
\end{bmatrix} dt \tag{15}
$$

where *ω<sup>n</sup> ib* and *<sup>f</sup> <sup>n</sup>* are the angular rate and specific force that measured by accelerometer and gyroscope; *<sup>p</sup><sup>n</sup> INS*,*tk* and <sup>ν</sup>*<sup>n</sup> INS*,*tk* represent the position and velocity, computed via INS; *Cn <sup>b</sup>*,*tk* is the transform matrix for attitude from body frame (*b*) to navigation frame (*n*); *<sup>C</sup><sup>n</sup> <sup>b</sup>* is the direction cosine matrix of attitude, which can be described based on the Euler angle;

, *tk tk*−<sup>1</sup> ()*dt* is the integral operation from epoch *tk*−<sup>1</sup> to *tk*; *<sup>ω</sup><sup>n</sup> in* and *<sup>ω</sup><sup>n</sup> ie* are the rotation angular rate of the *n* frame and ECEF frame in terms of inertial frame (*i*) projected in the *n* frame; *g<sup>n</sup>* is the gravity in the *<sup>n</sup>* frame. Then, the basic observational functions for SF-PPP/INS tight integration can be expressed as

⎡ ⎢ ⎣ *Ps INS*,*r*, *f*1 *Ls INS*,*r*, *f*1 *Ds INS*,*r*, *f*<sup>1</sup> ⎤ ⎥ <sup>⎦</sup> <sup>+</sup> ⎡ ⎢ ⎢ ⎣ *l s p l s p l s p l s ν* ⎤ ⎥ ⎥ ⎦= ⎡ ⎢ ⎢ ⎢ ⎣ - - -*p<sup>s</sup>* <sup>−</sup> - *pe INS* <sup>+</sup> *<sup>C</sup><sup>e</sup> nCn b l b INS*−*GNSS*- - - <sup>+</sup> <sup>Δ</sup>*P<sup>s</sup> <sup>f</sup>* - <sup>1</sup> - -*p<sup>s</sup>* <sup>−</sup> - *pe INS* <sup>+</sup> *<sup>C</sup><sup>e</sup> nCn b l b INS*−*GNSS*- - - <sup>−</sup> *<sup>λ</sup><sup>s</sup> f*1 *Ns <sup>r</sup>*, *<sup>f</sup>*<sup>1</sup> <sup>+</sup> <sup>Δ</sup>*L<sup>s</sup> <sup>f</sup>* - <sup>1</sup> - -ν*<sup>s</sup>* <sup>−</sup> <sup>ν</sup>*<sup>e</sup> INSC<sup>e</sup> n* - *ω<sup>n</sup> in*<sup>×</sup> *Cn b l b INS*−*GNSS* <sup>+</sup> *<sup>C</sup><sup>n</sup> b* - *l b INS*−*GNSS*× *ωb ib*- - - <sup>+</sup> <sup>Δ</sup>*D<sup>s</sup> f* 1 ⎤ ⎥ ⎥ ⎥ ⎦ (16)

where *p<sup>s</sup>* and *ν<sup>s</sup>* refer to position and velocity of a satellite in the *e* frame computed by the real-time satellite orbit and clock products; *p<sup>e</sup> INS* and <sup>ν</sup>*<sup>e</sup> INS* stand for the position at the IMU center in the geodetic coordinate system and velocity at the IMU center in *e* frame, which can be obtained from *p<sup>n</sup> INS* and <sup>ν</sup>*<sup>n</sup> INS*; *C<sup>e</sup> <sup>n</sup>* is the transform matrix from the *n* frame to the *<sup>e</sup>* frame; *<sup>l</sup> b INS*−*GNSS* is the lever arm measured from the IMU center to the GNSS receiver antenna phase center in *b* frame; Δ*P<sup>s</sup> f*1 , Δ*L<sup>s</sup> f*1 , and Δ*D<sup>s</sup> <sup>f</sup>*<sup>1</sup> represent the sum of the error corrections of the pseudorange, carrier phase, and doppler.

As illustrated above, the INS solutions and GNSS observations are in different frames (*n* frame for INS and *e* frame for GNSS). Therefore, to utilize the information on the same foundation, the following corrections on position and velocity can be introduced:

$$
\begin{bmatrix}
\delta p\_r \\
\delta \mathbf{v}\_r
\end{bmatrix} = \begin{bmatrix}
\mathsf{C}\_1 \delta p\_{\mathrm{INS}}^n + \mathsf{C}\_1 \left( \mathsf{C}\_b^n I\_{\mathrm{INS}-\mathrm{GNNS}}^b \times \right) \delta \theta \\
\mathsf{C}\_2 \delta p\_{\mathrm{INS}}^n + \mathsf{C}\_n^c \delta v\_{\mathrm{INS}}^n - \mathsf{C}\_n^c \gamma\_1 \delta \theta + \mathsf{C}\_n^c \mathsf{C}\_b^n \left( I\_{\mathrm{INS}-\mathrm{GNNS}}^b \times \right) \delta w\_{ib}^b
\end{bmatrix} \tag{17}
$$

where *<sup>C</sup>*<sup>1</sup> is the transform matrix between the geodetic coordinate frame and the *<sup>e</sup>* frame of the position increments; *<sup>C</sup>*<sup>2</sup> stands for the coefficient relevant to position derived from *δ*(*Ce nν<sup>n</sup> <sup>r</sup>* ) [21]; *γ*<sup>1</sup> is the coefficient related to attitude; *δω<sup>b</sup> ib* denotes the gyroscope errors (bias and scale factor). In general, this parameter can be expressed as [37]

$$
\begin{bmatrix}
\delta\boldsymbol{\omega}^{b}\_{\dot{}^{b}} \\
\delta\boldsymbol{f}^{b}
\end{bmatrix} = \begin{bmatrix}
\mathbf{S}\_{\mathcal{S}} & \mathbf{0} \\
\mathbf{0} & \mathbf{S}\_{a}
\end{bmatrix} \begin{bmatrix}
\boldsymbol{\omega}^{b}\_{\dot{}^{b}} \\
\boldsymbol{f}^{b}
\end{bmatrix} + \Delta t \begin{bmatrix}
\mathbf{B}\_{\mathcal{S}} \\
\mathbf{B}\_{a}
\end{bmatrix} \tag{18}
$$

where Δ*t* indicates the IMU observations interval.

The state vector of the multi-GNSS SF-PPP/INS model can be described as

$$\mathbf{X} = \begin{bmatrix} \delta \mathbf{p}\_{INS}^{\mathrm{n}} & \delta \mathbf{v}\_{INS}^{\mathrm{n}} & \delta \boldsymbol{\Theta} & \mathbf{B}\_{\mathcal{S}} & \mathbf{B}\_{a} & \mathbf{S}\_{\mathcal{S}} & \delta \mathbf{t}\_{r} & \mathbf{Z} \mathbf{V} \mathbf{D}\_{r} & \delta \mathbf{d}\_{r,f\_{1}}^{\mathrm{s}} & \delta \mathbf{N}\_{r,f\_{1}}^{\mathrm{s}} & \delta \mathbf{I}\_{r,f\_{1}}^{\mathrm{s}} \end{bmatrix} \tag{19}$$

where *δtr* and *δd<sup>s</sup> <sup>r</sup>*, *<sup>f</sup>*<sup>1</sup> refer to the parameters related to receiver clock error and receiver DCB errors; *δN<sup>s</sup> <sup>r</sup>*, *<sup>f</sup>*<sup>1</sup> and *<sup>δ</sup>I<sup>s</sup> <sup>r</sup>*, *<sup>f</sup>*<sup>1</sup> denote the ambiguities and ionospheric delays in the slant propagation path.

The PSI angle model [21] is adopted to describe the variations of position, velocity, and attitude in the temporal domain, which can be described as

$$
\begin{bmatrix}
\delta \dot{p}\_{\rm INS}^{\rm n} \\
\delta \dot{v}\_{\rm INS}^{\rm n} \\
\dot{\theta}
\end{bmatrix} = \begin{bmatrix}
\omega\_{\rm cn}^{\rm n} \times \delta p^{\rm n} + \delta v\_{\rm INS}^{\rm n} \\
\mathcal{f}^{\rm n} \times \theta + \mathcal{C}\_{b}^{\rm n} \delta \dot{f}^{\rm b} - \left(2\omega\_{\rm ic}^{\rm n} + \omega\_{\rm cn}^{\rm n}\right) \times \delta \mathbf{v}\_{\rm INS}^{\rm n} + \delta \mathbf{g}^{\rm n} \\
\end{bmatrix} \tag{20}
$$

where the meanings of the parameters are the same as those mentioned above. The variation of IMU errors can be described by the first-order Gauss–Markov process. Meanwhile, the random constant process is adopted to express the variation of float ambiguities. The random walk process is chosen as the dynamic model for the receiver-clock-related parameters and the atmosphere-related parameters.

Finally, the parameterized elements can be estimated by the Extended Kalman Filter (EKF) [38]

$$
\begin{bmatrix}
\mathbf{X}\_k \\
\mathbf{P}\_k
\end{bmatrix} = \begin{bmatrix}
\Phi\_{k,k-1}\mathbf{X}\_{k-1} + \mathbf{K}\_k(\mathbf{Z}\_k - \mathbf{H}\_k\boldsymbol{\Phi}\_{k,k-1}\mathbf{X}\_{k-1}) \\
(\mathbf{I} - \mathbf{K}\_k)\left(\boldsymbol{\Phi}\_{k,k-1}\mathbf{P}\_{k-1}\boldsymbol{\Phi}\_{k,k-1}^T + \mathbf{Q}\_{k-1}\right)(\mathbf{I} - \mathbf{K}\_k)^T + \mathbf{K}\_k\mathbf{R}\_k\mathbf{K}\_k^T
\end{bmatrix} \tag{21}
$$

where **I** is the unit matrix. The elements in **Φ***k*,*k*−<sup>1</sup> can be acquired from the state models mentioned above; *K* refers to the gain matrix; *Q* is the state noise variance.

#### *2.3. Implementation of SF-PPP/INS Tight Integration Model*

Based on the descriptions above, the structure of the proposed real-time multi-GNSS SF-PPP/INS tight integration model is presented in Figure 1. Velocity and angular increments are provided by IMU sensors. After initializing the system, the compensated IMU outputs are processed in the INS mechanization to supply position, velocity, and attitude information. Then, the information, along with real-time GNSS products, is utilized to obtain the GNSS observation predictions (pseudorange, carrier phase, and doppler). After this, the INS-predicted observations are fused with the original observations provided by GNSS in the EKF. Finally, the estimated IMU errors are fed back to the IMU outputs before INS mechanization. Meanwhile, the navigation information is corrected.

**Figure 1.** Implementation of real-time multi-GNSS SF-PPP/INS tight integration model.

#### **3. Tests, Results, and Discussions**

To validate the performance of the presented real-time GPS/BDS-2/BDS-3 SF-PPP/INS tight integration, the data of the GNSS original observations and IMU outputs were processed and analyzed. The first subsection demonstrates the positioning performance of GPS + BDS (G + B) SF-PPP and SF-PPP/INS tight integration based on IGS' final precise products under kinematic conditions. The second subsection is to evaluate the performance while using real-time orbit and clock products from CAS, GFZ, and WHU.

#### *3.1. Data Collection*

The original test data for GPS (L1) and BDS-2/BDS-3 (B1I) were gathered on 21 December 2021, at China University of Geosciences Beijing. Figure 2 (top left and top right) shows the typical scenery in data collection areas, which were mainly around large buildings and boulevards. In these environments, the GNSS signal is heavily blocked. Figure 2 (bottom left) shows the test platform and equipment. Figure 2 (bottom right) presents the mission route. The multi-GNSS multi-frequency PANDA PD318 receiver was adopted in this mission, the sampling rate of which was 1 Hz. IMU outputs were provided by POS320, and the output rate was 200 Hz. Table 1 presents the details of POS320.

**Figure 2.** Mission details of vehicle test on 21 December 2021, in Beijing, China. Typical scenery in data collection areas (top left and top right); test platform and equipment (bottom left); and mission trajectory (bottom right).

**Table 1.** Technical parameters of POS320.


For data processing, the final precise orbit and clock products were afforded by Wuhan University, and the real-time products were provided by CAS, GFZ, and WHU. The DCB product was from CAS. The satellite cutoff elevation was intercalated as 10◦. The satellite antenna phase center offsets were rectified by adopting igs14\_2076\_plus.atx. The slant ionospheric delay was corrected first using WHU's GIM data. Afterward, the residuals were parameterized as random walks. Receiver DCB, drifts and offsets of the receiver clock, and ISB were also modeled as random walk processes. Moreover, float ambiguities were parameterized as random constants. In INS data processing, coning offsets, rotational, and sculling effect engendered from inertial axes motion were rectified using the INS mechanization [39]. According to the research in [40], to reduce the impact of receiver Time Delay Bias (TDB) between BDS-2 and BDS-3, the weight ratio of BDS-2 and BDS-3 MEOs was set to 1:3. The weight ratio of BDS-2 GEOs and MEOs/IGSOs was set to 1:10. In the validation stage, the results provided by the smoothed RTK/INS tight integration were utilized as reference values.

Figure 3 describes the numbers of available satellites and relevant Position Dilution of Precision (PDOP) of GPS-only, BDS-only, and G + B modes. The numbers of satellites on average were 5.1, 12.4, and 16.2, respectively. The corresponding PDOPs on average were 3.35, 4.27, and 2.99, respectively. These results illustrate that frequent signal blocks occurred under the GPS-only mode. BDS satellites could remedy the loss of GPS satellites in most of these periods, but there was still a GNSS outage for the partially obstructed observation environment. According to the statistics, about 24 s satellite outages (from 2048 s to 2072 s) occurred during the test (the black rectangle in Figure 3); therefore, the effect of INS is mainly focused on this section.

**Figure 3.** Satellite number and PDOP of GPS-only, BDS-only, and G + B (the start time is 195,453 s (GPS time)); the black rectangle represents the GNSS outage period underG+B mode.

#### *3.2. Positioning Performance of PPP and PPP/INS Tight Integration*

Figure 4 presents the position differences ofG+B SF-PPP and G + B SF-PPP/INS in terms of the reference solutions, and the relevant RMS (Root Mean Square) values of the position differences are shown in Table 2. According to Figure 4, the performance of SF-PPP/INS was obviously higher than that of SF-PPP in both horizontal and vertical directions. The position RMS in the three directions is enhanced from 0.642 m, 0.649 m, and 1.331 m with SF-PPP to 0.303 m, 0.447 m, and 0.761 m with SF-PPP/INS tight integration, with improvements of 52.8%, 31.1%, and 42.8%, respectively. Figure 5 portrays the distributions of position differences of SF-PPP and SF-PPP/INS tight integration in the horizontal direction and vertical direction. Relevant statistics indicate that the horizontal position differences percentages within 0.3 m were 18.22% and 2.05% for SF-PPP/INS and SF-PPP, respectively. The vertical position differences percentages within 0.3 m were 12.90% and 2.08% for SF-PPP/INS tight integration and SF-PPP, respectively. The results also indicate that the percentage of the horizontal position differences from the SF-PPP/INS tight integration within 0.6 m reached 86.11%. However, for SF-PPP only, this percentage was 59.19%. After about 1800 s, the position accuracy of SF-PPP was strongly influenced by the surrounding observation environment. Nevertheless, the position accuracy of SF-PPP/INS tight integration was hardly influenced. Moreover, the SF-PPP/INS tight integration could still provide navigation solutions during GNSS outage periods. Therefore, INS can significantly increase position accuracy and continuity.

**Table 2.** RMS of position differences in the three directions of SF-PPP and SF-PPP/INS tight integration.


**Figure 4.** Position differences provided by G + B SF-PPP (**left**) and G + B SF-PPP/INS tight integration (**right**) in the three directions, in terms of the reference solution provided by RTK/INS tight integration (the start time is 195,453 s).

**Figure 5.** Distribution of position differences of G + B SF-PPP andG+B SF-PPP/INS tight integration in the horizontal direction (**left**) and vertical direction (**right**).

#### *3.3. Evaluation of Real-Time Orbit and Clock Products*

In this section, the accuracy of the real-time orbit and clock products provided by CAS, GFZ, and WHU are evaluated. Firstly, to validate the performance of each IGS center's products, the final orbit and clock products supplied by Wuhan University were adopted as reference values. Figure 6 presents the RMS of real-time orbit products compared to reference values. Relevant mean RMS values are enumerated in Table 3. According to the statistics, the accuracy of GPS real-time orbits provided by the three analysis centers were generally consistent with each other. For the products from WHU, the orbit accuracies of GPS satellites were 1.7 cm, 4.6 cm, and 3.6 cm, with improvements of 81.3%, 65.7%, and 78.7% compared to those of BDS satellites in the three directions. For BDS satellites, the orbit accuracy of WHU was higher than that of CAS and GFZ, with improvements of 32.1%, 51.8%, and 31.9% compared to that of CAS, respectively. Such improvements were up to 78.3%, 71.1%, and 73.1% compared to that of GFZ in radial, along, and cross directions. The orbit accuracies of the BDS MEO satellites were significantly higher than those of the GEO and IGSO satellites. When the GEO and IGSO satellites were detached, the RMSs of MEO-only in the three directions obtained about 45.1%, 31.3%, and 62.1% accuracy upgradation for the products from WHU. According to the work in [35], the accuracy of final precise orbits is 2.5 cm. Therefore, the orbit accuracies of the GPS real-time products were comparable, but those of the BDS MEO orbits were marginally lower.

**Figure 6.** Orbit RMS of real-time products of GPS satellites (**top**) and BDS satellites (**bottom**) from the three IGS analysis centers.

**Table 3.** Mean orbit RMS of GPS and BDS satellites in radial, along, and cross directions of each analysis center.


Figure 7 presents the RMS and STD of the real-time clock offsets from three IGS analysis centers with respect to the reference products. Table 4 illustrates the average values of RMS and STD for GPS and BDS satellite clock offsets. The results shows that the accuracies of the GPS clock offset products supported via the three centers were basically consistent with each other. The accuracies of the products provided by WHU were slightly higher than those for the other analysis centers. For the BDS satellites, the accuracy of WHU was lower compared to the others, especially for the BDS-2 satellites. The maximum clock offset RMS achieved 46.10 ns, which directly led to a larger RMS value, on average, for WHU. For the CAS and GFZ products, the accuracies of the BDS GEO and IGSO satellite clock offsets were significantly lower than those of BDS MEO satellites. When GEO and IGSO satellites were detached, the RMS of MEO-only was improved by 12.6% for the products from CAS, whereas such improvement was 43.1% for the products from GFZ.


**Table 4.** Average values of clock offset RMS and STD of GPS and BDS satellites of each analysis center.

**Figure 7.** Clock offset RMS and STD of real-time products of GPS satellites (**top**) and BDS satellites (**bottom**) from each analysis center.

#### *3.4. Performance of Real-Time PPP/INS Tight Integration*

To analyze the positioning and navigation performance of the presented real-time SF-PPP/INS tight integration, in addition to the real-time orbit/clock products, the real-time ionospheric data provided by Wuhan University (ftp address: igs.gnsswhu.cn), with an interval of 5 min, were also used.

Figure 8 reveals the position differences of real-time G + B PPP/INS tight integration in terms of the reference solutions provided by the smoothed RTK/INS tight integration. The RMS and STD are illustrated in Table 5. According to the results of STD, the realtime SF-PPP/INS tight integration could also provide stable and continuous positioning solutions. The position accuracies based on the real-time products from GFZ were higher in the horizontal and vertical directions compared with the results based on the products of CAS and WHU. The position RMSs of the GFZ-product-based solutions were 0.206 m, 0.542 m, and 0.368 m, which is about 65.5%, 14.9%, and 68.7% more accurate than those based on CAS products in the three directions. Compared to the solutions based on WHU products, the position improvements were more obvious, at about 50.1% and 73.1% in east and vertical directions. This is chiefly owing to the lower accuracies of the clock products from WHU, especially for BDS-2. Moreover, compared with the results based on WHU final products, the accuracies of the results based on GFZ's real-time products still present about 32.0% and 51.6% improvements in the north and vertical directions. This may be on account of a high rate of real-time products (orbit, clock offset, and ionospheric data). Figure 9

illustrates the distributions of position differences forG+B SF-PPP/INS tight integration adopting real-time products from CAS, GFZ, and WHU in horizontal and vertical directions. The statistics indicate that the horizontal position differences percentages within 0.3 m were 0.61%, 57.48%, and 0.15% for the results based on the products of CAS, GFZ, and WHU, respectively. The corresponding percentages of the vertical position differences within 0.3 m were 0.00%, 52.62%, and 3.36%, respectively. Accordingly, the positioning accuracy was principally affected by the real-time product quality of BDS MEO. The analysis of positioning performance and the accuracies of real-time products infer that the positioning accuracy is greatly impacted by the quality of the clock products under the premise of consistent orbit accuracy. When using higher-accuracy clock products, it will be easier to obtain positioning solutions with high accuracy.

**Table 5.** Position RMS and STD in the three directions of SF-PPP/INS tight integration using real-time products from CAS, GFZ, and WHU.


**Figure 8.** Position differences ofG+B SF-PPP/INS tight integration in the three directions using the real-time products from CAS, GFZ, and WHU (the start time is 195,453 s).

**Figure 9.** Distribution of position differences of G + B SF-PPP/INS tight integration in the horizontal direction (**left**) and vertical direction (**right**) using real-time products from CAS, GFZ, and WHU.

Other than the positioning solutions, the G + B SF-PPP/INS tight integration can also support real-time attitudes with regard to roll, pitch, and heading. The attitude offsets via comparing the attitude solutions with the reference values are portrayed in Figure 10. In the light of the RMSs enumerated in Table 6, attitude accuracy also illustrates a certain relationship to the accuracy of orbit/clock products. The attitude RMSs based on WHU's real-time products were 0.020◦, 0.019◦, and 0.523◦ in roll, pitch, and heading, with improvements of 37.5%, 40.6%, and 46.4% compared to the solutions based on CAS's products. Similarly, there were about 25.9% and 29.6% enhancements in roll and pitch compared to the results supplied by GFZ's products. Figure 11 presents the distributions of attitude offsets of the G + B SF-PPP/INS tight integration using the realtime products from CAS, GFZ, and WHU in roll, pitch, and heading. Relevant statistics indicate that the percentages of roll attitude offsets within 0.03◦ were 65.98%, 69.42%, and 91.20% for the results based on the products of CAS, GFZ, and WHU, respectively. The percentages of pitch attitude offsets within 0.03◦ were 62.23%, 66.93%, and 92.34%, respectively. In general, the offset distribution in roll and pitch was consistent for solutions based on CAS and GFZ. Additionally the percentages of heading attitude offsets within 0.5◦ were 32.99%, 66.32%, and 83.16%, respectively.

**Figure 10.** Attitude offsets ofG+B SF-PPP/INS tight integration in roll, pitch, and heading directions using the real-time products from CAS, GFZ, and WHU, in terms of reference solutions (the start time is 195,453 s).

**Figure 11.** Distribution of the offsets in roll (**top**), pitch (**middle**), and heading (**bottom**) of the G + B SF-PPP/INS tight integration using the real-time products from CAS, GFZ, and WHU.


**Table 6.** RMS of attitude and velocity offsets of SF-PPP/INS tight integration using the real-time products from CAS, GFZ, and WHU.

Theoretically, in GNSS/INS tight integration, attitudes are discernible when the vehicle makes maneuvers. In this condition, the accuracy of attitude estimation will be marginally affected by the positioning information change. In addition, the estimation accuracy of other parameters in the Kalman filtering could also slightly influence the estimation accuracy of attitude [21]. Therefore, the GNSS/INS tight integration based on different real-time products will make the attitude solutions a little different. Figure 12 illustrates the velocity differences ofG+B SF-PPP/INS tight integration. The distributions of velocity offsets of the G + B SF-PPP/INS tight integration using real-time products from CAS, GFZ, and WHU in horizontal and vertical directions are presented in Figure 13. The statistics indicate that percentages of the horizontal velocity offsets within 0.03 m/s were 61.23%, 62.04%, and 62.11% for the results based on the products of CAS, GFZ, and WHU, respectively. The corresponding percentages of the vertical velocity differences within 0.03 m/s were 98.09%, 99.89%, and 99.85%, respectively. In general, the distribution of velocity offsets based on the real-time products from three centers were consistent in horizontal and vertical directions. In terms of the statistics in Table 6, the velocity offset RMS was almost equivalent when adopting the real-time products from different centers (CAS, GFZ, and WHU). This is because (1) the velocity accuracy of GNSS/INS tight integration chiefly relies on the quality of doppler observations and the performance of IMU sensors, and (2) it has a weak relationship with positioning accuracy. Plotted in Figure 14 are the estimated biases and scale factors of the accelerometer and gyroscope of POS320 in the body frame. Additionally, the estimations present a visible relationship with the quality of real-time satellite orbit/clock products.

**Figure 12.** Velocity offsets of G + B SF-PPP/INS tight integration in the three directions using the real-time products from CAS, GFZ, and WHU, in terms of reference solutions (the start time is 195,453 s).

**Figure 13.** Distribution of velocity offsets in horizontal (**left**) and vertical (**right**) directions ofG+B SF-PPP/INS tight integration using the real-time products from CAS, GFZ, and WHU.

**Figure 14.** Accelerometer (**left**) and gyroscope (**right**) biases and scale factors in the body frame (Forward–Right–Down) by utilizing real-time products from CAS, GFZ, and WHU (the start time is 195,453 s).

#### **4. Conclusions**

In this contribution, a real-time GPS/BDS-2/BDS-3 SF-PPP/INS tight integration model is introduced. To assess its performance, we firstly dissect the positioning performance of SF-PPP and SF-PPP/INS tight integration. Then, the accuracy of real-time products afforded by CAS, GFZ, and WHU are evaluated. To present the performance using orbit and clock products from different analysis centers, a set of vehicle-borne data was processed using the same strategy. According to the results, it can be concluded that: (1) GPS + BDS SF-PPP/INS tight integration can obtain more accurate and continuous positioning solutions, especially during GNSS outage, compared to the solutions of the GPS + BDS SF-PPP. (2) The accuracy of GPS orbit and clock products provided by the three analysis centers are consistent with each other. For BDS satellites, the accuracy of orbits provided by WHU is higher, but the accuracy of the clocks is lower than the others, especially for BDS-2 satellites. The accuracy of BDS MEO satellite products is significantly superior to that of GEO and IGSO satellites. (3) The positioning RMS values based on GFZ's real-time products are better than those based on the products of CAS and WHU. (4) Owing to the

high rate of real-time products, the positioning accuracies in north and vertical directions based on GFZ real-time products are even higher compared to the solutions based on the IGS final products. In addition, the accuracy of attitude and the convergence of IMU sensor errors also present a visible relationship with the real-time orbit/clock products' accuracy.

With the development of multi-sensor fusion, other sensors such as LiDAR, camera, odometer, and Ultra Wide Band (UWB) can be utilized for positioning and navigation in a GNSS challenging environment.

**Author Contributions:** Conceptualization, J.L. and Z.G.; data curation, J.L., Q.X. and R.L.; funding acquisition, C.Y. and J.P.; investigation, J.L. and Z.G.; software, Z.G.; visualization, C.Y. and J.P.; writing—original draft preparation, J.L.; writing—review and editing, J.L. and Z.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partly supported by the National Key Research and Development Program of China (Grant No. 2020YFB0505802) and the National Natural Science Foundation of China (NSFC) (Grants No. 42074004).

**Data Availability Statement:** The datasets adopted in this paper are managed by the School of Land Science and Technology, China University of Geosciences, Beijing, and are available on request from the corresponding author.

**Acknowledgments:** The authors would like to thank anonymous reviewers who provided valuable suggestions that have helped to improve the quality of the manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.
