# Robust Kalman Filter Aided GEO/IGSO/GPS Raw-PPP/INS Tight Integration
## 2 Methods
### Geo/Igso/GPS Raw PPP Model  
In contrast to the conventional PPP model [1], the raw PPP mode has been proven to perform better in accelerating convergence speed and decreasing solution noise [32]. The basic observation models for un-differenced un-combined phase, code, and Doppler observations are  
\[P_{r,j}^{s}=\rho_{r}^{s}+c(t_{r}-t^{s})+c(d_{r,j}-d_{j}^{s})+T_{r}^{s}+I_{r,j}^ {s}+\varepsilon_{p,j} \tag{1}\]  
\[L_{r,j}^{s}=\rho_{r}^{s}+c(t_{r}-t^{s})+c(b_{r,j}-b_{j}^{s})+T_{r}^{s}-I_{r,j}^ {s}+\lambda_{j}N_{r,j}^{s}+\varepsilon_{L,j} \tag{2}\]  
\[D_{r,j}^{s}=(\dot{\rho}_{r}^{s}+c(\dot{t}_{r}-\dot{t}^{s})+c(\dot{d}_{r,j}- \dot{d}_{j}^{s})+\dot{T}_{r}^{s}+\dot{I}_{r,j}^{s}+\varepsilon_{D,j})/\lambda_ {j} \tag{3}\]  
where the indices \(r\), \(s\), \(j\) refer to receiver, GNSS satellite (GPS and BDS), and signal frequency (GPS: L1 and L2, BDS: B1 and B2), respectively; \(c\) is the light speed in vacuum; \(P\), \(L\), and \(D\) denote code, phase, and Doppler observations, respectively; \(\rho_{r}^{s}\) is the geometry distance between the antenna phase center of receiver and satellite; \(t_{r}\) and \(t^{s}\) are clock offsets of receiver and satellite; \(d\) and \(b\) denote the hardware delays on code and phase, where \(b\) will be absorbed in ambiguity in float PPP [7]; \(T\) and \(I\) indicate the tropospheric delay and ionospheric delay; \(\lambda\) and \(N\) are the wavelength and ambiguity; \(\varepsilon\) is the sum of observing noises and un-modelled errors; and \(\dot{(})\) denotes the variations of variables (e.g., geometry distance, clock offset of receiver and satellite, troposphere delay, and ionosphere delay). The troposphere delay, ionosphere delay, and the geometric distance \(\rho\) and its variation can be expressed, respectively, as  
\[T_{r}^{s}=m_{r,dry}^{s}z_{dry}+m_{r,wet}^{s}z_{wet} \tag{4}\]  
\[I_{r}^{s}=40.28\cdot VTEC/(f_{j}^{2}\cdot\cos\vartheta) \tag{5}\]  
Figure 2: Algorithm structure of R-EKF aided GPS/BDS PPP/INS tight integration. R-EKF is robust extended Kalman filter, PPP is precise point positioning, INS is inertial navigation system, IMU is inertial measurement unit, and PCO/PCV is phase center offset/variation. \[\rho_{r}^{s}=\|\mathbf{p}_{r,G/B}-\mathbf{p}^{s}\|\mathbf{\xi}\dot{\rho}_{r}^{s}=\|\mathbf{v}_{r,G/ B}-\mathbf{v}^{s}\| \tag{6}\]  
where \(m\) and \(z\) denote the troposphere mapping function (e.g., global mapping function [34]) and the zenith delay (e.g., Saastamoinen model [35]) of the dry (_dry_) and wet (_wet_) components of troposphere delay; _VTEC_, \(\theta\), and \(f\) are the vertical total electronic content, zenith angle, and phase frequency, respectively; \(\mathbf{p}_{r,G/B}\) and \(\mathbf{p}^{s}\) are the positions at antenna phase center of receiver and satellite; \(\mathbf{v}_{r,G/B}\) and \(\mathbf{v}^{s}\) are the corresponding velocities, and \(\|\ \|\) indicates the modular arithmetic. For raw PPP, the error terms are either corrected by using empirical models or are estimated as parameters, which is different from the processing method in the conventional PPP mode. Whereas, the earth rotation, relativity, antenna PCO/PCV, wind-up, solid tide, polar migration, dry troposphere delay, and code hardware delays on the satellite side are corrected by classic models and International GNSS Service (IGS) products [33], the receiver clock, code hardware delay of both GPS and BDS on the receiver side, ionosphere delays on frequencies L1 and B1, and ambiguity of each phase observation are estimated as parameters [32]. For the GPS + BDS case, there is inter-system bias (ISB) between GPS and BDS. Therefore, ISB is also estimated as a parameter. Thus, the estimated parameter vector is  
\[\mathbf{x}=\left[\mathbf{p}_{r,G/B},\mathbf{v}_{r,G/B},\mathbf{t}_{r},\dot{\mathbf{t}}_{r},z_{wet},b_{B-G},\mathbf{d}_{r,j},\mathbf{F}_{r,j},\mathbf{N}_{r,j}^{s}\right]^{T} \tag{7}\]  
where \(b_{B-G}\) is the ISB of BDS with respect to GPS. Note the fact that there is a strong correlation between the ionosphere delay and code hardware delay.