# Robust Kalman Filter Aided GEO/IGSO/GPS Raw-PPP/INS Tight Integration
## 2 Methods
### Robust Extended Kalman Filter Based PPP/INS Tight Integration  
As mentioned above, the positions provided by INS are related to the initial status. Therefore, in order to get high-accuracy position solutions, a high-accuracy initial position is needed. Here, the positions calculated by PPP in Equation (12) are utilized for position/velocity initializationand the static IMU data are adopted for attitude initialization [20]. As depicted in Figure 2, time synchronization between GPS/BDS observations and INS solutions is applied after the INS update. When both GPS/BDS and INS data are available, the tight integration function works. To compensate for the lever-arm (system offset) between INS and GPS/BDS, the INS position and velocity updates are transformed to the GPS/BDS receiver antenna phase center [20] by using  
\[\mathbf{p}_{r,G/B}^{n}=\mathbf{p}_{r,INS}^{n}+\Re\mathbf{C}_{b}^{n}\mathbf{l}^{b} \tag{21}\]  
\[\mathbf{v}_{r,G/B}=\mathbf{v}_{r,INS}-(\mathbf{\omega}_{ie}^{n}\times+\mathbf{\omega}_{en}^{n} \times)\mathbf{C}_{b}^{n}\mathbf{l}^{b}-\mathbf{C}_{b}^{n}\left(\mathbf{l}^{b}\times\right)\mathbf{ \omega}_{ib}^{b} \tag{22}\]  
where \(\mathbf{l}^{b}\) is the lever-arm of the GPS/BDS receiver antenna phase center with respect to the IMU center, which is measured accurately before data collection; \(\Re=\text{diag}\Big{(}\frac{1}{R_{M}+H},\frac{1}{(R_{N}+H)cosB},-1\Big{)}\) denotes the matrix to transform the lever-arm from n-frame to \(e\)-frame. The transformed position and velocity can be used together with the precise orbit/clock products to predict code (\(\mathbf{P}_{\text{INS}}\)), phase (\(\mathbf{L}_{\text{INS}}\)), and Doppler (\(\mathbf{D}_{\text{INS}}\)). Then, these predicted values and the observed data can be used to form the innovation vector of EKF for PPP/INS tight integration as  
\[\mathbf{Z}_{k}=\left( \begin{array}{c}\mathbf{P}_{j}-\mathbf{P}_{INS,j}\\ \mathbf{L}_{j}-\mathbf{L}_{INS,j}\\ \mathbf{D}_{j}-\mathbf{D}_{INS,j}\\ \mathbf{I}_{1}-\mathbf{I}_{prior,1}\end{array} \right),\;\; \begin{array}{c}GPS:j=L1 \&L2\\ BDS:j=B1\&B2\end{array} . \tag{23}\]  
The corresponding state parameter vector, which contains both PPP relative parameters and INS relative parameters, can be written as  
\[\mathbf{x}=\Big{[}\mathbf{p}_{r,G/B},\mathbf{v}_{r,G/B},\mathbf{\theta},\mathbf{B}_{\mathbf{f}},\mathbf{S} _{\mathbf{f}},\mathbf{B}_{\mathbf{\omega}},\mathbf{S}_{\mathbf{\omega}},t_{r},\dot{t}_{r},z_{wet}, b_{B-G},\mathbf{DC}\mathbf{B}_{r,G},\mathbf{DC}\mathbf{B}_{r,B},\mathbf{I}_{r,1}^{s},\mathbf{N}_{r,1}^{s}, \mathbf{N}_{r,2}^{s}\Big{]}^{T} \tag{24}\]  
where the symbols are the same as those mentioned above. The design matrix can be obtained by making derivation on Equation (23) with respect to Equation (24) (i.e., \(\partial\mathbf{Z}_{k}/\partial\mathbf{x}\)). The state transform matrix \(\mathbf{\phi}\) for PPP/INS tight integration is different from that used in Equation (10). In general, the PSI-angle model [25], instead of the constant velocity model and white noise model, is adopted to describe the behavior of position, velocity, and attitude; the first-order Gauss-Markov procedure [25] is used to indicate variations of IMU biases and scale factors. The other state functions remain the same as those in PPP. Then, by applying EKF [31; 38] as described in Equations (10) and (11), the integration solutions can be obtained. However, as the qualities of BDS GEO and IGSO are different, it is necessary to increase the benefit from IGSO and decrease the limitation of GEO. In this paper, the R-EKF [31] is adopted to detect the low quality observations and reduce the weight (e.g., by enlarging the variance). Thus, the equivalent weight function [39] instead of the empirical weight function [6] is used by  
\[\widetilde{\mathbf{R}}_{k}=\frac{1}{\mathbf{\alpha}}\mathbf{R}_{k}=\mathbf{\alpha}\mathbf{P}_{k}^{ -1} \tag{25}\]  
where \(\widetilde{\mathbf{R}}_{k}\) and \(\mathbf{\alpha}\) refer to the equivalent variance matrix and the robust-factor matrix.