# An Algorithm to Assist the Robust Filter for Tightly Coupled RTK/INS Navigation System
## 2 Materials and Methods
### The Update Step of the TC RTK/INS Integration
#### 2.3.3 The IGG-III Model  
The GNSS measurements are susceptible to multipath errors in urban areas, especially for the pseudo range. In the TC RTK/INS integration, the gross errors in the pseudo range can bias the float ambiguity estimation, which will degrade the performance of AR. Considering the resistance to the measurement outliers, we use the IGG-III model to inflate the components in the measurement noise covariance matrix [39; 77; 53]. The IGG-III model is based on the innovation test. The innovation vector \(\mathbf{s}\) in EKF can be defined as Equation (19) shows:  
\[\mathbf{s}_{k}=\mathbf{y}_{k}-\mathbf{h}(\mathbf{\hat{x}}_{k}^{-}), \tag{19}\]  
where \(\mathbf{y}_{k}\) denotes the measurement vector; \(\mathbf{\hat{x}}_{k}^{-}\) refers to the prior estimation at the \(k\)th epoch; \(\mathbf{h}(\cdot)\) represents the measurement function. In the absence of outliers, the innovation vector should be normally distributed with a mean of zero.The IGG-III model uses the innovation vector and the corresponding covariance to calculate the normalized innovation and determine the inflation factor. The \(\
u\)th component of the normalized innovation vector at the \(k\)th epoch, \(\widetilde{s}_{k,\
u}\), can be calculated as Equation (20) shows:  
\[\widetilde{s}_{k,\
u}=\frac{s_{k,\
u}}{\sqrt{(\mathbf{H}_{k}\hat{\mathbf{P}}_{k}^{-} \mathbf{H}_{k}^{\text{T}}+\mathbf{R}_{k})}\
u}, \tag{20}\]  
where \(s_{k,\
u}\) denotes the \(\
u\)th component of the innovation vector at the \(k\)th epoch; the subscript \({}^{\
u}\
u,\
u^{\prime\prime}\) refers to the component at the \(\
u\)th row and the \(\
u\)th column. The inflation factor can be formulated as Equation (21) shows:  
\[\eta_{\
u\
u}=\left\{ \begin{array}{l}1,|\widetilde{s}_{k,\
u}|\leq\xi_{ \text{ICGC},0}\\ \frac{\widetilde{s}_{k,\
u}}{\xi_{\text{ICGC},0}}(\frac{\xi_{\text{ICGC},1}- \widetilde{s}_{\text{ICGC},0}}{\xi_{\text{ICGC},1}-\widetilde{s}_{\text{ICIC},1}})^{2},\xi_{\text{ICGC},0}<|\widetilde{s}_{k,\
u}|<\xi_{\text{ICGC},1}\\ \infty,|\widetilde{s}_{k,\
u}|\geq\xi_{\text{ICGC},1}\end{array} \right., \tag{21}\]  
where \(\xi_{\text{ICGC},0}\) and \(\xi_{\text{ICGC},1}\) are empirical thresholds. In Equation (21), the first case means the corresponding measurement is normal. The second case means the weight of the corresponding measurement should be shrunk. The third case means a significant gross error has been added to the measurement, and the measurement should be discarded. As we mentioned in Section 2.3.2, the DD measurements are correlated. Considering the correlation, we can calculate the inflation factor related to the non-diagonal components of \(\mathbf{R}\), as Equation (22) shows:  
\[\eta_{\
u\mu}=\sqrt{\eta_{\
u\
u}\eta_{\mu\mu}}. \tag{22}\]  
Finally, we multiply the inflation factor with the corresponding components in \(\mathbf{R}\) to resist the outliers. Specifically, we assume the dimension of \(\mathbf{s}\) is \(w\). The corresponding \(\mathbf{R}\) is a matrix with \(w\) rows and \(w\) columns. We use \(\sigma\) with different subscripts to denote the elements in \(\mathbf{R}\). This is to say, \(\sigma_{\
u\mu}\) refers to the component at the \(\
u\)th row and the \(\mu\)th column. In this case, \(\mathbf{R}\) can be formulated as Equation (23) shows:\[R= \begin{bmatrix}\sigma_{11}&\dots&\sigma_{1v}&\dots&\sigma_{1w}\\ \vdots&\vdots&\vdots&\vdots&\vdots&\\ \sigma_{v1}&\dots&\sigma_{vv}&\dots&\sigma_{vw}\\ \vdots&\vdots&\vdots&\vdots&\\ \sigma_{w1}&\dots&\sigma_{uv}&\dots&\sigma_{ww}\\ \end{bmatrix} . \tag{23}\]  
The inflation factor will be combined with the corresponding components in \(R\) to obtain \(\overline{R}\) as Equation (24) shows:  
\[\overline{R}= \begin{bmatrix}\eta_{11}\sigma_{11}&\dots&\eta_{1v}\sigma_{1v}& \dots&\eta_{1w}\sigma_{1w}\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \eta_{v1}\sigma_{v1}&\dots&\eta_{vv}\sigma_{vv}&\dots&\eta_{vw}\sigma_{vw}\\ \vdots&\vdots&\vdots&\vdots&\\ \eta_{w1}\sigma_{w1}&\dots&\eta_{vv}\sigma_{uv}&\dots&\eta_{ww}\sigma_{ww}. \end{bmatrix} .