## 4 RMNCE-Based Adaptive UKF
### Expanded \(\mathbf{R}\) Design  
\(\mathbf{\hat{R}}_{k}^{RMANCE}\) is a statistic within a sliding window and therefore cannot keep up with the change of \(\mathbf{R}\) in real time, which will lead to improper estimate especially in case of multipath, disturbance and other sudden measurement noise changing circumstances. To cope with this indeterminate \(\mathbf{R}\), a performance based \(\mathbf{R}\) expanding algorithm is designed as below. Assuming that the expanded \(\mathbf{\overline{R}}_{k}\) can be defined by:  
\[\mathbf{\overline{R}}_{k}=\beta\mathbf{\hat{R}}_{k}^{RMANCE} \tag{25}\]  
where \(\beta\) is the expanding scale and can be expressed as \(diag\{\beta_{1}\cdots\beta_{2S_{N}}\}\). Then the estimated state vector \(\mathbf{\hat{X}}_{k}\) in the UKF can be expressed as:  
\[\left\{ \begin{array}{l}\mathbf{\hat{X}}_{k}=\mathbf{\hat{X}}_{k}^{-}+\mathbf{ K}_{k}\varepsilon_{k}\\ \mathbf{K}_{k}=\mathbf{P}_{xz,k}\big{(}\mathbf{P}_{z^{-}k}\ +\ \beta\mathbf{\hat{R}}_{k}^{RMANCE} \big{)}^{-1}\end{array} \right. \tag{26}\]  
where \(\mathbf{\hat{X}}_{k}^{-}\) is the state prediction, \(\mathbf{P}_{xz,k}\) is the covariance between \(\mathbf{\hat{X}}_{k}^{-}\) and the measurement prediction, and \(\mathbf{P}_{z^{-}k}\) is the covariance of the measurement prediction error. To control the influence of the undetermined measurement noise, we assume that \(\mathbf{K}_{k}\varepsilon_{k}\) should lie within a reasonable range:  
\[\mathbf{P}_{xz,k}\big{(}\mathbf{P}_{z^{-}k}+\beta\mathbf{\hat{R}}_{k}^{RMANCE} \big{)}^{-1}\varepsilon_{k}\leq\mathbf{C} \tag{27}\]  
where \(\mathbf{C}\) denotes the maximum state transition error and processing noise in one step. For an INS/GNSS integrated navigation system, \(\mathbf{C}\) is mainly decided by the performance of IMU. In order to avoid the computational complexity, sequential processing [41] is employed to estimate the boundary value of \(\beta\). In the \(l\)-th step of the sequential UKF at the \(k\)-th epoch, the corresponding expanding scale \(\beta_{l}\) can be calculated as:  
\[\left\{ \begin{array}{l}\beta_{l}=\bigg{[}\frac{\mathbf{P}_{xz,k}^{(l)}\cdot \mathbf{P}_{xz,k}^{(l)}}{\mathbf{P}_{xz,k}^{(l)}\cdot\mathbf{T}^{c}\mathbf{ \hat{c}}}\varepsilon_{k}^{(l)}\ -\ P_{z^{-}k}^{(l)}\bigg{]}\frac{1}{\hat{R}_{k}^{RMANCE,l} \big{)}}\\ \mathbf{P}_{xz,k}^{(l)}=\sum\limits_{i=0}^{2n}w_{(i)}^{(c)}\Big{(}\boldsymbol{ \chi}_{k,(i)}^{(l)}-\mathbf{\hat{X}}_{k}^{(l)}\Big{)}\bigg{\{}h^{(l)}\Big{[} \boldsymbol{\chi}_{k,(i)}^{(l)}\Big{]}\ -\ \sum\limits_{i=0}^{2n}\omega_{(i)}^{(\text{m})}h^{(l)}\Big{[} \boldsymbol{\chi}_{k,(i)}^{(l)}\Big{]}\bigg{\}}^{T}\\ P_{z^{-}k}^{(l)}=\sum\limits_{i=0}^{2n}\omega_{(i)}^{(c)}\bigg{\{}h^{(l)}\Big{[} \boldsymbol{\chi}_{k,(i)}^{(l)}\Big{]}\ -\ \sum\limits_{i=0}^{2n}\omega_{(i)}^{(\text{m})}h^{(l)}\Big{[} \boldsymbol{\chi}_{k,(i)}^{(l)}\Big{]}\bigg{\}}\bigg{\{}h^{(l)}\Big{[} \boldsymbol{\chi}_{k,(i)}^{(l)}\Big{]}\ -\ \sum\limits_{i=0}^{2n}\omega_{i}^{(\text{m})}h^{(l)}\Big{[} \boldsymbol{\chi}_{k,(i)}^{(l)}\Big{]}\bigg{\}}^{T}\end{array} \right. \tag{28}\]  
where \(\varepsilon_{k}^{(l)}\) is the innovation of the \(l\)-th observation, \(\Big{\{}\omega_{(i)}^{(\text{m})}\Big{\}}\) and \(\Big{\{}\omega_{(i)}^{(\text{c})}\Big{\}}\) are UT weights [30], \(\Big{\{}\boldsymbol{\chi}_{k,(i)}^{(l)}\Big{\}}\) denotes the sigma points, \(\mathbf{\hat{X}}_{k}^{-(l)}\) denotes the prediction in the \(l\)-th step, \(h^{(l)}[\cdot]\) is the \(l\)-th observation equation, \(\mathbf{P}_{xz,k}^{(l)}\) is a vector and \(P_{z^{-}k}^{(l)}\) is a scalar quantity. According to the performance vector \(\mathbf{C}\) and innovation, the expanding scale of every measurement can be determined but still not optimal due to the insufficient samples.