# Research on the SSUKF Integrated Navigation Algorithm Based on Adaptive Factors
## 3 INS/GNSS Integrated Navigation Algorithm Design
### ASSUKF Algorithm Design  
On this basis, the algorithm further introduces adaptive filtering operations to improve the accuracy and robustness of the filter. In Kalman Filtering, the measurement noise, \(\mathbf{R}_{k}\), plays a crucial role. Specifically, \(\mathbf{R}_{k}\) is the covariance matrix in the observation equation, which describes the magnitude and correlation of the measurement errors. It determines how the filter balances the relationship between the previous state estimate and the new measurement data. Therefore, accurately estimating the value of \(\mathbf{R}_{k}\) is essential for improving the filter's accuracy and stability.  
In Equation (15), the variation in \(\mathbf{R}_{k}\) is unknown, and a fixed value is often used, which can lead to significant errors. To overcome this limitation, an innovation vector is introduced, and the calculation formula is as follows [18]:  
\[\widetilde{\mathbf{Z}}_{\text{k/k-1}}=\mathbf{Z}_{\text{k}}-\mathbf{H}_{\text {k}}\widehat{\mathbf{X}}_{\text{k/k-1}}=\mathbf{H}_{\text{k}}\widetilde{ \mathbf{X}}_{\text{k/k-1}}+\mathbf{V}_{\text{k}} \tag{20}\]  
The covariance matrix of the innovation vector is as follows:  
\[\mathbf{S}_{\text{k}}=\text{E}\bigg{[}\widetilde{\mathbf{Z}}_{\text{k/k-1}} \widetilde{\mathbf{Z}}_{\text{k/k-1}}^{\text{T}}\bigg{]}=\mathbf{H}_{\text{k} }\mathbf{P}_{\text{k/k-1}}\mathbf{H}_{\text{k}}^{\text{T}}+\mathbf{R}_{\text {k}} \tag{21}\]  
By performing statistical analysis of the innovation vector, we can effectively estimate the actual measurement noise covariance. Next, we adjust the value of the measurement noise by introducing a matrix \(\mathbf{A}\). The matrix \(\mathbf{A}\) is derived from the innovation vector and the state covariance matrix, \(\mathbf{P}_{\text{XZ}}\), and is calculated as follows:  
\[\mathbf{A}=\widetilde{\mathbf{Z}}_{\text{k/k-1}}\widetilde{\mathbf{Z}}_{\text{ k/k-1}}^{\text{T}}-\mathbf{H}_{\text{k}}\mathbf{P}_{\text{XZ}} \tag{22}\]  
Then, we update the measurement noise covariance by detecting the diagonal elements of the matrix \(\mathbf{A}\). The update formula for the measurement noise covariance \(\mathbf{R}_{\text{k}}\) is given by [26]  
\[\left\{\begin{aligned} \mathbf{R}_{\text{k}}(\text{ii})=\text{min} \,\mathbf{A}_{\text{ii}}<\text{min}\\ \mathbf{R}_{\text{k}}(\text{ii})=(1-\beta)\mathbf{R}_{\text{k}}( \text{ii})+\beta\mathbf{A}_{\text{ii}}\text{min}<\mathbf{A}_{\text{ii}}<\text {max}\\ \mathbf{R}_{\text{k}}(\text{ii})=\text{max}\mathbf{A}_{\text{ii}}> \text{max}\end{aligned}\right. \tag{23}\]  
where \(\mathbf{R}_{\text{k}}(\text{ii})\) represents the diagonal elements of the measurement noise covariance, \(\mathbf{A}_{\text{ii}}\) is the diagonal element of the innovation vector's covariance matrix, and \(\beta\) is the adaptive factor that determines the rate of noise update. The min is taken as 0.01 times \(\mathbf{R}_{\text{k}}\) and the max is 100 times \(\mathbf{R}_{\text{k}}\). To prevent overly fast updates, the update rule for \(\beta\) is as follows:  
\[\beta=\frac{\beta}{\beta+\text{b}} \tag{24}\]  
where \(b\) is a constant used to control the rate of change in the adaptive factor, and in this study, \(b=0.5\). By iterating Equations (23) and (24) in the filter, the measurement noise covariance matrix, \(\mathbf{R}_{k}\), obtained from Equation (23) is substituted into Equation (15) to replace the originally fixed measurement noise covariance matrix.