# BDS Orbit Maneuver Detection Based on Epoch-Updated Orbits Estimated by SRIF
## 2 Mathematical Model
### Epoch-Updated Orbit Estimation  
The square root information filter is a parameter estimation method that is based on the standardization and Householder matrix transformations [16; 17; 18]. It provides an efficient and computationally stable approach to estimate the parameters. Unlike traditional methods, the SRIF operates directly on the square root of the information matrix, which avoids the need for matrix inversions and reduces computational complexity. By utilizing the square root factorization, the SRIF offers improved numerical stability and accuracy, making it well suited for applications in sensor fusion, navigation, and control systems. A brief explanation of the SRIF principle is provided in this section. \[\mathbf{Z}=\mathbf{Ax+V} \tag{1}\]  
where \(\mathbf{x}\) is the estimated parameters; \(\mathbf{A}\) is the design matrix; \(\mathbf{V}\) is the observation error; and \(\mathbf{Z}\) is the vector including observations and initial values. It is noted that Equation (1) contains the observation equation and the state constraint equation of the estimated parameters. Additionally, it needs to be normalized \((\mathbf{V}\sim N(\mathbf{0},\mathbf{I}))\). To minimize the mean square observation,  
\[\mathbf{J}(x)=\mathbf{V}\mathbf{V}^{T}=\left\|\mathbf{Z}-\mathbf{Ax}\right\|^{2}=min \tag{2}\]  
By applying the Householder orthogonal transformation and the minimum variance criterion for observation error [19], the latter can be derived as  
\[\mathbf{TA}= \begin{pmatrix}\mathbf{R}\\ \mathbf{0}\end{pmatrix} \quad\mathbf{TZ}= \begin{pmatrix}\mathbf{Z_{1}}\\ \mathbf{Z_{2}}\end{pmatrix}  \tag{3}\]  
where \(\mathbf{T}\) is the orthogonal transformation matrix; \(\mathbf{R}\) is the information matrix, which is also an upper triangular matrix; and \(\mathbf{Z_{2}}\) is the vector of the observation residual, respectively. Combining with Equation (2), it is obvious that \(\mathbf{x}\) can be derived as  
\[\mathbf{x}=\mathbf{R}^{-1}\mathbf{Z_{1}} \tag{4}\]  
SRIF has been successfully applied to precise orbit determination and clock estimation in GNSS satellites [20; 21; 22; 23], which includes orbit integration, parameter state updates,observation updates, quality control, and parameter solutions. The Runge-Kutta integration method was employed in this study [24; 25], and the detailed method of quality can be found in reference [20]. Due to the unknown duration of orbital maneuvers, weakly constrained parameter states could be employed at all times to absorb system errors caused by orbit maneuver thrust. The weakly constrained state solution estimates the orbit that is not highly precise, but the solution is sufficient to fulfill the requirements for maneuver detection. The state constraint equation for the position and velocity of the satellite can be shown,  
\[\mathbf{X}_{i}^{t_{s}}+\mathbf{x}_{i}^{t_{s}}=\mathbf{X}_{i-1}^{t_{e}}+\mathbf{\Phi}\mathbf{x}_{i-1 }^{t_{e}}+\mathbf{w} \tag{5}\]  
where \(\mathbf{X}\) denotes the initial position and velocity of satellites in the celestial reference system (CRS), with the subscript denoting the epoch symbol and the superscript denoting the corresponding time of the estimated parameters. The time of the estimated parameters at the current epoch is represented by \(t_{s}\), while the corresponding time of the next epoch is represented by \(t_{e}\), for example (\(t_{e,i}=t_{s,i+1}\)); \(\mathbf{x}\) denotes the correction values of the parameters to be estimated; \(\mathbf{\Phi}\) is the state transfer matrix; and \(\mathbf{w}\) is the process noise.