## 3 Proposed Tracking Algorithm
### Improved Tracking Algorithm  
It is considered that the covariance matrix \(\mathbf{P}_{k,k-1}\) is a symmetric matrix of the \(n\)th order. However, after enough large number of times of calculation, it no longer has non-negative and positive character. Suppose the rank is \(e\), \(0<e<1\), \(\mathbf{S}\) is a diagonal matrix, and they satisfy the semi-positive definite condition, whose eigenvalue is set as \(\sigma_{1}^{2},\ \ \sigma_{2}^{2},\ \ \cdots,\ \ \sigma_{n}^{2}\), and it holds that \(\sigma_{1}^{2}\geq\sigma_{2}^{2}\geq\cdots\geq\sigma_{n}^{2},\sigma_{r+1}^{2} =\cdots=\sigma_{n}^{2}=0\). Then, there must be an orthogonal matrix \(\mathbf{V}\) of order \(n\), making \(\mathbf{P}^{T}\mathbf{P}\) similar to a diagonal matrix, which can be expressed as follows [34]:  
\[\mathbf{V}^{T}(\mathbf{P}_{k,k-1}{}^{T}\mathbf{P}_{k,k-1})\mathbf{V}=diag(\ \sigma_{1}^{2},\ \ \sigma_{2}^{2},\ \ \cdots,\ \ \sigma_{r}^{2},\ \ 0,\ \ \cdots,\ \ 0\ )=\mathbf{S}_{n}^{2}=\left[ \begin{array}{cc}\mathbf{S}_{r}^{2}&0\\ 0&0\end{array} \right] \tag{25}\]  
where \(\mathbf{V}=\left[ \begin{array}{cc}\mathbf{V}_{1}&\mathbf{V}_{2}\end{array} \right]=[\ \ v_{1},\ \ \ v_{2},\ \ \cdots,\ \ v_{n}\ \ ]\), and \(\mathbf{P}_{k,k-1}\mathbf{V}_{2}=0\). Set \(\mathbf{U}_{1}=\mathbf{P}_{k,k-1}\mathbf{V}_{1}\mathbf{S}_{r}^{-1}\) to obtain \(\mathbf{U}_{1}^{T}\mathbf{U}_{1}=\mathbf{I}\), so the -column vectors of \(\mathbf{U}_{1}\) are unit orthonormal vectors, which can be extended to an orthonormal basis. There exists \(\mathbf{U}_{2}\) such that \(\mathbf{U}=\left[ \begin{array}{cc}\mathbf{U}_{1}&\mathbf{U}_{2}\end{array} \right]\) is an orthonormal matrix of order \(n\), and:  
\[\mathbf{U}^{T}\mathbf{P}_{k,k-1}\mathbf{V}=\left[ \begin{array}{cc}\mathbf{U}_{1}^{T}\mathbf{P}_ {k,k-1}\mathbf{V}_{1}&\mathbf{U}_{1}^{T}\mathbf{P}_{k,k-1}\mathbf{V}_{2}\\ \mathbf{U}_{2}^{T}\mathbf{P}_{k,k-1}\mathbf{V}_{1}&\mathbf{U}_{1}^{T}\mathbf{P}_{k,k-1}\mathbf{V}_{2} \end{array} \right] \tag{26}\]  
From \(\mathbf{U}_{1}=\mathbf{P}_{k,k-1}\mathbf{V}_{1}\mathbf{S}_{r}^{-1}\), we get \(\mathbf{P}_{k,k-1}\mathbf{V}_{1}=\mathbf{U}_{1}\mathbf{S}_{r}\), which yields to:  
\[\mathbf{U}_{1}^{T}\mathbf{P}_{k,k-1}\mathbf{V}_{1}=\mathbf{S}_{r}^{-1}\mathbf{V}_{1}^{T}\mathbf{P}_{k,k -1}\mathbf{}^{T}\mathbf{P}_{k,k-1}\mathbf{V}_{1}=\mathbf{S}_{r}^{-1}\mathbf{S}_{r}^{2}=\mathbf{S}_{r} \tag{27}\]  
\[\mathbf{U}_{2}^{T}\mathbf{P}_{k,k-1}\mathbf{V}_{1}=\mathbf{U}_{2}^{T}\mathbf{U}_{1}\mathbf{S}_{r}=0 \tag{28}\]  
Further, because \(\mathbf{P}_{k,k-1}\mathbf{V}_{2}=0\), then \(\mathbf{U}_{1}^{T}\mathbf{P}_{k,k-1}\mathbf{V}_{2}=\mathbf{U}_{2}^{T}\mathbf{P}_{k,k-1}\mathbf{V}_{2}=0\). Since is symmetric, it holds that:  
\[\mathbf{V}^{T}(\mathbf{P}_{k,k-1}{}^{T}\mathbf{P}_{k,k-1})\mathbf{V}=\mathbf{V}^{T}(\mathbf{P}_{k,k-1} \mathbf{P}_{k,k-1})\mathbf{V} \tag{29}\]  
So, \(\mathbf{V}^{T}\mathbf{U}=\mathbf{I}\) and \(\mathbf{V}=\mathbf{U}\), which yields to:  
\[\mathbf{P}_{k,k-1}=\mathbf{U}\left[ \begin{array}{cc}\mathbf{S}_{r}&0\\ 0&0\end{array} \right]\mathbf{U}^{T} \tag{30}\]  
By setting \(\mathbf{D}=\left[ \begin{array}{cc}\mathbf{S}_{r}&0\\ 0&0\end{array} \right]\), we obtain:  
\[\mathbf{K}_{k}=\mathbf{U}\mathbf{D}\mathbf{U}^{T}\mathbf{H}_{k}^{T}(\mathbf{H}_{k}\mathbf{U}\mathbf{D}\mathbf{U}^ {T}\mathbf{H}_{k}^{T}+\mathbf{R}_{cv,k})^{-1} \tag{31}\]  
The SVD is applicable to any matrix decomposition. It can express a complex matrix by the product of several relatively simple matrices (eigenvalues and eigenvectors), which can not only simplify the operation but also preserve the properties of the original matrix. In the classical kalman filter (KF) algorithm, the rounding error of the state covariance matrix gradually accumulates over time, leading to the loss of its non-negative property. Using the SVD can ensure its non-negative positive property so that the algorithm can go on and improve the operation accuracy.