## 1 Introduction
### Square-Root Continuous-Discrete Cubature Kalman Filter Algorithm
Step 2.5 Calculate the predicted square-root covariance: \(\boldsymbol{S}(t_{k})=\left[\boldsymbol{S}_{1}(t_{k})\quad\boldsymbol{S}_{2} (t_{k})\quad\cdots\quad\boldsymbol{S}_{2n}(t_{k})\right]\), where \(\boldsymbol{S}_{i}(t_{k})\) represents \(a\) \(\sin\) gle column vector, \(\boldsymbol{S}_{i}(t_{k})=\left(\boldsymbol{X}_{i}^{*}(t_{k})-\hat{\boldsymbol{ x}}_{k|k-1}\right)/\sqrt{n}\). Measurement update Step 3.1 Calculate the state cubature point: \(\boldsymbol{S}_{k|k-1}=\boldsymbol{S}(t_{k})\), \(\boldsymbol{X}_{i|k-1}=\boldsymbol{S}_{k|k-1}\xi_{i}+\hat{\boldsymbol{x}}_{k| k-1}\). Step 3.2 Measure the spread of cubature points: \(\boldsymbol{Z}_{i|k-1}=h(\boldsymbol{X}_{i|k-1},k)\). Step 3.3 Calculate the predicted value of the measurement: \(\hat{\boldsymbol{x}}_{k|k-1}=\frac{1}{2n}\sum\limits_{i=1}^{2n}\boldsymbol{Z}_ {i,k|k-1}\). Step 3.4 Construct the measurement weighted center matrix: \(\boldsymbol{Z}_{k|k-1}=\frac{1}{\sqrt{2n}}\Big{[}\boldsymbol{Z}_{1,k|k-1}- \hat{\boldsymbol{z}}_{k|k-1}\,,\cdots,\,\boldsymbol{Z}_{2n,k|k-1}-\hat{ \boldsymbol{z}}_{k|k-1}\Big{]}\). Step 3.5 Calculate the innovation covariance matrix: \(\boldsymbol{P}_{zz,k|k-1}=\boldsymbol{Z}_{k|k-1}\boldsymbol{Z}_{k|k-1}^{T}+ \boldsymbol{R}_{k}\). Step 3.6 Construct the state weighted center matrix: \(\boldsymbol{X}_{k|k-1}=\frac{1}{\sqrt{2n}}\Big{[}\boldsymbol{X}_{1,k|k-1}- \hat{\boldsymbol{x}}_{k|k-1}\,,\cdots,\,\boldsymbol{X}_{2n,k|k-1}-\hat{ \boldsymbol{x}}_{k|k-1}\Big{]}\). Step 3.7 Calculate the cross covariance matrix: \(\boldsymbol{P}_{xz,k|k-1}=\boldsymbol{X}_{k|k-1}\boldsymbol{Z}_{k|k-1}^{T}\). Step 3.8 The continuous-discrete cubature gain is: \(\boldsymbol{K}_{k}=\boldsymbol{P}_{xx|k-1}-\boldsymbol{P}_{zz,k|k-1}^{T}\). Step 3.9 Calculate the state estimate: \(\hat{\boldsymbol{x}}_{k|k}=\hat{\boldsymbol{x}}_{k|k-1}+\boldsymbol{K}_{k}( \boldsymbol{z}_{k}-\hat{\boldsymbol{z}}_{k|k-1})\). Step 3.10 Update the covariance matrix: \(\boldsymbol{P}_{k|k}=\boldsymbol{P}_{k|k-1}-\boldsymbol{K}_{k}P_{z,k|k-1} \boldsymbol{K}_{k}^{T}\). ```  
**Algorithm 1** SRCD-CKF. **Remark 1**. : _The above method gives the square root form of the prediction covariance and uses it for the measurement update process. The algorithm has effectively guaranteed the stability of the filtering. At the same time, a more thorough square-root version is also available._Here we redefine the cross-covariance matrix \(\mathbf{P}_{xz,k|k-1}\), the innovations covariance square-root \(\mathbf{R}_{e,k}^{1/2}\), and the filtering covariance square-root \(\mathbf{S}_{k|k}\) by QR decomposition:  
\[\text{qr} \begin{bmatrix}\mathbf{Z}_{k|k-1}&\mathbf{R}_{k}^{1/2}\\ \mathbf{X}_{k|k-1}&0\end{bmatrix} = \begin{bmatrix}\mathbf{R}_{e,k}^{1/2}&0\\ \mathbf{P}_{xz,k|k-1}&\mathbf{S}_{k|k}\end{bmatrix} , \tag{13}\]  
where \(\mathbf{0}\) is the zero-block of a proper size, \(\mathbf{R}_{k}^{1/2}\) is the measurement noise covariance square-root obtained by Cholesky factorization, and \(\mathbf{R}_{e,k}^{1/2}\) and \(\mathbf{S}_{k|k}\) are lower triangular matrices. On this basis, we can obtain the update state and covariance. In addition, the robust estimation methods under this square-root version can be obtained according to our idea.