# Vehicle Target Tracking Algorithm Based on Improved Strong Tracking Unscented Kalman Filter
## 3 Multidimensional Adaptive Factor-Based Strong Tracking UKF Algorithm
### MAST-UKF Algorithm Design  
Flowchart of the MAST-UKF algorithm steps, as shown in Figure 1.  
**Step 1:**: Initialize \(k=0\), and set \(D_{k}=I_{n\times n}\), \(J_{k+1}=I_{n\times n}\);
**Step 2:**: Use Equation (14) to calculate the sigma points based on a specific rule, selecting a total of \(2n+1\) points, where \(n\) is the dimensionality of the state:  
\[\left\{ \begin{array}{ll}\chi_{k|k}k^{(0)}=\hat{X}_{k|k}\\ \chi_{k|k}^{(i)}=\hat{X}_{k|k}+\alpha\Big{(}nP^{(i)}_{k|k}\Big{)}^{1/2}&,&i=1, \ldots,n\\ \chi_{k|k}^{(i)}=\hat{X}_{k|k}-\alpha\Big{(}nP^{(i)}_{k|k}\Big{)}^{1/2}&,&i=n+ 1,\ldots,2n\end{array} \right. \tag{14}\]  
where \(\hat{X}_{k|k}\) is the state estimate, \(P_{k|k}\) is the state estimation error covariance matrix, and \(\Big{(}P^{(i)}_{k|k}\Big{)}^{1/2}\) is the square root of \(P_{k|k}\); \(n\) is the dimensionality of the state vector. Afterward, use Equation (15) to calculate the corresponding weighting coefficients for these sigma points:  
\[\left\{ \begin{array}{ll}W_{0}^{(m)}=\big{(}\alpha^{2}-1\big{)}/\alpha^{2} \\ W_{0}^{(c)}=\big{(}\alpha^{2}-1\big{)}\big{(}\alpha^{2}+(1-\alpha^{2}-\beta) \big{)}\\ W_{i}^{(m)}=W_{i}^{(c)}=\frac{1}{2\alpha^{2}n}&,&i=1,2,\ldots,2n\end{array} \right. \tag{15}\]  
where \(m\) is the mean value, \(c\) is the covariance, and \(\alpha\) is a scaling parameter [25; 26]. For this study, we choose \(\alpha=0.001\), and \(\beta\) adjusts the accuracy of the higher-order terms, where \(\beta=2\). **Step 3:**: Substitute the sigma points into the nonlinear function in Equation (16) to calculate the mean value and covariance after the nonlinear transformation. Then, use Equation (17) to update the state estimate \(\hat{X}_{k+1|k}\), and use Equation (18) to calculate the theoretical residual error covariance \(P^{\prime}_{k+1|k}\):  
\[\hat{X}_{k+1|k}^{(i)}=f[\chi_{k|k}^{(i)}] \tag{16}\]\[\hat{X}_{k+1|k}=\sum_{i=0}^{2n}W_{i}^{(m)}\hat{X}_{k+1|k}^{(i)} \tag{17}\]  
\[P_{k+1|k}^{\prime}=\sum_{i=0}^{2n}W_{i}^{(c)}\Big{(}\hat{X}_{k+1|k}^{(i)}-\hat{X} _{k+1|k}\Big{)}\Big{(}\hat{X}_{k+1|k}^{(i)}-\hat{X}_{k+1|k}\Big{)}^{T} \tag{18}\]  
**Step 4:** Perform the second UT transformation. Use Equation (19) to observe the one-step predicted mean value \(\hat{Z}_{k+1|k}\), and calculate the residual \(\gamma_{k+1}\) through Equation (20):  
\[\hat{Z}_{k+1|k}=H_{k+1}\hat{X}_{k+1|k} \tag{19}\]  
\[\gamma_{k+1}=Z_{k+1}-\hat{Z}_{k+1|k} \tag{20}\]  
Here, \(H_{k+1}\) is the measurement matrix, and \(Z_{k+1}\) is the system measurement vector of size \(m\times 1\). **Step 5:** Use Equation (21) to compute the theoretical residual covariance \(P_{ZZ,k+1}\), and use Equation (22) to calculate the actual residual covariance \(C_{k+1}\):  
\[P_{ZZ,k+1}=H_{k+1}P_{k+1|k}H_{k+1}^{T}+R_{k+1} \tag{21}\]  
Figure 1: The flowchart of the MAST-UKF algorithm. \[C_{k+1}=\left\{ \begin{array}{cl}\gamma_{1}\gamma_{1}^{T}&,k=0\\ (pC_{k}+\gamma_{k+1}\gamma_{k+1}^{T})/(1+p)&,k=1,2,\ldots\end{array} \right. \tag{22}\]  
Here, \(R_{k+1}\) is the measurement noise covariance matrix, \(p\) is the forgetting factor, and the initial value is generally set to 0.95 [27,28]. **Step 6:** Compare \(C_{k+1}^{(ii)}\) and \(P_{ZZ,k+1}^{(ii)}\) for \(i=1,\ldots,m\). If \(C_{k+1}^{(ii)}>P_{ZZ,k+1}^{(ii)}\), set it to 1, and compute \(J_{k+1}^{(i)}\) using Equations (9) and (10). **Step 7:** Update \(P_{k+1|k}\) and \(P_{ZZ,k+1}\) using Equations (11) and (12), and compute the updated results using Equations (23) and (24):  
\[\left\{ \begin{array}{cl}K_{k+1}=P_{k+1|k}H_{k+1}^{T}/P_{ZZ,k+1}\\ \hat{X}_{k+1|k+1}=\hat{X}_{k+1|k}+K_{k+1}\Big{(}\hat{Z}_{k+1}-\hat{Z}_{k+1|k} \Big{)}\end{array} \right. \tag{23}\]  
\[P_{k+1,k+1}=P_{k+1|k}-K_{k+1}P_{ZZ,k+1}K_{k+1}^{T} \tag{24}\]  
where \(K_{k+1}\) is the Kalman gain matrix. **Step 8:** When \(k=10l\) (where \(l=3,4,5,\ldots\)), use Equation (6) to compute the ratio of \(C_{k+1}^{(ii)}>P_{ZZ,k+1}^{(ii)}\) and obtain the initial values of \(d_{k}^{(i)}\). Process the data in other dimensions similarly to obtain the initial values of \(d_{k}^{(i)}\).