# Vehicle Target Tracking Algorithm Based on Improved Strong Tracking Unscented Kalman Filter
## 3 Multidimensional Adaptive Factor-Based Strong Tracking UKF Algorithm
### Introduction of Multidimensional Adaptive Decay Factors
Due to the potential mismatch between the actual and theoretical residual covariance in different dimensions, there may be discrepancies. Therefore, it is necessary to review Equations (4) and (5) to judge the differences of each angle and dynamically adjust the weight factors for each dimension to ensure that the system adapts to the actual conditions of each dimension and adjusts the noise covariance accordingly. According to the strong tracking theory, increasing \(p_{ZZ,k+1}^{(ii)}\) is beneficial for dealing with the system's uncertainty; otherwise, it is not conducive to handling the system's uncertainty [23; 24]. When \(Q_{k}\) is large, the noise uncertainty of the system's state is higher, and the system's state becomes more unstable. In this case, a larger covariance matrix is required to handle the higher uncertainty. At this point, the probability of the situation described by Equation (4) is relatively low, while the probability of the situation described by Equation (5) is relatively high. Conversely, when \(Q_{k}\) is small, the probability of Equation (4) occurring is higher, while the probability of Equation (5) occurring is lower. In this case, the adjustment of \(Q_{k}\) can be determined by comparing the actual residual covariance with the theoretical residual covariance, so a threshold for decision-making is introduced. \[\frac{\sum\left(C_{k-28}^{(ii)}>P_{ZZ,k-28}^{(ii)}\cdots C_{k+1}^{(ii)}>P_{ZZ,k +1}^{(ii)}\right)}{30}<50\%,\left\{i=1,\ldots,m;k=10;l=3,4,5,\ldots\right\} \tag{6}\]  
Equation (6) shows that when \(k=10l\) (where \(l=3,4,5,\ldots\)), based on the empirical rules in engineering, a set of 30 groups of data (\(k-28\sim k+1\)) are selected for a joint comparison. This selection can effectively capture the correlation between the noise and uncertainty of the data. The choice of 30 datasets improves the ability to recognize the features of the data, which enhances the discriminatory ability, reduces the impact of noise on the results, ensures that important changes in the data are not lost, and, thus, makes the model more robust and improves its accuracy and stability. Finally, the degree of matching the actual residual covariance and theoretical residual covariance is required. If it is smaller than 50%, it indicates that \(Q_{k}\) should be reduced, and this corresponds to \(d_{k}^{(i)}\) values in \(D_{k}\), which should be taken from the range \([0.95,1.05]\) as the initial value. Conversely, it should be taken from \((1,1.05)\). The dimensional data of each axis are adjusted according to the corresponding data of each dimension. However, this simple judgment can only determine the initial range of \(d_{k}^{(i)}\), and the specific value is obtained through the UT transformation of the residual covariance and the predicted observation covariance, calculating the corresponding reduction factor \(D_{k}\). \[d_{k}^{(i)}=\min\left(d_{k|max^{\prime}}^{(i)},d_{k|min}^{(i)}+\Delta d_{k}^{(i )}\right) \tag{7}\]  
\[\Delta d_{k}^{(i)}=\left(d_{k|max}^{(i)}-d_{k|min}^{(i)}\right)\cdot\frac{ \text{tr}\left(P_{ZZ,k+1}^{(ii)}\right)}{\text{tr}\left(H_{k}P_{k|k-1}^{(ii)} H_{k}^{T}\right)} \tag{8}\]  
In these equations, \(d_{k|max}^{(i)}\) and \(d_{k|min}^{(i)}\) represent the maximum and minimum values of \(d_{k}^{(i)}\) within the current range, respectively. \(P_{k|k-1}^{(ii)}\) represents the predicted covariance matrix of the system. \(\text{tr}(\cdot)\) denotes the trace of a matrix, i.e., the sum of the diagonal elements of the matrix. Furthermore, the calculation is used to weigh the covariance matrix, measuring the difference between the model and the observed values. \(\Delta d_{k}^{(i)}\) is dynamically adjusted based on the ratio of residual covariance to predicted covariance. Next, we will discuss the construction of the matrix \(J_{k+1}\), which is similar to \(D_{k}\), and \(J_{k+1}\) is also a square matrix. When updating \(D_{k}Q_{k}\), it is necessary to satisfy the strong tracking conditions in Equation (1) along with Equations (4) and (5):  
\[I_{k+1}^{(i)}=\left\{