# 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  
The traditional strong tracking UKF algorithm is often affected by the uncertainty of state noise and observation noise when handling traffic scenarios, which can lead to a decrease in tracking filter accuracy. According to the sufficient conditions for the establishment of strong tracking theory [21; 22], combined with the prior state covariance update formula and the observation residual covariance calculation formula, the following result can be derived:  
\[H_{k+1}\Big{(}P_{k|k+1}^{\prime}+Q_{k}\Big{)}H_{k+1}^{\intercal}+R_{k+1}=C_{k+1} \tag{1}\]where \(H_{k+1}\) is the observation matrix; \(P_{k|k+1}^{\prime}\) is the state one-step prediction error covariance matrix; \(Q_{k}\) is the state noise covariance; \(R_{k+1}\) is the observation noise covariance; and \(C_{k+1}\) is the actual residual covariance. According to Equation (1), the sufficient condition for establishing the strong tracking UKF is that the theoretical residual covariance matrix matches the actual residual covariance matrix, etc. When the system is configured and the radar works stably, the theoretical residual covariance matrix matches the actual residual covariance matrix, satisfying the requirements of Equation (1). However, when the system contains model errors or noise uncertainties, the residual covariance matrix may deviate, leading to the failure of Equation (1). In traffic scenarios, when the target's speed and direction change abruptly, especially in dense traffic, the target's speed might suddenly change. The traditional strong tracking UKF algorithm passes through \(P_{k+1|k}\), and the prediction step leads to a large value in the predicted state error covariance matrix. The increased residual covariance, thus, increases the filtering error, preventing the system from quickly responding. However, the method proposed in this paper only updates \(P_{k+1|k}\) when it enters a one-dimensional adaptive matrix, \(\mu_{k}\), as shown in the following equation, and this avoids the high-dimensional adaptation of traditional methods. As a result, the method should be able to handle complex traffic scenarios where the system cannot quickly respond to target speed and direction changes, leading to increased tracking errors and inaccurate results. Consequently, we propose to introduce diagonal multidimensional adaptive factor matrices \(D_{k}\) and \(J_{k+1}\) in place of \(Q_{k}\) and \(R_{k+1}\), respectively, to enhance the convergence ability to sudden changes in the target speed and direction in the traffic scene:  
\[D_{k}=\text{diag}\Big{(}d_{k}^{(1)},d_{k}^{(2)},\dots,d_{k}^{(n)}\Big{)} \tag{2}\]  
\[J_{k+1}=\text{diag}\Big{(}I_{k+1}^{(1)},I_{k+1}^{(2)},\dots,I_{k+1}^{(m)}\Big{)} \tag{3}\]  
The function diag(...) is used to construct a diagonal matrix, where each element corresponds to different weighting factors. To determine the diagonal matrix \(D_{k}\) in the optimization process, and to meet the dynamic adjustment requirements of the system model, elements of \(D_{k}\) (i.e., \(d_{k}^{(i)},i=1,2,\dots,n\)) are expected to take values in the range of 0.95-1.05, depending on the actual dynamics of the system. Specifically, the required value should consider the experimental residual covariance matrix's matching and the allocation degrees as well as the evaluation of the change characteristics of the covariance matrix. In practical scenarios, the allocation degree can be divided into two types: one is when the residual covariance of the system model matches the allocation degree, and the other is when the residual covariance of the system is smaller, as shown in Equations (4) and (5):  
\[C_{k}^{(ii)}>P_{ZZ,k+1}^{(ii)},\quad i=1,\dots,m \tag{4}\]  
\[C_{k}^{(ii)}\leq P_{ZZ,k+1}^{(ii)},\quad i=1,\dots,m \tag{5}\]  
In the formula, \(C_{k}^{(ii)}\) and \(P_{ZZ,k+1}^{(ii)}\) represent the elements of the \(i\)-th row of the actual residual covariance and the theoretical residual covariance, respectively.