# An Interactive Transient Model Correction Active Sonar Target Tracking Method
## 2 Description of the TMC Tracking Method
### Tracking Filtering Method  
When the target is moving at a constant velocity, the KF algorithm is equivalent to the filtering method designed using the minimum mean squared error estimation criterion in steady state. However, during the transient process or when the target undergoes random maneuvering, the performance of KF is superior to other methods. Additionally, KF is a recursive algorithm that only requires the current time's prediction and measurement values to obtain the current time's state filtering estimate, without the need to transmit all historical data. Therefore, it has advantages such as small computational complexity, strong real-time performance, and easy engineering implementation. Based on the above considerations, KF is adopted as the tracking filtering method.  
In target tracking, the KF algorithm mainly implements the functions of prediction and filtering estimation. The filtering equation is:  
\[\mathbf{\hat{x}}(k|k)=\mathbf{\hat{x}}(k|k-1)+\mathbf{K}(k)\mathbf{d}(k). \tag{20}\]  
In the formula, \(\mathbf{\hat{X}}(k|k)\) represents the filtered estimate at time \(k\); \(\mathbf{\hat{X}}(k|k-1)\) is the priori filtered estimate (one-step advance prediction) of \(\mathbf{\hat{X}}(k-1|k-1)\) obtained by the measurement value \(\mathbf{Z}(k-1)\) at time \((k-1)\):  
\[\mathbf{\hat{X}}(k|k-1)=\mathbf{\Phi\hat{X}}(k-1|k-1), \tag{21}\]  
\(\mathbf{K}(k)\) represents the filtering gain matrix:  
\[\mathbf{K}(k)=\mathbf{P}(k|k-1)\mathbf{H}^{T}\Big{[}\mathbf{H}\mathbf{P}(k|k- 1)\mathbf{H}^{T}+\mathbf{R}(k)\Big{]}^{-1} \tag{22}\]  
\(\mathbf{P}(k|k-1)\) is the covariance matrix of one-step advance prediction error:  
\[\mathbf{P}(k|k-1)=\mathbf{\Phi P}(k-1|k-1)\mathbf{\Phi}^{T}+\mathbf{\Gamma Q}( k-1)\mathbf{\Gamma}^{T}, \tag{23}\]  
\(\mathbf{P}(k-1|k-1)\) is the covariance matrix of filtered estimation error:  
\[\mathbf{P}(k-1|k-1)=[\mathbf{I}-\mathbf{K}(k-1)\mathbf{H}]\mathbf{P}(k-1|k-2) \tag{24}\]  
\(\mathbf{d}(k)\) is the residual (innovation) vector:  
\[\mathbf{d}(k)=\mathbf{Z}(k)-\mathbf{\hat{Z}}(k|k-1), \tag{25}\]\(\mathbf{Z}(k|k-1)\) is the predicted measurement value:  
\[\hat{\mathbf{Z}}(k|k-1)=\mathbf{H}\hat{\mathbf{X}}(k|k-1) \tag{26}\]  
The covariance matrix of the residual vector \(\mathbf{d}(k)\) is:  
\[\mathbf{S}(k)=\mathbf{H}\mathbf{P}(k|k-1)\mathbf{H}^{T}+\mathbf{R}(k). \tag{27}\]  
The system state vector \(\mathbf{X}(t)\) contains the displacement and motion velocity of the target on the X- and Y-axis. Through the filter processing of Equation (20), the displacement and motion velocity of the target on the X- and Y-axis are constantly updated. After filtering, the distance and azimuth accuracy of the target is improved. The velocity component of the target movement is synthesized by the velocity component of the two directions. Further, if the measurement vector \(\mathbf{Z}(k)\) includes the speed measurement of the target, you can establish the state vector that contains the acceleration (the acceleration is no longer considered a random interference noise), and after filtering, you can obtain the motion acceleration components on the X- and Y-axis, and you can obtain the acceleration vector on the two-dimensional plane by the synthesis of the two acceleration components.