## 6 Tracking
### Track Estimation and Updating  
In the track estimation phase, the EKF [45; 46] was adopted to predict the state tracks \(S_{t-1}\) to the current frame \(t\), which is denoted as \(S_{t}^{\text{ref}}\). An EKF is a recursive linear filter used to determine the state of a dynamic system based on the time series of noisy observations. In addition, it features low computational complexity and a recursive structure and is resistant to measurement errors and correlations when dealing with multiple targets. Consequently, the radar community frequently uses a KF-based tracking technique [47]. Using the EKF when tracking a moving target will allow the system to detect the target even if it remains stationary, as well as to follow the target wherever it travels. In this paper, target tracking was performed utilizing distance and azimuth angle observations, rather  
Figure 5: Tracking process input and output. (**A**) The estimation of the track at a certain frame. (**B**) The continuous track estimation.  
than radial velocity observations. Most investigations in the literature included radial velocity observations in the model, which caused the system to become too non-linear to produce meaningful estimates using KF. Moreover, observing only the distance limits the ability to locate a target in 3D space. This limitation can be overcome by observing the angular positions of moving targets and eventually reconstructing the full track of the target.  
Therefore, the KF model observation vector consists of the detected target's radial distance, and the azimuth angle at frame \(t\) is defined as \(m^{\ t}=[r\,\varnothing\ ]^{t}\). A graphical representation of the radial distance and azimuth angle of the target from the ground station is shown in Figure 7. The current track state of the detected target at frame \(t\) is defined as \(S^{t}:=[x,y,z,v,\varnothing,\theta,l,h,w]^{t}\).  
The typical state-space representation of a non-linear time-discrete model is as follows:  
\[S^{t}=AS^{t-1}+u^{t}, \tag{13}\]  
\[M^{t}=h\big{(}T^{t}\big{)}+q^{t}, \tag{14}\]  
where Equation (13) is responsible for explaining the evolution of the target states through \(t\); Equation (14) is responsible for matching the target's state to the measurements. \(u^{t}\) and \(q^{t}\) are the white Gaussian process noise and measurement noise, respectively.  
Figure 6: The workflow of the proposed target tracker.  
Figure 7: The radial distance and the angular position of the detected target.  
\(h\big{(}S^{t}\big{)}=\left[\sqrt{x^{2}+y^{2}+z^{2}},\ tan^{-1}\big{(}\frac{y}{x} \big{)}\right]^{t}\) represents the non-linear measurement process. \(A\) is the state transition matrix given in the time-discrete model and is defined as:  
\[A=\begin{bmatrix}1&0&0&\Delta t&0&0\\ 0&1&0&0&\Delta t&0\\ 0&0&1&0&0&\Delta t\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{bmatrix}, \tag{15}\]  
To solve this non-linear measurement problem, a modified observation vector \(\hat{M_{t}}=[r\ cos\ \theta,\ r\ sin\ \theta]\) is obtained using Equations (13) and (14). The EKF is used to estimate the new position of the detected targets in two steps. In the first step, the new track state is predicted by the mean \(S_{est}^{t}\) and covariance \(P_{t}\) at time \(t\) and is defined as:  
\[S_{est}^{t}=AS^{t-1}+u^{t}, \tag{16}\]  
\[P^{T}=AP^{t-1}A^{T}+q^{t}, \tag{17}\]  
In the second step, the filter updates the first step state estimations using the Kalman gain \(K^{t}\), which is denoted as:  
\[K^{t}=P^{t}\ H^{t}T\big{(}H^{t}P^{t}H^{t}T+U\big{)}^{-1} \tag{18}\]  
\[S^{t}=S^{t-1}+K^{t}(M^{t}-h(P^{t})), \tag{19}\]  
\[P^{t}=(I-K^{t}\ H^{t})P^{t-1}, \tag{20}\]  
where \(H^{t}\) is the Jacobian matrix of the partial derivatives of \(h(\cdot)\), \(U\) is the noise covariance, and \(I\) is the identity matrix.