## 2 Model Establishment
### Target Motion Model  
For the motion model, we assume that it is fixed in structure and that maneuver is an interference caused by noise injection into this motion model, so the CS-jerk model can be used to match the target motion by adaptive modification. In this section, we start with a brief introduction of the CS-jerk model.  
Suppose the change rate of acceleration of the target is a time-dependent stochastic process with a non-zero mean, which can be expressed as follows [18],  
\[\dddot{x}(t)=\vec{j}+j(t), \tag{1}\]  
where \(\dddot{x}(t)\) is the time-dependent rate of change in maneuvering acceleration with non-zero mean, \(\vec{j}\) is the mean of \(\dddot{x}(t)\), \(j(t)\) is the exponentially-related rate of change of random acceleration with zero mean. Using the Wiener-Kolmogorov whitening algorithm, the random acceleration rate is expressed as result of driving by Gaussian white noises, and the discretized state model is obtained as follows:  
\[X_{k}=FX_{k-1}+U\vec{j}+w_{k},\]where \(X_{k}=[\begin{array}{cccc}x_{k}&\dot{x}_{k}&\ddot{x}_{k}&\dddot{x}_{k}\end{array}]^{\rm T}\) is the target state vector at time \(k\) which includes the position \(x_{k}\), velocity \(\dot{x}_{k}\), acceleration \(\ddot{x}_{k}\) and the rate of change of acceleration \(\dddot{x}_{k}\); \(F\) is the state transition matrix and \(U\) is the input control matrix. \(F\) and \(U\) are expressed as follows:  
\[F=\left[\begin{array}{cccc}1&T&T^{2}/2&(2-2\omega T+\omega^{2}T^{2}-2e^{- \omega T})/(2\omega^{3})\\ 0&1&T&(e^{-\omega T}-1+\omega T)/\omega^{2}\\ 0&0&1&(1-e^{-\omega T})/\omega\\ 0&0&0&e^{-\omega T}\end{array}\right] \tag{3}\]  
\[U=\left[\begin{array}{c}\Big{(}2T-\omega T^{2}+\omega^{2}T^{3}/3-(2-2e^{- \omega T})/\alpha\Big{)}/(2\omega^{2})\\ \Big{(}-T+\omega T^{2}/2+(1-e^{-\omega T})/\omega\Big{)}/\omega\\ T-(1-e^{-\omega T})/\omega\\ 1-e^{-\omega T}\end{array}\right], \tag{4}\]  
where \(T\) is the sampling interval, \(\omega\) is the random Jerk maneuver frequency, \(w_{k}\) is the white noise sequence with zero mean, and its covariance matrix can be expressed as follows:  
\[Q_{k}=\mathrm{E}\{w_{k}w_{k}^{\rm T}\}=2\omega\sigma_{j}^{2}\left[\begin{array} []{cccc}q_{11}&q_{12}&q_{13}&q_{14}\\ q_{21}&q_{22}&q_{23}&q_{24}\\ q_{31}&q_{32}&q_{33}&q_{34}\\ q_{41}&q_{42}&q_{43}&q_{44}\end{array}\right], \tag{5}\]  
where \(\sigma_{j}^{2}\) is the variance of random jerk. For details of the remaining parameter \(q\), see [14].  
During the tracking process, by using a filtering algorithm, the statistical characteristics of maneuver noises are corrected according to the estimated target motion state, and further, the noise covariance matrix \(Q_{k}\) is updated, so that the CS-jerk model can be adjusted to match the target-changing motion. For brevity, only the basic form of this CS-jerk model is given in this section; details of the CS-jerk model can be found in [17; 18].  
The above analysis is based on one-dimensional space. In three-dimensional motion space, the motion model needs to be extended. The target state vector will be expanded to \(X_{k}=[x_{k},\,\dot{x}_{k},\,\ddot{x}_{k},\,\ddot{x}_{k},\,y_{k},\,\dot{y}_{k },\,\ddot{y}_{k},\,\ddot{y}_{k},\,\ddot{y}_{k},\,\ddot{z}_{k},\,\dot{z}_{k}, \,\ddot{z}_{k},\,\ddot{z}_{k}]^{\rm T}\), where \(x,y,\) and \(z\) indicate the state in three directions. The state transition matrix \(F\) and the process noise covariance matrix \(Q_{k}\) are both expanded into a \(12\times 12\)-dimensional matrix.