# IMM Filtering Algorithms for a Highly Maneuvering Fighter Aircraft: An Overview
## 5 Measurement Model  
The AMTI radar measures the range, bearing (or azimuth), and radial velocity of the target [52]. Figure 1[51] depicts the true range, \(r_{k}\), azimuth, \(\beta_{k}\), and radial velocity, \(v_{r,k}\) of the target at time \(t_{k}\). Assume that the radar moves with CV. At time \(t_{k}\), the state vector of the AMTI radar is given by  
\[\mathbf{x}_{k}^{\mathrm{s}}=[x_{k}^{\mathrm{s}}\quad y_{k}^{\mathrm{s}}\quad \hat{x}_{k}^{\mathrm{s}}\quad\hat{y}_{k}^{\mathrm{s}}]^{\prime}. \tag{36}\]  
The position and velocity vectors of the target denoted by \(\mathbf{p}_{k}\) and \(\mathbf{v}_{k}\), respectively, at time \(t_{k}\) are specified by  
\[\mathbf{p}_{k}=[x_{k}\quad y_{k}]^{\prime}, \tag{37}\]  
\[\mathbf{v}_{k}=[\hat{x}_{k}\quad\hat{y}_{k}]^{\prime}. \tag{38}\]  
Let \(\mathbf{p}_{k}^{\mathrm{s}}\) denote the position vector of the AMTI radar at time \(t_{k}\):  
\[\mathbf{p}_{k}^{\mathrm{s}}=[x_{k}^{\mathrm{s}}\quad y_{k}^{\mathrm{s}}]^{ \prime}. \tag{39}\]  
The range vector of the target from the AMTI radar at time \(t_{k}\) is  
\[\mathbf{r}_{k}=\mathbf{p}_{k}-\mathbf{p}_{k}^{\mathrm{s}}=[x_{k}-x_{k}^{ \mathrm{s}}\quad y_{k}-y_{k}^{\mathrm{s}}]^{\prime}. \tag{40}\]  
Then, the range of the target from the AMTI radar at time \(t_{k}\) is given by  
\[r_{k}=\left\|\mathbf{p}_{k}^{\mathrm{t}}-\mathbf{p}_{k}^{\mathrm{s}}\right\| =\sqrt{\left(x_{k}-x_{k}^{\mathrm{s}}\right)^{2}+\left(y_{k}-y_{k}^{\mathrm{s} }\right)^{2}}. \tag{41}\]  
Let \(\mathbf{u}_{k}\) denote the unit vector along the radar line-of-sight (RLOS). Then,  
\[\mathbf{u}_{k}:=\mathbf{r}_{k}/r_{k}=\frac{\mathbf{p}_{k}-\mathbf{p}_{k}^{ \mathrm{s}}}{\left\|\mathbf{p}_{k}-\mathbf{p}_{k}^{\mathrm{s}}\right\|}. \tag{42}\]  
The radial velocity of the target \(v_{r,k}\) at time \(t_{k}\) is given by [24; 52]  
\[v_{r,k}=\frac{1}{r_{k}}[(x_{k}-x_{k}^{\mathrm{s}})\hat{x}_{k}+(y_{k}-y_{k}^{ \mathrm{s}})\hat{y}_{k}]. \tag{43}\]  
Hence, the radial velocity represents the component of the target's velocity that is projected onto the RLOS (Figure 1). At time \(t_{k}\), given the target speed \(v_{k}\) and the angle between the target velocity and the RLOS \(\alpha_{k}\), the radial velocity of the target \(v_{r,k}\) can be calculated as  
\[v_{r,k}=v_{k}\cos\alpha_{k}. \tag{44}\]Figure 1 and Equation (44) demonstrate that the radial velocity exhibits its peak values of \(v_{k}\) and \(-v_{k}\) when the target moves in the same or opposite direction as the RLOS. It vanishes to zero when the target's velocity is perpendicular to the RLOS. The true trajectories of the AMTI radar and aircraft moving at \(3g\), \(4g\), \(5g\), and \(6g\) are shown in Figure 2, where the red and blue lines indicate the trajectories of the aircraft and AMTI radar, respectively. An open circle and a diamond indicate the start and end points of a trajectory, respectively. Each aircraft trajectory encompasses a succession of motion phases, including segments of NCV, NCT+, another NCV, NCT-, and concluding with another NCV. In Figure 3, different segments of the trajectory of an aircraft moving at \(3g\) are presented, and the durations of various segments with start and end times are also given in Table 1.  
\[\mathbf{p}_{k}=[x_{k}\quad y_{k}]^{\prime}, \tag{37}\]0 Trajectories of an aircraft moving at (**a**) \(3g\), (**b**) \(4g\), (**c**) \(5g\), and (**d**) \(6g\) and AMTI sensor trajectories. At time \(t_{k}\), the radar measurement vector \(\mathbf{z}_{k}\) is defined as  
\[\mathbf{p}_{k}=[x_{k}\quad y_{k}]^{\prime}, \tag{37}\]1  
The measurement model for the AMTI radar is defined by [52]  
\[\mathbf{p}_{k}=[x_{k}\quad y_{k}]^{\prime}, \tag{37}\]2  
where \(\mathbf{h}\) is the nonlinear measurement function  
\[\mathbf{h}(\mathbf{x}_{k})= \begin{bmatrix}r_{k}\\ \beta_{k}\\ v_{r,k}\end{bmatrix} = \begin{bmatrix}\sqrt{\left(x_{k}-x_{k}^{s}\right)^{2}+ \left(y_{k}-y_{k}^{s}\right)^{2}}\\ \tan^{-1}\left(x_{k}-x_{k}^{s},y_{k}-y_{k}^{s}\right)\\ \frac{1}{t_{k}}[(x_{k}-x_{k}^{s})\dot{x}_{k}+(y_{k}-y_{k}^{s})\dot{y}_{k}] \end{bmatrix}