## 6 Numerical Simulation
### Tracking of High-Maneuvering Target  
Finally, the performance of the three filters is compared for high-maneuvering target tracking. We can predict that the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter achieves better performance for relatively complicated motion, including large jerk, compared with the \(\alpha\)-\(\beta\) and \(\alpha\)-\(\beta\)-\(\gamma\) filters. The aim of this simulation is to confirm the effectiveness of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter when the target has high-maneuvering motion. The measurement position \(x_{0,k}\) is obtained by adding white Gaussian noise with \(\sqrt{B_{\mathrm{x}}}=250\) m to the true position \(x_{\mathrm{t},k}\). The true position is set as  
\[x_{\mathrm{t},k}=\frac{1}{50}k^{2}\cos(\frac{\pi kT}{25}). \tag{54}\]  
Figure 5 shows the \(x_{\mathrm{t},k}\) and corresponding true snap of the target. Based on this motion, we set and calculate the optimal parameters. We evaluate the RMS prediction error using (51). To fairly compare the performance, the bias errors (tracking performance indices) that correspond to the maximum motion parameters of the targets are set to the same value for all the filters. We assume two cases of gain settings: a low gain case (\(\delta\) of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter is 0.01, \(\gamma\) of the \(\alpha\)-\(\beta\)-\(\gamma\) filter is 0.08, and \(\beta\) of the \(\alpha\)-\(\beta\) filter is 0.7) and a high gain case (\(\delta\) of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter is 0.05, \(\gamma\) of the \(\alpha\)-\(\beta\)-\(\gamma\) filter is 0.4, and \(\beta\) of the \(\alpha\)-\(\beta\) filter is 3.5).  
Figure 5(a) shows the simulation results for the low gain case. When the gains are low, the smoothing effect of the measurement noises increases. Thus, for all filters, when the parameters of the target motion (velocity, acceleration, jerk, snap, etc.) are relatively small, i.e., approximately \(kT<500\) s, the RMS errors converge. In this case, the tracking accuracy of the \(\alpha\)-\(\beta\) filter is larger than that of the other two filters. This result is consistent with that of the theoretical analysis shown in Figure 2. In contrast, for \(kT>500\) s, the RMS errors vary because of the bias errors caused by the complicated motion. When the snap of the target increases, the RMS error of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter is the smallest. However, when this is not the case, the RMS error of the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter increases due to the poor smoothing performance.  
Figure 5(b) shows the results for the high gain case. In this case, all the filters converge during the observation period. This is because the relatively large gain makes it possible to track complicated large motions. However, due to the poor tracking performance for the maneuvering motion, the \(\alpha\)-\(\beta\) and \(\alpha\)-\(\beta\)-\(\gamma\) filters generate larger RMS errors compared with those in Figure 5(a). In contrast, this type of deterioration in the tracking accuracy for the \(\alpha\)-\(\beta\)-\(\gamma\)-\(\delta\) filter is small owing to its better tracking performance for complicated motions including large jerk and snap.  
Figure 5: True target motion for simulations of high-maneuvering target. (a) True position; (b) True snap.