# Non-Gaussian Pseudolinear Kalman Filtering-Based Target Motion Analysis with State Constraints
## 5 Simulation
### Simulation Scenarios
Figure 4: Time-averaged RMSEs, BNorms and bearing noise standard deviation for the PL-MMSE, PLKF, BC-PLKF and IVKF algorithms, as well as, the PL-MMSE algorithm with linear constraints proposed in the paper. Figure 5: RMSEs, BNorms at different time scan for \(\sigma_{\theta}=7^{\circ}\) for the PL-MMSE, PLKF, BC-PLKF and IVKF algorithms, as well as, the proposed PL-MMSE algorithm with linear constraints in the paper. The RMSE performance of PL-MMSE-C (\(W=I\)) is RMSE\({}_{avg}^{pos}=1.781\) m and RMSE\({}_{avg}^{vel}=0.103\) m/s at large bearing noise level \(\sigma_{\theta}=7^{\circ}\), which are less than other constrained algorithms. Similarly, the BNorm performance of PL-MMSE-C (\(W=I\)) has \(\text{BNorm}_{avg}^{pos}=0.084\) m and \(\text{BNorm}_{avg}^{vel}=0.001\) m/s at large bearing noise level \(\sigma_{\theta}=7^{\circ}\) which are better than the others. The numerical results of PL-MMSE-C (\(W=P^{-1}\)) are similar to that of PL-MMSE-C (\(W=I\)), as observed from the table. The filters combined with constraints can achieve better performance from Figures 6 and 7 and Table 5. As shown in Table 5, the constraint algorithms with \(W=P^{-1}\) are not necessarily better than the corresponding filters with \(W=I\), where RMSE\({}_{avg}^{pos}=2.537\) m of IVKF-C (\(W=I\)) is less than RMSE\({}_{avg}^{pos}=2.842\) m of IVKF-C (\(W=P^{-1}\)) and RMSE\({}_{avg}^{pos}=2.890\) m of BC-PLKF-C (\(W=I\)) is greater than RMSE\({}_{avg}^{pos}=2.883\) m of BC-PLKF-C (\(W=P^{-1}\)). This difference is caused by the discrepancy between the actual distribution characteristics of the state estimate \(\hat{x}\) and \(P\). When the distribution is close to \(P\), the constraint algorithms with \(W=P^{-1}\) behave better than the corresponding constraint algorithms with \(W=I\). In the second scenario, the target moves on an arc, as shown in Figure 3b, with a nearly constant velocity magnitude \(V\). The turning center is \((R_{x},R_{y})\) with the radius \(R\). Hence, the constrained equation \(h(\cdot)\), the vector \(g\), the constrained matrix \(M\), the vector \(m\) and the variable \(m_{0}\), are given by  
\[h(x) = (x(1)-R_{x})^{2}+\big{(}x(2)-R_{y}\big{)}^{2}, \tag{76}\] \[g = R^{2},\] (77) \[M = \begin{bmatrix}1&0\\ 0&1\end{bmatrix} ,\] (78) \[m = -\big{[}R_{x}\ R_{y}\big{]}^{T},\] (79) \[m_{0} = R_{x}^{2}+R_{y}^{2}-R^{2}. \tag{80}\]  
In addition, the velocity constraint is introduced into the state estimation. The constrained velocity estimate \(\hat{v}\) is  
\[\hat{v}=\Big{(}\hat{v}^{T}\mu\Big{)}\mu \tag{81}\]  
where the unconstrained velocity estimate \(\hat{v}\) and constrained unit direction vector \(\mu\) are  
\[\hat{v} = [\hat{x}(3)\ \hat{x}(4)]^{T}, \tag{82}\] \[\mu = [-\sin\theta\ \cos\theta]^{T} \tag{83}\]  
with \(\theta=\tan^{-1}(\hat{x}(2)/\hat{x}(1))\). In the simulation, the sampling interval \(T\) is set to 0.1 s. The total time scan is the 20 s. The turning center \((R_{x},R_{y})\) is set to \((100,0)\) m. The turning radius \(R\) is 100 m. The initial position for the target is \([200\ 0]^{T}\) m. To compare the linear approximation with the second-order approximation method, two experiments are carried out, where the magnitude of \(V\) is \(10^{-4}\) m/s and 0.2 m/s, respectively. For \(V=10^{-4}\) m/s, the true initial state is \(x_{1}=\big{[}200\ 0\ 10^{-4}\big{]}^{T}\). The initial covariance matrix is \(P_{1|1}=\text{diag}([10^{-8}\ 10^{-8}\ 10^{-10}\ 10^{-10}])\). The process noise \(\omega_{k-1}\) and bearing noise \(e_{k}\) have the same composition as (74) and (75) where \(\mu_{x1}^{T}=\big{[}0\ \ 0\ \ 0\ \ 0\big{]}\), \(\mu_{x2}^{T}=\big{[}0\ \ 0\ \ 0\ \ 0\big{]}\), \(\mu_{x1}=0\), \(\mu_{x2}=0\), \(Q_{1}=\text{diag}[0\ \ 0\ 10^{-9}\ 10^{-9}]\), \(Q_{2}=\text{diag}([0\ \ 0\ 2\times 10^{-9}\ 2\times 10^{-9}])\), \(R_{1}=1.5\times 10^{-12}\), \(R_{2}=1.909\times 10^{-12}\) and \(\lambda=0.4\). The standard deviation \(\sigma_{\theta}\) of bearing noise for the corresponding \(\rho\) is set as shown in Table 6.