## 4 Simulations
### Bearings-Only Tracking Problem  
Bearings-only tracking is a typical nonlinear problem which estimates the position and the velocity of a target from noisy bearing measurements. As is shown in Figure 2, we assume that the target is in uniform linear motion with acceleration disturbance. The state vector at time index \(k\) can be expressed by \(\mathbf{x}_{k}^{t}=\left[x_{k}^{t},y_{k}^{t},\dot{x}_{k}^{t},\dot{y}_{k}^{t}\right] ^{T}\), where \(\mathbf{x}_{k}^{p}=\left[x_{k}^{t},y_{k}^{t}\right]^{T}\) is the position and \(\dot{x}_{k}^{p}=\left[\dot{x}_{k}^{t},\dot{y}_{k}^{t}\right]^{T}\) is the velocity vector. The true state of the observation platform is \(\mathbf{x}_{k}^{o}=\left[x_{k}^{o},y_{k}^{o},\dot{x}_{k}^{o},\dot{y}_{k}^{o}\right] ^{T}\), and the state vector in the relative own-ship reference at time index \(k\) can be defined as follows:  
\[\mathbf{x}_{k}\triangleq\mathbf{x}_{k}^{t}-\mathbf{x}_{k}^{o}=\left[x_{k},y_{k},\dot{x}_{ k},\dot{y}_{k}\right]^{T}\]  
Figure 1: The mean squared errors (MSEs) of various algorithms. CKF—cubature Kalman filter; SCKF—simplex CKF; OSCL-CKF—orthogonal simplex Chebyshev-Laguerre CKF. The system model is written as follows:  
\[\left\{ \begin{array}{c}\mathbf{x}_{k}=\mathbf{F}\mathbf{x}_{k-1}+\mathbf{v}_{k}-\mathbf{u}_{k-1,k}\\ z_{k}=h(\mathbf{x}_{k})=\arctan\frac{y_{k}}{x_{k}}+e_{k}\end{array} \right., \tag{34}\]  
where \(\mathbf{u}_{k-1,k}\) is the deterministic input and \(\mathbf{F}\) is the state transition matrix, which can be given by:  
\[\mathbf{u}_{k-1,k}=\left[ \begin{array}{c}x_{k}^{o}-x_{k-1}^{o}-\Delta\dot{x}_{k- 1}^{o}\\ y_{k}^{o}-y_{k-1,-o}^{o}-\Delta\dot{y}_{k-1}^{o}\\ \dot{x}_{k}-\dot{x}_{k-1}^{o}\\ \dot{y}_{k}^{o}-\dot{y}_{k-1}^{o}\end{array} \right],\,\mathbf{F}= \begin{array}{ cccc}1&0&\Delta&0\\ 0&1&0&\Delta\\ 0&0&1&0\\ 0&0&0&1\end{array} \]  
where \(\Delta=1\)min is the measurement interval. The standard deviation of measurement noise \(e_{k}\) is \(\sigma_{e}\), and \(\
u_{k}\) is the process noise vector with a covariance matrix of:  
\[\mathbf{Q}=\mathbf{\Gamma}^{T}\mathbf{\Gamma}\cdot q\]  
where \(q=10^{-10}\) km\({}^{2}\)/s\({}^{3}\) represents the intensity of the process noise, and \(\mathbf{\Gamma}\) is the noise drive matrix:  
\[\mathbf{\Gamma}=\left[ \begin{array}{cc}\Delta^{2}/2&0\\ 0&\Delta^{2}/2\\ \Delta&0\\ 0&\Delta\end{array} \right]\]  
A single observer is assumed to be maneuvering, where the start coordinate is (0,0). From 0 min to 12 min, the observer is in uniform linear motion, where the speed is 3 knots and the course is 2.44 rad (\"rad\"denotes radian). From 13 min to 17 min, the observer is in uniform turning motion, where the speed is 3 knots and the turn rate is 0.52 rad. From 18 min to 30 min, the observer is in uniform linear motion, where the speed is 3 knots and the course is 5.06 rad. The root mean square error in position (RMSEpos) at time index \(k\) is defined as follows:  
\[\text{RMSE}_{pos,k}=\sqrt{\frac{1}{L}\sum_{j=1}^{L}\Bigl{[}\left(\dot{x}_{k,j} ^{t}-x_{k,j}^{t}\right)^{2}+\left(\dot{y}_{k,j}^{t}-y_{k,j}^{t}\right)^{2} \Bigr{]}}, \tag{35}\]  
where \(L=100\). Similarly, the RMSE in velocity (RMSEvel) can be easily written according to the RMSEpos. The mean squared error in position (MSEpos) for each Monte-Carlo run is defined as follows:  
\[\text{MSE}_{pos,i}=\frac{1}{n}\sum_{k=1}^{n}\biggl{[}\left(x_{k}^{t}-\dot{x}_ {k|i,i}^{t}\right)^{2}+\left(\dot{y}_{k}^{t}-\dot{y}_{k|k,i}^{t}\right)^{2} \biggr{]} \tag{36}\]  
Figure 2: Model of bearings-only tracking. In the Monte-Carlo test, the target begins at the same state every time for a fair comparison. The scale parameter \(\kappa\) of the UKF is set as \(3-n_{x}\), and all the algorithms are set at the same initial conditions. The initial range estimate follows \(\overline{r}\sim N\big{(}r,\sigma_{r}^{2}\big{)}\), where \(r\) is the true range.