# Iterated Orthogonal Simplex Cubature Kalman Filter and Its Applications in Target Tracking
## 5 Simulations and Results
### Spacecraft Target Tracking Simulation  
#### 5.2.1 Spacecraft Orbit Dynamics Model  
For spacecrafts operating in low Earth orbit, Earth's J2 term non-spherical perturbation and atmospheric drag perturbation are the most important perturbations. In the J2000 geocentric inertial coordinate system (O-XYZ), considering the above two perturbations, the orbital dynamics model of the spacecraft is as follows:  
\[\left\{\begin{array}{l}\dot{x}=v_{x}\\ \dot{y}=v_{y}\\ \dot{z}=v_{z}\\ \dot{v}_{x}=\frac{\mu x}{r^{3}}\left[I_{2}\left(\frac{R_{x}}{r}\right)\left(7.5\frac{z^{2}}{r^{2}}-1.5\right)-1\right]+f_{ax}+f_{x}\\ \dot{v}_{y}=\frac{\mu y}{r^{3}}\left[I_{2}\left(\frac{R_{x}}{r}\right)\left(7.5\frac{z^{2}}{r^{2}}-1.5\right)-1\right]+f_{ay}+f_{y}\\ \dot{v}_{z}=\frac{\mu x}{r^{3}}\left[I_{2}\left(\frac{R_{x}}{r}\right)\left(7.5\frac{z^{2}}{r^{2}}-4.5\right)-1\right]+f_{ax}+f_{z}\\ r=\sqrt{x^{2}+y^{2}+z^{2}}\end{array}\right. \tag{37}\]  
where \(X=\left(x\quad y\quad z\right)^{\text{T}}\) is the position vector of the spacecraft in O-XYZ; \(\dot{X}=\left(v_{x}\quad v_{y}\quad v_{z}\right)^{\text{T}}\) is the velocity vector of the spacecraft in O-XYZ; \(j_{2}\) is the harmonic term coefficient; \(\mu\) is the gravitational constant of the Earth; \(R_{e}\) is the radius of the Earth; and \(\left(f_{x}\quad f_{y}\quad f_{z}\right)^{\text{T}}\) is the sum of the high-order non-spherical perturbation of the Earth, three-body gravitational perturbation, and solar luminous pressure perturbation on the three coordinate axes, which can be equivalent to Gaussian white noise with a zero mean value in this study. \(f_{a}=\left(f_{ax}\quad f_{ay}\quad f_{az}\right)^{\text{T}}\) is the atmospheric drag perturbation, and its specific expression is as follows:  
\[f_{a}=-\frac{1}{2}\frac{c_{d}A}{m}\rho_{d}v_{rel}\mathbf{v}_{rel} \tag{38}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Filters** & **Position RMSE/m** & **Velocity RMSE/(m/s)** \\ \hline CKF & 2.9379 & 0.1852 \\ SCKF & 2.3277 & 0.1643 \\ OSCKF & 2.3045 & 0.1622 \\ IOSCKF & 2.2929 & 0.1619 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Average RMSE of various filters.  
Figure 1: The RMSE of various filters. (**a**) The position RMSE. (**b**) The velocity RMSE.  
where \(c_{d}\) is the atmospheric drag coefficient, \(A/m\) is the surface mass ratio of the spacecraft, \(\rho_{d}\) is the atmospheric density, and \(\mathbf{v}_{rel}\) is the relative speed between the spacecraft and the atmosphere, assuming that the atmosphere rotates with the Earth, as follows:  
\[\mathbf{v}_{rel}=\dot{\mathbf{X}}-\mathbf{\omega}\times\mathbf{X} \tag{39}\]  
where \(\mathbf{\omega}=\omega_{e}\times\begin{pmatrix}0&0&1\end{pmatrix}^{\text{T}}\) is the angular velocity vector of the Earth rotation and \(\omega_{e}\) is the angular velocity of the Earth rotation. Using the fourth-order Runge-Kutta method, Equation (32) is written in the following discrete form:  
\[\mathbf{X}_{k}=f(\mathbf{X}_{k-1})+\mathbf{w}_{k-1} \tag{40}\]  
where \(\mathbf{X}_{k}=\begin{bmatrix}x_{k}&y_{k}&z_{k}&v_{x,k}&v_{y,k}&v_{z,k}\end{bmatrix}^ {\text{T}}\) is the track state at time \(k\), and \(\mathbf{w}_{k}\) is the system noise.