# A Multiple-Step, Randomly Delayed, Robust Cubature Kalman Filter for Spacecraft-Relative Navigation
## 6 Numerical Simulations
### Experimental Scenario Settings  
The installation positions of the optical beacons under the chief spacecraft body coordinate frame are shown in Table 1. Table 2 lists the initial orbital elements of the chief spacecraft. Table 3 summarizes the relevant parameters of the simulation. This paper  
Figure 6: Vision-based navigation system.  
assumes that the chief spacecraft gyro drift is relatively small, or it has been corrected by autonomous navigation. Therefore, chief spacecraft gyro drift \(\mathbf{\beta}_{c0}=0\) is adopted. The nominal trajectory of the chief spacecraft is depicted in Figure 7.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Beacon No.** & **x (m)** & **y (m)** & **z (m)** \\ \hline
1 & 0.5 & 0.5 & 0.0 \\
2 & \(-\)0.5 & 0.5 & 0.0 \\
3 & 0.5 & \(-\)0.5 & 0.0 \\
4 & \(-\)0.5 & \(-\)0.5 & 0.0 \\
5 & 0.2 & 0.5 & 0.1 \\
6 & 0.1 & 0.2 & \(-\)0.1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Installation positions of beacons.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Orbital Elements** & **Corresponding Value** \\ \hline Semi-major axis \(a\) & 26,555.137 km \\ Eccentricity \(e\) & 0.7395 \\ Orbit inclination \(i\) & 63.465\({}^{\circ}\) \\ Argument of perigee \(\omega\) & 274.163\({}^{\circ}\) \\ Right ascension of the ascending node \(\Omega\) & 115.024\({}^{\circ}\) \\ True anomaly \(v\) & 23.612\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Initial orbital elements of the chief spacecraft.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Corresponding Value** \\ \hline Number of Monte Carlo simulations & 100 \\ Discrete sampling period & 0.1 s \\ The update interval of camera & 0.2 s \\ Simulation time & 600 s \\ Perturbing parameter & 0.05 \\ Tuning parameters of kernel & 5 \\ Number of delay steps & 3 \\ Delay probability & 0.1 \\ Delay probability for each step & \(p_{0}=0.9,p_{1}=0.09,p_{2}=0.009,p_{3}=0.001\) \\ Initial relative position & \(\mathbf{\rho}_{0}=[0,-27.444,0]^{T}\) (m) \\ Initial relative velocity & \(\mathbf{v}_{0}=\left[0,-6.340\times 10^{-3},0\right]^{T}\)(m/s) \\ Initial attitude quaternion of chief spacecraft & \(\mathbf{q}_{c/L}=[0.0086,0.0086,0.0086,0.9999]^{T}\) \\ Initial relative attitude quaternion & \(\mathbf{q}_{d/c}=[0.0433,0.0142,0.0256,0.9986]^{T}\) \\ Initial generalized Rodrigues parameters & \(\mathbf{\delta}\mathbf{p}_{0}=[0,0,0)^{T}\) \\ Chief spacecraft angular velocity & \(\mathbf{\omega}_{c,0}=[0.0013,0.0013,0.0013]^{T}\)(rad/s) \\ Deputy spacecraft angular velocity & \(\mathbf{\omega}_{d,0}=[0.0020,0.0020,0.0020]^{T}\)(rad/s) \\ Gyro drift & \(\mathbf{\beta}_{d0}=[1,1,1]^{T}(^{\circ}/\text{h})\) \\ Angle random walk & \(\sigma_{u}=\sqrt{10}\times 10^{-8}\)(rad/s\({}^{1/2}\)) \\ Angular rate random walk & \(\sigma_{v}=\sqrt{10}\times 10^{-10}\)(rad/s\({}^{3/2}\)) \\ Power spectral density of perturbation acceleration & \(q_{\omega}=\sqrt{10}\times 10^{-6}\)\(\left(\left(\text{m}/\text{s}^{2}\right)^{2}/\text{Hz}\right)\) \\ Process noise covariance matrix & \(\mathbf{Q}_{k}=diag\left[\left(1^{\circ}\right)^{2}\mathbf{I}_{3\times 3},\sigma_{2}^{2} \mathbf{I}_{3\times 3},0_{3\times 3},q_{\omega}^{2}\mathbf{I}_{3\times 3}\right]\) \\ Initial state covariance matrix & \(\mathbf{P}_{0}=diag\left[\left(1^{\circ}\right)^{2}\mathbf{I}_{3\times 3},\left(1 ^{\circ}/\text{h}\right)^{2}\mathbf{I}_{3\times 3},\left(1\text{m}\right)^{2}\mathbf{I}_{3 \times 3},\left(1\text{m}/\text{s}^{2}\right)\mathbf{I}_{3\times 3}\right]\) \\ Covariance matrix of measurement noise & \(\mathbf{R}_{k}=(1.8\prime^{\prime\prime})^{2}\mathbf{I}_{12\times 12}\) \\ Covariance matrix of contaminated measurement noise & \(\mathbf{R}_{con}=7.5\mathbf{R}_{k}\) \\ Initial state vector true value & \(\mathbf{x}_{0}=\left[\delta p_{0}^{T},\mathbf{\beta}_{T0}^{T},p_{0}^{T},v_{0}^{T}\right]^ {T}\) \\ Initial state vector estimate & \(\mathbf{x}_{0}=\mathbf{x}_{0}+\sqrt{\mathbf{P}_{0}}\cdot\mathbf{N}_{12\times 1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulation parameters.