# A Multiple-Step, Randomly Delayed, Robust Cubature Kalman Filter for Spacecraft-Relative Navigation
## 3 Brief Review of DCS Kernel  
The DCS approach originates from the area of visual simultaneous localization and mapping (SLAM), and its core is a robust function, namely, the dynamic-covariance-scaling kernel. The DCS kernel enables L2 estimation to be optimal under Gaussian environment, and can completely eliminate the effect of relatively large residuals on the estimation results under non-Gaussian environment. Therefore, the DCS kernel is more robust than  
Figure 1: Schematic diagram of the system with multiple-step, randomly delayed measurements.  
Figure 2: The measurement noise of the two distributions.  
the Huber kernel and can reduce the loss of measurement information under Gaussian noise, as compared to the Gaussian kernel [34].  
Two highly efficient approaches, switchable constraint (SC) and dynamic covariance scaling (DCS), have been been rapidly developed in SLAM to handle outliers in images. [PERSON] [35] has derived the specific form of the cost function of SC, as follows:  
\[\rho(\xi)=s^{2}\xi^{2}+\eta(1-s)^{2} \tag{8}\]  
where \(s\) is the switchable variable, \(\eta>0\) is the kernel width, and \(\xi\) is the residual error.  
When the function \(\rho(\cdot)\) is continuous and bounded, the derivative of Equation (8) concerning \(s\) should satisfy the following conditions:  
\[\frac{\partial\rho(\xi)}{\partial s}=2s\xi^{2}-2\eta(1-s)=0 \tag{9}\]  
\[s=\frac{\eta}{\eta+\xi^{2}} \tag{10}\]  
By bringing Equation (10) into Equation (8), the specific form of the cost function of SC could be derived, as follows:  
\[\rho(\xi)=\left(\frac{\eta}{\eta+\xi^{2}}\right)^{2}\xi^{2}+\eta\left(1-\frac{ \eta}{\eta+\xi^{2}}\right)^{2}=\frac{\eta\xi^{2}}{\eta+\xi^{2}} \tag{11}\]  
The limit value of \(\rho(\cdot)\) at \(\pm\infty\) is determined by the following:  
\[\lim_{\xi\rightarrow\pm\infty}\rho(\xi)=\lim_{\xi\rightarrow\pm\infty}\frac{ \eta}{\eta/\xi^{2}+1}=\eta \tag{12}\]  
According to Equation (12), \(\eta\) is the upper bound of \(\rho(\cdot)\). We could determine the range of \(s\).  
\[s\in\left[0,\frac{2\eta}{\eta+\xi^{2}}\right] \tag{13}\]  
[PERSON] [36] suggested that the range of values for \(s\) is [0,1], and when this was combined with Equation (13), it resulted in the following:  
\[s=\min\left(1,\frac{2\eta}{\eta+\xi^{2}}\right) \tag{14}\]  
[PERSON] [35] suggested that \(s^{2}\) could be used as a form of weight function for M-estimation, and then the DCS weight function could be calculated, as follows:  
\[\psi_{D}(\xi)=s^{2}=\min\left(1,\frac{4\eta^{2}}{\left(\eta+\xi^{2}\right)^{2 }}\right)=\left\{\begin{array}{ll}1,&\text{for }\xi^{2}<\eta\\ \frac{4\eta^{2}}{\left(\eta+\xi^{2}\right)^{2}},&\text{for }\xi^{2}\geq\eta \end{array}\right. \tag{15}\]  
By integrating Equation (15), the cost function of DCS is as follows:  
\[\rho_{D}(\xi)=\int\xi\psi_{D}(\xi)d\xi=\left\{\begin{array}{ll}\frac{\xi^{2 }}{2},&\text{for }\xi^{2}<\eta\\ \frac{\eta\left(3\xi^{2}-\eta\right)}{2\left(\xi^{2}+\eta\right)},&\text{for }\xi^{2}\geq\eta \end{array}\right. \tag{16}\]  
The cost function and the weight function of a DCS kernel are shown in Figure 3.