## 3 Methodology of Adaptive Gaussian Mixture Model Using Virtual Sample Generation
### Virtual Sample Generation-Based Propagation Strategy
Therefore, tofurther reduce the expense, in this paper, up to \(n_{add}\) samples, with the larger variances, are added into the sample set at each iteration, for constructing the KRG model, where \(n_{add}\) is the user-defined parameter. In this paper, the KRG model is constructed using MATLAB toolbox SURROGATES. The procedure of this strategy is exhibited in detail as follows:  
_Step 1_: The parameters are determined, including the total number of cheap points \(n_{c}\), the number of superior cheap points added at each iteration \(n_{add}\), the maximal number of real samples \(n_{r}\), and the number of initial real samples \(n_{i}\). _Step 2_: Generate \(n_{c}\) points according to the current PDF \(p(\mathbf{x},t)\) as the candidate points set; denote as \(S_{c}^{0}=\{\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{n_{c}}\}\). Select \(n_{i}\) points from the candidate points set \(S_{c}^{0}\), note as \(S_{s}=\left\{\mathbf{x}_{i_{1}},\mathbf{x}_{i_{2}},\cdots,\mathbf{x}_{i_{n_{i}}}\right\}\). Evaluate the points in \(S_{s}\) using black-box dynamical model (2), and then responses are obtained. All of these \(n_{i}\) points and their corresponding responses make up the sample set. Construct an RBF model based on the existing samples. Renew the candidate points set as \(S_{c}^{1}=\left\{\mathbf{x}\mid\mathbf{x}\in S_{c}^{0}\text{ and }\mathbf{x}\
otin S_{s}\right\}\). The iteration counter \(k\) is set to be one, i.e., \(k\gets 1\). The current number of real samples \(n_{s}\) is set to be \(n_{s}=n_{i}\). _Step 3_: Evaluate all the points in candidate points set \(S_{c}^{k}\) using the KRG model. Sort these points according to their predicted variances. The top \(n_{add}\) superior points are selected, noted as \(S_{add}^{k}\). Evaluate these points and then add them to the sample set. Construct the RBF model. Remove the selected superior points from the candidate points set as \(S_{c}^{k+1}=\left\{\mathbf{x}\mid\mathbf{x}\in S_{c}^{k}\text{ and }\mathbf{x}\
otin S_{s}^{k}\right\}\), \(k\gets k+1\), \(n_{s}\gets n_{s}+n_{add}\). _Step 4_: If \(n_{s}<n_{r}\), return to _Step 3_; elsewise jump to _Step 5_. _Step 5_: Use the virtual sample evaluated by the KRG model to propagate. To further demonstrate the process of active sampling based KRG model, a one-dimensional numerical benchmark is tested. The benchmark describes the mapping relation between the final position error along the \(x\)-axis and the initial velocity error along the \(y\)-axis, which is given as:  
\[g\big{(}\Delta v_{y}(t_{0})\big{)}=\Delta r_{x}\Big{(}t_{f}\Big{)} \tag{24}\]  
The spacecraft moves on the equatorial plane and the dynamics considered the Earth's gravity and atmospheric drag, which is governed by [42]:  
\[ \begin{split}\mathbf{x}&=\left[\mathbf{r}^{T},\mathbf{v}^{T} \right]^{T}=\left[r_{x},r_{y},v_{x},v_{y}\right]^{T}\\ \dot{\mathbf{x}}&=\mathbf{f}(\mathbf{x})=\left[v_{x},v_{y},-\mu r _{x}r^{-3}-\rho(h)\beta v_{\text{rel}}\,v_{\text{rel},x}/2,-\mu r_{y}r^{-3}- \rho(h)\beta v_{\text{rel}}\,v_{\text{rel},y}/2\right]^{T}\end{split}  \tag{25}\]  
where \(r=\sqrt{r_{x}^{2}+r_{y}^{2}}\), \(\mathcal{V}_{rel,x}=\mathcal{V}_{x}-\omega r_{y}\), \(\mathcal{V}_{rel,y}=\mathcal{V}_{y}-\omega r_{x}\), and \(\omega\) is the angular velocity of the Earth. \(\rho(h)\) is the atmospheric density as a function of the altitude of the object. Additionally, it is assumed that the atmospheric density is described by an exponential atmosphere model via:  
\[\rho(h)=\rho_{0}\exp\{-(h-h_{0})/h_{s}\} \tag{26}\]  
with \(\rho_{0}=3.614\times 10^{-13}\) kg/m\({}^{3}\), \(h_{0}=700\) km and \(h_{s}=88.667\) km. \(\beta\) is the ballistic coefficient of the object, and it is taken to be 1.4 in this paper. The elements of the nominal orbit are illustrated in Table 3, and the propagation time is set to be one nominal orbit period, which is about 5331.288 s. The points are set to be evenly distributed in range.