# Mission-Oriented Real-Time Health Assessments of Microsatellite Swarm Orbits
## 3 Real-Time Health Assessment Algorithm
### Single Satellite Health Assessment Algorithm  
For the health assessment of a single satellite in a swarm system, it is necessary to first consider the health of the single satellite itself, and then the degree of its impact on the system performance. The former uses a reliability analysis [15] to model the health of single satellites; the latter uses the method of topological node importance [16] to describe the degree of association between single satellite and the health of the swarm. For a steady swarm topology diagram containing \(N\) satellites and \(M\) communication links given by \(G(t)\), the following definitions are given. **Definition 1**. : _Interstellar distance \(d_{ij}\): The shortest path between satellite \(v_{i}\) and satellite \(v_{j}\). When there is no link between \(v_{i}\) and \(v_{j}\), \(d_{ij}\rightarrow\infty\)._ \begin{table}
\begin{tabular}{c c c} \hline \hline
**Health Status Level** & **Healthiness Interval** & **Description** \\ \hline \multirow{6}{*}{Health (H)} & & The healthiness of the evaluation object is in the safe range, i.e., the single satellite can operate normally and stably/the \\  & [0.8000, 1.0000] & intersatellite communication is \\  & & smooth/the swarm function is normal, \\  & & and it is in a stable state, and the \\  & & probability of failure is small. \\  & & The health of the evaluated object reaches \\  & & the boundary of the safety range, and the \\  & & performance has a tendency to degrade, \\  & & but it does not considerably affect the \\  & & overall mission or function with a higher \\  & & probability of failure. \\  & & The health of the evaluation object is \\  & & considerably below the safety range with \\  & & abnormal function and in-orbit mission \\  & & incompletion, and failure occurs. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Health status level. **Definition 2**. : _Satellite tightness \(C_{i}\): The reciprocal of the sum of the distances of satellite \(v_{i}\) to all other satellites [15]._  
\[C_{i}=\frac{N-1}{\sum\limits_{j=1,j\
eq i}^{N}d_{ij}} \tag{3}\]  
The satellite tightness evaluates the satellite importance from a system-wide perspective, which reflects the ease of access to other satellites in the swarm system, and the satellite importance increases with the increasing tightness. **Definition 3**. : _Satellite mesonumber \(B_{i}\): The ratio of the number of satellites passing through the shortest path of all pairs of satellites in the swarm topology [15]._  
\[B_{i}=\frac{\frac{2\sum\limits_{i=1}^{N}\frac{N}{N}\frac{g_{jk}(i)}{S_{jk}}}{ \frac{N}{N}N(N-1)}}{j\
eq k\
eq i,j<k} \tag{4}\]  
_where \(g_{jk}\) is the number of shortest paths between satellite \(v_{j}\) and \(v_{k}\), \(g_{jk}(i)\) is the number of passing satellites \(v_{i}\) in \(g_{jk}\), and \(g_{jk}(i)/g_{jk}\) is the contribution of \(v_{i}\) to the number of satellites \(v_{j}\) and \(v_{k}\) mediators, which can reflect the satellite position information._  
**Definition 4**. : _Swarm importance contribution matrix (SICM) [15]._  
\[SICM=\left[ \begin{array}{cccc}1&\frac{C_{2}}{d_{12}}&\ldots&\frac{C_{N}}{d_ {N1}}\\ \frac{C_{1}}{d_{21}}&1&\ldots&\frac{C_{N}}{d_{2N}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{C_{1}}{d_{N1}}&\frac{C_{2}}{d_{N2}}&\ldots&1\\ \end{array} \right] \tag{5}\]  
_where element 1 on the diagonal of the matrix represents the value of the satellite's contribution to its importance; element \(C_{j}/d_{ij}\) represents the value of the \(v_{i}\) satellite's contribution to the importance of other satellites in the swarm topology._  
**Definition 5**. : _The importance \(I(v_{i})\) of satellite \(v_{i}\): depends on the location information of the satellite and its correlation with other satellites; thus, the satellite importance model consists of the satellite mesonumber \(B_{i}\) and the satellite \(v_{i}\) contribution \(C_{j}/d_{ij}\) together._  
\[I(v_{i})=B_{i}\times\sum\limits_{j=1,j\
eq i}^{N}\frac{B_{j}C_{j}}{d_{ij}} \tag{6}\]  
The normalization process leads to the following values of satellite importance weights.