## 3 Link Cost Metric With Reliability Assurance
### Satellite Mobility  
First, on the basis of Assumption (4) in Section 2, each satellite in the SCN has a network position matrix \(V_{[N,3]}\) and its corresponding velocity matrix \(U_{[N,3]}\) in ECI at time \(t\). Then, in order to describe the mobility of satellites in detail, it is necessary to change positions and velocity in ECI to a relative coordinate system and obtain the relative position matrix \(V^{r}\) and relative velocity matrix \(U^{r}\), as shown in Figure 4, where the relative position and velocity matrix \([V^{r},U^{r}]\) meet the following equation:  
\[[V^{r},U^{r}]^{{}^{\prime}}=([V^{r}_{[i,3]},U^{r}_{[i,3]}]^{{}^{\prime}}-[V^{ r}_{[j,3]},U^{r}_{[j,3]}]^{{}^{\prime}})A \tag{3}\]  
where \(A\) is the coordinate transformation matrix [35].  
Let the time of \(V^{r}\) and \(U^{r}\) be the initial time, i.e., \(t=0\), \(V^{r}=[x_{0},y_{0},z_{0}]\), \(U^{r}=[\dot{x}_{0},\dot{y}_{0},\dot{z}_{0}]\). According to Assumption (5), satellites run in a near-circular orbit and ignore any disturbance. Then, from the literature [27], for any two satellites, \(i\) and \(j\), the relative position from \(i\) to \(j\) meets the following equations:  
\[\left\{\begin{aligned} & x(t)=-(3x_{0}+2\dot{y}_{0}/\omega)\cos \omega t+\dot{x}_{0}/\omega\sin\omega t+2(2x_{0}+2\dot{y}_{0}/\omega)\\ & y(t)=2(3x_{0}+2\dot{y}_{0}/\omega)\sin\omega t+2\dot{x}_{0}/ \omega\cos\omega t+y_{0}-2\dot{x}_{0}/\omega-3(\dot{y}_{0}+2\omega x_{0})t\\ & z(t)=z_{0}\cos\omega t+\dot{z}_{0}/\omega\sin\omega t\end{aligned}\right. \tag{4}\]  
where:  
\[\omega=\sqrt{\mu a_{j}(1-e_{j})}/|V_{[j,3]}| \tag{5}\]  
Figure 4: Illustration of transforming position and velocity in ECI to the relative coordinate system of satellite \(j\).  
and \(\omega\) is the orbit instantaneous angular rate of \(j\), \([x_{0}\,y_{0}\,z_{0}\,\dot{x}_{0}\,\dot{y}_{0}\,\dot{z}_{0}]\) is the initial relative position and velocity for \(i\) to \(j\), \(|V_{[j,3]}|=\sqrt{x_{j}^{2}+y_{j}^{2}+z_{j}^{2}}\) is the geocentric distance of \(j\), \(\mu\) is the standard gravitational parameter of the Earth, a constant value of \(3.986\times 10^{14}\) m\({}^{3}\)/s\({}^{2}\), and \(a_{j}\), \(e_{j}\) is the orbit's semi-major axis and eccentricity of \(j\), respectively.  
Additionally, from the Definition 1 of SCNs, satellites in SCNs must runs with others in a bounded range, and this means that the equations above do not have the offset of time [27], i.e., \(-3(\dot{y}_{0}+2\omega x_{0})t=0\Rightarrow\dot{y}_{0}=-2\omega x_{0}\). Then, Equation (4) can be simplified as:  
\[\left\{\begin{aligned} x(t)&=-(3x_{0}+2\dot{y}_{0}/ \omega)\cos\omega t+\dot{x}_{0}/\omega\sin\omega t\\ y(t)&=2(3x_{0}+2\dot{y}_{0}/\omega)\sin\omega t+2 \dot{x}_{0}/\omega\cos\omega t+y_{0}-2\dot{x}_{0}/\omega\\ z(t)&=z_{0}\cos\omega t+\dot{z}_{0}/\omega\sin\omega t \end{aligned}\right. \tag{6}\]  
Thus, the relative motion of every two satellites in the SCN can be defined as Equation (6), which can be used to predict any other satellites' positions, since each satellite can obtain all others' positions through cyclic navigation information sharing.  
Meanwhile, by the analysis of Equation (6), the period \(T\) of \(j\) orbiting around can be calculated as:  
\[T_{ij} =2\pi/\omega \tag{7}\] \[=2\pi|V_{[j,3]}|/\sqrt{\mu a_{j}(1-e_{j}^{2})}\]  
In Equation (7), both \(a_{j}\) and \(e_{j}\) can be regarded as constants for the reference satellite \(j\), since the satellites orbit without any orbit control and any perturbations. Thus, the period \(T\) of relative motion is fixed for every two satellites, and also in the relative coordinate system of satellite \(j\), as other satellites' periods of flying around \(j\) are equivalent.