# Distributed Orbit Determination for Global Navigation Satellite System with Inter-Satellite Link
## 3 Fundamental Equations for Measurement and Motion  
### Equations for Measurement  
The position vectors, \(\overrightarrow{r}^{i}=[ \begin{array}{cccc}x^{i}&y^{i}&z^{i}\end{array} ]^{T}\), velocity vectors, \(\overrightarrow{v}^{-i}=[ \begin{array}{cccc}v_{x}^{i}&v_{y}^{i}&v_{z}^{i} \end{array} ]^{T}\), and dynamic parameter vector, \(x_{D}^{i}\), constitute the state vector which needs to be estimated:  
\[\mathbf{X}^{i}=\left[\overrightarrow{r}^{i}T\overrightarrow{v}^{i}T\ x_{D}^{i}T \right]^{T} \tag{1}\]  
where the superscript \(T\) denotes the transpose of a matrix, and the superscript \(i\) denotes that it is for the _ith_ satellite. The reference values for the state vector are stored in \(\mathbf{X}^{i*}\), and the improving values for the state vector are written in:  
\[\delta\mathbf{x}^{i}=\mathbf{X}^{i}-\mathbf{X}^{i*} \tag{2}\]  
After correcting the hardware delay, ionospheric delay, relativistic effect, multi-path effect, and the antenna phase center offset [4], two one-way ISL PRs between the _ith_ satellite and _jth_ satellite need to be translated into the same measurement epoch (e.g., ranging frame epoch \(t\)). The PR equations are:  
\[\rho^{i\to j}(t)=c\cdot\left[\delta t^{j}(t)-\delta t^{i}(t)\right]+d(\mathbf{X}^ {i}(t),\mathbf{X}^{i}(t))+\varepsilon^{i\to j}(t) \tag{3}\]  
\[\rho^{j\to i}(t)=c\cdot\left[\delta t^{i}(t)-\delta t^{j}(t)\right]+d(\mathbf{X}^ {i}(t),\mathbf{X}^{i}(t))+\varepsilon^{j\to i}(t) \tag{4}\]  
where, \(c\) is the speed of light, \(\delta t^{i}(t)\) and \(\delta t^{j}(t)\) are clock errors for the _ith_ satellite and the _ith_ satellite, respectively, \(\varepsilon^{i\to j}(t)\) and \(\varepsilon^{j\to i}(t)\) are the measurement errors, and \(d(\mathbf{X}^{i}(t),\mathbf{X}^{i}(t))=\sqrt{\left(x^{i}-x^{j}\right)^{2}+\left(y^{i}- y^{j}\right)^{2}+\left(z^{i}-z^{j}\right)^{2}}\) is the geometric distance between the two satellites. Combining Equations (3) and (4), the distance measurement equation that only contains orbit parameters is derived as:  
\[\rho^{ij}(t)=\left[\rho^{j\to i}(t)+\rho^{i\to j}(t)\right]/2=d(\mathbf{X}^{i}(t),\mathbf{ X}^{i}(t))+\varepsilon^{ij}(t) \tag{5}\]  
where, \(\varepsilon^{ij}(t_{k})=[\varepsilon^{i\to j}(t_{k})+\varepsilon^{j\to i}(t_{k}) ]/2\). Subtracting Equation (4) from Equation (3), the time measurement equation that only contains clock error parameters is deduced as:  
\[\rho^{ij}_{clock}(t)=\left[\rho^{j\to i}(t)-\rho^{i\to j}(t)\right]/2=c \cdot\left[\delta t^{j}(t)-\delta t^{i}(t)\right]+\left[\varepsilon^{j\to i}(t)- \varepsilon^{i\to j}(t)\right]/2 \tag{6}\]  
Following the steps above, the distance measurements and the clock bias measurements are decoupled. The orbit ephemeris and clock offsets can therefore be estimated independently. Linearizing Equation (5) with Taylor expansion at the reference state vector \(\mathbf{X}^{i*}\) and \(\mathbf{X}^{j*}\) yields:  
\[\rho^{ij}(t)=d\left(\mathbf{X}^{i*},\mathbf{X}^{j*}\right)+\left.\frac{\rho^{ij}(t)}{ \partial\mathbf{X}^{i}}\right|_{\mathbf{X}^{i*}}\delta x^{i}+\left.\frac{\rho^{ij}(t) }{\partial\mathbf{X}^{j}}\right|_{\mathbf{X}^{j*}}\delta x^{i}+\cdots+\varepsilon^{ij} (t) \tag{7}\]  
After that, Equation (7) is converted into a linear measurement equation:  
\[z^{ij}(t)=\mathbf{H}^{i}\delta\mathbf{x}^{i}(t)+\mathbf{H}^{i}\delta\mathbf{x}^{i}(t)+ \varepsilon^{ij}(t) \tag{8}\]  
where \(z^{ij}(t)=\rho^{ij}(t)-d\left(\mathbf{X}^{i*},\mathbf{X}^{j*}\right)\) is the innovation. \(\mathbf{H}^{i}\) and \(\mathbf{H}^{i}\) are the measurement matrices. \[\mathbf{H}^{i}=\left.\frac{\rho^{ij}(t)}{\partial\mathbf{X}^{i}}\right|_{\mathbf{X}^{i*}} =\left[ \begin{array}{cc}\frac{x^{i}-x^{i}}{d\left(\mathbf{X}^{i*},\mathbf{X}^{j*} \right)}&\frac{y^{i}-y^{i}}{d\left(\mathbf{X}^{i*},\mathbf{X}^{j*}\right)}&\frac{z^{i} -z^{j}}{d\left(\mathbf{X}^{i*},\mathbf{X}^{j*}\right)}\\ \end{array}