## 3 Time-Frequency Compensation and Measurement Model
### Unidirectional Orbit-Determination Motion and Observation Equation  
After the extended user completes link establishment using a limited number of inter-satellite link time slots, the unidirectional measurements are initialized and processed in batch, state and observation equations are defined, and the parameters to be estimated are calculated and corrected. The process iterates until convergence criteria are met, enabling the determination of satellite orbit and clock bias parameters. The algorithm flow is shown in Figure 3[24; 25; 26]. The geocentric inertial coordinate system is an inertial coordinate system that remains stationary in inertial space. The origin of the coordinate system is the center of the Earth; the fundamental plane is the equatorial plane of the Earth; the Z-axis is the axis of rotation perpendicular to the equatorial plane, pointing to the north pole of the Earth; the X-axis points to the vernal equinox; the Y-axis is perpendicular to the X-axis; and X, Y and Z follow the right-hand spiral rule. According to [PERSON]'s first law, the mass of a satellite can be considered negligible compared to the mass of the Earth. Assuming that the Earth is a uniform sphere with an orbital eccentricity of 0, the acceleration of a satellite can be given by [PERSON]'s law of universal gravitation. The motion equation of the satellite in the Earth-centered inertial (ECI) frame is as follows [27; 28; 29]:  
\[\left\{ \begin{array}{l}\dot{\vec{R}}=\vec{V},\\ \dot{\vec{V}}=-\frac{GM_{\oplus}}{R^{2}}\frac{\vec{R}}{R}+\vec{P}\left(\vec{R},\vec{V},\vec{P}_{d},t\right)\equiv\vec{A},\\ \vec{R}(t_{0})=\vec{R}_{0},\dot{\vec{R}}(t_{0})=\vec{V}(t_{0}).\end{array} \right. \tag{7}\]  
The Equation (7) is as above, where \(\vec{R}\) denotes the satellite's center-of-mass position vector; \(\vec{V}\) denotes the satellite's center-of-mass velocity vector; \(\vec{P}\) denotes the perturbing acceleration acting on the satellite; \(\vec{A}\) denotes the total acceleration acting on the satellite; \(\vec{P}_{d}\) denotes the vector of dynamics parameters to be estimated, which may include the solar radiation pressure coefficient, etc. ; and \(GM_{\oplus}\) denotes the Earth's gravitational constant. Parameters that do not appear in the satellite's motion equation are referred to as geometric parameters, denoted as \(\vec{P}\). Definition of the state vector:  
\[X=\left[ \begin{array}{cccc}\vec{R}^{T}&:\vec{V}^{T}&:\vec{P}_{d}^{T}&:\vec{ P}_{g}^{T}\end{array} \right]^{T}. \tag{8}\]At the time of observation \(t\), the state variable \(X\) can be expressed as a differential equation with respect to time. The motion Equation (7) can be rewritten as the state equation:  
\[\left\{ \begin{array}{c}\dot{X}=F(X,t)=\left[\ddot{V}\dddot{A}0\right],\\ X(t_{0})=X_{0}.\end{array} \right. \tag{9}\]  
This is the dynamic equation of the satellite, which is established based on [PERSON]'s second law, where \(V\) is the velocity vector of the satellite's center of mass; \(A\) is the total acceleration experienced by the satellite. The actual state vector \(X\) of the satellite cannot be known precisely in advance and can only be obtained as a reference value \(X^{*}\). It needs to be continuously optimized using the constraints from the measurements, making it as close as possible to the true value. The  
Figure 3: Extended user unidirectional orbit-determination algorithm flow. observation quantity for unidirectional ranging is the distance \(\rho_{0}\) from the satellite to the BeiDou satellite, and its calculated value is as follows:  
\[\rho_{C}=\sqrt{\left(\overset{\rightharpoonup}{r}-\overset{\rightharpoonup}{r} ^{sta}\right)^{T}\left(\overset{\rightharpoonup}{r}-\overset{\rightharpoonup}{r} ^{sta}\right)}. \tag{10}\]  
In Equation (10), \(\overset{\rightharpoonup}{r}\) and \(\overset{\rightharpoonup}{r}^{sta}\) represent the position vectors of the BeiDou satellite and the satellite in the Earth-fixed coordinate system, respectively.