## 3 The Agent-Weighted RLS Positioning Framework  
This paper proposes an Agent-Weighted RLS Positioning Framework (AWR-PF) for LEO SOP positioning to achieve real-time epoch dynamic weight decision-making and establish temporal dependencies between epochs. The innovation lies in designing a neural network-based agent that comprehensively evaluates each epoch in real-time based on multiple characteristics (ephemeris age, elevation angle, azimuth angle, satellite-to-receiver distance, SNR, Doppler frequency, and epoch rank within its satellite). The agent additionally incorporates current observation matrix GDOP information. The evaluation result is expressed as a dynamically generated weight, which participates in RLS iteration to enhance positioning accuracy. To train the agent, we formulated an MDP model including state, environment, action, reward, and state transition process. This MDP model integrates all Iridium epoch characteristics and iteratively updates based on selected action. Using the DDQN-EW algorithm, we trained the agent on an extensive real-world dataset comprising 107 h of continuous observations, ensuring full 24 h coverage and representative sampling of all characteristic Iridium signal conditions. The trained agent effectively learns epoch credibility evaluation across dynamic scenarios, enabling comprehensive and robust positioning accuracy optimization. As demonstrated in Figure 5, the framework initially computes the preliminary position estimation, observation matrix, and covariance matrix using epochs accumulated during an initialization period. Then it performs epoch-wise recursive estimation, where an agent dynamically weights each \(n\)-th epoch based on its multi-dimensional features and current matrix conditions, and subsequently updates the position estimate and observation matrix through weighted RLS (Algorithm 1) until all epochs are processed to obtain the final position estimation. Figure 4: The simulation results of temporal evolution of both Doppler frequency and Doppler rate. (**a**) Doppler frequency and Doppler rate as functions of time; (**b**) the simulated sky map of temporal evolution. ```
Input: Initial position, LEO satellite epochs. Output: Recursive position estimation. 1 //Initialize via LSM (LSM denotes least squares method):
2 Construct observation equation \(\mathbf{G}_{0}\). //Equations (11) and (12)
3 Compute initial position estimation \(\overset{\wedge}{\mathbf{x}}_{0}\). //Equation (13)
4 Compute initial covariance matrix \(\mathbf{P}_{0}\). //Equation (14)
5 //Iterations:
6 foreach new epoch \(n\)do
7 Obtain the weight \(w_{n}\) through the agent. 8 Obtain new observation \(\Delta\Delta F_{n}\) and observation matrix \(\Delta\mathbf{G}_{n}\). 9 //Update covariance matrix and GDOP. 10 \(\mathbf{P}_{n}=\mathbf{P}_{n-1}-\mathbf{P}_{n-1}\cdot\Delta\mathbf{G}_{n}^{T} \left(\mathbf{I}\cdot\frac{1}{w_{n}}+\Delta\mathbf{G}_{n}\cdot\mathbf{P}_{n-1} \cdot\Delta\mathbf{G}_{n}^{T}\right)^{-1}\cdot\Delta\mathbf{G}_{n}\cdot \mathbf{P}_{n-1}\). 11 \(GDOP_{n}=\sqrt{\text{Trace}(\mathbf{P}_{n})}\). 12 //Compute gain matrix of this epoch. 13 \(\mathbf{K}_{n}=\mathbf{P}_{n}\cdot\Delta\mathbf{G}_{n}^{T}\). 14 //Update position estimation. 15 \(\overset{\wedge}{\mathbf{x}}_{n}=\overset{\wedge}{\Delta\mathbf{x}}_{n-1}+w_{ n}\cdot\mathbf{K}_{n}\left[\Delta\mathbf{G}_{n}^{T}\cdot\Delta\Delta F_{n}- \Delta\mathbf{G}_{n}\cdot\overset{\wedge}{\Delta\mathbf{x}}_{n-1}\right]\). 16
17 endfor
17 return the final position estimation. ```  
**Algorithm 1** Agent-Weighted RLS Positioning Framework  
Before applying the RLS for position estimation, an initial position estimate is required. This is to ensure that the recursion starts with a reasonable initial state, allowing it to recursively update the target position as new observations become available. In this article, a small number of observations are first processed using the least squares method (LSM) to obtain an initial positioning result.