## 3 Simulations and Results
### Orbit Determination Simulation
#### 3.2.1 Effects of the Measurement Errors  
In this study, the impact of measurement errors is reflected in two main aspects. First, the simulated observations are generated by intentionally introducing measurement errors. Second, this noise is used to calculate the covariances in Equation (15), which subsequently affects the performance of the filter. The magnitude of initial state errors caused by different levels of measurement noise is estimated first. Measurement noise is applied at seven levels. The standard deviations for range \(\rho\), azimuth \(Az\), and elevation \(El\) are set as follows: [50 m, 50 arcsec, 50 arcsec] for Level-1; [100 m, 100 arcsec, 100 arcsec] for Level-2; [200 m, 200 arcsec, 200 arcsec] for Level-3; [300 m, 300 arcsec, 300 arcsec] for Level-4; [500 m, 500 arcsec, 500 arcsec] for Level-5; and [1000 m, 1000 arcsec, 1000 arcsec] for Level-6. For every case, the Monte Carlo method is applied to analyze the distribution of errors between the estimated initial states and the true orbital value. Figure 7 shows the distribution of initial state estimation errors for 1000 simulation runs of IOD under various measurement noise levels. A clear upward trend in the median, upper, and lower quartiles can be observed as the noise level increases. This indicates that higher measurement noise not only leads to less accurate orbit initialization but also introduces greater uncertainty in the estimates. Taking into account the capabilities of modern observation technology, as well as potential differences in observational accuracy among various devices, the measurement noise of Levels 2-4 was used to simulate 3 sets of pseudo-radar observations. The initial state errors (deviations of (\(r_{\text{initial}}\), \(v_{\text{initial}}\)) from (\(r_{0}^{true}\), \(v_{0}^{true}\))) calculated values by each set of simulated observations are as follows:  
* Case 1: \(\Delta r_{0}=[-195.4272,\,915.1713,\,-364.8435]\) m, \(\Delta v_{0}=[0.4716,\,-3.7691,\,-1.9408]\);
* Case 2: \(\Delta r_{0}=[1295.5440\,\,-863.3109\,-1545.5358]\) m, \(\Delta v_{0}=[-6.1658\,3.6929\,6.2951]\);
* Case 3: \(\Delta r_{0}=[585.9460\,\,2818.3729\,-1410.7564]\) m, \(\Delta v_{0}=[2.6370\,-14.4218\,-18.6149]\). The OD algorithm was tested using numerical analysis with four different levels of measurement noise for each case: [50 m, 50 arcsec, 50 arcsec], [100 m, 100 arcsec, 100 arcsec], [300 m, 300 arcsec, 300 arcsec], and [1000 m, 1000 arcsec, 1000 arcsec]. In all tests, covariance values for the initial position and velocity uncertainties are set to [(10,000 m)\({}^{2}\), (10,000 m)\({}^{2}\), (10,000 m)\({}^{2}\), (10 m/s)\({}^{2}\), (10 m/s)\({}^{2}\)], (10 m/s)\({}^{2}\)]. As suggested in [27], for LEO mega-constellations equipped with low-thrust electric propulsion systems, the low-thrust  
Figure 7: Box chart of initial state errors caused by different levels of measurement noise: (a) Box chart of initial position error; (b) Box chart of initial velocity error. often has a magnitude of \({f_{2}}^{2}\). Based on their use of a prior thrust error, the initial covariance for the along-track acceleration component \(\alpha_{\rm T}\) is assumed to be (\(10^{-4}\) m/s\({}^{2}\))\({}^{2}\). Figure 8a-c illustrate the iterative histories of position errors and along-track thrust acceleration (\(\alpha_{\rm T}\)) estimates at the chosen epoch for Cases 1-3, respectively. The solid lines with different markers represent the position errors, while the dashed lines with corresponding markers denote the estimated values of \(\alpha_{\rm T}\). Each color is associated with a specific measurement noise level. It can be seen that all numerical test cases meet the convergence criterion within 10 iterations. Using the same set of observational data (with identical initial errors), the final position errors and acceleration estimates converge to similar values after the iterations, even when different measurement noise levels are applied during orbit determination.