## 5 Test and Simulation
### Hardware-in-the-Loop Simulation for SFF Relative Navigation
The performance in this situation was not good, as the estimation error was divergent gradually over time, especially during perigee periods (bolded red lines in Figure 11), and converged slightly near apogee (blue lines in Figure 11). The range estimation error was up to 80 cm around perigee, and reduced to about 10 cm during apogee flight. The range rate error achieved a maximum of 1 mm/s during perigee, and 0.03 mm/s during apogee. Similar estimation performance of elevation, azimuth angle, and their rates was obtained, which will be provided later in detail. Clearly, this is not an optimal result for engineering application. The reason for the filter divergence is that the formation dynamic model used in the filter (i.e., radar model) is not sufficiently accurate for the prediction of states through recursive calculation at each sampling time. The process noise values of \(w_{x},w_{y},w_{z}\) change consistently according to the orbit perturbations, and need to be adjusted through an adaptive approach from measurement updates. For the purpose of comparison, the data analyzed in Figure 11 are redrawn in Figures 12 and 13 with new results. The green line in Figure 12 demonstrates the relative ranging errors while using the adaptive filter introduced in Section 4, from BeiDou/SBR equipment. It can be seen that the estimation accuracy notably improved during both perigee and apogee orbit time, as a result of using the process noise adaptive filter algorithm. The range estimation error was  
Figure 10: Block diagram of the simulation platform, mainly including a high-precision 6-DOF moving platform, optical/S-band ranging (SBR)/MWR/BeiDou payloads, a SPIRENT GSS9000 simulator, central control computer, and standard CAN bus. less than 10 cm around perigee, and reduced to 1 cm during apogee. The range rate estimation errors were also significantly reduced, as can be seen in Figure 13, achieving less than 15 \(\upmu\)m/s around apogee. Figure 11: Range and range rate estimation errors from BeiDou/SBR equipment. The red lines denote the estimation errors, and the short blue lines denote the same data during apogee time. KF: traditional Kalman filter. Figure 12: Relative range estimation errors from different payloads and filters. The red line shows the relative range estimation errors of BeiDou/SBR payload using the traditional Kalman filter algorithm (the same data as during apogee time in the blue lines); the green line represents the estimation errors of the BeiDou/SBR payload using the adaptive filter (AF); the black lines show the estimation errors of the MWR payload using the adaptive filter during apogee time. However, centimeter-level ranging accuracy is not sufficient for the real HEFFD mission. The black lines in Figure 12 and 13 illustrate the MWR results using the adaptive approach during apogee periods in the simulation. With data collection, statistics, and analysis after simulation, the final accuracy of MWR was 35 \(\upmu\)m in range and 8.5 \(\upmu\)m/s in range rate. Table 4 provides the maximum estimation error and root mean square (RMS) error during three orbiting periods, for the full states, using different relative filter algorithms and ranging equipment. The results clearly demonstrate the effectiveness of an adaptive filter that incorporates process noise uncertainty, and submillimeter-level ranging accuracy for formation flight in HEO using MWR technology.