## 5 Test and Simulation
### Hardware-in-the-Loop Simulation for SFF Relative Navigation  
A hardware-in-the-loop (HIL) simulation was conducted, with all of the deployed payloads fixed in the 6-DOF platform, aiming for an in-depth analysis of the relative navigation filter performance using real MWR equipment. When considering the sharp orbit drift during perigee time, mission designers proposed a mature S-band ranging (SBR) equipment, in addition to a BeiDou dual-frequency receiver, as a backup payload for both relative measurement data transfer and TT&C payloads. SBR is a reliable medium-accuracy measurement payload that operates at the amateur frequency of 2450 MHz \(\pm\) 50 Hz. The ranging accuracy of SBR is 1 m(1\(\sigma\)) in pseudo-noise (PN) code measurement mode, 0.01 m(1\(\sigma\)) in carrier phase measurement mode, and the data rate achieved a maximum of 1 Mbit/s. The whole ranging process during the HEFFD mission is scheduled as follows: (1) The BeiDou B1C/B2a dual-frequency receiver is operated during the whole flight, which provides relative carrier phase differential measurement of centimeter-level accuracy, and provides a synchronized MWR time tag. (2) The SBR payload is used for reliable medium-accuracy distance measurement and data transfer, which is arranged to operate during the whole flight time. (3) The relative ranging values from the BeiDou receiver and SBR are simply optimally fused at the output level. This will be regarded as baseline formation distance data in HEFFD mission. (4) The MWR payload is especially developed for the HEFFD mission, and it will only be used during the apogee orbit period for submillimeter-level ranging and precise GNC technology validation. (5) Dual-frequency interference optical ranging payload is also used with a ranging accuracy of 1 \(\upmu\)m(1\(\sigma\)), which provides the reference distance data for the validation process. The platform was precisely installed with transmitting/receiving payloads and antennas of SBR, MWR, and optical equipment, with a guaranteed accuracy of a fixed position deviation within two microns. Figure 10 demonstrates the block diagram of the whole simulation platform used in this paper. The whole simulation system was coordinated by a central control computer, which conducted the HEFFD mission formation scenario design, high-precision moving platform control, payload operation, data collection, and analysis. Note that a state-of-the-art high-fidelity SPIRENT GSS9000 simulator was used for the generation of BeiDou B1C/B2a signals and real HEFFD mission scene. Dual-frequency  
Figure 9: MWR ranging rate errors of 2400 samples, with data collected from a 6-DOF moving platform under a relative longitudinal velocity of 5 \(\upmu\)m/s. The ranging rate errors were less than 1.6 \(\upmu\)m/s during the test. BeiDou receivers from both formation spacecrafts were connected directly to the simulator for the purpose of differential carrier phase measurement and precise time synchronization with an accuracy of 0.3 ns(1\(\sigma\)). Some of the parameters used in this simulation were as follows: initial date of simulation: 8 April 2021, 14:15:00 (GMT+08:00), sample time interval: 1 s. The spectral densities of the process noise components \(w_{x},w_{y},w_{z}\) in Equation (19), which are each given by \(\sqrt{5}\times 10^{-11}\) m/s\({}^{3/2}\)[20]. The individual standard deviations for initial states \(\boldsymbol{\eta}\) and \(\dot{\boldsymbol{\eta}}\) were 0.01 km for \(\varrho\), 0.1 deg for \(\theta\), 0.1 deg for \(\phi\), 1 \(\times\) 10\({}^{-4}\) km/s for \(\dot{\varrho}(0)\), 0.01 deg/s for \(\dot{\theta}(0)\), and 0.01 deg/s for \(\dot{\phi}(0)\), 0.05 and 0.05 deg/s\({}^{2}\) for leading orbit angular velocity and its rate, and 50 m and 0.01 m/s for the leading orbit radius and its rate. Figure 11 illustrates the estimation error (the differential of estimated values and real values from the model in Table 1) of HEFFD's relative range and range rate by using BeiDou/SBR measurement and the traditional Kalman filter algorithm.