Orbit Determination Methods For LEO Satellites From Probabilistic Analysis, Circular Motion Model And Single Pass Doppler Measurements

The Doppler measurements of the telemetry radio signals nanosatellite CubeBel-l for a single pass over the Belarusian State University ground station were carried out. Two methods for orbit determination of a small satellite are considered. The first method is based on the SGP4 model and requires additional information from the NORAD TLE catalog of the satellite orbital parameters. An unknown small satellite is identified using the NORAD TLE catalog based on a probabilistic estimation of the elevation angle and the Doppler frequency shift of receiving telemetry signals. The second method is based on processing experimental measurements of the Doppler frequency of the telemetry radio signals and Keplerian circular motion model for the small satellite. It does not require additional information from the NORAD database of satellite orbital parameters.

Another way to obtain initial data for USS prediction models is to measure of the orbit parameters for its pass over the UGS using the telemetry or command radio signals [13][14][15][16]. The measured parameters for the UGS are the time and the Doppler shift of the frequency of the received radio signal [16][17][18]. In paper [13], a general method for determining the satellite orbit was proposed, based on improving approximations for the initial components of position and velocity by means of successive differential corrections from Doppler observations. In the work [19,20], a method for calculating satellite orbit was developed on the basis of a model of circular perturbed motion and measurements of the Doppler frequency shift of telemetry signals. Based on a probabilistic estimate of the elevation angle and Doppler frequency shift from 10-20 measurements on multiple passes of a small satellite, a set of orbital parameters are determined for the estimated time of receiving telemetry signals. Based on the analysis of the database of orbital parameters of low-orbit spacecraft, including satellites of the Cubesat standard, the applicability and correctness of the circular motion model with a reduced number of unknown orbital parameters (independent variables) from six to four is justified.
In this paper, two methods for orbit determination of a SS are considered on the example of processing experimental measurements of the Doppler frequency of the telemetry radio signals nanosatellite CubeBel-1 for a single pass over UGS. The first method is based on the SGP4 model and requires additional information from the NORAD TLE catalog of the satellite orbital parameters. A small satellite is identified using the NORAD TLE catalog based on a probabilistic estimation of the elevation angle and the Doppler frequency shift of receiving telemetry signals. The second method is based on processing experimental measurements of the Doppler frequency of the telemetry radio signals and Keplerian circular motion model (nonperturbed motion) for single pass of SS over the UGS. It does not require additional information from the NORAD database of satellite orbital parameters.

II. SMALL SATELLITE RADIO SIGNAL RECEIVING AND PROCESSING. ORBIT DETERMINATION BASED ON THE SGP4
MODEL AND THE NORAD TLE DATABASE The nanosatellite CubeBel-1 telemetry receiving and processing on a Belarusian State University GS is additionally equipped with an orbit measurement and determination system with time synchronization [7]. The GS hardware consists of: 435-438 MHz band Yagi-Uda antennas with circular polarization, receiving system based on the IC-9100 transceiver; receiving system based on the software defined radio receiver, YAESU G-5500 azimuthelevation rotator with a control interface. The GS software includes: satellite orbital and radio signal parameter prediction software, simulation and visualization of cooperative GS scenarios and express calculation of standard ballistic information software, telemetry receiving and processing software. The orbit measurement and determination system with time synchronization for UGS consists of: GPS receiver, module for frequency and time measurements of the received radio signal based on a microcontroller for time processing and a two-channel digital oscilloscope, software for processing measurements; software for orbit determination and correction.
Measurements of the orbit parameters for a single nanosatellite CubeBel-1 pass over a UGS were performed at several points in the orbit. The satellite pass interval over a UGS is about 10 minutes. Therefore, the simplest Keplerian motion model for satellite can be used to process single pass measuring data. The nanosatellite CubeBel-1 orbit determination was made using 20 measurements of the reception time ti and telemetry signal frequency Doppler frequency shift of the radio telemetry signal is given by: An unknown SS is identified using the NORAD TLE catalog based on a probabilistic estimation of the elevation angle and the Doppler frequency shift of receiving telemetry signals. The identification of the orbital parameters of the small satellite using the TLE database of the NORAD system was carried out according to the following algorithm. For each j-th (j = 1, 180) Cubesat standard satellite of the orbital parameters were uploaded from the NORAD TLE catalog for 01.11.2019 (UTC). Using the SGP 4 model the elevation angle el and the Doppler frequency shift The probability of success β1j for the j-th satellite by the data analysis on elevation angle el > 0 (strategy 1) was calculated as where K1j is the number of passes for the j-th satellite with the elevation angle el > 0.
The satellite with β1j > 50% are sorted and selected for further analysis on the deviation of the calculated Doppler frequency shift where K2j is the number of passes for the j-th satellite with the elevation angle el > 0 and the deviation of the calculated Doppler frequency shift The mathematical model of a small satellite motion is based on the Keplerian circular orbit approximation for single pass of small satellite over the university ground station. The motion in a circular orbit is determined by the following parameters of the state vector X: where i is the orbit inclination, T is the orbital period, u is the latitude argument, and Ω is the longitude of the ascending node at time t.
The determination of the orbit of an unknown SS was carried out by the UGS based on N measurements of telemetry radio signals on single pass where ti is the time of receiving and exp i f ∆ is the Doppler frequency shift of the received telemetry radio signals.
For the calculated point in time t0, the SS state vector was found X0 = (T0, i0, u0, Ω0), which best meets the measurement results according to three criteria: • the elevation angle of the SS above the UGS at the times of measurements ti must be positive (the SS relative to the GS should be above the horizon): the sign of the time derivative of the Doppler frequency shift corresponds to the pattern of change in the experimental Doppler curve: According to the results of measurement processing, the ranges of variation ΔX = (ΔT, Δi, Δu, ΔΩ) of the parameters of the state vector X0 are determined. Next, for each obtained state vector X at the measurement time ti the Keplerian circular motion model is used to calculate the elevation angle eli of the SS above the GS and the Doppler frequency shift lc i ca f ∆ of the telemetry radio signal using the following algorithm. At the time of measurements ti, we find the orbital parameters T(ti ) = T0, i(ti ) = i0, Ω(ti ) = Ω0 and u(ti). In the calculations in the Keplerian circular motion model, the three parameters T, i, and Ω remain unchanged for all time points of the measurements ti, while the latitude argument u changes according to where u0 is the value of the latitude argument at time t0.
Then find the coordinates and projections of the velocity vector in the orbital coordinate system (CS) and the geocentric inertial CS. According to the coordinates of the UGS (φ = 53°54'27" north latitude, λ = 27°33'52" east longitude, altitude H = 230 m) at the time of measurement ti we determine the radius vector of the slant range ρ and the rate of its change in the OXYZ coordinate system and the vector of the slant range ρ and the projection of its velocity in the topocentric rectangular CS. Then we find the slant range ρ, the elevation angle el and the azimuth β, as well as the rate of their change at the time ti and calculate the Doppler frequency shift where c is the speed of light in vacuum, f0 is the nominal frequency of the onboard SS transceiver. The probability of success βj of the given set of orbital parameters (T0, i0, u0, Ω0) for the time t0 by estimating only the calculated elevation angle el (j = 1) and the elevation angle and the Doppler frequency shift Δf calc (j = 2) at the moment of receiving the telemetry radio signal was found according to where Nj is the number of calculated points that satisfy the criterion (6) for j = 1 and (6-8) for j = 2 and N is the total number of measurement points at which the numerical simulation was carried out for a given set of orbital parameters (T0, i0, u0, Ω0). The probability of success of β2= 50% means that the N2 = Ntotal/2 conditions (6) el > 0 and (4) |Δf exp − Δf calc | < 200 Hz are satisfied only in half of the experimental data.

IV. RESULTS AND DISCUSSION
The identification of a SS from the NORAD TLE catalog was carried out by processing 20 measurements of the reception time and telemetry signal frequency on single pass for the period from 05:54:00 to 06:03:30 for 01.11.2019 (UTC) over the UGS. First, based on the NORAD TLE catalog of the Cubesat standard satellites and SGP 4 model for each reception time ti (i = 1…20) of telemetry signal, the single pass parameters over the UGS (elevation el, azimuth az, range ρi, range rate dρi/dt and Doppler frequency shift c i alc f ∆ ) were calculated for each jth (j = 1, 180) satellite. An unknown SS was identified based on a probabilistic estimation of the elevation angle and the Doppler shift in the frequency of receiving telemetry signals. Second, the probability of success β1j for the j-th satellite by the data analysis on el > 0 for elevation angle was calculated. The satellite with β1j > 50% are sorted and selected for further analysis on the deviation of the calculated Doppler frequency shift   estimation. Based on the Keplerian circular motion model for single pass of small satellite numerical simulation was carried out for the time of reception and the Doppler shift of radio telemetry at the N =10 and N =20 measuring points on single pass for the period from 05:54:00 to 06:03:30 for 01.11.2019 (UTC) over the University ground station (Fig. 3). The ranges of change of the orbital period Т was selected from 93 to 100 min with the step 1 s, the inclination i was from 97.13° to 98.43° with the step of 0.01°, the argument of latitude u was from 20° to 180° with the step of 1° and the longitude of ascending node Ω was from 0 to 360° with the step of 1°. The probability of success β1 and β2 of each set of orbital parameters (T, i, u, Ω) was calculated for four estimated time points t0 = 5:56:00; 5:57:00; 5:58:00; 5:59:00 (Fig. 3)   It was found from Table 1, that the ranges of variation of the orbital period T and orbital inclination i with the probability of success 100% for estimated time point t0 = 5:56:00 based on Strategy 2 in the comparison in Strategy 1 are reduced by 55% from 200 to 90 s and from 0.62° to 0.27°, respectively. While the ranges for the latitude u and longitude of ascending node Ω with the probability of success 100% for estimated time point t0 = 5:56:00 based on an estimate of the elevation angle and the Doppler frequency shift in the comparison only elevation el estimation are reduced by 92% from 13° to 1° and reduced by 94% from 53° to 3°, respectively.
When the estimated time t0 approaches the center of the Doppler curve (the Doppler frequency shift approaches zero, which corresponds to the minimum range between the satellite and the ground station), the number of possible sets of orbital parameters Norb with β2 = 100% decreases, and the ranges of changes in the orbital period T and inclination i become smaller. It was found that the ranges of variation of the orbital period T and orbital inclination i based on Strategy 2 with the probability of success 100% for estimated time point t0 = 5:59:00 in the comparison estimated time point t0 = 5:56:00 are reduced by 74% from 90 to 24 s and from 0.27° to 0.07°, respectively. As you can see from Table 2 1 are reduced by 53% from 200 to 94 s and from 0.62° to 0.28°, respectively. While the ranges for the latitude u and longitude of ascending node Ω with the probability of success 100% for estimated time point t0 = 5:56:00 based on an estimate of the elevation angle and the Doppler frequency shift in the comparison only elevation el estimation are reduced by 94% from 16° to 1° and reduced by 95% from 58° to 3°, respectively.
For the Strategy 2, when the estimated time t0 approaches the center of the Doppler curve the number of possible sets of orbital parameters Norb with β2 = 100% decreases, and the ranges of changes in the orbital period T and inclination i become smaller. It was found that the ranges of variation of the orbital period T and orbital inclination i for estimated time point t0 = 5:59:00 in the comparison with estimated time point t0 = 5:56:00 are reduced by 74% from 94 to 24 s and from 0.28° to 0.07°, respectively. The numerical simulation of orbital parameters based on the Keplerian circular motion model for single pass of small satellite over the University ground station and N =10 and N=20 measurements of the time of reception and the Doppler shift of radio telemetry allows us to conclude that the latitude u and longitude of ascending node Ω are defined unambiguously using analysis the elevation and the Doppler frequency shift estimation. While the orbital period T and orbital inclination i were estimated in range. It was also obtained that if the estimated time t0 approaches the center of the Doppler curve the number of possible sets of orbital parameters and the ranges of changes in the orbital period T and inclination i decreases.
In conclusion, we have demonstrated the potential of Belarusian State University GS for orbit determination of the unknown SS. Orbital parameters are theoretically investigated using NORAD TLE database and Keplerian circular motion model. If using the NORAD TLE catalog based on a probabilistic estimation of the elevation angle and the Doppler frequency shift of receiving telemetry signals was allowed to determine the nanosatellite CubeBel-1 unambiguously, then the method based on Keplerian circular motion model was allowed only to calculate the average state vector unknown satellite. Finally, corrected state vector in the geocentric inertial CS was obtained based on differential correction method.