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A spherical microsatellite, aimed at technology verification of upper atmospheric density detection and precision orbit determination in low Earth orbit (LEO), has been developed by Tsinghua University and is scheduled to launch in the second half of 2019. In order to reduce the influence of other nonconservative forces, the area-to-mass ratio of the satellite was designed to ensure that the aerodynamic drag of the satellite is more than one to two orders of magnitude greater than the solar pressure perturbation. The influence of the components of the antenna, separation support, and solar cell arrays (SCAs) on the spherical structure for the area-to-mass ratio is reduced by reasonable design of the satellite flight attitude. In addition, a novel method has been developed to estimate and correct the parameters for atmospheric density (correlated to the drag) calculation, using precise orbit data (POD) obtained from an onboard dual-frequency global position system (GPS) receiver. The method is used to obtain the partial derivative of the total acceleration of the satellite to the initial state (position, velocity) and Harris-Priester (HP) density model parameters (antapex and apex density). Further, the least squares estimation method is used to solve the overdetermined linear equations and obtain the change of initial state of the satellite and the HP density model parameters relative to change of the satellite state. The validity of this method has been verified through numerical simulations with the parametric setup of the satellite. The estimated density precision was found to be higher than two to three orders of the initial HP density model with continuous iteration.

Estimation and correction of upper atmospheric semi-empirical density models have attracted substantial interest over recent decades due to their wide applications for the prediction of spacecraft motion, precise orbit determination, debris collision avoidance, and location estimation of re-entry spacecraft. Density decreases exponentially with orbit height but is affected by sunshine and geomagnetic activity, among others, resulting in uncertainties of 10–30% [

Currently, empirical models of upper atmospheric density mainly rely on in-orbit observation data from upper atmospheric density detection satellites such as Atmospheric Neutral Density Experiment (ANDE) [

Whether using atmospheric density detection satellites or near-Earth space science mission data, the main payloads for atmospheric density detection include mass spectrometers [

Using satellite orbit data to determine atmospheric density is also important for atmospheric density estimation and correction. Doornbos [

In this study, we used the aerodynamic shape design of a spherical satellite and its precision orbit data obtained by onboard GPS receivers of commercial off-the-shelf products to propose a method for estimating and correcting the upper atmospheric density model and verified the proposed method via numerical simulation. The magnitude of the atmospheric drag and the solar pressure at a 400–600 km altitude with different area-to-mass ratios is analyzed in the next section and provides reference for aerodynamic shape design of spherical satellites. The design of a spherical satellite shape to ensure a small error caused by aerodynamic shape and attitude change is outlined in Section

The motion equation for a satellite in low orbit can be written as

The atmospheric drag and the solar pressure perturbation (calculated using Equation (

Simulation conditions.

Simulation time (s) | Initial orbital element | |||||
---|---|---|---|---|---|---|

Orbital height (km) |
Eccentricity | Inclination (°) | Right ascension of ascending node (°) | Argument of perigee (°) | True anomaly (°) | |

1500 | 412.18 | 0 | 96.8213 | 340.251 | 0 | 0 |

512.18 | 0 | 96.8213 | 340.251 | 0 | 0 | |

612.18 | 0 | 96.8213 | 340.251 | 0 | 0 |

Atmospheric drag at heights of (a) 400 km, (b) 500 km, (c) 600 km, and (d) solar pressure for different area-to-mass ratios.

In this section, we outline the design of a spherical microsatellite, including the platform and primary payload, considering the influence of separation supports, antennas, and the installed SCAs on the aerodynamic shape of the satellite to ensure that the area-to-mass ratio experiences minimal changes.

The satellite (mass: 23 kg, diameter: 626 mm, height: 510 mm) has a payload of a dual-frequency commercial off-the-shelf (COTS) GPS receiver (Figure

(a) Dual-frequency GPS, (b) precision orbit determination technology based on GNSS, (c) upper atmospheric density detection, and (d) long-wave gravity field detection.

The satellite [

Main components of the satellite.

Subsystem | Main components | Key specifications |
---|---|---|

Structure | Hemispherical frame (HF), payload cabin (PC), central connection ring (CCR), releasing device (RD) | 2A12-H112 aluminum alloy |

Payload | Dual-frequency GPS (D-GPS), single-frequency GPS (S-GPS), micro camera (MC) | Real-time positioning (velocity) precision: less than 20 m (0.2 m/s) |

Attitude control | Inertial measurement unit (IMU), integrated full field sun sensor (IFFSS), magnetometer, flywheel, magnetorquer | Attitude determination precision: 0.5° |

Attitude control precision: 2° | ||

Power | Solar cell array (SCA), power controller unit (PCU), battery (BAT) | SCA: 931 cells of triple-junction GaAs (29.5%) |

BAT capacity: 8.8 Ah | ||

Bus voltage: 6.9–8.2 V | ||

Secondary voltage: 12 V, 3.3 V, 5 V | ||

Power consumption: 11.45 W (steady state) | ||

Integrated electronic | Onboard computer (OBC), attitude control computer (ACC), integrated electronic circuit board (IECB) | Processing chip: LPC2294 |

Telemetry, telecontrol, and data transition | Unified S-band (USB), industrial scientific medical (ISM) band, power amplifier (PA), combiner | USB: Telemetry: 2000 bps |

Telecontrol: 4096 bps | ||

Data transition: 0.5 Mbps | ||

ISM: 115.2 kbps |

Exploded view of the satellite.

The satellite uses a central load-bearing device (CCR and PC) with two hemispherical frames to assemble the configuration. A releasing device enables separation of satellite and rocket via a locking and ejecting mechanism on the CCR. The heavier components and the large power consumption devices are directly installed in the ninth palace configuration, which improves the mechanical performance and heat dissipation environment. The light components on the satellite are connected through the IECB (Figure

D-GPS.

The primary payload of the satellite is the 80 mm long, 52 mm wide, 20 mm high, dual-frequency COTS GPS receiver (Figure

Antenna of D-GPS.

Thermal vacuum test for D-GPS.

Although the satellite is spherical, its characteristics will be diluted owing to separation of the support, antenna, and SCAs. In this section, the influence of the satellite antenna and separation support on the surface quality ratio of the satellite is analyzed. The flying attitude design and SCA installation are considered for reducing the error of the area-to-mass ratio caused by the aerodynamic configuration of the satellite. Part of the spherical satellite design has been granted patent authorization [

(a) Components at satellite surface and (b) flying attitude of the satellite.

The SCAs are another major factor affecting the spherical structure of the satellite. In the initial design of the satellite, the installation of the soleplate was designed as a plane (Figure

(a) Prototype of satellite, (b) new surface for the satellite, (c) the polyimide film, and (d) a sample of the SCAs.

When a low-orbit satellite is in orbit, the inaccuracy of the empirical atmospheric density model leads to deviation between the real orbit and the reference orbit integrated with the dynamic model. The current empirical model is mainly based on the hypothesis of hydrostatic balance, and the parameters of model are fitted by track data, mass spectrometer data, and acceleration data. In this section, a method for modifying the parameters of the empirical density model based on the precision orbit data is proposed, and the parameter correction model of the Harris-Priester (HP) density model is derived using this method. Finally, the validity of the corrected model is verified by a simulation test.

Montenbruck and Gill [

The satellite state vector is also relative to the initial state, dynamics parameters, and time and is defined as

According to Figure

Comparison of the reference orbit and real orbit.

At time

We define

Further, Equation (

We define

We then create variational equations:

The derivation equation (Equation

In order to achieve the effective solutions, the number of observation equations should be much greater than the number of unknowns to be solved. Then, the solution to Equation (

The parameters of density are obtained from the density model and influence the accuracy of the model. The precise orbit data can be applied to derive density data by estimating the aerodynamic scale factor parameters [

For a given orbit altitude,

Then,

Then,

Thus,

Equations (

The real orbit should be represented by observation data produced by precision orbit data. However, in order to validate the method of parameter estimation and correction, we assume the following: the real orbit is an orbit added into the error-based reference orbit; the nonspherical and atmospheric drag perturbations are considered both in the reference orbit and in the real orbit (additionally, three-body gravitational perturbations and solar radiation perturbation are considered as a constant disturbance force of 10^{−9} m/s^{2}); the initial HP density model in the real orbit is accurate; the same magnitude bias is added to the HP density model for the reference orbit as to the initial HP density model for the real orbit; a fix drag coefficient [^{2}/kg is adapted for the simulations (the design parameters of the spherical satellite).

Because calculation of the dynamic inversion method is very large and time-consuming, the parallel computation method is considered. The computation progress is shown in Figure

Computation progress of the dynamic inversion.

Simulation parameters for density estimation.

Orbit height (km) | Error of orbit determination (cm) | Sample time/total time (s) | Order of the gravity field | Arc length (s) (parallel program) | Total simulation times (s) | Number of iterations |
---|---|---|---|---|---|---|

512 | 5 | 1 | 30 | 400 | 9600 | 6 |

8 | ||||||

10 | 1 | 30 | 800 | 14400 | 6 | |

8 |

Comparison of corrected parameters and initial parameters for an arc length of 400 s and a height of 500 km.

^{−12} kg/m^{3}) |
^{−12} kg/m^{3}) |
^{−12} kg/m^{3}) |
^{−12} kg/m^{3}) | |||||
---|---|---|---|---|---|---|---|---|

Bias HP model | 0.8916 | 1.5042 | 0.7819 | 1.105 | ||||

No. of iterations | Corrected HP | Initial HP [ |
Corrected HP | Initial HP [ |
Corrected HP | Initial HP [ |
Corrected HP | Initial HP [ |

6 | 0.7531 | 0.3916 | 1.9648 | 2.042 | 0.2861 | 0.2819 | 1.9127 | 1.605 |

8 | 0.3927 | 2.0369 | 0.2823 | 1.6081 |

Comparison of corrected parameters and initial parameters for an arctime of 800 s and a height of 500 km.

^{−12} kg/m^{3}) |
^{−12} kg/m^{3}) |
^{−12} kg/m^{3}) |
^{−12} kg/m^{3}) | |||||
---|---|---|---|---|---|---|---|---|

Bias HP model | 0.8916 | 1.5042 | 0.7819 | 1.105 | ||||

No. of iterations | Corrected HP | Initial HP [ |
Corrected HP | Initial HP [ |
Corrected HP | Initial HP [ |
Corrected HP | Initial HP [ |

6 | 0.5073 | 0.3916 | 2.2606 | 2.042 | 0.3384 | 0.2819 | 1.6452 | 1.605 |

8 | 0.3916 | 2.0403 | 0.2820 | 1.6063 |

Initial density model (red dotted line) and the corrected density model (green dashed line) indicating the corrected density minus the initial density with (a and b) 6 and (c and d) 8 iterations for a 5 cm orbit error.

Initial density model (red dotted line) and the corrected density model (green dashed line) indicating the corrected density minus the initial density with (a and b) 6 and (c and d) 8 iterations for a 10 cm orbit error.

The parallel calculation is used here in order to improve the effectiveness of calculation. The total simulation times were divided into some equal time (arc length) which is assigned to different CPU processors for solving the orbital integral and differential equations. Arc length is an important computation parameter, because atmospheric correction is a nonlinear process, and currently we can only use the linear approximation method. Arc length has an influence on the linearization error and thus the corrected density model. Here, different arc lengths were considered to reflect the performance of the corrected density model. First, the 5 cm orbit error was considered and the initial bias value for the HP model and the final correction of the density parameter for six and eight iterations are shown in Table

In this paper, the design of a spherical microsatellite, which is aimed at upper atmospheric density and precision orbit determination, was presented. The satellite will be launched in 2019 as a secondary payload. The rational aerodynamic shape and flight attitude design of the satellite reduces any interference of the area-to-mass ratio change in atmospheric density estimation precision. Thus, using a spherical microsatellite to detect the upper atmosphere density is superior to the traditional satellite. The precision orbit data obtained by GPS can improve the temporal resolution of the density model estimation. The estimation and correction method achieves density detection, without accelerometer data, and establishes the relationship between the orbit change and density model parameters and numerical simulation results, illustrating the effectiveness of the proposed method. Note that the orbit determination error, arc length, and number of iterations affect the performance of parameter correction in the HP density model. From the perspective of orbit simulation, the magnitude of the orbit error directly affects the number of iterations; for the same number of iterations, a longer integral arc length produces a better model correction result.

The Harris-Priester model data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This satellite mission and research was supported by the National Natural Science Foundation of China (No. 11002076). The author is grateful to the many colleagues and associates who helped with this research.

The expression of the Harris-Priester (HP) model which is mentioned in Dynamic Method of Atmospheric Density Model Estimation and Correction.