Formation-flying Missions Research Articles

Precise orbit determination of the LuTan satellite using GPS, BDS-2, and BDS-3 signals

Abstract The LuTan satellite is the first formation-flying mission of China enabling synthetic aperture radar interferometry in the L-band. As a key mission requirement, the absolute position of each satellite must be determined with an accuracy of 5 cm in three dimensions (3D). To fulfill this demand, each satellite carries a Global Positioning System/Beidou Navigation Satellite System (GPS/BDS) dual-constellation Global Navigation Satellite System receiver that can simultaneously track the GPS L1 and L2 signals, the BDS-2 B1I and B3I signals (BD2), and the BDS-3 B1C and B2a signals (BD3). In this study, we assess the performances of precise orbit determination with different signals and their combinations. For that purpose, we produce a total of five sets of orbit solutions, i.e., GPS-based, BD2-based, BD3-based, GPS/BD2 combined, and GPS/BD3 combined solutions. To evaluate the orbit solutions, comparisons in the orbital overlaps and with the GPS-based orbit are adopted. The orbital overlap analysis shows that the GPS-based solution has the smallest root mean square (RMS) of overlap orbit differences than other single-constellation solutions with 3.7 mm in 3D. The BD2- and BD3-based orbits are suffered from large errors and the overlap RMS differences are 24.7 and 18.9 mm, respectively. Furthermore, all of the combined orbits (both 3.1 mm for the GPS/BD2 and GPS/BD3 combined orbits) based on the variance component estimation method can obtain improvement than GPS-based solutions. For orbit comparisons, the GPS-based solution is selected to serve as the reference. The RMS values of the orbit differences are 29.9 and 15.4 mm for the BD2- and BD3-based solutions, respectively. For the GPS/BD2 and GPS/BD3 combined solutions, the values are 4.1 and 4.8 mm. In general, the results agree with those obtained in the overlap analysis. These results indicate that the BD3-based orbit is superior to the BD2-based orbit, while both of them are inferior to the GPS-based orbit. We show that inaccurate ephemerides of the BDS satellites can partly explain the degraded performance of the BDS-derived orbits. Nevertheless, the orbits still fulfill the 5 cm accuracy demand and can be readily used for subsequent precise baseline determination.

Relative Orbit Control Algorithms and Scenarios for the Inertial Alignment Hold Demonstration Mission by CubeSat Formation Flying

CANYVAL-C is a formation-flying mission that demonstrates a coronagraph utilizing two CubeSats. The coronagraph is a space telescope that blocks sunlight to examine the overcast regions around the sun. It is composed of optical and occult segments. Two spacecraft were aligned with respect to an inertial system to configure a virtual telescope using inertial alignment hold technology. The relative orbit control scenario for this mission involves rendezvous, differential air drag control, and inertial alignment holding. Orbit control algorithms and simple strategies that can be automatically constructed onboard have also been developed. For each maneuver, the control performance under the errors from navigation, attitude determination and control, and propulsion systems were assessed via Monte Carlo simulation, taking into account the hardware specifications and operations. In addition to the algorithm and strategy of this mission, the relative orbit control scenario was evaluated for its practicability using Monte Carlo simulations. The feasibility of this mission is ensured by a statistical analysis of the prospect of its success during its operation.

Passively Safe Spacecraft Motion Using Reachable Sets and Orbital Element Differences

The problem of passive safety between spacecraft is studied in this paper by means of backward reachable sets and orbital element differences. These reachable sets characterize the initial states that enter a swept region around a designated spacecraft within some time horizon, leading to collision or conjunction events. We provide an in-depth passive safety analysis using such representations and discuss how the motion of spacecraft can be constrained to satisfy such safety constraints. The work herein is particularly relevant for future formation-flying missions but can be extended to certain constellations, for example, Walker constellations. We demonstrate how to compute these sets for general dynamics models, which typically require numerical methods, but provide analytical and closed-form solutions for the Keplerian case as well as when first-order secular perturbations are considered with a mean orbital element difference description. Additionally, we show how the passive safety constraint based on such sets can be further simplified (or reduced) by considering projections or slices of the derived reachable sets. Finally, we propose an algorithm that finds passively safe configurations when considering multiple spacecraft. Simulations verify the passive safety constraints derived in this paper and their utility for a number of design cases in formation flying and a -perturbed low-Earth-orbit Walker constellation. In the latter case, long-term passive safety is shown even when subjected to unmodeled perturbations.

Optimal On-Orbit Inspection of Satellite Formation

In a formation-flying mission where multiple spacecraft must cooperate and maintain a prescribed relative separation, the early detection of possible anomalies is a primary requirement. This is possible, for example, by employing an inspector spacecraft whose aim is to monitor the condition of the formation members with an on-orbit inspection. This paper analyzes a rest-to-rest multiple-impulse transfer that the inspector spacecraft must accomplish to visit all of the formation members. The problem is studied using the linearized Hill–Clohessy–Wiltshire equations and is solved in an optimal framework by minimizing the total velocity variation along the transfer trajectory. The solution algorithm implements a two-step procedure that combines differential evolution algorithms and Nelder–Mead simplex method-based routines. A case study is thoroughly investigated where a formation of six satellites covers a circular orbit of altitude 300km over Earth. The proposed algorithm could efficiently find a solution and with reduced computational times.

Satellite Formation Flying for Space Advertising: From Technically Feasible to Economically Viable

The paper presents a feasibility study of satellite formation-flying missions for space advertising. To estimate a space advertising mission viability, the global population coverage model is designed and the demonstration schedule with a focus on larger cities is optimized for the formation of small satellites deployed in repeat ground track Sun-synchronous orbits. Monetization of an image demonstration over a city depends on the city population, outdoor advertising cost, and parameters limiting the number of potential advertising observations. Formation lifetime expressed in terms of fuel consumption for image reconfigurations and maintenance is one of the key factors and is analyzed via numerical simulation of satellite formation-flying dynamics and control.

Pattern Reconfigurable Antenna for Formation Flying Missions

In this paper, an innovative compact pattern reconfigurable antenna is proposed and developed to establish an inter-satellite communication link between the Formation flying spacecraft. The reconfigurable beam switching antenna comprises a driven Monopole at the center, encompassed with four Reconfigurable Selective Reflectors (RSR) which act as RF radiators and reflectors for different switching conditions. PIN diodes are used to electronically control the switch states of RSR and beam switching is achieved with uniform gain along the azimuth plane without affecting the antenna’s overall performance. The peculiar demands of Formation Flying Mission (FFM) satellites such as small size and low power for communication system are resolved by the proposed technique where a single antenna with different patterns communicates with multiple satellites reducing the antenna space and power consumption. Furthermore, it also offers a solution to the problem encountered in existing FFM antenna, like microstrip patch, which serves only one satellite at a time. The antenna is designed at 2.45GHz center frequency. Simulated results show good impedance matching and performance with 13 to 15% bandwidth and wide beamwidth. The experimental validation shows that the simulation results of the antenna agree well with the measured results.

Enhanced orbit determination for formation-flying satellites through integrated single- and double-difference GPS ambiguity resolution

The formation-flying technique is a fundamental concept for earth observing satellite missions, which usually require both absolute and relative orbit accuracies. Their precise orbit determinations are usually exclusively performed based on spaceborne GNSS data, where integer ambiguity resolution (IAR) plays a crucial role in achieving the best orbit accuracy. However, it is found that single-receiver IAR by resolving the single-difference (SD) ambiguities between GNSS satellites at each individual formation-flying satellite cannot achieve the same relative orbit accuracy that is attained by double-difference (DD) IAR between formation-flying satellites. To unravel this problem, 1 year of GPS data collected by the Gravity Recovery and Climate Experiment (GRACE) mission are used and four types of orbits are derived for comparison: (1) orbits where no ambiguities are fixed; (2) orbits where SD IAR is performed for both satellites; (3) orbits where only DD ambiguities between the twin GRACE satellites are resolved; (4) and orbits where SD IAR is carried out on only GRACE A while DD IAR is further accomplished between the twin satellites, namely the integrated SD IAR and DD IAR solutions. They are then evaluated through comparison to the reduced-dynamic orbit generated at the Jet Propulsion Laboratory, residual analysis of satellite laser ranging (SLR), and K-Band ranging (KBR) measurements. As expected, the integrated SD IAR and DD IAR solutions can achieve the highest absolute and relative orbit accuracies simultaneously. Specifically, SLR residuals in case of the integrated IAR are reduced by at least 25% for the kinematic orbit, when compared to the case of DD IAR. KBR residuals in case of the integrated IAR are reduced by 35 and 16% for the dynamic and kinematic orbit, respectively, when compared to those of SD IAR. Importantly, we find that errors in GPS clocks and/or narrow-lane fractional-cycle biases are in part responsible for the deteriorated relative accuracy of SD IAR achieved orbits. Therefore, we suggest that the integrated SD IAR and DD IAR scheme should be implemented for the best orbit solutions of formation-flying missions.

Output Regulation Control for Satellite Formation Flying Using Differential Drag

This paper proposes a new approach of using differentials in aerodynamic drag in combination with thrusters to control satellite formation flying in low Earth orbits. Parameterized output regulation theory for formation-flying missions with combined control action is developed based on the Schweighart–Sedwick relative dynamics equations. The theory is implemented to precisely track the different trajectories of reference relative motion and eliminates the effects of the perturbations. The parametric Lyapunov algebraic equation is proposed to ensure the stability of the linear relative model subject to saturated inputs. The main goal of this study is to approve the viability of using the differentials in aerodynamic drag to precisely control different formation-flying missions. Numerical simulations using a high-fidelity relative dynamics model and a high-precision orbit propagator are implemented to validate and analyze the performance of the proposed control algorithm in comparison with the linear quadratic regulator algorithm based on actual satellite models.

Principle of Fredholm image reconstruction in the vignetting zone of an externally occulted solar coronagraph

Aims.This study is carried out in the context of data processing of the solar coronagraph ASPIICS of the future formation-flying mission Proba-3, which is expected to provide images of the corona very close to the limb. There will be a transition zone of the order of 100 arcsec close to the limb, where the telescope aperture suffers a strong vignetting by the external occulter (a disc of 1.42 m at 144 m). The instrument response in this region will vary rapidly both in shape and in integrated intensity, the latter being particular to the external occultation. The aim of this paper is to propose a technique to recover as much as possible of the image of the corona very close to the limb in the vignetting zone.Methods.The object image relationship in this zone is not defined by the usual convolution but by the more general Fredholm integral of the first kind. Theoretical aspects of the problem are detailed in the context of a matrix formalism for the inversion of the Fredholm integral, formalism that we maintain up to the end of the numerical simulations, which is specific to the present work. The iterative Richardson-Lucy algorithm, specially written for the non-constant integrated intensity of the responses is used here for reconstruction. A study of the effect of noise on a photodetected image is made.Results.An important part of the work consisted in calculating the elements of the transfer matrix between the object and the image for a simulation on a small region of size 100 × 100 arcsec sampled over 128 × 128 pixels. This is obtained propagating the light through the system using a previously published approach. On a toy object, the reconstruction is excellent down to about 60 arcsec from the limb, corresponding to a vignetting of 50%. The drawback is that the recovery of aN × Nobject requires the handling of aN2 × N2matrix, i.e. a 16384 × 16384 transfer matrix here. However, taking into account radial symmetries of the experiment, we propose the use of a transformation from Cartesian to polar coordinates which allows to apply the same procedure all around the sun as for a small region.

Analytical Solution for Formation Flying Problem near Equatorial-Circular Reference Orbit

The relative motion between multiple satellites is a developed technique with many applications. Formation-flying missions use the relative motion dynamics in their design. In this work, the motion in invariant relative orbits is considered under the effects of second-order zonal harmonics in an equatorial orbit. The Hamiltonian framework is used to formulate the problem. All the possible conditions of the invariant relative motion are obtained with different inclinations of the follower satellite orbits. These second-order conditions warrantee the drift rates keeping two, or more, neighboring orbits from drifting apart. The conditions have been modeled. All the possibilities of choosing mean elements of the leader satellite orbit and differences in momenta between leader and follower satellites’ orbits are presented.

GPS Relative Navigation for the CanX-4 and CanX-5 Formation-Flying Nanosatellites

In November 2014 the CanX-4 and CanX-5 spacecraft became the first nanosatellites to demonstrate autonomous formation control with error less than 1 m. This feat was accomplished both in along-track formations at 1000 and 500 m range and projected circular orbit formations at 100 and 50 m. This control performance was enabled through carrier-phase differential GPS navigation techniques, providing online relative state estimates typically accurate to better than 10 cm. It was an important milestone on the road to regular and fully operational formation-flying missions. This paper provides an overview of the relative positioning algorithm design, presents an independent assessment of the receiver performance, and assesses the absolute and relative navigation results. The mission’s on-orbit results are compared with an independently determined orbit solution computed using the GPS High Precision Orbit Determination Software Tools at the German Aerospace Centre.

Relative positioning of formation-flying spacecraft using single-receiver GPS carrier phase ambiguity fixing

In recent years, differential carrier phase-based relative positioning (or “baseline determination”) with precision at the millimeter and submillimeter levels has been demonstrated for the GRACE, TanDEM-X and Swarm missions in offline processing. Specific features of such missions have included the use of spacecraft of similar shapes placed in almost identical orbits as well as the use of consistent geodetic-class GPS receivers. These elements have proven to be advantageous for the computation of baseline solutions with such precision levels. Particularly, they have allowed to fully leverage the use of differential GPS techniques, including the estimation and use of carrier phase integer ambiguities. Similarly, the aforementioned spacecraft and orbit characteristics have made it possible to tightly constrain the relative dynamics of formations in the generation of reduced-dynamic solutions. Other than the former examples, prospective formation-flying mission proposals, such as SAOCOM-CS and PICOSAR, may comprise spacecraft with very different characteristics, including dissimilar GPS/GNSS receivers. Such cases may no longer provide favorable conditions for relative orbit determination strategies. As an alternative, absolute orbit solutions may be computed individually for each spacecraft and used for the generation of precise baseline products. This study aims at the assessment of the potential of single-receiver ambiguity fixing for the generation of precise baseline solutions. Results using flight data from the GRACE, TanDEM-X and Swarm missions exhibit baseline accuracy better than 5 mm (3D RMS) for a one-month test period in June 2016. As such, the presented solutions may be considered for prospective formation-flying remote sensing missions with baseline precision requirements at the subcentimeter level. Likewise, the method is considered of particular interest for future multi-spacecraft formations and swarms that require efficient determination of a large number of individual baselines.

Theoretical performance of serrated external occulters for solar coronagraphy

Context. This study is made in the context of the future solar coronagraph ASPIICS of the ESA formation-flying mission Proba-3. Aims. In the context of solar coronagraphy, we provide a comparative study of the theoretical performance of serrated (or toothed) external occulters by varying the number and size of the teeth, which we compare to the sharp-edged and apodized disks. The tooth height is small (a few centimeters), to avoid hindering the observation of the solar corona near the limb. We first analyze the diffraction pattern produced by such occulters. In a second step, we compute the umbra profile by integration over the Sun. Methods. We explored a few methods to compute the diffraction pattern. Two of them were implemented. The first is based on 2D fast Fourier transformation (FFT) routines and a multiplication by the Fresnel filter of the form exp(−iπλzu2). Simple rules were derived and discussed to set the sampling conditions. The Maggi–Rubinowicz representation is then proposed as an alternative method, and is proven to be very efficient for this study. Results. Serrated occulters tend to create a two-level intensity pattern, the inner being the darker, which perfectly matches a previously reported geometrical prediction. The diffraction in this central region is lower by two to four orders of magnitude when compared to the sharp-edged disk. The achieved umbra level at the center ranges from 10−4 to below 10−7, depending on the geometry of the teeth. Conclusions. Our study shows that serrated occulters can achieve a high rejection and can almost reach the performance of the apodized disk when very many teeth are used. We prove that shaped occulters must be preferred to simple disks in solar and stellar coronagraphy.

Survey on Guidance Navigation and Control Requirements for Spacecraft Formation-Flying Missions

Survey on Guidance Navigation and Control Requirements for Spacecraft Formation-Flying Missions

Analysis of Nanosatellite Formation Establishment and Maintenance Using Electric Propulsion

Nanosatellite clusters are one of the current and future trends in space technology. To maintain a satellite cluster over a long period of time, the nanosatellites need to mitigate the alongtrack drift created by the initial orbit injection. In the mass range of 1–10 kg, CubeSats have strict constraints on allowed mass, volume, and electrical power; and they are equipped with only limited sensor and actuator capabilities. A state-of-the-art miniaturized electric propulsion system is one option to realize the required orbit control capability. Due to the low thrust provided by an electric propulsion system, long orbit control maneuvers are required. Therefore, mission design is highly affected by long-term attitude and power constraints. Assuming a cyclic thrust controller, four methods are developed to allocate the satellite power and attitude resources for orbit control. Two time division methods are presented that allocate dedicated time slots for each of the satellite’s main tasks. Alternatively, two cosine methods are presented that use an error cone, around the required attitude, as a criterion for selecting when to operate the thruster. It is demonstrated that the proposed methods are capable of maintaining a realistic CubeSat formation-flying mission using low-power electric propulsion. For a CubeSat with deployable solar panels, the double cosine method shows the best performance.

Improved GPS-based Satellite Relative Navigation Using Femtosecond Laser Relative Distance Measurements

This study developed an approach for improving Carrier-phase Differential Global Positioning System (CDGPS) based realtime satellite relative navigation by applying laser baseline measurement data. The robustness against the space operational environment was considered, and a Synthetic Wavelength Interferometer (SWI) algorithm based on a femtosecond laser measurement model was developed. The phase differences between two laser wavelengths were combined to measure precise distance. Generated laser data were used to improve estimation accuracy for the float ambiguity of CDGPS data. Relative navigation simulations in real-time were performed using the extended Kalman filter algorithm. The GPS and laser-combined relative navigation accuracy was compared with GPS-only relative navigation solutions to determine the impact of laser data on relative navigation. In numerical simulations, the success rate of integer ambiguity resolution increased when laser data was added to GPS data. The relative navigational errors also improved five-fold and two-fold, relative to the GPS-only error, for 250 m and 5 km initial relative distances, respectively. The methodology developed in this study is suitable for application to future satellite formation-flying missions.

Determination and Estimation of Differential Mean Orbital Elements for Formation-Flying Missions

Differential mean orbital elements are often taken as control states for relative-motion operations in spacecraft formation-flying missions. This paper investigates the problem of estimating differential mean orbital elements for formation-flying missions. Semi-analytical solutions for the perturbed Earth orbits with a dimensionless and near-circular modified form are introduced. The differential mean orbital elements are estimated from the absolute positions and velocities of the chief spacecraft, as well as the relative positions and velocities between the dual spacecraft. Three representative approaches, including numerical computation, extended Kalman filter, and square-root unscented Kalman filter, are presented in detail for this purpose. Numerical simulations are provided to test the performance of each approach.

Very High-Resolution Solar X-Ray Imaging Using Diffractive Optics

This paper describes the development of X-ray diffractive optics for imaging solar flares with better than 0.1 arcsec angular resolution. X-ray images with this resolution of the \geq10 MK plasma in solar active regions and solar flares would allow the cross-sectional area of magnetic loops to be resolved and the coronal flare energy release region itself to be probed. The objective of this work is to obtain X-ray images in the iron-line complex at 6.7 keV observed during solar flares with an angular resolution as fine as 0.1 arcsec - over an order of magnitude finer than is now possible. This line emission is from highly ionized iron atoms, primarily Fe xxv, in the hottest flare plasma at temperatures in excess of \approx10 MK. It provides information on the flare morphology, the iron abundance, and the distribution of the hot plasma. Studying how this plasma is heated to such high temperatures in such short times during solar flares is of critical importance in understanding these powerful transient events, one of the major objectives of solar physics. We describe the design, fabrication, and testing of phase zone plate X-ray lenses with focal lengths of \approx100 m at these energies that would be capable of achieving these objectives. We show how such lenses could be included on a two-spacecraft formation-flying mission with the lenses on the spacecraft closest to the Sun and an X-ray imaging array on the second spacecraft in the focal plane \approx100 m away. High resolution X-ray images could be obtained when the two spacecraft are aligned with the region of interest on the Sun. Requirements and constraints for the control of the two spacecraft are discussed together with the overall feasibility of such a formation-flying mission.

State Transition Matrix Approximation Using a Generalized Averaging Method

This paper presents a method for approximating the state transition matrix for orbits around a primary body and subject to arbitrary perturbations. A generalized averaging method is employed to isolate the high- and low-frequency regions of the perturbation terms and to construct a functional form of the approximate state transition matrix composed only of elementary analytic functions. The resulting state transition matrix is expressed with a small number of constant parameter matrices and osculating orbit parameters at an initial epoch and is valid for tens of orbital revolutions without having to update the parameters. Numerical simulations show that this method is valid for arbitrary-eccentricity orbits with semimajor axes ranging from low Earth orbit up to around 10 Earth radii when applied to Earth orbits. This method has been developed for implementation onboard spacecraft for high-accuracy formation-flying missions. Furthermore, it is shown that the symplectic property, which is a fundamental mathematical structure of Hamiltonian systems, can be incorporated into the method. This not only reduces the number of parameters required for approximations, but also preserves the physically true structure of the state transition matrix and provides some important properties that are useful for practical onboard computation.

Cryogenic Heat Pipe for Cooling High-Temperature Superconducting Coils

An emerging method of propellantless formation flight is the use of electromagnets coupled with reaction wheels called electromagnetic formation flight. To create a large magnetic field necessary for actuating formation-flying spacecraft, electromagnetic formation flight uses high-temperature superconducting wire. To achieve superconductivity, the high-temperature superconducting wire requires a consumable-free cryogenic thermal control system to maintain the wire temperature below the critical temperature throughout the entire electromagnetic formation-flight coil, which could be as large as 2 m in diameter. The research in this paper investigates a consumable-free method of maintaining isothermalization for a large-scale high-temperature superconducting coil. The high-temperature superconducting coil resides inside a thermally conductive jacket that is used for isothermalization. Both a solid conductor and a heat pipe were investigated for use as the thermally conductive jacket. In this paper, a proof-of-concept circular cryogenic heat pipe was tested in a 2-m-diam toroidal vacuum chamber. This system showed the potential for high-temperature superconducting cooling. The experiments in this paper demonstrate the feasibility of operating large high-temperature superconducting coils for future formation-flying missions.