Circular Low Earth Orbit Research Articles

Study on trajectory optimization of satellite rapid orbit formation flying for space object

This paper proposes a method of collaborative trajectory planning that utilizes convex optimization and IFPSO to tackle the issue of rapid formation reconstruction by multiple satellites in a low Earth circular orbit. The trajectory optimisation in this study uses convex optimisation and takes into account thrust limitations and safety distance. The time optimal solution is replaced by the path optimal solution, resulting in a suboptimal solution. The time-suboptimal trajectory solving method for formation flying using the CVX software is designed. The IFPSO is used to allocate configurations and waypoints for formation reconstruction, with the aim of balancing fuel consumption across the fleet and reducing the overall fuel consumption of the satellite. The effectiveness of the proposed approach is confirmed by numerical simulation. Despite sacrificing some computational efficiency, the method offered can solve the issues of configuration allocation and trajectory optimization.

Collision-Avoiding Model Predictive Rendezvous Strategy to Tumbling Launcher Stages

This paper considers the situation where a small satellite shall autonomously rendezvous with a tumbling object in a circular low Earth orbit (LEO) and derives a path-based model predictive controller that uses the docking point state and position of the chaser to guide it to a safe docking autonomously. The strategy embeds collision avoidance elements and reduces the computational effort for calculating the pulses to be provided by the thrusters through opportune algebraic manipulations, a Runge–Kutta 4 propagation method using linearized state transition matrices, and implicit embedding of dynamically equivalent thrust models, leading to constant state propagation matrices. Furthermore, the inputs design optimization problem and the embedded collision avoidance scheme are modeled and explicitly crafted as convex problems, contributing positively to low computational requirements. The docking and collision avoidance capabilities of the proposed scheme are extensively tested in an environment that accounts for all the perturbations relevant to LEO frameworks, for realistic thrust schemes, and for uncertainties in the measurement. Numerical results assess which tumbling objects can be docked or not by means of the proposed schemes as a function of the tumbling rates versus the thrust capabilities and hardware uncertainty of the docker.

An analysis tool for collision avoidance manoeuvres using aerodynamic drag

Aerodynamic collision avoidance manoeuvres provide an opportunity for satellites in Low Earth Orbits to reduce the risk during close encounters. With rising numbers of satellites and objects in orbit, satellites experience close encounters more frequently. Especially those satellites without thrusting capabilities face the problem of not being able to perform impulsive evasive manoeuvres. For satellites in Low Earth Orbits, though, perturbing forces due to aerodynamic drag may be used to influence their trajectories, thus offering a possibility to avoid collisions. This work introduces a tool for the analysis of aerodynamic collision avoidance manoeuvres for satellites in circular Low Earth Orbits. Current space-weather data are employed to estimate the density the satellite encounters. Achievable in-track separation distances following a variation of the ballistic coefficient through a change in attitude are then derived by evaluating an analytical equation from literature. Considering additional constraints for the attitude, e.g., charging phases, and uncertainties in the used parameters, the influence of a manoeuvre on the conjunction geometry and the collision probability is examined. The university satellite Flying Laptop of the University of Stuttgart is used as an exemplary satellite for analysis, which show the general effectiveness of evasive manoeuvres employing aerodynamic drag. First manoeuvring strategies can be deducted and the influence of parameter uncertainties is assessed.

OPTIMAL TWO-IMPULSE INTERPLANETARY TRAJECTORIES: EARTH-VENUS AND EARTH-MARS MISSIONS

This work describes several models to design optimal interplanetary trajectories. The transfer problem consists in transferring a space vehicle from a circular low Earth orbit (LEO) to a circular low orbit around a destiny planet (Venus or Mars). Models based on the two-body, four-body, and five-body problems are considered. Also, several versions of the patched-conic approximation are utilized including a detailed version that designs a lunar swing-by maneuver. The results show that the optimal trajectories for Earth-Mars and Earth-Venus missions collide with the Moon if a lunar swing-by maneuver with an unspecified altitude of the closest approach is included in the trajectory design; however, sub-optimal trajectories that do not collide with the Moon exist, presenting a smaller fuel consumption than the trajectories without lunar swing-by and with no greater changes in the time of flight.

Rapid Evaluation of the Decay Time of a Plasma Brake-Based CubeSat

The plasma brake is a propellantless device conceived for de-orbiting purposes. It consists of an electrically charged thin tether that generates a Coulomb drag by interacting with the ionosphere. In essence, a plasma brake may be used to decelerate an out-of-service satellite and to ensure its atmospheric re-entry within the time limits established by the Inter-Agency Space Debris Coordination Committee. Moreover, since it only needs a small amount of electric power to work properly, the plasma brake is one of the most cost-effective systems for space debris mitigation. This paper exploits a recent plasma brake acceleration model to construct an iterative algorithm for the rapid evaluation of the decay time of a plasma-braked CubeSat, which initially traced a circular low Earth orbit. The altitude loss at the end of each iterative step was calculated using the linearized Hill–Clohessy–Wiltshire equations. It showed that the proposed algorithm, which was validated by comparing the approximate solution with the results from numerically integrating the nonlinear equations of motion, reduced computational time by up to four orders of magnitude with negligible errors in CubeSat position.

Retrospect and prospect of ionospheric weather observed by FORMOSAT-3/COSMIC and FORMOSAT-7/COSMIC-2

FORMOSAT-3/COSMIC (F3/C) constellation of six micro-satellites was launched into the circular low-earth orbit at 800 km altitude with a 72-degree inclination angle on 15 April 2006, uniformly monitoring the ionosphere by the GPS (Global Positioning System) Radio Occultation (RO). Each F3/C satellite is equipped with a TIP (Tiny Ionospheric Photometer) observing 135.6 nm emissions and a TBB (Tri-Band Beacon) for conducting ionospheric tomography. More than 2000 RO profiles per day for the first time allows us globally studying three-dimensional ionospheric electron density structures and formation mechanisms of the equatorial ionization anomaly, middle-latitude trough, Weddell/Okhotsk Sea anomaly, etc. In addition, several new findings, such as plasma caves, plasma depletion bays, etc., have been reported. F3/C electron density profiles together with ground-based GPS total electron contents can be used to monitor, nowcast, and forecast ionospheric space weather. The S4 index of GPS signal scintillations recorded by F3/C is useful for ionospheric irregularities monitoring as well as for positioning, navigation, and communication applications. F3/C was officially decommissioned on 1 May 2020 and replaced by FORMOSAT-7/COSMIC-2 (F7/C2). F7/C2 constellation of six small satellites was launched into the circular low-Earth orbit at 550 km altitude with a 24-degree inclination angle on 25 June 2019. F7/C2 carries an advanced TGRS (Tri Gnss (global navigation satellite system) Radio occultation System) instrument, which tracks more than 4000 RO profiles per day. Each F7/C2 satellite also has a RFB (Radio Reference Beacon) on board for ionospheric tomography and an IVM (Ion Velocity Meter) for measuring ion temperature, velocity, and density. F7/C2 TGRS, IVM, and RFB shall continue to expand the F3/C success in the ionospheric space weather forecasting.

Detection of the third innermost radiation belt on LEO CORONAS-Photon satellite around 2009 solar minimum

We analyze variations of high energy charged particle populations filling various magnetospheric regions under, inside and outside of the Van Allen inner and outer electron radiation belts in May 2009. The study is based on the experimental data obtained from the STEP-F and the SphinX instruments placed close to each other aboard the low-Earth circular orbit CORONAS–Photon satellite. Data analysis of particle fluencies collected from the highly sensitive STEP-F device indicates the presence of a persistent electron belt at L ≈ 1.6, i.e., beneath the well-known Van Allen electron inner radiation belt of the Earth’s magnetosphere. The electron energy spectrum in this “new” belt is much steeper than that of the inner belt, so that the electrons with energies Ee ≥ 400 keV were almost not recorded on L ≈ 1.6 outside the South Atlantic Anomaly (SAA). We introduce the concept of effective lowest threshold energies for X-ray detectors used in the solar soft X-ray spectrophotometer SphinX and define their values for two regions: the SAA and in the Van Allen outer belt. Different values of lowest threshold energies are directly associated with different slopes of particle energy spectra. Cross-analyses of data obtained from the STEP-F and SphinX instruments initially built for various purposes made it possible to detect the highly anisotropic character of the spatial electron distribution in radiation belts in both Southern and Northern hemispheres. We detected also the presence of low-energy electrons at all latitudes during the main phase of a weak geomagnetic storm.

Trajectories Derived from Periodic Orbits around the Lagrangian Point L1 and Lunar Swing-Bys: Application in Transfers to Near-Earth Asteroids

To present a set of trajectories derived from the retrograde periodic orbits around the Lagrangian equilibrium point L1, this paper considers the Circular Restricted Three-body Problem with Earth-Moon masses (CR3BP), the Restricted Bicircular, and Full Four-Body Sun-Earth-Moon-spacecraft Problems (BCR4BP and FR4BP, respectively). These periodic orbits are predicted by the dynamics of the CR3BP. To generate the trajectories of this set, first, slightly different increments of velocity (∆Vs) from those needed to generate periodic orbits around L1 are applied to a spacecraft in circular low Earth orbits in the same direction of their motion when the Earth, the spacecraft, and the Moon are aligned in this order. Thus, translunar trajectories derived from the periodic orbits are obtained and they will lead the spacecraft to the vicinity of the Moon. Depending on the values of the |∆Vs|, which are also functions of the relative positioning between the Sun, the Earth, and the Moon, three types of trajectories of interest are found: Collision with the Moon, escape, and geocentric orbits with large semi-major axes. For a well-defined interval of the |∆Vs|, the trajectories accomplish swing-bys with the Moon and obtain energy to escape from the Earth–Moon system and reach Near-Earth Asteroids (NEAs) between the orbits of Venus and Mars. This procedure reduces the costs of inserting spacecraft into transfer trajectories to a set of NEAs in terms of the required |∆V| by up to 5% when compared to Lambert’s problem, for example. This work also presents analyses of examples of transfers to the NEAs 3361 Orpheus, 99942 Apophis, and 65803 Didymos, from 2025 on.

The CYGNSS Mission: On-Going Science Team Investigations

In 2012, the National Aeronautics and Space Administration (NASA) selected the CYclone Global Navigation Satellite System (CYGNSS) mission coordinated by the University of Michigan (UM) as a low-cost and high-science Earth Venture Mission. The CYGNSS mission was originally proposed for ocean surface wind speed estimation over Tropical Cyclones (TCs) using Earth-reflected Global Positioning System (GPS) signals, as signals of opportunity. The orbital configuration of each CYGNSS satellite is a circular Low Earth Orbit (LEO) with an altitude ~520 km and an inclination angle of ~35°. Each single Delay Doppler Mapping Instrument (DDMI) aboard the eight CYGNSS microsatellites collects forward scattered signals along four specular directions (incidence angle of the incident wave equals incidence angle of the reflected wave) corresponding to four different transmitting GPS spacecrafts, simultaneously. As such, CYGNSS allows one to sample the Earth’s surface along 32 tracks simultaneously, within a wide range of the satellites’ elevation angles over tropical latitudes. Following the Earth Science Division 2020 Senior Review, NASA announced recently it is extending the CYGNSS mission through 30 September 2023. The extended CYGNSS mission phase is focused on both ocean and land surface scientific investigations. In addition to ocean surface wind speed estimation, CYGNSS has also shown a significant ability to retrieve several geophysical parameters over land surfaces, such as Soil Moisture Content (SMC), Above Ground Biomass (AGB), and surface water extent. The on-going science team investigations are presented in this article.

Circular orbit transfer employing recursive nonlinear state transformations and nonlinear feedback

A recursive application of approximate feedback linearization is applied to low-Earth circular orbit rendezvous utilizing continuous low thrust propulsion. A family of solutions is established up to the $$\rho $$ -th degree utilizing the null space that appears at each step in the computations. A convergence analysis is presented to guarantee that the recursive process converges as $$\rho \rightarrow \infty $$ . During the recursive computations, patterns were detected as part of the solution process that illuminated a family of exact nonlinear control solutions obtained by judiciously selecting the null space coefficients. Examples of various analytic solutions are presented. The relationship between the $$\rho $$ -th degree exact solution obtained through the recursive approximations and a known exact solution is illustrated where it is shown that different exact solutions can have different performance in terms of fuel usage leading to the possibility of optimization considerations in selecting the desired exact solution.

Performance Analysis of Space Surveillance Using Space Based Optical Sensors

This paper aims to describe the analysis of the performance of an electro-optical space-based sensor for space surveillance purposes and space debris detection in the geostationary (GEO) ring. Such sensor is considered to be operating on a dawn–dusk Sun-synchronous, circular low Earth orbit at an altitude of 630 Km, while its optical characteristics have been taken from those of the Space-Based Visible (SBV) sensor. Two main simulations have been carried out through the use of the MATLAB software. The first simulation deals with the detection capability of the sensor, which is discussed in terms of detectable visual magnitude when the target of the observation is a diffuse sphere orbiting in the geostationary (GEO) orbit; its minimum detectable size is then determined. In addition, the relative geometry between the Sun, the sensor and the target has also been studied along with the configurations which can limit the visibility of the sensor over the target. The second simulation has been used to evaluate the performance of the sensor in terms of number of detectable GEO targets and duration of the observation when a certain pointing strategy is adopted. In such strategy, two SBV-like sensors are placed on the same orbit, thus creating a constellation in which each sensor points towards a fixed location in the inertial space. These locations have been chosen to be the geosynchronous pinch points.

Analytic Model for Low Earth Orbit Satellite Solar Power

An analytic model for calculating the instantaneous and average orbital power available to a satellite in a circular low Earth orbit is presented. By accounting for the effects of angle of solar incidence and eclipsing, for both stabilized and tumbling satellites, this model allows for fast, simple, accurate calculations of available power without the use of complex, often expensive, numerical simulation software. In addition, this method provides the value of maximum and minimum solar power available.

A method based on Jacobi Integral variational equation for computing Earth-Moon trajectories in the four-body problem

This work proposes an alternative method for solving the two-point boundary value problem concerning to Earth-Moon bi-impulsive trajectories in the dynamics of the planar bi-circular restricted four-body problem, which describes the motion of a space vehicle subjected to the gravitational attraction of Earth, Moon and Sun. Initially, the space vehicle is at a circular low Earth orbit (LEO) with prescribed altitude. After applying the first impulsive velocity increment, the space vehicle is inserted into a transfer trajectory. The second velocity increment is applied to decelerate and circularize the movement of the space vehicle at a circular low Moon orbit (LMO) with prescribed altitude. To solve this problem, a new two-point boundary value problem (TPBVP) is formulated, which includes an unknown value of the Jacobi integral at the departure time, and, a prescribed value at the arrival time. Since the Jacobi integral is not a first integral for the four-body problem, it is taken as additional state variable, and, its variational equation is added to the system of differential equation in the description of the dynamics of the space vehicle. Taking into account the boundary conditions, expressions for the velocity increments are deduced from the Jacobi integral computed at the initial and final times. Based on this new TPBVP, a numerical procedure is proposed to obtain different families of Earth-Moon trajectories with decreasingly fuel consumption.

Density-Based Feedback Control for Earth Orbiting Swarms via Velocity Fields

In this paper, we consider controlling a spacecraft swarm in a circular low Earth orbit (LEO) via controlling its density distribution. We build on our previous research on decentralized density feedback control, which generates smooth velocity fields as a function of time and position. Performing kernel density estimation, each spacecraft compares desired density profiles with local density estimates, and invokes heat equation based feedback controller to generate the velocity commands to be tracked. Governed by Clohessy-Wiltshire equations, each spacecraft uses the velocity commands to calculate the next vehicle state and two consecutive ∆V commands to transition between states described in the local-vertical local-horizontal (LVLH) frame. The continuity of the velocity field facilitates smooth and organized behavior for spacecraft and as the numerical results illustrate, the swarm density distribution converges to the desired density distributions.

Model Predictive Control for Continuous Low Thrust Satellite Formation Flying

This paper presents a low thrust guidance and control strategy for distributed spacecraft systems flying in proximity in nearly circular Low Earth Orbit (LEO) using Model Predictive Control (MPC). It is designed to converge to arbitrary three dimensional (3D) force free formation topologies taking different constraints and system limitations into account. However it is also applicable to forced formations with the same given constraints. Its performance is evaluated using simulations of different typical 3D formation topologies.

A study of the influence of the sun on optimal two-impulse Earth-to-Moon trajectories with moderate time of flight in the three-body and four-body models

In this work, a study about the influence of the Sun on optimal two-impulse Earth-to-Moon trajectories for interior transfers with moderate time of flight is presented considering the three-body and the four-body models. The optimization criterion is the total characteristic velocity which represents the fuel consumption of an infinite thrust propulsion system. The optimization problem has been formulated using the classic planar circular restricted three-body problem (PCR3BP) and the planar bi-circular restricted four-body problem (PBR4BP), and, it consists of transferring a spacecraft from a circular low Earth orbit (LEO) to a circular low Moon orbit (LMO) with minimum fuel consumption. The Sequential Gradient Restoration Algorithm (SGRA) is applied to determine the optimal solutions. Numerical results are presented for several final altitudes of a clockwise or a counterclockwise circular low Moon orbit considering a specified altitude of a counterclockwise circular low Earth orbit. Two types of analysis are performed: in the first one, the initial position of the Sun is taken as a parameter and the major parameters describing the optimal trajectories are obtained by solving an optimization problem of one degree of freedom. In the second analysis, an optimization problem with two degrees of freedom is considered and the initial position of the Sun is taken as an additional unknown.

Satellite Constellation Orbit Design Optimization with Combined Genetic Algorithm and Semianalytical Approach

This paper focuses on maximizing the percent coverage and minimizing the revisit time for a small satellite constellation with limited coverage. A target area represented by a polygon defined by grid points is chosen instead of using a target point only. The constellation consists of nonsymmetric and circular Low Earth Orbit (LEO) satellites. A global optimization method, Genetic Algorithm (GA), is chosen due to its ability to locate a global optimum solution for nonlinear multiobjective problems. From six orbital elements, five elements (semimajor axis, inclination, argument of perigee, longitude of ascending node, and mean anomaly) are varied as optimization design variables. A multiobjective optimization study is conducted in this study with percent coverage and revisit time as the two main parameters to analyze the performance of the constellation. Some efforts are made to improve the objective function and to minimize the computational load. A semianalytical approach is implemented to speed up the guessing of initial orbital elements. To determine the best parametric operator combinations, the fitness value and the computational time from each study cases are compared.

Finite-time continuous sliding mode magneto-coulombic satellite attitude control

A finite-time continuous sliding mode control is proposed and designed for the attitude control of an earth-pointing magneto-Coulombic satellite actuated using Coulomb shells and orbiting in a circular low earth orbit. A new nonlinear sliding manifold is proposed in terms of errors in quaternions and angular velocity and is used for average finite-time continuous sliding control formulation and closed loop stability analysis of averaged system dynamics. A real and finite-time continuous sliding control is proposed for real-time control of the system. and it is proved that this is reduced to the average finite-time continuous sliding control. Using the Lyapunov theorem, it is proved that the averaged closed loop system is globally stable and converges in finite time to the proposed sliding manifold in the presence of disturbance and destabilizing gravity gradient torque, which is followed by fast and finite-time convergence to the desired orientation and angular velocity (origin) along the sliding manifold. Simulation results of the sliding control are compared against those obtained using proportional-differential control, showing superiority of the sliding control in terms of less charge requirement and better convergence of states of the system. It is also shown that changing the distance between the shells and keeping moments of inertia constant does not alter the results qualitatively or quantitatively except for scaling of charge.

The Altitude Attainment and Inclination Alignment for the Satellite Launched from Kourou Site

The geostationary orbit is a circular orbit 35786 kilometers above the Earth's equator and following the direction of the Earth's rotation. Communications satellites are often placed in the geostationary orbit. The rockets are used to place the satellite in geostationary orbit. The main goal of the launching process is to enable the satellite to acquire the desired geostationary orbit space parameters. There are two methods applied for putting satellites in the geostationary orbit, the first one is when a rocket takes the spacecraft to a low Earth circular orbit and then towards geostationary transfer orbit, and by the second method, the low circular orbit is skipped and the satellite goes straight to geostationary transfer orbit. Because of too large distance from the Earth’s surface this placement to geostationary transfer orbit is not done at once. The location of Kourou at French Guiana is the launching site of France and also shared with ESA (European Space Agency). The launch process from Kourou site firstly places the satellite at geostationary transfer elliptical orbit, and then the orbit geostationarity is achieved through two phases, the first phase is the altitude attainment at 35786km and the second is the inclination to be aligned with equatorial plane. During the whole process, the needed satellite’s propellant mass must be minimized. Three thrusts apogee and nodal thrust method applied for the altitude attainment and then the inclination alignment for the satellite to be consolidated in geostationary orbit, which is launched from Kourou site, is simulated and analyzed within this paper.

Adaptive control for two-spacecraft electromagnetic formation keeping

Electromagnetic spacecraft formation flying is a new type of spacecraft formation using electromagnetic forces between spacecraft to control relative motion. Primary principles of the relative motion of a two-craft electromagnetic formation were discussed. Analytical solutions of control magnetic moments according to commanded control forces for the case of two-spacecraft electromagnetic formation were developed considering the energy consumption equilibrium. Nonlinear relative motion dynamic models for a two-spacecraft electromagnetic formation were derived based on polar coordinates. Parameter uncertainties of the equations because of the approximation of the electromagnetic model and unknown disturbed forces were analyzed. Nonlinear adaptive feedback control laws for formation keeping were then designed,and the closed-loop dynamic procedure of a two-spacecraft electromagnetic formation in a low earth circular orbit was simulated. Results indicate that the relative motion equations and the adaptive control laws are valid. The formation can effectively converge to desired states and the unknown parameters are estimated accurately. The simulations demonstrate that electromagnetic forces between spacecraft can be used to realize spacecraft formation keeping and the electromagnetic formation flying is feasible.