2009. № 34

The algorithm developed in Keldysh Institute of Applied Mathematics was successfully used for many years to calculate rendezvous task for the piloted and automated vehicle and orbital station [1,2]. This algorithm has become the main choice for calculating maneuvers of European ATV owing to being simple, reliable and efficient. The ideas that proved their efficiency in practice were used to develop the algorithm of calculating maneuvers of satellite constellation deployment [3,4]. This paper describes the ways of improving the algorithm, referred to in [4], for finding difference in the numbers of revolutions of the chaser vehicle and target vehicle and also provides algorithms of numeric-analytical definition of maneuver parameters themselves.

Keywords: Optimal maneuvers, rendezvous task, satellite constellation, satellite constellation deployment, formation flying

Existing RNF and GNSS OC description system provides accuracy, availability, integrity, continuity. These parameters are insufficient for the RNF and OC comparative analysis.

Keywords: radionavigation field; radionavigation field parameters and characteristics; navigation services; degradation areas

Russia and USA both have Moon and Mars exploration in their plans.  Exploration will require specific planetary infrastructure, with positioning, navigation and timing as its crucial component.
Therefore, the design of new GNSS and the creation of the appropriate design instruments are high priority tasks of the global growth.

Keywords: Advanced Global Navigation Satellite Systems; multifunctional systems

At present, the projects with constellations (and formation flying) attract great interest for various reasons. Firstly, they can represent various configurations. And these architectures are extremely interesting from the point of view of space applications. Their structural nature can be used to create a flexible and efficient range of measurements, to synthesize virtual instrument. Here is the possibility of receiving of wide variety: methods (options) for given problems & data.
The semi-analytical theory named THEONA provides the satellite motion prediction with high rapidity and good accuracy. For the problem examined here, an efficiency of the theory increases even more: it has developed additional methods with united structures of integrals and special functions of THEONA. Accordingly, it leads to higher speed computations, relative error decreases to 10-12. So, new version of THEONA permits to provide almost all necessary operative and research calculations for the tasks with constellations and formation flying.
I propose several examples of accuracy estimations of orbit propagation based on THEONA. It is considered absolute and relative motion of constellations for different classes of orbits. This analysis shows an efficient use of THEONA for the study of relative motion evolution of constellations.

Keywords: Satellite constellations, formation flying, orbit prediction, semi-analytical satellite theory, perturbations, special functions

The tasks of optimization of nominal orbital structures of satellite systems are examined in the paper. Attention is given to the ways of formulating the tasks whose major components are models of dynamically stable orbital structures under optimization and quality criteria aimed to attain different goals. The paper discusses the results obtained by other authors researching into the tasks of optimization of nominal orbital structures. Further research directions are outlined in the paper.
The tasks of optimization of the processes of flexible correction of satellite systems are examined. Flexibility of correction means that its purpose is to bring the parameters of relative satellites motion to nominal values. For the cases when correction is performed by means of low-thrust engines, analytical solutions of tasks for different types of flexibility have been found by methods of the group presentation theory.

Keywords: Satellite system; orbit; optimization of motion

In order to keep Formation Flying it is necessary to eliminate the secular drift in relative motion. With an allowance for nonspherical Earth if the satellites have different orbits there are the secular drifts in longitude of ascending node and in argument of perigee. In the present paper the ability to nullify the secular drifts for two satellites in perturbed gravitational field is performed. The deputy satellite has low thrust propulsion and passive magnetic attitude control system.

Keywords: Small satellite; passive magnetic attitude control system; Formation Flying; relative position control

A short historical survey of satellite constellation design for continuous coverage is presented. Comparison results of the major types circular orbit constellations for continuous global coverage are described. A table classification of satellite constellations based on an interface between concepts «constellation type» and «coverage type» is proposed. New constellation design methods for the complex coverage are also briefly described. Two type of continuous coverage are considered. First, it is a traditional simple continuous coverage connected with a visibility points on the Earth’s surface. Second, it is a more complex coverage for arbitrary geographic regions. If implies that, at any time, the region is completely or partially within the instantaneous access area of a satellite of the constellation. The key idea of the methods is based on common two-dimensional maps of satellite constellations and coverage requirements. The space dimensions are right of ascension of ascending node (in an inertial space) and time. In the space, visibility requirements of each region can be presented as a polygon and satellite constellation motion as a uniform moving grid. At any time, at least one grid vertex must belong to the polygon. The optimal configuration of the satellite constellation corresponds to the maximum sparse grid. Algorithmic basis for the methods is modern computational geometry. As an example, satellite constellation in circular orbit for continuous coverage of full geographic region by a satellite from the constellation is presented. The methods are extended to design of MOLNIYA type elliptic orbit constellations. Kinematically regular elliptic orbit constellations for continuous coverage of geographic latitude belts is considered. An analysis method of single track constellations for continuous coverage of arbitrary geographic areas by a satellite from the constellation is described. As examples, satellite constellations on elliptical orbits with periods ~12 and ~24 hours are considered.

Keywords: Satellite constellations; constellation design; continuous coverage; region coverage; MOLNIYA type orbit

An approach is submitted to a research based on orbital dice for GLONASS SV ephemeris determination on condition that intersatellite range measurements are used.  The formulation and proof is made for sufficient conditions of uniqueness of the orbital dice improvement using intersatellite range measurements.  In addition, mathematical proof is made for satisfiability of sufficient conditions in case of GLONASS system.  Analytical estimations are derived for orbital dice relative coordinate improvement errors subject to intersatellite range measurements errors.  Derived results allow to relate characteristics of moving orbital dice with observability of broadcast ephemeris improvement problem and as well as with processing algorithm effectiveness for intersatellite range measurements.

Keywords: Basic constellations; intersatellite measurements; orbital dice; GLONASS system; a chain of constellations; ephemeris

The quasi-rigid body formulation views a formation of satellites as a single entity by attaching a coordinate frame to the formation that captures its overall orientation. To begin, this paper reviews the formulation and presents the quasi-rigid body equations of motion. Then, a nonlinear controller for a formation is derived using Lyapunov stability theory. System performance criteria can be met by tuning the proportional and derivative feedback gains. Because the controller is based upon the quasi-rigid body equations of motion, it can be used to regulate orientation errors or track a time-varying trajectory for the attitude of the formation; this is shown through simulation results. Finally, the formulation is used to design two constrained reorientation trajectories for formations in deep space, one of which is a single Euler axis rotation, while the second is motivated by axisymmetric rigid body dynamics. It is shown that the second approach provides as much as 11% fuel savings over the Euler axis rotation.

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Low Earth orbit constellations deserve several advantages with respect to geostationary platforms, i.e. lower costs for satellite development and launch, increased imaging resolution, as well as reduced power requirements and signal time delays. This research is concerned with an original method for constellation design, based on the use of a correlation function. All satellites are placed in repeating ground track orbits, and two conflicting requirements are considered: the maximization of the maximum continuous coverage and the minimization of the maximum revisit time of a target area located on Earth surface. A suitable way of determining the related optimal constellation configurations is based on avoiding overlapping between visible passes of distinct satellites. With this intent, an analytic expression can be derived for the correlation function, which is employed to evaluate the overlapping between visible passes. Then an algorithmic search for the zeros of this function allows determining some constellation configurations with the desired characteristics. This heuristic method turns out to be a successful approach for constellation design and several results are reported with reference to different latitudes of the target area and distinct repeating orbits.

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This paper considers the problem of orbital maneuvers that use a low thrust sub-optimal control to perform station-keeping in a satellite that is part of a constellation. The main idea is to assume that the position of a satellite that belongs to a constellation is given, as well as a nominal orbit for this satellite, not far from the initial orbit. Then, it is necessary to maneuver this satellite from its current position to the nominal specified orbit. The control available to perform this maneuver is the application of a low thrust to the satellite and the objective is to perform this maneuver with minimum fuel consumption. A sub-optimal approach will be used, to allow a simple implementation of the hardware.  

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A bit more than ten years after the deployment of the first constellations of satellites in the field of telecommunications, there is a need to re-populate them with a second generation to ensure the continuity of service. Thales Alenia Space is aware of this need as it has been strongly involved in this field in the late 90's and has always kept an active leading part in the reflections on constellations. In that context, this article presents a review of possible replacement strategies of telecom constellations, after having exposed the complexity of the process by enumerating some of the constraints that apply to this critical phase in terms of service.

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There has been impending interest in the formation flying with many satellites. Multiple satellite system enhances the missions' flexibility with less total mass and cost, and realizes some missions that were impossible with a single satellite. At the Institute of Space and Astronautical Science (ISAS/JAXA), the plasma and magnetic field observation missions with several satellites is under investigation. The mission under consideration is designated as SCOPE. The observation area of the SCOPE mission is twenty or thirty earth radii away fr om the center of the earth wh ere the geomagnetic field has interaction with the energetic particles from the sun. Therefore its orbit becomes highly elliptic. This paper first discusses the design method for spontaneous maintaining the formation geometry on the elliptic orbits. In the observation aspect, the formation of plural satellites is requested to constitute a polygon that assures the high spatial resolution observation. This study next show the orbital design method for the SCOPE mission. The frozen property that maintains high spatial resolution near the apogee is found feasible for elliptic orbit. Numerical examples are presented with practical illustrations.

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Imaging satellites utilize optical, infrared, or radar sensors to conduct the Earth observations, which are now extensively applied to our daily life. In the beginning we considered one-satellite mission, which is required to implement with high-resolution sensors, short revisit cycle, and global coverage. Now we considered mission conducted with multiple satellites, which may be operated by same or different countries, to increase the spatial coverage and to reduce the revisit cycle. Equatorial coverage can be achieved with multiple satellites, but the polar coverage can be done with a large field of regard. Daily revisit and global coverage can be both satisfied by selection of affordable number of satellites with certain field of regard. When taking into account operation complexity and polar coverage, it is recommended to use a constellation with shorter repeat cycle and large field of regard. One can also combine it with an equatorial orbit of 0-deg inclination or a polar orbit of 90-deg inclination.

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Since nineties, the GAUSS (Group of Astrodynamics of the «Sapienza» University of Roma) has started UNISAT program with the aim to design, manufacture and launch University microsatellites, completely built and operated in orbit by students of the school of Aerospace engineering. In the framework of this program four satellites have been launched, two years apart one from the other, starting in 2000, and a fifth satellite is nowadays under construction.

In the same period, GAUSS has also been involved in optical space debris surveillance, participating to the IADC (Inter-Agency Space Debris Coordination Committee) observation joint campaign and, more recently, by manufacturing the first Italian observatory completely dedicated to space debris monitoring.

Combining these two experiences GAUSS students, researchers and professors are analysing the feasibility of a formation flight mission in order to detect space debris, taking advantage of an in situ observation above Earth’s atmosphere.

The paper deals with this space debris detection mission focussing on LEO orbital regime. As a matter of fact, these orbits are very crowded. Small (from 1 to 10 centimetres) debris in these orbits are difficult to detect and track by ground based optical systems. A constellation of satellites boarding optics and sensors sensible to infrared or visible radiation could monitor the LEO space debris environment through in-situ measurements. Moreover, a constellation of satellites flying in small formations permits to achieve not only debris detection but also preliminary orbit determination, thus simplifying follow-up ground base observations.

The paper gives an overview of the proposed constellation of satellites flying in small formations, highlighting main constraints, achievements and flaws of a few possible solutions.

The proposed mission configuration is made up of three satellites in a trailing formation. One satellite, the central one, will be responsible of detection and fast orbit determination of LEO objects. The other two smaller satellites will be able to track the object in order to allow a larger number of pictures and a better orbit determination.

Numerical simulations of these control strategies have been carried out taking into account different sensor accuracies; the results of some study cases are reported.

Two orbit control strategies are analyzed in order to keep the satellites within the formation constraints: the use of periodic maneuvers with chemical propulsion or a continuous counterbalance of the main perturbation (i.e. the aerodynamic one) with the use of Pulsed Plasma Thrusters.

In the end, sensors and optics feasible for this mission are analysed and two different image processing methods have been studied in order to achieve automatic in-orbit debris detection.

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On the 3rd of April, 2008, the Automated Transfer Vehicle Jules Verne (ATV-JV) successfully docked to the International Space Station (ISS). One of the mission objectives was to demonstrate the new technology developed to perform autonomous rendezvous (RDV) to the ISS with particular focus on functionalities critical for guaranteeing the ISS safety. ATV RDV innovative technology has been in flight demonstrated. Some of those technologies have potential application to formation flight: phasing manoeuvres to fly ATV to a precise position in proximity of ISS, accurate relative GPS autonomous navigation with proximity link communication, innovative RDV sensors, RDV guidance via station keeping points and final approach.
An ESA-lead team based at ATV-CC in Toulouse has performed a near real-time assessment of GNC critical functions of RDV during two dedicated demonstration days.
Successful trilateral reviews of the reports, involving teams from ESA, CNES, Astrium in ATV-CC, ISS Mission Control Centre in Houston and in Moscow have confirmed the good behaviour of ATV GNC resulting in a GO from ISS Management Team for the continuation of ATV-JV mission. The proposed paper presents a summary of results of in-flight experience which are related to the assessment of RDV technology with emphasis on observed performance of those functionalities which could be applied to formation flights.

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SIMBOL-X is a hard X-ray space-based observatory to be launched in 2013 onto a High Elliptical Orbit (HEO). This new generation telescope covers by a single instrument a continuous energy range starting at classical X-rays and extending to hard X-rays (i.e. from 0.5 to 80 keV). For mass and size reasons, a classical monolithic instrument cannot be used and this Franco-Italian mission will thus consist of two satellites. The Mirror spacecraft (MSC) will be in free flight on a HEO orbit and will target very precisely the source to observe, focusing the hard X-ray emission thanks to its mirror module. At the focal point, 20 meters behind the Mirror satellite, the Detector spacecraft (DSC) maintains its position on a forced orbit. This paper aims at presenting some of SIMBOL-X GNC challenges. The first part will introduce main formation flying phases and related system requirements. In the next section focus will be made on DSC GNC design: algorithms and equipment adapted to the previously introduced phases will be presented. The last section deals with the line-of-sight (LOS) restitution problematic: for scientific purposes the line-of-sight between the two spacecrafts shall be reconstructed a posteriori very precisely. The last section will thus describe the chosen methodology to address this challenging problem.

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This paper deals with how the formation flight target is expressed as trivial and fixed in some appropriate coordinates, instead of an orbital target. This fixed bin/slot property is essential for the formation maintenance control. A new coordinate transformation is introduced for the case of circular reference trajectory, and compared with conventional C-W coordinate, the advantage of the new coordinate transformation is shown in terms of control performance index and formation maintenance property. Then, the discussion is extended to the case of general elliptic reference orbit, where regularization and Levi-Civita non-linear coordinate transform is introduced as a formation design methodology. Pseudo-constant distance formation around elliptic orbit is designed as an example and the effectiveness of the proposed method is numerically verified.

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There is a growing interest in formation flight of satellites and its control strategy. The centralized control architecture is optimal when all information is shared by every member in the formation, however, it possibly shows unstable behavior in a realistic situation where the information exchange cost (for example, communication load) is large and the information can be only partially shared. This paper proposes a new decentralized control strategy for multi-vehicle formation control, which effectively propagates the information between adjacent agents and uses the information in the control calculation. The strategy can overcome the drawback of the centralized approach while maintaining the communication load very low. The effectiveness of the proposed strategy is numerically verified by applying to one-dimensional car train control problem and two-dimensional in-plane satellite formation control problem.

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The coordination and control of a constellation of spacecraft, flying a few meters from one another, dictates several interesting design requirements, including efficient architectures and algorithms for formation acquisition, reorientation and resizing. The spacecraft must perform these transitions without interfering or colliding into each other. Furthermore position keeping is fundamental for formation efficiency. This paper presents an optimal deployment of the DARWIN formation using the potential function control technique in the vicinity of the Sun-Earth L2 point. The method hinges on defining a potential function from the geometric configuration of the constellation together with any collision avoidance requirement. A review of the fundamentals of relative motion and dynamics is presented before describing the features of the different control algorithms and validating the method using Lyapunov’s theorem. The potential function method has been used to control both translational and rotational control. Obstacles, in the shape of other satellites and constrained payload pointing directions have been included. Finally it will be shown that the attitude control algorithm can successfully used to avoid plume impingement that can have catastrophic consequences for the mission.

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Electromagnetic formation flight (EMFF) for satellites in LEO is discussed. EMFF is a technique to control the satellites’ relative position using electromagnetic force without any propellants. It is estimated that the superconductive magnets have capability to produce required magnetic force for formation keeping. The problem to use EMFF in LEO is the huge amount of disturbance torque, caused by enormous magnetic moment and earth magnetic field. Sinusoidal driving of the superconductive coil is proposed for this issue, and novel method is also proposed for magnetic force control using phase difference between the magnetic moments. Proposed methods are evaluated with experiments with actual superconductive coil, and hardware in the loop simulations is also carried out to demonstrate the relative position control capability of proposed system.

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Future space missions such as virtual telescopes and interferometers require precision formation flying, i.e., a highly precise control of the relative positions of spacecrafts. This paper discusses a strategy to suppress the relative position variation during one orbit. It proposes a new approach to design a state feedback impulsive controller. By estimating the disturbance fr om the result of the previous control cycle and compensating for the estimated disturbance in a feed-forward manner, stable control can be achieved under uncertainties of natural disturbances. An along-track formation control in a low earth orbit is taken as an example, wh ere the J2 term and air drag are considered as representative natural disturbances. Numerical simulations are performed to validate the proposed controller.

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PROBA-3 is an ESA technology demonstration mission devoted to the validation of the novel technologies required by future Formation Flying (FF) missions. The PROBA-3 mission and in particular its Guidance, Navigation and Control (GNC) architecture are being defined within the currently running Bridging Step after phase A. The Bridging Step includes evaluation and consolidation of the results of the two previous phase A studies before the start of the project phase B foreseen for the end of 2008.
This paper describes PROBA-3 GNC including requirements, GNC architecture and design, and to a lesser extent, the calibration process. A key feature of the GNC is a design that allows the verification of different types of control architectures (functions allocations), facilitating the change in the level of centralisation of the control. The paper also includes the definition and description of the different GNC modes, the sensors and actuators to be used at SC and at FF level, and the way they are going to be used. Emphasis is put on the flexibility of the system for maximisation of technology demonstration.

Keywords: Proba-3; Formation Flying; GNC

Interferometric missions with an instrument distributed on several vehicles in orbit allow deep space observations with an accuracy never reached, but demand a formation flying Guidance Navigation and Control (GNC) with a ultra-high level of precision. One ESA mission of that class, dedicated to exo-planets detection and characterization is nowadays identified: DARWIN. The different GNC stages, from the initial closed loop engagement at the end of deployment to the final control of the Optical Path Difference (OPD) and intensity mismatch used in science mode will be described.
First, a focus on the Navigation Process Unit (NPU) based on the Radio-Frequency (RF) sensor and the modes developed to achieve the coarse formation acquisition, the FDIR Collision Avoidance Mode (CAM) and the recovery after a failure will be given. The RF subsystem is developed by TAS under a CNES contract. Its aim is to provide a coarse relative position sensor and the Inter-Satellite Link (ISL), necessary for the finest mode. It covers a 30 km distance range, but will be adapted up to a 100 km range. It has 3 functional modes and is bi-frequency. The coarse mode covers all the directions of space and is based on the received powers measurements of several antennas scattered around each vehicle. The interferometric mode Before Ambiguity Raising (BIAR) measures the widelane phase shift and provides intermediate accuracy measurements in a privileged direction. The interferometric mode After Ambiguity Raising (AIAR) measures the carrier phase and provides 1 cm on inter-vehicles distance and 1 deg on Line of Sight accuracy measurements in the same direction, after a GNC maneuver. The two last modes require a triplet of antennas forming two orthogonal baselines. The CAM and the reconfiguration are based on the RF coarse mode. The CAM is used in a decentralized way to assure the robustness to formation level failures, whereas the reconfiguration is centralized in one vehicle to have a better coordination. The first control stage is based on the fine RF mode. These modes are generic to all formation flying missions.
Then, an overview of the EMMA configuration of DARWIN, a novel concept proposed by TAS, will be described from the system and GNC points of view. The Darwin GNC contains two more control stages in addition to the first one based on the RF sensor. The second stage based on optical coarse and fine metrology and ionic or Field Electrical Emission Propulsion (FEEP) actuates the Collecting Spacecrafts (CS) relatively to the Beam Combiner Spacecraft (BCS) and acts on the formation's geometry. The third stage, internal to the BCS, acts directly on the scientific beam, adding an optical path with an Optical Delay Line (ODL) and changing its orientation with corrective tip/tilt mirrors based on piezo-actuators. The joint use of these two stages allows to meet the science requirements imposed on the OPD (<1nm) and the intensity mismatch.

Keywords: Radio-Frequency Sensor; Collision Avoidance; Reconfiguration; Darwin Mission; Emma; Guidance Navigation Control

Formation flying is commonly identified as the collective usage of two or more cooperative spacecraft to exercise the function of a single monolithic virtual instrument. The distribution of tasks and payloads among fleets of coordinated smaller satellites offers the possibility to overcome the classical limitations of traditional single-satellite systems. The science return is enhanced through observations made with larger, configurable baselines and an improved degree of redundancy can be achieved in the event of failures. Different classes of formation flying missions are currently under discussion within the European engineering and science community: technology demonstration missions (e.g. PRISMA, PROBA-3), synthetic aperture interferometers and gravimeters for Earth observation (e.g. TanDEM-X, postGOCE), dual spacecraft telescopes which aim at the detailed spectral investigation of astronomical sources (e.g., XEUS, SIMBOL-X), multi-spacecraft interferometers in the infrared and visible wavelength regions as a key to new astrophysics discoveries and to the direct search for terrestrial exoplanets (e.g., DARWIN, PEGASE). These missions are characterized by different levels of complexity, mainly dictated by the payload metrology and actuation needs, and require a high level of on-board autonomy to satisfy the continuously increasing demand of relative navigation and control accuracy.
In order to respond to this demand the DLR’s German Space Operations Center (GSOC) is pursuing a dedicated autonomous formation flying research and development roadmap since 1998. The research work has largely been motivated by the conviction that only the development, deployment and on-orbit validation of innovative Guidance, Navigation and Control (GNC) techniques can bring formation flying to the forefront and enable the definitive transfer of this revolutionary technology to space. As a result the GSOC’s contributions to TanDEM-X and PRISMA (both launches expected in 2009) will demonstrate, for the first time in Europe, autonomous fuel-efficient formation keeping and reconfiguration on a routine basis, with minimum collision risk. After a comprehensive introduction on the state-of-the-art of the formation flying technology in Low Earth Orbit (LEO), the paper addresses the design, implementation and testing of the DLR/GSOC’s GNC subsystems for TanDEM-X and PRISMA demonstration missions. An overview of the developed subsystems is provided, highlighting communalities and differences of the two parallel developments. Furthermore key results from the validation of the guidance strategy, of the real-time GPS-based navigation and of the impulsive relative orbit control functions are presented.
A technological gap clearly exists between the remote sensing LEO formations, yet to be demonstrated, and the planned outer space distributed telescopes in high elliptical orbits or in the vicinity of the Lagrange points. It is not only given by the envisaged three-order-of-magnitude improvement of the required metrology and actuation needs, but is also driven by the necessity of implementing navigation systems at altitudes above the GNSS constellations. The final part of the paper is thus devoted to the identification of the major discrepancies between present and next generation formation flying. An attempt is made to define the way forward and offer an outlook beyond the first European technology demonstration missions.

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The geostationary circular orbit (GSO) (at the height of approximately 36,000 km with zero inclination) proposed by Arthur C. Clark in the middle of the 20-th century has become very popular for creation of different satellite systems for communication, TV broadcasting and some other purposes. Insertion one or more satellites to geostationary orbits is the best way to provide coverage of big equatorial parts of the globe. In spite of obvious GSO advantages it is clear now that GSO implementation has some difficulties connected with existing limitation for insertion of GSO satellites as well as impossibility to cover the non-equatorial regions of the globe.
The elliptical orbits of Molniya type firstly used for Russian communication satellites of the same name do not have the GSO disadvantages mentioned above. But this type of orbits (with 63,4 deg inclination and perigee and apogee of nearly 500 km and 40,000 km accordingly) can not pretend for universal use.
The locally geostationary orbits (LGO) invented by first author of the present paper in 1987 give the possibility for design of the satellite constellations in more general situations connected with the Earth local coverage. Mathematical description of the LGO includes GSO and Molniya orbits as the particular cases (herewith the GSO is the only circular orbit in the locally geostationary, generally elliptical, class of orbits).
Taking into account that implementation of LGO based on emulation of geostationary observation of the Earth using elliptical orbits is becoming nowadays very popular, the paper contains some new aspects of LGO parameters calculation for different implementations.

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Design of satellite constellations is associated with continuous or periodical coverage of an Earth region of interest depending on the concrete task of the space mission. Satellite constellations for continuous and periodical coverage are principally differed by orbit parameters as well as by methods of constellation design. In this study route theory for design of constellations for circular low Earth orbits providing periodical coverage with minimum revisit time criteria is described. The route theory includes: 1) consideration of so called route orbital pattern as mathematical abstraction of arbitrary constellation; 2) analytical solution for calculation problem of distribution of revisit time values on Earth surface for one satellite and multi-satellite route pattern; 3) formulating several regularities for revisit time as a function of satellites positions in constellation; 4) development of method for optimal design of constellation under given criterion and requirements. Using of route orbital patterns is practically not connected with restriction for type of constellation. It is also shown that optimal constellations calculated using the route theory are in general not worse and in many cases much better (in respect of criteria mentioned above) comparing with the analogs found using other well-known methods for periodical coverage.

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