Flower Constellations Research Articles

GPS radio occultation constellation design with the optimal performance in Asia Pacific region

The growing desire for better spatial and also temporal distribution of radio occultation data is a motivation for extensive researches considering either number of GNSS/receiver satellites or better optimization tools resulting in better distributions. This paper addresses the problem of designing a global positioning system-only radio occultation mission with the optimal performance in Asia Pacific region. Constellation Patterns are discussed and 2D-lattice and 3D-lattice flower constellations are adopted to develop a system with circular and elliptical orbits, respectively. A perturbed orbit propagation model leading to significantly more accurate pre-analysis is used. Emphasizing on the spatial and also temporal distribution of radio occultation events for the first time, distribution norm is provided as a volumetric distribution measure using Voronoi diagram concept in a 3D space consisting temporal and spatial intervals. Optimizations are performed using genetic algorithm to determine optimal constellation design parameters by the suitable fitness function and constraints devised. The resulted constellation has been evaluated by a regional comparison to the globally distributed FORMOSAT-3/COSMIC in terms of the distribution norm, number of radio occultation events and also coverage as an additional point-to-point distribution measure. Although it is demonstrated that the optimal 3D-lattice enjoys better performance than FORMOSAT-3, the design approach results in a 2D-lattice flower constellation which is superior to other constellations in regional emphasis of radio occultation events. Its global performance is discussed and it is demonstrated that using multi-GNSS receiver to increase satellites may not guarantee a good distribution of radio occultation data in some aspects.

Seeking GDOP-optimal Flower Constellations for global coverage problems through evolutionary algorithms

In this paper, given a certain number of satellites (Nsat), which is limited due to the sort of mission or economical reasons, the Flower Constellation with Nsat satellites which has the best geometrical configuration for a certain global coverage problem is sought by using evolutionary algorithms. In particular, genetic algorithm and particle swarm optimization algorithm are used. As a measure of optimality, the Geometric Dilution Of Precision (GDOP) value over 30000 points randomly and uniformly distributed over the Earth surface during the propagation time is used. The GDOP function, which depends on the geometry of the satellites with respect to the 30000 points over the Earth surface (as ground stations), corresponds to the fitness function of the evolutionary algorithms used throughout this work. Two different techniques are shown in this paper to reduce the computational cost of the search process: one that reduces the search space and the other that reduces the propagation time. The GDOP-optimal Flower Constellations are obtained when the number of satellites varies between 18 and 40. These configurations are analyzed and compared. Owing to the Flower Constellation theory we find explicit examples where eccentric orbits outperform circular ones for a global positioning system.

Lattice-preserving Flower Constellations under $$J_2$$ J 2 perturbations

2D Lattice Flower Constellations (2D-LFCs) are stable in the Keplerian model. This means that a flower constellation maintains its structure (the lattice) at any instant of time. However, this is not necessarily true when the $$J_2$$ harmonic is included in the gravitational potential of the Earth. This paper deals with the new theory of Lattice-preserving Flower Constellations, which shows how 2D-LFC can be designed in such a way that the relative displacement of the orbital parameters of its satellites is invariant even under the presence of the $$J_2$$ effect. This is achieved following two different procedures: the first consists of the modification of the semi-major axis of all the satellites in a 2D-LFC slightly to control their orbital period, and the second consists of the modification of the values for the eccentricity and inclination, so that the perturbations result in motion that still preserves the lattice of the flower constellation. The proposed theory of Lattice-preserving Flower Constellations validates the theory of 3D Lattice Flower Constellations and has a wide range of potential applications.

Design of flower constellations using necklaces

This paper introduces a new approach in the design of optimal satellite constellations comprising few satellites and characterized by symmetric distribution. Typical examples are missions requiring persistent observation of sites or regions for intelligence or science. To minimize the number of satellites, the mathematical necklace theory is applied to the two-dimensional Lattice Flower Constellation framework. The necklace theory identifies in the phasing space all satellite subsets characterized by symmetric distributions. Mathematically, these subsets are parameterized by necklaces, identifying the actual satellite locations in the first orbital plane and a shifting parameter governing phasing between subsequent orbital planes. This article specifically targets the emerging importance of small satellite formations and constellations.

The 3-D lattice theory of Flower Constellations

Flower Constellations (FCs) have been extensively studied for use in optimal constellation design. The Harmonic FCs (HFCs) subset, representing the symmetric configurations, have recently been reformulated into 2-D Lattice Flower Constellations (2D-LFCs), encompassing the complete set of HFCs. Elliptic orbits are generally avoided due to the deleterious effects of Earth’s oblateness on the constellation, but here we present a novel concept for avoiding this problem and enabling more effective global coverage utilizing elliptic orbits. This new 3D Lattice Flower Constellations (3D-LFCs) framework generalizes the 2D-LFCs, Walker constellations, elliptical Walker constellations, and many of Draim’s global coverage constellations. Previous studies have shown FCs can provide improved performance in global navigation over existing Global Navigation Satellite Systems (GNSS). We found a 3D-LFC design that improved the average positioning accuracy by 3.5 % while reducing launch \(\varDelta v\) requirements when compared to the existing Galileo GNSS constellation.

The 2-D lattice theory of Flower Constellations

The 2-D lattice theory of Flower Constellations, generalizing Harmonic Flower Constellations (the symmetric subset of Flower Constellations) as well as the Walker/ Mozhaev constellations, is presented here. This theory is a new general framework to design symmetric constellations using a \(2\times 2\) lattice matrix of integers or by its minimal representation, the Hermite normal form. From a geometrical point of view, the phasing of satellites is represented by a regular pattern (lattice) on a two-Dimensional torus. The 2-D lattice theory of Flower Constellations does not require any compatibility condition and uses a minimum set of integer parameters whose meaning are explored throughout the paper. This general minimum-parametrization framework allows us to obtain all symmetric distribution of satellites. Due to the \(J_2\) effect this design framework is meant for circular orbits and for elliptical orbits at critical inclination, or to design elliptical constellations for the unperturbed Keplerian case.

Reliable Global Navigation System using Flower Constellation

For many space missions using satellite constellations, symmetry of satellites distribution plays usually a key role. Symmetry may be considered in space and/or in time distribution. Examples of required symmetry in space distribution are in Earth observation missions (either, for local or global) as well as in navigation systems. It is intuitive that to optimally observe the Earth a satellite constellation should be synchronized with the Earth rotation rate. If a satellite constellation must be designed to constitute a communication network between Earth and Jupiter, then the orbital period of the constellation satellites should be synchronized with both Earth and Jupiter periods of revolution around the Sun. Another example is to design satellite constellations to optimally observe specific Earth sites or regions. Again, this satellites constellation should be synchronized with Earth’s rotational period and (since the time gap between two subsequent observations of the site should be constant) also implies time symmetry in satellites distribution. Obtaining this result will allow to design operational constellations for observing targets (sites, borders, regions) with persistence or assigned revisit times, while minimizing the number of satellites required. Constellations of satellites for continuous global or zonal Earth coverage have been well studied over the last twenty years, are well known and have been well documented [1], [2], [7], [8], [11], [13]. A symmetrical, inclined constellation, such as a Walker constellation [1], [2] provides excellent global coverage for remote sensing missions; however, applications where target revisit time or persistent observation are important lead to required variations of traditional designs [7], [8]. Also, few results are available that affect other figures of merit, such as continuous regional coverage and the systematic use of eccentric orbit constellations to optimize“hang time” over regions of interest. Optimization of such constellations is a complex problem and the general-purpose constellation design methodology used today is largely limited to Walker-like constellations. As opposed to Walker Constellations [1], [2], which were looking for symmetries in inertial reference frame, Flower Constellations [11] were devised to obtain symmetric distributions of satellites on rotating reference frames (e.g., Earth, Jupiter, satellite orbit). Since the theory of Flower Constellations has evolved with time the next section is dedicated to the summary of the theory up to the current status. The FCs solution space has been recently expanded with the Lattice theory [13], [14], encompassing all possible symmetric solutions.

New Insights on Flower Constellations Theory

The theory of Flower Constellations consists of a general methodology to design axial-symmetric satellite constellations whose satellites all move on the same trajectory with respect to a reference frame rotating at assigned constant angular velocity. Due to the complexity, the Flower Constellations theory as introduced by the authors, is incomplete, leaving open several problems, such as the equivalency and the similarity. In this article, the foundations of the Flower Constellations theory are revisited from a mathematical perspective, and three new important invariants of these constellations are found constituting fundamentals toward the complete theory. These are: the -space, the flower anomaly, and the configuration number. Using these elements, we provide a theorem giving the necessary and sufficient conditions for the equivalency problem in the -space and an algorithm allowing to generate all the similar Flower Constellations to a given one.

Design of Flower Constellations for Telecommunication Services

Satellite constellation designers take into consideration the entire telecommunication network and their design choices are influenced by many factors including: number of satellites, orbital characteristics, coverage area, network interconnections, system cost, and complexity. When the satellite constellation design method does not include a large number of possible configurations, then the final result of the design process is a suboptimal solution. Flower Constellation (FC) design provides a relatively new design approach which can overcome this problem. The time evolution of the FC theory is here summarized and the fundamental mathematics allowing the constellation design is provided. In particular, the theory is applied to the design and optimization of constellations maximizing the global coverage and the network connectivity via intersatellite links. Performance results are compared to the classical type of satellite constellations, i.e., the Walker Constellations. The performance improvement provided by FC design with respect to Walker Constellation design is shown. Finally, considerations regarding the cost for deployment and orbital control are also provided.

Flower elliptical-orbit constellation exploiting millimetre-wave radiometry and radio occultation for meteo-climatological applications

Abstract. This paper reports on the potential of combining elliptical-orbit Flower constellations with millimeter-wave radiometry and radio-occultation, a mission concept briefly named FloRad2. The advantages of flower constellation with respect to conventional orbits are discussed, including the flexibility ensuring increasing coverage with separate launches. Millimeter-wave radiometry and radio-occultation receivers provide the advantage to design fairly compact payloads that comply well with current technology of mini-satellites. Millimeter-wave radiometry and radio-occultation techniques are somewhat complementary and an optimal combination of these observations results in atmospheric products with enhanced vertical and horizontal resolutions. Thus, the combination of small, light payloads employing millimeter-wave radiometry and radio-occultation with Flower elliptical-orbit constellations may result in an optimal compromise between retrieval performances and system complexity that is ideal for continued long-term missions with meteorological and climatological applications.

High-Repetition Millimeter-Wave Passive Remote Sensing of Humidity and Hydrometeor Profiles from Elliptical Orbit Constellations

AbstractThe potential of an elliptical-orbit Flower Constellation of Millimeter-Wave Radiometers (FLORAD) for humidity profile and precipitating cloud observations is analyzed and discussed. The FLORAD mission scientific requirements are aimed at the retrieval of hydrological properties of the troposphere, specifically water vapor, cloud liquid content, rainfall, and snowfall profiles. This analysis is built on the results already obtained in previous works and is specifically devoted to evaluate the possibility of (i) deploying an incremental configuration of a Flower constellation of six minisatellites, optimized to provide the maximum revisit time over the Mediterranean area or, more generally, midlatitudes (between ±35° and ±65°); and (ii) evaluating in a quantitative way the accuracy of a one-dimensional variational data assimilation (1D-Var) Bayesian retrieval scheme to derive hydrometeor profiles at quasi-global scale using an optimized set of millimeter-wave frequencies. The obtained results show that a revisit time over the Mediterranean area (latitude 25° 45′, longitude −10° 35′°) of less than about 1 and 0.5 h can be obtained with four satellites and six satellites in Flower elliptical orbits, respectively. The accuracy of the retrieved hydrometeor profiles over land and sea for a winter and summer season at several latitudes shows the beneficial performance from using a combination of channels at 89, 118, 183, and 229 GHz. A lack of lower frequencies, such as those below 50 GHz, reduces the sounding capability for cloud lower layers, but the temperature and humidity retrievals provide a useful hydrometeor profile constraint. The FLORAD mission is fully consistent with the Global Precipitation Mission (GPM) scope and may significantly increase its space–time coverage. The concept of an incremental Flower constellation can ensure the flexibility to deploy a spaceborne system that achieves increasing coverage through separate launches of member spacecrafts. The choice of millimeter-wave frequencies provides the advantage of designing compact radiometers that comply well with the current technology of minisatellites (overall weight less than 500 kg). The overall budget of the FLORAD small mission might become appealing as an optimal compromise between retrieval performances and system complexity.

Flower Constellation of Millimeter-Wave Radiometers for Tropospheric Monitoring at Pseudogeostationary Scale

In this paper, the design of a minisatellite FLOwer constellation (FC), deploying millimeter-wave (MMW) scanning RADiometers, namely, FLORAD, and devoted to tropospheric observations, is analyzed and discussed. The FLORAD mission is aimed at the retrieval of thermal and hydrological properties of the troposphere, specifically temperature profile, water-vapor profile, cloud liquid content, and rainfall and snowfall rate. The goal of frequent revisit time at regional scale, coupled with quasi-global coverage and relatively high spatial resolution, is here called pseudogeostationary scale and implemented through a FC of three minisatellites in elliptical orbits. FCs are built on compatible (resonant) orbits and can offer several degrees of freedom in their design. The payload MMW channels for tropospheric retrieval were selected following the ranking based on a reduced-entropy method between 90 and 230 GHz. Various configurations of the MMW radiometer multiband channels are investigated, pointing out the tradeoff between performances and complexity within the constraint of minisatellite platform. Statistical inversion schemes are employed to quantify the overall accuracy of the selected MMW radiometer configurations.

Flower constellation set theory. Part I: Compatibility and phasing

Flower Constellations are special satellite constellations whose satellites follow the same 3-dimensional space track with respect to assigned rotating reference frame. This paper presents the theoretical foundation of compatibility and phasing of the Flower Constellations. Compatibility is the synchronization property of a Flower Constellation with respect to a rotating reference frame while phasing dictates the satellite distribution property. Compatibility and phasing, which are ruled by a set of five independent integer parameters, constitute the two main properties of the Flower Constellations. In particular, the dual-compatible Flower Constellations theory, which allows a simultaneous synchronization of the Flower Constellation dynamics with two independent rotating reference frames, is introduced. Meaningful examples and potential applications are briefly discussed.

Flower constellation set theory part II: Secondary paths and equivalency

In our previous research, the Flower Constellation set theory was introduced but specific details left out. In this work, the particular phasing theory that we have adopted is discussed in full. As a consequence of this choice of parametrization, a new class of orbit theory has emerged: secondary paths (SPs). The theory of SPs is developed and proved in this work. Examples of SPs are presented and discussed. Furthermore, we discuss the equivalency of Flower Constellations and resolve how certain disparate choices of integer parameters can generate identical satellite distributions.

Two-Way Orbits

This paper introduces a new set of compatible orbits called “Two-Way Orbits,” whose ground track path is a closed-loop trajectory that intersects at certain points with tangent intersections. The spacecraft passes over these tangent intersections once in a prograde mode and once in a retrograde mode. Motivations are found for the need to have simultaneous observations of the same target area in both Earth observation and reconnaissance systems. The general mathematical model to design a Two-Way Orbit is presented for the specific case where the tangent points are experienced at the orbit extremes, perigee and apogee. As for the general case, Two-Way Orbit conditions are formulated and numerically solved. Results show that, in general, Two-Way Orbits could be formed over any point on Earth. Since Two-Way Orbits use compatible orbits, the theory of Flower Constellations can be applied to them. Using these Two-Way Orbits, this paper also introduces the Two-Way Flower Constellations that have one spacecraft prograde and one retrograde passing simultaneously over the tangent intersection.

The Flower Constellations

This paper introduces a methodology to design a set of satellite constellations, called the Flower Constellations, which is generally characterized by repeatable ground tracks and a suitable phasing mechanism. A Flower Constellation, which can be complete or restricted, is identified by a number of parameters. Three are integer parameters: the number of petals (Np), the number of sidereal days to repeat the ground track (Nd), and the number of satellites (Ns). The others include the phasing parameters and three orbit parameters equal for all satellites: the argument of perigee (ω), the orbit inclination (i), and the perigee altitude (hp). Flower Constellations present beautiful and interesting dynamical features that allow us to explore a wide range of potential applications that include: telecommunications, Earth and deep space observation, global positioning systems, distributed space systems, and new kind of formation flying schemes. A Flower Constellation can be reoriented arbitrarily; however, the repeating ground track property is lost. To demonstrate their potential, some specific Flower Constellations are briefly described and discussed.