Космические сервисы на основе спутниковых формаций: математические модели, планирование миссий, оптимизация орбитального движения тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Биктимиров Шамиль Насимович

• Биктимиров Шамиль Насимович
• кандидат наук
• 2023

Оглавление диссертации кандидат наук Биктимиров Шамиль Насимович

Contents

Page

Introduction

1. Mission Design

1.1 Reference Frames

1.2 Pixel Image Visibility Requirements

1.3 Target Orbit Selection

1.3.1 Orbit selection method validation

1.4 Solar Reflector Sizing

1.4.1 Reflector sizing yielding a pre-defined pixel magnitude

1.4.2 Defining the worst-case pixel magnitude

1.5 Conclusion to the Chapter

2. Satellite Formation Design and Control Algorithms

2.1 Orbital Configuration

2.2 Control Algorithms

2.2.1 Impulsive maneuvers

2.2.2 Continuous maneuvers

2.2.3 Optimized reconfiguration

2.3 Numerical Study

2.3.1 Short-term mission

2.3.2 Long-term mission

2.4 Conclusion to the Chapter

3. Space service efficiency evaluation

3.1 Coverage model

3.2 Demonstration price

3.3 Coverage calculation and optimization

3.4 Space advertising mission feasibility analysis

3.5 Conclusion to the Chapter

Conclusion

References

Appendix A. Disturbing forces analysis

Appendix B. Satellite relative motion equations

Appendix C. Initial conditions for orbital configurations

Appendix D. LQR gain matrices

List of figures

1 Artist's view of the Olympic rings demonstration in the sky above

Moscow

1.1 Reference frames

1.2 Reflection geometry

1.3 Orbit geometry

1.4 Single pixel visibility conditions satisfaction

1.5 Demonstration parameters

1.6 Ground track of the target orbit

1.7 Reflector sizing geometry

1.8 Reflector sizing

1.9 Pixel magnitude m(t) during image demonstration above Moscow

for different reflector areas Ar

1.10 Reflector parameters

2.1 Orbital configuration design method illustration

2.2 Benchmark orbital configuration and reference trajectories

2.3 Orbital configuration for the Eiffel tower image demonstration

2.4 Orbital configuration for the Olympic rings image demonstration

2.5 General scheme of the formation control algorithms

2.6 Reconfiguration scenario example

2.7 Concept of operations of the demonstration mission

2.8 Position error ôp for all formation satellites during the short-term mission

2.9 Velocity error 5v for all formation satellites during the short-term mission

2.10 Formation's orbital configurations during image demonstrations

2.11 Maximin optimization results

2.12 Cumulative graph of fuel consumption during the short-term mission

2.13 LQR gains optimization

2.14 Satellite dynamics and control at deployment to the farthest

reference trajectory

2.15 Position error ôp for all formation satellites during the short-term mission

2.16 Velocity error 5v for all formation satellites during the short-term mission

2.17 Cumulative graph of fuel consumption during the mission

2.18 Pareto front from maximin optimization

2.19 a) Reconfiguration cost S(â), b) Images' relative orientation for à*

2.20 Fuel consumption estimates for satellite formation reconfiguration

and maintenance

2.21 Fuel consumption estimates

3.1 Coverage geometry

3.2 Coverage visualization

3.3 Average CPM in world countries in

3.4 The seasonal coefficient

3.5 The cloud interference coefficient

3.6 OTS discounting coefficients comparison

3.7 Price demonstration map of the world

3.8 Demonstration price distribution over the Earth regions

3.9 Ground track for different initial conditions

3.10 Daily price distribution by month

3.11 Cumulative demonstrations price

3.12 Space advertising system performance analysis

A.1 Disturbances analysis

B.1 Satellite relative motion illustration

List of tables

1 Target orbit parameters

2 Slots for image demonstrations above Moscow

3 Mission Parameters

4 Simulation parameters for the demonstration mission on December

22,

5 Coverage optimization results

6 Initial conditions for the Eiffel tower demonstration

7 Initial conditions for the Olympic rings demonstration

Введение диссертации (часть автореферата) на тему «Космические сервисы на основе спутниковых формаций: математические модели, планирование миссий, оптимизация орбитального движения»

Introduction

Advances in technology have led to a shift in the complex space systems design paradigm. Large satellites carrying all instruments required for their mission can now be replaced with distributed space systems consisting of small spacecraft cooperating to fulfil the same mission requirements. Multi-satellite space systems that require relative state acquisition and control to maintain certain geometrical arrangements of the satellites in orbit are said to perform formation flying.

The problem of satellites' relative motion has been studied since the beginning of the space era. It was initially considered in the problems of satellite rendezvous [1], for describing motion of the Moon with the respect to the rotating Earth-centric frame [2], to analyse cosmonaut's motion during space walk [3]. In the works the relative motion equations were derived in Cartesian coordinates. The linearized equations of relative motion are usually called as the Hill-Clohessy-Wiltshire (HCW) equations and extensively used for various satellite formation flying missions design.

The equations of relative motion can also be derived in curvilinear coordinates as was proposed in [4; 5]. It describes the relative motion of spacecraft in more natural way and hence leads to a smaller deviation from reference trajectories in comparison to the relative motion equations derived in Cartesian coordinates. In [6] the relative motion dynamics was described using relative orbital elements (ROE). The approach can be used to derive a dynamics model of satellite formation flying in elliptical orbits as was proposed in [7]. The studies of relative motion dynamics incorporating J2 perturbation have been presented in [8; 9]. In [10] a comprehensive survey and analysis of different models describing satellites' relative motion dynamics is made.

Formation-flying missions are considered for various tasks such as: distributed astrophysical observations [11], space telescopes for detection and study of exoplanets [12] or small solar system bodies exploration [13], synthetic aperture radar interferometry [14], distributed space-borne antennas [15], instruments for Earth observation [16] including photogrammetric observations of clouds [17], study of the Earth's gravity [18], magnetic field [19; 20], ionosphere [21], multipoint environmental measurements as a real-time service [22], and space advertising [23—

26]. Relative motion control methods can also be employed for on-orbit servicing tasks [27], which are becoming vital nowadays owing to the deployment of new space-based services to be provided by satellite mega-constellations [28].

Space missions necessitating satellite formation flying can pose different requirements for the orbital configuration geometry, reference orbits' tracking precision, formation's reconfiguration periodicity, and lifetime. For example, according to [29], the LEO distributed synthetic aperture radar interferometer requires reference orbit tracking accuracy ranging from 100 m to 1 m while deep space multiple spacecraft telescope would require a millimeter precision of reference orbits maintenance. On the other hand the missions requiring spatial orbital configuration may require controlling a certain quality factor characterizing the formation's orbital configuration instead of maintaining individual reference orbits. For example, a tetrahedron spacecraft formation is proposed for geomagnetic field dynamics study [30; 31].

There are applications that require a group of satellites flying in formation whose orbital configuration should be changed periodically to ensure certain mission goals. It usually requires precise maintenance of reference trajectories as well. For example, the synthetic radar interferometry requires different operation phases meaning different orbital configurations that have to be deployed periodically [32; 33]. The space advertising application [24—26; 34] assumes demonstration of different images to the observers on the ground thus requiring periodic reconfigurations of the formation's orbital configuration.

Space advertising is a promising although still futuristic concept for outdoor advertising and the subject of arguments about rational and sustainable space exploration. The existing approaches for space advertising can be divided into singletime events and multiple demonstration event missions. There are many examples of the former such as logos on board a rocket [35], branded food delivery to the International Space Station [36], or even a car launched to space [37]. However, in these examples, space advertising was merely a side-issue in a major space mission, whereas the idea proposed in the thesis assumes a dedicated space system.

A long-term space advertising mission would rely on complex satellite system orbiting the Earth and demonstrating pixel images to observers on the ground. In this case, an advertisement appears as a constellation of bright artificial stars

formed into an image that can be observed in clear night sky for several minutes. Development of such missions has become a point of interest for a few space startups because the approach provides global Earth coverage and thus allows showing an advertisement to regions of high-demand multiple times [38; 39]. In book [23] various engineering aspects of space advertising system design are discussed.

Despite the fact that there are no successful examples of long-term space advertising missions, there were several attempts to launch them. First attempts were carried out in the last century. In 1989, to celebrate the Eiffel Tower centennial, it was planned to deploy a string of a hundred solar reflectors in the low-Earth orbit (LEO) to form a ring of light, visible throughout the world [40]. Another space advertising campaign was dedicated to the Olympics in Atlanta in 1996 [41]. The idea was to launch a big reflective sheet with a length of a mile and width of a quarter-mile that would be visible on Earth. Both missions, however, were to be devoted to a single event and relied on a space structure rather than on a satellite formation to display the graphics.

There are two major options for producing space advertising in terms of a payload: solar reflectors and lasers. The former is used to reflect sunlight to a point of interest (POI) on Earth. It requires relatively large sunlight reflectors with an area of about 30 square meters [25] for LEO orbits to ensure the required pixel brightness as well as keeping the required reflectors' attitude to illuminate the required region on Earth. The latter gives more flexibility on satellite attitude during image demonstration, but requires additional power supply. The study considers the space advertising performed with the aid of solar reflectors. However, the approach delineated in the study can be applied for the laser-based as well.

Space-based solar mirrors have been discussed ever since the beginning of space era starting from Oberth [42], who suggested their usage for night illumination of cities, weather control, supply of solar power plants, etc. In work [43] the basic concept of a solar reflector satellite located at GEO orbit was outlined and the parametric model for reflected sunlight intensity on the ground was introduced. A series of studies have been performed in NASA in 70-80s to assess technical feasibility of space mirrors utilization for power generation on Earth [44—46]. In work [46] the improved parametric model of reflected sunlight intensity incorporating different loses into the model was proposed.

The research on space mirrors is still in demand nowadays. Several studies were performed by Lewis Fraas and his group devoted to the utilization of space mirrors for power generation on Earth [47—49]. Different engineering aspects of solar reflectors utilized for northern regions illumination are studied in [50—52]. Interesting applications of space mirrors helping surviving the Lunar night and for terraforming Mars were proposed in [53; 54]. A comprehensive overview of space-based mirrors and their applications can be found in [55].

An experiment to prove the concept of using a space-borne mirror to illuminate the Earth was performed within the project Znamya, during which a 20 m diameter circular mylar mirror was successfully deployed from the MIR space station [56] and produced a spot of light of about 5 km in diameter moving across the Earth's surface. Iridium flares [57] known to be caused by reflected sunlight also corroborate that a single satellite can be seen from the Earth's surface as a bright star. It then remains to formulate the visibility conditions and requirements to the formation geometry for a group of spacecraft to be observed as a pixel image from specific points of interest on Earth.

To produce an image in the sky (as shown in Fig. 1), each satellite can be appointed such initial conditions that it moves along a special trajectory with respect to a certain orbital reference frame so that all pixels are synchronized and form an image rotating about the origin of the chosen frame as a rigid body in accordance with the orbital dynamics laws (when such motion is considered in the central gravity field).

These reference trajectories can be obtained with the aid of the analytical solutions to the HCW equations [1; 2] describing relative motion of satellites. However, control is required for all satellites to achieve their pre-defined relative trajectories, track them throughout the mission in the presence of disturbing forces not accounted for in the HCW equations, and transfer to a new set of relative trajectories when reconfiguration is called for.

One of the possible approaches for deploying the required orbital configuration for the mission is to use tethered satellite formations. The important results in the topic of dynamics and control of motion of space tether system were obtained by Beletskii V.V., Levin E.M., Alpatov A.P., Ivanov V.A., Aslanov V.S., Sidorenko V.V., Zabolotnov Yu. M., Lorenzi E.C., Misra A.K., Williams P., Huang P., and

Figure 1 — Artist's view of the Olympic rings demonstration in the sky above

Moscow

others in [58—66]. Nevertheless, owing to the considered class of formation flying missions requiring a few dozen of small spacecraft that should be reconfigured frequently the approach may lead to additional difficulties in the control algorithms and hence is not considered in the thesis.

The satellite relative motion control algorithms can be divided, in general, into the impulsive and continuous one. The derivation of the impulsive maneuvers assumes instantaneous change of satellite velocity vector V while having fixed position vector R. The idealization is adequate if satellite thruster firing lasts for a short period of time At negligible in comparison to the satellite orbital period T. In this case the satellite position vector R changes slightly and the maneuver yields very close satellite's state correction to the one that derived analytically.

Most analytical impulsive control schemes are derived from equations describing the dynamics of osculating orbital elements with disturbing force given in the orbital reference frame [67; 68] also known as the Gauss Variational Equations (GVE) [69]. The classical impulsive scheme [70] derived from GVE corrects the control error expressed through the mean classical orbital elements. The two-impulse scheme proposed in [71] is considered for deployment and reconfiguration of satellite formations in circular orbits. It employs equinoctial orbital elements to express the

difference between current and required orbital elements. The resulting solution for control impulses is derived under the assumption that satellites are at the circular orbits with the same period. An elegant approach for formation flying dynamics representation was proposed in [72]. The authors proposed using an eccentricity and inclination vectors separation for formation flying design. The method was initially developed for geostationary satellites [73]. In comparison to the previously mentioned sets of relative orbital elements, the geometrical representation gives an insight into characteristics of a closed relative motion between satellites. The relative motion dynamics representation is used to design passively stable and safe satellite formation flying missions. The study [74] extends the GVE-based impulsive control approach to a continuous control. In the work the reconfiguration of satellite formation orbiting in near-circular orbits is addressed. The numerical study shows better control accuracy of finite-thrust based control comparing to the impulsive one. In the work [75] the GVE-based analytical solution for continuous control of satellite formations in circular orbits was proposed. The presented analytical control scheme is derived for relative motion dynamics parameterized in terms of relative orbital elements (ROE) taking into account J2 perturbation. A comprehensive overview of impulsive control approaches for spacecraft formation flying is presented in [76].

Impulsive control allows correcting reference orbit within a short period of time which is typically equal to 2-3 orbital periods depending on the maneuver sequence. This becomes important when the mission requirements are such that several image reconfigurations are scheduled during the day to display different images over different locations. Another advantage is that the impulsive maneuvers are derived analytically and therefore do not require significant computational resources and can be implemented in CubeSats autonomous missions.

A prevalent approach to continuous low-thrust control algorithms for satellite formation flying is based on the linearized relative motion dynamics and utilizes linear quadratic regulator (LQR) [77] for relative state control [78; 79]. The LQR-based continuous control algorithms' performance can be improved by adding major disturbing forces acting on satellite into the linearized relative motion dynamics model as was proposed in [80; 81]. Nevertheless, the models incorporating J2-perturbation has no significant advancements for the case of satellite formation flying in near-circular orbits [82]. State-Dependent Riccati Equation (SDRE) based

continuous control technique proposed in [83] advances the classical LQR-based formation control by introducing the state dependent linearized dynamics matrix. It has advantages in terms of convergence time and fuel consumption for non-linear relative motion dynamics. The Lyapunov-based control is also frequently applied for non-linear relative motion dynamics [84; 85].

An interesting approach for propellant-free relative motion control algorithms were proposed in [86; 87]. The recent study [26] analyzed how a satellite formation can be configured into a predefined image by decentralized differential aerodynamic drag-based control. The prescribed relative trajectories are attained by adjusting attitude of spacecraft's reflectors with respect to the incoming airflow. The results of the numerical simulations showed that the approach can be applied for control of formations orbiting at relatively low orbits (up to 350 km) where atmospheric drag force is significant. However, differential drag control requires greater reflectors area, which, especially in low orbits, notably decreases mission's lifetime.

Let us note that the work does not consider the problems of satellite attitude control for thruster orientation during maneuvering and reflector pointing during the demonstration. It is assumed that the formation satellites can be equipped with reaction wheel-based attitude control system [88; 89] that can fulfil the mission needs. Also the study does not consider the problem of the satellite state estimation which contributes to the accuracy of the controller input. The control error in reallife missions is estimated from the sensors' measurements, which may be additionally processed by an algorithm to reduce the measurement and process noise. The position and velocity of a spacecraft is usually determined by a GPS-receiver, and the relative positions of spacecraft in a formation flying mission can be enhanced with the aid of the Differential GPS technique [90]. For example, The Radio Aurora eXplorer (RAX) CubeSat mission [91] launched two CubeSats containing a GPS subsystem. The position accuracy (standard deviation errors) was found to have a mean of 2.89 m and a maximum of 4.02 m. The CanX 4&5 mission [92] with similar or worse position accuracy has demonstrated relative position control with a sub-meter accuracy. Recent developments of DGPS [93] show the promise of providing nanosatellite-based distributed space systems with centimeter-level relative position accuracy. The state estimation accuracy and corresponding control errors demonstrated in the aforementioned missions are within the image

demonstration mission requirements, determined by what can be observed as a misalignment of the formation's geometry by a human eye. Therefore, the study assumes the consideration of ideal position and velocity signals and neglect the measurements noise in spacecraft formation flying dynamics and control simulations.

Aims and objectives

The principal aim of this work is to study the dynamics and control of large formations of spacecraft in LEO comprising a few dozen of satellites, whose missions imply frequent orbital reconfigurations with stringent requirements to the relative positions' accuracy. The results of the study are to be illustrated by an example space mission to form and display by reflected sunlight predefined sequences of pixel images from the sky.

The thesis objectives are divided into three major parts:

- To design a parametric model that projects the high-level mission requirements onto the choice of the formation base orbit and relates it to the sizing of the sunlight reflectors that need to be mounted on the spacecraft, which requires:

— formulating and parameterizing the requirements for the visibility of pixel images formed in space by a recipient located at a point of interest on earth;

— obtaining and parameterizing the families of orbits where the formations to display pixel images can be deployed;

— relating the sunlight reflector size to the choice of orbits.

- To design a general control scheme for large formations of spacecraft with frequent reconfigurations, which requires:

— defining sets of relative trajectories for individual spacecraft in each orbital configuration and analyzing major natural disturbances along such trajectories;

— analyzing state-of-the-art formation establishment and maintenance algorithms and selecting those to be implemented in the formation control loop;

— combining the selected control algorithms into a general control scheme with a goal of maximizing the formation lifetime, i.e.,

optimizing the fuel consumption while meeting the mission requirements.

— To develop a method for space service performance evaluation based on the metrics of consumer involvement and apply it to the example mission. This requires:

— implementing the service coverage model based on the mission geometry and the population density model;

— analyzing and introducing various factors (i.e. demography-based, climate-based, etc) that may affect the interaction between the service and the population;

— using the developed models, performance metrics, and control scheme to optimize the performance of the example space advertising mission.

Scientific Novelty and Practical Importance

The thesis presents new results in space mirror-based mission design, control algorithms for large formations of LEO spacecraft requiring frequent reconfigurations, and methods for space services performance evaluation.

The parametric model for space mirror-based missions design is relevant to various projects utilizing space mirrors such as reflected solar power generation on the ground, northern regions illuminations, and space advertising.

A general control scheme for multiple spacecraft formation flying missions requiring frequent reconfigurations and precise reference orbits tracking is developed. An approach for satellite formation fuel consumption minimization during reconfiguration based on the combinatorial optimization is proposed. The control scheme for large formations of spacecraft in LEO is valuable for enabling novel applications of microsatellite formation flying requiring frequent reconfigurations.

The proposed method for space service performance evaluation based on the metrics of consumers involvement is universal and can be applied to various kinds of space systems, e.g. telecommunication mega-constellations, Earth observation systems, space advertising, and others.

Propositions for the Defence

1. A parametric model that links high-level solar reflector-based mission requirements, formation base orbit parameters, and solar reflector parameters;

2. A general control scheme for multiple spacecraft formation flying in LEO requiring frequent reconfigurations and high relative orbits control accuracy; The method for formation fuel consumption minimization during reconfiguration via assignment problem solution with the aid of combinatorial optimization; The efficiency of the proposed control scheme is illustrated in the numerical study of dynamics and control of 50 spacecraft formation flying for pixel image demonstration in the sky;

3. A method for space service efficiency evaluation via calculating and optimizing Earth coverage by the space service; The method for designing the space service's consumer model representing revenue for providing a space service at a particular location on Earth depending on the geographic location of the region, its population, service's unit price at the region and various factors limiting consumer's access to the service; The feasibility analysis of the space advertising mission is performed using the proposed method;

Presentations and Validation of Research Results

The results reported in the thesis have been published in the following peer-reviewed journals and conference proceedings:

1. Biktimirov S., Belyj G., and Pritykin D., Satellite formation flying for space advertising: From technically feasible to economically viable. Aerospace, 9(8), 2022. (WoS, Scopus)

2. Biktimirov S., Ivanov D., and Pritykin D., A satellite formation to display pixel images from the sky: Mission design and control algorithms. Advances in Space Research, 69(11):4026-4044, 2022 (WoS, Scopus)

3. Kharlan A., Biktimirov S., Ivanov A., Prospects for the Development of Global Satellite Communication Constellations in the Context of

New Services in the Telecommunications Market. Cosmic Research, 58(5):402-410, 2020 (WoS, Scopus)

4. Biktimirov S., Ivanov D., Sadretdinov T., Omran B., and Pritykin D., A Multi-Satellite Mission to Illuminate the Earth: Formation Control Based on Impulsive Maneuvers. Advances in the Astronautical Sciences, 173:463-474, 2020 (Scopus)

5. Afanasev A. and Biktimirov S., CubeSat formation architecture for small space debris surveillance and orbit determination. Informacionno-upravlyayushchie sistemy, (4 (113)):37-46, 2021 (Scopus)

6. Biktimirov S., Solodovnikova N., Afanasev A., and Pritykin D. A CubeSat-Based Space System to Monitor Space Debris Population in LEO. In Proceedings of the International Astronautical Congress, IAC, 2021 (Scopus)

7. Ivanov D., Biktimirov S., Chernov K., Kharlan A., Monakhova U., and Pritykin D., Writing with Sunlight: Cubesat Formation Control Using Aerodynamic Forces. In Proceedings of the International Astronautical Congress, IAC, 2019 (Scopus)

The results of the thesis were presented and discussed at the following scientific conferences and seminars:

1. 11th International Workshop on Satellite Constellations and Formation Flying, Milano, Italy, June 2022

2. XLV, XLVI Academic Space Readings "Korolev Readings", Moscow, Russia, April 2021, January 2022

3. 62nd,63rd International MIPT Scientific Conference, Dolgoprudny, Russia, November 2020, November 2021

4. The 72nd International Astronautical Congress, Dubai, UAE, October 2021

5. AA-AAS SciTech Forum 2020 Cyber Edition, Moscow, Russia, December 2020

6. 5th IAA Conference on University Satellite Missions and CubeSat Workshop, Rome, Italy, January 2020

7. 12th All-Russian Congress on Fundamental Problems of Theoretical and Applied Mechanics, Ufa, August 2019

8. Seminar organized by the program committee of the Doctoral program "Engineering Systems"at Skolkovo Institute of Science and Technology (Skoltech), June 2022

9. Scientific seminar of Theoretical mechanics department at MIPT led by Dr.Sc. (Phys.-Math.) Sokolov S.V., October 2022

Personal Contribution

The main thesis content and results are obtained by the author. The problem statements, methods, and results were discussed with scientific supervisor and coauthors.

Methodology and Research Methods

The research methods utilized in the thesis are standard method of theoretical mechanics, control theory and methods for analytical and numerical solution to the systems of ordinary differential equations.

Thesis Structure

The dissertation consists of an introduction, 3 chapters, list of references, and appendix. The full dissertation is 107 pages, including 46 figures, 7 tables, and 133 references.

The first chapter is devoted to mission design. The chapter starts with introducing main reference frames used in the study. The geometrical image demonstrations requirements are stated further. The geometrical requirements are used to derive the type of LEO orbits suitable for space mirror operation. The second part of the mission design is solar reflector sizing method. It allows selecting solar reflector parameters yielding the required pixel brightness. The procedure can be used for reverse problem where a worst-case pixel brightness is calculated for the given formation's target orbit and reflector's parameters.

The second chapter is devoted to satellite formation design and control algorithms. It starts with the orbital configuration design method utilizing closed-form solutions to the linearized equation of satellites' relative motion. Secondly, the general control scheme for large formations of spacecraft in LEO is proposed. The control scheme is implemented for two control loops - hybrid and continuous.

The hybrid control loop is based on impulsive maneuvers and continuous control. It uses impulse maneuvers for correcting relative orbital elements error during reconfiguration while an LQR-based continuous control algorithm is applied for relative orbits maintenance. The continuous control loop uses an LQR-based control algorithm for both reconfiguration and maintenance. A comparison of the hybrid and continuous control loops is performed. The methods for optimizing the fuel consumption of formation satellites are introduced. The assignment problem is solved using combinatorial optimization. Also, the linear-quadratic regulator gain matrix tuning procedure is proposed. Lastly, the formation lifetime estimates for long-term space advertising missions are made. The lifetime estimates are important for assessing space advertising missions' feasibility.

In the third chapter the method for space service efficiency evaluation is developed. It based on the service coverage model utilizing mission geometry and population density distribution. The method is applied for to assess the feasibility of space advertising mission. For the purpose the daily revenue of the mission is calculated and optimized for different system parameters and compared with its cost.

Conclusion outlines the main results obtained the thesis.

The reported study was funded by Russian Foundation for Basic Research (RFBR), project number 20-31-90115.

Заключение диссертации по теме «Другие cпециальности», Биктимиров Шамиль Насимович

Conclusion

The problem of small satellite formation flying in the frame of missions requiring frequent reconfigurations and precise reference orbits tracking has been addressed in the thesis. Following the formation flying requirements and growing interest to the space advertising topic the application of pixel image demonstration in the sky was chosen for the proposed relative motion control algorithms analysis.

A target orbit design method was derived based on the image demonstration requirements. The target orbits are near-circular Sun-synchronous repeat ground track orbits that should also be oriented close to the Earth terminator plane. The proposed formation's target orbit ensures daily demonstrations in an arbitrary city at least two times per day at the dawn and dusk. The solar reflector sizing method utilizing reflected sunlight intensity model was proposed to find a reflector area satisfying requirement on pixel brightness during image demonstration. It was demonstrated that current level of solar sailing technologies is sufficient enough for space advertising purposes.

Two control loops were proposed for deployment, maintenance, and reconfiguration of satellite formations - hybrid and continuous. The efficiency of control algorithms has been demonstrated in the possible mission scenario derived in the mission design chapter. Numerical study of the controlled orbital motion dynamics of the formation flying satellites demonstrated that both control algorithms satisfies concept of operation of the example missions. However, the hybrid one leads to a smaller fuel consumption and takes longer time for reconfiguration. On the other hand, the continuous one allows performing faster reconfigurations while having a greater fuel consumption. The latter also allows tuning the control algorithms performance in terms of reconfiguration time and fuel consumption. The method for bi-objective LQR weight matrices optimization was proposed for the purpose. A method for formation lifetime estimation was proposed and applied for a test image demonstration mission. The method allows calculating upper boundary value for formation lifetime based on the mean fuel consumption of formation satellites for reconfiguration corresponding to the optimal solution to the assignment problem and mean fuel consumption for relative orbits maintenance.

An approach for calculation and optimization the Earth coverage proposed and applied to assess the economic feasibility of image demonstration mission. For the purpose the cost model is built characterizing image demonstration cost at large world cities. The numerical analysis of the Earth coverage for space advertising mission was performed. An unrealistic idea as it may first seem, space advertising turns out to have a potential for commercial viability. The analysis indicates that the two key factors affecting the effectiveness of the mission are satellite footprint area and formation lifetime. The former requires a diffusive solar reflector with a scattering angle greater than the angular size of the Sun. It follows from our simulations that technically feasible solar reflectors of the size that has already been used in CubeSat missions are sufficient for space advertising missions to be economically viable.

Список литературы диссертационного исследования кандидат наук Биктимиров Шамиль Насимович, 2023 год

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