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

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## Abstract

**:**

## 1. Introduction

## 2. Reference Frames

- The orthonormal basis $OXYZ$ is the Earth-centered inertial reference frame (ECI J2000) denoted by ${\mathcal{F}}^{I}$. ECI frame has its origin at the Earth’s center, its X-axis is pointed to the Sun at vernal equinox, Z-axis is aligned with the Earth’s rotation axis, Y-axis completes the reference frame to the right-hand triad.
- The orthonormal basis $O\xi \eta \zeta $ denoted by ${\mathcal{F}}^{E}$ is the Earth-centered Earth-fixed reference frame (ECEF) which is a geocentric coordinate system fixed with the rotating Earth.
- The orthonormal basis ${o}^{\prime}xyz$ is the orbital reference frame denoted by ${\mathcal{F}}^{O}$ that has its origin at the target orbit, its z-axis is aligned with the local vertical, y-axis is directed along the target orbit angular momentum vector ${\mathbf{h}}_{0}$, and x-axis completes the reference frame to the right-hand triad.

## 3. Target Orbit Selection

## 4. Spacecraft Payload Sizing

## 5. Formation Lifetime

#### 5.1. Orbital Configuration

#### 5.2. LQR-Based Continuous Control Algorithms

#### 5.2.1. LQR Gains Optimization for Reconfiguration

#### 5.3. Assignment Problem

#### 5.4. Formation Lifetime Estimates

## 6. Earth Coverage

#### 6.1. Coverage Model

#### 6.2. Demonstration Price

- The seasonal coefficient. The total number of the so-called impressions (i.e., contacts with the advertisement) varies significantly with the season change. For example, the probability of observing an outdoor demonstration during the calendar winter reduces owing to weather conditions. Therefore, it is necessary to take into account the month in which the space advertising demonstration takes place as well as the POI’s distance to the tropical belt. Areas located close to the tropical zones have mild and comfortable climate, which contributes to the frequent presence of advertising consumers outdoors. It leads to an increase of probability of the demonstration to be noticed. In these areas, the weather differences are not pronounced by seasons. At the same time, the visibility depends on the level of natural light: the lower it is, the less a person’s attention is scattered (people become more determined on the choice of the road). The overall level of illumination depends on the average length of the day, which, in turn, has an annual cycle and is expressed by the cosine function. The maximum of this function can be considered June, the minimum is December in the Northern hemisphere.The formula that takes into account the influence of the month of observation and the location of the object with respect to the tropical zones was obtained in [29]. The expression for the seasonal coefficient a for the city at latitude $\varphi $ is given by:$$a=\frac{1}{1+{e}^{-\lambda}},\phantom{\rule{1.em}{0ex}}\lambda =\alpha +\beta \xb7(1-\varphi /{\varphi}_{T})\xb7cos\left(2\pi \frac{\tau +6}{12}\right),$$
- The cloud interference coefficient.The cloud interference is expressed as the part of the Earth’s surface covered by clouds, relative to the part of the Earth not covered by clouds. The data on the cloud interference are taken from the MODIS Cloud Product [30]. These data represent the monthly values of the cloud fraction averaged from cloud groupings of 5 km × 5 km in size with maximum pixel resolution. These values are approximations based upon the scaled range of satellite images. The cloud fraction value for a specific i-th city is denoted by ${b}_{i}\in [0,1]$. To estimate the coefficients, data for different months of 2021 are taken from [30]. Thus, for each city, the coefficient representing the cloud fraction above city ${b}_{i}$ is varying depending on the month defined. Figure 15 illustrates the evolution of the cloud interference coefficient $b\left(t\right)$ within a year for a list of cities with the greatest demonstration price ${C}_{i}$ at different Earth regions.
- The demographic coefficient. Individual differences between message recipients can lead to discrepancies in how people respond to advertisements. Thus, taking into account the characteristics of the population parameters allows constructing a more realistic model of economic viability for advertising missions. Residents older than 15 years are considered to be the solvent audience; thus, our statistics take into account people aged 15 years and older. The audience is divided into 3 age groups: 15–24, 25–64, 65+.The approach to describe how people of different age respond to outdoor advertising is described in [29]. The overall response is expressed as a weighted sum of age-group fractions, with weights selected to approximate the statistical survey data [31]. The expression for the demographic coefficient c (determined for each country) has the form:$$c=\frac{1}{1+{e}^{-s}},\phantom{\rule{1.em}{0ex}}s=\mu +\gamma \xb7(2\xb7{n}_{15-24}+4\xb7{n}_{25-64}+5\xb7{n}_{65+}),$$

#### 6.3. Coverage Calculation and Optimization

## 7. Results

## 8. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Linear Quadratic Regulator Gain Matrix K for Satellite Formation Reconfiguration

## Appendix B. Linear Quadratic Regulator Gain Matrix K for Satellite Formation Maintenance

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**Figure 1.**Reference frames [8].

**Figure 2.**Orbit geometry [8].

**Figure 8.**(

**a**) Reconfiguration cost $S\left(\tilde{\alpha}\right)$, (

**b**) image’s relative orientation for ${\tilde{\alpha}}^{*}$.

${\mathit{m}}_{\mathit{req}}$ | ${\mathit{\gamma}}_{\mathit{beam}}$, arcmin | ${\mathit{A}}_{\mathit{fp}}$, km${}^{2}$ |
---|---|---|

−5 | 15.6 | 51.9 |

−4 | 24.7 | 130.3 |

−3 | 39.2 | 327.2 |

−2 | 62.1 | 821.7 |

−1 | 98.4 | 2063.4 |

Paramater | Value | Units |
---|---|---|

Target orbit | ||

Altitude | 895.45 | km |

Inclination | 98.98 | deg |

Satellite parameters | ||

Mass | 18 | kg |

${T}_{max}$ [25] | 0.4 | N |

${I}_{sp}$ [25] | 285 | s |

Admissible control errors at reconfiguration | ||

${\epsilon}_{\rho}$ | 50 | m |

${\epsilon}_{v}$ | 0.5 | m/s |

Admissible control errors at maintenance | ||

${\epsilon}_{\rho}$ | 1 | m |

${\epsilon}_{v}$ | 0.1 | m/s |

Epoch (UTC) | $\mathit{h}\phantom{\rule{3.33333pt}{0ex}}\left[\mathbf{km}\right]$ | $\mathit{e}\phantom{\rule{3.33333pt}{0ex}}[-]$ | $\mathit{i}\phantom{\rule{3.33333pt}{0ex}}\left[\mathbf{deg}\right]$ | $\mathit{\Omega}\phantom{\rule{3.33333pt}{0ex}}\left[\mathbf{deg}\right]$ | $\mathit{\omega}\phantom{\rule{3.33333pt}{0ex}}\left[\mathbf{deg}\right]$ |
---|---|---|---|---|---|

22/12/2022, 00:00:00 | 895.45 | 0 | $98.98$ | $0.1$ | 0 |

${\mathit{A}}_{\mathbf{fp}}$, km${}^{2}$ | m | ${\mathit{u}}_{\mathbf{opt}}$, deg | ${\mathit{N}}_{\mathbf{demo}}$ | P, mln USD |
---|---|---|---|---|

51.9 | −5 | 35.20 | 491 | 0.35 |

130.3 | −4 | 39.96 | 498 | 0.61 |

327.2 | −3 | 37.31 | 493 | 1.07 |

821.7 | −2 | 37.31 | 493 | 1.50 |

2063.4 | −1 | 37.31 | 493 | 1.93 |

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**MDPI and ACS Style**

Biktimirov, S.; Belyj, G.; Pritykin, D.
Satellite Formation Flying for Space Advertising: From Technically Feasible to Economically Viable. *Aerospace* **2022**, *9*, 419.
https://doi.org/10.3390/aerospace9080419

**AMA Style**

Biktimirov S, Belyj G, Pritykin D.
Satellite Formation Flying for Space Advertising: From Technically Feasible to Economically Viable. *Aerospace*. 2022; 9(8):419.
https://doi.org/10.3390/aerospace9080419

**Chicago/Turabian Style**

Biktimirov, Shamil, Gleb Belyj, and Dmitry Pritykin.
2022. "Satellite Formation Flying for Space Advertising: From Technically Feasible to Economically Viable" *Aerospace* 9, no. 8: 419.
https://doi.org/10.3390/aerospace9080419