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A Satellite Formation to Display Pixel Images from the Sky: Mission Design

and Control Algorithms

Shamil Biktimirov, Danil Ivanov, Dmitry Pritykin

PII: S0273-1177(22)00204-6

DOI: https://doi.org/10.1016/j.asr.2022.03.018

Reference: JASR 15810

To appear in: Advances in Space Research

Received Date: 16 August 2021

Revised Date: 28 January 2022

Accepted Date: 16 March 2022

Please cite this article as: Biktimirov, S., Ivanov, D., Pritykin, D., A Satellite Formation to Display Pixel Images

from the Sky: Mission Design and Control Algorithms, Advances in Space Research (2022), doi: https://doi.org/

10.1016/j.asr.2022.03.018

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© 2022 Published by Elsevier B.V. on behalf of COSPAR.

Spacecraft formations can form and display graphics in the sky by reflected

sunlight

Spacecraft and orbits requirements are derived based on the demonstration

constraints

Formation control algorithms combine impulsive maneuvers and low-thrust

control

The proposed control scheme allows fast deployment and reconfiguration

Reconfigurations are optimized to maximize the remaining fuel in the spacecraft

Highlights

A Satellite Formation to Display Pixel Images from the

Sky: Mission Design and Control Algorithms

Shamil Biktimirov1,∗, Danil Ivanov2, Dmitry Pritykin3

Abstract

A concept of deploying spacecraft formations to display advertisement images

from space is considered. The spacecraft are equipped with sunlight reﬂectors,

which under certain lighting conditions and given the right attitude can be

observed from Earth as bright stars in the sky. When deployed into appropriately

selected orbits the spacecraft can be grouped into a pixel image. The problem of

formation deployment and maintenance to demonstrate two particular images

over a chosen point of interest is addressed in this study. The images visibility

conditions in terms of reﬂector size and proper lighting are formalized and

relative orbits of the formation spacecraft are chosen. This brings the study to a

fully formulated set of problems: formation deployment after launch, formation

keeping to maintain the image geometry and formation reconﬁguration to change

the demonstrated images. The problem is solved by a hybrid control strategy

comprising impulsive maneuvers and low-thrust control taking into account fuel

consumption and collision avoidance. The proposed approach can be used in

design studies of space advertising missions to evaluate the number of images to

be displayed along the selected orbit, the amount of fuel required for formation

maintenance and reconﬁguration, and the mission lifetime.

Keywords: satellite formation ﬂying, image demonstration, hybrid control

∗Corresponding author:

email – Shamil.Biktimirov@skoltech.ru

1Space Center, Skolkovo Institute of Science and Technology, Bolshoy Boulevard 30, bld.

1, Moscow, Russia

2Keldysh Institute of Applied Mathematics RAS, Miusskaya square 4, Moscow, Russia

3Moscow Institute of Physics and Technology, Moscow, Russia

Preprint submitted to Advances in Space Research January 19, 2022

algorithms, single-impulse maneuvers, LQR-based continuous control, optimal

reconﬁguration

1. 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 spacecraft in orbit are said to perform

formation ﬂying. Formation-ﬂying missions are considered for various tasks such

as: distributed astrophysical observations (Scala et al., 2020), space telescopes

for detection and study of exoplanets (Fridlund, 2004) or small solar system

bodies exploration (D’Arrigo and Santandrea, 2006), synthetic aperture radar

interferometry (Gill and Runge, 2004), distributed space-borne antennas (She

et al., 2019), instruments for Earth observation (Krieger et al., 2013), study of

the Earth’s gravity (Tapley et al., 2004), magnetic ﬁeld (Fuselier et al., 2016;

Nogueira et al., 2015), ionosphere (Fish et al., 2014), multipoint environmental

measurements as a real-time service (Afanasev et al., 2021), and graphic de-

monstration (Biktimirov et al., 2020; Ivanov et al., 2019). Relative motion

control methods developed for these applications can also be employed for on-

orbit servicing tasks (Sabatini et al., 2020), which are becoming vital nowadays

owing to the deployment of new space-based services to be provided by satellite

mega-constellations.

This study explores yet another application of formation ﬂying, which became

a driving idea for recent space startups (Chow, 2019), space advertising. It can

be implemented by forming an image of a logo or an inscription in the sky by a

group of satellites ﬂying over a point of interest and reﬂecting sunlight upon this

point. Space-based solar mirrors have been discussed ever since the beginning

of space era starting from Oberth (Oberth, 2019), who suggested their usage

2

for night illumination of cities, weather control, supply of solar power plants,

etc. A comprehensive overview of space-based mirrors and their applications

can be found in Lior (2013). An experiment to prove the concept of using a

space-borne mirror to illuminate Earth was the project “Znamya”, during which

a 20 m diameter circular mylar mirror was successfully deployed from the MIR

space station (Semenov et al., 1994) and produced a spot of light of about

5 km in diameter moving across the Earth’s surface. Iridium ﬂares (Maley and

Pizzicaroli, 2003) 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.

At least two space advertising projects were discussed in the 20th century. A

string of 100 reﬂectors to form a ring of light, visible throughout the world, could

be sent into orbit in 1989 to mark the centennial of the Eiﬀel Tower (REUTERS,

1986). A few years later in the 90s the city of Atlanta investigated a Space

Billboard concept (a mile-wide, quarter-mile tall reﬂective sheet that would

be visible from Earth) for the 1996 Olympics (Rossen, 2018). Both missions,

however, were to be devoted to a single event and relied on a space structure

rather than on a spacecraft formation to display the graphics. In contrast

to this, our study is focused on the technical feasibility of using formation-

ﬂying missions for advertising. We propose to carry out orbital conﬁguration

design and establish the visibility requirements from which the size of solar

reﬂectors can be derived. We also propose to consider the control algorithms

to be employed for deployment and maintenance of the formations displaying

images from the sky. Another important step for an advertising mission is image

reconﬁguration as diﬀerent graphics can be displayed over diﬀerent points of

interest along the orbit. Choice of the control algorithms entails the amount

of fuel the satellites spend for the required maneuvers, which helps relating the

mission lifetime to the spacecraft size for trade-of analysis.

The proposed approach will be illustrated by an example mission whose

3

objective is to display two images over Moscow on September 24, 2021, ﬁrst

demonstration to take place in the morning, and the second in the evening. Our

choice of images to be displayed is prompted by the two advertising missions that

were considered in the past. The ﬁrst image is that of the Eiﬀel tower (whose

anniversary it was contemplated to celebrate by the “ring of light” project), and

the second image is the ﬁve Olympic rings (as Atlanta Organizing Committee

considered the “space billboard” for the Olympics opening ceremony).

To produce an image in the sky 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 co-

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 tra jectories can be obtained with the aid of the analytical

solutions to Hill-Clohessy-Wiltshire (HCW) equations (Wiltshire and Clohessy,

1960; Hill, 1878) 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. Formation control algorithms

are based on the combination of continuous low-thrust actuation and single

impulsive control actions. The latter ensures fast although coarse control of

relative orbits while the low-thrust closed-loop continuous control is used for

the ﬁner maneuvers that meet the demanding requirements to the formation

geometry. Such hybrid approach was implemented in the Canadian formation-

ﬂying technology demonstration mission CanX-4&5 (Bonin et al., 2015).

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 maneu-

vers are derived analytically and therefore do not require signiﬁcant computational

4

resources and can be implemented in CubeSats autonomous missions. Preliminary

study (Biktimirov et al., 2019) showed that the image demonstration mission

requirements can be met by 12U CubeSat platform. For this platform, impulsive

control can be performed with the aid of propulsion modules (Dawn Aerospace;

Busek) with low power consumption of about 10-15 W that ﬁt into power budget

of typical CubeSat missions.

Most analytical impulsive control schemes are derived from Gauss Variational

Equations (GVE) (Battin, 1999) that describe the dynamics of six orbital elements

with acceleration input given in orbital reference frame. The classical impulsive

scheme (Schaub and Alfriend, 2001) derived from GVE corrects the control error

expressed through the mean classical orbital elements. The two-impulse scheme

proposed in Vaddi et al. (2005) 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 D’Amico and Montenbruck (2006). The authors proposed using an eccentricity

and inclination vectors separation for formation ﬂying design. 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. Larbi and

Stoll (2018) extend 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 (Di

Mauro et al., 2018) 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 J2perturbation.

5

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) for relative state control (Palmerini and Sabatini,

2007). However, to improve the algorithm performance the linearized relative

motion dynamics should take into account main perturbing forces acting on

satellites at low Earth orbits such J2perturbation, atmospheric drag, and

solar pressure. Our prior study (Ivanov et al., 2019) 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.

The paper has the following structure. The introduction is followed by

the image demonstration requirements section that addresses single satellite

visibility conditions and formation geometry requirements. Third section is

devoted to mission design. It introduces the general approach and analytical

method to choose a target orbit for the mission. We show how the target

orbit is chosen for the test mission with the goal to perform demonstrations of

two diﬀerent images above Moscow within one day in the morning and in the

evening. Then solar reﬂector sizing is discussed and reﬂector area satisfying the

single pixel brightness requirements is calculated for the test mission. Finally, a

method to design the formation’s orbital conﬁguration is presented. Two orbital

conﬁgurations are then designed based on parameters of the target orbit and

images to be demonstrated. The fourth section presents relative motion control

algorithm used for the mission. Section ﬁve demonstrates the simulation results

for the proposed mission to deploy, maintain, and reconﬁgure a formation of

solar reﬂector-equipped satellites for image demonstration above Moscow. The

paper ends with conclusion section where the advantages and limitations of the

6

control scheme employed for the tasks of multi-satellite formations deployment,

maintenance, and reconﬁguration in the frame of image demonstration mission

are discussed.

2. Image demonstration requirements

Image demonstration requirements can be divided into two groups: single-

pixel visibility and formation geometry requirements.

The requirements for single satellite visibility during the demonstration are

deﬁned as follows:

•(Req1.1) The satellite shall be in the direct line of sight (LOS) from both

the Sun and the point of interest (POI) on Earth. Moreover, the satellite’s

elevation angle as seen from POI should be greater than some threshold

value. We shall further assume that demonstration takes place if the

satellite’s elevation angle θsat is greater than 10 degrees (see Fig 1);

•(Req1.2) The demonstration shall be performed at certain lighting conditions

at POI expressed through the Sun elevation angle θSun. We assume that

the Sun elevation angle during demonstration should not be greater than

−5 degrees. The maximum elevation angle for demonstration is chosen

according to Kishida (1989), where suﬃcient lighting conditions at POI

for stars observations are discussed;

•(Req1.3) The pixels must be clearly visible with the naked human eye.

We stipulate that the pixel magnitude mshall be such that the pixels are

brighter than Iridium satellite ﬂares (Maley and Pizzicaroli, 2003) at the

observed extrema whose magnitude ranges from −6 to −8;

The requirement to the formation geometry during the demonstration is

deﬁned as follows:

•(Req2) Pairwise distances between any formation satellites shall be such

that each satellite is distinguished as an independent pixel from the POI

7

Figure 1: Sunlight reﬂection geometry

during demonstration. The distance is thus derived for the chosen orbit

altitude from the angular resolution of the human eye, which is known to

be approximately one arc-minute (Yanoﬀ and Duker, 2009);

3. Mission design

This section describes a benchmark mission design with the objective of

deploying a satellite formation to show two diﬀerent images above Moscow

(latitude ϕ= 55.76◦, longitude λ= 37.62◦) on the day of an annual advertising

festival (September 24, 2021). The ﬁrst demonstration was supposed to take

place in the morning and the second – in the evening, both in the same lighting

conditions. The image for the morning demonstration is the Eiﬀel tower, the

evening image – the Olympic rings. The choice of demonstrated images is a

tribute to the two missions that were considered in Paris and Atlanta, but

never took place.

8

We shall ﬁnd a set of target orbits where all geometrical demonstration

requirements are satisﬁed for both demonstrations. This yields an estimate of

the minimum reﬂector area that meets the magnitude requirement. Finally,

we shall design orbital conﬁgurations for the formation ﬂying mission to ensure

demonstrations of both images above Moscow on the chosen day.

3.1. Reference frames

The following reference frames to describe the motion of the formation

spacecraft are used (Fig. 2):

1. OXY Z denoted by FIis the Earth-centered inertial frame (J2000 ECI

frame) with the origin at the Earth center O;

2. Oξηζ denoted by FEis the Earth-centered Earth-ﬁxed reference frame

(ECEF) which is a geocentric coordinate system ﬁxed with the rotating

Earth;

3. o′xyz is the orbital reference frame (FO) whose origin o′travels along

the formation target orbit. As shown in Fig. 2 z-axis is aligned with the

local vertical, y-axis coincides with angular momentum vector of the target

orbit, and the x-axis completes the reference frame to the right-handed

triad.

It is assumed that a virtual satellite is placed at the target orbit with a state

vector X0= [R⊤

0,V⊤

0]⊤as in Fig. 2. Then the formation satellite designated

by the index i(i= 1, .., Nsat) has state vector Xi= [R⊤

i,V⊤

i]⊤given in FI,

whereas its relative state is described by xi= [ρ⊤

i,v⊤

i]⊤in FO(Fig. 2). The

formation control algorithms are to deploy, maintain, and reconﬁgure a satellite

formation with a pre-deﬁned orbital conﬁguration represented by a set of relative

trajectories e

xi(t) = [e

ρ⊤

i(t), e

v⊤

i(t)]⊤whose geometric median (as computed from

the required trajectories) will coincide with the virtual satellite position R0(t)

given in FI.

The transition matrix Abetween FOand FIis given by A= [exeyez],

where

ez=R0

R0

,ey=R0×V0

|R0×V0|,ex= [ey×ez].

9

Figure 2: Reference frames

The transition matrix Arelates the absolute and relative state vectors as

follows (Goldstein et al., 2002):

ρi=A−1(Ri−R0),

vi=A−1(Vi−V0)−n×ρi,

(1)

where n= [0,qµ

R3

0

,0]⊤is the mean motion of the virtual satellite projected onto

the orbital reference frame, µis the Earth standard gravitational parameter.

3.2. Target orbits

A target orbit is selected to ensure that all image demonstration requirements

are satisﬁed during the scheduled performances. Image demonstration require-

ments leave us with a narrow range of points in time and orbital positions

where image demonstration is possible. It stands to reason that at the time of

demonstration the formation must be relatively close to the terminator plane.

10

In this case, all formation spacecraft are lit by the Sun and are able to reﬂect

sunlight upon those regions on the Earth that are beyond the terminator line.

Thus the principal mission design idea is to set up the orbit close to the

terminator line.

The target orbit for the image demonstration mission is assumed to be

circular Sun-synchronous orbit (SSO) as in our previous studies (Ivanov et al.,

2019). The fact that the orbit is circular ensures the same angular size of

demonstrated image at diﬀerent POIs on Earth which is important for a mission

with demonstrations in multiple locations. Furthermore, the RAAN change rate

˙

Ω corresponding to an SSO ensures that the target orbit remains close to the

terminator line. The right ascension of the ascending node (RAAN) secular rate

˙

Ω for a circular orbit is given by (Vallado, 2001):

˙

Ω = −3nR2

⊕J2

2R2

0

cos(i),(2)

where R⊕= 6378.1363 km is the mean equatorial radius of the Earth, nis the

mean motion of the formation target orbit, R0is orbit radius, J2= 1.082 ×10−3

is second order zonal harmonic of Earth potential, iis the orbit inclination.

Despite the fact that RAAN will not change signiﬁcantly within one day mission

if the orbit is not Sun-synchronous, the SSO orbit will be inevitable while

designing long term missions for image demonstration with the aid of solar

reﬂectors-equipped satellites.

For the proposed test mission we consider circular target orbits with altitudes

ranging from 500 km to 1000 km. The lower bound is chosen because of

the atmospheric drag which is notable in the lower orbits and leads to an

extremely short ballistic lifetime especially seeing that spacecraft with solar

reﬂectors of considerable area are involved. The upper bound is selected because

of the potentially large distance between the reﬂector and POI during image

demonstration necessitating the use of very large reﬂectors to ensure the required

pixel brightness. In accordance with the Eq. (2) for the given range of altitudes,

the inclination of corresponding circular Sun-synchronous orbits varies from

11

97.39◦to 99.47◦.

Taking into account the aforementioned assumptions we use the following

procedure to select the target orbit parameters for the test image demonstration

mission above Moscow.

Step 1. Target orbit RAAN value. On one hand it has been made clear that

it is desirable that the target orbit runs close to the terminator plane. On the

other hand, the inclination range for circular SSO is limited and thus the target

orbits cannot be made co-planar with the terminator plane (with the exception

of very few speciﬁc epochs). However, the proper choice of the RAAN makes

it possible for the target orbit to have its ascending node at the line where the

terminator plane and the equatorial plane intersect.

The right ascension of ascending node Ω can thus be found as follows:

Ω = cos−1(eX·Υ),(3)

where Υis the unit vector pointing to the ascending node of the target orbit:

Υ=[eZ×e

eSun ]

|[eZ×e

eSun ]|,(4)

where e

eSun deﬁnes the proper direction of the normal to the terminator plane 1:

e

eSun =−sign(eZ

Sun)·sign(ϕ)·eSun ,(5)

where eSun = [eX

Sun, eY

Sun, eZ

Sun]⊤is the unit vector (in FI) of the Sun direction

as seen from Earth, ϕis the POI latitude.

Fig. 3 illustrates the orbit selection idea. The case when POI is located in

Northern Hemisphere and the Sun has negative declination, i.e. eZ

Sun <0 is

shown. The case corresponds to the considered image demonstration mission

1In case when eZ

Sun = 0 and POI located either in Northern or Southern Hemisphere

e

eSun =−sign(ϕ)·eSun . If the POI is located in the equator i.e. ϕ= 0 the normal to the

terminator plane is deﬁned as e

eSun =−eSun

12

which is designed for September 24, 2021 with the POI located in the Northern

Hemisphere. This yields

eSun = [−0.999,−0.0139,−0.006]⊤,

and the corresponding right ascension of ascending node Ω = 270.8◦. It should

be noted that the intermediate vector e

eSun coincides with the unit Sun position

vector eSun in the considered case according to Eq. (5) and hence is not displayed

in Fig. 3. The vector hSSO shown in the ﬁgure represents normal to the orbital

plane.

Equation (5) allows choosing a proper node as ascending taking into account

the seasonal variation of the terminator plane inclination. The goal is to align

the orbit closer to the terminator plane such that formation satellites’ ground

tracks pass at the side of the Earth which is not lit by the Sun at the moment

(which is not always possible to ensure) while lying close to the POI during the

image demonstration.

Figure 3: Orbit design geometry

Step 2. Considering the test mission with two demonstrations above the same

POI produced in one day, the initial conditions deﬁning orbital motion should

13

univocally deﬁne the argument of latitude ugiven at a certain epoch. Under

the assumption of near-polar orbits and assuming that POI is located near the

sub-satellite point during the demonstration the arguments of latitude uat the

mid-points of morning and evening demonstrations can be deﬁned as1:

•for POI in the Northern Hemisphere (ϕ > 0):

(u1, u2) =

(ϕ, π −ϕ),for eZ

Sun ≥0

(π−ϕ, ϕ),for eZ

Sun <0

(6)

•for POI in the Southern Hemisphere (ϕ < 0):

(u1, u2) =

(π−ϕ, 2π+ϕ),for eZ

Sun ≥0

(2π+ϕ, π −ϕ),for eZ

Sun <0

(7)

where u1,u2are arguments of latitude for the target orbit at the mid-points of

the morning and evening demonstrations.

Moscow with latitude ϕ= 55.76◦is a POI for the mission and eZ

Sun =−0.006

at the demonstration date. Therefore, according to the equations (6)–(7) the

arguments of latitude u1and u2corresponding to the morning and evening

demonstrations are 124.4◦and 55.76◦, respectively. The angles u1and u2are

depicted in the illustration to the orbit selection procedure (see Fig. 3).

Step 3. Figure 4 shows the Sun elevation angle θSun above Moscow on the

chosen day (September 24, 2021). Let us recall the Req1.2establishing that

both shows take place at the time when the Sun elevation angle is below −5◦.

In the plot of ﬁgure 4 two points are marked for the evening and morning

demonstration corresponding to the Sun elevation of −6◦. The two points

are the mid-points for the two events and their choice allows the complete

demonstration to meet the Req1.2of having the Sun at a lower elevation than

1For a POI located in equator, i.e. ϕ= 0 the arguments of latitude for morning and

evening demonstrations are u1=ϕ,u2=π−ϕindependently on Sun declination

14

04:00 06:00 08:00 10:00 12:00 14:00 16:00 18:00 20:00

Local time at POI (UTC+3) Sep 24, 2021

-20

-10

0

10

20

30

40

Sun, deg

Sun elevation

Sun elevation at the morning demo mid-point

Sun elevation at the evening demo mid-point

X Sep 24, 2021, 05:42

Y -6

X 24 Sep 2021 18:59:42

Y -6

Figure 4: Sun elevation at Moscow on the demonstration day

−5◦. The choice of the demonstration mid-points epochs determines the time

diﬀerence between them, which entails the value of the orbital period according

to

Torb =∆t

N+∆u

2π

,(8)

where N is the number of complete revolutions between the mid-points of the

two demonstrations, ∆u=u2−u1if u2> u1and ∆u= 2π+u2−u1if u2< u1.

Given the orbital period we obtain the semi-major axis aof the target orbit:

a=3

sµTorb

2π2

,(9)

and its inclination iis now uniquely deﬁned by (2).

Going through steps 1-3 as explained above, we arrive at the complete set

of orbital parameters for the target orbit, which is summarized in the following

table:

Table 1: Target orbit parameters

Epoch (UTC+2) h[km]e[−]i[deg] Ω [deg]ω[deg]u[deg]

24/09/2021,00:00:00 867.2 0 98.88 270.8 0 358.86

15

We have run a simple simulation to verify that the designed target orbit

meets the conditions of Req1.1and Req1.2for the spacecraft and Sun elevation

angles at POI during the demonstration. Figure 5 shows how the two angles

change during the evening and the morning session. It is seen that the morning

demonstration starts at 05:38:18 by Moscow time and lasts for 9 minutes and

4 seconds, whereas the evening demonstration starts at 18:57:45 and lasts for 8

minutes and 56 seconds.

05:38 05:40 05:42 05:44 05:46 05:48

time (UTC+3) Sep 24, 2021

-6.5

-6

-5.5

-5

Sun, deg

morning demonstration

05:38 05:40 05:42 05:44 05:46

time (UTC+3) Sep 24, 2021

10

15

20

25

sat, deg

18:58 19:00 19:02 19:04 19:06

time (UTC+3) Sep 24, 2021

-7

-6.5

-6

-5.5

Sun, deg

evening demonstration

18:58 19:00 19:02 19:04 19:06

time (UTC+3) Sep 24, 2021

10

15

20

25

sat, deg

Figure 5: Variation of the Sun elevation angle θSun and the formation elevation angle θsat

during the two planned demonstrations

3.3. Solar reﬂector sizing

The goal of the solar reﬂector sizing procedure is to calculate the minimum

reﬂector area Arensuring the required pixel magnitude during both demonstra-

tions. The magnitude of the reﬂector mis calculated as the ratio of incident

light intensity Iat POI to a reference intensity:

m=−2.5·log I

Iref ,(10)

where the reference intensity Iref = 2.56 ×10−6lux (Cox, 2000). The intensity

of the incident light Iat the POI is given by (Canady and Allen, 1982):

16

I=I0Arρτ cos(γ) sin(θsat)

4d2tan(αSun

2)2,(11)

where I0= 1360 W/m2is the average intensity of solar energy at the Earth

distance, Aris the area of the satellite’s reﬂector, ρis the mirror reﬂectivity

coeﬃcient, γis the incident angle of solar rays, θsat is the elevation angle

of the satellite measured at POI, dis the distance between the reﬂector and

POI, scattering angle αSun = 0◦32′(Canady and Allen, 1982) is the mean

included angle of the Sun measured from the Earth, and τis the atmospheric

transmissivity (Canady and Allen, 1982):

τ= 0.1283 + 0.7559e−0.3878 sec(π/2−θsat).(12)

Mylar ﬁlm coated with aluminum is considered as the reﬂector material

because of low weight and high reﬂectivity coeﬃcient ρ= 0.92 (Canady and

Allen, 1982). The reﬂector, which is deployed and maintained by a rigid support

structure, is assumed to be of square shape (similar to previous solar sail-

projects (Biddy and Svitek, 2012; Palla et al., 2017)).

To estimate the size of the solar reﬂector suﬃcient for the required pixel

magnitude during both demonstrations we analyze the following expression that

characterizes the geometrical conditions of a particular image demonstration:

I∗(t) = τ(θsat (t)) cos(γ(t)) sin(θsat(t))

d(t)2.(13)

The minimum required reﬂector area Aris then expressed as follows:

Ar=4Ireq tan(αsun/2)2

I0I∗

minρ,(14)

where Ireq is the incident light intensity corresponding to the required pixel

magnitude m(Eq. (10)), I∗

min is the minimum value of I∗for both demonstrations

corresponding to the worst case geometrical conditions yielding the lowest pixel

brightness. The curves I∗(t) for morning and evening demonstrations are shown

in Fig. 6a.

17

The minimum required reﬂector areas Arcorresponding to pixel magnitudes

mof −6 and −8 are evaluated to be 36 m2and 225 m2, respectively. For the

chosen reﬂector areas the corresponding magnitudes of a single pixel during

morning and evening demonstration are presented in Fig. 6b and Fig. 6c. It

can be seen that the pixel magnitude meets the requirement Req2for both

demonstrations.

Figure 6: Solar reﬂector sizing and validation. a) I∗(t) curves for morning and evening

demonstrations; b) Pixel magnitude m(t) during morning demonstration for mreq values of

-8 and -6; c) Pixel magnitude m(t) during evening demonstration for mreq values of -8 and -6

Let us note that the required reﬂector size is obtained for the orbit that

runs close to the terminator plane (as explained in Section 3.2). It allows

performing demonstrations in the morning and in the evening at the same

lighting conditions but degrade geometrical conditions expressed in Eq. (13) and

decrease the demonstration duration. For the considered mission the satellite

elevation θsat marginally reaches 25 degrees as shown in Fig. 5. A signiﬁcantly

smaller sunlight reﬂector can be used for a demonstration passing its POI in

zenith in the mid-point. For example, a satellite in zenith of the same orbit

18

(see Table 1) with incident angle of solar rays γ=π/4 with the magnitudes

-6 and -8 will require the reﬂector side of about 80 and 30 cm, respectively.

Furthermore, it has been demonstrated by LightSail 2 CubeSat mission that it

is possible to deploy solar sail with area of 32 m2and utilize it for orbital motion

control (Spencer et al., 2021). Thus, our reﬂector area estimate Ar= 36 m2

corresponding to the required magnitude m=−6 is technically feasible.

3.4. Orbital conﬁguration

We shall now assume that the satellites can be assigned to periodical closed

relative trajectories in the orbital reference frame as proposed in Vaddi et al.

(2005); Ivanov et al. (2019). In this case, the produced image rotates with an

orbital period relative to the orbital reference frame FOmoving along the target

orbit.

To model relative motion of closely orbiting satellites in the central gravity

ﬁeld in near-circular orbits we employ the Hill-Clohessy-Wiltshire equations

(Wiltshire and Clohessy, 1960), which are extensively used for formation ﬂying

analysis. The linearized equations describing formation spacecraft relative motion

with respect to the orbital reference frame FOare:

¨xi+ 2n˙zi=ui

x,

¨yi+n2yi=ui

y,

¨zi−2n˙xi−3n2zi=ui

z,

(15)

where xi,yi,ziare the components of i-th satellite position vector ρigiven in

FO,ui=ui

x, ui

y, ui

z⊤is the thrust-induced acceleration term.

In case of free motion, i.e. for u= 0, equations (15) admit analytical solution

of the following form (Wiltshire and Clohessy, 1960):

xi(t) = −3Ci

1nt + 2Ci

2cos(nt)−2Ci

3sin(nt) + Ci

4,

yi(t) = Ci

5sin(nt) + Ci

6cos(nt),

zi(t) = 2Ci

1+Ci

2sin(nt) + Ci

3cos(nt).

(16)

19

Setting to zero the values of Ci

1and Ci

4, which are responsible for the constant

drift term and x-axis oﬀset, respectively, the equations Eq.(16) can be written

in the following form:

xi(t) = e

Ci

1cos(nt +e

Ci

3),

yi(t) = e

Ci

2sin(nt +e

Ci

4),

zi(t) = e

Ci

1

2sin(nt +e

Ci

3),

(17)

where e

Ci

1,e

Ci

2constants set the relative orbit size, shape and orientation, and

constants e

Ci

3and e

Ci

4deﬁne the phases for the reference trajectories in in-plane

and out-of-plane motion. The phase constants can also be used to set a certain

spatial orientation of a relative satellite orbit. For example, in (Afanasev et al.,

2021) the Eq. (17) are used for satellite formation ﬂying mission with tetrahedral

orbital conﬁguration.

For the image demonstration mission, the projection of the demonstrated

image onto the local horizontal plane should be circular. In this case, the image

has the same shape for a point of view on Earth independently from the image

orientation. From the expression x2

i+y2

i=ϱ2

i, where ϱiis the radius of the

i-th satellite’s relative orbit projection onto the local horizontal plane, it is clear

that the required constants for projected circular orbits (PCO) are deﬁned as

e

Ci

1=e

Ci

2=ϱi.

Due to the fact the only projection of the reference trajectory to the local

horizontal plane is important for the mission we simplify the form of closed

relative trajectories by taking the same phases for the trajectory in the in-plane

and out-of-plane directions. Finally, the solutions to the HCW equations used

for the mission design purpose are written as follows:

xi(t) = ϱicos(nt +αi),

yi(t) = ϱisin(nt +αi),

zi(t) = ϱi

2sin(nt +αi),

(18)

The concept of a PCO is illustrated by Figure 7.

20

Figure 7: Target orbit and the relative motion in PCO

Having designed a circular target orbit (shown in red color) and given the

initial conditions for an example spacecraft according to (18), we have this

spacecraft moving in a circular orbit (shown in blue), which is close to the

target orbit. However, in the orbital reference frame with the origin moving

along the target orbit, the spacecraft relative motion is a periodical trajectory,

which projects onto xy-plane as a circle (hence the name PCO). The period of

the relative orbit coincides with the period of the target orbit, thus the spacecraft

makes exactly one circle around the origin of the orbital reference frame, whilst

it circles the Earth. Placing a number of spacecraft in diﬀerent PCO with

respect to the same orbital reference frame makes a natural formation which

is seen from the ground as a number of co-rotating pixels forming an image.

The phase argument in (18) determines the locations of individual spacecraft in

their respective relative trajectories, and having ﬁxed those individual phases,

we can add a constant to them, thus controlling the required orientation of the

whole image.

We shall deﬁne the phase of the i-th satellite in its relative orbit as αi=

αi

0+α∗, where αi

0depends on the individual pixel location within the image

21

and phase α∗is used to set the required image orientation at demonstration.

To design an orbital conﬁguration for a graphic to be displayed we determine

the positions of all pixels in the polar coordinates ϱi, αi

0with respect to the

geometrical median of the pixels. The coordinates are then scaled to satisfy the

requirement to the minimum admissible inter-pixel distance IP D during image

demonstration (see Req2):

I P Dmin = 2 tan(βmin /2)dmax,(19)

where βmin = 1′is the angular resolution of human naked eye, dmax is the

maximum distance between satellite and POI during demonstrations. Finally,

the image should be mirrored with respect to y-axis to have the appropriate

view from the point of interest.

The initial image phase α∗is chosen to ensure the required orientation of

the image during demonstration. Assuming that the projected image is aligned

along x-axis of the orbital reference frame FOat the midpoint of demonstration

the required phase of an image α∗is deﬁned as follows:

α∗= 2π−uk,(20)

where ukis the argument of latitude of the target orbit at the mid-point of k-th

demonstration.

The required reference trajectories are deﬁned by pixels’ polar coordinates

ϱi, αi

0and the image phase angle α∗using equations (18) and taking into account

the relation e

Ci

1=e

Ci

2=ϱi. The state vector e

Xito describe the required orbit

in the inertial frame FIis then calculated using equations (1).

Figures 8,9 show the orbital conﬁgurations for the morning and evening

demonstrations with respect to FO. The formation for the designed mission

consists of 50 satellites. The initial conditions ϱi, αi

0, α∗for both orbital conﬁg-

urations are presented in Appendix A.

22

Figure 8: Orbital conﬁguration for the morning demonstration of Eiﬀel tower (as seen from

POI)

4. Formation control algorithm

The proposed control scheme operates in three regimes – reconﬁguration,

maintenance, and standby. Reconﬁguration takes place when the formation

is commanded to assume a new pattern (e.g. directly after deployment or

whenever there must be a change in the displayed image). Maintenance is

to preserve an existing line-up with a given accuracy. Finally, standby regime

implies no control and is used, for instance, during image demonstrations, when

all formation spacecraft should ensure the required attitude of reﬂectors.

The centralized approach we describe, assumes that all satellites control their

relative trajectories individually with respect to the virtual satellite placed at

the target orbit. Orbital dynamics of each spacecraft is described in the inertial

23

Figure 9: Orbital conﬁguration for the evening demonstration of Olympic rings (as seen from

POI)

reference frame FIby the following equation:

¨

Ri=−µRi

R3

i

+aJ2

i+ui,(21)

where Riis position vector of ith satellite, aJ2

istands for external disturbance

caused by Earth oblateness, uiis the control thrust vector applied to ith satellite

in formation. In all subsequent simulations only the eﬀect of J2is taken into

account because altitude of the considered target orbit is about 870 km where

other perturbing forces are much smaller and can be neglected. Acceleration

caused by J2eﬀect is given by:

aJ2

i=3µJ2R2

⊕

2R5

i5Z2

i

R2

i−1Ri−2Zi,(22)

where Zi= [0,0, Z]⊤.

24

The required reference trajectories of formation spacecraft are deﬁned with

respect to the orbital frame of the virtual satellite according to the analytical

solution to the HCW equations (15). These reference trajectories are given by

e

xi(t)=[e

ρ⊤

i(t),e

v⊤

i(t)]⊤, and according to the notations introduced by Eq. (1),

their expressions in the FIare:

e

Ri=R0+Ae

ρi,

e

Vi=V0+A[n×e

ρi] + Ae

vi.

(23)

Formation reconﬁguration is carried out with the use of both impulsive

and continuous control. First, the impulsive maneuvers are applied for coarse

relative orbits correction while the continuous control complements the impulsive

one to ensure the required positioning accuracy after impulsive control. The

maintenance regime is based on continuous control and its objective is to keep

the relative orbits within the required tolerances ε=|x(t)−e

x(t)|.

4.1. Impulsive maneuvers

The impulsive control scheme consists of 4 single-impulse maneuvers, which

consecutively correct the error δebetween the required erand the current orbital

elements ecof a spacecraft. The ﬁrst two maneuvers to correct the semi-major

axis and eccentricity are given by (Schaub and Alfriend, 2001):

dV p

θ

dV p

h

dV p

ρ

=

naη

4(δa

a+δe

1+e)

0

0

,

dV a

θ

dV a

h

dV a

ρ

=

naη

4(δa

a−δe

1−e)

0

0

,(24)

where dV i

θ,dV i

h,dV i

ρare tangential, out-of-plane, and radial components of the

dV vector, superscripts pand adenote perigee and apogee according to the

impulse application points, η=√1−e2. Note, that δa and δe are the error in

semimajor axis and eccentricity expressed in the mean orbital elements.

It was decided earlier that the designed image demonstration mission assumes

circular Sun-synchronous orbits. Therefore, we need to relate the required

reference orbits to nonsingular orbital elements. Following work of Vaddi et al.

25

(2005) we utilize the equinoctial orbital elements eeq = [a, q1, q2, i, Ω, λ]⊤, where

ais semi-major axis, q1=ecos(ω), q2=esin(ω), iis orbit inclination, Ω is

longitude of ascending node, λ=ω+M,ωis argument of perigee, Mis mean

anomaly.

Thus, the two remaining maneuvers that complete the 4-impulse scheme and

correct the errors in all remaining orbital elements are given by (Vaddi et al.,

2005):

dV 1

θ

dV 1

h

dV 1

ρ

=

0

1

γqδi2+δΩ2·sin2(i)

1

2γpδq2

1+δ2

2

,

dV 2

θ

dV 2

h

dV 2

ρ

=

0

0

−1

2γpδq2

1+δ2

2

,(25)

where dV i

θ,dV i

h,dV i

ρare tangential, out-of-plane, and radial components of the

dV vector, γ=qa

µ. The impulses are applied at argument of latitudes θ1and

θ2deﬁned as follows:

θ1= tan−1δΩ sin(i)

δi , θ2=θ1+π.

Taking into account that impulsive scheme is derived using linearized var-

iational equations for orbital elements and the fact that orbital elements are

strongly coupled to each other, each impulsive correction will slightly inﬂuence

the rest of orbital elements and hence has limited precision. Therefore, considering

long-term operations the continuous control for ﬁne relative orbits correction has

to be implemented.

4.2. Continuous maneuvers

The low-thrust continuous control algorithm for satellite relative trajectory

adjustment is based on the linear-quadratic regulator. The state-space repre-

sentation of the Hill-Clohessy-Wiltshire model (15) is:

˙x =Ax +Bu,(26)

26

where Ais corresponding dynamic state matrix

A=

O3×3E3×3

Acentr Acor

,Acentr =

0 0 0

0−n20

0 0 n2

,

Acor =

0 0 −2n

0 0 0

2n0 0

.

The control matrix Bis given by

B=

O3×3

E3×3

.

The linear equations describing dynamics of the deviation from a desired

reference trajectory can be expressed as follows:

d

dtδx=Aδx+Bu.(27)

The linear quadratic regulator is the feedback control u=Kδxthat ensures

the minimum of the functional Jalong a relative satellite trajectory:

J=Z∞

0

(δx⊤Qδx+u⊤Ru)dt, (28)

where Qand Rare the positive deﬁnite diagonal weight matrices that determine

the weight of errors for the state vector δxand the weight of the control resource

consumption, respectively.

The static gain matrix Kis deﬁned as follows:

K=−R−1B⊤P,(29)

where Pmatrix is obtained as a solution of the Riccati equation

A⊤P+PA −PBR−1B⊤P+Q= 0.(30)

The matrix algebraic Riccati equation (30) is used to ﬁnd the matrix Pfor

speciﬁc weight matrices Qand Rensuring optimal gain matrix K. The matrices

27

Qand Rare the tuning parameters for the continuous control algorithm that

characterize the transient processes.

The control vector saturation is taken into account. If the magnitude of

the computed control vector uis greater that the maximum one allowable by

onboard thruster the controller scales it and returns control vector uwith the

magnitude equal to the maximum possible control vector umax.

4.3. Optimized reconﬁguration

Reconﬁguration procedure is called to change the current orbital conﬁguration

of a formation represented by a set of reference trajectories to a new set of

trajectories. Fig. 10 presents a simple example of three satellites with diﬀerent

levels of remaining fuel and three required trajectories the satellites must be

assigned to.

Figure 10: Reconﬁguration scenario example

Retaining all formation satellites in operation requires keeping track of fuel

consumption. The satellites perform maneuvers that have diﬀerent cost in

terms of the consumed fuel, and the maneuver costs need to be distributed

among the group so as not to lose any satellite by spending all fuel it has. This

28

brings us to the idea of solving the combinatorial optimization problem for each

reconﬁguration to ﬁnd a scenario yielding maximization of minimum remaining

fuel among the formation satellites. The optimization problem is the standard

assignment problem (Duﬀ and Koster, 2001) that takes into account the level

of remaining fuel and is deﬁned as follows.

Given two sets of equal size Nsat: spacecraft Sand trajectories Tand also

a weight function C:S×T→R, ﬁnd a bijection f:S→Tsuch that the cost

function N

X

i=1

C((i), f (i))

is minimized. The weight function is deﬁned in terms of the cost matrix Cgiven

by:

Cij =Ctransf er

ij +Cmaintenance

j,(31)

where Ctransf er

ij (i, j =1:Nsat), represents the cost of a transfer of a satellitei

to a new trajectoryjin terms of amount fuel spent, and Cmaintenance

jis the

cost of maintenance of jth trajectory for a speciﬁc period of time which is

predeﬁned based on image demonstration mission concept of operations. The

maintenance cost Cmaintenance

jis calculated via simulation of the formation

satellites dynamics and control performed with the aid of continuous maneuvers.

The LQR-based control algorithm proposed in the Section 4.2 is used. Thruster

saturation is taken into account in the simulations.

Let us also denote by F= [F1, F2, Fi, ..., FNsat]⊤the formation fuel state

vector representing the amount of fuel in each satellite.

The algorithm to perform the required optimization is then carried out in the

steps shown below. Our numerical experiments indicate that it takes just a few

iterations for the algorithm to converge to the desired assignment (owing to a

good initial approximation given by the assignment corresponding to the overall

minimum fuel consumption). Let us note, however, that this reconﬁguration

algorithm is greedy, and for missions that require several consecutive reconﬁgu-

rations the algorithm must be adjusted so as to maximize the level of remaining

29

Algorithm 1: Spacecraft assignment to trajectories

1initialization

Fmin = min

iF

while Fmin >0do

2compute the cost matrix Cfor the maneuver;

3solve the assignment problem that minimizes the overall fuel

consumption (using the matchpairs Matlab function Duﬀ and

Koster (2001)):

N

X

i=1

Ci,f(i)→min

f;

4recompute the formation fuel state vector Faccording to the

solution obtained in the previous step;

5ﬁnd the minimal amount of fuel among the formation spacecraft

Fmin = min

iF;

6change the cost matrix according to the following rule:

Ci,j =

Fsum,if Fi−Ci,j ≤Fmin

Ci,j ,otherwise

,

where Fsum is a suﬃciently large number (for instance, equal to

the total amount of fuel in all formation satellites in the beginning

of the mission), which signiﬁes that the {i, j}pair is excluded from

the assignment problem solution in the following iterations;

7end

fuel at the end of the whole sequence instead of doing it for each step of the

sequence.

30

4.4. Control Scheme Summary

Fig. 11 shows the general scheme of the hybrid formation control algorithm

that can be applied for multiple image demonstrations mission. The scheme

represents how the satellite formation is operated to perform k-th graphic dem-

onstration above a given POI on Earth. Parameters Tstart

demok,Tend

demokdesignate

pre-computed moments in time when k-th demonstration starts and terminates.

In order to perform the demonstration the required orbital conﬁguration has to

be deployed in advance. For the purpose the formation is commanded to start

reconﬁguration at time Treconfk. During the maintenance stage the relative

error is controlled to be within the threshold of ε0(which is broken into two

separate thresholds for position ερand velocity εv).

Figure 11: Hybrid formation control algorithm general scheme

Thus, the proposed algorithm starts its operation in reconﬁguration regime.

At the stage, the formation satellites should be gathered into a certain orbital

conﬁguration required to produce the graphics. The assignment problem is

31

ﬁrstly solved that allows allocating formation satellites to the relative trajectories

in a way minimizing total spent fuel for reconﬁguration and further maintenance

of the conﬁguration and maximizing amount of fuel for a satellite within formation

with least fuel level. The solutions are checked for collision occurrences and if the

reconﬁguration scenario simulation shows that some spacecraft come within a

certain threshold distance from each other denoted as safe inter-satellite distance

(ISDsaf e ), the solution is either discarded or the time proﬁle of the scenario is

changed (to start some of the maneuvers with a delay). The deployed orbital

conﬁguration is then maintained up to the time of the image demonstration and

after it until the next orbital conﬁguration is assigned to the satellite formation.

The algorithm performance is illustrated in the following section where

simulation results are discussed.

32

5. Simulation results

This section demonstrates the hybrid formation control algorithms perfor-

mance for the mission whose objective to demonstrate two pixel images in the

sky over Moscow on September 24, 2021. The concept of operation is deﬁned

in accordance with the previously described mission design (Fig. 12). Following

the sequence of events as given by Fig. 12, we model the controlled dynamics of

satellite formation consisting of 50 satellites equipped with sunlight reﬂectors.

Table 2 lists the simulation parameters.

Figure 12: Concept of operation of the demonstration mission

It is assumed that all satellites are delivered to the target orbit by a cluster

launch and deployed consecutively with a short time step such that we can

consider that all satellites are located at the same orbit at the beginning of

the numerical simulation. When all formation satellites are released from an

orbital deployer they are assigned to reference trajectories corresponding to the

ﬁrst orbital conﬁguration (see Fig. 8 and Appendix A, Table 3). When the

deployment starts, the formation operates in the reconﬁguration regime that

ﬁrstly corrects relative orbits of formation satellites using impulsive maneuvers

and then completes the deployment procedure with LQR-based continuous

control.

Figures 13 and 14 show position vectors error magnitude δρ for all formation

satellites during the ﬁrst and the second half of the mission. Figure 13 shows

that the deployment phase lasts for 163 minutes (after which all satellites have

converged to the required trajectories within given tolerances). Two inserted

frames in Figure 13 detail the level of position error after the reconﬁguration

33

Table 2: Simulation parameters for the demonstration mission on September 24, 2021

Parameter Value Units

Satellite Parameters

Mass 18 kg

Dimensions 340 ×200 ×200 mm

Tmax (Busek) 180 mN

Isp (Busek) 214 s

Formation Parameters

Number of satellites 50 -

Treconf [00:00:00, 12:00:00]⊤UTC+3

Tstart

demo [05:38:18, 18:57:45]⊤UTC+3

Tend

demo [05:47:22, 19:06:41]⊤UTC+3

Control parameters

Maintenance tolerance ερ1 m

Maintenance tolerance εv0.01 m/s

ISDsaf e 30 m

Q matrix diag([10−7,10−7,10−7,10−9,10−9,10−9]⊤) -

R matrix E3x3-

(ﬁrst zoomed plot) and right after the ﬁrst demonstration has started (second

zoomed plot). The orbital conﬁguration after deployment is shown in Fig. 15.

Figure 14 portrays the transition to the second demonstration and also contains

two zoomed insertions showing the level of position error after the reconﬁguration

and after the start of the demonstration.

When deployment is complete the ﬁrst orbital conﬁguration is maintained

using continuous LQR-based continuous control until the ﬁrst demonstration

above Moscow. The ﬁrst demonstration is performed at the time period (see

Fig. 12) identiﬁed by mission design study. The demonstration phase assumes

standby operation regime where no control is applied because all formation

satellites have to ensure the required attitudes of solar reﬂectors. The ﬁrst

34

Figure 13: Position error δρ for all formation satellites during the ﬁrst half of the mission

Figure 14: Position error δρ for all formation satellites during the seconds half of the mission

orbital conﬁguration is maintained up to the time when the reconﬁguration to

the second orbital conﬁguration is scheduled.

The same routine is applied for the second part of the mission where the

35

Figure 15: Orbital dynamics of ﬁrst orbital conﬁguration for Eiﬀel tower demonstration

Figure 16: Orbital dynamics of second orbital conﬁguration for the Olympic rings

demonstration

evening demonstration is to be performed above Moscow. However, reconﬁguration

starts with optimization problem that allows to assign satellite to a new set

of reference trajectories (see Appendix A,Table 4) in a way to minimize total

fuel spent for reconﬁguration and maximize minimum amount of fuel among

formation satellites. For the considered mission the reconﬁguration procedure

lasts for 209 minutes. The resulted orbital conﬁguration after reconﬁguration

36

is presented in Fig. 16. The animation of controlled dynamics of the formation

satellites within the modeled mission is presented in Biktimirov et al.

Figure 17 shows the iterations of the algorithms that assigns spacecraft

to trajectories for the second reconﬁguration (the ﬁrst conﬁguration is not

presented as it starts from the same level of fuel in all spacecraft and is thus not

illustrative of the algorithm). Note that the ﬁrst assignment problem solution

yields the minimum overall fuel consumption (represented by a point in the left

bottom corner of the plot, which corresponds to the average fuel consumption

of 31.2 g). The lowest fuel level over the formation spacecraft corresponding

to this solution is 859.6 g. The algorithm converges to the maximin solution

in eleven iterations (the point in the upper right corner of the plot), the ﬁnal

solution increases both the average consumption (by 1 g) and the lowest level

of remaining fuel (by 8 g).

Let us note that we employed a greedy version of the reconﬁguration algorithm

as we are concerned with a single reconﬁguration maneuver. The solution

portrayed in Figure 17 is such that the 50th satellite is assigned the maneuver

with the lowest possible fuel consumption. However, our numerical experiments

with diﬀerent orbital conﬁgurations showed that a satellite may come to an

orbit, reconﬁguration maneuvers from which are costly in terms of consumed

fuel. Minimizing the one-step fuel consumption of such satellites by a greedy

algorithm may lead to jumping between the orbits with costly maneuvers and

thus spending all available fuel much faster than other formation spacecraft.

Such would be the fate of satellite 50 if we were to repeat the same sequence of

two reconﬁgurations. In order to avoid this and evenly distribute the amount of

consumed fuel among the formation satellites, the reconﬁguration assignment

algorithm should analyze all reconﬁgurations which are to take place in the

future (or at least have a certain horizon of several forthcoming maneuvers).

Thus, satellite 50 could spend more fuel at the second reconﬁguration to move

to a less expensive orbit thus saving more fuel in the future reconﬁgurations.

Figure 18 shows cumulative fuel level distribution of formation satellites

during the modeled mission. The mission timeline depicted in the bottom of the

37

31 31.2 31.4 31.6 31.8 32 32.2

Average fuel consumption, g

859

860

861

862

863

864

865

866

867

868

Fmin, g

Last iteration of maximin optimization

Satellite 50 is assigned to trajectory 2

Figure 17: Maximin optimization results

Figure 18: Cumulative graph of fuel consumption during the mission

ﬁgure shows the sequence of the formation operation stages during the mission,

where diﬀerent color represents diﬀerent control regimes. The reconﬁguration

regime is marked with the red color, maintenance regime with the yellow, and

gray color is used for the standby.

38

6. Conclusion

We have outlined the general mission design procedure that can be applied

to various image demonstration missions. The proposed routine can be used

to assess technical feasibility of image demonstration missions, evaluate their

characteristic lifetime, and estimate the proposed hybrid controller performance.

The example mission with two images displayed over Moscow is characterized

by relatively short reconﬁguration time (163 minutes for the ﬁrst conﬁguration

and 209 minutes for the second one), proving that the proposed hybrid formation

control allows demonstrating multiple images even within one day mission while

meeting the image quality requirements. For the simulated mission, the relative

position error ερdoes not exceed 10 meters for all formation satellites during

both image demonstrations. The lifetime estimations showed that for the spacecraft

parameters presented in Table 2 the maximum monthly consumption over for-

mation satellites while maintaining the ﬁrst orbital conﬁguration is 199.6 g and

230.8 g for the second orbital conﬁguration. If estimating mission lifetime for the

mission with two reconﬁgurations per day (with the timeline as in the modeled

mission but repeated constantly), it allows showing 7 diﬀerent images within

4 days with a fuel mass of 1 kg for each spacecraft. However, the lifetime can

be increased by adjusting the assignment optimization algorithm allowing it to

take into account the whole sequence of formation reconﬁgurations. It appears

that the further the reference trajectory goes from the target orbit, the more

fuel it requires for reconﬁguration and maintenance. Owing to this one might

adopt a strategy of breaking up elongated images (such as long words) into

smaller pieces (such as individual letter), whose centers belong to the target

orbit and the phasing angles of all relative trajectories are tuned for all pieces

to synchronize their orientation during the image demonstration.

One of the obvious applications for image demonstration missions is space

advertising. An interesting venue to explore might lie in using the proposed

mission design framework to evaluate its economic feasibility. This study can

be carried out with the use of the Earth population density distribution model

39

as in Kharlan et al. (2020), which makes it possible to estimate how many people

come into visual contact with the displayed images during its display.

Finally, more work can be done on the algorithmic level. One obvious

improvement of the proposed control scheme is seeing whether additional control

can be applied with the use of diﬀerential aerodynamic forces in LEO in order to

reduce the fuel consumption. Another interesting problem that comes to mind as

future work is the study of decentralized control approaches with communication

constraints applied to the maintenance of the proposed formation.

7. Acknowledgments

The presented study except for Section 3.4 was funded by the Russian

Foundation for Basic Research (RFBR), project number 20-31-90115. Section

3.4 is supported by the Moscow Center for Fundamental and Applied Mathematics,

Agreement with the Ministry of Science and Education of the Russian Federation,

number 075-15-2019-1623.

The authors are grateful to Vladilen Sitnikov for fruitful discussions that

contributed to initiation of this study.

40

References

Afanasev, A., Shavin, M., Ivanov, A., Pritykin, D., 2021. Tetrahedral Satellite

Formation: Geomagnetic Measurements Exchange and Interpolation.

Advances in Space Research 67, 3294–3307.

Battin, R.H., 1999. An Introduction to the Mathematics and Methods of

Astrodynamics, Revised Edition. American Institute of Aeronautics and

Astronautics.

Biddy, C., Svitek, T., 2012. LightSail-1 Solar Sail Design and Qualiﬁcation.

Proceedings of the 41st Aerospace Mechanisms Symposium .

Biktimirov, S., Ivanov, A., Kharlan, A., Mullin, N., Pritykin, D., 2019.

Deployment and Maintenance of Solar Sail-Equipped CubeSat Formation in

LEO. In AIAC18: 18th Australian International Aerospace Congress (2019):

HUMS-11th Defence Science and Technology (DST) International Conference

on Health and Usage Monitoring (HUMS 2019): ISSFD-27th International

Symposium on Space Flight Dynamics (ISSFD) , p. 983.

Biktimirov, S., Ivanov, D., Pritykin, D., . Animation of Dynamics

and Control of Satellite Formation Flying for Pixel Image

Demonstration in the Sky. https://drive.google.com/drive/folders/

1bewWDhQ5rQ6G3lsafP7KQ6DiFyiRDHDA?usp=sharing.

Biktimirov, S., Ivanov, D., Sadretdinov, T., Omran, B., Pritykin, D., 2020. A

Multi-Satellite Mission to Illuminate the Earth: Formation Control Based on

Impulsive Maneuvers. Advances in the Astronautical Sciences 173, 463–474.

Bonin, G., Roth, N., Armitage, S., Newman, J., Risi, B., Zee, R.E., 2015. CanX–

4 and CanX–5 Precision Formation Flight: Mission Accomplished! 29th

Annual AIAA/USU Conference on Small Satellites .

Busek, . BGT-X1 Green Monopropellant Thruster. http://www.busek.com/

technologies__greenmonoprop.html.

41

Canady, J.E., Allen, J.L., 1982. NASA Technical Paper 2065: Illumination from

Space with Orbiting Solar-Reﬂector Spacecraft. Technical Report. Langley

Research Center.

Chow, D., 2019. This Russian startup wants to put

huge ads in space. Not everyone is on board with

the idea. https://www.nbcnews.com/mach/science/

startup-wants-put-huge-ads-space-not-every-one-board-idea- ncna960296.

Cox, A.N. (Ed.), 2000. Allen’s Astrophysical Quantities. New York: AIP Press;

Springer. 4 edition.

D’Amico, S., Montenbruck, O., 2006. Proximity Operations of Formation-Flying

Spacecraft Using an Eccentricity/Inclination Vector Separation. Journal of

Guidance, Control, and Dynamics 29, 554–563.

Dawn Aerospace, . CubeSat Propulsion Modules. https://www.

dawnaerospace.com/products/cubedrive.

Di Mauro, G., Bevilacqua, R., Spiller, D., Sullivan, J., D’Amico, S., 2018.

Continuous maneuvers for spacecraft formation ﬂying reconﬁguration using

relative orbit elements. Acta Astronautica 153, 311–326.

Duﬀ, I.S., Koster, J., 2001. On Algorithms For Permuting Large Entries to

the Diagonal of a Sparse Matrix. SIAM Journal on Matrix Analysis and

Applications 22, 973–996.

D’Arrigo, P., Santandrea, S., 2006. Apies: A mission for the exploration of the

main asteroid belt using a swarm of microsatellites. Acta Astronautica 59,

689–699.

Fish, C., Swenson, C., Crowley, G., Barjatya, A., Neilsen, T., Gunther,

J., Azeem, I., Pilinski, M., Wilder, R., Allen, D., et al., 2014. Design,

Development, Implementation, and on-Orbit Performance of the Dynamic

Ionosphere Cubesat Experiment Mission. Space Science Reviews 181, 61–

120.

42

Fridlund, C., 2004. The Darwin Mission. Advances in Space Research 34, 613–

617.

Fuselier, S., Lewis, W., Schiﬀ, C., Ergun, R., Burch, J., Petrinec, S., Trattner,

K., 2016. Magnetospheric Multiscale Science Mission Proﬁle and Operations.

Space Science Reviews 199, 77–103.

Gill, E., Runge, H., 2004. Tight Formation Flying for an Along-Track SAR

Interferometer. Acta Astronautica 55, 473–485.

Goldstein, H., Poole, C., Safko, J., 2002. Classical mechanics.

Hill, G.W., 1878. Researches in the Lunar Theory. American journal of

Mathematics 1, 5–26.

Ivanov, D., Biktimirov, S., Chernov, K., Kharlan, A., Monakhova, U.,

Pritykin, D., 2019. Writing with Sunlight: Cubesat Formation Control

Using Aerodynamic Forces, in: Proceedings of the International Astronautical

Congress, IAC.

Kharlan, A., Biktimirov, S.N., Ivanov, A., 2020. Prospects for the Development

of Global Satellite Communication Constellations in the Context of New

Services in the Telecommunications Market. Cosmic Research 58, 402–410.

Kishida, Y., 1989. Changes in Light Intensity at Twilight and Estimation of the

Biological Photoperiod. Japanese Agricultural Research Quarterly , 22–247.

Krieger, G., Zink, M., Bachmann, M., Br¨autigam, B., Schulze, D., Martone,

M., Rizzoli, P., Steinbrecher, U., Walter Antony, J., De Zan, F., Hajnsek,

I., Papathanassiou, K., Kugler, F., Rodriguez Cassola, M., Younis, M.,

Baumgartner, S., L´opez-Dekker, P., Prats, P., Moreira, A., 2013. TanDEM-

X: A Radar Interferometer with Two Formation-Flying Satellites. Acta

Astronautica 89, 83–98.

Larbi, M.K.B., Stoll, E., 2018. Spacecraft formation control using analytical

ﬁnite-duration approaches. CEAS Space Journal 10, 63–77.

43

Lior, N., 2013. Mirrors in the Sky: Status, Sustainability, and Some Supporting

Materials Experiments. Renewable and Sustainable Energy Reviews 18, 401–

415.

Maley, P.D., Pizzicaroli, J.C., 2003. The Visual Appearance of the Iridium®

Satellites. Acta Astronautica 52.

Nogueira, T., Scharnagl, J., Kotsiaros, S., Schilling, K., 2015. NetSat-4G A Four

Nano-Satellite Formation for Global Geomagnetic Gradiometry, in: 10th IAA

Symposium on Small Satellites for Earth Observation.

Oberth, H., 2019. Wege zur Raumschiﬀahrt. Oldenbourg Wissenschaftsverlag.

Palla, C., Kingston, J., Hobbs, S., 2017. Development of Commercial Drag-

Augmentation Systems for Small Satellites. 7th European Conference on

Space Debris .

Palmerini, G.B., Sabatini, M., 2007. Dynamics and Control of Low-Altitude

Formations. Acta Astronautica 61, 298–311.

REUTERS, F., 1986. Europe Plans to Orbit Ring of Light to Hail Eiﬀel Tower.

Rossen, J., 2018. Ad Astra: The Time Earth Almost Got a

Space Billboard. https://www.mentalfloss.com/article/557485/

when-earth-almost-got-space-billboard. [Online; accessed 19-July-

2021].

Sabatini, M., Volpe, R., Palmerini, G., 2020. Centralized Visual Based

Navigation and Control of a Swarm of Satellites for on-Orbit Servicing. Acta

Astronautica 171, 323–334.

Scala, F., Zanotti, G., Curzel, S., Fetescu, M., Lunghi, P., Lavagna, M.,

Bertacin, R., 2020. The HERMES Mission: A CubeSat Constellation For

Multi-Messenger Astrophysics, in: 5th IAA Conference on University Satellite

Missions and CubeSat Workshop, pp. 1–17.

44

Schaub, H., Alfriend, K.T., 2001. Impulsive Feedback Control to Establish

Speciﬁc Mean Orbit Elements of Spacecraft Formations. Journal of Guidance,

Control, and Dynamics 24, 739–745.

Semenov, Y., Branec, V., Grigorev, Y., Zelenshchikov, N., Koshelev, V.,

Melnikov, V., Platonov, V., Sevastianov, N., Syromyatnikov, V., 1994.

Kosmicheskij Eksperiment po Razvyortyvaniyu Plyonochnogo Beskarkasnogo

Otrazhatelya D = 20 m (Znamya-2) [Space Experiment Znamya 2 on the

Deployment of 20 Meters Diameter Frameless Film Reﬂector]. Kosmicheskie

issledovaniya 32, 186–193.

She, Y., Li, S., Wang, Z., 2019. Constructing a Large Antenna Reﬂector

via Spacecraft Formation Flying and Reconﬁguration Control. Journal of

Guidance, Control, and Dynamics 42, 1372–1382.

Spencer, D.A., Betts, B., Bellardo, J.M., Diaz, A., Plante, B., Mansell, J.R.,

2021. The LightSail 2 Solar Sailing Technology Demonstration. Advances in

Space Research 67, 2878–2889.

Tapley, B.D., Bettadpur, S., Watkins, M., Reigber, C., 2004. The Gravity

Recovery and Climate Experiment: Mission Overview and Early Results.

Geophysical Research Letters 31.

Vaddi, S., Alfriend, K.T., Vadali, S., Sengupta, P., 2005. Formation

Establishment and Reconﬁguration Using Impulsive Control. Journal of

Guidance, Control, and Dynamics 28, 262–268.

Vallado, D.A., 2001. Fundamentals of Astrodynamics and Applications.

volume 12. Springer Science & Business Media.

Wiltshire, R., Clohessy, W., 1960. Terminal Guidance System for Satellite

Rendezvous. Journal of the Aerospace Sciences 27, 653–658.

Yanoﬀ, M., Duker, J.S., 2009. Opthalmology , 54.

45

Appendix A: Initial conditions for orbital conﬁguration

Table 3,4 presents the initial conditions ϱi, α0

ifor the required reference

trajectories for morning and evening demonstrations. The initial phases α∗

deﬁning a proper image attitude during morning and evening demonstrations

are 234.95◦and 301.46◦, respectively.

Table 3: Initial conditions for the morning demonstration

Traj ϱ[m] α0[deg] Traj ϱ[m] α0[deg] Traj ϱ[m] α0[deg]

1 6878 319.4 18 2690 236.3 35 1492 90

2 5827 309.8 19 3730 216.9 36 2110 135

3 5004 296.6 20 2690 326.3 37 2359 71.6

4 5004 243.4 21 1668 296.6 38 2359 108.4

5 5827 230.2 22 1668 243.4 39 3076 76

6 6878 220.6 23 2690 213.7 40 3076 104

7 5827 320.2 24 1668 333.4 41 3804 78.7

8 4777 308.7 25 746 270 42 3730 90

9 4777 231.3 26 1668 206.6 43 3804 101.3

10 5827 219.8 27 1492 0 44 4476 90

11 4777 321.3 28 0 58.6 45 5222 90

12 3730 306.9 29 1492 180 46 5968 90

13 3730 233.1 30 1668 26.6 47 6756 83.7

14 4777 218.7 31 1055 45 48 6756 96.3

15 3730 323.1 32 1055 135 49 7460 90

16 2690 303.7 33 1668 153.4 50 8206 90

17 2238 270 34 2110 45

46

Table 4: Initial conditions for the evening demonstration

Traj ϱ[m] α0[deg] Traj ϱ[m] α0[deg] Traj ϱ[m] α0[deg]

1 8166 34.3 18 2577 286.6 35 4275 245

2 9458 24.8 19 946 321 36 5629 231.8

3 9859 14.2 20 1980 27.9 37 6471 217.7

4 9499 3.50 21 4481 91.1 38 6538 202.9

5 8387 354.6 22 4199 65.4 39 6266 187.3

6 6858 349.7 23 3614 41 40 5225 171.1

7 5163 351.6 24 2719 13.1 41 8080 146.3

8 4100 8.60 25 1763 331.9 42 6195 141.5

9 4724 31.8 26 1377 269.7 43 4521 148.9

10 6245 38.1 27 1915 208.3 44 4031 171.8

11 3645 23.2 28 2923 167.9 45 5145 188.8

12 5288 8.80 29 3750 141.5 46 6886 190.9

13 6308 354.6 30 4255 116.2 47 8366 185.7

14 6712 338.4 31 3715 157.9 48 9394 176.7

15 6483 323.7 32 1783 154.4 49 9778 166.2

16 5590 308.3 33 951 225.1 50 9297 155.2

17 4182 294.7 34 2645 254.4

47

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships

that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered

as potential competing interests:

Declaration of Interest Statement