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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.
Journal Pre-proofs
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. oxyz is the orbital reference frame (FO) whose origin otravels 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=A1(RiR0),
vi=A1(ViV0)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 ×103
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
97.39to 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 = cos1(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
(πϕ, ϕ),for eZ
Sun <0
(6)
for POI in the Southern Hemisphere (ϕ < 0):
(u1, u2) =
(πϕ, 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= 55.76is 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 124.4and 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=πϕindependently 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=u2u1if u2> u1and ∆u= 2π+u2u1if 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 ×106lux (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 = 032(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 + 0.7559e0.3878 sec(π/2θsat).(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 Ifor 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,
¨zi2n˙xi3n2zi=ui
z,
(15)
where xi,yi,ziare the components of i-th satellite position vector ρigiven in
FO,ui=ui
x, ui
y, ui
zis 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 αis 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 = 1is 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 αis 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 αis defined as follows:
α= 2πuk,(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 αusing 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, αfor 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
i5Z2
i
R2
i1Ri2Zi,(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
aδe
1e)
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=1e2. 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, , 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+δ2·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= tan1δΩ 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
0n20
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
(δxQδx+uRu)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=R1BP,(29)
where Pmatrix is obtained as a solution of the Riccati equation
AP+PA PBR1BP+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×TR, find a bijection f:STsuch 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 FiCi,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([107,107,107,109,109,109]) -
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
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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 234.95and 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
... 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 [8] 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. ...
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