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29th International Symposium on Space Flight Dynamics (ISSFD)
22-26 April 2024 at ESA-ESOC in Darmstadt, Germany.
Active feedback control on post-mission disposal compliance to limit the space debris proliferation
Martina Rusconi(1), Camilla Colombo(2)
(1)Politecnico di Milano
Milan, Italy
martina.rusconi@polimi.it
(2) Politecnico di Milano
Milan, Italy
camilla.colombo@polimi.it
Abstract – Space utilisation faced unforeseeable
changes in the last decades. However, the policy
definition for debris mitigation has not matched the
rapid growth of the inert population on orbit. The
interdisciplinary framework proposed in the
GREEN SPECIES project, funded by the European
Research Council, aims at providing scientific
support to the reactive definition of regulations and
at systematic investigating debris mitigation
strategies. In this respect, this paper focusses on the
concurrent development of a propagator of the
objects’ dynamics with sources, sinks and mitigation
measures and of a feedback controller acting on the
population. The objects orbiting low-Earth space are
modelled as a fluid with continuous properties. A
deposition profile is modelled along with a term
emulating post-mission disposal of objects. As a first
approach, a feedback, proportional and linear
control logic automatically selects the post-mission
disposal compliance of the deposited objects, to limit
the growth of the inert population on orbit. An
example of the methodology is provided, and the
results discussed in terms of validity of the approach.
I. INTRODUCTION
The problem of an uncontrolled debris proliferation has
been discussed since the ‘70s, when Kessler and Cour-
Palais raised attention on the exponential generation of
small particles in the next future [1]. Since then, the
focus of the scientific community has been set on the
definition of models to better understand the debris
problem and to predict the evolution of the inert space
population. As of today, many agencies and institutions
developed their own debris propagator, such as the
European Space Agency’s (ESA) DELTA [2], the
LEGEND model of the National Aeronautics and Space
Administration (NASA) [3] or SDM of the Italian
National Research Council [4]. Following simulation
results, a first attempt to limit the debris proliferation
was made with the Inter-Agency Debris Committee
(IADC) formulation of mitigation guidelines in 2002,
that provided requirements for mission design and
operations to limit the probability of debris generation
[5]. However, utilisation of the space environment faced
unforeseeable changes in the last decades.
Miniaturisation of technologies and more affordable
access to space caused increased launch traffic, with
private actors gaining a major role in the current space
economy. These factors contribute to the dramatic
growth of inert objects in-orbit, with consequent rise of
the probability of collision with active satellites [6]. In
this new scenario, a common outcome of the many years
of research and despite the different modelling
techniques, is that the current exploitation of the orbital
environment is not sustainable, even with wider
adoption of the IADC mitigation guidelines [6].
Consequently, it is necessary to revisit and update the
guidelines and define policies for mission design that are
up to date with the current space exploitation. ESA took
the lead in this direction formulating a new set of
internal requirements for sustainability compliance of its
missions, following years of research and data [7].
However, the slowness in policy definition is an issue
when facing the fast-evolving environment that became
the space sector. Thinking back at the early 2000s, the
number of objects launched nowadays would have been
unpredictable and the same unpredictability applies to
future activities.
Aim of the GREEN SPECIES project, funded by the
European Research Council, is the development of an
adaptive tool to support the prompt definition of
guidelines to enforce a sustainable utilisation of space.
As a starting point, similar complex systems are
considered, such as climate change evolution and
epidemics spread, which face quick and dramatic
changes in their evolution, involve large spatial scales
and require reactive policies [8][9][10]. Taking example
from the techniques applied in those fields, the project
proposes to automate as much as possible the process of
analysis and definition of regulations by developing an
interdisciplinary framework for modelling and control
of the debris population. Core activity is the integration
of a simulator for the evolution of the space objects’
population with an active feedback controller on the
environment, a schematic of which is provided in Fig. 1.
Previous works in this direction for the debris problem
can be found in literature [11][12], that dealt with
automatic feedback definition of Active Debris
Removal (ADR) annual rate based on simplified models
of the debris population.
The project is in its development phase and, in the
following work, the preliminary one-dimensional model
of in-space objects’ density propagation and its
29th International Symposium on Space Flight Dynamics (ISSFD)
22-26 April 2024 at ESA-ESOC in Darmstadt, Germany.
integrated linear feedback controller are presented. In
Section II the model and control blocks’ current versions
are described, in Section III a simple example of the
method application is provided. Finally, in Section IV
the main conclusions are summarised.
II. METHODOLOGY
In the GREEN SPECIES project three main building
blocks are connected to create a new interdisciplinary
framework for debris evolution and its control. As
depicted in Fig. 1 a model of the space objects’
population is developed along with an active controller
of its evolution; the outcomes of these blocks will then
be processed to support definition of policies and
regulations for debris mitigation. The work described in
the following focusses on the model and the control
tasks, which are being developed concurrently to
enforce correct integration of the two elements. At the
end of the work a versatile controller will act on
simplified models of the debris environment for fast and
preliminary analyses, and on complex systems for more
accurate results. Moreover, robustness will be enforced
to deal with the unavoidable uncertainties of
evolutionary models.
In this early development phase, a one-dimensional
propagator of the population density is controlled with a
proportional linear feedback logic.
A. The model
The model exploits a density-based formulation, in
which the population of objects is described as a flow
with continuous properties evolving in time and space.
This approach has been extensively validated in
literature, and its benefits in terms of computational cost
and the method’s accuracy have been analysed and
documented. The first to propose the application of
continuum mechanics to small non-interacting particles
was Heard in the 70’s [13]. Later, McInnes was the first
to develop a fully analytical solution for debris density
propagation in Low Earth Orbit (LEO), under the effect
of atmospheric drag, considering some simplifying
assumptions such as quasi-circular orbits for the objects
[14]. Then, in [15] Letizia et al. developed the CiELO
suite, exploiting the analytical method by McInnes for
the propagation of in-orbit fragmentations. The problem
was also extended to multiple dimensions and different
force models, adding the J2 perturbation and solar
radiation pressure to the drag effect [16]. The Starling
suite developed by Frey et al. [17] put the basis for
debris cloud propagation under any nonlinear dynamics
and in multiple dimensions; then Giudici et al. carried
on the work with a fully probabilistic model that
numerically propagates the density of a continuous flow
in all the orbital elements and physical properties of the
objects, and in any dynamical regime [18].
The continuum approach was also applied to the
propagation of the whole debris population by Colombo
et al. [19] and extended to consider for feedback effect
of fragmentations in Duran et al. [20]. A multi-
dimensional model based on the continuity equation was
then proposed by Giudici et al., and it is embedded in
the COMETA tool, developed at Politecnico di Milano,
for the future projection of the in-orbit population under
the effect of objects’ sources and sinks, and mitigation
actions [18].
These models have the advantage of being agnostic to
the number of fragments considered and the reduced
dimensionality of the problem makes them suited to the
application of an external control. As previously stated,
even if multi-dimensional complex models for debris
population’s evolution are available at Politecnico di
Milano [18], the preliminary phase of control
development is built on a one-dimensional system in
orbital radius. Leveraging on the work by McInnes and
Letizia et al. [14][15], the spatial domain is binned in
spherical shells, each delimited by an upper and a lower
radius. The model exploits the conservation of the
number of objects in time in terms of spatial density .
In every fixed volume the integral form of the
continuity equation (1) is propagated in time, where the
term accounts for the density flux flowing in and
out the shell surfaces with radial velocity , and the
terms and are source and sink density rates that
directly add or remove objects in the considered volume.
(1)
A graphical representation of the domain is given in Fig.
2. This approach is defined as the finite volume method
[21]. The work in [22] was adapted to the one-
dimensional case and the governing equation (1)
obtained applying the divergence theorem to the
differential form of the same, i.e. integrating over the
spherical shell volume.
The natural dynamics term comes from the rate of
loss energy due to atmospheric drag [14], and is
modelled as in (2). is the gravitational parameter of the
Earth and the Earth’s radius, is the orbital radius
associated to the volume, which is taken as the value at
the centre of the shell,
is the average area to mass ratio
of the objects whose density is propagated, is the
drag coefficient, typically considered equal to 2.1 from
Fig. 1. Block scheme of the model-control system that is
being developed within the GREEN SPECIES project.
29th International Symposium on Space Flight Dynamics (ISSFD)
22-26 April 2024 at ESA-ESOC in Darmstadt, Germany.
flat plates assumption, and are the local air density
and the reference scale height, respectively, which are
modelled exploiting the superimposition of exponential
functions described in [23].
(2)
The deposition and removal terms and include all
the effects changing the rate of density that are not
caused by the dynamical flow of the same. In the debris
problem, these contributions may account for launches,
ADR or Post-Mission Disposal (PMD) of objects.
Equation (3) describes the currently modelled dynamics
of the density, it applies to each spherical shell
delimited by lower and upper orbital radii values,
and , resulting in a system of ordinary differential
equations. Since only one species of objects is accounted
for in the preliminary model described, it is assumed that
it represents only the intact objects population. The fast-
evolving effects include a source term , that is
defined as a continuous function in time and orbital
radius for the deposition of new objects per unit time and
unit radius, whose volume integral provides the density
rate in that shell. It models launches of new intact
objects in space, that contribute to the debris population,
similar to what was done in [12]. The second
contribution accounts for removal and deposition
of objects emulating a PMD profile in the domain. A
percentage of the objects eligible for disposal
in each higher altitude shell with is
removed and added to the first available volume
completely below the compliance limit . To set a
value for , a time for disposal is assigned to
the population. identifies the time span in which
the disposed objects are required to fall below the
limiting re-entry radius due to drag, which is set at
km altitude. This translates into a requirement in the
maximum orbital radius to be compliant, that is obtained
solving the implicit relation proposed by King-Hele (4),
where is the re-entry limit previously defined and
equal to km altitude, and is the compliance
limit, assuming circular orbits for the objects subject to
drag effect only and assuming constant air density and
scale height computed at : and [24].
The function, that identifies the objects per unit
time and unit radius that reached end of life at the current
time and might manoeuvre for re-entry, is defined as in
(3).
is the number of objects
added to the control volume at time ,
where is the current instant and is the time the
population of deposited objects spends orbiting under
the drag effect before manoeuvring for end-of-life
disposal. The control volume at time is identified
by the orbital radii limiting the current shell
backpropagated up to deposition time as
. In other
words, assuming to follow a shell of the domain back in
time, its shape changes from to under drag effect.
Considering that the deposited objects are assumed to
have a residual orbiting life before manoeuvring for
PMD, similarly to the approach in [12], the objects
added to that same volume at are the ones reaching
end-of-life at in the control volume of the domain.
Since only objects added through the source are
considered for disposal, the term becomes active from
time , with initial instant.
(3)
where:
shell with
•
•
•
(4)
In the preliminary model discussed, and are
continuous functions in time and orbital radius.
Different profiles of deposition and removal scenarios
can be investigated, and their parameters controlled.
This would allow analyses of different contributions and
regulations for acting on the in-orbit objects population,
both in time and space. The dynamics in (3) is a first step
to a complete definition of the debris population
environment. Collisions and explosions contributions
Fig. 2. Graphical representation of the space
environment domain model. The arrows in and out of
the shells represent deposition, removal, and motion of
objects.
29th International Symposium on Space Flight Dynamics (ISSFD)
22-26 April 2024 at ESA-ESOC in Darmstadt, Germany.
may be regarded to be part of the source term as
functions of time and radius only. However,
fragmentation models based on the collision probability
of the objects typically introduce nonlinearities in the
dynamics [12][20] and will be accounted for in future
work.
B. The controller
The literature on active control applied to the space
debris problem is reduced. A first approach to the
problem is found in [11], where the author developed an
adaptive strategy for the definition of the annual number
of ADR. Similarly to a model predictive approach, a
simplified plant was used to investigate the outcomes of
different ADR rates in the next future and to select the
best strategy in terms of number of population objects.
Then, Somma et al. [12] took on from the previous work
and integrated a statistical model of the objects’
population with a proportional feedback control to
adequately tune the ADR rate based on the error of the
overall population density with respect to a target one.
Concerning similar applications, the active control of
population growth has been widely applied to epidemics
models, where feedback controls are exploited to
investigate the effects of vaccination campaigns. In
these works, susceptible and infected individuals
interact through disease transmission rates and the
system can be controlled by introducing vaccines [10] or
biological control [9] of the population. It is a problem
of proliferation containment, the more the number of
infected patients, the more the epidemy spreads;
similarly, the larger the number of debris, the more the
collisions that generate fragments.
As clear from literature, a feedback approach is widely
used due to its continuous adaptation to the current state
of the plant, which makes it adequate to deal with a fast-
evolving scenario. Moreover, a proportional law is
typically the first choice to be investigated when
defining a controller. Consequently, in this work a
proportional linear feedback control is considered for
the preliminary version of the plant.
From (3) is chosen as the input variable of the system,
so the controller acts on the ratio of objects eligible for
disposal that successfully manoeuvre to lower altitudes.
The system of ordinary differential equations generated
from (3) is reformulated in a linear time-variant state-
space representation of the type (5), where the state is
the vector of the densities of the shells, is the
control, and is the given initial density profile.
in (5) is the state matrix, which is constant and accounts
for the drag dynamics. is the control matrix that
maps the input variable to its effect on the state. In
Section III is treated both as a vector, meaning
different compliance for each of the shells with
, for which takes the matrix shape in
(5), and as a single value, meaning same compliance for
all the higher shells, for which is adequately shaped
in vector form of dimension . Finally, is the
disturbance vector that includes all the sources that do
not depend on the state, meaning the deposition term in
(3).
,
given
(5)
where:
for shell with
A linear feedback control law acts on the system (5)
changing the value to reach a target density profile in
orbital radius, in a fixed time span. The control problem
deals with the minimisation of the cost function in (6),
made up of a quadratic form in the terminal error of the
state () with respect to the target () and an integral
from initial to final times, and , of quadratic terms
of the control and state error variables. and are
weight matrices defining acceptable levels of and in
time, is a weight matrix adding penalty on the final
state.
29th International Symposium on Space Flight Dynamics (ISSFD)
22-26 April 2024 at ESA-ESOC in Darmstadt, Germany.
where
(6)
The only constraints to the problem are provided by the
dynamics (3) for all . Applying the sweep method
[25], the control law (7) is obtained. is the time-
dependant gain matrix computed backpropagating in
time the differential Riccati equation, the vector is
added to the control to bring the state to a target different
from zero and the vector is required for disturbance
rejection [26].
(7)
For the system to be controllable it is required that the
control action is capable of transferring any initial state
to the final desired state within the given time span [27].
This has been checked with the matrix sequence method
[28]. Indeed, being and and times
differentiable, respectively, the matrix series (8) can be
built and for the problem at hand it exists a time
for which (8) has rank equal to the state
dimension, which is a sufficient condition for
controllability, as stated by the theorems reported in
[28].
for
(8)
Finally, a saturation block is added after the controller
to keep within its feasibility limits .
In Section III the presented approach is used on simple
examples showing applicability of the same, its
versatility and novelty, with little attention to the realism
of the simulations. Both the model and the control block
will undergo future updates and extensions adding
realism to the results.
III. PRELIMINARY RESULTS
An application of the model-control system described in
Section II is now shown. The example has no claims on
realism but will validate the methodology and provide
insights on its potential. A simple initial Gaussian
density profile is considered, of the form (9) and shown
in Fig. 4, similar to the one in [14], with , and
constants (see Table 1).
(9)
The initial state does not represent a real scenario,
however in future applications the profile can be
customized to the purpose and a sampled density
condition, based on real objects distribution, can be
obtained through binning or spatial density computation
in each shell. A single homogeneous population is
considered with an average value for the area to mass
ratio, and its evolution is simulated for 1000 days. The
domain spans from 200 km in altitude up to 1000 km
(from to in Table 1) and it is divided in 8
spherical shells of width 100 km. The density value
associated to the bin is computed at the central radius of
each volume. The system of equations (3) describes the
dynamics of the problem: the initial profile evolves
under the effect of atmospheric drag, deposition source
and PMD-like effect. To keep generality of the analysis
a simple linear function in and is considered for the
term, of the form (10).
(10)
Fig. 3 shows the evolution of the initial state spatial
distribution at time snapshots recorded every 100 days,
including only the deposition effect. As expected, the
Fig. 4. Initial density profile in the altitude domain
considered. The dots are the sampled values at the centre
of each shell.
Fig. 3. Density values associated to the spherical shells
of the domain captured every 100 days for 1000 days.
The dynamics is affected by atmospheric drag and
deposition of new objects.
29th International Symposium on Space Flight Dynamics (ISSFD)
22-26 April 2024 at ESA-ESOC in Darmstadt, Germany.
density at high altitudes increases largely in time, both
because the number of objects added to the environment
grows linearly with and (10) and because the drag
effect is much weaker than at lower shells. The same
source profile enters in the definition of the PMD-like
term in (3).
In Table 1 are listed the other model inputs required: the
simulation time is set at 1000 days, at 100 days, the
limit at 100 days, which translates in a compliance
radius limit of about 400 km altitude.
Table 1. List of inputs to the analyses of Section III.
Simulation time
Initial state inputs
Domain definition inputs
Drag inputs
PMD inputs
Control inputs
Case 1
Case 2
shape
shape
shape
shape
The behaviour in Fig. 3 brought to the control
parameters definition in Table 1. Indeed, the desired
final state after 1000 days is set equal to a uniform
profile in space of 2 m-3, to limit the density growth due
to . The weight matrices , , and have been set
constant in time and such as to enforce a maximum
control equal to 1, a maximum acceptable error value on
the evolving state equal to 1 m-3 and a maximum
acceptable error on the final state equal to 0.5 m-3. The
feedback control will continuously adapt the term in
the volumes above the compliance radius limit in (5) to
reach the target profile. Consequently, the weight
matrices are defined so that the upper shells only are
brought to the target. First, the control history in time is
obtained applying the sweep method from [25], then its
saturated effect is added to the evolution of the density
profile.
As clear from Table 1, two cases have been considered.
First, is varied independently in each shell, is a
vector and is in its matrix form (5). Then, is one
single value equal in all the controlled volumes and
is a vector. In Fig. 6 the controlled density profile in
space is provided at time snapshots every 100 days, and
the saturated control profile is in Fig. 5. The objects are
removed from the upper shells and moved to the first
available volume compliant with PMD requirements. As
desired, the densities of the shells above 400 km in Fig.
6 settle around the target value of 2 m-3. At the beginning
of the simulation the percentage of moved objects is
closer to zero, allowing the density to get larger; with
time the profile gets closer to the target and the number
of removed objects must increase to face the growing
deposition rate. However, in the final 200 days the
saturated control above 500 km cannot face the source
term and the density profile rises inevitably. The
required removal rate is also diversified in space and
is larger with increasing altitudes. This is due to two
effects: the deposition profile linearly grows with
29th International Symposium on Space Flight Dynamics (ISSFD)
22-26 April 2024 at ESA-ESOC in Darmstadt, Germany.
orbital radius requiring more objects removals in the
uppermost shells to reach the same target density than
the lower ones; indeed, the drag effect weakens with
altitude causing accumulation of objects with rising
radius, while in the lower shells a smaller percentage of
removals combined with a stronger drag is enough to
face the deposition of new objects and reach the target.
In Fig. 7 and Fig. 8 the density profile and control input
evolution resulting from Case 2 are provided. The
control variable is reduced to one single value and at
every instant the same percentage of objects is removed
from all the shells above 400 km. considering time steps
of 100 days, is null for the initial 200 days, since the
target density is larger than the initial condition in all the
controlled shells. Then the removal action increases and,
until 400 days have passed, the required is larger than
the corresponding ones in Case 1 (see Fig. 8) in all the
domain. From 500 days on the control is completely
saturated everywhere to face the growing density at the
highest altitudes, while in Case 1 the control below 700
km does not saturate for longer.
A. Discussion
The resulting density profile is similar in the two
analysed cases. However, the condition of same removal
percentage in all the high-altitude domain over-
estimates the number of objects to move to reach the
target. As clear from Fig. 5, the required in the
uppermost shells is larger than the one in the lower
domain, due to the accumulation effect caused by weak
drag combined with a deposition rate linearly increasing
Fig. 6. Case 1 - Density profile evolution in space
recorded every 100 days for 1000 days. Each dot
represents the density associated to one shell. The
dynamics is affected by atmospheric drag, deposition of
new objects and the PMD-like term with
controlled
independently in each shell.
Fig. 5. Case 1 - Control input evolution in space
recorded every 100 days for 1000 days. Each dot
represents the
associated to one shell. The control
input is active only from the shell at 450 km on.
Fig. 7. Case 2 - Density profile evolution in space
recorded every 100 days for 1000 days. Each dot
represents the density associated to one shell. The
dynamics is affected by atmospheric drag, deposition of
new objects and the PMD-like term with one controlled
value equal for all the shells from 450 km on.
Fig. 8. Case 2 - Control input evolution in space
recorded every 100 days for 1000 days. Each dot
represents the
associated to one shell (equal for all the
controlled shells). The control input is active only from
the shell at 450 km on.
29th International Symposium on Space Flight Dynamics (ISSFD)
22-26 April 2024 at ESA-ESOC in Darmstadt, Germany.
with orbital radius. For the same reasons this
contribution influences largely the cost (6) and the
control definition. Consequently, in Case 2 more objects
are removed than necessary and deposited in the
compliant volume and the saturation limit is enforced
from 400 days on, while in Case 1 saturation is required
in the upper domain after 500 days. This effect is also
visible between 400 km and 600 km altitude in Case 2,
where the final profile is lower than Case 1 and also
lower than the targeted value. Similarly, at final time the
density of the high-altitude shells in Fig. 7 is within the
acceptable error, while in the lowest ones it is not, so the
required settles at zero in all the domain to try and
compensate for the missing objects below 600 km.
The simple example of this Section shows the capability
of the feedback control to automatically tune the
percentage of removed objects in time and space to
reach a target. Provided a set of inputs and a final
scenario, the controller selects the most suited inputs to
fulfil the requirements. As shown, the proposed
approach enables fast and critical comparison of
different strategies. In fact, even if applied to unrealistic
scenarios, the benefits of two different compliance
profiles, aiming at the same final state, have been
observed and quantified.
Different controls and final targets can be analysed
exploiting the versatility of the approach. These
preliminary results put the basis to future extensions and
complexity addition in both model and control blocks.
IV. CONCLUSIONS
A model for in-space objects propagation and a feedback
controller for population containment are concurrently
under development within the GREEN SPECIES
project. The preliminary versions of the two blocks have
been presented. A simplified one-dimensional density-
based model is exploited for the propagation of the
population as function of the orbital radius. The density
term evolves under the effect of drag dynamics and is
affected by a source term and a PMD-like effect. A
linear feedback control is applied to the linear time-
varying system, and it actively tunes the percentage of
objects to remove from the high-altitude shells to reach
a target density profile. An example of the approach has
been provided that showed the capability of the
controller to define a suitable strategy in time to reach
the desired final state. Moreover, critical comparison of
two alternative mitigation procedures has been carried
out, providing evidence of the versatility of the method.
Both the model and the controller will undergo
extensions and updates in future work: fragmentations
and multiple species will be added to the debris
population definition and more advanced control logics
will be investigated. Moreover, the potential of the
method will be analysed considering different control
inputs and target scenarios.
Aim of the project is to build an adaptive tool to
systematically investigate different strategies for debris
containment. The tool will be valuable in promptly
reacting to the fast-evolving space situation and in
providing scientific support to the definition of
mitigation policies.
V. ACKNOWLEDGEMENT
The research received funding from the European
Research Council (ERC) under the European Union’s
Horizon Europe research and innovation program as part
of the GREEN SPECIES project (Grant agreement No
101089265) and was partially supported by the ICSC—
Centro Nazionale di Ricerca in High Performance
Computing, Big Data, and Quantum Computing funded
by European Union—NextGenerationEU.
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29th International Symposium on Space Flight Dynamics (ISSFD)
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