DESIGN AND GUIDANCE OF A MULTI -ACTIVE DEBRIS REMOVAL MISSION Minduli C. Wijayatunga Roberto Armellin Harry Holt Laura Pirovano_2

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DESIGN AND GUIDANCE OF A MULTI-ACTIVE DEBRIS

REMOVAL MISSION

Minduli C. Wijayatunga , Roberto Armellin , Harry Holt, Laura Pirovano

The University of Auckland,

Auckland 1010,

New Zealand

mwij516@aucklanduni.ac.nz

Aleksander A. Lidtke

Astroscale Japan Inc.,

1-16-4-16 Kinshi, Sumida-ku,

Tokyo, Japan

a.lidtke@astroscale.com

ABSTRACT

Space debris have been becoming exceedingly dangerous over the years as the number of objects in

orbit continues to rise. Active debris removal (ADR) missions have garnered signiﬁcant attention as

an effective way to mitigate this collision risk. This research focuses on developing a multi-ADR

mission that utilizes controlled reentry and deorbiting. The mission comprises two spacecraft: a

Servicer that brings debris down to a low altitude and a Shepherd that rendezvous with the debris

to later perform a controlled reentry. A preliminary mission design tool (PMDT) is developed to

obtain time or fuel optimal trajectories for the proposed mission while taking the effect of

, drag,

eclipses, and duty ratio into account. The PMDT can perform such trajectory optimizations within

computational times that are under a minute. Three guidance schemes are also studied, taking the

PMDT solution as a reference, to validate the design methodology and provide guidance solutions for

this complex mission proﬁle.

Keywords

Active Debris Removal

Trajectory Design

Optimization

Autonomous Guidance

Low-thrust Electric

Propulsion ·∆v-Law guidance ·Q-Law guidance

1 Introduction

The space environment in low Earth orbit (LEO) is increasingly populated with space debris. As a result, the average rate

of debris collisions has increased to four or ﬁve objects per year [

]. Most debris are artiﬁcial objects, including derelict

satellites, discarded rocket stages, and fragments originating from collisions. As satellites become increasingly essential

to daily life, more and more satellites are added to expand space-enabled services. However, additional launches

increase the risk of collision for all satellites as they further saturate space with objects, thereby endangering the critical

space infrastructure. A collision in space can create debris that can collide with other space objects and generate more

debris. This cascading effect is known as the “Kessler Syndrome”, named after D.J. Kessler [

]. Kessler et al. [

]

discussed the frequency of collisions and their consequences, describing standard mitigation techniques for the ﬁrst

time. Then, Pelton [

] discussed the cascading effect of collisions and the international standards for debris mitigation

and space trafﬁc management. He also gave estimates for the number of orbital debris at the time to be around six

metric tonnes in mass and 22000 in number. Several events in recent history have caused signiﬁcant additions to the

space debris population. These include the anti-satellite missile tests in 2007 and 2021, and the collision of Iridium 33

and Kosmos 2251 in 2009 [5, 6].

Active debris removal (ADR) is the process of removing derelict objects from space, thus minimizing the build-up of

unnecessary objects and lowering the probability of on-orbit collisions that could fuel the “collision cascade ”[

ADR has gained traction in the past two decades, leading to numerous studies and implementations of potential debris

removal missions and technologies. The ELSA-d mission designed by Astroscale was launched in March 2021 and

has successfully tested rendezvous algorithms needed for ADR and a magnetic capture mechanism needed to remove

objects carrying a dedicated docking plate at the end of their missions

. The RemoveDebris mission by the University

1https://astroscale.com/astroscales-elsa-d-mission-successfully-completes-complex-rendezvous-operation/

arXiv:2210.11701v1 [math.DS] 21 Oct 2022

Design and Guidance of a Multi-Active Debris Removal Mission

of Surrey is another project that demonstrated various debris removal methods, including harpoon and net capture [9].

The CleanSpace-1 mission by the European Space Agency (ESA) aims to deorbit a 112 kg upper stage of the Vega

rocket 2.

While individual removals are essential stepping stones towards ADR implementation, a deployment on a larger scale,

targeting more objects, might be necessary [

]. In order to make it ﬁnancially feasible, each ADR Servicer might

need to remove more than one object and use mass-efﬁcient low-thrust electric propulsion (EP). This combination of

long, EP-based transfers and complex vehicle paths rendezvousing with multiple moving targets presents a difﬁcult

optimization challenge that has to be addressed at the design stage of ADR missions. Due to the large number of

potential ADR targets to be visited, transfers between consecutive mission orbits need to be analyzed quickly to enable

design iteration and parametric studies.

This paper introduces a novel multi-ADR removal mission concept that involves a two-spacecraft system. On request,

the system is able to provide contact-based debris removal through a rendezvous and deorbit process. One spacecraft

- called the Servicer - is reused for multiple debris, allowing the mission costs to remain low. The other spacecraft

- the Shepherd - performs coupled reentry with the debris, so the reentry process can be controlled adequately, thus

complying with the ground-casualty risk requirements. The majority of the mission utilizes electric propulsion. This

paper is dedicated to discussing the proposed mission in detail and developing a mission design tool to simulate

multi-ADR tours accurately and efﬁciently.

To this end, a preliminary mission design tool (PMDT) is developed to optimize both fuel consumption and time of

ﬂight of multi-target missions while taking the effect of

, eclipses, and duty ratio into account. PMDT utilizes

achieve RAAN changes, in order to reduce the fuel consumption of the mission.

The PMDT extends the traditional Edelbaum method by introducing the contribution of drag and duty ratio. Then drift

orbits are used for matching RAAN when required, as discussed in [

]. Lastly, the altitude and inclination of the drift

orbits are optimized to obtain either time or fuel optimal trajectories. The sequence of targets can also be treated as an

optimization variable in the PMDT, however, it was treated as a constant for the examples given in this study.

Our approach shares similarities with the Multidisciplinary desigN Electric Tug tool (MAGNETO) developed in [

]

as well as the work by Viavattene et al. in [

]. However, it takes the presented models further by taking duty ratios

into account and a more accurate description of eclipses and drag. Furthermore, the tool considers mission-speciﬁc

constraints and uses an optimizer to perform rapid design iterations and parametric studies of the proposed multi-ADR

mission.

Three guidance laws are introduced to assess the accuracy of the models adopted in the PMDT. Ruggiero et al. [

]

developed a series of closed-loop guidance laws based on the Gauss form of Lagrange Planetary laws. Locoche [

]

developed a guidance law based on Lyapunov feedback control known as the

∆v

-Law to supplement preliminary

mission design tools. Finally, Petropoulos [16] developed one of the most versatile and well-known control laws - the

Q-Law - which is also based on Lyapunov control. These three approaches are here used to optimally track the transfers

computed by the PMDT, thus validating its key assumptions and providing a possible way to ﬂy the missions.

The remaining sections of the paper are organized as follows. Firstly, the mission concept of operations is presented.

Then, the design of PMDT is discussed and is used to generate optimal debris removal trajectories at high computational

speeds. Thirdly, guidance schemes implemented on the PMDT outcomes are discussed. The results section provides

an example trajectory optimization solution for both a time and mass optimal multi-ADR mission. Lastly, the paper’s

outcomes are summarized, and conclusions are drawn regarding the method’s usefulness.

2 Concept of Operations of the Multi-ADR Mission

The proposed multi-ADR mission architecture is shown in Figure 1, where two spacecraft are involved in the debris

removal process. A Servicer is used to approach and rendezvous with the debris. Once rendezvoused, the Servicer

brings the object down to a low altitude orbit (

≈350

km). The debris is then handed over to a Reentry Shepherd, which

docks with the debris and performs a controlled reentry on its behalf. Controlled reentry reduces the casualty risk

posed by removing the debris, which is desirable because the ADR targets are, by deﬁnition, large and thus contain

components likely to survive the reentry. The Servicer shall be reused for several debris removals, while each Reentry

Shepherd can only be used once as it burns up while deorbiting the debris.

The proposed mission architecture can perform multi-ADR services signiﬁcantly cheaper than those that use coupled

deorbiting and controlled reentry systems. When the deorbiting and reentry functionality are installed on a single

spacecraft, it cannot be reused, which leads to higher mission costs. Furthermore, depending on the debris features and

2https://www.esa.int/Space_Safety/ClearSpace-1

Design and Guidance of a Multi-Active Debris Removal Mission

Figure 1: Mission architecture of the multi-ADR mission

the governing regulations, the requirements of controlled reentry for each debris will differ. The Reentry Shepherd

allows the required ﬂexibility for controlled reentry of large-scale objects.

The separation of the two systems ensures that the Servicer is not obligated to carry re-entry-related hardware, thus

saving mass for fuel. It also warrants that the Shepherd does not need to perform extensive orbital changes or have a

long lifetime in space which, in turn, reduces its size and cost.

Trajectory optimization for this mission through traditional means is non-intuitive, as multiple transfer arcs and

targets are involved. Hence, a trajectory optimization tool capable of developing suboptimal solutions with limited

computational capacity is developed in the following section.

3 Methodology

3.1 Design of the Preliminary Mission Design Tool

In the 1960s, an analytical solution for the transfer between two inclined circular orbits under continuous thrust was

developed by Theodore N. Edelbaum [

]. While the transfer arcs developed were both time and fuel optimal, they were

obtained under the assumption of continuous thrust and the lack of other perturbations such as

and air resistance.

Several studies were conducted following Edelbaum’s work to include the effect of discontinuous thrust and orbital

perturbations on the problem dynamics. Colasurdo and Casalino [

] extended Edelbaum’s analysis to compute optimal

quasi-circular transfers while considering the effect of the Earth’s shadow, and Kechichian [

] developed a method

for calculating coplanar orbit-raising maneuvers taking eclipses into account while constraining the eccentricity to

zero. However, both [

] could only provide suboptimal solutions, as they utilized thrust steering to maintain zero

eccentricity. In 2011, Kluever [

] further extended Kechichian’s method into a semi-analytic method that considers

the effect of

and Earth-shadow arcs. This method used Edelbaum-based orbital elements to compute the Earth

shadow arc during the transfer. However, it failed to consider the effect of air resistance, which is of crucial value

for LEO transfers. In 2019, Cerf [

] proposed utilizing

to achieve right ascension of ascending node (RAAN)

changes during transfers to reduce fuel consumption while keeping the time of ﬂight constant and not taking the effect

of eclipses and air resistance into account. The PMDT is developed to unite the ideas given in [

] and take

them a step further by considering air resistance and duty ratio.

The PMDT ﬁrst calculates the time of ﬂight and fuel expenditure of a single transfer using Edelbaum’s method described

in [

]. Then, additions to the classical Edelbaum method- creating the Extended Edelbaum method (Algorithm 1)- are

made such that the effect of atmospheric drag, engine duty ratio, and solar eclipses are taken into account. Thirdly,

a RAAN matching algorithm (Algorithm 2) that does not utilize fuel to make RAAN adjustments is implemented to

make transfers cheaper. This is achieved by introducing an intermediate drift orbit where the Servicer can utilize the

effect of

perturbations to reach the desired RAAN. Lastly, this process is introduced into an optimization scheme

(Figure 3) where the launch time and the drift orbits involved can be optimized to achieve the minimum time of ﬂight or

the minimum fuel expenditure.

Design and Guidance of a Multi-Active Debris Removal Mission

3.1.1 Extended Edelbaum Method

The Extended Edelbaum method is a version of the classical Edelbaum method adapted to take the effect of atmospheric

drag, solar eclipses, and duty ratios into account. This method is detailed in Algorithm 1. Note that the Extended

Edelbaum method only ensures that a desired semi-major axis and inclination are reached.

Algorithm 1 Extended Edelbaum Method

Require: Initial and ﬁnal orbital velocity (V0, Vf), change in inclination (∆i), thrust acceleration (f)

Calculate ∆vand mission time of ﬂight (tf) using Eq. (1) and (2).

∆vtotal =qV2

0+V2

f−2V0Vfcos(π/2∆i)(1)

tf=∆vtotal

f(2)

Calculate the initial yaw steering angle β, deﬁned in the plane normal to the orbit plane, using Eq. (3).

tan β0=sin (π/2∆i)

Vf−cos(π/2∆i)(3)

Discretise the time of ﬂight (

) into N segments and compute the semi-major axis, inclination and

∆v

per segment

using Eq. (4), (5), and (2), respectively.

a(t) = µ

0+f2t2−2V0ft cos(β0)(4)

i(t) = i0+sgn(if−i0)2

πtan−1ft −V0cos β0

V0sin β0+π

2−β0(5)

for k=1:Ndo

Calculate sunlit time during a single orbit (wecl), using eclipse time formulation in [21].

Calculate the fraction of thrust time per orbit (w).

w= min[DR, wecl]where DR :Duty Ratio (6)

Compute the new transfer time using Eq. (7).

tk+1 =tk+∆vk+1 −∆vk

fkwk

(7)

Calculate drag acceleration at tk+1 and tkusing Eq. (8).

adrag =−1

ρCdAv2

m(8)

ρ, Cd, A, v and mrepresent the air density, drag coefﬁcient, frontal area, velocity, and mass.

Calculate ak+1,drag and ak,drag corresponding to each drag acceleration using Eq. (4).

if |ak+1,drag −ak,drag|>∆athen

ak+1 =ak+1 +|ak+1,drag −ak,drag|

Go back to the ﬁrst step of this algorithm and repeat the procedure from tk+1 to tf.

end if

Propagate the RAAN using Eq. (9) and (10)

Ω = −3

2J2rµ

a3Re

a2

cos i(9)

Ωtk+1 = Ωtk+˙

Ω(tk+1 −tk)(10)

end for

3.1.2 RAAN Matching Method

This method builds on the Extended Edelbaum method such that RAAN changing transfers can be optimized. In

this method, orbital precession is used to achieve a target RAAN by drifting at an intermediate drift orbit as done in

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DESIGNANDGUIDANCEOFAMULTI-ACTIVEDEBRISREMOVALMISSIONMinduliC.Wijayatunga,RobertoArmellin,HarryHolt,LauraPirovanoTheUniversityofAuckland,Auckland1010,NewZealandmwij516@aucklanduni.ac.nzAleksanderA.LidtkeAstroscaleJapanInc.,1-16-4-16Kinshi,Sumida-ku,Tokyo,Japana.lidtke@astroscale.comABSTRACTSpacedebri...

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