Content uploaded by Yasuhiro Yoshimura
Author content
All content in this area was uploaded by Yasuhiro Yoshimura on Apr 27, 2022
Content may be subject to copyright.
Contactless Attitude Control of an Uncooperative
Satellite by Laser Ablation
Daisuke Sakaia, Yasuhiro Yoshimuraa, Toshiya Hanadaa, Yuki Itayab,
Tadanori Fukushimab
aKyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
bSKY Perfect JSAT Corporation, 8-1 Akasaka 1-chome, Minato-ku, Tokyo 107-0052,
Japan
Abstract
An active debris removal method using a laser is a promising technology for
its advantage in contactless operations. This paper deals with the attitude
control of an uncooperative target by a laser, which is an important phase
before deorbiting. The difficulty of attitude control by the laser stems from
the torque directional constraint because the laser generates thrust along the
normal vector of the irradiated face irrespective of the irradiating direction.
Thus, the control torque along the normal vector cannot be generated, which
makes the attitude control with the laser torque challenging. To tackle this
problem, this paper first designs a reference controller that assumes arbitrary
control torques are available. Then, a method for determining the irradiating
point is proposed so that the difference between the reference torque and the
actual one is minimized. Although the proposed controller does not guaran-
tee theoretical convergence to the desired attitude, the effectiveness of the
proposed controller is numerically verified for a box-type object. Further-
more, the robustness to the uncertainties of thrust magnitude and direction
is also examined by Monte Carlo simulations.
Preprint submitted to Acta Astronautica April 20, 2022
Keywords: Space Debris, Laser Ablation, Attitude Control, Active Debris
Removal
1. Introduction
The dramatic increase of space debris in near-Earth orbit is a threat to
satellite operation, and active debris removal (ADR) has been intensively
studied in recent years. ADR is a promising technology in which multiple
spacecraft operations are required. ADR methods using robotic arms [1, 2],
nets [3], or electrodynamic tethers [4] have been proposed, which require
contact operations. Such contact operations may have potential risks to
the collision between a removal satellite and the target. Moreover, flexible
cables such as nets or tethers have complicated dynamics, making the analysis
and planning of ADR missions difficult. On the other hand, ADR methods
using a laser [5, 6, 7], ion beam [8, 9], thruster plume [10], and electrostatic
force [11, 12] have an advantage in contactless operations. The ADR by
a laser has a lower risk of functional loss of the removal satellite due to
accidental collisions.
For an effective deorbit, the thrust forces by a laser or thrust plume must
be generated in a desired direction, and the rotational motion of the target
should be stabilized in advance. To this end, attitude stabilization methods
using contactless external forces have been studied. Vetrisano [13, 14] deals
with laser-induced deorbit for asteroids. The mechanism of laser ablation in
deorbit: the spot size varied with the relative distance and velocity between
the target and the laser satellite is investigated. An effective attitude con-
troller is proposed by assuming the shape of the asteroid to be an ellipsoid.
2
Kumar et al. [5] derive a control law that can be adapted to Envisat, which
has a cylindrical shape. Borelli [15] has adopted the control algorithm in [5]
to a box-wing type satellite considering plume impingement as a control in-
put. Most of these methods are designed for detumbling a target object
assuming the laser or thrust plume is irradiated at spatially discrete points.
The irradiation point is determined so that the most desirable torques are
generated on the target at each time instance. In other words, all possible
irradiating points and their induced torques are calculated, and the determi-
nation method of the irradiating point is not explicitly formulated. Moreover,
existing methods consider attitude stabilization, and the attitude angle con-
trol is not considered. For deorbiting by a laser, the attitude control of a
target is significant for efficient deorbiting because the thrust direction due
to laser ablation is perpendicular to the irradiation plane. That is, control-
ling the target attitude enables generating efficient ∆Vparallel to the orbit
plane [16].
In this context, this paper proposes a control method to drive an unco-
operative target’s angular velocity and attitude angle to the desired ones by
laser ablation. The difficulty of controlling the attitude with laser ablation
thrust stems from the fact that ablation causes thrust along the normal vec-
tor of the irradiating point irrespective of the irradiating direction [17]. The
control torque around the normal vector thus cannot be generated. More-
over, since the target is rotating and the laser can irradiate to the target’s
surface where the laser satellite can see, the laser ablation torque depends
on the target attitude at each time instance. To tackle this problem, this
paper first designs a reference controller that assumes that arbitrary control
3
torques are available. Then, the cross-product law to determine the irradiat-
ing point is proposed, which minimizes the difference between the reference
torque and the actual one. Although the proposed controller does not guar-
antee theoretical convergence to the desired attitude, the effectiveness of the
proposed controller is numerically verified. Furthermore, the robustness to
uncertainties on thrust magnitude and direction is examined by Monte Carlo
simulations.
2. Preliminaries
The equations of motion of a satellite and laser ablation model are sum-
marized in this section. This study assumes that a laser satellite and the
target satellite travel along the same circular orbit. For deorbiting the target
satellite, the laser satellite should be orbiting forward of the target. The laser
ablation model shows that the ablation thrust is along the negative normal
direction of the irradiated face, which makes the attitude control by laser
ablation challenging.
2.1. Dynamics and kinematics of a satellite
This paper uses three coordinate frames as illustrated in Fig. 1. The
inertial frame {x, y, z}has its origin at the Earth’s center. The xaxis is
along the vernal equinox direction, the zaxis is along the rotational axis of
Earth, and the yaxis completes the right-handed frame. The orbital frame
{xo, yo, zo}has its origin at the center of mass of the target. The xoaxis
lies in the outward direction from the Earth, the zoaxis is along the orbital
angular momentum direction, and the yoaxis completes the right-handed
frame. The body-fixed frame {xb, yb, zb}also has its origin at the center of
4
Figure 1: Inertial frame, orbital frame, and body-fixed frame.
mass of the target, and the axes are assumed to coincide with the principal
axes of inertia.
The orbital motion of the target is described as
¨
rpos =−µ
krposk3rp os (1)
where rpos is the target position in the inertial frame and µis the Earth’s
gravitational constant. For simplicity, external disturbances such as atmo-
spheric drag and solar radiation pressure are ignored in this paper.
The attitude motion of the target satellite is described with a quaternion
5
as
˙
q=1
2˜
ω⊗q(2)
=1
2
0ωz−ωyωx
−ωz0ωxωy
ωy−ωx0ωz
−ωx−ωy−ωz0
q(3)
J˙
ω=−ω×Jω+τ(4)
where q= [q1, q2, q3, q4]Tis the quaternion vector whose scalar part is q4,
ω= [ωx, ωy, ωz]Tis the angular velocity vector in the body-fixed frame,
J= diag(Jx, Jy, Jz) is the satellite’s inertia tensor, and τis the control
torque. In Eq. (2), ˜
ωis an augmented angular velocity ˜
ω= [ωT,0]Tso
that the quaternion product ⊗can be calculated. Note that the quaternion
and the angular rate in Eqs. (3) and (4) are described with respect to the
inertial frame. Since this paper uses three coordinate frames, the subscripts
of the variables specify the frames. For example, ωb/o means that the angular
velocity of the body-fixed frame with respect to the orbital frame.
In Section 4, the numerical simulation results are evaluated using the 3-
2-1 sequence of the Euler angle (φ, θ, ψ). The conversion between the Euler
angle and the quaternion is obtained using the following rotational matrix
6
from the inertial frame to the body-fixed frame.
Rb/i =
q2
1−q2
2−q2
3+q2
42 (q1q2+q3q4) 2 (q1q3−q2q4)
2 (q1q2−q3q4)−q2
1+q2
2−q2
3+q2
42 (q2q3+q1q4)
2 (q1q3+q2q4) 2 (q2q3−q1q4)−q2
1−q2
2+q2
3+q2
4
(5)
=
cos θcos φcos θsin φ−sin θ
sin θsin ψcos φ−cos ψsin φsin θsin ψsin φ+ cos ψcos φcos θsin ψ
sin θcos ψcos φ+ sin ψsin φsin θcos ψsin φ−sin ψcos φcos θcos ψ
(6)
For example, the Euler angle can be obtained as
tan φ=R1,2
R1,1
(7)
sin θ=−R1,3(8)
tan ψ=R2,3
R3,3
(9)
where Rj,k means the jth row and kth column component of the rotational
matrix Rb/i. Also, the following small-angle approximation can be used [18].
2q1≈ψ(10)
2q2≈θ(11)
2q3≈φ(12)
In this paper, the desired attitude and angular velocity of the target
satellite is to coincide with the orbital frame. Thus, describing the quaternion
and its kinematics with respect to the orbital frame is useful. Let qb/o and
ωb/o be the quaternion and the angular rate with respect to the orbital frame.
7
The kinematics is described in the same form in Eq. (2) as [19]
˙
qb/o =1
2˜
ωb/o ⊗qb/o (13)
The angular velocity ωb/o has the relation ωb/o =ωb/i −ωo/i.
2.2. Laser ablation model
The ablation model used in this paper is briefly described, and more
detailed descriptions are found in [20]. When a laser lands on the surface of
a target satellite, a phenomenon called ablation occurs in which the incident
laser evaporates the material of the satellite body. The thrust force generated
by the ablation also causes torque, which is the control input for the target
attitude control in this paper.
The force generated on the target surface fis given by the product of the
velocity of the ejected gas ¯
vand the mass flow rate of the ablated material
˙mas
f=λ¯
v˙m(14)
where λis a constant scatter factor used to account for the non-unidirectional
expansion of the ejecta gas. Figure 2 illustrates the uniform diffusion of
ejecta gasses generated by laser ablation around the landing spot. Thus, the
thrust force is directed in the direction of the internal normal vector of the
local surface. In other words, the thrust force is always generated along the
internal normal vector of the surface, which does not depend on the laser
irradiation direction. This is a unique property for the system that uses
laser ablation as a thrust force. Consequently, the control torque τby laser
ablation is described with the position vector rat the laser landing point
8
Figure 2: Geometry of the ejecta gas.
and the thrust force vector fas
τ=r×f(15)
=−kfkr×n(16)
The ablation thrust and its torque in Eqs. (14) and (16) are ideal for-
mulations. In practice, other factors, such as the spot size of the laser and
the relative distance between the laser satellite and the target, affect the
magnitude of the ablation thrust [20]. Moreover, since the thrust direction
depends on the normal vector of the face, the modeling error of the target
shape would yield the ablation thrust in undesired directions. This paper
considers these uncertainties on the magnitude and direction of the ablation
thrust in numerical examples. The magnitude uncertainty of the thrust is
defined as the multiplicative uncertainty. That is, the thrust magnitude with
uncertainty kˆ
fkis written as
kˆ
fk=ηkfk(17)
9
Figure 3: Thrust directional uncertainty
where ηis the magnitude uncertainty coefficient and is assumed to follow
the normal distribution N(mf, σ2
f) where mfis the mean and σ2
fis the vari-
ance. Figure 3 defines the thrust force under directional uncertainty. The
directional uncertainty is expressed with the angle θfthat follows the normal
distribution N(mθ, σ2
θ) and the angle φfthat follows the uniform distribution
from 0 to 360 deg.
3. Methodology
To tackle the input directional constraint, a reference controller based
on quaternion feedback is firstly designed. Secondly, the laser irradiation
point on the target surface is determined using the cross-product law as
explained in Section 3.2. If the determined irradiation point would be out
of the range of the target surface, the irradiation point is corrected to be
within the maximum length of the target surface, which is also described in
Section 3.2.
3.1. Reference controller
A reference controller is designed with a quaternion feedback controller [21].
The quaternion feedback consists of proportional and derivative parts with
10
respect to a desired attitude as
τref =−kqqe−kωωe(18)
where qeis the error quaternion, ωeis the angular velocity error, and kq
and kωare control gains. The error between the current attitude or angular
velocity and the desired one is calculated as
qe=qb/i ⊗q−1
d(19)
ωe=ωb/i −ωd(20)
where the subscript dmeans the desired value of the variable. This paper
sets the desired attitude of the target satellite to coincide with the orbital
frame, i.e., qd=qo/i and ωd=ωo/i.
3.2. Cross-product-based determination of irradiation point
Although the reference controller in Eq. (18) is designed assuming that
arbitrary torques can be generated, such a reference torque cannot be realized
due to the directional constraint on the ablation thrust. The thrust direction
depends on the normal direction of the irradiation point, and the irradiation
point should be determined so that the generated torques are as close to the
reference torques as possible. To this end, the following cross-product law is
proposed.
r=c1
kfk2(f×τref ) (21)
where ris the laser irradiation position vector expressed in the body-fixed
frame and c1is the correction coefficient of the irradiation point which is
11
discussed later. Using the cross-product law in Eq. (21), the actual torque
generated is written as
τ=r×f(22)
=c1
kfk2(f×τref )×f(23)
=c1τref −c1
τT
ref f
kfk2f(24)
The first term in Eq. (24) is equal to the reference torque and the second
term is a disturbance torque that stems from the input directional constraint.
If the reference torque is ideally perpendicular to the force direction, i.e.,
τT
ref f= 0, the second term vanishes and the reference torque is purely realized
by the cross-product law. Such a situation, however, rarely occurs, and the
second term is considered as a disturbance. Thus, the cross-product law to
determine the irradiation point does not guarantee the convergence to the
desired attitude. Nevertheless, because the magnitude of the ablation thrust
is small and the rotational motion is periodic, the reference torque is expected
to be realized averagely, even if the disturbance term exists.
The irradiation position rmust be on the target’s surface. Offsetting a
vector in the opposite direction to the thrust force vector fis required as
follows.
r=c1
kfk2(f×τref ) + c2
f
kfk(25)
where c2is the offset coefficient and c2=nTrfor a convex body. Note that
adding the offset correction term does not affect the control torque generated
because f×f= 0. The coefficient c1is calculated to limit the irradiation
position vector within the maximum length of the target surface.
12
4. Numerical examples
Numerical simulations are conducted to verify the proposed attitude con-
troller. Four simulation conditions are examined: 1) under no thrust uncer-
tainty, 2) under thrust magnitude uncertainty, 3) under thrust directional
uncertainty, and 4) under both thrust magnitude and directional uncertain-
ties. Although the proposed controller does not guarantee theoretical con-
vergence, 100 Monte Carlo runs in each case are conducted for the numerical
verification in this section. It is noted that the first case does not consider
thrust uncertainty, and the initial attitude and angular velocity are random
variables for each Monte Carlo run.
4.1. Simulation condition
The box shape and size illustrated in Fig. 4 are considered as the tar-
get configuration, and its moment of inertia is assumed to be (Jx, Jy, Jz) =
(32.6,73.7,79.9) kgm2assuming a 150 kg class satellite. The target and laser
satellite are assumed to be in the same inclined circular orbit at an altitude
of 1200 km. The inclination is 87.9 deg, and the other orbital elements such
as the right ascension of the ascending node, the argument of perigee, and
the argument of latitude are zero. Orbital perturbations and their torques
are not considered in this paper, and these orbital elements determine the
desired attitude with respect to the inertial frame. This paper focuses on the
target’s attitude control, and the orbital decay due to the ablation thrust is
also ignored.
The initial quaternion is random and the initial angular velocity is ran-
domly distributed with kωb/ik= 6 deg/s, corresponding to 1 rpm. The same
13
Figure 4: Box-type target
Table 1: Uncertainties
thrust magnitude mf, σfthrust direction mθ, σθ[deg]
mean 1.0 0.0
standard deviation 0.1 15.0
simulation condition is used for all four cases. The ablation thrust force is
assumed to be 0.72 mN when no uncertainties exist, supposing a 20µNs/J
impulse and a laser power of 1 J at 36 Hz [22]. The mean and standard
deviation of the uncertainties are summarized in Table 1. It is noted again
that the thrust magnitude uncertainty is multiplicative. The control gains
are empirically determined and set to kq= 1.0×10−3and kω= 5.0×10−2.
4.2. Results under no uncertainty
Figure 5 represents one of the time history of the angular rate error from
100 Monte Carlo runs, where the initial condition is qb/i = [0,0,0,1]Tand
ωb/i = [3.00,−3.00,4.24]Tdeg/s in this case. The angular rate errors about
all three axes converge to zero, which means that the angular velocity ωb/i
14
corresponds to the orbital angular rate ωo/i. Figure 6 shows the time history
of the Euler angle with a 3-2-1 sequence (φ, θ, ψ) for better understanding.
The attitude is also controlled to the desired one, but θhas slow convergence.
This is because the attitude is controlled to make the body-fixed frame cor-
respond to the orbital frame, and in this situation, the laser satellite orbiting
forward can irradiate +ybface of the target alone. That is, the ablation
thrust causes along the ybaxis, and the attitude angle around the ybaxis is
almost uncontrollable. The attitude angle around the ybaxis, however, can
gradually converge because of the coupling between the orbit and attitude
motion.
In all Monte Carlo runs, the target attitude is successfully controlled.
Table 2 summarizes the mean and standard deviation of the residual attitude
angle error and the angular rate error for 100 Monte Carlo runs. It is noted
that these values are calculated for the final time step of all Monte Carlo
runs and not averaged over time. The attitude angle error in Table 2 is also
described with the 3-2-1 Euler angle. Although the attitude angle around
the ybaxis, θ, has the largest mean error and standard deviation, the other
attitude angles have the mean and standard deviation less than 1 deg. The
mean and standard deviation of the angular rate error also have small errors,
which verifies the effectiveness of the proposed controller.
4.3. Results with thrust magnitude uncertainty
Figures 7 and 8 describe the time histories of the angular velocity error
and the Euler angle error under the thrust magnitude uncertainty, respec-
tively. The initial attitude and angular rate are the same as the simulation
in Figs. 5 and 6. Both the angular rate error and attitude error converge
15
Figure 5: Time history of the angular rate error under no uncertainty
Table 2: Monte Carlo simulation results under no uncertainty
Euler angle error Angular rate error
φ, θ, ψ [deg] ωe,b/i [deg/s]
mean 0.7939, 4.3637, 0.3791 0.0049, 0.0038, 0.0005
standard deviation 0.1953, 5.1217, 0.5414 0.0037, 0.0028, 0.0013
to zero and mean that the target attitude is successfully controlled even if
the ablation thrust has uncertainty in the thrust magnitude. Compared to
Fig. 6, the attitude convergence time in Fig. 8 is almost the same even under
uncertainty. The target attitude is successfully controlled in all Monte Carlo
runs. Table 3 describes the mean and standard deviation of the attitude
angle error and angular rate error. The attitude control accuracy is almost
same as the result in Table 2 and has the small mean error and standard
deviation. This result means that the uncertainty on the thrust magnitude
16
Figure 6: Time history of the Euler angle error under no uncertainty
Table 3: Monte Carlo simulation results with thrust magnitude uncertainty
Euler angle error Angular rate error
φ, θ, ψ [deg] ωe,b/i [deg/s]
mean 0.1796, 4.7668, 0.4708 0.0046, 0.0036, 0.0006
standard deviation 0.6613, 3.9575, 0.3837 0.0034, 0.0028, 0.0022
does not degrade the accuracy of the attitude control.
4.4. Results with thrust directional uncertainty
Figures 9 and 10 represent the time histories of the angular velocity error
and the Euler angle error under the thrust directional uncertainty, respec-
tively. The initial attitude and angular rate are also the same as in the
previous simulation. The target attitude is controlled to the desired one un-
der the uncertainty on the thrust direction. The attitude convergence time
17
Figure 7: Time history of the angular rate error under thrust magnitude uncertainty
in Fig. 10 is almost the same as the one shown in Fig. 6. The results from 98
Monte Carlo runs indicate the convergence of the target attitude and only two
runs show the attitude divergence. Table 4 describes the mean and standard
deviation of the attitude angle error and angular rate error. The attitude
angle θhas a larger mean error compared to the previous two cases. The
standard deviations also have larger values, which indicate that the thrust
directional uncertainty impacts the attitude control accuracy more than the
uncertainty on the thrust magnitude.
4.5. Results with thrust magnitude and directional uncertainties
Figures 11 and 12 respectively show the time histories of the angular
velocity error and the Euler angle error under both the thrust magnitude and
directional uncertainties. The initial attitude is the same as in the previous
cases. Even if there are combined uncertainties in thrust magnitude and
direction, the proposed controller can drive the target attitude to the desired
18
Figure 8: Time history of the Euler angle error under thrust magnitude uncertainty
Table 4: Monte Carlo simulation results with thrust directional uncertainty
Euler angle error Angular rate error
φ, θ, ψ [deg] ωe,b/i [deg/s]
mean 0.3481, 5.8038, 0.6489 0.0056, 0.0043, 0.0008
standard deviation 1.0955, 4.8929, 0.6176 0.0047, 0.0034, 0.0020
one. However, the attitude convergence time in Fig. 12 is the longest, which
is a reasonable result due to uncertainties. Only two Monte Carlo runs show
the attitude divergence, and the other 98 runs indicate the convergence of the
target attitude. The mean and standard deviation of the attitude angle error
and angular rate error are tabulated in Table 5. The standard deviation of
both the attitude error and angular rate error have the largest value compared
to the other cases. This result indicates that, although the accuracy of the
19
Figure 9: Time history of the angular rate error under thrust directional uncertainty
attitude control is degraded, the robustness of the proposed controller is
useful for practical mission situations under the thrust uncertainties.
5. Conclusions
This paper deals with the attitude control of an uncooperative target by
laser ablation thrust. Because the ablation thrust is generated along the
Table 5: Monte Carlo simulation results with thrust magnitude and directional uncertain-
ties
Euler angle error Angular rate error
φ, θ, ψ [deg] ωe,b/i [deg/s]
mean 0.5236, 4.1946, 0.5852 0.0052, 0.0046, 0.0010
standard deviation 2.9810, 4.9346, 0.9467 0.0072, 0.0050, 0.0047
20
Figure 10: Time history of the Euler angle error under thrust directional uncertainty
normal direction of the irradiating point, the attitude control torque direc-
tion is constrained regardless of the irradiating direction. To tackle this
difficulty, the reference controller is designed using quaternion feedback, and
the method of determining the irradiating point is proposed, minimizing the
difference between the reference torque and the actual torque generated. Nu-
merical simulations with 100 Monte Carlo runs verify the effectiveness of the
proposed controller. Furthermore, its robustness to the uncertainties on the
thrust magnitude and direction is examined in the Monte Carlo simulations,
which indicates that the thrust directional uncertainty impacts the attitude
control accuracy more than the uncertainty on the thrust magnitude.
References
[1] S. Nishida, S. Kawamoto, Y. Okawa, F. Terui, S. Kitamura, Space
debris removal system using a small satellite, Acta Astronautica 65
21
Figure 11: Time history of the angular rate error under thrust magnitude and directional
uncertainties
(2009) 95–102.
[2] S. Nishida, S. Kawamoto, Strategy for capturing of a tumbling space
debris, Acta Astronautica 68 (2011) 113–120.
[3] M. Shan, J. Guo, E. Gill, Contact dynamic models of space debris
capturing using a net, Acta Astronautica 158 (2019) 198–205.
[4] S. Kawamoto, Y. Ohkawa, S. Kitamura, S.-i. Nishida, Strategy for active
debris removal using electrodynamic tether, Transactions of the Japan
Society for Aeronautical and Space Sciences, Space Technology Japan 7
(2009) Pr 2 7–Pr 2 12.
[5] R. Kumar, R. J. Sedwick, Despinning orbital debris before docking using
laser ablation, Journal of Spacecraft and Rockets 52 (2015) 1129–1134.
22
Figure 12: Time history of the Euler angle error under thrust magnitude and directional
uncertainties
[6] Y. Matsushita, Y. Yoshimura, T. Hanada, Y. Itaya, T. Fukushima,
Risk assessment of a large constellation of satellites in low-earth or-
bit, TRANSACTIONS OF THE JAPAN SOCIETY FOR AERO-
NAUTICAL AND SPACE SCIENCES, AEROSPACE TECHNOLOGY
JAPAN 20 (2022) 10–15.
[7] S. Fukii, D. Sakai, Y. Yoshimura, Y. Matsushita, T. Hanada, Y. Itaya,
T. Fukushima, Assessing collision probability in low-thrust deorbit,
Journal of Space Safety Engineering (2022).
[8] V. S. Aslanov, A. S. Ledkov, Attitude motion of cylindrical space debris
during its removal by ion beam, Mathematical Problems in Engineering
2017 (2017) 1–7.
[9] V. S. Aslanov, A. S. Ledkov, V. G. Petukhov, Spatial dynamics and
23
attitude control during contactless ion beam transportation, Journal of
Guidance, Control, and Dynamics (2021) 1–6.
[10] Y. Nakajima, H. Tani, T. Yamamoto, N. Murakami, S. Mitani, K. Ya-
manaka, Contactless space debris detumbling: A database approach
based on computational fluid dynamics, Journal of Guidance, Control,
and Dynamics 41 (2018) 1906–1918.
[11] T. Bennett, H. Schaub, Contactless electrostatic detumbling of axi-
symmetric geo objects with nominal pushing or pulling, Advances in
Space Research 62 (2018) 2977–2987.
[12] Detumbling an uncontrolled satellite with contactless force by using an
eddy current brake, volume 2013 IEEE/RSJ International Conference
on Intelligent Robots and Systems, IEEE, 2013.
[13] M. Vetrisano, N. Thiry, M. Vasile, Detumbling large space debris via
laser ablation, in: Detumbling large space debris via laser ablation,
volume 2015 IEEE Aerospace Conference, IEEE, 2015, pp. 1–10.
[14] M. Vetrisano, C. Colombo, M. Vasile, Asteroid rotation and orbit control
via laser ablation, Advances in Space Research 57 (2016) 1762–1782.
[15] G. Borelli, G. Gaias, C. Colombo, Rotational control with plume im-
pingement to aid rigid capture of an uncooperative failed satellite, in:
2020 AAS/AIAA Astrodynamics Specialist Conference, American Insti-
tute of Aeronautics and Astronautics (AIAA), 2020.
[16] T. Fukushima, D. Hirata, K. Adachi, Y. Itaya, J. Yamada, K. Tsuno,
T. Ogawa, N. Saito, M. Sakashita, S. Wada, End of life deorbit service
24
with a pulsed laser onboard a small satellite, in: Proceedings of the 8th
European Conference on Space Debris, 2021.
[17] H. Chang, X. Jin, W. Zhou, Experimental investigation of plume expan-
sion dynamics of nanosecond laser ablated al with small incident angle,
Optik 125 (2014) 2923–2926.
[18] J. L. Crassidis, J. L. Junkins, Optimal estimation of dynamic systems,
Chapman and Hall/CRC, 2004.
[19] F. Celani, Quaternion versus rotation matrix feedback for spacecraft
attitude stabilization using magnetorquers, Aerospace 9 (2022) 24.
[20] M. Vasile, A. Gibbings, I. Watson, J.-M. Hopkins, Improved laser
ablation model for asteroid deflection, Acta Astronautica 103 (2014)
382–394.
[21] B. Wie, P. M. Barba, Quaternion feedback for spacecraft large an-
gle maneuvers, Journal of Guidance, Control, and Dynamics 8 (1985)
360–365.
[22] K. Tsuno, S. Wada, T. Ogawa, T. Ebisuzaki, T. Fukushima, D. Hirata,
J. Yamada, Y. Itaya, Impulse measurement of laser induced ablation in
a vacuum., Opt Express 28 (2020) 25723–25729.
25
