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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.

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