dynamics and control of unmanned spacecraft rendezvous and ...1110762/fulltext01.pdf · rendezvous...

DEGREE PROJECT IN SPACE TECHNOLOGY, SECOND CYCLE, 30 CREDITSSTOCKHOLM, SWEDEN 2017

Dynamics and Control of Unmanned Spacecraft Rendezvous and Docking

HELMI MAAROUFI

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF INDUSTRIAL ENGINEERING AND MANAGEMENT

DEGREE PROJECT REPORT, SECOND LEVEL

1

Dynamics and Control of Unmanned Spacecraft

Rendezvous and Docking

Helmi Maaroufi

Master Programme in Aerospace Engineering

Royal Institute of Technology - KTH

Stockholm, Sweden 2016

[email protected]

Abstract—This paper presents a study on spacecraft

Rendezvous and Docking (RvD) through a comprehensive

literature review, in addition to investigate the main phases and

possible control methods during the space rendezvous and docking.

It presents a study of different energy efficient far range

rendezvous (e.g. Hoffman, bi-elliptic, phasing maneuver etc.). A

set of formation flying models (i.e. relative navigation) for two

spacecraft operating in close proximity are examined. One

approach to depict the relative orbit’s dynamics model for close

proximity operations is to mathematically express it by the

nonlinear equations of relative motion (NERM), that present the

highest accuracy and are valid for all types of orbit eccentricities

and separations. Another approach is the Hill-Clohessy-Wiltshire

(HCW). This method is only valid for two conditions, when the

target satellite orbit is near circular and the distance between the

chaser and target is small. The dynamics models in this paper

describe the spacecraft formation flying in both unperturbed and

perturbed environment, where only the Earth oblateness

perturbations are being treated.

Furthermore, this paper presents a design of a control, guidance,

and navigation (GNC) based on the aforementioned dynamic

model. This will enable the chaser satellite to autonomously

approach towards the target satellite during the close proximity

navigation using some control techniques such Linear Quadratic

Regulation (LQR) and Linear Quadratic Gaussian (LQG). These

control techniques aim to reduce both the duration and the ∆V

cost of the entire mission.

Sammanfattning—I denna rapport utreds olika faser av

rymdfarkosters rendezvous och docknings manövrar (RvD) samt

undersöks en rad matematiska modeller för formationsflygning,

nämligen de icke-linjära ekvationerna av relativ rörelse (NERM)

och Hill-Clohessy-Wiltshire-ekvationer (HCW). Dessa dynamiska

systemmodeller beskriver formationsflygningen i både ostörd och

störd omgivning. Vidare undersöks alternativa reglermetoder och

en filtreringsmetod såsom Linjär Kvadratisk Regulator (LQR),

Linjär Kvadratisk Gaussisk (LQG) och utökad Kalman filtrering

(EKF), för att reducera både tid och ∆V-kostnaden för hela rymd-

uppdraget. Målen med dessa reglermetoder är att rymdfarkosten

självständigt skall kunna utföra rendezvous och dockning.

I. INTRODUCTION

Spacecraft rendezvous always imposes challenges to

aerospace engineers, due to the fact that this process is risky and

involves complex procedures. When an unmanned spacecraft is

performing a rendezvous maneuver to meet up with a target

vehicle, it is very important to plan all the mission’s phases

carefully such as launch, phasing, up to mating (docking or

berthing), Fig.1, in order to reduce the fuel consumption. The

relative position of the chaser spacecraft with respect to the

target spacecraft during space rendezvous is very important

during the close-range maneuver to the capture point. This type

of formation is referred as formation flying.

Formation flying means that two or more satellites are operating

in close proximity and are able to autonomously interact with

one another. Mathematical models used to describe formation

flying are the nonlinear equations of relative motion, Hill-

Clohessy-Wiltshire, Tschauner-Hempel, etc. The objectives of

these models are to estimate where the satellite will go after

executing a small maneuver, to describe the relative motion

between two satellites operating in neighboring orbit or for

trajectory keeping in satellite formation flying.

Choosing an appropriate control technique, such Linear

Quadratic Regulator (LQR) or Linear Quadratic Gaussian

(LQG), to counter unexpected perturbations must be thoroughly

done to autonomously maintain the optimal pre-calculated

trajectory or formation flying.

Figure 1 - Space rendezvous phases, [2].


2

II. ORBITAL MECHANICS

A. Classical Orbital Elements (COE)

Kepler's laws are used to determine the motion of a celestial

object of mass m orbiting the Earth under the influence of

gravity. In the case of elliptical orbits, six parameters must be

defined to determine the position of an orbiting object. The first

two, the semi-major axis a and eccentricity e define the ellipse

in a plane. The right ascension of the ascending node (RAAN),

, the argument of perigee and inclination i define the

orientation of the orbital plane space. Finally, the true anomaly

defines the satellite position within an orbit.

When visualizing a celestial body’s local orbit with respect

to the Earth Centered Inertial (ECI) frame, it is more

advantageous to use the orbital elements, see Fig.2.

B. Orbit Equations

At time t, the trajectory is determined by three position

coordinates and three velocity components. The state vector

thus includes six parameters. In the case of a Kepler orbit, it is

more appropriate to describe the trajectory of a moving

celestial body as function of the true anomaly , [1].

The position state vector of the spacecraft in the orbit perifocal

frame (PQW), Fig.2:

0

sin

cos

cos

1)(

2

hpqwr (1)

The velocity state vector of the spacecraft in the orbit perifocal

frame (PQW):

0

cos

sin

)(2

e

hpqwv (2)

where h is the angular momentum is obtained by taking the

vector product of velocity and position vrh

reh )cos1( (3)

cos1

)1( 2

e

ear

(4)

In order to solve the true anomaly t , the eccentric anomaly

E as function of the mean motion M must be solved, one need

to solve the inverse of the Kepler equation.

2

)(tan

1

11tan2)(

tE

e

et (5)

EeEM sin (6)

Where, M is the mean anomaly:

)( pttnM (7)

n is the orbit mean motion, given by:

Tn

2 (8)

T is the orbital period, given by:

3

2a

T (9)

Where, is the standard gravitational parameter

)/skm64.398600( 23

C. Coordinates Transformation from Orbital Frame (PQW) to

Geocentric Frame (ECI)

We define a reference linked to the orbit and centered at the

origin (i.e. the center of the Earth). With these assumptions p is

directed from the origin towards the perigee w is a unit vector

normal to the orbit and pwq is in the plane of the orbit. If

one also notes (X, Y, Z) the coordinates of a celestial body in

the equatorial geocentric reference (Galilean) and (x,y,z)

coordinates of a celestial body in the reference related to the

orbit.

Hence, the transformation matrix from the spacecraft orbit

frame to geocentric frame is defined as following, [1]:

cos

sin

0

sin

cos

0

0

0

1

xM

1

0

0

0

cos

sin

0

sin

cos

zM (10)

)( )( )( = zxz MMMT iECIPQW (11)

PQWECIPQWECI rTr (12)

PQWECIPQWECI VTV (13)

D. Perturbed Motion

The previously presented orbital elements apply to the

satellite in the case of a non-disturbed Kepler orbit, Fig. 3.

However, in reality, the satellite is subject to a number of

external/perturbing forces.

At low Earth orbit (LEO), these disturbances/external forces

may be caused by the asymmetry of the ground magnetic field,

the atmospheric drag, solar pressure, etc.

These different types of perturbations act on the satellite's

movement at different temporal scales and of different

Figure 2 - Classical orbital elements.


3

important levels, which requires that the Keplerian elements

have to be updated on regular basis. To be noted here, only the

secular J2 effect and the atmospheric drag will be considered in

this study.

a) Perturbation due to Secular J2 Effect

The three orbital parameters affected by J2 are right

ascension, argument of perigee, and mean anomaly, [3]:

)( )sin2

5(2)(1

2

3)()( 0

22220 tt n i e

a

R J+t= ωtω

2

)(cos)(12

3)()( 0

222

20 tti n e a

R Jt=t

(14)

)(sin2

31)1(

2

3)()( 0

22322

2000 tt n ie a

R Jt =MtM

The perturbation due to the oblateness of Earth yield two

interesting effects. First, a nodal precession due to a torque

which rotates the orbit in the equatorial plane. The second

effect, the perigee drift ω , i.e. rotation of semi-major axis, due

to the non-central geopotential J2 effect. The effect due to J2

perturbation can be seen in Fig. 4.

b) Perturbation due to Atmospheric Drag

The atmospheric drag is a non-gravitational disturbance

type and is considered a non-constant, since the density of the

atmosphere at high altitude varies enormously depending on the

time of day, season and solar activity. The effect of the

atmospheric drag on the satellite trajectory is more notable for

orbits of low altitude (altitude from 100 km to 1000 km). The

effect of the atmospheric drag on satellites can be calculated

and presented as:

ea2

2

1V

m

ACρ = D

drag (15)

where e.g. 05.1DC for a CubeSat,and

00 exp

H

hρ= ρ (16)

where V is the spacecraft velocity with respect to the rotating

atmosphere, A is the spacecraft projected cross-sectional area,

and e is a unit vector in the relative velocity direction.

III. ORBITAL MANEUVERS AND SPACECRAFT

RENDEZVOUS

A. Launch Phase/Orbit Injection

Deciding the launch window of a spacecraft to rendezvous

with another is the most crucial part of the launch phase. In

order to achieve an orbit launch, two things must be considered.

The first is the target satellite orbit and the second is the

launching site. In the case where the launch site and the target

satellite trajectory intersect it is then then possible to launch at

any time of the day. However, it is more efficient to launch

when the target satellite orbit passes directly overhead from

launch site, [4].

Once the chaser satellite is injected into the parking orbit, the

first step of space rendezvous consists of performing an orbit

transfer maneuver towards the target orbit.

Assume that we need to transfer a satellite from a parking

orbit to higher/lower orbit altitude, the simplest way to change

its altitude is to make the satellite go through a temporary

transfer orbit. It is Walter Hohmann, in the 1920, who first

issued the concept of a transfer with minimum energy

consumption. This trajectory orbit is tangent to the two orbits

of departure and arrival.

B. Hohmann transfer

A Hohmann transfer, [1], [4], is a two-impulsive maneuver

positioned 180 degrees apart, Fig. 5, and considered as the most

energy efficient maneuver in the case of coplanar orbital

transfer. The first impulsive maneuver is required to enter the

semi-elliptic transition orbit, and the second one is applied at

the end of the transition orbit to enter the final circular orbit,

Fig. 6.

2

21 rratr

(17)

1tr11

r

μ

a

μ

r

2μΔV (18)

tr222

a

μ

r

2μ

r

μΔV (19)

The total ∆V budget is thus

2ΔV 1ΔVΔV= (20)

Figure 3 - Unperturbed orbit.

Figure 4 – J2-perturbed orbit.


4

Figure 7 - Hohmann transfer with plane change – Case 1.

Figure 6 - Semi-major axis change during Hohmann

impulsive maneuver.

Figure 5 - Hohmann orbit maneuver.

C. Hohmann Transfer with Plane Change

Case 1: The satellite is first transferred to another coplanar

orbit of radius equal to that of the target orbit (which

corresponded to a line of nodes, where the two orbits intersect),

and this is done by applying two tangential maneuvers, Fig. 7.

Then, an out-of-plane thrust is applied at the point where the

two orbits intersect, which corresponds to the line of nodes, in

order to enter the final orbit, [1], [4].

Case 2: A second case scenario is similar to the

aforementioned case. However, instead of transferring to a

coplanar orbit, the spacecraft is transferred to an intermediate

orbit with an inclination less than the arrival orbit, Fig. 8.

D. Bi-elliptic Hohmann Transfer

The bi-elliptic transfer, [1], consists of two semi-elliptic

temporary trajectories, Fig. 9. It is usually done by transferring

to a higher orbit altitude, then to transfer back to the final orbit

as depicted in Fig. 10. In certain cases, the bi-elliptic transfer

presents a more energy efficient method than the Hohmann

transfer, thus requires less V .

Figure 9 - Bi-elliptic Hohmann orbit transfer.

Figure 8 - Hohmann transfer with plane change – Case 2.


5

Figure 10 - Semi-major axis change during bi-

elliptic impulsive maneuver.

E. Phasing Maneuvers

The phasing maneuver, [1], aims to reduce the phasing angle

between the chaser and the target satellite.

We consider two cases where two spacecraft destined to

rendezvous are in the same orbit.

Case 1: The chaser satellite is ahead of the target spacecraft. In

order to achieve the rendezvous with the target satellite, it is

more appropriate to transfer the chaser satellite to a higher orbit.

Case 2: The target satellite is ahead of the chaser. It is then

more appropriate to transfer to a lower orbit and then back to

the chaser orbit. This phasing maneuvers is also called two

impulse Hohmann maneuver initiated at the perigee.

Each of these correction maneuvers will have a specific

duration, and thus a different V . Hence, new orbital

parameters must be calculated.

Now, to make a proper estimation of a phase angle given at the

launch of a spacecraft, there are two approaches:

Calculate the exact time of the launch according to the

target’s orbit.

For a specific launch date, adjust the orbit of the chaser.

IV. EQUATION OF RELATIVE MOTION AND FORMATION FLYING

Consider two satellites flying in close proximity and

arbitrarily elliptic orbits. The relative position and velocity

vector can be written as:

Tzyxzyx

ρ

ρ (21)

The relative position of the target satellite is written as:

ρrr TC (22)

where Tr denotes the target satellite radius and Cr the chaser

radius. The spacecraft relative motion is represented in a Local

Vertical/Local Horizontal (LVLH) frame, see Fig. 11.

The LVLH coordinate system, (co-moving frame xyz), is

commonly used to describe the relative motion of the chaser and

target spacecraft. The origin of the frame is located at the center

of mass (CoM) of the target. The x-axis is radially oriented from

the center of the Earth towards the target satellite, the z-axis is

perpendicular, (local vertical), to the orbit plane, in the direction

of the orbit angular momentum vector. The y-axis, (local

horizontal), is oriented in the direction of the target satellite

velocity vector.

A. Coordinates Transformation from to Geocentric Frame

(ECI) to Local Vertical/Local Horizontal Frame (LVLH)

The transformation matrix LVLHT can be obtained as follow.

We define first, the unit vector in the x-direction:

T

T

r

ri ˆ (23)

The unit vector in z-direction:

T

T

h

hk ˆ (24)

The unit vector in y-direction is orthogonal to i and k :

ikj ˆˆˆ (25)

The transformation matrix is therefore:

kjiT ˆˆˆLVLHECI (26)

The transformed relative position and velocity from the

Geocentric frame (ECI) to the Local Vertical/Local Horizontal

frame (LVLH) can be obtained as follows:

XYZLVLHECI rTr xyz (27)

and

XYZLVLHECI vTv xyz (28)

Figure 11 – Coordinate frames in relative motion analysis.


6

B. Nonlinear Equations of Relative Motion (NERM)

The nonlinear differential equations of relative motion

present the highest accuracy for all types of separation Fig. 14,

Fig. 15 and orbit eccentricity Fig. 12.

The NERM can be obtained by the following steps, [16]. First,

we derive the relative position of the chaser satellite relative to

the target in the LVLH frame, Fig. 11.

TTTC zyxr ρrr (29)

Then, the relative acceleration in the LVLH frame

ρρ2ωρ)(ωωρωrr TC (30)

where ρω is the acceleration due to a changing of frame

rate, ρ2ω is the Coriolis acceleration, and ρ)(ωω is the

centripetal acceleration.

ω the inertial angular acceleration vector of the LVLH frame

kω ˆ2T

T

r

r (31)

and ω is the angular velocity of the frame rotating with respect

to the ECI frame:

kω ˆ2

Tr

h (32)

The chaser radius can be expressed as:

222zyxrr TC (33)

By substituting equations (31) and (32) into equation (30) and

omitting some steps, we obtain the full nonlinear equation of

relative motion as

22/3222

2

)(

)(2

TT

T

rzyxr

rxyyxx

2/3222

2

)(

2

zyxr

yxyxy

T

(34)

2/3222)( zyxr

zz

T

2

2

T

TTr

rr

C. Linear Equations of Relative Motion (LERM)

The NERM, [16], is valid for large initial separation

between the target and the chaser satellite, Fig. 13. However, in

the case of small separation the NERM can be further reduced,

and with the following assumptions:

Tr and 02

Tr

Equation (29) becomes

CT rr (35)

which yields the linear differential equations of motion

3

2 22

Tr

xyyxx

3

2 2

Tr

yxyxy

(36)

3Tr

zz

Figure 12 – Impact of eccentricity change on NERM.

Figure 13 – Relative orbit (LERM) – Case 4.


7

Figure 14 – Relative orbit (NERM) – Case 2. Figure 15 – Relative orbit (NERM) – Case 4.

D. Hill-Clohessy-Wiltshire (HCW)

When the target orbit eccentricity is zero, i.e. the case of a

perfectly circular orbit, the linear equations of relative motion

can be reduced to Hill-Clohessy-Wiltshire equations, [1] and

[16]:

ynnxx 23

xny 2 (37)

znz 2

where, n denotes the mean motion of the target’s orbit, Eq. (8).

The homogeneous solution of HCW is expressed as

00 )()()( xxx ttt rrrr

00 )()()( xxv ttt rrrr (38)

c

snt

ctrr

34

0

)(6

0

0

0

0

1

)(

c

s

c

s

c

trr 0

2

0

0

2

0

43

)(

n

s

n

c

n

s

n

c

tn

s

trr 0

)1(2

0

0

)1(20

34

)(

ns

cn

nstrr

3

0

)1(6

0

0

0

0

0

)(

where s = sin(i) and c = cos(i).

The offset and secular drift in Fig. 16, can be eliminated by

adjusting the initial state vector of the relative velocity in both

x-direction and y-direction, see Fig.17.

)0(2)0(

)0(2

)0(

xny

yn

x

(39)

The HCW equations are valid if two conditions are satisfied:

The distance between the chaser and the target is small

compared to the distance between the target and the

center of Earth.

The target orbit is near circular.

E. Tschauner-Hempel (TH)

The Tschauner-Hempel motion dynamics generalize the

HCW method and is based on a true anomaly variation. Its

derivation and applications are similar to the Clohessy-

Wiltshire method, [7].

From a mathematical viewpoint the TH method presents a

weakness, as the orbit eccentricity increases the method

becomes inaccurate, due to the fact that the denominator

approaches to zeros in the case of e = 1.

Figure 17 - Relative position (HCW) – Case 2.

Figure 16 - Relative orbit (HCW) – Case 2.


8

The equations of motion for the TH method are

ye

eny

r

ex

re

enx

TT

2/32

2

33

2

2/32

2

)1(

)cos1(2

sin22

)1(

)cos1(

xe

enx

r

ey

re

eny

TT

2/32

2

33

2

2/32

2

)1(

)cos1(2

sin2

)1(

)cos1(

(40)

zr

z

T3

F. Virtual Chief (VC)

The virtual chief method, Fig. 18, consists of using a third

virtual satellite with zero eccentricity. This virtual satellite uses

the same orbit elements as the target orbit. Both the target and

the chaser satellite are considered here as chasers. The

advantage of using the VC method is to avoid an existing

elliptical chief model, [17], by taking advantage of the

simplicity of the HCW model (six-dimensional equations). The

VC method depends on true anomaly variation.

Since the fictional satellite orbit is circular (e = 0), both the

target and the chaser satellite motion can be expressed using the

HCW approach, [17]:

yyAxAx T 24241

xyAxAy T 25251 (41)

znz 2

where

)(cos 3+ = 222241 TTTTT MnnA

)sin( )cos( 3 =2

42 TTTTTT MMnA

)cos( )sin( 3 =2

51 TTTTTT MMnA

)( sin 3+ = 222252 TTTTT MnnA

22/32 )cos1()1( eenTT

G. Root Mean Square Error Method (RMS)

In order to compare the accuracy of the different methods for

relative motion, an error measure must be specified.

The root mean square (RMS) method is frequently used in

model evaluation studies to compare the differences between

two measurements that may vary, where the measurements are

done on the same scale with the same units as x . The smaller

RMS results the better is the relative motion model.

N

i

iiT

iiRMS xxxxN

1

)ˆ()ˆ(1

(42)

The RMS method will be used here to compare the result

accuracy of the different relative motions methods with respect

to the NERM, performed on the same time scale.

H. Test Case, Results and Discussions

Five cases are selected for the formation flying analysis,

Table I, were the spacecraft is being tested for high elliptical

orbit and zero eccentric orbit. In this analysis, the effect of the

orbit inclination on the formation flying method is examined.

TABLE I. TEST CASES

Case aT (km) aC (km) eT eC iT (deg) iC (deg)

1 7800 7800.2 0.001 0 0 0

2 7800 7800.2 0.001 0 10.2 10

3 7800 7800.2 0.101 0.1 0 0

4 7800 7800.2 0.101 0.1 10.2 10

TABLE II. RMS ERROR (KM)

Case LERM HCW VC TH

1 0.0020 0.0018 0.0080 0.0024

2 0.0209 0.0153 0.0249 0.0207

3 0.0032 0.4263 0.4626 15686.80

4 0.0263 0.4014 0.4880 14993.97

The RMS results of each simulated relative motion model,

demonstrate that in the case of low orbit eccentricity e<<1 both

HCW and LERM show better accuracy than VC and TH.

In the case of high orbit eccentricity, only the LERM shows

the lowest RMS error among the other models. This is due to

the fact that the LERM model is just a further simplification of

the NERM by dropping the relative position term and assuming

that the separation is negligible in comparison with distance

from the center of Earth to the target satellite. However, the TH

model shows the largest RMS error, which makes of it non-

valid for elliptical orbit analysis and confirms the hypothesis

made in section E, i.e. that the model becomes unreliable since

the denominator tends to zero as the orbit eccentricity increases.

In the upcoming sections, only the HCW model will be

considered for the analysis of perturbed relative motion and the

controlled relative motion, where the orbit eccentricity is near

circular or perfectly equal to zero. Figure 18 - Relative orbit (VC) – Case 4.


9

Figure 19 - J2 perturbed relative orbit (MHCW) – Case 2.

V. MODIFIED HILL-CLOHESSY-WILTSHIRE (MHCW)

EQUATIONS INCLUDING PERTURBATIONS

A. J2 Perturbation

A set of modified differential equations of the HCW model

were used to propagate the relative motion of the chaser satellite

in the LVLH frame. These equations incorporate the Earth

oblateness effect [11].

The MCHW derivation can be obtained by following a

procedure similar to that of the Hill’s equations by introduction

the J2 acceleration vector to the regular two-body equation:

2)( Jgr rrel (43)

where rg2

)(r

r

(44)

and

sincossin2

cossinsin2

sin31

2

3 2

2

4

22

2

ii

i

i

r

RJ eJ (45)

the full set of differential equations can be expressed as

zncz

ktir

RJnxncy

iii

r

RJnyncxncx

T

e

T

e

22

222

22

222

2222

23

)(sinsin2

32

8

2cos31sinsin

2

3

2

13225

(46)

where

ir

RJnck

T

e 2

27

22 cos

2

3

sc 1 (47)

2

22

8

2cos313

T

er

iRJs

It is well known that the solution of the Hill-Clohessy-

Wiltshire-equations consists of a circular trajectory, Fig. 19,

with long term drift in the y direction, where the chaser satellite

is drifting away from the target satellite, Fig. 20.

In formation flying, the motion of the chaser should remain

bounded with respect to the target in the presence of J2 forces

so that the formation flying is maintained and no secular drift is

experienced. Hence, to eliminate offset and secular drift, we

apply the no-drift constraint to the initial conditions as follow:

)0()1(4

252)0(

)0()1(

2)0(

22

3

2

xsc

ccny

yc

snx

(48)

Using these new initial conditions, we remove completely any

long-term drift in the y direction as seen in Fig. 21.

VI. RELATIVE NAVIGATION CONTROL TECHNIQUES

In this section, several control methods are presented that

could be used to reduce the relative position between the chaser

satellite and the target satellite before executing the final

approach maneuver. The control methods study in this section

are the linear quadratic regulator (LQR), LQR Pursuit/Evasion

game and the linear quadratic Gaussian (LQG) control method.

By reducing the relative position, the chaser satellites enters the

target’s orbit and thus has the same orbital elements.

The control methods take as input, the position and velocity

of a chaser satellite relative to a target satellite in the LVLH

coordinate frame.




10

A. LQR method

Once the state matrix of the spacecraft is defined, the

automated maneuver can be started. The LQR method provides

an optimal control concerned with operating a dynamic system

at a minimum cost, [6], [18].

Consider the following linear dynamic system model

Cxy

BuAxx (49)

where x and u denote the state variable vectors. The pair

matrices A and B are controllable. We define the Performance

Index (cost function):

ft

t

TT dtJ0

)(2

1RuuQxx (50)

where the weighting matrix Q is symmetric and positive semi-

definite, and R is symmetric and positive definite. There is no

straightforward method to select the weighting matrices Q and

R, Eq. (51). It is more based on a trial and error method.

nq

q

1

Q

mr

r

1

R (51)

The Q and R matrices have the following properties :

0iq and 0ir .

A small value of ir or iq imply large control commands

)(tu high gain control and fast convergence of y to

zero.

A large value of ir or iq imply a smaller control

commands )(tu low gain control, and slow

convergence of y to zero.

is the chosen trade-off regulation versus control

effort.

Next, is to design the optimal state feedback law Kxu ,

that minimizes PI, Eq. (50).

To obtain the control gain matrix K, the Algebraic Riccati

Equation (ARE) needs to be solved for ft :

0QBPRPBPAPA 1TT (52)

where P is symmetric and positive definite solution, and the

gain matrix

PBRKT1 (53)

The closed-loop dynamic is guaranteed to be asymptotically

stable and is defined as follows:

BK)x(Ax (54)

B. LQR Pursuit/Evasion Game Method

The pursuit/evasion game is a method based on the

interception avoidance approach where two satellites will play

the zero-sum differential game, [18]. By applying the

pursuit/evasion game onto the LQR control method, some

updates must be made by introducing new control laws,

defining a new dynamic system model, and hence, a new PI,

Eq. (56).

Cxy

uBuBAxx TTCC (55)

where TC xxx , BB C and BB T

The main objective of using the LQR(PE) is that the

chaser/pursuiter satellite tries to intercept/rendezvous, while the

target satellite/evader tries to delay the rendezvous. Both the

satellites try minimize the cost function.

The new PI of the evader/target is assumed to be the

opposite of the pursuiter/chaser:

ft

tTTCCC dtJ

0

TT

2

1RuuRuuQxx

T (56)

CT JJ

The new optimal control laws can be written as follows:

PxBRu CC1 (57)

PxBRu TT1

2

1

(58)

where 2 = 2. The value of depends on the time of simulation

and could also be determined by trial and error, [18].

The new ARE is extended and written as follows:

0QPBRB1

BRBPPAPA

T

TTTcc

T 1

2

1

γ (59)

C. Extended Kalman Filter method (EKF)

The Extended Kalman Filter (EKF) can be implemented in

two steps, Prediction and Update, [6].

The dynamic model can be written as

wBuAxx

vxHz kk (60)

where kH is the measurement matrix, w is referred to as the

process noise and v is referred as the measurement noise. They

are also referred to as zero-mean white Gaussian noise with

known covariance matrices Qk and Rk,

where TkkE wwQ k and T

kkE vvR k .

The first step involves the prediction part based on the last

estimate, where both the initial state estimate 1kx and the

initial error covariance 1kP are projected forward in time. This

step is called time update.

The second step involves correcting the project state

estimate and error covariance predicted in the first step, this


11

step is called measurement update or filtering update and

consists of computing the Kalman gain, update estimate and

update the covariance matrix respectively, see Tab. III.

The Extended Kalman filter is a recursion that provides an

estimate of the state vector. It is used in applications where it

provides updated information about the state of a dynamic

system when some information is corrupted by noise.

TABLE III. EXTENDED KALMAN FILTER

D. Linear Quadratic Gaussian method (LQG)

The LQG control combines both the concept of LQR and

the EKF method for state estimation, [6].

The design process starts by checking the controllability of the

pairs (A, B), and the observability of the pairs (A, C), see

Eq.(63).

First, a good approximation of the state vector and error

covariance matrix must be made.

e,00e0000 )(,ˆ)(ˆ,)( PPxxxx ttt (61)

The next step in design process, is to find the optimal control

law LQGu . The crucial and difficult part here is to find the

weighting matrices Q and R to obtain the optimal LQR gain

matrix LQRK

xKu LQRLQG (62)

The linear dynamic model is expressed as:

v)h(xy

wuBAxx

k

LQG (63)

where the process noise w and the measurement noise v are

Gaussian zero mean white noises. The optimal state estimation

of Kalman gain:

1 kTke,kk

Tke,kk RHPHHPK (64)

Update:

kkkkk xˆˆ hyKxx (65)

ke,kke, PHKIP (66)

Propagate:

LQGˆˆ BuxAx (67)

wT

QAPPAP (68)

In the coming sections, the aforementioned control techniques

will be used to examine a controlled relative navigation

problems which are unperturbed or perturbed.

E. Control of Unperturbed Relative Navigation

In this section, the J2 perturbation is neglected and only an

ideal case scenario is considered were no external forces are

acting on the satellites. The aim of this analysis is to reduce the

relative position between the chaser and the target using the

control methods LQR and LQG. The control methods will

enable the chaser satellite to approach autonomously towards

the targets satellite during the close proximity navigation.

The analysis of control methods is based on data from case

test 2, Tab. I in section IV.

1) LQR/LQG results

The results of the LQR and LQG control implementation.

As it can be seen in Fig. 23, the ∆V versus time is converges

to nearly constant value as the chaser spacecraft approaches

the target satellite, Fig. 24.

2) LQR(PE) results

From Fig. 26, it can be seen the LQR(PE) produces almost

same result as the LQG and the LQR controller. However, Fig.

25 shows a slight increase of the ∆V requirements during the

first 1000 seconds.

Time Update Measurement Update

Project the state ahead:

kkk BuxAx

1ˆ

Project the error covariance:

kkT

kk QAPAPP

1

Kalman gain: 1)( k

Tkkk

Tkkk RHPHHPK

Update estimate with measurement:

)ˆ(ˆˆ kkkkkk xHzKxx

Update error covariance: kkkk PHKIP )(

Figure 23 – ∆V cost – Case 2.

Figure 24 – LQR & LQG Relative orbit – Case 2.


12

Figure 26 – LQR (PE) Relative orbit – Case 2.

3) Extended Kalman Filter (EKF) results

Figure 27 shows an estimated ∆V cost for space rendezvous

during close proximity navigation provided by the EKF

estimator is higher than has been obtained by the

aforementioned control methods. The obtained result is based on

the assumption that the initial state estimated is 3/2 higher than

the initial state vector. As result, the estimated trajectory of the

chaser satellite using the EKF estimator, Fig. 28, is different

than what was obtained from LQR controller, Fig. 26.

F. Control of Perturbed Relative Navigation under J2-effect

Sabatini and Giovanni make use of the Hill-Clohessy-

Wiltshire linearized equations of relative motion to introduce an

improved model by transforming the linearized equations into a

state matrix [10], [13]. The newly introduced dynamic model

will be used to investigate the relative navigation based on the

same control methods used in section VI, subsection E.

However, in this case the J2 effect is included to the dynamic

model.

The state matrix of the linearized model can be written as

follows:

0

2

0

2

0

2

0

2

0

1

0

0

0

1

0

0

0

1

0

0

0

0

0

0

0

0

0

63

53

43

62

52

42

61

51

41

x

x

z

z

a

a

a

a

a

a

a

a

a

A (69)

where, 5

2e26J

r

μRK

) sin sin 31(+ 2

+= 22

3

2z41 θiK

r

μωa

)2sin sin(+ = 2z42 θiKωa

)sin (sin2 + = zx43 iKa

)sin2 (sin+ = 2z51 iKa

2

1sin

4

7 sin

4

1+ + = 22

3

2252

iK

ra xz

cos sin2

4

1+ = x53 iKa

sin sin2+ = x61 iKa z

cos sin2

4

1+ = x62 iKa

2

1sin

4

5 sin+

4

3+ = 22

3

2x63

iK

ra

Figure 28 - EKF Estimated relative orbit – Case 2.

Figure 27 – ∆V cost results – Case 2.

Figure 25 – LQR (PE) ∆V cost– Case 2.


13

Figure 31 – LQG ∆V cost (with J2) – Case 2.

Figure 32 – LQG Relative orbit (with J2) – Case 2.

Figure 29 - LQR ∆V cost (with J2) – Case 2.

Figure 33 – LQR(PE) ∆V cost (with J2) – Case 2.

1) LQR-method

Figure 29 shows the ∆V cost obtained from the LQR

controller. The ∆V cost is converging to its minimum value in

the xy-direction as the chaser approaches the end of the close-

range rendezvous phase. The chaser trajectory is different than

what obtained in the unperturbed case, see Fig. 24. In the

perturbed case the chaser satellite experience a drift in

trajectory due to the J2 perturbation, Fig. 30.

2) LQG -method

The ∆V cost obtained in Fig. 31 is the result of combining

the EKF estimator with the LQR controller. As it can be seen in

Fig. 32, the chaser satellite trajectory is similar to what was

obtained with the LQR method, see Fig. 30.

3) LQR(PE)-method

As it can be seen in Fig. 34 using the LQR(PE) controller

produces the same results as with the LQG and the LQR

controller. However, the ∆V cost, Fig. 33, is higher than the

other control techniques. This is due to the nature of the

LQR(PE) controller where the target tries to delay the

rendezvous and the chaser tries to intercept the target satellite.

4) EKF – method

Figure 35 shows the estimated ∆V cost due to J2 perturbation

is similar to what obtained in the unperturbed case. Thus, the

estimated trajectory is found in Fig. 36.

Figure 30 - LQR Relative orbit (with J2) – Case 2.

Figure 34 -LQR(PE) Relative orbit (with J2) – Case 2.


14

Figure 35 – ∆V cost (with J2) results – Case 2.

G. Results and Discussion

TABLE IV. CONTROL TECHNIQUES ACCURACY ANALYSIS

1: the state estimate is 3 times larger than the initial state input.

2: the state estimate is 3/2 times larger than the initial state

input.

The RMS analysis is based on a comparison of LQR(PE),

Fig. 34, and LQG, Fig. 32, control methods with respect to the

LQR-method, Fig. 30, see Tab. IV, for both perturbed and

unperturbed case.

In the case of the LQG controller, we considered two cases

with two different state estimates, in order to examine the effect

of the Kalman filter on the LQG. The initial guess of the state

estimates in the first case is approximated to be 3 times larger

than the actual relative orbit input, and 3/2 of the actual input in

the second case. The Gaussian white noises w and v can be

generated through the MATLAB command rand(n,1), where n

is the number of the row in white noises vector. From Tab. IV,

one can conclude that making a good guess of the initial state

estimate and the error covariance has a considerable influence

on LQG accuracy, as viewed from the RMS value in Tab. IV.

Figure 27 and Fig. 28 show an estimation of the relative

orbit and ∆V cost using EKF. By combining both concepts of

LQR for full state feedback and EKF for state estimation, it can

be seen the results produced by the LQG are similar to those of

LQR for both perturbed and unperturbed cases. The results of

using the LQG controller are depicted in Fig. 31 and Fig. 32 for

a perturbed case. For an unperturbed case see Fig. 23 and Fig.

24.

The difference between the LQR and LQG control method is

that the LQG .uses measurement or output feedback and hence K

is dynamic, while the LQR uses state feedback .and hence K is

just a constant matrix.

From Fig. 34, it can be seen that the relative orbit of the

LQR(PE) produces similar results to those of the LQR and LQG

model, Fig. 30 and Fig. 32. However, on a short timescale the

analysis accuracy is less than the LQG controller, see Tab. IV. It

can also be noticed that the ∆V cost, Fig 35, during the first

1000 seconds is slightly higher than the other control techniques

for both perturbed and unperturbed cases. This is due to the

nature of the Pursuit/Evasion controller, where the chaser try to

intercept/rendezvous with the target satellite, while target

satellite tries to delay or avoid the rendezvous. This type of

control method is more suitable in the case of space rendezvous

with noncooperative target, i.e. space debris.

I. APPROACH, ATTITUDE AND DOCKING CONTROL

A. Satellite Mating

In this section, we will discuss about the guidance,

navigation and control (GNC) architecture and algorithms

presented to focus on the docking of the non-tumbling satellites.

The basic problem which brings GNC into effect is the

spacecraft which loses control about its own axis and starts to

tumble in free space. There are various types of missions in this

category. It focuses on the terminal phase of a docking process

that refers to the last 100 meters before maneuvers are executed

for physical mating of the docking ports. The algorithms that

provide the autonomous system the ability to consider obstacles

and dock from any initial configuration of the two spacecraft.

Testing of this autonomous control system is done for the

synchronized position hold engage and reorient Satellites

(SPHERES) tested aboard the international space station (ISS).

The spacecraft has fixed docking ports which does not allow

any movement along the body axis of the satellite during the

docking. There are two types of docking scenarios to the target.

First is the non-tumbling and the second one is referred as

rotational tumble, [15]. In non-tumbling dynamics, the

spacecraft tries to maintain its own attitude during docking and

in rotational tumble the satellite performs a constant rotation,

i.e. circular motion along its angular vector. When the

spacecraft performs the rotational tumble, two motions takes

place. Supposedly the angular rate vector is perpendicular to

the docking port axis, a circular motion takes place. In this type

of scenario axis sweeps a cone which is known as coning, see

Fig. 37.

Time

(orbits) Case

LQG1 LQG2 LQR(PE)

1.5 without J2 4.590 ∙ 10-9 5.488 ∙ 10-10 6.429 ∙ 10-05

with J2 1.875 ∙ 10-9 1.880 ∙ 10-10 6.365 ∙ 10-05

3 without J2 5.297 ∙ 10-10 2.026 ∙ 10-10 3.046 ∙ 10-10

with J2 3.102 ∙ 10-10 1.130 ∙ 10-10 1.102 ∙ 10-10

9 without J2 8.172 ∙ 10-11 1.206 ∙ 10-11 1.576 ∙ 10-10

with J2 6.560 ∙ 10-11 4.076 ∙ 10-11 1.838 ∙ 10-11

Figure 36 – EKF Estimated relative orbit (with J2) – Case 2.


15

Figure 37 - Docking scenarios, [15].

There are two prospects considered i.e. facing forward and

facing backwards. The only difference between these

possibilities are that the docking port face directly downwards

to chaser in forward facing and flip 180 degree in second. The

rotation of the docking port along the axis sweep generate two

type of rotation that are rotating in and rotating out of plane.

There are following docking possibilities, [15], Fig. 37.

Docking to fixed:

a) non-tumbling target facing forwards.

b) non-tumbling target in-plane .

c) coning target facing forwards.

d) non-tumbling target facing backwards.

e) non-tumbling target out of plane.

f) coning target facing backwards.

1) Docking

In a docking process [2], the GNC system of the chaser controls

the spacecraft mating process. The docking operations during

mating consist of two types of operations; one is pressurized (in

which astronauts are involved) and the other one is

unpressurised (which is un-manned). The following functions

that are performed during docking operations are reduction of

velocity of the approaching vehicle and removal of all

misalignments, reception of the target and chaser spacecraft

when it comes in docking range and the attenuation of the

impact when the docking operation has been done to reduce the

shock of impact and decrement of the rebound speed and

protect it from rebounding. In free space, capture is the most

important part of docking; it is the main part of the mating

process in which two spacecraft are linked through ports. The

capture of the chaser satellite in the docking process requires

that the chaser approach is actively controlled and guided until

it reaches the capture interface on the target satellite. Some

safety measures must be considered during this process, is that

the chaser applies the final braking burn at a predetermined

distance, far enough so the gas temperature is sufficiently

reduced to avoid any damage to the capture interface.

2) Berthing

For berthing [2], the chaser satellite must remain within the

reception range of the manipulators arm to attempt the capture

of the spacecraft. The capture of the satellite must be

accomplished in the short time before the two satellites rebound

and separate again.

There are many similarities and differences in docking and

berthing operation such as collection of the capture area,

shutting down the capture equipments, pressurization and the

opening of the hatches are some of the basic examples of the

common features of both the operations. In docking the major

part is played by the chaser. The chaser is commenced

automatically by various ways, one of them is by active loading

of the spring latches dropping them into the target vehicle ports

whereas in berthing operation the purpose of the capture is to

just make a link to the interfaces of the two vehicles then a

manipulator controls grab the capture of the other vehicle.

B. Final Approach Maneuver

During the final approach, before docking, the GNC system

must achieve some additional conditions, [2]:

Determination of the approach velocity

Lateral alignment of the nominal docking axis, Fig. 38

Angular alignment

Lateral and angular rates for docking

Some important types of maneuvers used for the trajectory

correction and the removal of misalignments with respect to

the docking axis in rendezvous approaches are:

Figure 38 - Alignment of the nominal docking axis, [2].


16

1) Impulsive change ∆Vx

)3sin4(1

)( ntntVn

tx x (70)

)1(cos2

)( ntVn

tz x (71)

2) Impulsive change ∆Vy

ntVn

ty y sin1

)( (72)

3) Impulsive change ∆Vz

)cos1(2

)( ntVn

tx z

ntVn

tz z sin1

)( (73)

4) Straight Line V-bar tVtx x)(

xx Vn 2 (74)

where x , is the force per mass unit.

5) Straight Line R-bar

tVtz z)( (75)

xx Vn 2

tVn xz 23

C. Attitude Control

To properly dock with the target satellite, an attitude

determination and control system, (ADCS), is required to

achieve capture condition in terms of attitude and angular rate,

i.e. to control the orientation of the satellite with respect to its

inertial frame and hence. Thus, to give the satellite a stable

attitude.

1) Spacecraft Moment of Inertia

Let consider a CubeSat of a given size (W✕H✕L) and with a

mass m, to be homogenous and a rigid body. We assume also

that the CubeSat axes are aligned with the principal axes. The

moment of inertia about the x-y-z axis are obtained as follows:

)(12

1 22 HLmIxx

)(12

1 22 HWmI yy (76)

)(12

1 22 LWmIzz

While the off-diagonal products of inertia yxxy II , zxxz II ,

zyyz II are equal to zero. The matrix of the mass moment of

inertia about its principal axes is then written as follows:

zz

yy

xx

I

0

0

0

I

0

0

0

I

I (77)

2) Euler Angles to Direct Cosine Matrix

The Euler angles are the spatial orientation of an object in a

three-dimensional frame.

The Direct Cosine Matrix (DCM), also called the rotation

matrix, is a linear operator used to transform coordinates

between different reference frames, i.e. from the stationary

frame to the rotated frame. Let the following three elementary

rotation matrices Mi describe the rotation about the satellite

axes.

1

1

1

11

cos

sin

0

sin

cos

0

0

0

1

)(

M

2

2

2

2

2

cos

0

sin

0

1

0

sin

0

cos

)(

M

1

0

0

0

cos

sin

0

sin

cos

)( 3

3

3

3

3

M (78)

where M1 denotes the rotation about the x-axis (Roll), M2

rotation about y-axis (Pitch) and M3 denotes the rotation about

the z-axis (Yaw).

There are 12 possible Euler angles sequences to define the

orientation of a spacecraft. In this paper the Euler angles

sequence be (3-2-1) has been considered for the attitude and

docking control maneuver.

)( )( )( = )( 332211 MMMC (79)

21212

321313213132

321313213132

)(

cccss

ssccssssccsc

cscsscsssccc

C

3) DCM to Quaternion

The DCM can be transformed to the principal rotation

elements e, , [14]. This can found by calculating the principal

rotation angle, Eq. (80) and (81).

Principal Rotation Angle

1

2

1cos 332211

1 CCC (80)

2112

1331

3223

sin2

1

CC

CC

CC

e (81)

The quaternion, q, the Euler parameter vector, is defined in

terms of the principle rotation elements,

2

cos=0

q

2sine= 11

q (82)

2

sine= 22

q

2

sine= 33

q

4) Quaternion to Euler angles

The DCM can be defined in terms of the quaternion, from

which one can derive the Euler angles based on the Euler

sequence that was chosen, [14]:


17

23

22

21

20

1032

2031

1032

23

22

21

20

3021

2031

3021

23

22

21

20

2

2

2

)(2

2

2

qqqq

qqqq

qqqq

qqqq

qqqq

qqqq

qqqq

qqqq

qqqq

C (83)

Since we previously chosen the Euler sequence 3-2-1, the

rotation angles can be defined as follows:

33

3211 tan

C

C

311

2 sin C (84)

11

2113 tan

C

C

5) Magnetic Torquer

The magnetic torque is defined as following,

bb

bmT

coils

coils

where b denotes the dipole geomagnetic field, Eq. (85).

itn

i

itn

atb

tb

tb

t

T

Tf

sinsin2

cos

sincos

)(

)(

)(

)(3

3

2

1 b (85)

15109.7 f Wb.m, is the field’s dipole strength, Tn is target

mean motion, i is the orbit inclination, a is the semi-major axis.

6) Satellite Kinematics

The quaternion rate of change can be expressed as

z

y

x

q

q

q

q

q

q

q

q

q

q

q

q

q

q

q

q

0

2

1

0

1

2

3

1

0

3

2

2

3

0

1

3

2

1

0

q (86)

where Tzyx ω , is the angular velocity of the spacecraft.

Which is equivalent to

qωSq2

1

0

0

0

0

2

1

3

2

1

0

q

q

q

q

x

y

z

x

z

y

y

z

x

z

y

x

(87)

where ωS , is the skew symmetric matrix.

7) LQR-Control of Approach and Attitude Determination

The dynamic system describes the attitude control,

quaternion and path correction during the final approach. The

chaser and target satellites are assumed to be stable, non-

tumbling and facing each other. The analysis is performed in an

undisturbed environment, i.e. J2 perturbation is excluded. The

final approach starts at 250 m before the docking process.

The linearized system dynamics can be written as

BuAxx (88)

where,

v

x

q

ω

x

v

x

q

ω

x

thrust

coils

T

mu (89)

0

0

0

0

0

2

0

2

0

0

0

0

0

0

0

0

3

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

1

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

5.0

0

0

0

5.0

0

0

5.0

0

0

0

5.0

0

0

0

5.0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

6

0

0

0

6

0

0

0

0

0

0

0

2

2

2

n

n

n

n

n

n

n

nn

n

y

xx

z

A

(90)

11

3322

I

IIx

22

1133

I

IIy

33

2211

I

IIz

m

m

m

T

/1

0

0

0

/1

0

0

0

/1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0)()(1

b

bSbSI

B (91)

where m is the satellite mass.

D. Results and Discussion

TABLE V. CHASER AND TARGET SATELLITE PARAMETERS

Chaser Target

Dry mass [kg] 2 2

ω [rad/s] [0 0 0] [0 0 0]

Dimension [m] [0.2 0.1 0.2] [0.2 0.1 0.1]

Euler angles [deg] [10 –20 –10] [0 0 0]

The control method used for both attitude and trajectory

control is the LQR. Where K1, Eq.(91), is the gain matrix

generated by the LQR controller for the attitude control, and K2,

Eq.(93), is the gain matrix for the CubeSat translation and lateral

alignment part.

424.1

0

399.0

0

703.58

0

735.0

0

539.1

837.752

002.0

274.62

004.0

05.1636

019.0

13.987

0.019

380.052

1021K (92)

418.10

0

0

0

644.10

205.0

0

205.0

616.11

006.0

0

0

0

007.0

003.0

0

003.0

013.0

10 32K (93)


18

Figure 41 - Euler angles control. Figure 42 - Relative position control.

Figure 39 - Angular velocity control. Figure 40 - Quaternions control.

The GNC was designed in such a way the angular velocity’s

stability, Fig. 39, and the angular alignment, Fig. 41, were

achieved few minutes before the lateral alignment of the

satellite, Fig. 42, to be able to dock virtually to the target

satellite. The test case we considered in this analysis is called

the rest-to-rest simulation, meaning that at the beginning of the

analysis we assume that the chaser and the target satellites are

stable and non-tumbling and that both the satellites are facing

each other during the final approach.

In this analysis, the magnetorquer is used as the main actuator

for attitude control and to rotate the CubeSat to the desired

angles. The Euler angles converge within 5600 seconds, and

the lateral alignment is achieved 2400 seconds later. The total

time of the docking phase is estimated to 2 hours 13 minutes,

which is considered reasonable, if compared with the docking

time between the Kosmos-186 and Kosmos-188 that continued

for 3 hours and 30 minutes back in 1976, or compared to Soyuz

MS-01 docking with the ISS that continued for 2 hours and 20

minutes, [19], [20].

II. CONCLUSION AND FUTURE WORK

In this paper, various scenarios of formation flying methods

have been investigated. Additionally, several of control methods

have been used to evaluate and simulate the relative motion

under the J2-perturbation during the last phase of the satellites

rendezvous, using the HCW model. Furthermore, in the last

chapter of this thesis, a docking scenario has been investigated

and simulated called, rest-to-rest simulation. The case scenario

assumes that both satellites are stable and facing each other,

before the docking maneuver is initiated.

While the thesis objective is achieved, there is always a

room for future work in space rendezvous and docking that

includes investigating the different docking maneuvers

mentioned in section XII. A more detailed/advanced research

and simulation

of the final approach that covers the dynamics of an autonomous

berthing using automated robot arms. Spline trajectory

following controller during orbit transfer or when the satellite is

operating in its orbits to avoid an unexpected obstacle.

ACKNOWLEDGMENT

I would like to express, my gratitude to my supervisor

Dr. Gunnar Tibert, Associate Professor at the Department of

Aeronautical and Vehicle Engineering at KTH, for the useful

comments and guidance throughout the work involved in, this

thesis.

A tremendous thanks is also extended to my family, and,

friends for their support and encouragement during the process

of earning, my Master's Degree.

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19

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[19] http://spaceflight101.com/soyuz-ms-01/upgraded-soyuz-

successfully-docks-with-iss/, 23-12-2016.

[20] http://gagarin.energia.ru/en/past-future-en/146-the-world-s-first-

fully-automatic-docking-of-two-spacecraft.html, 21-12-2016.

www.kth.se

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