Master of Science in Space Engineering Department …...The space race started during the Cold War, when USA and USSR wanted to demonstrate their power. It started in 1958 with the

POLITECNICO DI MILANOMaster of Science in Space Engineering

Department of Aerospace Engineering

School of Industrial and Information Engineering

OPTIMAL DOCKING FOR AN UNDER-ACTUATED

12U CUBESAT

Supervisor: Prof. James Douglas Biggs

MSc Thesis by:

Filippo Menegolo

ID Number: 852292

Academic Year 2016-2017

Acknowledgment

The first person to thank is Professor James Douglas Biggs for his great support

during this thesis. He has always been available whenever I needed any help and,

with his great knowledge, he helped me to overcome all the problems I encountered

during this work.

I must express my enormous gratitude to my parents and my sister Alice, who

made these five years of study possible and always supported me, also in the

difficult moments. If I am where I am, I just owe it to them.

An enormous thank to Chiara, who has made these years pass very quickly. She has

given me a great emotional support and made me live a lot of beautiful moments.

Without you, everything would have been more difficult.

A big, big thank to all my friends with which I had great moments during these

university years, in particular Andre, Chris, Fede, Fritz, Lele, Marcone, Mauro,

Meph, Palmi, Tommy and Uilli.

Many thanks to my family, in particular to my grandmother Caterina and my

great-aunts Romana and Luigina. I am very happy to have shared these moments

with you.

A special thought goes to my grandfather Gianni, who passed away the summer

before the start of my studies. I would had shared these important events with

you.

I would like to thank all those people that shared also a single moment with me

during these five years. You have all contributed to complete my path in the

university.

I

Ringraziamenti

La prima persona da ringraziare e il Professore James Douglas Biggs per il

suo grande supporto durnate questa tesi. E sempre stato disponibile per ogni

chiarimento e, con la sua esperienza, mi ha aiutato a superare tutti i problemi

incontrati durante questo progetto.

Devo esprimere la mia piu profonda gratidune per i miei genitori e per mia sorella

Alice, che hanno reso questi cinque anni possibili e mi hanno sempre supportato,

anche nei momenti piu difficili. Se sono dove sono, lo devo solo a voi.

Un enorme grazie a Chiara, che ha fatto volare questi cinque anni. Mi e sempre

stata di supporto e mi ha fatto passare dei momenti fantastici. Senza di te sarebbe

stato tutto piu difficile.

Un grandissimo grazie a tutti i miei amici con i quali ho passato dei bellissimi

momenti, in particolare Andre, Chris, Fede, Fritz, Lele, Marcone, Mauro, Meph,

Palmi, Tommy e Uilli.

Grazie ai miei famigliari, in particolare alla nonna Caterina e alle prozie Romana

e Luigina. Sono contento di aver condiviso questi momenti con voi.

Un pensiero particolare va a mio nonno Gianni che e venuto a mancare l’estate

prima dell’inizio dell’universita. Avrei tanto voluto passare questi momenti con

te.

Vorrei inoltre ringraziare tutte le persone che hanno condiviso anche solo un singolo

momento con me. Mi avete aiutato a completare questo importante percorso.

II

Abstract

CubeSats, small spacecrafts with a mass lower than 10 kilograms, enable access

to space at a significantly reduced cost compared to conventional spacecrafts. As

a result of their constrained volume and mass, precision guidance and control is

highly challenging and, in particular, a limited amount of fuel is available on-board.

Furthermore, the drive towards miniaturization means that future CubeSats will

likely be under-actuated in control.

In this thesis, the problem of optimally guiding an under-actuated 12 U CubeSat

to dock with a target body is addressed. This problem is formulated as a con-

strained non-linear optimal control problem and solved using PSOPT, a pseudo

spectral optimal control software. In particular, this thesis concentrates on the

optimization of the total thrust needed to perform these maneuvers in order to

minimize the amount of fuel used per maneuver.

The main analyses for general docking maneuvers were performed assuming only

a planar motion. This work, instead, presents the solution to the 6-DOF prob-

lem which includes the problem of under-actuation and which minimizes fuel. The

analysis considers the motion of a CubeSat and a rotating target subject to Earths

gravitational field. The effect of air drag on the docking maneuver is also consid-

ered.

III

Contents

Acknowledgment I

Ringraziamenti II

Abstract III

List of Figures V

List of Tables VI

Nomenclature IX

1 Introduction 1

1.1 CubeSat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 6-DOF Relative Motion Dynamics 8

2.1 Translational Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Restricted Three Body Problem . . . . . . . . . . . . . . . . 9

2.1.2 Clohessy-Wiltshire Equations . . . . . . . . . . . . . . . . . 12

2.1.3 Translational Dynamics Equations . . . . . . . . . . . . . . 14

2.2 Rotational Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Attitude Representation . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Attitude Error Kinematics . . . . . . . . . . . . . . . . . . . 20

2.2.3 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . 21

IV

2.2.4 Rotational Dynamics Equations . . . . . . . . . . . . . . . . 23

2.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4 Confguration of the Spacecraft . . . . . . . . . . . . . . . . . . . . . 27

2.5 Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Implementation 32

3.1 State Space Formulation and Optimization Problem . . . . . . . . . 32

3.1.1 Theoretical Review . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.2 State Space Formulation of the 6-DOF Relative Motion System 34

3.2 PSOPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Translational Example . . . . . . . . . . . . . . . . . . . . . 39

3.2.2 Rotational Dynamics Example . . . . . . . . . . . . . . . . . 49

3.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.4 Air Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4 Optimal Docking Maneuvers 59

4.1 Physical Properties of the CubeSat . . . . . . . . . . . . . . . . . . 60

4.2 Pareto Front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Simulations’ Results - Unperturbed . . . . . . . . . . . . . . . . . . 63

4.3.1 Case 1: 400 km Circular Orbit . . . . . . . . . . . . . . . . . 65






4.4 Unperturbed Results’ Output . . . . . . . . . . . . . . . . . . . . . 95

4.5 Air Drag Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.5.1 Comparison 1: 400 km Circular Orbit . . . . . . . . . . . . . 98






V

4.6 Perturbed Results Analysis . . . . . . . . . . . . . . . . . . . . . . . 108

5 Conclusion 110

5.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Bibliography v

Appendix A vi

VI

List of Figures

2.1 Three Body Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Newton’s Law of Universal Gravitation . . . . . . . . . . . . . . . . 9

2.3 Restricted Three Body Problem System . . . . . . . . . . . . . . . 10

2.4 Satellites Relative Motion System . . . . . . . . . . . . . . . . . . . 12

2.5 LVLH Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Euler Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 12U CubeSat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.8 Docking Cone (left) and Docking Probe (right) . . . . . . . . . . . 29

2.9 Docking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.1 Translational Example - Trajectory . . . . . . . . . . . . . . . . . . 47

3.2 Translational Example - Relative Position Profile . . . . . . . . . . 48

3.3 Translational Example - Relative Velocity Profiles . . . . . . . . . . 48

3.4 Translational Example - Thrust Profiles . . . . . . . . . . . . . . . 49

3.5 Rotational Example - MRP Profiles . . . . . . . . . . . . . . . . . . 53

3.6 Rotational Example - Angular Velocity Profiles . . . . . . . . . . . 53

3.7 Rotational Example - Control Torque Profiles . . . . . . . . . . . . 54

3.8 Bounded Motion of the Chaser Satellite . . . . . . . . . . . . . . . . 55

3.9 Hatten and Russel minimum and maximum density profiles [1] . . . 57

4.1 Pareto Front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Case 1 - Orbits Underactuated . . . . . . . . . . . . . . . . . . . . . 66

4.3 Case 1 - Orbits Fully Actuated . . . . . . . . . . . . . . . . . . . . 66

4.4 Case 1 - Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.5 Case 1 - Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.6 Case 1 - Force Profiles . . . . . . . . . . . . . . . . . . . . . . . . . 69

VII



4.9 Case 2 - Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.10 Case 2 - Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73




4.14 Case 3 - Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.15 Case 3 - Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78


4.17 Case 4 - Orbits Under-Actuated . . . . . . . . . . . . . . . . . . . . 81

4.18 Case 4 - Orbits Fully-Actuated . . . . . . . . . . . . . . . . . . . . 81

4.19 Case 4 - Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.20 Case 4 - Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83




4.24 Case 5 - Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.25 Case 5 - Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88




4.29 Case 6 - Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.30 Case 6 - Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


4.32 Case 1 - Relative Orbits Comparison . . . . . . . . . . . . . . . . . 99

4.33 Case 1 - Relative Distances Comparison . . . . . . . . . . . . . . . 99







VIII





IX

List of Tables

1.1 CubeSat Missions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4.1 Thrusters Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.2 Spacecraft Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3 Boundary Conditions Case 1 . . . . . . . . . . . . . . . . . . . . . . 65

4.4 Results Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65


4.6 Results Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70


4.8 Results Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75


4.10 Results Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80


4.12 Results Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85


4.14 Results Case 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.15 Unperturbed Resulsts’ Summary . . . . . . . . . . . . . . . . . . . 95

4.16 Unperturbed Resulsts’ Summary - Maneuvers . . . . . . . . . . . . 97

4.17 Comparison Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.18 Comparison Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.19 Comparison Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.20 Comparison Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.21 Comparison Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.22 Comparison Case 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.23 Air Drag Resulsts’ Summary for Under-Actuated Configuration . . 108

X

Nomenclature

i x-axis Versor

j y-axis Versor

k z-axis Versor

σ Modified Rodrigues Parameters Vector

a Acceleration Vector

F Force Vector

I Inertia Matrix

M External Moment Vector

q Quaternions Vector

r Position Vector

v Velocity Vector

δ(...) Relative

µ⊕ Earth’s Standard Graviational Parameter

Ω Angular Orbital Velocity

φ Roll Angle

Ψ Yaw Angle

XI

θ Pitch Angle

Favg Average Force During the Maneuver

Fi Control Thrus Component

g0 Gravitational Acceleration at the surface of the Earth (9,81 m/s2)

Mapp Apparent Moment

Mci Target Inertial Torque

Mg Gyroscopic Torque

mman Mass of Fuel per Maneuver

R⊕ Radius of the Earth

Ti Control Torque Component

Vman Volume per Maneuver

6-DOF Six Degrees of Freedom

ADN Ammonium Di-Nitramide

AOCS Attitude and Orbit Control System

CW Clohessy-Wiltshire

DAE Dynamic Algebraic Equation

ECI Earth-Centered Inertial

FEEP Field-Electric Emission Propulsion

G Universal Gravitational Constant

GNC Guidance and Navigation Control

h Orbit Altitude

IDVD Inverse Dynamics in the Virtual Domain

XII

Isp Specific Impulse

ISS International Space Station

J Cost Function

LEO Low Earth Orbit

m Mass

MRP Modified Rodrigues Parameters

NLP Non Linear Programming

ODE Ordinary Differential Equation

PSOPT Pseudo Spectral OPTimization

RW Reaction Wheels

SGRA Sequential Gradient-Restoration Algorithm

T Period of the Orbit

XIII

Chapter 1

Introduction

The space race started during the Cold War, when USA and USSR wanted to

demonstrate their power. It started in 1958 with the launch by the Soviet Union

of the Sputnik 1, the first satellite to orbit around the Earth: one year later the

United States launched Explorer 1. The challenge extended to an higher level: in

1961 Jurij Gagarin was the first human to flight in space, and was followed by

other compatriots in the following years. The US answered few days after the first

USSR human flight in space with the mission MR-3 that took the first American

man in space.

What seemed to be a challenge for the domination of the world resulted in the

start of the space revolution, that would have changed the world as it was known it

in the following decades. Nowadays, the results of this process are clearly visible:

spacing from television broadcasts to meteorological forecasts, going from navi-

gations systems to telecommunications, ranging from military devices to science

studies, everybody is surrounded by instruments and services that use thousands

of in-orbit satellites as a platform. It was also possible to explore Mercury, Venus,

Mars, Jupiter and the borders of the solar system with the mission New Horizons

that provided the first ever high quality images of the surface of Pluto. Thanks to

Hubble telescope, scientists were able to explore the deep space and, with the fu-

ture launch of the James Webb Telescope they will be capable of seeing the deepest

part of the universe that has never been reached by any optical instrument.

1

1.1 CubeSat

Year by year, the technological level increased, and not only in the space sector:

just think to the computers and mobile phones that were on the market 20 years

ago: with todays technology, smartphones have a computational power that the

first satellites could never reach. In general, in this years a miniaturization of the

components has happened, going from large devices to always smaller.

This is what has happened also with the satellites. During the years, the average

launch mass of a spacecraft increased, peaking with the 8.2 tons of Envisat, a

Earth Observation satellites developed by the European Space Agency [2]. The

reason for an increase in the mass, as explained by Poghosyan [3], was done to

reduce the cost per kg of payload launched and to incorporate multiple payload

instruments in a single satellite. However, micro-vibrations and electromagnetic

issues could create problems in the integration process, as it happened for the

Envisat [4]. These complications, together with the development of the technology,

led to the miniaturizations of satellites, with a new concept called CubeSat. The

first CubeSat standard was defined at Stanford and California Polytechnic State

Universities in 1999 [5, 6], and defines the unit of a cubesat as 10x10x10 cm3 with

a mass of 1.33 kg [7]. The first launches of nanosatellites (not already known as

CubeSats but with comparable dimensions) are dated in the early ’60s, exactly at

the beginning of the space revolution presented before. After that, there was a

very long period with no launch of small satellites until early 2000s were, after the

study of CubeSats cited above, the launches increased drastically [5].

Due to their lower cost with respect to the traditional satellites, CubeSats are

widely used by Universities to perform scientific experiments in space. However,

also bigger organizations such as national agencies and companies are helping the

development of this sector [3]. In Table 1.1 some of the most recent and future

missions involving Cubesats are presented, with the aim of showing that CubeSats

can have, just to show that the field of operations is very various and that both

universities, national agencies and military forces are pushing in the direction of

this new technology.

2

Mission Size Organization Objective Launch

AENEAS [8] 3U University ofSouthern Califor-nia

Track cargo con-tainers in theopen ocean

2012

Firefly [9] 3U NASA Explore relation-ships betweenlightnings andTGFs

2013

Sense SV1/SV2 [10] 3U US Air Force Investigate spaceweather

2013

Perseus M1/M2 [11] 6U Aquia Space,Dauria Aerospace

Detect AIS trans-mitted by vessels

2014

Aoxiang-Sat [12] 12U NorthwesternPloytechnicalUniversity, China

Detect skylightpolarization pat-terns and measuregravity

2016

EON-MW [13] 12U MIT Scanning 22-channel from23 to 183 GHzhigh-resolutionmicrowave spec-trometer

2017

iSat [14] 12U NASA Test CubeSat Ma-neuverability

2017

Lunar Flashlight [15] 6U NASA JPL andMarhsall SpaceFlight Center

Locating icedeposits in themoons perma-nently shadowedcrater and detect-ing its properties

2019

EQUULEUS [16] 6U JAXA and Uni-versity of Tokyo

Study of theradiation envi-ronment in theregion of spacearound Earth

2020

Table 1.1: CubeSat Missions

3

What is done, typically, is to launch a constellation of CubeSats able to per-

form the mission. Due to their limited coverage and power, they would not be

able to achieve their goals alone. This is the case of the SWEET mission [17],

that need a global coverage for the monitoring of the water quality and water

level for African countries. Another example of multiple CubeSats launch is repre-

sented by AAReST (Autonomous Assembly of a Reconfigurable Space Telescope),

where the different modules combine directly in orbit to form a space telescope

[18]. CubeSats are widely used in planetary science missions and are a cheaper

instrument with respect to typical satellites for universities who wants to test their

systems directly in space. One of the first nanosatellite launched by a university

was QuakeSat [19], a 6U CubeSat deployed in 2003 by Stanford University with

the aim of detecting low frequency magnetic field signals using AC magnetome-

ters. After that, many universities developed their own projects: this is the case

of the University of Illinois in 2006, with its ION [3],with the mission of measur-

ing molecular oxygen airglow emissions from the Earths mesosphere and in 2008

of the University of Toronto, that developed a 3U CubeSat named CanX-2 [20]

that has the goal of measuring the total amount of electrons and water vapor

contents in the atmosphere as a function of the altitude. This new technology

has influenced also national agencies, such as NASA (USA) and JAXA (Japan).

It is the case, for example, of ASTERIA [3], a 6U CubeSat that has to look for

transits of identified RV planets and conduct high cadence stellar photometry over

long durations. OMOTENASHI (Outstanding MOon exploration TEchnologies

demonstrated by NAno Semi-Hard Impactor) [21, 22], that will be launched in

2020, will be the smallest moon lander ever. It provides attitude control and puts

itself into a impact trajectory. Before reaching the surface, the Cubesat spins up

and deploys the 0.7 kg surface probe featuring an inflatable airbag and a solid

rocket motor. Then the airbag is inflated and, just before impact, the solid rocket

motor is fired, performing a semi-hard landing. This mission could open the pos-

sibilities for nanosatellites, giving them the opportunity to act as a spacecraft and

as a lander. Also private companies, such as Lockheed Martin, Ball Aerospace

and many others, are investing in this field. Lockheed Martin, one of the most

important company in the aerospace and military sector, will launch in 2019 the

SKyfire, a 6U CubeSat with the aim of performing a lunar flyby and taking in-

4

frared sensor data for surface characterization, remote sensing, and site selection

as well as collecting data on thermal environments [3, 23]. Ball aerospace will

launch CIRiS (Compact Infrared Radiometer in Space), a 6U CubeSat equipped

with an infrared radiometer designed for high radiometric performance for land

and water resource management, research, and modeling [24].

However, CubeSats are not limited only to planetary missions. The Interplane-

tary NanoSpacecraft Pathfinder In Relevant Environment (INSPIRE), launched in

summer 2014, opened the field of interplanetary missions for CubeSats [25]. The

goal of this nanosatellite was to demonstrate its ability to communicate, operate

and be navigated remotely from earth at 1AU from earth, that is the distance of

Mars from our planet, for three months, together with the testing of CubeSats

systems that were never used before on interplanetary missions. In May 2018,

MarCO (Mars Cube One) CubeSats [26] will be launched together with the In-

Sight mission, that has to bring a lander on Mars. The two nanosatellites will

not be an active part of the original lander mission, on which they were added

subsequently, but they will be experimental entities. They, in fact, test the ability

of a 6U CubeSat of transmitting data from Mars to the Earth through a high-gain

antenna and of collecting solar power from the sun at 17 W. These nanosatellites

will rely only on the InSight final stage of Atlas V, that will insert them in the

interplanetary trajectory to Mars: their propulsion system consists only in atti-

tude control. In 2019, EQUULEUS [16, 27] will be launched and it will analyze

the Earth-Moon surrounding space, measuring the distribution of plasma to help

the scientists understand the radiation environment. It will also test low-energy

trajectory control techniques within the Earth-Moon Lagrangian points. In the

same year, on the Exploration Mission 1 (EM-1), 13 CubeSats payloads will be

launched, including EQUULEUS and: Lunar Flaslight [15] that with solar sail

will light the shadowed crater and help the detection of ice deposits; Near-Earth

Asteroid Scout [28], a solar sail CubeSat that will encounter near-earth asteroids

and, thanks to a high-resolution camera, retrieve information from them; Lunar

Ice Cube [29] that will search for the presence of ice from a low lunar orbit; and

many others.

As can be understood from this section, CubeSats can be used in every field

(imaging, data handling, science, communication) and by any organization (pub-

5

lic, private and universities). It is an area that is still developing, and with the

improvement of the technology that is leading to always smaller and more efficient

components, CubeSats can become the reference for space systems.

1.2 Problem

The main problem regarding CubeSats concerns the attitude and orbit de-

termination and control [5]. The major function of AOCS (Attitude and Orbit

Control System) in nano-satellites is to apply a simple rotational damping to re-

duce the spin rate of the satellite, that is important if CubeSats that need to

orient their antenna or sensor in some directions are taken into account. Com-

pared with the bigger satellites system, the control of the rotation is more coarse,

and a rough pointing of the instrument should be expected. An optimal control of

the nano-satellite is difficult because at least a two-axis system for the orientation

is required, and the lack of space in the typical 3U CubeSats used in multiple mis-

sions represents a strong limitation. However, in the last years the attitude control

technology has drastically improved, thanks to the miniaturization of orbit deter-

mination star trackers[3]: moreover, some companies have developed GNC systems

that fits in only 0.5U [30]. Strictly related to the attitude and orbit control, there

is the docking problem. It involves two bodies, one of them being the target and

the other the chaser. The typical example for space application is the ISS, where,

periodically, supplies and crew members berth with their module; the maneuvers

can either be performed autonomously or by members of the ISS/module. In

general, the interactions between different spacecrafts can have the most various

applications, and this work will exploit the docking scenario.

Ciarcia and Romano firstly [31] developed a near-optimal planar proximity maneu-

ver coupling two methods, IDVD (Inverse Dynamics in the Virtual Domain) and

SGRA (Sequential Gradient-Restoration Algorithm): the computations were done

considering a chaser spacecraft docking to a non-cooperative target, and both time

and energy optimizations were considered. In a subsequent work [32], they imple-

mented the same combined methods but in a planar translational only problem in

the Clohessy-Wiltshire environment. After these, they exploited experimentally

the IDVD-SGRA using a Floating Spacecraft Simulators testbed to perform au-

6

tonomous cooperative planar docking maneuvers between two spacecrafts [33, 34].

Inverse dynamics method is widely used, and also Biggs and Caubet in their work

[35] on the sub-optimal motion planning method for attitude maneuvers take ad-

vantage of the same technique, with quaternions describing the orientation of the

spacecraft. These works, as could be noticed, perform computations for planar

maneuvers or attitude-only, and they consider fully-actuated bodies as the tar-

gets of the projects. However, the typical configuration of a CubeSat, mainly due

to lack of space, is with one thruster only (under-actuated): for this reason, an

in-depth analysis of this technology must be performed.

1.3 Methodology

The motion planning of an under-actuated CubeSat in the 3D space has not

been evaluated yet, and in this work an in-depth analysis of this technology will

be produced. In particular, the motion planning that will be analyzed regards the

docking maneuver of a chaser spacecraft with a target satellite.

First of all, the equations characterizing the problem are presented, followed by

a presentation of the instrument used to carry out the solutions. After that, the

physical properties of the spacecraft that is going to be considered will serve as a

basis for the following presentation of the results. The simulations, on which the

results will be based, are 6, and they cover almost the whole LEO environment,

going from 400 kilometers (the height at which ISS is orbiting) to 900 kilometers.

A comparison between under-actuated and fully-actuated spacecrafts will be done

in the analysis, in order to show the main differences between a real-life system and

an ideal one. The results will comprehend both numerical values of the CPU time

(all computations have been carried out with a machine working with an Intel Core

i7-6500 that has a base frequency of 2.50 GHz with the possibility to boost it up

to 3.10 GHz), cost function value, average propulsion force during the maneuver

and the graphs with the absolute orbits, relative distances and velocities and the

thrust profiles. At the end, the results for the real life system, the under-actuated,

will be compared with the case of a perturbed environment caused by atmospheric

drag.

7

Chapter 2

6-DOF Relative Motion Dynamics

In this chapter, the fundamental mathematical and physical concepts necessary

for this work are presented. For the translational part, the R3BP is stated and then

the Clohessy-Wiltshire equations of relative dynamics deriving from it are devel-

oped. This part provides the 3-DOF equations of the relative translational motion

of two bodies. For the rotational part, instead, three different way to represent

the attitude of a spacecraft are expressed as follows: Euler angles, quaternions

and modified Rodrigues parameters. These three concepts are fundamental for

the next sections that are Euler equations, first, and attitude relative dynamics,

second. This last part will provide the 3-DOF relative rotational equations of a 2

satellite system. At the end, the translational and rotational part are coupled and

the optimal control problem that needs to be solved will be stated.

2.1 Translational Dynamics

This section presents the fundamental translational dynamics equations neces-

sary for the statement of the optimization problem. First of all, the more general

formulation of the R3BP is presented, then, from it, the Clohessy-Wiltshire equa-

tion of relative motion are derived. At the end, the set of the 3 dynamics equations

of relative motion are stated.

8

2.1.1 Restricted Three Body Problem

The general three body problem was first stated by Isaac Newton in 1687

when in his ”Principia” he presented the problem of the movement of three mas-

sive bodies subjected to their mutual gravitational perturbations. This problem

can be represented in the following way:

Figure 2.1: Three Body Problem

Each body interacts with the other two under the Newton’s Law of Universal Grav-

itation, that states that two masses attracts each other with a force directed along

the intersection of the two bodies, being this force proportional to the product

of the two masses and inversely proportional to the square of the distance. The

gravitational law is represented in the picture below

Figure 2.2: Newton’s Law of Universal Gravitation

with G = 6.67 · 10−11Nm2kg−2 being the universal gravitational constant.

Using the Newton’s second law, the general three body problem can be stated as

follows. At a certain time, for a mass mi and its vector position ri, the three

9

differential equations governing the problem are:

r1 = −Gm2r1 − r2|r1 − r2|3

−Gm3r1 − r3|r1 − r3|3

r2 = −Gm3r2 − r3|r2 − r3|3

+Gm1r1 − r2|r1 − r2|3

r3 = Gm1r1 − r3|r1 − r3|3

+Gm2r2 − r3|r2 − r3|3

(2.1)

A closed solution for this system of equations does not exist but it is possible to

retrieve an approximated one using numerical integration techniques.

In the restricted three body problem, instead, it is possible to find a closed solution.

Figure 2.3: Restricted Three Body Problem System

As shown in Figure 2.3 the two main bodies, m1 and m2, are moving in a circular

orbit of radius r12 under the action of their mutual gravitational attraction. G is

the centre of mass of the two body systems and it is coincident with the centre of

the xyz non-inertial comoving reference frame with the axis directed as shown in

the figure. The reference frame is rotating at a constant angular velocity Ω given

by

Ω = Ωk (2.2)

10

where

Ω =2π

T(2.3)

and T is the period of the circular orbit.

Given a rotating vector in the xyz frame, in this case the vector connecting the

centre of the reference system and the third body m,

r = xi+ yj + zk (2.4)

its first derivative is given by

r = vG + Ω× r + vrel (2.5)

where vG is the velocity of the centre of the reference frame and vrel = xi+yj+zk.

The second derivative is given by

r = aG + Ω× r + Ω× (Ω× r) + 2Ω× vrel + arel (2.6)

where arel = xi+yj+ zk. As stated above, the angular velocity Ω of the reference

frame is constant, so its time derivative is null. Also the velocity of the center of

mass is constant, as a consequence aG = 0. At the end, the second derivative of

the position vector r can be expressed as follows:

r = Ω× (Ω× r) + 2Ω× vrel + arel (2.7)

The third body m is subjected to the gravitational attractions of both m1 and m2

with the total force F being

F = −Gmm1r1r31−Gmm2

r2r32

(2.8)

From the second law of dynamics, combining equations (2.7) and (2.8), the for-

mulation of the restriced three body problem is obtained:

m[Ω× (Ω× r) + 2Ω× vrel + arel] = −Gmm1r1r31−Gmm2

r2r32

(2.9)

11

2.1.2 Clohessy-Wiltshire Equations

Now that the restricted three body problem has been stated, the next step will

be the derivation of the Clohessy-Wiltshire equations of relative motion.

From Figure 2.3 it is possible to define ρ1 and ρ2 as the vectors connecting the

center of mass to m1 and m2 respectively. Recalling equation (1.9) it can be said

r1 = r − ρ1 r2 = r − ρ2

To adapt the R3BP to the problem of a chaser spacecraft m and a target spacecraft

m2 both orbiting around a central body m1 (typically much bigger than m and

m1), let’s consider m and m1 infinitesimal. From this assumption derives that

ρ1 = 0 → r = r1

Moreover it can said that r2 = R is the position vector of the chaser with respect

to the target satellite and the angular velocity Ω is given by

Ω2 =Gm1

ρ32→ Ω2ρ32 = Gm1

The resulting systems is represented in the figure below:

Figure 2.4: Satellites Relative Motion System

After these assumptions, the equation of the chaser spacecraft can be written as

follows:

Ω× [Ω× (R+ ρ2)] + 2Ω× dR

dt+d2R

dt2= −Ω2ρ32

r3r (2.10)

This differential equation is clearly non-linear due to term 1/r3, but thanks to the

Taylor series expansion it can transformed in a linear equation. First of all it can

be defined

12

x =R

ρ2

and then it can be written

r2

ρ22=

(ρ2 +R)(ρ2 +R)

ρ22=ρ22 + 2ρ2 ·R+R2

ρ22= x2 + 2iξ · iRx+ 1

with iξ and iR being the versor of ρ2 and R respectively. The Taylor series

expansion leads to

ρ2r

= (1 + 2iξ · iRx+ x2)−

1

2 = 1− iξ · iRx+ . . .

Therefore, taking the third power of this last equation and discarding the second

and higher order terms brings to

ρ32r3

= 1− 3iξ · iRR

ρ2+O

(R2

ρ22

)(2.11)

Thus, equation (2.10) can be rewritten as follows

d2R

dt2+ 2Ω× dR

dt+ Ω× [Ω× (R+ ρ2)] = −Ω2

[1− 3(iξ ·R)

1

ρ2

](ρ2 +R) (2.12)

Since (iξ ·R) ·R = O(R2), the previous equation can be reduced to

d2R

dt2+ 2Ω× dR

dt+ Ω× (Ω×R) = −Ω2R+ 3Ω2(iξ ·R) · iξ +O(R2) (2.13)

For this relative motion problem it is useful to define a proper reference frame on

the target satellite as follows: x−axis coincident with the radial direction, y−axiswith the direction of motion and z − axis resulting from the cross-product of the

previous two. This is called LVLH (Local Vertical Local Horizon) frame (Figure

2.5).

As a consequence R can be expressed as

R = xiθ + yir + ziz Ω = Ωiz iξ = ir

At the end, after these assumptions, and after having solved the triple vectorial

product, equation (2.13) can be rewritten in vectorial form as follows

d2R

dt2+ 2Ω× dR

dt= −Ω2ziz + 3Ω2xir (2.14)

13

Figure 2.5: LVLH Frame

or, in scalar form,

d2x

dt2− 3Ω2x− 2Ω

dy

dt= 0

d2y

dt2+ 2Ω

dx

dt= 0

d2z

dt2+ Ω2z = 0

(2.15)

The system (2.15) represents the Clohessy-Wiltshire equations. These three cou-

pled, second order differential equations with constant coefficients are capable of

providing an exact solution.

2.1.3 Translational Dynamics Equations

Now that the review of the relative motion equation has been presented, the

final step is to develop the translational dynamics equations that will be used to

solve the problem. Considering the general case of a chaser satellites with thrusters

acting together on x − y − z directions of the target satellite, the system (1.15)

14

can be rewritten as follows x− 3Ω2x− 2Ωy =

Fxm

y + 2Ωx =Fym

z + Ω2z =Fzm

(2.16)

where m is the mass of the chaser and the dots represent the time derivative of

the variables. System (2.16) represents the final translational dynamics problem.

Later in this chapter, these three equations will be coupled with the rotational

dynamics’ ones to form the complete mathematical problem. It must be kept in

mind that this Ω represents the angular velocity of the orbit, not the angular

velocity of the body around its axes.

15

2.2 Rotational Dynamics

In this section, the relative rotational dynamics equations of the chaser with

respect to the target are presented. First of all, a review of the possible attitude

representations will be outlined, then, after choosing the proper representation,

the equations will be developed, starting from the more general Euler equations,

passing through the attitude error kinematics and finally coming to the set of three

dynamics equations.

2.2.1 Attitude Representation

There are different way to represent the rotation of a spacecraft around its main

axis. The first is by means of Euler angles, the most intuitive one but also the most

computational-demanding too. The second one is by using quaternions, composed

by a vectorial part and a scalar part, which provide good computational results

with and avoid singularities that could be encountered using Euler angles. The last

ones that is presented in this document are the Modified Rodrigues Parameters,

derived from quaternions with a reduction in the dimensions (3 MRP with respect

to the 4 quaternions). The most important aspect of these 3 formalism that will

be analysed is the Direction Cosine Matrix, which represent the orientation of the

chaser with respect to the target reference frame and which is needed for the final

coupling of translational and rotational dynamics.

Euler Angles

Euler angles represent the orientation of a rigid body with respect to a fixed

reference frame. In the aerospace field, the rotations around the three axes are

called (in the order around x, y and z) roll, pitch and yaw (Figure 2.6). This

formalism is called Tait-Bryan angles, but it is known with the general name of

Euler angles. It is mostly used in aeronautical applications but it can be extended

to the aerospace field. Typically, roll, pitch and yaw are represented with the

greek letters φ, θ and Ψ.

The direction cosine matrix is produced in the following way: the complete ro-

tation of a spacecraft is done by means of three subsequent rotations around the

16

Figure 2.6: Euler Angles

three main axes in whatever order (xyz, zyx, zxy, yzx, ...) and from each of these

rotations, a rotation matrix is generated. Here these matrices are shown:

Ax =

1 0 0

0 cos(φ) − sin(φ)

0 sin(φ) cos(φ)

(Roll)

Ay =

cos(θ) 0 sin(θ)

0 1 0

− sin(θ) 0 cos(θ)

(Pitch)

Az =

cos(Ψ) − sin(Ψ) 0

sin(Ψ) cos(Ψ) 0

0 0 1

(Y aw)

17

The resulting DCM is built in accordance with the order of the rotation. In fact, if

the sequence was xyz, then Axyz = AzAyAx, so the matrices must be multiplied

in the opposite order of the sequence of rotations. The resulting DCM from a xyz

sequence of rotation is

Axyz =

c(θ)c(Ψ) −c(φ)s(Ψ) + s(φ)s(θ)c(Ψ) s(φ)s(Ψ) + c(φ)s(θ)c(Ψ)

c(θ)s(Ψ) c(φ)c(Ψ) + s(φ)s(θ)s(Ψ) −s(φ)c(Ψ) + c(φ)s(θ)s(Ψ)

−s(θ) s(φ)c(θ) c(φ)c(θ)

where c and s stand for cosine and sine. As can be seen from the matrix above,

the terms are highly non linear, and once they are inserted in the equations the

computational effort results to be very demanding.

Quaternions

Quaternions are mathematical entities defined by a vectorial part and a scalar

part:

q = (q0, q1, q2, q3) = (q0, qv)

According to Euler’s rotation theorem, the rotation or sequence of rotations of a

body around its axes can be represented through a single rotation Θ around a

proper axis, called Euler axis, which in this case is defined by the unit vector qv.

This rotation can be represented by the following quaternion:

q =

cos

(Θ

2

), sin

(Θ

2

)qv

with qv = qvxi, qvyj, qvzk

being i, j and k the versor of the reference cartesian frame. The rotation of

any vector v by means of quaternions can be done representing the vector in the

following way:

a = 0,v

The final rotation of the vector v is given by the vectorial part of the following

conjugation of a by q

a′ = qaq−1 where q−1 =

cos

(Θ

2

),− sin

(Θ

2

)qv

18

The direction cosine matrix expressed using quaternion’s component is

Aq =

q20 + q21 − q22 − q23 2(q1q2 + q0q3) 2(q1q3 − q0q2)2(q1q2 − q0q3) q20 − q21 + q22 − q23 2(q2q3 + q0q1)

2(q1q3 − q0q2) 2(q2q3 + q0q1) q20 − q21 − q22 + q23

It is easy to notice that this matrix is easier to handle for any machine and this is

the great advantage of using quaternions instead of Euler angles.

Modified Rodrigues Parameters

The quaternions normalization constraint, that needs to be done cycle by cycle

when solving a problem, has shown to produce singularities in control problems.

The three-dimensional parametrization of the attitude representation can overcome

this problem, and this is why in control problems there are still 3D models [36].

However, Euler angles produces matrices and equations heavy to handle; this is

why modified Rodrigues parameters (MRP) gained importance in the time.

MRP are defined as follows:

σ =qv

1 + q0= n tan

Θ

4where n = i, j,kT

As can be noticed, MRP present a singularity for Θ = 2π, which corresponds to a

complete rotation around the axes until the initial position.

The DCM expressed in terms of the MRP can be shown to be as follows [37]:

Aσ=1

(1+σTσ)2

4(σ21−σ2

2−σ23)+Σ2 8σ1σ2+4σ3Σ 8σ1σ3−4σ2Σ

8σ1σ2−4σ3Σ 4(−σ21+σ

22−σ2

3)+Σ2 8σ2σ3+4σ1Σ

8σ1σ3+4σ2Σ 8σ2σ3−4σ1Σ 4(−σ21−σ2

2+σ23)+Σ2

(2.17)

Σ = 1− σTσ

As for the quaternions’ DCM, the MRP’s direction cosine matrix is easier to han-

dle from the computational point of view with respect to the Euler angles’ one.

19

In this paper the use of the MRP will be exploited because it has been proven

their advantages in the optimal problem solutions [36] and because Euler angles

and quaternions have demonstrated computational and singularities issues respec-

tively.

2.2.2 Attitude Error Kinematics

The problem at the base of this project is a relative one, therefore the relative

expression of the MRP is needed, and in particular their time derivatives: this

argument was previously treated by Younes in 2013 [38]. ”Relative” can be sub-

stituted with ”error” without change in the meaning.

As defined above in the document,

σ =qv

1 + q0= n tan

Θ

4

The inverse transformations can be written as

q0 =1− σ2

1 + σ2qv =

2σ

1 + σ2

where

σ2 = σTσ

It can be done the same for the error MRP (expressed with the prefix δ), leading

to

δσ =δqv

1 + δq0(2.18)

with the inverse transformations expressed as

δq0 =1− δσ2

1 + δσ2δqv =

2δσ

1 + δσ2(2.19)

The time variation of the error MRP can be derived from equation (2.18), and it

is given by

δσ =δqv

1 + δq0− δq0δqv

(1 + δq0)2(2.20)

20

where

δqv =1

2−([δω×] + 2[ωt×])δqv + δq0δω (2.21)

At the end, putting equations (2.19) and (2.21) in (2.20) leads to

δσ =1

4−2([δω×] + 2[ωt×])δσ + (1− δσ2)δω+

1

2(δωT δσ)δσ (2.22)

where

δω = ωc − ωt (2.23)

and the notation [A×] indicates

[A×] =

0 −A3 A2

A3 0 −A1

−A2 A1 0

2.2.3 Euler Equations

Euler equations are used to describe the rotation of a body around through a

reference of frame whose axes are coincident with the principal axes of inertia of

the body. In vectorial form the eqation is stated as follows

Iω + ω × (Iω) = T (2.24)

where I is the inertia matrix, ω the angular velocity vector of the body and T

the applied control torques. In the considered case of principal axes of inertia, I

is represented by the following matrix

I =

Ix 0 0

0 Iy 0

0 0 Iz

where Ix, Iy and Iz are the inertias of the body on its principal axes. Developing

the vectorial equations, the following set of scalar differential Euler equations is

obtained

21

ωx =TxIx

+Iy − IzIx

ωyωz

ωy =TyIy

+Iz − IxIy

ωxωz

ωz =TzIz

+Ix − IyIz

ωxωy

(2.25)

where the subscripts x y z on the angular rates and on the torques mean that

those velocities and external solicitations are acting on the principal axes of inertia

of the body. System (2.24) provides the Euler equations, but this is valid only for

a single body. This problem deals with relative dynamics, and this also involves

relative attitude. Therefore it is needed to derive the relative attitude equations

between target and chaser as done by Mauro Massari in his paper [39].

The more complete vectorial Euler equations of the chaser (subscript c) is:

Icωc + ωc × (Icωc) = T c +M c (2.26)

where the disturbance moment M c acting on the chaser has been added to the

previous equation. The following definitions will be useful for the derivation of the

final relative Euler equations:

Relative angular velocity: ωr = ωc − Γωt

Relative angular acceleration: ωr = ωc − Γωt + ωr × (Γωt)

Note that all the terms with the subscript t are referred to the target satellite.

Substituting the first previous equation in the second brings to

ωr = ωc − Γωt + ωc × (Γωt)− (Γωt)× (Γωt)︸︷︷︸=0

which, expressed for ωc, can be rewritten as

ωc = ωr + Γωt − ωc × (Γωt) (2.27)

Keeping in mind that ωc = ωr + Γωt, and substituting equation (2.27) in (2.26)

leads to

Ic(ωr + Γωt − ωc × Γωt) + (ωr + Γωt)× Ic(ωr + Γωt) = M c + T c

22

which can be developed and rewritten as follows:

Icωr+IcΓωt−Ic(ωc×Γωt)+ωr×Icωr+ωr×IcΓωt+Γωt×Icωr+Γωt×IcΓωt= M c+T c

After moving some terms to the right and condensing some parts, the equation

can be rearranged as

Icωr + ωr × Icωr = −M g −M ci +M app +M c + T c (2.28)

where

M gcoup = ωr × IcΓωt + Γωt × Icωr

M gc = Γωt × IcΓωt

M g = M gc +M gcoup

M ci = IcΓωt

M app = Icωr × Γωt

M g represents the gyroscopic torques, M ci the target inertial torques and M app

the apparent torques. All these terms influence the relative attitude dynamics of

the two bodies. This vectorial equation describes the time variation of the relative

attitude of the target and chaser with all the disturbances acting on the system.

2.2.4 Rotational Dynamics Equations

As it was illustrated in the Rotational Dynamics section, non only dynamic

was taken into account. Later in this document, the relevance of the kinematic

equations of the MRP will be discovered during the formulation of the complete

problem. Now the fundamental equation regarding the relative rotational part will

be shown, as a brief summary of what was done above. Starting from the Modified

Rodrigues Parameters, their kinematic can be expressed, in vectorial form, by the

following equation

δσ =1

4−2([δω×] + 2[ωt×])δσ + (1− δσ2)δω+

1

2(δωT δσ)δσ

23

or, in scalar form,δσ1 =

1

2[δσ2(δω3 + 2ωt3)−δσ3(δω2 + 2ωt2)]+

1

4(1− δσ2)δω1+

1

2Aδσ1

δσ2 =1

2[δσ3(δω1 + 2ωt1)−δσ1(δω3 + 2ωt3)]+

1

4(1− δσ2)δω2+

1

2Aδσ2

δσ3 =1

2[δσ1(δω2 + 2ωt2)−δσ2(δω1 + 2ωt1)]+

1

4(1− δσ2)δω3+

1

2Aδσ3

(2.29)

where

A = δωTδσ = δω1δσ1 + δω2δσ2 + δω3δσ3

and δω and δσ are defined in equations (2.23) and (2.18) respectively.

For what concerns the dynamic equations, i.e. the Euler equations expressed in

terms of relative motion, their vectorial form is given by

Icωr + ωr × Icωr = −M g −M ci +M app +M c + T c

or, expressed in scalar form,Ic1δω1 + (Ic3 − Ic2)δω2δω3 = −Mg1 −Mc1 +Mapp1 +Mci1 + Tc1

Ic2δω2 + (Ic1 − Ic3)δω1δω3 = −Mg2 −Mc2 +Mapp2 +Mci2 + Tc2

Ic3δω3 + (Ic2 − Ic1)δω1δω2 = −Mg3 −Mc3 +Mapp3 +Mci3 + Tc3

In order to uniform the notations, ωr has been substituted with δω.

In the first instance, the disturbance moment M c, which comprehends solar pres-

sure, J2 effect (due to the non-sphericity of the Earth), drag effect and third body

perturbation, will be be ignored. These parameters, in fact, have a huge effect in

the orbit propagation, but they can be neglected during short maneuvers like the

ones analyzed in this work. Having said that, the previous system can be rewritten

as follows:Ic1δω1 + (Ic3 − Ic2)δω2δω3 = −Mg1 −Mci1 +Mapp1 + Tc1

Ic2δω2 + (Ic1 − Ic3)δω1δω3 = −Mg2 −Mci2 +Mapp2 + Tc2

Ic3δω3 + (Ic2 − Ic1)δω1δω2 = −Mg3 −Mci3 +Mapp3 + Tc3

(2.30)

24

The reference 1, 2 and 3 are used in place of x, y and z in order to avoid confusion

with all the other letters present in the equations.

However, at the end of the simulations the air drag will be taken into account

and the perturbed results will be compared with the unperturbed ones, to see

how the software behaves with an higher level of difficult for the equations and

how the chaser-target system operates under this disturbance. Systems (2.29)

and (2.30) present the fundamental equations needed to represent the relative

rotational dynamic of a rigid body.

2.3 Problem Statement

Before stating the problem there is one last step that needs to be evaluated, i.e.

the coupling. What have been presented till now are just decoupled equations of

translational and rotational dynamics, but in the system considered, translation

and rotation take place simultaneously. What is needed is to couple these two

parts, and this can be done through the control forces acting on the chaser. These

forces, both thrusts and torques, relate translation and rotation through the DCM.

In fact, since they belong to the chaser body frame they need to be rotated and

decomposed along the three main axes of the system on which the equations derived

previously are expressed.

Recalling equation (2.17), the DCM expressed in terms of the MRP can be seen

as

Aσ =

A11 A12 A13

A21 A22 A23

A31 A32 A33

Being F b and T b the forces and torques acting on the chaser satellite expressed in

the body reference frame, their transformation in the global LVLH frame can be

done as follows:

25

F LV LH = AσF b =

A11Fx + A12Fy + A13Fz



T LV LH = AσT b =

A11Tc1 + A12Tc2 + A13Tc3

A21Tc1 + A22Tc2 + A23Tc3

A31Tc1 + A32Tc2 + A33Tc3

(2.31)

Now it is possible to derive the proper dynamic formulation of the problem.

In order to keep the equations light and easy to understand, the symbols δ and

r that stood for ”relative” have been removed. Recalling the systems (2.16),

(2.29) and (2.30), it is possible to proceed with the presentation of the differential

scalar equations that represent the system. Note that the notation (t) will be

removed without any loss of meaning. The complete system of relative dynamics

26

and kinematic equations is given by:

δx = δvx

δy = δvy

δz = δvz

δvx − 3Ω2δx− 2Ωδy =A11Fx + A12Fy + A13Fz

m

δvy + 2Ωδx =A21Fx + A22Fy + A23Fz

m

δvz + Ω2δz =A31Fx + A32Fy + A33Fz

m

δσ1 =1

2[δσ2(δω3 + 2ωt3)−δσ3(δω2 + 2ωt2)]+

1

4(1− δσ2)δω1+

1

2Aδσ1

δσ2 =1

2[δσ3(δω1 + 2ωt1)−δσ1(δω3 + 2ωt3)]+

1

4(1− δσ2)δω2+

1

2Aδσ2

δσ3 =1

2[δσ1(δω2 + 2ωt2)−δσ2(δω1 + 2ωt1)]+

1

4(1− δσ2)δω3+

1

2Aδσ3

Ic1δω1+(Ic3−Ic2)δω2δω3 =(A11Tc1+A12Tc2+A13Tc3)−Mg1−Mci1+Mapp1



(2.32)

where


and the terms of the matrix Aσ can be retrieved from equation (2.17). This is the

final set of coupled differential equations that will be written inside the solver and

used to find the solutions of the problem. It can now be seen that the problem

is fully coupled, since in the thrust forces in the translational part there is the

presence of DCM terms that are composed by the MRP of the rotational part.

2.4 Confguration of the Spacecraft

The spacecraft considered for the analysis is a 12U CubeSat, with dimensions

23x24x36 cm. Typically, a satellite with these dimensions has only one thruster

27

Figure 2.7: 12U CubeSat

along a direction: this is called “under-actuated”satellite. A typical representation

of this system is shown in Figure 2.7, with a 2x2x3 disposition of the units. The

“fully actuated”one, instead, is a configuration in which all the three axis of the

spacecraft present actuators in both directions. Clearly, since each thruster usu-

ally occupies one unit of the CubeSat, the fully-actuated case does not represent a

feasible configuration for small-dimensions satellite: the thrusters, in fact, occupy

one unit of the CubeSat each. If it is considered that 2/3 units are destined to

the reaction wheels, the satellite would have only 3/4 units where to store the fuel

and the electronics of the satellite. This problem can be avoided if this concept

is applied to bigger spacecrafts, where there is more space where to fit all these

systems. However, the fully actuated structure will be used as a reference, being

the best controllable spacecraft with all the 6-DOF covered by an actuator: it will

represent the reference for the results, showing how much the under-actuated case

differs from the fully-actuated. The single-thruster configuration, instead, will be

28

evaluated to show the possible results of the current technology. In the analyses

the thrusters will be considered as non-steerable, even if on the market there is

the possibility to find steerable thruster that would give to the overall system an

higher controllability. Typically, for a docking mission like the one performed in

this project, the berthing mechanism is placed on the opposite face of the thruster

in order to have more control during the final maneuvers, as shown in Figure 2.9.

In order to perform docking between two satellites, two mechanisms are needed: a

docking probe on the chaser and a docking cone on the target. A typical design of

these part is shown in Figure 2.8, with a slender part that will be accommodated

in the housing at the bottom of the cone. This configuration is used on multiple

spacecrafts, and also on the Soyuz when it has to dock with the Russian or Inter-

anation Space Station.

Figure 2.8: Docking Cone (left) and Docking Probe (right)

The docking cone allows a small error in the final position of the probe, because it

will guide it in its housing thanks to its shape. However, a good precision in the

orientation of the chaser is needed, because the probe is slender and it can easily

get damaged.

What typically happens is that, at the beginning of the docking maneuver, the

probe is not aligned with the target cone, as shown in Figure 2.9. Moreover, the

initial velocity of the chaser with respect to the target could be different from zero.

The docking problem solved in this work has the goal of bringing the chaser, with

certain initial conditions, to match the orientation and position of the probe with

the cone, having a final value of rotational and translational velocity equal to zero

in order to avoid any stress on the mechanism.

29

Figure 2.9: Docking

2.5 Constraint

Another important aspect that needs to be treated is the the one regarding

the constraints. They are a fundamental part of the overall problem, since they

impose physical and mathematical limitations to the solution.

Physical constraints regard the control torques and forces. For what concerns the

propulsion systems, they have an intrinsic limit that is the maximum amount of

thrust they can provide (wthis value will be called α): it will be considered the

same propulsion system for each thruster and their limits will be expressed as an

inequality, and, whether it presents a fully or under actuated configuration, the

constraints can be written as0 < Fx < α

Fy = 0

Fz = 0

(UNDER− ACTUATED)

30

−α < Fx < α

−α < Fy < α

−α < Fz < α

(FULLY − ACTUATED)

Regarding the control torques, the reaction wheels are the main systems used for

the maintenance of the attitude on CubeSats. The problem rising from exceeding

the limits of the reaction wheels is called saturation: once the physical limit has

been reached (it will be called β the maximum amount of torque they can provide),

the wheels are no more able to provide the control required by the spacecraft. As

for the control thrusts, since the RW can rotate in both directions, the constraint

on the torques for both the fully and under-actuated case can be expressed as−β < Tx < β

−β < Ty < β

−β < Tz < β

These are the main physical constraint on the system, driven by the properties

of the chaser spacecraft. The last thing needed for the solution, since differential

equations are dealt in this work, are initial and final conditions. Whereas the

initial conditions are arbitrary, the finals are not since a docking problem is con-

sidered. All the equations express the relative dynamics of two bodies, so, the final

conditions that represent the docking is to have the final relative position, angular

velocity and attitude equal to zero. This means that at the end of the maneuver,

chaser and target spacecraft will have the same position, orientation and velocity

in space. For what concerns the translational velocities, they do not have to be

necessarily zero. In fact, in some docking problems a small difference in the ve-

locities is needed in order to be able to dock to the target satellite. The relative

angular velocities, instead, must be zero, because during the docking maneuver

undesired moments can damage the mechanism.

31

Chapter 3

Implementation

In this chpater, the implementation of the kinematics and dynamics equation

in the software will be outlined.

First of all, an overview of the state space formulation, necessary to develop the

problem in the software, will be presented, with all the concepts relative to the

optimization: after that, the state space formulation of the system considered will

be derived together with the optimization function needed. Then, a presentation of

the software, PSOPT, will be given and it will be shown how it operates, with the

help of some examples. After these parts, that are an overview of the instruments

needed for the solution, all the assumptions made in the problem will be stated.

With these hypotheses, the complete problem that is required to be written in

PSOPT will be presented and it will be explained how the code is build.

The aim of this chapter is to make the reader understand how the software operates

and how the solution to the problem is computed.

3.1 State Space Formulation and Optimization

Problem

The software used in this project works with the state space formulation, a

typical way of expressing a large dimension problem with high order differential

equations. Before stating the optimization problem, that is the aim of this section,

a brief review of the optimal control theory will be presented.

32

3.1.1 Theoretical Review

The following example will show how the state space formulation works. Given

a general second order differential vectorial equation

Ms(t) +Cs(t) +Ks(t) = F (t)

M , C andK are matrices, F is the matrix of external forces and s(t) is the vector

containing the physical coordinates that vary in time (in this case it could be the

position, the velocity, the angular rate). From this equation a state vector can be

derived, and it has the following form

x(t) =

s(t)

s(t)

=

x1(t)

x2(t)

The advantage of the state vector, and in general of the state space formulation, is

that the higher size problem analyzed, can be reduced to a single order differential

equation problem. The firs equation can be rewritten in terms of u(t) and stated

as follows

s(t) = −M−1Cs(t)−M−1ks(t) +M−1F (t)

Using the definition of state vector, the problem can be written asx1(t) = x2(t)

x2(t) = −M−1Cx2(t)−M−1kx1(t) +M−1F (t)

The most known state space formulation, the matrix one, is given by

x(t) = Ax(t) +Bu(t) (3.1)

where u represents the input matrix that can contain both control and disturbance

forces. Moreover

A =

[0 I

−M−1K −M−1C

]

33

B =

[0

M−1u

]Optimal control problems aim at the minimization of the cost functional, that

formulated in the following way

J = ϕ[x(t0), t0,x(tf ), tf ] +

∫ tf

t0

Λ[x(t),u(t), t] dt (3.2)

where x are the states, u the controls, ϕ the endpoint cost function and Λ the

Lagrangian cost function. The goal of the mission is to minimize the total thrust

during the maneuver in order to save the maximum amount of fuel possible: for

this reason, the endpoint cost function will be zero whereas the integrand cost

function part is expressed as follows:

Λ = uTW uuu (3.3)

Mind that vector u contains both control thrusts and torques and that the propul-

sion system is the only part that needs to be considered in the analyses. The matrix

W uu is a weighting matrix used to give more or less importance in the minimiza-

tion process to certain parameters. In this project no term has to be decreased in

importance, so the matrix W FF will be equal to an identity matrix.

3.1.2 State Space Formulation of the 6-DOF Relative Mo-

tion System

Now that the dynamic problem is completely stated and that the state space

formulation has been presented, the formulation of the 6-DOF relative motion

problem can be stated. As seen in the previous section, the first step is to define

a state vector, that in, for the case considered, is given by

x(t) =δx δy δz δvx δvy δvz δσ1 δσ2 δσ3 δω1 δω2 δω3

T(3.4)

Keeping in mind what was done in the theoretical section of the state space formu-

lation and recalling the equations of system (2.32), the formulation of the problem

34

can be written as follows

δx = δvx

δy = δvy

δz = δvz

δvx = 3Ω2δx+ 2Ωδy +A11Fx + A12Fy + A13Fz

m

δvy = −2Ωδx+A21Fx + A22Fy + A23Fz

m

δvz = −Ω2δz +A31Fx + A32Fy + A33Fz

m

δσ1 =1

2[δσ2(δω3 + 2ωt3)−δσ3(δω2 + 2ωt2)]+

1

4(1− δσ2)δω1+

1

2Aδσ1

δσ2 =1

2[δσ3(δω1 + 2ωt1)−δσ1(δω3 + 2ωt3)]+

1

4(1− δσ2)δω2+

1

2Aδσ2

δσ3 =1

2[δσ1(δω2 + 2ωt2)−δσ2(δω1 + 2ωt1)]+

1

4(1− δσ2)δω3+

1

2Aδσ3

δω1 =(Ic2−Ic3)Ic1

δω2δω3+(A11Tc1+A12Tc2+A13Tc3)

Ic1+Mapp1−Mg1−Mci1

Ic1




Ic2




Ic3

(3.5)

where


The last thing that needs to be outlined is the cost function relative to the prob-

lem. As said in the previous section, the inputs of this function will only be the

thrust force. Recalling equations (3.1) and (3.2), the integrand cost function of

the problem can be written as

J =

∫ tf

0

(F 2x + F 2

y + F 2z ) dt (3.6)

where tf is the time required by the maneuver.

System (3.5) shows how the equations must be written in order to make the solver

able to retrieve a solution for the problem and equation (3.5) is the general opti-

35

mization function of the problem considered in this work. The reader has to re-

member that both the under and fully actuated case will be analyzed, and whether

it is used one ore the other, one or three components of the force vector will be

present.

In the next section it will be shown how the software works and how the code of

the system is structured.

3.2 PSOPT

As previously said in the introduction to this chapter, the software used for

the resolution of the problem is PSOPT.

PSOPT (Pseudo Spectral OPTimization) is an open source software written in

C++ aimed at the solution of optimization problems using polynomials to ap-

proximate the time dependant variables: this method is called “Direct Collocation

Method”and it is used for the discretization of differential equations that form the

problem. The discretization is carried out choosing a number of grid points that are

placed in the most relevant points of the functions by a mesh refinement method.

There are different nodes collocation methods built in the software, and they are:

Trapezoidal, Hermite-Simpson, Chebyshev and Legendre (the default one). For

the problem considered, all these four methods have been tested and the results

showed that the trapezoidal and Hermite-Simpson were the fastest converging

ones. Even if the trapezoidal has an accuracy of O(h2) whereas Hermite-Simpson

has O(h4), where h is the local distance between grid points, actual computations

outlined that, since small optimal integral function values are dealt in this work,

the difference between the results was negligible. Anyway, only the trapezoidal

one was used in all the computations because it displayed more robustness with

respect to the Hermite-Simpson for what concerns higher difficult problems.

Once the equations are discretized, a “Non Linear Programming”(NLP) program

is formulated, and it is then solved numerically using a nonlinear sparse program-

ming solver (IPOPT for PSOPOT) that carries out local optimal solution rather

than the absolute one. It is important to understand this aspect, because for such

difficult problems the absolute optimal solution can not be granted.

Apart from PSOPT, one of the most used software on the market is Matlab, which

36

solves differential equations with built-in functions called “ODE”. These systems

work with iterative methods, such as the widely used Runge-Kutta. The reason

why PSOPT rather than Matlab was used for this work must be sought in the dis-

cretization methods. The practice showed that pseudo spectral, in fact, presents

some advantages in the solution of optimization problems [40]:

• Faster rate of convergence

• Better accuracy

• Best suited for optimal control problems

The general optimal control problem in PSOPT is formulated as follows:

Find the control profiles u(t), the state profiles x(t), the static parameters p, with

t ∈ [t0, tf ], and the times t0 and tf to minimize the following cost function

J = ϕ [x(tf ), p, tf ] +

∫ tf

t0

Λ [x(t), u(t), p, t] dt

subjected to the differential equations

x(t) = f [x(t), u(t), p, t], t ∈ [t0, tf ]

the path constraints

hL ≤ h[x(t), u(t), p, t] ≤ hU , t ∈ [t0, tf ]

the event constraints

eL ≤ e [x(t0), u(t0), x(tf ), u(tf ), p, t0, tf ] ≤ eU

and the bound constraints

uL ≤ u(t) ≤ uU , t ∈ [t0, tf ]

xL ≤ x(t) ≤ xU , t ∈ [t0, tf ]

pL ≤ p ≤ pU

37

This formulation is valid for a single phase problem, that is the one considered

in this project. For a multiphase one, instead, the PSOPT manual presents the

general formulation that is similar to the one presented here.

One further clarification on how the code is implemented must be done. In the

analyses, the time is re-parametrized from t0 = 0 to tf = 1: this procedure is widely

used in mathematical problems because it increases the rate of convergence. So,

the problem is reformulated in the following way: given the general differential

equationd

dt(...) = f

the time steps used for the integration are

t0 = 0

tf = T

t = [0, ..., T ]

and the re-parametrized time vector is given by

t = [0, ..., 1]

The integration time can be rewritten as follows

t = T t

The differential of t is given by

dt = T dt

Being said that, the re-parametrized equations can be written as

d

dt(...) =

d

T dt(...) = f → d

dt(...) = T ∗ f

This is how the equations will be written in PSOPT.

Now that a general review of the software has been given, two examples will be

presented to show how the code is implemented.

38

3.2.1 Translational Example

Let’s assume that the satellite has a mass of mass m = 300 kg equipped with

thrusters able to provide the force in all the verses of the main axes xyz. The

total thrust during a maneuver of 10 minutes where the vehicle go from a point

A to a point B, with a certain initial and final velocities, needs to be optimized.

For simplicity, the forces of the body will be considered aligned with the axes of

the inertial reference frame. The maximum amount of force each thrusters can

provide is 1N .

First of all, PSOPT requires the definition of the endpoint cost function and the

integrand cost function. In this case the endpoint will be zero, whereas the inte-

grand, considering the parametrization of the time, is given by

J =

∫ 1

0

T (F 2x + F 2

y + F 2z ) dt

where

T = 600 s

In PSOPT, the code is implemented as follows:

adouble i n t e g r and co s t ( adouble ∗ s t a t e s , adouble ∗ cont ro l s ,

adouble ∗ parameters , adouble& time , adouble ∗ xad , int iphase ,

Workspace∗ workspace )

adouble Fx = con t r o l s [ CINDEX(1) ] ;

adouble Fy = con t r o l s [ CINDEX(2) ] ;

adouble Fz = con t r o l s [ CINDEX(3) ] ;

double T = 600 . 0 ;

return T∗(Fx∗Fx + Fy∗Fy + Fz∗Fz ) ;

The next step is to define the set of differential equations that describe the system

(DAEs). The satellite is moving under the Neton’s second law, with no distur-

bances for simplification. These equationsvexpressed using the state space formu-

39

lation are

x = vx

y = vy

z = vz

vx =Fxm

vy =Fym

vz =Fzm

First of all, each state and control variable is assigned to its command, then

the derivatives of the states (the differential equations) are listed. The code is

produced as follows

void dae ( adouble ∗ de r i v a t i v e s , adouble ∗ path , adouble ∗ s t a t e s ,

adouble ∗ cont ro l s , adouble ∗ parameters , adouble& time ,

adouble ∗ xad , int iphase , Workspace∗ workspace )

adouble Fx = con t r o l s [ CINDEX(1) ] ;

adouble Fy = con t r o l s [ CINDEX(2) ] ;

adouble Fz = con t r o l s [ CINDEX(3) ] ;

adouble x = s t a t e s [ CINDEX(1) ] ;

adouble y = s t a t e s [ CINDEX(2) ] ;

adouble z = s t a t e s [ CINDEX(3) ] ;

adouble vx = s t a t e s [ CINDEX(4) ] ;

adouble vy = s t a t e s [ CINDEX(5) ] ;

adouble vz = s t a t e s [ CINDEX(6) ] ;

double m = 300 . 0 ;

// //////DERIVATIVES OF THE STATES///////////

double T = 600 . 0 ;

d e r i v a t i v e s [ CINDEX(1) ] = T∗( dxdot ) ;d e r i v a t i v e s [ CINDEX(2) ] = T∗( dydot ) ;

40

d e r i v a t i v e s [ CINDEX(3) ] = T∗( dzdot ) ;d e r i v a t i v e s [ CINDEX(4) ] = T∗(Fx/m) ;

d e r i v a t i v e s [ CINDEX(5) ] = T∗(Fy/m) ;

d e r i v a t i v e s [ CINDEX(6) ] = T∗(Fz/m) ;

After that, the events (initial and final conditions) must be assigned to their com-

mand before being defined. This is a typical procedure used in C++ where a new

parameter need to be inserted in the code. Initial and final conditions are set on

all the states, whereas none on the control forces. Thus, the code is expressed as

follows

void events ( adouble ∗ e , adouble ∗ i n i t i a l s t a t e s ,

adouble ∗ f i n a l s t a t e s , adouble ∗ parameters , adouble& t0 ,

adouble& t f , adouble ∗ xad , int iphase , Workspace∗ workspace )

adouble x0 = i n i t i a l s t a t e s [ CINDEX(1) ] ;

adouble y0 = i n i t i a l s t a t e s [ CINDEX(2) ] ;

adouble z0 = i n i t i a l s t a t e s [ CINDEX(3) ] ;

adouble vx0 = i n i t i a l s t a t e s [ CINDEX(4) ] ;

adouble vy0 = i n i t i a l s t a t e s [ CINDEX(5) ] ;

adouble vz0 = i n i t i a l s t a t e s [ CINDEX(6) ] ;

adouble x f = f i n a l s t a t e s [ CINDEX(1) ] ;

adouble y f = f i n a l s t a t e s [ CINDEX(2) ] ;

adouble z f = f i n a l s t a t e s [ CINDEX(3) ] ;

adouble vxf = f i n a l s t a t e s [ CINDEX(4) ] ;

adouble vyf = f i n a l s t a t e s [ CINDEX(5) ] ;

adouble vz f = f i n a l s t a t e s [ CINDEX(6) ] ;

//INITIAL AND FINAL CONDITIONS//

e [ CINDEX(1) ] = x0 ;

e [ CINDEX(2) ] = y0 ;

e [ CINDEX(3) ] = z0 ;

e [ CINDEX(4) ] = vx0 ;

e [ CINDEX(5) ] = vy0 ;

e [ CINDEX(6) ] = vz0 ;

e [ CINDEX(7) ] = xf ;

e [ CINDEX(8) ] = yf ;

41

e [ CINDEX(9) ] = z f ;

e [ CINDEX(10) ] = vxf ;

e [ CINDEX(11) ] = vyf ;

e [ CINDEX(12) ] = vz f ;

The next part of the code contains the call to the solver and the files that need to

be generated. The relevant part of this section is the one were the information of

the system need to be defined, that are number of states (6), number of controls

(3), number of events (12), number of paths (0) and number of nodes (120). The

code is written in the following way:

problem . phases ( 1 ) . n s t a t e s = 6 ;

problem . phases ( 1 ) . n con t r o l s = 3 ;

problem . phases ( 1 ) . nevents = 12 ;

problem . phases ( 1 ) . npath = 0 ;

problem . phases ( 1 ) . nodes = ” [ 1 2 0 ] ” ;

p s op t l e v e l 2 s e t up ( problem , a lgor i thm ) ;

Subsequently the initial and final values and the upper and lower bounds of the

states and controls must be inserted. The following bounds will be considered:

−∞ ≤ x ≤ ∞

−∞ ≤ y ≤ ∞

−∞ ≤ z ≤ ∞

−∞ ≤ vx ≤ ∞

−∞ ≤ vy ≤ ∞

−∞ ≤ vz ≤ ∞

−100N ≤ Fx ≤ 100N

−100N ≤ Fy ≤ 100N

−100N ≤ Fz ≤ 100N

that means no limits are given to the values the states can assume, in order to give

less constraints to the solver and make the problem easier to solve: the physical

42

limit of the thrusters, instead, is stated. Mind that these bounds are not the typical

values for a CubeSat system, but they are valid for this example only. Table 4.2

presents the classic rates expected for a nanosatellite.

Regarding the initial conditions, considering that all the values are expressed in

km and km/s, the following parameters for point A are assumed:

x0 = 0

y0 = 0

z0 = 0

vx0 = 0.02

vy0 = 0.05

vz0 = −0.03

The final conditions for point B, instead, are

xf = 1.2

yf = 2.3

zf = 0.8

vxf = 0

vyf = 0

vzf = 0

These values are assigned to the events and to the states and controls defined above

in the code, so that the source representing these bounds is written as follows:

double x0 = 0 ;

double y0 = 0 ;

double z0 = 0 ;

double vx0 = 0 . 0 2 ;

double vy0 = 0 . 0 5 ;

double vz0 = −0.03;

double xf = 1 . 5 ;

double yf = 2 . 3 ;

double z f = 0 . 8 ;

double vxf = 0 . 0 ;

double vyf = 0 . 0 ;

double vz f = 0 . 0 ;

43

double l owe r l im i t Fx = −0.1; //LOWER AND UPPER PHYSICAL//

double upper l im i t Fx = 0 . 1 ; //LIMITS OF THE THRUSTERS x ///

double l owe r l im i t Fy = −0.1; //LOWER AND UPPER PHYSICAL//

double upper l im i t Fy = 0 . 1 ; //LIMITS OF THE THRUSTERS y///

double l owe r l im i t Fz = −0.1; //LOWER AND UPPER PHYSICAL//

double uppe r l im i t Fz = 0 . 1 ; //LIMITS OF THE THRUSTERS z ///

/////LOWER AND UPPER BOUNDARIES OF THE STATES////////

problem . phases ( 1 ) . bounds . lower . s t a t e s (1 ) = − i n f ;






problem . phases ( 1 ) . bounds . upper . s t a t e s (1 ) = i n f ;






/////LOWER AND UPPER BOUNDARIES OF THE CONTROLS////////

problem . phases ( 1 ) . bounds . lower . c on t r o l s (1 ) = lowe r l im i t Fx ;

problem . phases ( 1 ) . bounds . lower . c on t r o l s (2 ) = lowe r l im i t Fy ;

problem . phases ( 1 ) . bounds . lower . c on t r o l s (3 ) = l owe r l im i t Fz ;

problem . phases ( 1 ) . bounds . upper . c on t r o l s (1 ) = upper l im i t Fx ;

problem . phases ( 1 ) . bounds . upper . c on t r o l s (2 ) = upper l im i t Fy ;

problem . phases ( 1 ) . bounds . upper . c on t r o l s (3 ) = uppe r l im i t Fz ;

///BOUNDARIES ON THE INITIAL AND FINAL CONDITIONS OF THE STATES////

problem . phases ( 1 ) . bounds . lower . events (1 ) = x0 ;

problem . phases ( 1 ) . bounds . lower . events (2 ) = y0 ;

problem . phases ( 1 ) . bounds . lower . events (3 ) = z0 ;

problem . phases ( 1 ) . bounds . lower . events (4 ) = vx0 ;

44

problem . phases ( 1 ) . bounds . lower . events (5 ) = vy0 ;

problem . phases ( 1 ) . bounds . lower . events (6 ) = vz0 ;

problem . phases ( 1 ) . bounds . lower . events (7 ) = xf ;

problem . phases ( 1 ) . bounds . lower . events (8 ) = yf ;

problem . phases ( 1 ) . bounds . lower . events (9 ) = z f ;

problem . phases ( 1 ) . bounds . lower . events (10) = vxf ;

problem . phases ( 1 ) . bounds . lower . events (11) = vyf ;

problem . phases ( 1 ) . bounds . lower . events (12) = vz f ;

problem . phases ( 1 ) . bounds . upper . events (1 ) = x0 ;

problem . phases ( 1 ) . bounds . upper . events (2 ) = y0 ;

problem . phases ( 1 ) . bounds . upper . events (3 ) = z0 ;

problem . phases ( 1 ) . bounds . upper . events (4 ) = vx0 ;

problem . phases ( 1 ) . bounds . upper . events (5 ) = vy0 ;

problem . phases ( 1 ) . bounds . upper . events (6 ) = vz0 ;

problem . phases ( 1 ) . bounds . upper . events (7 ) = xf ;

problem . phases ( 1 ) . bounds . upper . events (8 ) = yf ;

problem . phases ( 1 ) . bounds . upper . events (9 ) = z f ;

problem . phases ( 1 ) . bounds . upper . events (10) = vxf ;

problem . phases ( 1 ) . bounds . upper . events (11) = vyf ;

problem . phases ( 1 ) . bounds . upper . events (12) = vz f ;

/////LOWER AND UPPER BOUNDARIES OF THE TIME////////

problem . phases ( 1 ) . bounds . lower . StartTime = 0 . 0 ;

problem . phases ( 1 ) . bounds . upper . StartTime = 0 . 0 ;

problem . phases ( 1 ) . bounds . lower . EndTime = 1 . 0 ;

problem . phases ( 1 ) . bounds . upper . EndTime = 1 . 0 ;

As can be seen from the part of code above, the forces have been inserted with

the proper unit of measure: since all the dimensions are in km, the forces need to

be considered in kN (kg km s−2). Note that also the time has been bounded. The

software could have solved the problem for whatever temporal bounds. In this

case the complete time interval has to be considered, going from 0 to 1.

Then, the guess of the states and controls must be given to the software. Typically

what is done is to give PSOPT a vector with all the values that assumes a certain

parameter. For example, for the time a vector t = [0, ..., 1] will be provided, for

the first state variable δx = [δx0, ..., 0], and so on. So, the code looks like this:

45

DMatrix s t a t e gu e s s = ze ro s ( ns tate s , nnodes ) ;

DMatrix c on t r o l g u e s s = ze ro s ( ncontro l s , nnodes ) ;

DMatrix t ime gues s = l i n s p a c e ( 0 . 0 , 1 . 0 , nnodes ) ;

s t a t e gu e s s (1 , co lon ( ) ) = l i n s p a c e ( x0 , xf , nnodes ) ;

s t a t e gu e s s (2 , co lon ( ) ) = l i n s p a c e ( y0 , yf , nnodes ) ;

s t a t e gu e s s (3 , co lon ( ) ) = l i n s p a c e ( z0 , z f , nnodes ) ;

s t a t e gu e s s (4 , co lon ( ) ) = l i n s p a c e ( vx0 , vxf , nnodes ) ;

s t a t e gu e s s (5 , co lon ( ) ) = l i n s p a c e ( vy0 , vyf , nnodes ) ;

s t a t e gu e s s (6 , co lon ( ) ) = l i n s p a c e ( vz0 , vzf , nnodes ) ;

c on t r o l g u e s s (1 , co lon ( ) ) = l i n s p a c e ( l ower l im i t Fx ,

upper l imit Fx , nnodes ) ;

c on t r o l g u e s s (2 , co lon ( ) ) = l i n s p a c e ( l ower l im i t Fy ,

upper l imit Fy , nnodes ) ;

c on t r o l g u e s s (3 , co lon ( ) ) = l i n s p a c e ( l owe r l im i t Fz ,

upper l imi t Fz , nnodes ) ;

Last, but not of least importance, the algorithm informations must be stated.

These mainly consist in the maximum number of iterations, tolerances and collo-

cation method. For this problem, the following parameters have been selected:

a lgor i thm . n lp i t e r max = 10000 ;

a lgor i thm . n l p t o l e r a n c e = 1 . e−5;a lgor i thm . nlp method = ”IPOPT” ;

a lgor i thm . s c a l i n g = ”automatic ” ;

a lgor i thm . d e r i v a t i v e s = ”automatic ” ;

a lgor i thm . co l l o ca t i on method = ” t r ap e z o i d a l ” ;

a lgor i thm . mesh ref inement = ”automatic ” ;

a lgor i thm . ode t o l e r anc e = 1 . e−5;

In this problem the values of the ODE and NLD tolerances are kept small, but

typically the one of the ODE is lower with respect to the one of the NLP because

if the software is dealing with highly non-linear equations, a strict tolerance on

the ODE would increase the rate of convergence and the number of iterations by

a lot.

The solution to this problem gives a cost function of J = 5.082347 achieved in

4.330056e seconds with a maximum relative local error of 3.189276e-06.

Figure 3.1 shows the resulting trajectory of the satellite. From this picture it is

46

not clear if the boundary conditions are satisfied, but from figure 3.2 and 3.3 it

is possible to see that the solver matched the initial and final conditions on both

position and velocity. The last one, figure 3.4, displays the resulting force profile

for each thruster needed to obtained the desired result.

Figure 3.1: Translational Example - Trajectory

47

Figure 3.2: Translational Example - Relative Position Profile

Figure 3.3: Translational Example - Relative Velocity Profiles

48

Figure 3.4: Translational Example - Thrust Profiles

3.2.2 Rotational Dynamics Example

The second example consists in the reorientation of a rigid body around its

principal axes of inertia. The total torque used by the attitude control of satellite

to rotate it needs to be minimized. The spacecraft has dimensions of 6x4x4 meters

and a mass of m = 5000 Kg: as a consequence, the inertia matrix is given by

I =

13333.33 0 0

0 21666.67 0

0 0 21666.67

Kgm2

The maximum torque that can be provided is 100 Nm and the time required for

the rotation is 60 seconds.

As in the previous section, the first step is to define the integrand cost function

(no endpoint because only the torque over the maneuvering time needs to be

49

evaluated), which in this case is given by

J =

∫ 1

0

T (T 21 + T 2

2 + T 23 ) dt

where

T = 60 s

The code responsible for the implementation of this cost function is written as

follows:




adouble T1 = con t r o l s [ CINDEX(1) ] ;



double T = 60 . 0 ;

return T∗(T1∗T1 + T2∗T2 + T3∗T3 ) ;

The rotation will be described using MRP in order to prepare the reader to the

problem of this document. The states vector, for this example, is composed by

MRPs and angular velocities, both considered on the principal axes of inertia. The

time derivative of the MRPs, in the more compact matrix form, can be written as

[37]:

σ =

σ1

σ2

σ3

=1

4

(1 + σ2

1 − σ22 − σ2

3) 2(σ1σ2 − σ3) 2(σ1σ3 + σ2)

2(σ1σ2 + σ3) (1− σ21 + σ2

2 − σ23) 2(σ2σ3 − σ1)

2(σ1σ3 − σ2) 2(σ2σ3 + σ1) (1− σ21 − σ2

2 + σ23)

ω1

ω2

ω3

50

The derivatives of the angular velocities, instead, are expressed through the Euler

Equations, and are given by

ω1 =T1I1

+I2 − I3I1

ω2ω3

ω2 =T2I2

+I3 − I1I2

ω1ω3

ω3 =T3I3

+I1 − I2I3

ω1ω2

The DAE in PSOPT are generated as follows




double T = 60 . 0 ;




adouble P1 = s t a t e s [ CINDEX(1) ] ;



adouble om1 = s t a t e s [ CINDEX(4) ] ;



double I1 = 13333 . 33 ;

double I2 = 21666 . 67 ;

double I3 = 21666 . 67 ;

d e r i v a t i v e s [ CINDEX(1) ] = T∗ ( 0 . 25∗ (om1∗(1.0+P1∗P1−P2∗P2−P3∗P3)+2.0∗om2∗(P1∗P2−P3)+2.0∗om3∗(P1∗P3+P2 ) ) ) ;

d e r i v a t i v e s [ CINDEX(2) ] = T∗ ( 0 . 2 5∗ ( 2 . 0∗om1∗(P1∗P2+P3)+om2∗(1.0−P1∗P1+P2∗P2−P3∗P3)+2.0∗om3∗(P2∗P3−P1 ) ) ) ;

d e r i v a t i v e s [ CINDEX(3) ] = T∗ ( 0 . 2 5∗ ( 2 . 0∗om1∗(P1∗P3−P2)+2.0∗om2∗(P2∗P3+P1)+om3∗(1.0−P1∗P1−P2∗P2+P3∗P3 ) ) ) ;

d e r i v a t i v e s [ CINDEX(4) ] = T∗ ( ( (T1/ I1 )−(( I3−I2 )/ I1 )∗om2∗om3 ) ) ;


51


After the equations of the problem are stated, the boundary conditions must be

stated. The initial conditions for this problem are:

σ1(0) = 0

σ2(0) = 0

σ3(0) = 0

ω1(0) = 0.05 rad/s

ω2(0) = 0.05 rad/s

ω3(0) = 0.05 rad/s

whereas the finals are given by

φ(tf ) =π

6→ σ1(tf ) = tan

(φ

4

)θ(tf ) =

4π

3→ σ2(tf ) = tan

(θ

4

)Ψ(tf ) = −3π

4→ σ3(tf ) = tan

(Ψ

4

)ω1(tf ) = 0

ω2(tf ) = 0

ω3(tf ) = 0

These limits, together with the limits on the maximum control torque, can be

easily written in PSOPT as done in the previous example.

The results to this example give an optimal cost function value of J = 2.514952e+

05 carried out with a computational time of 3.600409 seconds and with a relative

local error of 1.646516e-05.

The resulting MRPs profiles are shown in figure 3.5, whereas the angular velocities

in 3.6. From picture 3.7 it is possible to see the torques profiles during the ma-

neuver. As can be seen, all the boundary conditions are respected and the limits

on the maximum torque available are not exceeded.

52

Figure 3.5: Rotational Example - MRP Profiles

Figure 3.6: Rotational Example - Angular Velocity Profiles

53

Figure 3.7: Rotational Example - Control Torque Profiles

3.3 Assumptions

Typically, when real life problems need to be solved on electronic machines,

some assumptions and simplifications are made. This is done because sometimes

the problem is too difficult and wide to be represented or because the computer is

not able to solve it. What is done is to try to reduce the difficult of the equations

without losing too much details that would give a solution that differs a lot from

the exact one. This has be done also during the production of this work: in this

section all the main changes and hypothesis made are presented.

The first assumption, already presented in chapter 2, is that target and chaser

satellites are on a circular orbit: from this guess, the Clohessy-Wiltshire equa-

tions, the basis for the translational dynamics, were derived.

As you could have read in the presentation of the equations, the perturbations

have been neglected. Disturbances such as third body, solar pressure and J2 effect

become of relevant importance in orbit propagation problems, where the trajec-

tory of the satellites last from several days to years. The maneuvers analyzed

will last maximum one hour, so these effects would not produce a huge effect on

the system. For what concerns the drag effect, the orbit where the spacecraft is

54

inserted is not too low to have huge effect on the body, and this effect would not

anyway be relevant for fast maneuvers.

One important aspect that has not been presented yet is the initial condition of

the chaser at the start of the maneuver. It has been assumed that the CubeSat is

inserted in a periodic bounded orbit around the central target satellite.

Figure 3.8: Bounded Motion of the Chaser Satellite

As explained by Djojodihardjo in his paper [41], the set of solutions to the CW

equations that define a family of bounded periodic trajectories of the chaser satel-

lite is given by x(t) = A0 sin(ωt+ α)

y(t) = 2A0 cos(ωt+ α) + C0

z(t) = B0 sin(ωt+ α)

where A0, B0andC0 are integration constants and α is the phase angle, all depend-

ing on the initial conditions. It can also be said that B0 = 2A0 and, for simplicity,

C0 = 0. in this way, the initial conditions on the position of the chaser satellite

are given by x0 = A0 sin(α)

y0 = 2A0 cos(α)

z0 = 2A0 sin(α)

55

Deriving the first system of equation, the velocities equations for a bounded orbit

of the chaser can be found, and they are given byx(t) = A0ω cos(ωt+ α)

y(t) = −2A0ω sin(ωt+ α)

z(t) = 2A0ω cos(ωt+ α)

that leads to the following initial conditions on the initial velocities:x0 = A0ω cos(α)

y0 = −2A0ω sin(α) = −2ωx0

z0 = 2A0ω cos(α)

In conclusion, the main assumptions made in this work are

• Circular orbit around the Earth

• No perturbations

• Periodic bounded motion of the chaser with respect to the target

In this project A0 = 0.5km, which corresponds to a maximum distance from the

target of 1 km.

3.4 Air Drag

As mentioned in section 2.2.4, air drag will be taken into account at the end of

the unperturbed simulations, in order to provide a comparison between disturbed

and undisturbed model, both for the software and the system itself. For this

reason, a model of the air drag, and how it has been included in the analysis, will

be presented.

The general equation of the drag is given by

D =1

2ρ v2 S CD (3.7)

56

where ρ is the atmosphere density, v the velocity, S the surface and CD the drag

coefficient. The density plays an important role in the drag: even if the spacecraft is

orbiting in space, there is a small amount of particles belonging to the atmosphere

that interfere with the spacecrafts. It is well known that the ISS has to adjust

regularly its orbit due to a decay caused by air drag. However, there is no direct

way to measure the air density as on the Earth, so there is the need to rely

on mathematical models. In their paper [1], Hatten and Russel improved the

robustness and smoothness of the already existing Harris-Priester atmospheric

density model. The output of their work is to provide a air density that is function

of the altitude and solar activity, with the minimum and maximum values that ρ

can assume, and it is shown in picture 3.9.

Figure 3.9: Hatten and Russel minimum and maximum density profiles [1]

For what concerns the velocity, the constant speed of the circular orbit where both

target and chaser are will be considered. As you will see from the results, in fact,

the difference in velocity between the target (that is following a perfect circular

orbit) and the chaser (that is orbiting around the target) is in the order of some

meters per second, three order of magnitude lower than the orbital velocity that

is around 7.5 kilometers per second.

Regarding the drag coefficient, the CD for a parallelepiped-shaped body depends

on the surface considered. The CubeSat considered has a front face (in the x

direction) that is squared and the lateral (y and z directions) that are rectangular,

that give the following values:

• CD Front = 1.05

• CD Lateral = 2.05

57

In order to represent the worst case scenario, the value that will be added to the

dynamics equations as perturbation corresponds to the maximum drag force that

can be generated on each surface. For the disturbance moment deriving from

air drag, the force described above will be considered as applied at the edges of

the body, generating the maximum moment possible on every axes. It has been

used this method because adding the surface as a function of time (the system is

rotating, so the face influenced by the drag is not constant) would have increased

enormously the computational effort.

58

Chapter 4

Optimal Docking Maneuvers

In this chapter the results of the optimization will be presented and discussed.

First, an overview of the hardware of the chaser CubeSat is done in order to define

the spacecraft on which the analysis are performed. Then, a quick evaluation of the

time used in the simulations is done, together with the explanation of the choices.

In order to compare the results, in fact, the different times for the under-actuated

and fully-actuated cases are kept constant for each simulation. Two different times

have been used because, as it was predictable, the under-actuated spacecraft take

much more time to perform the maneuver with respect to the fully-actuated one:

in this work the optimization of the total thrust is performed, but a reasonable

time has been considered. It could be in fact used a very large time for both cases,

reducing the total propulsion consumption, but this would have not led to real life

applications outputs with maneuvers lasting tens of days. The circular orbits on

which the target-chaser system is moving vary for each example: they go from an

altitude of 400 km up to 900 km, a range where most of LEO satellites are orbiting.

In this way, the performance of the system will be outlined in its most suitable

environment. For each altitude, all the results are presented and the resulting

orbits are displayed. Note that the reference frame where the results are plotted

is the LVLH of the target satellite, not the ECI: this means, together with the

assumption of absence of perturbations, that these results can be assumed valid

for every inclination of the circular orbit. Moreover, the target satellite has been

considered as it is performing an earth observation mission, so the only rotation he

59

is having is around its perpendicular-to-orbital-plane axis at the orbital angular

velocity, so that he is always pointing to the earth. This is the typical mission

of LEO spacecrafts: the most common are communications and science of the

earth. This means that the boundary condition on the relative angular velocities

indicate that the chaser satellite is rotating, whereas the target is performing only

its rotation; but since a relative dynamics problem is considered, this could also

represent an unstable rotating target satellite and a still chaser CubeSat at the

beginning of the maneuver.

4.1 Physical Properties of the CubeSat

Parallel to the exponential growth of the CubeSats in the last years, the market

of nanosatellites propulsion systems has developed different alternatives, the most

known of whom are:

• Hall Thrusters: thrust generated by creating and accelrating ionized gas

through magnetic and electrostatic fields.

• FEEP: Electrostatic propulsion system where a liquid metal is ionized with

an electric field and emitted at high velocities.

• Green Monopropellant Thrusters: Hydrazine (toxic) substituted with a

new propellant, AF-M315E (Hydroxylammonium Nitrate), which has higher

density and can provide a better specific impulse than hydrazine.

• ADN-Vacco System: ADN is another substitute of the hydrazine. Vacco

systems use this greener propellant: they are really reliable due to their

simple design.

The main characteristic needed for these systems are: specific impulse, maximum

thrust, weight and density of the propellant. For every satellite category, going

from CubeSats to huge spacecrafts, these numbers can vary a lot. In the following

table the data relative to nano-satellites propulsion systems is provided.

All these systems occupy one unit of a CubeSat, so the dimension of each one is

60

Thruster Isp [s] F [mN] Mass [kg] Power [W] Density [g cm-3]Hall 1500 10-80 1.2 200 0.005366

FEEP 9000 0.01-0.5 1 40 1.84-8GMT 225 50-500 1.5 20 1.84

ADN-Vacco 1000 10-100 1.8 15 1.81

Table 4.1: Thrusters Comparison

not a parameter for comparison. Looking at the power consumption, it can be seen

that the Hall system is the most demanding one, with five times or above the maxi-

mum power request of the others: since nano-satellites that have a limited amount

of power are dealt in this project, Hall thrusters are the less indicated. FEEP

has the highest specific impulse and the lowest mass, but it also has the lowest

maximum thrust: this is the drawback of this system, because if under-actuated

satellites are considered, it is not possible to rely on thrusters that provide a so

small amount of power. Any mission can be achieved with the FEEP, but with this

amount of energy the mission would take a lot of time. GMT and ADN, instead,

are very similar. They both utilize green propellant that substitute hydrazine, the

power demanding is similar, as for the mass, but the specific impulse and, mostly,

the force range are different. The higher specific impulse of the ADN could make

the choice obvious, but if consider the type of spacecraft that is the target of this

work, a wider propulsion power range (this will give more flexibility to the system)

and a save, even if small, in mass make the Green Monopropellant Thruster the

best available option.

For what concerns the attitude control, a set of reaction wheels will be respon-

sible of this aspect. Since the second aim of the mission, after the optimization

of the total thrust, is to limit the time of te maneuver, RWs need to be powerful

enough to reorient the satellite in a limited amount of time. The typical value for

the maximum torque provided by a CubeSat RW is in the order of 2 mN m, with

best performing systems that can achieve up to 20 mN m, which is the value used

in the simulations. Whether the smallest RWs needed a lot of time to reorient the

spacecraft, the biggest ones had great performances and where able to reduce the

time by an order of magnitude.

61

Table 4.2 presents all the physical properties of the chaser satellite that are

used in the simulations.

Dimensions [cm] 23x24x36Mass [kg] 24.42

Isp [s] 225Ix,Iy,Iz [kg m2] 0.1656/0.2671/0.2643

Max F [mN] 500Max T [Nm] 0.02

Table 4.2: Spacecraft Properties

4.2 Pareto Front

As said in the presentation of this chapter, the analysis will be done considering

a fixed time of maneuver for the under-actuated and fully-actuated case. In order

to decide what is the most reasonable time, the Pareto front of the cost function

value versus the maneuver time has been performed. What has been done is to

consider a maneuver with initial and final conditions equal to zero (this was done

in order to reduce the amount of time required by the simulation, that with this

method already took some days), except for the distance and the corresponding

velocity (A0 = 0.5) and to vary at each step the time (+10 s); the lowest time

was computed using PSOPT defining the endpoint cost function equal to the final

time; the reference orbit around the Earth has height 750 km, an intermediate

value that represent low earth orbits, the one dealt in this work. In Figure 4.1

the results of the iterations are displayed. As it could be expected, the lower the

time, the higher the cost function value, and vice versa. As it can bee seen from

the figures above, for lower time, the integrand cost function values do not follow

exactly the theoretical hyperbolic profile, because the solver finds it harder to solve

the problem. This is easier to seen, not only for low times, for the under-actuated

case: the presence of only one thruster, in fact, makes the system harder to be

controlled. Given these profiles, it can be seen, for the one thruster case, that the

results present a huge decrease in the cost function value until 2800 s, whereas,

62

(a) Under-Actuated (b) Fully-Actuated

Figure 4.1: Pareto Front

after that, the decay is really small and not a great difference is worth such an

increase in maneuvering time; the same can be said for 600 s in the fully actuated

case. For these reasons, the time used in the analysis are 600 seconds for the

fully-actuated case and 2800 seconds for the under-actuated.

4.3 Simulations’ Results - Unperturbed

This section is structured in the following way: for each case, the boundary

conditions are stated (these have been chosen with the aim of representing the

most various situations possible), then the results for both under-actuated and

fully actuated are shown. At the end of all the cases, a general outline of the

results is presented.

For all the next analysis some parameters have been kept constant during all the

simulations: these are are the physical properties of the controls (thrusters force

and RW torque). For the under-actuated case (left) and fully actuated (right)

63

0 ≤ Fx ≤ 500 mN

Fy = 0 mN

Fz = 0 mN

−0.02 ≤ T1 ≤ 0.02 Nm

−0.02 ≤ T2 ≤ 0.02 Nm

−0.02 ≤ T3 ≤ 0.02 Nm

−500 ≤ Fx ≤ 500 mN

−500 ≤ Fy ≤ 500 mN

−500 ≤ Fz ≤ 500 mN

−0.02 ≤ T1 ≤ 0.02 Nm

−0.02 ≤ T2 ≤ 0.02 Nm

−0.02 ≤ T3 ≤ 0.02 Nm

For the simulations, the number of nodes (120), the NLP tolerance (1e-5) the

ODE tolerance (1e-2) and the maximum number of iterations (10000) have been

kept constant. A larger number of nodes would for sure have produced a smoother

results, but due to the complexity of the problem and its differential equations the

value has not been increased. This resulted in not smooth profiles, especially for

the under-actuated where these nodes have to be distributed on a larger time span

(2800 s) with respect to the fully actuated case (600 s). Concerning the tolerances,

the ODE one is smaller than the NLP because putting a strong constraint on such

difficult equations would have required too much iterations and time. The results

shown in the picture have been produced using MATLAB because the built-in plot

software of PSOPT had problems in cutting axes and in resolution of the images.

64

4.3.1 Case 1: 400 km Circular Orbit

In this first case, the chaser and target satellites are orbiting on a 400 km circu-

lar orbit around the earth. The initial difference in the roll-pitch-yaw orientation

is -30°, 120° and 40°. The phase angle of the bounded orbit of the chaser is 0°whereas the initial relative angular velocity is 0.05 rad/s for each axis. As a recap,

all the boundary conditions of case 1 are listed in Table 4.3.

δx(0) 500 m δvx(0) 0 m/s δσ1(0) -0.1317 δω1(0) 0.05 rad/sδy(0) 0 m δvy(0) -1.1314 m/s δσ2(0) 0.5774 δω2(0) 0.05 rad/sδz(0) 1000 m δvz(0) 0 m/s δσ3(0) 0.1763 δω3(0) 0.05 rad/sδx(tf ) 0 m δvx(tf ) 0 m/s δσ1(tf ) 0 δω1(tf ) 0 rad/sδy(tf ) 0 m δvy(tf ) 0 m/s δσ1(tf ) 0 δω2(tf ) 0 rad/sδz(tf ) 0 m δvz(tf ) 0 m/s δσ1(tf ) 0 δω3(tf ) 0 rad/s

Table 4.3: Boundary Conditions Case 1

In Table 4.4 the numerical results of the simulation are presented, both for

the under-actuated and fully-actuated cases. Moreover, the average thrust force

during the maneuver has been computed and will be used later in this project

together with the other results.

Under-Actuated Fully-ActuatedtCPU [s] 37.38 31.19

J 2.650e-05 8.339e-06Favg [mN] 66.38 104

Table 4.4: Results Case 1

In the next pages the resulting orbits and profiles of states and controls for case 1

are displayed.

65

Figure 4.2: Case 1 - Orbits Underactuated

Figure 4.3: Case 1 - Orbits Fully Actuated

66

(a) Position Profiles Under-Actuated

(b) Position Profiles Fully-Actuated

Figure 4.4: Case 1 - Position

67

(a) Velocity Profiles Under-Actuated

(b) Velocity Profiles Fully-Actuated

Figure 4.5: Case 1 - Velocity

68

(a) Force Profile Under-Actuated

(b) Force Profiles Fully-Actuated

Figure 4.6: Case 1 - Force Profiles

69


In this second case, the chaser and target satellites are orbiting on a 500 km

circular orbit around the earth. The initial difference in the roll-pitch-yaw orien-

tation is 60°, -20° and 200°. The phase angle of the bounded orbit of the chaser

is 0° whereas the initial relative angular velocity is 0.04 rad/s for each axis. As a

recap, all the boundary conditions of case 1 are listed in Table 4.5.

δx(0) 500 m δvx(0) 0 m/s δσ1(0) 0.2679 δω1(0) 0.04 rad/sδy(0) 0 m δvy(0) -1.1068 m/s δσ2(0) -0.0875 δω2(0) 0.04 rad/sδz(0) 1000 m δvz(0) 0 m/s δσ3(0) 1.1918 δω3(0) 0.04 rad/sδx(tf ) 0 m δvx(tf ) 0 m/s δσ1(tf ) 0 δω1(tf ) 0 rad/sδy(tf ) 0 m δvy(tf ) 0 m/s δσ1(tf ) 0 δω2(tf ) 0 rad/sδz(tf ) 0 m δvz(tf ) 0 m/s δσ1(tf ) 0 δω3(tf ) 0 rad/s


In Table 4.6 the numerical results of the simulation are presented, both for

the under-actuated and fully-actuated cases. As for the previous case, the average

force of the whole maneuver has been calculated.


J 4.099e-05 7.102e-06Favg [mN] 76.69 95.58



are displayed.

70



71




72




73




74


In this third case, the chaser and target satellites are orbiting on a 600 km


tation is 0°, 0° and 180°. The phase angle of the bounded orbit of the chaser is 30°whereas the initial relative angular velocity is 0 rad/s for each axis. As a recap,

all the boundary conditions of case 1 are listed in Table 4.7.

δx(0) 433.01 m δvx(0) -0.2707 m/s δσ1(0) 0 δω1(0) 0 rad/sδy(0) -500 m δvy(0) -0.9379 m/s δσ2(0) 0 δω2(0) 0 rad/sδz(0) 866.02 m δvz(0) -0.5415 m/s δσ3(0) 1 δω3(0) 0 rad/sδx(tf ) 0 m δvx(tf ) 0 m/s δσ1(tf ) 0 δω1(tf ) 0 rad/sδy(tf ) 0 m δvy(tf ) 0 m/s δσ1(tf ) 0 δω2(tf ) 0 rad/sδz(tf ) 0 m δvz(tf ) 0 m/s δσ1(tf ) 0 δω3(tf ) 0 rad/s


In Table 4.8 the numerical results of the simulation are presented, both for the

under-actuated and fully-actuated cases. As for the previous cases, the average



J 1.581e-04 7.729e-06Favg [mN] 200.18 102.51



are displayed.

75



76




77




78




79


In this fourth case, the chaser and target satellites are orbiting on a 700 km


tation is -20°, -40° and 80°. The phase angle of the bounded orbit of the chaser

is 330° whereas the initial relative angular velocity is 0.02 rad/s for x-axis, 0.03

rad/s for y-axis and 0.01 rad/s for z-axis,. As a recap, all the boundary conditions

of case 1 are listed in Table 4.9.

δx(0) 433.01 m δvx(0) 0.2651 m/s δσ1(0) -0.0875 δω1(0) 0.02 rad/sδy(0) 500 m δvy(0) -0.9182 m/s δσ2(0) -0.1763 δω2(0) 0.03 rad/sδz(0) 866.02 m δvz(0) 0.5301 m/s δσ3(0) 0.3640 δω3(0) 0.01 rad/sδx(tf ) 0 m δvx(tf ) 0 m/s δσ1(tf ) 0 δω1(tf ) 0 rad/sδy(tf ) 0 m δvy(tf ) 0 m/s δσ1(tf ) 0 δω2(tf ) 0 rad/sδz(tf ) 0 m δvz(tf ) 0 m/s δσ1(tf ) 0 δω3(tf ) 0 rad/s






J 1.111e-05 5.635e-06Favg [mN] 48.77 85.35



are displayed.

80

Figure 4.17: Case 4 - Orbits Under-Actuated

Figure 4.18: Case 4 - Orbits Fully-Actuated

81




82




83




84




tation is 50°, 230° and 130°. The phase angle of the bounded orbit of the chaser

is 10° whereas the initial relative angular velocity is 0.05 rad/s for each axis. As a


δx(0) 492.4 m δvx(0) -0.09 m/s δσ1(0) 0.2217 δω1(0) 0.05 rad/sδy(0) -173.65 m δvy(0) -1.0224 m/s δσ2(0) 1.5697 δω2(0) 0.05 rad/sδz(0) 984.81 m δvz(0) -0.1803 m/s δσ3(0) 0.6371 δω3(0) 0.05 rad/sδx(tf ) 0 m δvx(tf ) 0 m/s δσ1(tf ) 0 δω1(tf ) 0 rad/sδy(tf ) 0 m δvy(tf ) 0 m/s δσ1(tf ) 0 δω2(tf ) 0 rad/sδz(tf ) 0 m δvz(tf ) 0 m/s δσ1(tf ) 0 δω3(tf ) 0 rad/s






J 6.754e-05 7.756e-06Favg [mN] 89.7 99.79



are displayed.

85



86




87




88




89




tation is 190°, 10° and 30°. The phase angle of the bounded orbit of the chaser is

230° whereas the initial relative angular velocity is 0.05 rad/s for each axis. As a


δx(0) -321.39 m δvx(0) 0.389 m/s δσ1(0) 1.0913 δω1(0) 0.05 rad/sδy(0) 766.04 m δvy(0) 0.654 m/s δσ2(0) 0.0437 δω2(0) 0.05 rad/sδz(0) 642.79 m δvz(0) 0.779 m/s δσ3(0) 0.1317 δω3(0) 0.05 rad/sδx(tf ) 0 m δvx(tf ) 0 m/s δσ1(tf ) 0 δω1(tf ) 0 rad/sδy(tf ) 0 m δvy(tf ) 0 m/s δσ1(tf ) 0 δω2(tf ) 0 rad/sδz(tf ) 0 m δvz(tf ) 0 m/s δσ1(tf ) 0 δω3(tf ) 0 rad/s






J 9.814e-06 8.625e-06Favg [mN] 42.88 106.34



are displayed.

90



91




92




93




94

4.4 Unperturbed Results’ Output

Altitude [km]Under-Actuated Fully-Actuated

tCPU [s] J Favg [mN] tCPU [s] J Favg [mN]

400 37.38 2.650e-05 66.38 31.19 8.339e-06 104

500 72.43 4.099e-05 76.69 20.19 7.102e-06 95.58

600 36.54 1.581e-04 200.18 22.89 7.729e-06 102.51

700 37.87 1.111e-05 48.77 24.48 5.635e-06 85.35

800 67.04 6.754e-05 89.7 37.15 7.756e-06 99.79

900 31.86 9.814e-06 42.88 20.19 8.625e-06 106.34

Average 47.19 5.234e-05 87.43 26.02 7.531e-06 98.93

Table 4.15: Unperturbed Resulsts’ Summary

From the results obtained, summarized in Table 4.15, it is possible to outline

the general behavior of the CubeSat during these missions. As can be seen from

all the resulting orbits, it is clear that for the fully-actuated cases the resulting

trajectory is more straight with respect to the under-actuated, due to the 3-axis

thrusters that can provide more control on all the directions. The single-thruster

CubeSat has a more rounded orbit, mostly due to the initial spiral velocity of the

bounded orbit around the target: this initial velocity, in fact, is difficult to control

having only a thruster in one direction and for this reason the path to the target

results to be longer.

It can can be noticed that the resulting cost functions have similar values (same

order of magnitude or one of difference) and do not change a lot on the different

orbits; the fact that the under-actuated cost function is always higher than the

fully-actuated one is caused by the choice of the maneuver time done in the “Pareto

Front”section, where comparable cost results but with different time were wanted.

For what concerns the average force during the maneuvers, the results outlined a

similar behavior for all the altitudes and for all the various initial condition, with

a small fluctuations of the values. Whereas the cost function is smaller for the

fully-actuated spacecraft, the average force is smaller for the under-actuated: this

outline that a saving in time and a more controllable system do not correspond to

95

a saving in the propulsion power. The six thrusters of the fully-actuated system

works at a lower average force values with respect to the one thruster CubeSat,

but combined all together they generate a higher amount of power.

The most power demanding maneuver, both for under-actuated and fully-actuated

case corresponds to case 3: in this simulation, in fact, the most difficult situation

for the chaser has been depicted, that at the beginning is oriented in the opposite

direction with respect to the target and has no initial angular velocity that could

have been used to easily re-orient the CubeSat.

What is important to extract from these results is the amount of fuel consumed for

each maneuver. As said before, Green Monopropellant Thrusters, that work with

AF-M315E (Hydroxylammonium Nitrate), which has a density of % = 1.84 g/cm3,

are considered. Being said that the results do not vary a lot in the simulations, an

average value of the forces for both the under and fully-actuated cases across the

LEO altitudes can be extracted, which are:

• Under-Actuated: F undavg = 87.43 mN

• Fully-Actuated: F fulavg = 98.93 mN

The specific impulse of the thruster is 225. From the definition of the specific

impulse

Isp =F

m g0

it is possible to retrieve the mass flux of the propellant, which is

• Under-Actuated: mund = 3.961e-5 kg/s

• Fully-Actuated: mful = 4.482e-5 kg/s

From the mass flux, given time and density of the propellant, it is straight forward

to retrieve the volume of fuel consumed for a single maneuver. This is given by

• Under-Actuated: Vund = 60.27 cm3

• Fully-Actuated: Vful = 14.61 cm3

So, even if the under-actuated satellites require less propulsion power, the higher

time for the maneuver cause an increase of fuel consumption with respect to the

96

fully-actuated one. As said previously in this project, the fully-actuated case was

used as a comparison and it is a more feasible systems on bigger spacecrafts, not

on CubeSats that have a limited amount of space.

The unit of a CubeSat has dimensions of 10× 10× 10 cm typically, equivalent to

a volume of 1000 cm3. If it is assumed, for the under-actuated case, that one unit

of a CubeSat is occupied by a tank of fuel, it could perform approximately 16 of

these maneuvers, whereas an hypothetical fully-actuated satellite could achieve 68

maneuvers with only one unit of fuel. The results, summarized in Table 4.16, show

an higher thrust rate for the fully-actuated case, but the time of the maneuver plays

a big role in the consumption: the one-thruster spacecraft, in fact, takes more than

four times the time taken by the six-thruster satellite to complete the mission.

Under-Actuated Fully-Actuated

Favg [mN] 87.43 98.93

Vman [cm3] 60.27 14.61

mman [g] 110.89 26.88

Maneuvers* 16 68

* Number of maneuvers fo 1U (1000 cm3) of CubeSat

dedicated to the fuel.

Table 4.16: Unperturbed Resulsts’ Summary - Maneuvers

97

4.5 Air Drag Effects

In this section, the effects caused by air drag (presented in section 3.4) will

be displayed for the simulations performed in the previous section of the under-

actuated case only. The only difference with those computations is the addition

of the translational and rotational disturbance due to the presence of atmospheric

particles.

4.5.1 Comparison 1: 400 km Circular Orbit

Unperturbed PerturbedtCPU [s] 37.38 23.07

J 2.650e-05 3.488e-04Favg [mN] 66.38 339.71

Table 4.17: Comparison Case 1

As can be seen from the resulting graphs below and from the results in the

table above, the problem regarding absolute and local optimal solution, outlined

in Section 3.2, emerges. Even if the atmospheric density is really low, this change

in the dynamics of the problem is big enough to make the software diverge from

the local solution found for the unperturbed case, leading to a completely different

result that do not represent the real behavior of the system affected by air drag.

This could also be understood by the fact that the computational time used by

the perturbed case is much lower than the unperturbed case: typically, in fact,

the opposite happens, because some terms are inserted in the already complicated

system of equations. Moreover, the average thrust used during the maneuver

present a huge difference that could no be addressed only to the air drag.

This drastic change in the results is not caused by a wrong choice of tolerances:

they have been in fact tested for multiple values, without any change in the results.

98

Figure 4.32: Case 1 - Relative Orbits Comparison

Figure 4.33: Case 1 - Relative Distances Comparison

99



J 4.099e-05 1.696e-05Favg [mN] 76.69 58.11


The same problem presented in the Comparison 1 seems to happen for case 2,

but this visible difference in the results could not be addressed to either a software

different local minimum or a direct effect of air drag.The local optimal solution

computed by the software is much closer to the unperturbed case with respect

to case 1. This is determined by the decrease of the atmospheric density due to

the increase of the altitude, and, subsequently, in the reduction of the disturbance

forces acting on the system. In this case, it seems like the drag is helping the

chaser to close the docking maneuver on the target. The reader has to remember

that the under-actuated CubeSat performs, in all the cases, a spiral-like maneuver

to dock to the target. This trajectory is forced by the initial bounded orbit that

the chaser has around the reference satellite, and it is hard for it to reduce it due

to the presence of one thruster only. The drag, in this case, could have helped the

CubeSat to reduce the amplitude of the spiral in absence of thrusters on the other

axes. In conclusion, there is no possibility to know if this result is close to the

real-life behavior or if it is caused by the choice of a different local optimization

point.

100



101



J 1.581e-04 1.796e-04Favg [mN] 200.18 217.7


Looking at Figure 4.36 it could seem that the same problem of Comparison 2

is outlined. Figure 4.37, instead, shows that the differences are not as explicit as

in the previous section, where it peaked with variations of 500 meters. Differently

from Comparison 2, now the CubeSat on the perturbed case needs more propulsive

power to perform the maneuver.


102




J 1.111e-05 1.076e-05Favg [mN] 48.77 44.82


At this altitude, the atmosphere is thinner and, consequently, the drag effects

are lower. As can be seen from the results, in fact, the orbits tend to coincide and

the numerical results of the propulsive power are more similar. As in Comparison

2, the drag seems to help the reduction of the spiral amplitude for the docking

maneuver.

103



104



J 6.754e-05 6.319e-05Favg [mN] 89.71 86.29


The presence of particles at 800 kilometers is almost naught, and this aspect is

reflected in the results presented in Figure 4.40 and 4.41. Now the orbits almost

coincide and the average thrust is practically the same of the unperturbed case.


105




J 9.814e-06 9.593e-06Favg [mN] 42.88 42.99


The same of Comparison 5 can be said for Case 6 at 900 kilometers.Here the

particles are even more rarefied, and almost no difference can be seen between the

perturbed and unperturbed case.

106



107

4.6 Perturbed Results Analysis

Altitude [km]Unperturbed Perturbed

tCPU [s] J Favg [mN] tCPU [s] J Favg [mN]

400 37.38 2.650e-05 66.38 23.07 3.488e-04 339.71

500 72.43 4.099e-05 76.69 52.59 1.696e-05 58.11

600 36.54 1.581e-04 200.18 49.53 1.796e-04 217.7

700 37.87 1.111e-05 48.77 62.12 1.076e-05 44.82

800 67.04 6.754e-05 89.7 82.46 6.319e-05 86.29

900 31.86 9.814e-06 42.88 37.62 9.593e-06 42.99

Average 47.19 5.234e-05 87.43 56.86* 5.602e-05 * 89.98*

* 400 km not considered because the software provided a non-comparable result.

Table 4.23: Air Drag Resulsts’ Summary for Under-Actuated Configuration

As it could be expected, the influence of air drag decreases as the altitude in-

creases. However, this could not be said for sure for the lower case considered (400

and 500 kilometers), because probably the software played a big role in changing

drastically the results. So, if it is possible to say that air drag is negligible for al-

titudes higher than 600 kilometers, this could not be said for the inferior, because

it is not possible to know if the software considered the same local minima for

both computations. Apart from Case 1, where no valid data can be extracted for

the perturbed case, in all the other situations air drag can be neglected without

any huge loss in the meaning of the results, as can be seen from the summary

of the results in Table 4.23, where the average force used in a perturbed LEO

environment differs of only 2 mN with respect to the unperturbed case. The com-

putational time, as it could be expected, is higher for the perturbed case, since

the disturbance terms have been added to the system’s equations.

It could also be noticed that, depending on the initial conditions, the drag either

helps or obstruct the chaser to reach the target. This translates in a decrease or

increase, respectively, of the total thrust power.

It has to be remembered that these computations are done for the worst case sce-

nario. The variation of the disturbance forces and moments was not a function of

108

the actual attitude of the spacecraft (that would have been the best representation

of the real-life behavior) because it would have complicated too much the problem,

with the software not being able of solving the cases.

109

Chapter 5

Conclusion

The main goal of this project was to optimize the total thrust, and, conse-

quently, the fuel consumption, of a 12U under-actuated CubeSat during a docking

maneuver with a spinning body orbiting around the Earth. This was carried out

using a pseudospectral optimal control method, which use polynomials to approx-

imate the differential equations and the cost function. What was shown in the

examples was the starting point of the project: the first step was to evaluate the

simplest problem possible (simple translational dynamics using the second law

of dynamics and reorientation of a rigid body using Euler equations).Then, the

complexity was increased incrementally, first by coupling the translational and

rotational parts and, after that, adding the relative dynamics. The results have

outlined that the controlled under-actuated CubeSat is able to perform docking

maneuvers in a limited amount of time and with a relevant saving in fuel. Typi-

cally the docking process takes a lot of time: the Space Shuttle, for example, took

around 40 minutes to dock when it was 150 meters far from the target, that is

approximately the same time needed by the Soyuz to dock with the ISS. Moreover,

the fuel needed to perform a single docking maneuver is around 150 kilograms for

the Soyuz module. These values are reasonable if the dimensions of the space-

crafts are taken into account: a maximum accuracy, in fact, is needed, because

even a small error can seriously damage the chaser and the target. From a great

accuracy derives a higher time for the maneuver, and, consequently, a greater fuel

consumption. Moreover, the mass of these spacecrafts requires a huge amount of

110

control power to perform the maneuvers, and this coincide with an increase in the

fuel consumption. This problems do not affect CubeSat systems. With their small

dimensions, in fact, they are easier to control and the required time and fuel drop

significantly, with maneuvers lasting around 40 minutes with 110 grams of fuel

consumed per maneuver for an under-actuated CubeSat 1 kilometer away from

the target. For a fully-actuated nanosatellite, time and fuel consumption further

reduce because the system is easier to control, with 10 minutes and 26.8 grams of

fuel burned per maneuver.

From the analyses performed, it can be seen how the fully actuated system is more

efficient with respect to the under-actuated, being easier to control. Tables 4.15

and 4.16 show that, even with a similar force profile during the maneuver, the

fully-actuated provides a huge saving in mass of fuel. Moreover, always for the

easiness of control of the system, the CPU time is lower, being the computational

time of the under-actuated case, however, relatively small. If we consider only the

under-actuated case, which is the most important for CubeSats applications, the

system can perform 16 maneuvers in LEO for each unit of CubeSat dedicated to

the fuel, that correspond to 1.84 kilograms of fuel, almost 10 of the total mass

of the spacecraft. Comparing the unperturbed with the perturbed environment,

it is clear that there is not a huge difference in the results, as it could be expected

being the maneuvers limited in time. The CubeSats, in fact, has a force profile

that is only 2 mN higher with respect to the unperturbed case. Moreover, the

resulting orbits are very similar, except for Case 1 where the software appeared to

have found a different local minimum. This can lead to neglecting air drag per-

turbation, as it was done in the first analyses: it in fact will not affect the general

results of multiple simulations.

5.1 Applications

This project was focused on CubeSats because they can represent a cheaper

solution for actual problems. The field of application of this work is very wide: it

is possible to think of CubeSats that docks to space debris, whether of compara-

ble dimensions or larger. The NASA Orbital Debris Quarterly News of July 2010

estimated 14 000 debris with dimensions greater than 10 cm in LEO, which also

111

includes dead satellites, rocket bodies or payloads. With a CubeSat it would be

possible to dock to these bodies and de-orbit them with their propulsion system.

This could take a lot or less time, depending on the dimensions of the target body.

Another application could be the re-fueling of in-orbit satellites that have to per-

form interplanetary missions or simply that need fuel to end their mission or for

other purposes. If it is considered that two units would be dedicated to the propul-

sion system and the fuel tank, two to the reaction wheels and two to the electronic

and power system, the CubeSat would have 6 liters of remaining space where to

store the target fuel.

These are the main operations that can be done after this study, but of course

there are a lot of applications for space missions.

5.2 Future Works

This is only the first step of a very wide area of work. The main aspect that

can be noticed in this work is that, at this stage, the state of the target must

be known a priori. Once it is known, the optimal maneuver can be computed on

Earth and upload to the on-board system of the CubeSat with a passage on the

ground station or the states can be uploaded from ground to the nanosatellite,

that will perform the computations needed for the maneuver. Thus, the next step

to improve this study and to create a system that is able to provide a real time

attitude estimation of the target would be the tracking of the target body and

the determination of its attitude and position. Scanning Laser Radar [42] allows

the attitude determination of a non-cooperative target and its line-of-sight range

thanks to a laser with a field of view of 30° × 30° and a range of approximately

55 kilometers. Once the attitude and position of the target is determined by

the chaser, the translational and angular velocities can be determined thanks to

multiple measurements, knowing the attitude and position of the chaser. When

the initial conditions are known, the algorithm, already uploaded on the on-board

system of the CubeSat, could perform the computations needed to provide the

thrust and torque profiles of the control systems to perform the optimal docking

maneuver.

Since a docking problem is considered, one work could be focused on the berthing

112

mechanism, in particular how to attach to the target body and the design of

a universal interface for CubeSats and targets, whether re-fueling system, data

exchange or simple physical docking are evaluated.

The simulations performed during this work, as explained in the previous chapter,

used a limited amount of nodes due to the limited computational power of the

machine used: for the same reason, and also because they could be neglected

without any big difference in the results, perturbations were not included in the

dynamics equations. These analysis could be done on very powerful machines

that could provide better results. Note that these results are not far from the

real behavior of the system: they differ because the method used for solving the

equations approximate them using polynomials but the suggestion is to provide

better results and, with a more powerful systems, more simulations from where

significant data could be extracted.

Moreover, only the circular case have been considered in this work, and typically

a lot of satellites have elliptic orbits. That is why the next work could be the

extension of this algorithm to the elliptical case. This will involve a change in the

Clohessy-Wiltshire equations that, as used in this thesis, were based on circular

trajectories.

The quality of the results can be further increased with a better model of air drag,

that takes into account the attitude of the spacecraft. This will translate in a drag

force/torque that will be a function of the time and attitude, increasing by a lot

the computational effort.

As could be seen from the results, Case 1 of the under-actuated case, perturbed

environment, presented a strange behavior. It has to be remembered, in fact, that

the software doe not grant that the solution we have found is the absolute optimal,

but more likely it is a local minimum. For the 400 kilometers case, the presence of

the perturbation, even if small, seems to drive the software to a different minimum

with respect to the one of the unperturbed case, generating a completely different

solution. Thus, this work can be improved with the use of different softwares, in

order to provide more results and compare them to see what the global optimal

solution could be.

113

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v

Appendix A

The code used in this work is provided down below for anyone who may need

it. The notation ????? must be substituted with the user’s values.

#include ”psopt . h”

// ////////////////////////////////////////////////////////////////

// ///////// Define the end p o i n t ( Mayer ) c o s t f u n c t i o n //////////

// ////////////////////////////////////////////////////////////////

adouble endpo in t co s t ( adouble ∗ i n i t i a l s t a t e s , adouble ∗ f i n a l s t a t e s ,

adouble ∗ parameters , adouble& t0 , adouble& t f ,


return 0 . 0 ;

// //////////////////////////////////////////////////////////////

// /////// Define the i n te gran d ( Lagrange ) c o s t f u n c t i o n //////

// //////////////////////////////////////////////////////////////




adouble Fx = con t r o l s [ CINDEX(1) ] ; ///CONTROL THRUST///////

adouble Fy = con t r o l s [ CINDEX(2) ] ; ///CONTROL THRUST///////

adouble Fz = con t r o l s [ CINDEX(3) ] ; ///CONTROL THRUST///////

double T = ????? ;

vi

return T∗(Fx∗Fx+Fy∗Fy+Fz∗Fz ) ; //COST FUNCTION//

// //////////////////////////////////

// ////// Define the DAE’ s /////////

// //////////////////////////////////




double T = ????? ; // time reparametr i zed

adouble Fx = con t r o l s [ CINDEX(1) ] ; // //////////////////////

adouble Fy = con t r o l s [ CINDEX(2) ] ; ///CONTROL THRUSTS//////

adouble Fz = con t r o l s [ CINDEX(3) ] ; // //////////////////////

adouble T1 = con t r o l s [ CINDEX(4) ] ; // /////////////////////

adouble T2 = con t r o l s [ CINDEX(5) ] ; ////CONTROL TORQUES////

adouble T3 = con t r o l s [ CINDEX(6) ] ; // /////////////////////

adouble dx = s t a t e s [ CINDEX(1) ] ; // ///////////////////

adouble dy = s t a t e s [ CINDEX(2) ] ; /////POSITION////////

adouble dz = s t a t e s [ CINDEX(3) ] ; // ///////////////////

adouble dxdot = s t a t e s [ CINDEX(4) ] ; // /////////////////////

adouble dydot = s t a t e s [ CINDEX(5) ] ; // ////VELOCITY/////////

adouble dzdot = s t a t e s [ CINDEX(6) ] ; // /////////////////////

adouble dP1 = s t a t e s [ CINDEX(7) ] ; // ///////

adouble dP2 = s t a t e s [ CINDEX(8) ] ; ///MRP///

adouble dP3 = s t a t e s [ CINDEX(9) ] ; // ///////

adouble dom1 = s t a t e s [ CINDEX(10) ] ; // //////////////////////

adouble dom2 = s t a t e s [ CINDEX(11) ] ; ///ANGULAR VELOCITIES///

vii

adouble dom3 = s t a t e s [ CINDEX(12) ] ; // //////////////////////

double I1 = ????? ; //PRINCIPAL INERTIA//

double I2 = ????? ; // /////OF THE////////

double I3 = ????? ; // /////CHASER////////

double m = ????? ; //MASS OF THE CHASER//

double mu = ????? ; //STANDARD GRAVITATIONAL PARAMETER OF THE

PLANET CONSIDERED//

double R = ????? ; //RADIUS OF THE PLANET CONSIDERED//

double h = ????? ; //HEIGHT OF THE CIRCULAR ORBIT//

double omega = sq r t (mu/( (R+h )∗ (R+h )∗ (R+h ) ) ) ; //ANGULAR VELOCITY

OF THE ORBIT//

//DIRECTION COSINE MATRIX//

adouble dPsquare = dP1∗dP1+dP2∗dP2+dP3∗dP3 ;adouble A = dom1∗dP1 + dom2∗dP2 + dom3∗dP3 ;adouble P3orb = tan (omega∗ time / 4 . 0 ) ;

adouble den = 1.0/( (1 .0+dP1∗dP1+dP2∗dP2+dP3∗dP3)∗(1 .0+dP1∗dP1+dP2∗dP2+dP3∗dP3 ) ) ;

adouble sigma = 1.0−dP1∗dP1+dP2∗dP2+dP3∗dP3 ;adouble a11 = den ∗( sigma∗ sigma+4.0∗(dP1∗dP1−dP2∗dP2−dP3∗dP3 ) ) ;adouble a12 = den ∗ (8 . 0∗dP1∗dP2−4.0∗dP3∗ sigma ) ;

adouble a13 = den ∗ (8 . 0∗dP1∗dP3+4.0∗dP2∗ sigma ) ;


adouble a22 = den ∗( sigma∗ sigma+4.0∗(−dP1∗dP1+dP2∗dP2−dP3∗dP3 ) ) ;adouble a23 = den ∗ (8 . 0∗dP3∗dP2−4.0∗dP1∗ sigma ) ;


adouble a32 = den ∗ (8 . 0∗dP3∗dP2−4.0∗dP1∗ sigma ) ;

adouble a33 = den ∗( sigma∗ sigma+4.0∗(−dP1∗dP1−dP2∗dP2+dP3∗dP3 ) ) ;

double omegat1 = ????? ; // ////////////////////////////

double omegat2 = ????? ; //ANGULAR RATE OF THE TARGET//

double omegat3 = ????? ; // ////////////////////////////

// //////DERIVATIVES OF THE STATES///////////

viii

d e r i v a t i v e s [ CINDEX(1) ] = T∗( dxdot ) ;d e r i v a t i v e s [ CINDEX(2) ] = T∗( dydot ) ;d e r i v a t i v e s [ CINDEX(3) ] = T∗( dzdot ) ;d e r i v a t i v e s [ CINDEX(4) ] = T∗ ( ( (Fx∗a11+Fy∗a12+Fz∗a13 )/m)+

2.0∗ omega∗dydot+3.0∗omega∗omega∗dx ) ;d e r i v a t i v e s [ CINDEX(5) ] = T∗ ( ( (Fx∗a21+Fy∗a22+Fz∗a23 )/m)−

2 .0∗ omega∗dxdot ) ;d e r i v a t i v e s [ CINDEX(6) ] = T∗ ( ( (Fx∗a31+Fy∗a32+Fz∗a33 )/m)−

omega∗omega∗dz ) ;d e r i v a t i v e s [ CINDEX(7) ] = T∗ ( 0 . 25∗ ( dP2∗ (2 . 0∗dom3+4.0∗omegat3)−

dP3∗ (2 . 0∗dom2+4.0∗omegat2 )+(1.0−dPsquare )∗dom1)+

0 . 5∗ (A∗dP1 ) ) ;d e r i v a t i v e s [ CINDEX(8) ] = T∗ ( 0 . 25∗ ( dP1∗(−2.0∗dom3−4.0∗omegat3)+


0 . 5∗ (A∗dP2 ) ) ;d e r i v a t i v e s [ CINDEX(9) ] = T∗ ( 0 . 25∗ ( dP1∗ (2 . 0∗dom2+4.0∗omegat2)−


0 . 5∗ (A∗dP3 ) ) ;d e r i v a t i v e s [ CINDEX(10) ] = T∗ ( ( ( (T1∗a11+T2∗a12+T3∗a13 )/ I1)−

( ( I3−I2 )/ I1 )∗dom2∗dom3 ) ) ;

d e r i v a t i v e s [ CINDEX(11) ] = T∗ ( ( ( (T1∗a21+T2∗a22+T3∗a23 )/ I2)−( ( I1−I3 )/ I2 )∗dom1∗dom3 ) ) ;

d e r i v a t i v e s [ CINDEX(12) ] = T∗ ( ( ( (T1∗a31+T2∗a32+T3∗a33 )/ I3)−( ( I2−I1 )/ I3 )∗dom1∗dom2 ) ) ;

// //////////////////////////////////////////////////////////////////

// ///////////// Define the e v e n t s f u n c t i o n ////////////////////////

// //////////////////////////////////////////////////////////////////

void events ( adouble ∗ e , adouble ∗ i n i t i a l s t a t e s , adouble ∗ f i n a l s t a t e s ,

adouble ∗ parameters , adouble& t0 , adouble& t f , adouble ∗ xad ,

int iphase , Workspace∗ workspace )

adouble dxi = i n i t i a l s t a t e s [ CINDEX(1) ] ;

adouble dyi = i n i t i a l s t a t e s [ CINDEX(2) ] ;

adouble dz i = i n i t i a l s t a t e s [ CINDEX(3) ] ;

ix

adouble dvxi = i n i t i a l s t a t e s [ CINDEX(4) ] ;

adouble dvyi = i n i t i a l s t a t e s [ CINDEX(5) ] ;

adouble dvz i = i n i t i a l s t a t e s [ CINDEX(6) ] ;

adouble dP1i = i n i t i a l s t a t e s [ CINDEX(7) ] ;



adouble dom1i = i n i t i a l s t a t e s [ CINDEX(10) ] ;



adouble dxf = f i n a l s t a t e s [ CINDEX(1) ] ;

adouble dyf = f i n a l s t a t e s [ CINDEX(2) ] ;

adouble dz f = f i n a l s t a t e s [ CINDEX(3) ] ;

adouble dvxf = f i n a l s t a t e s [ CINDEX(4) ] ;

adouble dvyf = f i n a l s t a t e s [ CINDEX(5) ] ;

adouble dvzf = f i n a l s t a t e s [ CINDEX(6) ] ;

adouble dP1f = f i n a l s t a t e s [ CINDEX(7) ] ;



adouble dom1f = f i n a l s t a t e s [ CINDEX(10) ] ;



//INITIAL AND FINAL CONDITIONS//

e [ CINDEX(1) ] = dxi ;

e [ CINDEX(2) ] = dyi ;

e [ CINDEX(3) ] = dz i ;

e [ CINDEX(4) ] = dvxi ;

e [ CINDEX(5) ] = dvyi ;

e [ CINDEX(6) ] = dvz i ;

e [ CINDEX(7) ] = dxf ;

e [ CINDEX(8) ] = dyf ;

e [ CINDEX(9) ] = dz f ;

e [ CINDEX(10) ] = dvxf ;

e [ CINDEX(11) ] = dvyf ;

e [ CINDEX(12) ] = dvzf ;

e [ CINDEX(13) ] = dP1i ;



x

e [ CINDEX(16) ] = dom1i ;



e [ CINDEX(19) ] = dP1f ;



e [ CINDEX(22) ] = dom1f ;



// /////////////////////////////////////////////////////////////////

// //////////// Define the phase l i n k a g e s f u n c t i o n ////////////////

// /////////////////////////////////////////////////////////////////

void l i n k ag e s ( adouble ∗ l i nkage s , adouble ∗ xad , Workspace∗ workspace )

// S i n g l e phase problem

// ////////////////////////////////////////////////////////////////

// //////////// Define the main r o u t i n e //////////////////////////

// ////////////////////////////////////////////////////////////////

int main (void )

// /////////////////////////////////////////////////////////////

// ///////////////// Declare key s t r u c t u r e s ///////////////////

// /////////////////////////////////////////////////////////////

Alg algor i thm ;

Sol s o l u t i o n ;

xi

Prob problem ;

// /////////////////////////////////////////////////////////////

// ///////////////// R e g i s t e r problem name ///////////////////

// /////////////////////////////////////////////////////////////

problem . name = ”Rodrigues Parameters ” ;

problem . out f i l ename = ”Solution6T −6. txt ” ;

// //////////////////////////////////////////////////////////////////

// ////// Define problem l e v e l c o n s t a n t s & do l e v e l 1 se tup ////////

// //////////////////////////////////////////////////////////////////

problem . nphases = 1 ;

problem . n l i nkage s = 0 ;

p s op t l e v e l 1 s e t up ( problem ) ;

// ////////////////////////////////////////////////////////////////

//// Define phase r e l a t e d in format ion & do l e v e l 2 se tup //////

// ////////////////////////////////////////////////////////////////

problem . phases ( 1 ) . n s t a t e s = 12 ;

problem . phases ( 1 ) . n con t r o l s = 6 ;

problem . phases ( 1 ) . nevents = 24 ;

problem . phases ( 1 ) . npath = 0 ;

problem . phases ( 1 ) . nodes = ” [ 1 2 0 ] ” ;

p s op t l e v e l 2 s e t up ( problem , a lgor i thm ) ;

// //////////////////////////////////////////////////////////////

// /////////// Enter problem bounds in format ion ////////////////

// //////////////////////////////////////////////////////////////

double mu = ????? ;

double R = ????? ;

xii

double h = ????? ;

double omega = sq r t (mu/( (R+h )∗ (R+h )∗ (R+h ) ) ) ;

double A0 ????? ; //INITIAL CONSTANT THAT DETERMINES INITIAL POSITION

AND VELOCITY ( see below ) //

double alpha = ????? ; //PHASE ANGLE OF THE RELATIVE CW ORBIT//

double dx0 = A0∗ cos ( alpha ) ; // ///////////////////////////////////

double dy0 = −2.0∗A0∗ s i n ( alpha ) ; //INITIAL CONDITIONS THAT CORRESPONDS

double dz0 = 2.0∗A0∗ cos ( alpha ) ; ///TO A CHASER SATELLITE ORBITING////

double du0 = −A0∗omega∗ s i n ( alpha ) ; ///AROUND A FIXED TARGET ONE//////

double dv0 = −2.0∗omega∗dx0 ; // //////////////////////////////////////

double dw0 = −2.0∗A0∗omega∗ s i n ( alpha ) ; // ///////////////////////////

double dxe = 0 . 0 ; // //////////////////////////////////

double dye = 0 . 0 ; ///END CONDITIONS THAT CORRESPONDS//

double dze = 0 . 0 ; ///TO DOCKING //////////////////////

double due = 0 . 0 ; // //////////////////////////////////

double dve = 0 . 0 ; // //////////////////////////////////

double dwe = 0 . 0 ; // //////////////////////////////////

double d r o l l 0 = ????? ; // //////////////////////////////

double dpitch0 = ????? ; // //////////////////////////////

double dyaw0 = ????? ; ////INITIAL AND FINAL DIFFERENCE///

double d r o l l e = ????? ; //IN THE RELATIVE ATTITUDE//

double dpi tche = ????? ; ////OF THE TWO SPACECRAFTS,//

double dyawe = ????? ; ////FIRSTE EXPRESSED IN //

double dP10 = tan ( d r o l l 0 / 4 . 0 ) ; //TERMS OF EULRANGLES

double dP20 = tan ( dpitch0 / 4 . 0 ) ; //AND THEN TRANSFORMED INTO/

double dP30 = tan (dyaw0 / 4 . 0 ) ; //INTO MODIFIED RODRIGUES//

double dP1e = tan ( d r o l l e / 4 . 0 ) ; // ////PARAMETERS//////

double dP2e = tan ( dp i tche / 4 . 0 ) ; // /////////////////////////

double dP3e = tan (dyawe / 4 . 0 ) ; // //////////////////////

double dom10 = ????? ; // ///////////////////////////////////

double dom20 = ????? ; //INITIAL DIFFERENCE IN THE ANGULAR/

double dom30 = ????? ; // //////////////RATE////////////////

double dom1e = ????? ; // //////////////////////////////////////

xiii

double dom2e = ????? ; //FINAL DIFFERENCE IN THE ANGULAR RATE//

double dom3e = ????? ; // //////////////////////////////////////

double l ower l im i t om = ????? ; //LOWER AND UPPER LIMITS OF THE//

double upper l imit om = ????? ; // ////ANGULAR VELOCITIES////

double l owe r l im i t rw = ????? ; //LOWER AND UPPER PHYSICAL//

double uppe r l im i t rw = ????? ; //LIMITS OF THE REACTION WHEELS//

double l owe r l im i t Fx = ????? ; //LOWER AND UPPER PHYSICAL//

double upper l im i t Fx = ????? ; //LIMITS OF THE THRUSTERS x ///

double l owe r l im i t Fy = ????? ; //LOWER AND UPPER PHYSICAL//

double upper l im i t Fy = ????? ; //LIMITS OF THE THRUSTERS y///

double l owe r l im i t Fz = ????? ; //LOWER AND UPPER PHYSICAL//

double uppe r l im i t Fz = ????? ; //LIMITS OF THE THRUSTERS z ///










problem . phases ( 1 ) . bounds . lower . s t a t e s (10) = lower l im i t om ;










xiv



problem . phases ( 1 ) . bounds . upper . s t a t e s (10) = upper l imit om ;



problem . phases ( 1 ) . bounds . lower . c on t r o l s (1 ) = lowe r l im i t Fx ;

problem . phases ( 1 ) . bounds . lower . c on t r o l s (2 ) = lowe r l im i t Fy ;

problem . phases ( 1 ) . bounds . lower . c on t r o l s (3 ) = l owe r l im i t Fz ;

problem . phases ( 1 ) . bounds . lower . c on t r o l s (4 ) = l owe r l im i t rw ;



problem . phases ( 1 ) . bounds . upper . c on t r o l s (1 ) = upper l im i t Fx ;

problem . phases ( 1 ) . bounds . upper . c on t r o l s (2 ) = upper l im i t Fy ;

problem . phases ( 1 ) . bounds . upper . c on t r o l s (3 ) = uppe r l im i t Fz ;

problem . phases ( 1 ) . bounds . upper . c on t r o l s (4 ) = upper l im i t rw ;



problem . phases ( 1 ) . bounds . lower . events (1 ) = dx0 ;

problem . phases ( 1 ) . bounds . lower . events (2 ) = dy0 ;

problem . phases ( 1 ) . bounds . lower . events (3 ) = dz0 ;

problem . phases ( 1 ) . bounds . lower . events (4 ) = du0 ;

problem . phases ( 1 ) . bounds . lower . events (5 ) = dv0 ;

problem . phases ( 1 ) . bounds . lower . events (6 ) = dw0 ;

problem . phases ( 1 ) . bounds . lower . events (7 ) = dxe ;

problem . phases ( 1 ) . bounds . lower . events (8 ) = dye ;

problem . phases ( 1 ) . bounds . lower . events (9 ) = dze ;

problem . phases ( 1 ) . bounds . lower . events (10) = due ;

problem . phases ( 1 ) . bounds . lower . events (11) = dve ;

problem . phases ( 1 ) . bounds . lower . events (12) = dwe ;

problem . phases ( 1 ) . bounds . lower . events (13) = dP10 ;



problem . phases ( 1 ) . bounds . lower . events (16) = dom10 ;



xv

problem . phases ( 1 ) . bounds . lower . events (19) = dP1e ;



problem . phases ( 1 ) . bounds . lower . events (22) = dom1e ;



problem . phases ( 1 ) . bounds . upper . events (1 ) = dx0 ;

problem . phases ( 1 ) . bounds . upper . events (2 ) = dy0 ;

problem . phases ( 1 ) . bounds . upper . events (3 ) = dz0 ;

problem . phases ( 1 ) . bounds . upper . events (4 ) = du0 ;

problem . phases ( 1 ) . bounds . upper . events (5 ) = dv0 ;

problem . phases ( 1 ) . bounds . upper . events (6 ) = dw0 ;

problem . phases ( 1 ) . bounds . upper . events (7 ) = dxe ;

problem . phases ( 1 ) . bounds . upper . events (8 ) = dye ;

problem . phases ( 1 ) . bounds . upper . events (9 ) = dze ;

problem . phases ( 1 ) . bounds . upper . events (10) = due ;

problem . phases ( 1 ) . bounds . upper . events (11) = dve ;

problem . phases ( 1 ) . bounds . upper . events (12) = dwe ;

problem . phases ( 1 ) . bounds . upper . events (13) = dP10 ;



problem . phases ( 1 ) . bounds . upper . events (16) = dom10 ;



problem . phases ( 1 ) . bounds . upper . events (19) = dP1e ;



problem . phases ( 1 ) . bounds . upper . events (22) = dom1e ;



problem . phases ( 1 ) . bounds . lower . StartTime = 0 . 0 ;

problem . phases ( 1 ) . bounds . upper . StartTime = 0 . 0 ;

problem . phases ( 1 ) . bounds . lower . EndTime = 1 . 0 ;

problem . phases ( 1 ) . bounds . upper . EndTime = 1 . 0 ;

xvi

// /////////////////////////////////////////////////////////

// //// R e g i s t e r problem f u n c t i o n s ///////////////////////

// /////////////////////////////////////////////////////////

problem . i n t e g r and co s t = &in t e g r and co s t ;

problem . endpo in t co s t = &endpo in t co s t ;

problem . dae = &dae ;

problem . events = &events ;

problem . l i n kag e s = &l i nkag e s ;

// /////////////////////////////////////////////////////////

// //// Define & r e g i s t e r i n i t i a l guess ///////////////////

// /////////////////////////////////////////////////////////

int nnodes = problem . phases ( 1 ) . nodes ( 1 ) ;

int ncon t r o l s = problem . phases ( 1 ) . n con t r o l s ;

int ns t a t e s = problem . phases ( 1 ) . n s t a t e s ;

DMatrix s t a t e gu e s s = ze ro s ( ns tate s , nnodes ) ;

DMatrix c on t r o l g u e s s = ze ro s ( ncontro l s , nnodes ) ;

DMatrix t ime gues s = l i n s p a c e ( 0 . 0 , 1 . 0 , nnodes ) ;

s t a t e gu e s s (1 , co lon ( ) ) = l i n s p a c e ( dx0 , dxe , nnodes ) ;

s t a t e gu e s s (2 , co lon ( ) ) = l i n s p a c e ( dy0 , dye , nnodes ) ;

s t a t e gu e s s (3 , co lon ( ) ) = l i n s p a c e ( dz0 , dze , nnodes ) ;

s t a t e gu e s s (4 , co lon ( ) ) = l i n s p a c e (du0 , due , nnodes ) ;

s t a t e gu e s s (5 , co lon ( ) ) = l i n s p a c e ( dv0 , dve , nnodes ) ;

s t a t e gu e s s (6 , co lon ( ) ) = l i n s p a c e (dw0 , dwe , nnodes ) ;

s t a t e gu e s s (7 , co lon ( ) ) = l i n s p a c e ( dP10 , dP1e , nnodes ) ;



s t a t e gu e s s (10 , co lon ( ) ) = l i n s p a c e ( dom10 , dom1e , nnodes ) ;



c on t r o l g u e s s (1 , co lon ( ) ) = l i n s p a c e ( l ower l im i t Fx ,

xvii

upper l imit Fx , nnodes ) ;

c on t r o l g u e s s (2 , co lon ( ) ) = l i n s p a c e ( l ower l im i t Fy , upper l imit Fy ,

nnodes ) ;

c on t r o l g u e s s (3 , co lon ( ) ) = l i n s p a c e ( l owe r l im i t Fz , upper l imi t Fz ,

nnodes ) ;

c on t r o l g u e s s (4 , co lon ( ) ) = l i n s p a c e ( l owe r l im i t rw , upper l imi t rw ,

nnodes ) ;


nnodes ) ;


nnodes ) ;

problem . phases ( 1 ) . guess . s t a t e s = s t a t e gu e s s ;

problem . phases ( 1 ) . guess . c on t r o l s = con t r o l g u e s s ;

problem . phases ( 1 ) . guess . time = t ime gues s ;

// /////////////////////////////////////////////////////

///// Enter a l gor i thm o p t i o n s ///////////////////////

// /////////////////////////////////////////////////////

a lgor i thm . n lp i t e r max = ????? ;

a lgor i thm . n l p t o l e r a n c e = ????? ;

a lgor i thm . nlp method = ”IPOPT” ;

a lgor i thm . s c a l i n g = ”automatic ” ;

a lgor i thm . d e r i v a t i v e s = ”automatic ” ;

a lgor i thm . co l l o ca t i on method = ” t r ap e z o i d a l ” ;

a lgor i thm . mesh ref inement = ”automatic ” ;

a lgor i thm . ode t o l e r anc e = ????? ;

// ///////////////////////////////////////////////////////

// /////// Now c a l l PSOPT to s o l v e the problem ////////

// ///////////////////////////////////////////////////////

psopt ( so lu t i on , problem , a lgor i thm ) ;

xviii

// ///////////////////////////////////////////////////////

// Extrac t r e l e v a n t v a r i a b l e s from s o l u t i o n s t r u c t u r e ///

// ///////////////////////////////////////////////////////

DMatrix s t a t e s , c on t ro l s , t , pos , ve l , x , y , z , F , om,

P, u ;

s t a t e s = s o l u t i o n . g e t s t a t e s i n p h a s e ( 1 ) ;

c on t r o l s = s o l u t i o n . g e t c o n t r o l s i n ph a s e ( 1 ) ;

t = s o l u t i o n . g e t t ime in pha s e ( 1 ) ;

pos = s t a t e s ( co lon (1 , 3 ) , co lon ( ) ) ;

x = pos (1 , co lon ( ) ) ;

y = pos (2 , co lon ( ) ) ;

z = pos (3 , co lon ( ) ) ;

v e l = s t a t e s ( co lon (4 , 6 ) , co lon ( ) ) ;

F = con t r o l s ( co lon (1 , 3 ) , co lon ( ) ) ;

om = s t a t e s ( co lon (10 , 12 ) , co lon ( ) ) ;

P = s t a t e s ( co lon (7 , 9 ) , co lon ( ) ) ;

u = con t r o l s ( co lon (4 , 6 ) , co lon ( ) ) ;

// ////////////////////////////////////////////////////////

// ////// Save s o l u t i o n data to f i l e s i f d e s i r e d /////////

// ////////////////////////////////////////////////////////

s t a t e s . Save ( ” s t a t e s . dat” ) ;

c on t r o l s . Save ( ” c on t r o l s . dat” ) ;

t . Save ( ” t . dat” ) ;

// //////////////////////////////////////////////////////////

// ////// Plo t some r e s u l t s i f d e s i r e d ( r e q u i r e s g n u p l o t ) //

// //////////////////////////////////////////////////////////

double T = ????? ;

p lo t3 (x∗1000 , y∗1000 , z ∗1000 , ” Tra jec tory ” , ”x (m) ” , ”y (m) ” ,

”z (m) ” ) ;

mu l t ip l o t (T∗ t , v e l ∗1000 , ” Ve loc i ty ” , ” time [ s ] ” , ”dvx (m/ s )

dvy (m/ s ) dvz (m/ s ) ” , ”dvx dvy dvz” ) ;

xix

mul t ip l o t (T∗ t ,F∗1000000 , ”Contro l Thrust ” , ” time [ s ] ” , ”Fx [mN]

Fy [mN] Fz [mN] ” , ”Fx Fy Fz” ) ;

mu l t ip l o t (T∗ t ,P, ”Rodrigues Parameters ” , ” time [ s ] ” , ”dP1 dP2

dP3” , ”dP1 dP2 dP3” ) ;

mu l t ip l o t (T∗ t , om, ”Angular V e l o c i t i e s ” , ” time [ s ] ” , ”dom1 dom2

dom3” , ”dom1 dom2 dom3” ) ;

mu l t ip l o t (T∗ t , u , ”Control Torques ” , ” time [ s ] ” , ”T1 T2 T3” ,

”T1 T2 T3” ) ;

p lo t3 (x∗1000 , y∗1000 , z ∗1000 , ” Tra jec tory ” , ”x [m] ” , ”y [m] ” ,

”z [m] ” , ”png” , ” Tra jec tory . png” ) ;

mu l t ip l o t (T∗ t , pos ∗1000 , ” Pos i t i on ” , ” time [ s ] ” , ”dx [m] dy [m]

dz [m] ” , ”dx dy dz” ,3 ,1 , ”png” , ” Po s i t i on s . png” ) ;

mu l t ip l o t (T∗ t , v e l ∗1000 , ” Ve loc i ty ” , ” time [ s ] ” , ”dvx [m/ s ]

dvy [m/ s ] dvz [m/ s ] ” , ”dvx dvy dvz” ,3 , 1 , ”png” , ” V e l o c i t i e s . png” ) ;

mu l t ip l o t (T∗ t ,F∗1000000 , ”Contro l Thrust ” , ” time [ s ] ” , ”Fx [mN]

Fy [mN] Fz [mN] ” , ”Fx [mN] Fy [mN] Fz [mN] ” ,3 , 1 , ”png” ,

”Contro l Thrust . png” ) ;

mu l t ip l o t (T∗ t ,P, ”Rodrigues Parameters ” , ” time [ s ] ” ,

”dP1 dP2 dP3” , ”dP1 dP2 dP3” ,3 , 1 , ”png” ,

”Rodrigues Parameters . png” ) ;

mu l t ip l o t (T∗ t , om, ”Angular V e l o c i t i e s ” , ” time [ s ] ” ,

”dom1 [ rad/ s ] dom2 [ rad/ s ] dom3 [ rad/ s ] ” , ”dom1 dom2 dom3” ,

3 ,1 , ”png” , ” Angu l a r v e l o c i t i e s . png” ) ;

mu l t ip l o t (T∗ t , u , ”Control Torques ” , ” time [ s ] ” , ”T1 [Nm] T2 [Nm]

T3 [Nm] ” , ”T1 T2 T3” ,3 , 1 , ”png” , ”Control Torques . png” ) ;

// //////////////////////////////////////////////////////////

// //////////// END OF FILE ////////////////////////

// //////////////////////////////////////////////////////////

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