Thesis Submitted to Florida Institute of Technology in Partial Fulfllment of the Requirements for the Degree Of

Home , NEXTSat, Orbital Express

Dynamics of Spacecraft Orbital Refueling

by Casey Clark

Bachelor of Aerospace Engineering Mechanical & Aerospace Engineering College of Engineering 2016

A thesis submitted to Florida Institute of Technology in partial fulfllment of the requirements for the degree of

Master of Science in Aerospace Engineering

The author grants permission to make single copies. We the undersigned committee hereby approve the attached thesis

Dynamics of Spacecraft Orbital Refueling by Casey Clark

Dr. Tiauw Go, Sc.D. Associate Professor. Mechanical & Aerospace Engineering Committee Chair

Dr. Jay Kovats, Ph.D. Associate Professor Mathematics Outside Committee Member

Dr. Markus Wilde, Ph.D. Assistant Professor Mechanical & Aerospace Engineering Committee Member

Dr. Hamid Hefazi, Ph.D. Professor and Department Head Mechanical & Aerospace Engineering ABSTRACT Title: Dynamics of Spacecraft Orbital Refueling Author: Casey Clark Major Advisor: Dr. Tiauw Go, Sc.D.

A quantitative collation of relevant parameters for successfully completed experimental on-orbit fuid transfers and anticipated orbital refueling future missions is performed. The dynamics of connected satellites sustaining fuel transfer are derived by treating the connected spacecraft as a rigid body and including an internal mass fow rate. An orbital refueling results in a time-varying local center of mass related to the connected spacecraft. This is accounted for by incorporating a constant mass fow rate in the inertia tensor. Simulations of the equations of motion are performed using the values of the parameters of authentic missions in an endeavor to provide conclusions regarding the efect of an internal mass transfer on the attitude of refueling spacecraft. The efect of a nonzero internal mass fow rate is qualitatively investigated and a parametric analysis is performed in order to validate the conclusion. A damping efect occurs as a result of the internal mass transfer; implying that the control efort required to stabilize spacecraft sustaining fuel transfer could be less than what is necessary in the absence of an internal mass fow rate.

iii Table of Contents

Abstract iii

List of Figures vi

List of Tables ix

Abbreviations x

Acknowledgments xv

Dedication xvi

1 Introduction 1 1.1 Motivation ...... 2 1.2 Research Objective ...... 2 1.3 Outline ...... 3 1.4 On-Orbit Refueling Missions and Patents ...... 3 1.4.1 ETS-VII ...... 4 1.4.2 Orbital Express ...... 5 1.4.3 Robotic Refueling Mission ...... 7 1.4.4 Restore-L ...... 9 1.4.5 Recent Patents ...... 11

iv 2 Dynamics of Connected Spacecraft Sustaining Fuel Transfer 12 2.1 Point Mass Model ...... 13 2.1.1 Center of Mass Location ...... 14 2.1.2 Time Derivative of the Inertia Tensor ...... 16 2.1.3 Newton-Euler Formulation ...... 18 2.1.4 Euler Transformation19 ...... 20 2.2 Spacecraft Model ...... 23 2.2.1 Center of Mass Location ...... 24 2.2.2 Modifcation of the Inertia Tensor ...... 25 2.2.3 Modifcation of Newton-Euler Formulation ...... 27 2.3 Orbital Disturbances ...... 28 2.3.1 Atmospheric Drag ...... 28 2.3.2 Solar Radiation Pressure ...... 32 2.3.3 Thrust Plume Impingement ...... 33 2.3.4 Gravity Gradient28 ...... 34

3 Simulations 36 3.1 Astro and NextSat ...... 38 3.2 TRMM ...... 42 3.3 Centaur II ...... 49

4 Conclusion 56

Bibliography 58

Appendices 62

v List of Figures

1.1 Hikoboshi and Orihime12 ...... 4 1.2 ASTRO and NextSat11 ...... 6 1.3 RRM Servicer Module1 ...... 7 1.4 Visual Inspection Poseable Invertebrate Robot33 ...... 8 1.5 Restore-L Servicer29 ...... 10

2.1 Model of Connected Spacecraft Undergoing Fuel Transfer ...... 13 2.2 Point Mass Model of Connected Spacecraft in the Space Set . . . . 13 2.3 Point Mass Model in the Body Set Centered at the COM ...... 14 2.4 Space and Body Set ...... 14 2.5 Rotation from One Cartesian Coordinate System to Another . . . . 20 2.6 Sequencing of Euler Rotations ...... 21 2.7 Rigidly Connected Spacecraft in the Body Set ...... 23 2.8 Projected Area Experiencing Drag ...... 29 2.9 Adversely Orientated Connected Spacecraft ...... 30

23 2.10 Coefcient of Drag CD vs. Altitude for Diferent Projected Areas 30 2.11 Nominal Density and Scaling Height based on Altitude24 ...... 31 2.12 Solar Radiation Pressure Force on Surface Area25 ...... 32 2.13 Thrust Plume Impingement25 ...... 33

3.1 Hard Docking scenario of Cubic Spacecraft (b = 0) ...... 37

vi 3.2 Astro and NextSat - System Response ...... 39 3.3 Astro and NextSat - Diference between System Responses . . . . . 40 3.4 Torque Experienced by Astro and Next Sat ...... 41 3.5 TRMM - Orientation and Angular Velocity for 100% Fuel Transfer . 43 3.6 TRMM - Orientation and Angular Velocity - Diference between Simulations for 100% Fuel Transfer ...... 44 3.7 TRMM - Torque due to Atmospheric Drag and Gravity Gradient for 100% Fuel Transfer ...... 45 3.8 TRMM - Orientation and Angular Velocity for 55% Fuel Transfer . 46 3.9 TRMM - Orientation and Angular Velocity - Displacement between Simulations for 55% Fuel Transfer ...... 47 3.10 TRMM - Torque due to Atmospheric Drag and Gravity Gradient for 55% Fuel Transfer ...... 48 3.11 Spacecraft 2 with a Cylindrical Body ...... 49 kg 3.12 Centaur II - System Response form ˙ = 0.5 ...... 50 s kg 3.13 Centaur II - Diference between System Responses form ˙ = 0.5 s ...... 51 kg 3.14 Centaur II - Torque form ˙ = 0.5 ...... 52 s kg 3.15 Centaur II - System Response form ˙ = 10 ...... 53 s kg 3.16 Centaur II - Diference between System Responses form ˙ = 10 s ...... 54 kg 3.17 Centaur II - Torque form ˙ = 10 ...... 55 s

1 TRMM - Orientation and Angular Velocity for 85% Fuel Transfer . 62

vii 2 TRMM - Orientation and Angular Velocity - Displacement between Simulations for 85% Fuel Transfer ...... 63 3 TRMM - Torque due to Atmospheric Drag and Gravity Gradient for 85% Fuel Transfer ...... 64 4 TRMM - Orientation and Angular Velocity for 70% Fuel Transfer . 65 5 TRMM - Orientation and Angular Velocity - Displacement between Simulations for 70% Fuel Transfer ...... 65 6 TRMM - Torque due to Atmospheric Drag and Gravity Gradient for 70% Fuel Transfer ...... 66 kg 7 Centaur II - Orientation and Angular velocity withm ˙ = 1 . . . 67 s 8 Centaur II - Orientation and Angular velocity - Displacement be- kg tween Simulations form ˙ = 1 without Mass Transfer ...... 68 s 9 Centaur II - Torque due to Atmospheric Drag and Gravity Gradient kg form ˙ = 1 ...... 69 s kg 10 Centaur II - Orientation and Angular velocity withm ˙ = 2.5 . . 70 s 11 Centaur II - Orientation and Angular velocity - Displacement be- kg tween Simulations form ˙ = 2.5 without Mass Transfer ...... 71 s 12 Centaur II - Torque due to Atmospheric Drag and Gravity Gradient kg form ˙ = 2.5 ...... 72 s

viii List of Tables

3.1 Astro and NextSat Simulation Parameters ...... 38 3.2 TRMM Simulation Parameters ...... 42 3.3 Centaur II Simulation Parameters ...... 49

ix List of Symbols, Nomenclature or Abbreviations

(OE) Orbital Express (ETS-VII) Engineering Test Satellite Seven (DARPA) Defense Advanced Research Projects Agency (ASTRO) Autonomous Space Transport Robotic Operations vehicle (NextSat) Next Generation Serviceable Satellite (TRL) Technology Readiness Level (NASDA) National Space Development Agency of Japan (ESA) European Space Agency (CRL) Communication Laboratory (DLR) Deutsches Zentrum f¨urLuftund Raumfahrt (ETL) Electrotechnical Laboratory (NAL) National Aerospace Laboratory (6-DOF) Six-Degree of Freedom (ORU) Orbital Replacement Unit (RRM) Robotic Refueling Mission (Dextre) Special Purpose Dexterous Manipulator robot

x (CSA) Candian Space Agency (VIPIR) Visiual Inspection Poseable Invertebrate Robot (CORD) Cryogen Orbital Resupply Domonstration (LEO) Low Earth Orbit (SSCO) Satellite Servicing Capabilites Ofce (GSFC) Goddard Space Flight Center (COM) Center of Mass

xi S(x, y, z) \The space set reference inertial reference frame m1 \Mass of spacecraft or point 1 m2 \Mass of spacecraft or point 2 F \Total body force vector ρ \Distance vector from the time-varying COM to the line of action of the line of action of the force S r1(0) \Initial position vector from the origin of the space set to the COM of pacecraft or point 1 S r2(0) \Initial position vector from the origin of the space set to the COM of spacecraft or point 2 S rCM (0) \Initial position vector from the origin of the space set to the local COM of the connected spacecraft or points S Θ \Orientation of the time-varying COM in the space set S ω \Angular velocity of the time-varying COM in the space set M \Total mass of connected spacecraft or points m10 \Initial mass of spacecraft or point 1 m20 \Initial mass of spacecraft or point 2 m˙ 1 \Mass fow rate out of spacecraft or point 1 m˙ 2 \Mass fow rate into spacecraft or point 2 m˙ \Positive and constant mass fow rate from spacecraft or point 1 to spacecraft or point 2 B(x′, y′, z′) \The principal body set axes fxed at the time-varying COM S r˙CM (0) \Initial, constant time rate of change of the distance vector from the origin of the space set to the time-varying local COM of the connect -ed spacecraft or points S rCM (t) \Time-varying vector from the origin of the space set to the time -varying local COM of the connected spacecraft or points D \The constant position vector in the space set representing the difer -ence between the distances of spacecraft or points from the local COM D \Constant distance along the x’ axis representing the diference between the distances of spacecraft or points from the local COM T \Total time of mass transfer ′ x1(t) \Time-varying distance from the COM of spacecraft or point 1 along the principal x’ axis ′ x10 \Initial distance from the COM of spacecraft or point 1 along the principal x’ axis ′ x2(t) \Time-varying distance from the COM of spacecraft or point 2 along the principal x’ axis ′ x20 \Initial distance from the COM of spacecraft or point 2 along the principal x’ axis B ˙ ICM (t) \Time-varying inertia tensor about the COM in the body set

xii I1 \Principal moment of inertia about the x’ axis I2 \Principal moment of inertia about the y’ axis I3 \Principal moment of inertia about the z’ axis ˙ I1 \Time derivative of principal moment of inertia about the x’ axis ˙ I2 \Time derivative of principal moment of inertia about the y’ axis ˙ I3 \Time derivative of principal moment of inertia about the z’ axis ′ m1(t) \Time-varying mass of spacecraft or point 1 ′ m10 \Initial mass of spacecraft or point 1 ′ m2(t) \Time-varying mass of spacecraft or point 2 ′ m20 \Initial mass of spacecraft or point 2 a \Side length of spacecraft b \Distance between spacecraft B I˙ \Constant time-rate of change of the inertia tensor in the body set S r¨CM \Acceleration of the time-varying COM in the space set S ¨ Fi \Individual body force vectors in the space set B ¨ Fi \Individual body force vectors in the body set B ρ¨i \Distances vectors from the time-varying COM to the line of action of the individual body force vectors L \Angular momentum vector N \Torque vector N1 \Torque about the principal x’ axis N2 \Torque about the principal y’ axis N3 \Torque about the principal z’ axis φ \General Euler rotation angle about any axis r \An inconsequential position vector ϕ \Euler angle regarding the counterclockwise rotation about the z axis to arrive in the set ξηζ θ \Euler angle regarding the counterclockwise rotation about the ξ axis to arrive in the set ξ′η′ζ′ ψ \Euler angle regarding the counterclockwise rotation about the ζ′ axis to arrive in the body set B(x′, y′, z′) A \Transformation matrix from the body set to the set ωx′ \Angular velocity component about the x’ axis ωy′ \Angular velocity component about the y’ axis ωz′ \Angular velocity component about the z’ axis ϕ˙ \The spin rate of the Euler angle ϕ about the z axis θ˙ \The spin rate of the Euler angle θ about the ξ axis ψ˙ \The spin rate of the Euler angle ψ about the ζ′ axis A˜ \Transformation matrix from the space set to the body set FD \Atmospheric drag force A \Projected area experiencing the drag force xiii h \Altitude of spacecraft CD \Coefcient of Drag based on altitude and area shape ρ \Atmospheric mass density v \Spacecraft speed with respect to the atmosphere

µE \Gravitational constant of Earth p \Radiation momentum fux mf \Mass to be transferred η \Ratio of fuel mass to system mass

xiv Acknowledgements

In the words of the man, Sir Isaac Newton, if I have seen further than it is because I stood upon the shoulders of giants.

Thank you, Dr. Go and Dr. Wilde, for your perpetual guidance and for all of the time you allotted to me over the years.

Thank you, Dr. Kovats, for always ofering me your time when I needed it most.

Thank you, Dr. Kirk, for serving as the PI for this research and for your as- sistance in acquiring the funding provided by FSGC.

Thank you, Dr. Hefazi, for your distinctive patience in dealing with the faculty and students (including me).

Thank you, Jennifer, for everything!

Also, Euler was a demigod and we would all be lost without the extensive description of his perspective that he left behind.

xv Dedication

To my father, who would have studied space robotics if he had the chance.

xvi Chapter 1

Introduction

Launcher failure was once the primary cause of failure in satellites. In recent years, on-orbit failures have exceeded launch failures and cumulatively have resulted in the loss of billions of dollars.1 Aside from anomalous failures, the end of a satellite’s 10 to 15 year life span is typically dictated by the depletion of the fuel load. An economic model from 2011 claims that the ability to refuel 10 to 12 of these end of life satellites in one mission would extend their longevity by 3 to 5 years, saving $100-$250 million by delaying the outlay needed to replace them.2 United Launch Alliance (ULA) has developed an interest in orbital refueling of the upper stages of their rockets. The fuel mass of the upper stages of rockets are substantially greater than that of an average satellite and drastically greater than the on-orbit refueling experiments conducted thus far. For clarifcation, on-orbit, in-orbit, and orbital are all adjectives defning the same thing.

1 1.1 Motivation

Propellant transfer systems are critical in further developing a robust in-orbit satellite servicing capability.3 There have been various space missions conducted on refueling spacecraft as well as patents fled regarding the technology required to carry out future missions.2,4−7 However, the dynamics of connected satellites undergoing a substantial fuel transfer has not yet been investigated. The connected spacecraft have not adversely spun during experiments performed thus far as the mass transferred from one satellite to the other was unsubstantial compared with the total mass of the satellites. In reality, a fuel transfer of a substantial mass could produce an atypical and convoluted spinning motion as the center of mass is changing with time. This could lead to a loss of communication with ground support if the orientation of the body of the spacecraft with respect to the ground station.

1.2 Research Objective

In orbit, two spacecraft in a connected confguration have a local center of mass with a relative angular velocity about the center of Earth. The addition of a mass transfer between the connected spacecraft results in a time-varying local center of mass, as the moment of inertia also varies with time. This is a phenomenon, which has ostensibly not been investigated prior to the development of this research, in general or with regard to spacecraft. The proper dynamics to describe such a scenario are derived in detail within this paper using a Newton-Euler approach. Time, size and mass scaling for the mission simulations was based on real situations that have either occurred or are scheduled

2 to occur. This paper shows how the addition of the mass fow rate efects the orbital perturbations for mission parameters including: altitude, system mass, fuel mass, mass fow rate, dimensions and spacecraft body shape.

1.3 Outline

The equations of motion of connected spacecraft sustaining fuel transfer are derived by frst simplifying the spacecraft as rigidly connected point masses. The location of the center of mass (COM) varies with time as does the inertia tensor and are solved for accordingly. The point mass system is replaced by cubic spacecraft of diferent initial masses as well as a cylindrical target spacecraft. Euler angles and their respective rates are used to show the transformation from the space coordinates to the body axes. Realistic orbital disturbances are discussed and included in the model. The equations of motion are simulated to analyze how the orientation changes due to the mass transfer in the presence of orbital disturbances based upon the scenarios for the aforementioned refueling missions. Potential future work regarding this research is considered followed by a conclusion regarding its signifcance.

1.4 On-Orbit Refueling Missions and Patents

The successful and future orbital refueling missions are discussed in this section. Orbital Express (OE) , Experimental Test Satellite 7 (ETS-VII), Robotic Refueling Mission (RRM)-1 and RRM-2 have been completed. RRM-3 will launch this year and Restore-L will launch in 2020. The missions conducted thus far have not transferred an amount of propellant comparable with that required for refueling of

3 a real satellite, spacecraft or upper stage of a rocket.

1.4.1 ETS-VII

The frst documented satellite refueling mission is titled Engineering Test Satel- lite No. 7 (ETS-VII) funded by National Space Development Agency of Japan (NASDA). The satellite was launched on November 28th, 1997 into an elliptical orbit with perigee of 350 km and apogee of 550 km. ETS-VII was a twin satellite that would separate into two vehicles in orbit, the chaser named Hikoboshi and the target, Orihime, depicted in Figure 1.1.5

Figure 1.1: Hikoboshi and Orihime12

The masses of Hikoboshi and Orihime were 2267 kg and 363 kg, respectively. Ori- hime used nitrogen thrusters to maintain its attitude while Hikoboshi was equipped

4 with three reaction wheels and hydrazine thrusters, along with a 2 m long, six- degree of freedom (6-DOF) manipulator arm used to exchange the Orbital Re- placement Unit (ORU) in order to demonstrate that components of a satellite can be replaced on orbit.5 The mission objectives were to demonstrate autonomous rendezvous/docking between the chaser and target and to perform various robot experiments including the transfer of water from one vehicle to the other. The robot experiments were conducted over a two-year period by numerous organizations including Kyoto Uni- versity, Tokyo Institute of Technology, European Space Agency (ESA), Communi- cation Research Laboratory (CRL), Deutsches Zentrum f¨urLuft und Raumfahrt (DLR), Electrotechnical Laboratory (ETL) and National Aerospace Laboratory (NAL).12 Attitude correction was necessary during motion of the robot arm in order to maintain communication with the data relay satellite.13 Signal transmis- sion as well as fuel transfer between the main satellite and ORU were successfully demonstrated.12

1.4.2 Orbital Express

A mission titled Orbital Express (OE) involved two satellites in the mated confguration launched aboard an Atlas-V rocket into a 492 km, 46 degree inclined and circular orbit from the Cape Canaveral Air Force Station on March 8, 2007.4,8,9 The mission was funded by Defense Advanced Research Projects Agency (DARPA) in order to demonstrate several on-orbit servicing scenarios including the transfer of hydrazine and helium.4,10 This program sought to prove that the technical obsta- cles that were once used as an excuse to prevent the execution of satellite servicing missions were indeed surmountable.4

5 Figure 1.2: ASTRO and NextSat11

The experimental satellites are depicted in Figure 1.2. Boeing provided the servicing vehicle known as Autonomous Space Transport Robotic Operations vehicle (ASTRO) while Ball Aerospace provided the client referred to as Next Genera- tion Serviceable Satellite (NextSat).8 NextSat was 3-axis stabilized using reaction wheels while ASTRO was equipped with a propulsion system consisting of 16 hydrazine thrusters providing 4 N each.4,11 The fuid transfer pump between ASTRO and NextSat permitted fow rates from 0.014 to 0.095 kg · s−1, transferring a total of approximately 22 kg on average (from 20 kg to 24 kg).11 The mission had a 95% success rate over a lifetime of 135 days where as many as 45 contacts were executed in a 24 hour planning period amounting to a total of 3000 attempted contacts.8 The successful demonstration elevated propellant resupply of orbiting spacecraft missions out of the realm of high risk development by bringing the Technology Readiness Level to a TRL 7 (demonstration of a system prototype in an operational environment).9

6 1.4.3 Robotic Refueling Mission

The fnal mission of the Space Shuttle program included the Robotic Refueling Mission (RRM). The mission was developed and managed by NASA Goddard’s Satellite Servicing Capabilities Ofce and was successfully installed on the ISS Truss after its launch onboard STS-135 Atlantis in July 2011.3,14 The Canadian Special Purpose Dexterous Manipulator robot, also known as Dextre, was used with RRM tools to demonstrate an array of fueling tasks on the ISS in cooperation with the Canadian Space Agency (CSA).

Figure 1.3: RRM Servicer Module1

The 3.5 m, 2 armed 6-DOF robot, shown in Figure 1.3, was required to use spe- cially designed RRM tools to cut and manipulate thermal blankets and wires, then unscrew caps to access valves prior to transferring fuid. After the fuid transfer had been completed a new cap was placed for future refueling activities.1,3 RRM demonstrated that the tasks needed to service a satellite in orbit could be performed by a robot.15

7 The frst use of the specialized tools for experimental on-orbit satellite servicing occurred on March 8, 2012. The RRM multifunction tool was used by Dextre to remove launch locks securing RRM tool adapters. This was followed by cutting wires and removing gas fttings required to fll the spacecraft with propellant.14 The RRM module is 1.1 m x 1.1 m x 0.8 m consisting of robotic servicing tools including a fuid transfer system. Although, only 1.7 L of ethanol was transferred, RRM demonstrated a successful on-orbit fuel transfer in January of 2013.30,31 The transfer mass of RRM is drastically smaller than that of Orbital Express as 1.7 L of ethanol translates to a mass of less than 2 kg. The frst phase of RRM operations concluded in May 2013.3

Figure 1.4: Visual Inspection Poseable Invertebrate Robot33

The Visual Inspection Poseable Invertebrate Robot (VIPIR), shown in Figure 1.4,

8 was a new inspection tool used during the follow-on experiment RRM-2. VIPIR was designed for near to mid-range inspection in space and was successfully tested in May 2015.3 RRM-3, also known as Cryogenic Orbital Resupply Demonstration (CORD), was originally scheduled for 2017, however it has been pushed back to late 2018.3,16 The third phase of the mission will demonstrate cryogenic fuid transfer as well as xenon replenishment.3 RRM is part of a long range plan to develop the capability to service legacy satellites that were not originally designed for robotic servicing.16

1.4.4 Restore-L

A newly developed servicing mission by NASA plans to demonstrate that robotic satellite servicing technologies are operational. Originally formulated in May 2016, the Restore-L mission plans to refuel a government-owned satellite in Low Earth Orbit (LEO) in 2020 with a free-fying robotic servicer. NASA’s Space Technology Mission Directorate is managing the mission itself, while the Satellite Servicing Ca- pabilities Ofce (SSCO) is managing the development, execution and operations of the mission. The performing center is NASA’s Goddard Space Flight Center (GSFC).7 Landsat 7, the satellite to be refueled, was not designed with the intention of being serviced.7 Unlike it’s predecessors that had much more broad mission scopes, Restore-L will focus on refueling and repositioning services. The Restore-L servicing spacecraft, displayed in Figure 1.5, will have a total mass of approximately 4,000 kg.17

9 Figure 1.5: Restore-L Servicer29

The vehicle will employ the 7 DOF, 2 m NASA Servicing Arm, which is capable of executing tasks with high-speed motion and high accuracy while providing torque control. Utilizing specialized tools, the vehicle will gain access to the fll/drain valves of Landsat 7 and transfer hydrazine propellant up to 122 kg of hydrazine.7 The successful completion of the mission will demonstrate to owners of satellites that the lifespan could be extended by additional years, thus delaying the need to launch a new satellite. A robotic refueling capability would allow satellite manufacturers to include more instruments by eliminating the need to devote as much launch mass to fuel.

10 1.4.5 Recent Patents

Several patents have been fled by MacDonald Dettwiler & Associates Madison Inc. regarding on-orbit satellite refueling. The systems, methods and apparatuses presented in each patent maintain a strong design emphasis on robotic refueling of uncooperative targets.2,6,18 The equipment described in these patents were used to successfully conduct the fuel transfer during the aforementioned Orbital Express program.2,4,6 The frst patent fled in 2011 presented a system that could be either launched directly from Earth, or alternatively it can be stored on the space station until needed. The system includes a robotic arm, along with tools which can be mounted to the end-efector in order to refuel a satellite. The tools are required for accessing, opening and closing the fuel fll valve on the satellite being serviced. The system is controlled remotely by an operator located on Earth, the ground station or in the space station. Cameras on the end efector allow images to be transmitted to the operator in order to control the refueling procedure. If necessary, the system can be confgured to operate autonomously under computer control.2

11 Chapter 2

Dynamics of Connected Spacecraft Sustaining Fuel Transfer

The dynamics of connected spacecraft sustaining fuel transfer are derived in order to investigate the efect that an internal mass fow rate will have. The model is frst simplifed as a point mass system with a constant mass transfer, leading to a time-varying center of mass location. Subsequently, the equations of motion of the point mass model are reformulated using the geometry for cubic Spacecraft of identical size with a uniform mass distribution. The connected spacecraft will experience disturbances aside from the mass transfer including atmospheric drag, solar radiation pressure, gravity and thruster plume impingement during formation fight. The dynamic model of the connected spacecraft sustaining fuel transfer includes these perturbations. The efects of fuel slosh are outside the scope of this paper and therefore are not included in the formulation of the dynamic model. A

12 rudimentary model is illustrated in Figure 2.1.

Figure 2.1: Model of Connected Spacecraft Undergoing Fuel Transfer

2.1 Point Mass Model

A point mass model can be developed in the inertial coordinate system deemed the space set S(x, y, z) for two connected point masses m1 and m2 with body forces F as shown in Figure 2.2. The origin of the space set for the development of the point mass model is arbitrary.

Figure 2.2: Point Mass Model of Connected Spacecraft in the Space Set

13 2.1.1 Center of Mass Location

The body set is described in Figure 2.3 where the principal axes B(x′, y′, z′) are aligned with with principal moments of inertia and centered at the time-varying COM.

Figure 2.3: Point Mass Model in the Body Set Centered at the COM

From this model, the COM in the body set of the connected point masses is defned as the origin,

B rCM = 0. (2.1)

Figure 2.4 displays the body set and the space set orientated relative to each other.

Figure 2.4: Space and Body Set

14 From this model, the COM in the space set of the connected point masses is defned as S S S m10 r1 + m20 r2 rCM = (2.2) m10 + m20

S S where r1 and r2 are the respective locations of the point masses in the space set,

while m10 and m20 are the respective initial masses of point 1 and point 2. The total mass of the system M is simply

M = m1 + m2. (2.3)

The derivative of Equation (2.3) w.r.t. time of is

m˙ =m ˙ 2 = −m˙ 1 (2.4) wherem ˙ is the constant mass fow rate from point 1 to point 2. For a total mass transfer time T , m˙ 1 ∝ . (2.5) M T

The mass transfer is restricted to occur only along the x′ principal axis. The following equations describe the distance from the time-varying center of mass to the respective point masses

′ ′ t x1(t) = x10 + D T (2.6) t x′ (t) = x′ − D 2 20 T

′ ′ as the location of point mass one x1(t) is increasing from x10 while the the location

15 ′ ′ of point mass two x2(t) is decreasing from x20 with

′ ′ D = x20 − x10 (2.7)

2.1.2 Time Derivative of the Inertia Tensor

The moment of inertia of a system can be written as19

{ 0 j ̸= k ∑ 2 Ijk = mi(δjkri − rijrjk) where δjk = (2.8) i 1 j = k

Applying (2.13) to the body set yields the inertia tensor of the connected point masses in the body set:

⎡ ⎤ I1 0 0 ⎢ ⎥ B ⎢ ⎥ ICM (t) = ⎢ 0 I 0 ⎥ , (2.9) ⎢ 2 ⎥ ⎣ ⎦ 0 0 I3 with principal moments of inertia,

I1 = 0 (2.10) ( ′ )2 ( ′ )2 I2 = I3 = m1(t) x1(t) + m2(t) x2(t)

B where ICM (t) is a function of time as both points have varying masses and locations relative to the local COM w.r.t. time. The equations for the point masses

16 as a function of time are simply

m1(t) = m10 − mt˙ (2.11)

m2(t) = m20 +mt. ˙

The existence of an internal mass transferm ˙ yields a non-zero time rate of change B ˙ of the inertia tensor ICM (t), which is peculiar as these terms normally do not appear in the equations of motion. Including this in the formulation takes into account that the relative Bω is occurring at a diferent spin axis at each time step as the COM of the system is changing with time. Applying the product rule when taking the derivative of (2.9) w.r.t. time yields

⎡ ⎤ 0 0 0 ⎢ ⎥ B ˙ ⎢ ⎥ ICM (t) = ⎢ 0 I˙ 0 ⎥ . (2.12) ⎢ 2 ⎥ ⎣ ⎦ ˙ 0 0 I2

The expanded form of the principal moments of inertia are required and evaluated by substituting Equations (2.6) and Equations (2.11) into Equation (2.10) yielding

t ( t )2 I = I = m (x′ )2 + 2m x′ D + m D 2 3 10 10 10 10 T 10 T t ( t )2 − mt˙ (x′ )2 − 2x′ mt˙ D − mt˙ D 10 10 T T (2.13) t ( t )2 + m (x′ )2 + 2m x′ D + m D 20 20 20 20 T 20 T t ( t )2 +mt ˙ (x′ )2 + 2x′ mt˙ D +mt ˙ D 20 20 T T

17 which reduces to

[ t t ] I = I =mt ˙ (x′ )2 + 2x′ D − (x′ )2 − 2x′ D 2 3 20 20 T 10 10 T [ t ( t )2 ] + m (x′ )2 + 2x′ D + D (2.14) 10 10 10 T T [ t ( t )2 ] + m (x′ )2 + 2x′ D + D . 20 20 20 T T

Now the time rate of change of the principal moments of inertia are calculated to their reduced form as

[ t t ] I˙ = I˙ =m ˙ (x′ )2 + 4x′ D − (x′ )2 − 4x′ D 2 3 20 20 T 10 10 T [ 1 t ] + m 2x′ D + 2 D2 (2.15) 10 10 T T 2 [ 1 t ] + m 2x′ D + 2 D2 . 20 20 T T 2

2.1.3 Newton-Euler Formulation

Newton’s second law of motion can be applied to the point mass model yielding

S ∑ S M( r¨CM ) = Fi (2.16) i

S where Fi represent the external forces on the connected point masses in the space set. The Newtonian equation of motion relative to the rotating body axes is derived from Euler’s conservation of angular momentum19

d BL + Bω × BL = BN (2.17) dt

18 where BL is the angular momentum defned as

BL = IBω, (2.18) with angular velocity vector Bω and torque vector BN. Carrying out the time derivative and cross product in Equation (2.17) leads to Euler’s equations of motion for a rigid body

I1ω˙1 − ω2ω3(I2 − I3) = N1

I2ω˙2 − ω3ω1(I3 − I1) = N2 (2.19)

I3ω˙3 − ω1ω2(I1 − I2) = N3 where Ii are the principal moments of inertia and Ni are the torques corresponding to the Euler angles of rotation. The Equations (2.19) are modifed with

B L˙ = IBω˙ + I˙Bω. (2.20)

Incorporating Equation (2.20) with Equation (2.17) leads to a new set of equations

˙ I1ω˙1 + I1ω1 − ω2ω3(I2 − I3) = N1 ˙ I2ω˙2 + I2ω2 − ω3ω1(I3 − I1) = N2 (2.21) ˙ I3ω˙3 + I3ω3 − ω1ω2(I1 − I2) = N3, which account for the time-varying inertia tensor. The equations of motion describing the position and orientation are derived in terms of the space and body axes, respectively. A transformation is required to solve these equations simultaneously.

19 2.1.4 Euler Transformation19

Figure 2.5 below depicts a rotation φ about one axis of a Cartesian coordinate system to another.

Figure 2.5: Rotation from One Cartesian Coordinate System to Another

The expression for the axes of the new coordinate system are

′ x1 = x1 cos φ + x2 sin φ

′ x2 = −x1 sin φ + x2 cos φ (2.22)

′ x3 = x3

A transformation matrix A can extracted from Equation (2.22) as19

⎡ ⎤ cos φ sin φ 0 ⎢ ⎥ ⎢ ⎥ A = ⎢− sin φ cos φ 0 ⎥ . (2.23) ⎢ ⎥ ⎣ ⎦ 0 0 1

The vector r, as shown in Figure 2.5, can be transformed into the prime coordinate system as r′ = Ar. (2.24)

20 The space set to the body set can be described by an orthogonal transformation matrix A expressed in terms of the set of Euler angles (θ, ϕ, ψ) as follows19

⎡ ⎤ cψcϕ − cθsϕsψ cψsϕ + cθcϕsψ sψsθ ⎢ ⎥ ⎢ ⎥ A = ⎢−s c − c s c −s s + c c c c s ⎥ , (2.25) ⎢ ψ ϕ θ ϕ ψ ψ ϕ θ ϕ ψ ψ θ⎥ ⎣ ⎦ sθsϕ −sθcϕ cθ where c and s signify cosine and sine, respectively. Referring to Figure 2.6 below,

Figure 2.6: Sequencing of Euler Rotations the sequence starts by rotating the space set S(x, y, z) by an angle ϕ about the z axis to arrive in the set ξηζ. This is followed by a rotation about the ξ axis by an angle θ, arriving in the set ξ′η′ζ′. Finally, a rotation is made about the ζ′ axis by an angle ψ to arrive in the body set B(x′, y′, z′).19 The inverse transformation from the body coordinates to space axes is given by

21 r = (A)−1r′. (2.26) where the inverse of the transformation matrix is simply the transpose

⎡ ⎤ cψcϕ − cθsϕsψ −sψcϕ − cθsϕcψ sθsϕ ⎢ ⎥ −1 T ⎢ ⎥ A = A = A˜ = ⎢c s + c c s −s s + c c c −s c ⎥ (2.27) ⎢ ψ ϕ θ ϕ ψ ψ ϕ θ ϕ ψ θ ϕ⎥ ⎣ ⎦ sψsθ cψsθ cθ due to the orthogonality of the matrix A. The components of the angular velocity in the body set Bω can be expressed in terms of the Euler angles and their corresponding spin rates as

˙ ˙ ωx′ = ϕ sin θ sin ψ + θ cos ψ (2.28)

˙ ˙ ωy′ = ϕ sin θ cos ψ − θ sin ψ (2.29)

˙ ˙ ωz′ = ϕ cos θ + ψ. (2.30)

Multiplying equation (2.28) by cos ψ and equation (2.29) by − sin ψ then summing the results yields the equation for the spin rate θ˙ about the ξ axis

˙ θ = ωx′ cos ψ − ωy′ sin ψ. (2.31)

Multiplying equation (2.28) by sin θ sin ψ and equation (2.29) by sin θ cos ψ then summing the results yields the spin rate ϕ˙ about the z axis

˙ ϕ = ωx′ sin θ sin ψ + ωy′ sin θ cos ψ. (2.32)

Substituting equation (2.32) into equation (2.30) and solving for the spin rate ψ˙

22 about the ζ′ axis results in

˙ ( ) ψ = ωz′ − ωx′ sin θ sin ψ + ωy′ sin θ cos ψ cos θ. (2.33)

The Euler angles are solved for by integrating equations (2.31-2.33) w.r.t. time as

˙ θ(t) = θt + θ0 ˙ ϕ(t) = ϕt + ϕ0 (2.34) ˙ ψ(t) = ψt + ψ0. where θ0, ϕ0, and ψ0 are the initial Euler angles at the start of the mass transfer.

2.2 Spacecraft Model

The formulation of the dynamics of the point mass model can be modifed in order to model the dynamics of a fuel transfer scenario of two, rigidly connected spacecraft identical in size with a massless connection arranged and aligned faces as shown in Figure 2.7 below.

Figure 2.7: Rigidly Connected Spacecraft in the Body Set

23 For the derivation of the dynamic model, the spacecraft are considered to be rigidly connected cubes and of uniform mass with a uniform and constant mass transferm ˙ occurring along the principal x′ axis. The line of action of the body force vectors F are separated from the COM by the position vectors ρ. Restricting the masses to be uniform and the mass fow rate to be constant allows the connected spacecraft to be treated as a rigid body, rendering Newton-Euler applicable.

2.2.1 Center of Mass Location

The formulation for the COM of set of the connected satellites is identical to that of the previously shown point mass model described by Equation (2.2)

S S S m10 r1 + m20 r2 rCM = (2.35) m10 + m20

S S where r1 and r2 are the respective locations of the spacecraft in the space set,

while m10 and m20 are the respective initial masses of the spacecraft at the start of the fuel transfer. Equations (2.3-2.7) and Equation (2.11) for the point mass model are valid for the spacecraft model:

t x′ (t) = x′ + D 1 10 T t x′ (t) = x′ − D 2 20 T ′ ′ (2.36) D = x20 − x10

m1(t) = m10 − mt˙

m2(t) = m20 +mt. ˙

24 2.2.2 Modifcation of the Inertia Tensor

The principal moments of inertia defned for the point masses in Equation (2.10) are modifed in order to account for the geometry connected cubic spacecraft by incorporating the known moment of inertia of a square face as

a2 I1 = M 6 (2.37) a2 a2 I = I = m (t) + m (t) + m (t)(x (t))2 + m (t)(x (t))2, 2 3 1 6 2 6 1 1 2 2 written in expanded form as

a2 I = M 1 6 a2 a2 I = I = m (t) + m (t) 2 3 1 6 2 6 [ t t ] +mt ˙ (x′ )2 + 2x′ D − (x′ )2 − 2x′ D 20 20 T 10 10 T (2.38) [ t ( t )2 ] + m (x′ )2 + 2x′ D + D 10 10 10 T T [ t ( t )2 ] + m (x′ )2 + 2x′ D + D . 20 20 20 T T

The derivative of Equation (2.38) w.r.t. time yields the same time rates of change of the principal moments of inertia for the point mass model:

[ t t ] I˙ = I˙ =m ˙ (x′ )2 + 4x′ D − (x′ )2 − 4x′ D 2 3 20 20 T 10 10 T [ 1 t ] + m 2x′ D + 2 D2 (2.39) 10 10 T T 2 [ 1 t ] + m 2x′ D + 2 D2 . 20 20 T T 2

25 It would be more appropriate to calculate the principal moments of inertia of spacecraft using cylindrical bodies for a scenario involving the refueling of the upper stage of a rocket. The inertia tensor for an orbital refueling of a rocket with a cylindrical servicer is

r2 I = M R 1 2 1 1 I = I = m (t)(3r2 + H2) + m (t)(3r2 + H2) (2.40) 2 3 12 1 R 12 2 R ( )2 ( )2 + m1(t) x1(t) + m2(t) x2(t)

where rR and h represent the radius and height of the rocket, respectively. The equation can be similarly expanded as

r2 I = M R 1 2 1 1 I = I = m (t)(3r2 + H2) + m (t)(3r2 + H2) 2 3 12 1 R 12 2 R [ t t ] +mt ˙ (x′ )2 + 2x′ D − (x′ )2 − 2x′ D 20 20 T 10 10 T (2.41) [ t ( t )2 ] + m (x′ )2 + 2x′ D + D 10 10 10 T T [ t ( t )2 ] + m (x′ )2 + 2x′ D + D . 20 20 20 T T

The derivative of Equation (2.41) w.r.t. time is identical to Equation (2.39),

[ t t ] I˙ = I˙ =m ˙ (x′ )2 + 4x′ D − (x′ )2 − 4x′ D 2 3 20 20 T 10 10 T [ 1 t ] + m 2x′ D + 2 D2 (2.42) 10 10 T T 2 [ 1 t ] + m 2x′ D + 2 D2 . 20 20 T T 2

26 2.2.3 Modifcation of Newton-Euler Formulation

The equation of motion describing the translation of the spacecraft in the space set is identical to that of the point mass model

S ∑ S M( r¨CM ) = Fi. (2.43) i

A satellite can be represented as a gyroscope, making Euler’s angular momentum law applicable.20 Equations (2.26) are also identical to that of the point mass model

˙ I1ω˙1 + I1ω1 − ω2ω3(I2 − I3) = N1 ˙ I2ω˙2 + I2ω2 − ω3ω1(I3 − I1) = N2 (2.44) ˙ I3ω˙3 + I3ω3 − ω1ω2(I1 − I2) = N3 with the updated principal moments of inertia and their time derivatives, described by Equation (2.37) and Equation (2.39) for cubic spacecraft and Equation (2.40) and Equation (2.42) for cylindrical spacecraft. The torque vector BN can be expanded into

B ∑ B B B N = ( ρi × Fi) + τ (2.45) i

B B where ρi is the distance vector from the line of action of force Fi to the COM and Bτ is torque applied by a controller. Equation (2.44) describes the orientation BΘ of two connected Spacecraft sustaining a constant fuel transferm ˙ .

27 2.3 Orbital Disturbances

The existence of the mass transfer alone is not enough to generate rotations. A mass transfer scenario modeled without disturbances will not change the orientation of the spacecraft. However, the mass transfer has a direct efect on the angular velocity and orientation in the presence of orbital disturbances. An orbital refueling scenario would occur over a time span that would result in the spacecraft’s altitude changing a negligible amount. Consequently, the position in the space set is assumed to be constant. A comparison between the orbital disturbances is made in order to determine which among them are appropriate to include within the simulations.

2.3.1 Atmospheric Drag

A spacecraft on a circular orbit will experience a constant drag force due to the interaction of the body with the surrounding molecules. The lift force and sideslip force due to interaction with the atmosphere is generally assumed to be absent at the altitudes of most spacecraft with the exception of reentry studies. The drag force experienced by a spacecraft in orbit can be represented by23

1 F = ρAC v2 (2.46) D 2 D

where A represents the projected area in the direction of motion, CD is the drag coefcient, ρ is the atmospheric mass density and v is the spacecraft speed w.r.t. the atmosphere. The torque in the body axes due to the drag is

B S ND = A ND (2.47)

28 S where the torque due to the atmospheric drag in the space set ND is solved for as ⎡ ⎤ 0 ⎢ ⎥ S ⎢ ⎥ ND = ⎢−λ F ⎥ (2.48) ⎢ z D⎥ ⎣ ⎦ λyFD where λ is the distance from the center of mass to the center of pressure on the projected area experiencing the atmospheric drag along the x axis, depicted in Figure 2.8, where the origin of the inertial coordinate system S(x, y, z) is the center of Earth.

Figure 2.8: Projected Area Experiencing Drag

The shape of the projected area in the direction of motion depends on the orientation of the connected spacecraft in the space set as depicted in Figure 2.9.

29 Figure 2.9: Adversely Orientated Connected Spacecraft

The coefcient of drag CD is dependent on the projected area perpendicular to the direction of motion and the altitude; it can be inferred from Figure 2.10.

23 Figure 2.10: Coefcient of Drag CD vs. Altitude for Diferent Projected Areas

30 The atmospheric density ρ is a function of the altitude above sea level h

( ) h − h0i − Hi ρ(h) = ρ0i e (2.49)

where for section i, h0i is the base altitude, ρ0i is the nominal density and Hi represents the scale height. Each quantity for section i can be extrapolated from Figure 2.11.

Figure 2.11: Nominal Density and Scaling Height based on Altitude24

The spacecraft speed v w.r.t. the atmosphere in a circular orbit about Earth is equivalent to the orbital velocity defned by

√µ v = E (2.50) r where µ is the gravitational constant of Earth and r is radius from the center of the earth to the spacecraft defned by

r = RE + h (2.51) where RE is the radius of Earth.

31 2.3.2 Solar Radiation Pressure

The solar radiation force on a spacecraft in the Sun-satellite direction can be defned as25

FSP = −p · A · uˆS (2.52) where A is the cross sectional area of the satellite, m is the mass of the satellite andu ˆS is the Sun-satellite direction unit vector, while the radiation momentum fux p is p = 4.38 × 10−6 N/m2 at aphelion (2.53) p = 4.68 × 10−6 N/m2 at perihelion.

The force will be intermittent as the solar radiation pressure is zero when the satellite is in the shadow of the Earth.25 As can be observed in Figure 2.12, the force exerted on the satellite by the Sun is dependent on the orientation of the satellite’s surface with respect the Sun.

Figure 2.12: Solar Radiation Pressure Force on Surface Area25

The acceleration due to solar radiation pressure is about two to three orders of magnitude lower than that of atmospheric drag at an altitude of 400 km.25

32 2.3.3 Thrust Plume Impingement

The efect of plume impingement from the thrusters of a spacecraft become an important disturbance when spacecraft are operating in close proximity.25

Figure 2.13: Thrust Plume Impingement25

Referring to Figure 2.13, the forces due to a thruster plume must be integrated over the various surfaces based upon the orientation of those surfaces direction w.r.t. the thrust direction:

dF = −P (r, θ) cos γdS (2.54) where the thruster plume P (r, θ) has been modeled by Desplats as34

θ2 − ϕ0 2 P (r, θ) = e 2θ0 . (2.55) r2

The plume angle or half cone angle θ0 represents the 1σ value of the gas jet assuming a Gaussian distribution, while θ is the angle between the center line of the thruster nozzle and force experienced on the the surface due the thruster plume.

33 The value of the fux constant ϕ0 is dependent on the thruster characteristics and can be calculated as25 F 1 ϕ = 0 (2.56) 0 π E with ∫ π θ2 − 2 E = e 2θ0 sin 2θdθ (2.57) 0 where typical values of the half cone angle θ0 are of the order of 13 degrees. The accelerations due to thruster plume impingement are capable of reaching one to two orders of magnitude higher than that of atmospheric drag.25

2.3.4 Gravity Gradient28

A large spacecraft in LEO may experience signifcant gravity-gradient torques. The torques arise from the diferent attraction of gravity across the satellite. The gravitational torque on a spacecraft is given by

∫ Ng = r × ag dm (2.58) where the gravitational acceleration is

R + r a = −µ . (2.59) g |R + r|3

The position vector R locates the COM of the spacecraft as seen from Earth, while r represents the distance from the COM of the spacecraft to mass element dm. Notice that if r r is set to zero, the variation in mass over the satellite is ignored and there will be no torque generated. Evaluating the integral in Equation (2.66) in the body frame yields the equations for the gravity-gradient torque generated

34 about each principal axis:

3µ N = R ′ R ′ (I − I ) g1 R5 y z 3 2 3µ N = R ′ R ′ (I − I ) (2.60) g2 R5 x z 1 3 3µ N = R ′ R ′ (I − I ), g3 R5 x y 2 1 where R is the distance from the center of Earth, µ is the gravitational parameter of Earth, I1, I2 and I3 are the principal moments of inertia and Rx′ , Ry′ and Rz′ are the values of the radius vector in the body set given by

BR = A˜ S R (2.61)

The gravity-gradient torque leads to oscillations about the equilibrium orientation for asymmetric bodies. Before the demise of Skylab, the gravity-gradient torque caused its long axis to remain pointed at Earth.

35 Chapter 3

Simulations

The connected spacecraft are modeled as cubes with uniform mass and identical side lengths. For each simulation, appropriate altitude, mass fow ratem ˙ and mass to be transferred mf were estimated. The total time of the transfer was calculated based on those parameters as m T = f . (3.1) m˙

The simulation end time was fxed for every simulation to be three times the total transfer time with a circle representing the end of the fuel transfer. A ratio of the fuel mass to the system mass is expressed as

m η = f (3.2) M in order to qualitatively compare the diferent refueling scenarios simulated. The connected spacecraft are assumed to be in a circular orbit at diferent altitudes according to the mission and the body axes are set to be initially orientated with the space set. The initial angular velocities about the y′ and z′ axes are set to

36 be 10−5 rad/s. The velocity is aligned with the positive x′ direction. A circle is used on relevant plots to represent the end of the fuel transfer. The efects of solar radiation pressure are ignored as they are two to three orders of magnitude lower than the efects of atmospheric drag at this low of an altitude. Atmospheric drag acts against the direction of motion, thus it acts along the negative x′ direction. Drag is modeled with a surface area equivalent to one and a half times the face of the cube for cubic space and an average of the area exposed for the cylindrical spacecraft. The refueling of a satellite would most likely be in a hard dock scenario to minimize the moment arm as illustrated in Figure 3.1.

Figure 3.1: Hard Docking scenario of Cubic Spacecraft (b = 0)

This implies that the thrusters would not invoke plume impingement on the spacecraft; thus, efects of thruster plume are ignored. Neither the mass transfer nor the body forces discussed will cause a rotation about the x′ axis. However, torsions will occur about the y′ and z′ axes due to the gravity-gradient and atmospheric drag torque; they will be altered by the addition of the mass transfer. The simulation is executed under diferent scenarios to show how the addition of the mass transfer afects the orientation. The scenario is fxed for Astro and NextSat with a mass fow rate chosen based upon the values mentioned within the OE sub-

37 section of the Introduction. A simulation for the Restore-L mission was executed for a satellite with a larger reservoir for fuel as the η value is comparable to that of Astro and NextSat using the intended satellite to be refueled, Landsat 7. The satellite chosen to simulate is Tropical Rainfall Measurement (TRMM) satellite as it has the largest fuel capacity of all unclassifed satellites operating in LEO.27 The simulations for the Restore-L servicer refueling TRMM are parametrized based upon the fuel tank level with a fxed mass fow rate, while the simulations involving an orbital refueling of Centaur II upper stage are performed assuming an entirely depleted propellant tank with a variable mass fow rate. Each simulation transpires over diferent time intervals, resulting in diferent magnitudes of torques, orientation and angular velocity; thus, the trends are plotted in individual graphs.

3.1 Astro and NextSat

The chaser and target of the OE mission, Astro and NextSat, were modeled in an attempt to show the diference in stability between the experiments performed in the past and those that will be performed in the future to service spacecraft of substantial mass and size.

Table 3.1: Astro and NextSat Simulation Parameters

h Altitude 500 km

m10 Initial Mass of Servicer 1,091 kg

m20 Initial Mass of Client 227 kg mf Mass to be Transferred 22 kg m˙ Mass Flow Rate 0.02 kg/s T Total Time of Fuel Transfer 18.33 min a Side Length 1.8 m η Fuel to System Mass Ratio 0.017

38 Quantities for the parameters were chosen according to the aforementioned infor- mation in Chapter 1 and are represented in Table 3.1. The size and mass of these spacecraft are unsubstantial compared to the satellites and spacecraft intended to be serviced in the future.

Figure 3.2: Astro and NextSat - System Response

It can be observed from Figure 3.2 that the time-span of the fuel transfer is too short for the orientation to signifcantly change. This agrees with NextSat On- Orbit Experiences, which states that ”the performance of fuid transfer scenarios required the least amount of active monitoring by the NextSat operators.”26 Com-

39 munications would not be afected during a time-span this short involving an unsubstantial propellant mass transfer shown by the OE mission. The change in the orbital perturbations due to the addition of the mass transfer for vehicles of this size and mass is inconsequential. However, in the case of transferring a large mass over a considerably large period of time, the connected spacecrafts’ orientation will noticeably shift.

Figure 3.3: Astro and NextSat - Diference between System Responses

As shown in Figure 3.3, the diference between the simulations is minuscule due to the combined efect of OE’s short transfer time and unsubstantial ratio of transfer mass to the total mass of Astro and NextSat. OE was purely a conceptual demonstration and transferred on average a mere 50 lb over less than an hour, attributing to why the fuid transfer scenarios required the least amount of active monitoring. The value of η needs to be higher in order for more notable dynamics from the mass transfer to be present.

40 Figure 3.4: Torque Experienced by Astro and Next Sat

Referring to Figure 3.4, the addition of the mass fow causes the atmospheric drag torque about the y′ axis to decrease in magnitude during the fuel transfer as the relevant moment arm is decreasing with the changing center of mass. There is nothing that forbids gravity-gradient torques from causing oscillations about a stable equilibrium.28 The tendency of the gravity gradient torque to align the body with it’s principal axes is perspicuously depicted in the scenarios with an absent mass transfer.

41 3.2 TRMM

According to data published by the U.S. geological survey, the satellite to be refueled in the Restore-L mission, Landsat-7 had a total mass of 2,200 kg including 122 kg of hydrazine monopropellant stored in a single tank.21 The ratio of the fuel mass being transferred to the total spacecraft mass η is comparable to that of Astro and NextSat and would again yield nugatory dynamics. Therefore, a simulation for the Restore-L mission was executed for a satellite with a larger reservoir for fuel. The Restore-L servicer is set to be approximately a cube with side length 2 m.29 The satellite chosen for the refueling simulation is the Tropical Rainfall Measurement (TRMM) satellite as, according to The New SMAD, it is has the largest propellant depot of all current operating satellites in LEO and is able to hold 899 kg of hydrazine and pressurant, though spy satellites have much more. It operates in a circular orbit at 350 km.27 The other relevant parameters for the simulation have been tabulated below in Table 3.2. Table 3.2: TRMM Simulation Parameters

h Altitude 350 km

m10 Initial Mass of Servicer 4,899, 4,764, 4,629 and 4,494 kg

m20 Initial Mass of Client 2,605, 2,740, 2,875 and 3,010 kg mf Mass to be Transferred 100%, 85%, 70% and 55% of 899 kg m˙ Mass Flow Rate 0.1 kg/s T Total Time of Fuel Transfer 2.5, 2.12, 1.75 and 1.37 hr a Side Length 2 m η Fuel to System Mass Ratio 0.120, 0.102, 0.084 and 0.066

It is noted that this satellite possesses symmetric arrays to balance drag torques; the simulation assumes that they are absent. The properties of the Restore-L servicer have been chosen for the simulation as there have not been any completed

42 and documented missions involving a propellant transfer of over 55 lb.

Figure 3.5: TRMM - Orientation and Angular Velocity for 100% Fuel Transfer

Observed from Figure 3.5, the trend of the angular velocity about the y′ axis is similar to the case of Astro and NextSat shown by Figure 3.2, while the orientation changes more signifcantly for TRMM both with and without a mass transfer. The positions and angular velocities about the z′ axis are sinusoidal and bounded responses with gradually decaying amplitudes. It can be discerned from Figure 3.5 that the amplitudes of these responses decrease, while the periods increase for the case of a nonzero mass fow rate. This is indicative of a damping efect occurring

43 as a result of the internal mass transfer, implying that the control efort required to stabilize spacecraft sustaining fuel transfer could be less than what is necessary in the absence of an internal mass fow rate. This curious phenomenon becomes more evident during the simulations for Centaur II, involving a substantial transfer mass and time-span.

Figure 3.6: TRMM - Orientation and Angular Velocity - Diference between Sim- ulations for 100% Fuel Transfer

Shown by Figure 3.6, the diference in the orientation about the y′ axis is more signifcant than that of Astro and NextSat. However, it is still not of concern as the correction of a rotation of this magnitude during this time span is a trivial control problem. Nevertheless, there is a displacement of the orientation about both x′ and y′ axes; thus, some control efort would be extricated in this case. The connected spacecraft did not experience a prominent damping efect about the z′ axis during the simulations for Astro and NextSat for two reasons: diminutive time scale and minuscule transfer mass. A trend is shown when comparing these later in this section to a scenario involving a fuel mass of 55%, rather than the 100%

44 being analyzed currently.

Figure 3.7: TRMM - Torque due to Atmospheric Drag and Gravity Gradient for 100% Fuel Transfer

Referring to Figure 3.7, form ˙ = 0, the gravity gradient torque about the y′ and z′ axis are both oscillatory responses. This is due to the gravity gradient’s tendency to align the body with it’s principal axes. This tendency is augmented with a decrease in amplitude and increase in period when the simulation is ran with a non-zero mass fow rate. The torque due to the atmospheric drag is purely sinusoidal about the z′ axis without mass transfer as there is no tendency of drag

45 to drive a body to be aligned with it’s principal axes. The drag torque about the y′ axis is inauspiciously increasing and does not appear to be changing it’s direction after 7 and a half hours for simulations with and without the mass transfer. When comparing this with Figure 3.5, it is clear that the orientation about the y′ axis is unstable regardless of the existence of an internal mass fow rate. The addition of the mass transfer changes the dynamics favorably, though not necessarily in a stabilizing manner.

Figure 3.8: TRMM - Orientation and Angular Velocity for 55% Fuel Transfer

46 A more resolved transition can be made by incorporating the graphs provided in the Appendices for 85% and 70% fuel transfers. The same general trend is observed in the system response involving the refueling of a completed depleted propellant depot. Shown by Figure 3.8, the orientation with respect to the z′ axis is clearly being damped, just at a more gradual rate compared to the scenario for 100% fuel transfer, implying that a larger transfer mass for a given mass fow rate results in better damping. The magnitudes of the response are smaller while the damping from the mass fow results is less efective.

Figure 3.9: TRMM - Orientation and Angular Velocity - Displacement between Simulations for 55% Fuel Transfer

Referring to Figure 3.9, the diference in orientation about the y′ axis is again notable and would be advantageous from a control efort’s perspective. It is clear when comparing Figure 3.9 to Figure 3.6 that there would be more control efort saved for scenarios involving a larger fuel mass.

47 Figure 3.10: TRMM - Torque due to Atmospheric Drag and Gravity Gradient for 55% Fuel Transfer

In view of Figure 3.10, the fuel transfer completes prior to half of the period of the torque generated by gravity gradient about the y′ and z′ axis. The magnitude of the torque due to the atmospheric drag decreases more rapidly during the fuel transfer than it does withm ˙ = 0.

48 3.3 Centaur II

Future orbital refueling missions are being considered by United Launch Alliance (ULA) for refueling the upper stages of their rockets. A refueling scenario involving the Centaur II stage of the Atlas V Rocket as the target with a servicer similar to that of the Restore-L mission has therefore been considered. The Centaur II simulations are executed with diferent mass fow rates assuming a completely depleted fuel tank. A basic model is shown in Figure 3.11.

Figure 3.11: Spacecraft 2 with a Cylindrical Body

The relevant parameters are displayed in Table 3.3.22,32

Table 3.3: Centaur II Simulation Parameters

h Altitude 400 km

m10 Initial Mass of Servicer 26,830 kg

m20 Initial Mass of Client 2,247 kg mf Mass to be Transferred 22,830 kg m˙ Mass Flow Rate 0.5, 1, 2.5 and 10 kg/s T Total Time of Fuel Transfer 12.68, 6.34, 2.54 and 0.634 hr

rR Radius of Centaur II 1.5025 m H Height of Centaur II 12.68 m η Fuel to System Mass Ratio 0.785

Notice the diference in magnitudes of the parameters for the Centaur II simulation

49 compared to the simulation parameters for Astro & NextSat and TRMM. The remodeled spacecraft 2 introduces a new parameter H that signifcantly changes the principal moments of inertia about the y′ and z′ axes due to its relatively large magnitude compared to cubic spacecraft with similar dimensions. These adjusted moments of inertia, coupled with the heavy spacecraft masses, cause stronger torques from the orbital perturbations. Centaur II possesses a fuel tank capable of containing 22,830 kg of fuel.22 A fuel transfer involving this magnitude of mass produces curious dynamics.

kg Figure 3.12: Centaur II - System Response form ˙ = 0.5 s

50 It can be seen in view of Figure 3.12 that for an absent fuel transfer the orientation and angular velocity about the z′ axis are sinusoidal responses. The orientation and the angular velocity about the y′ axis approaches are unbound for the simulation withm ˙ = 0. These trends compare with that of those observed for the simulations for TRMM. Furthermore, the magnitudes for the orientation are larger for Centaur II, as predicted. The body rotates nearly 180 degrees about the y′ axis for an absent mass transfer due to the large mass and size coupled with a relatively long transfer time. A value of η this large coupled with the relatively low mass fow rate results in an incontestable damping efect about the y′ axis. However, the mass transfer stops at approximately 12 hours and the efects of atmospheric drag and gravity-gradient cause the spacecraft to deviate from a stable orientation.

kg Figure 3.13: Centaur II - Diference between System Responses form ˙ = 0.5 s

Analyzing Figure 3.13, the diference of the orientation about the y′ axis between the simulations with and without mass transfer is signifcant. Insinuating that the control efort maintaining the y′ axis would be decreased drastically for a

51 controller maintaining the stability of connected spacecraft sustaining a constant fuel transfer. The damping does not occur about the z′ axis for a mass fow rate this slow, however it can be seen later that increasing the mass fow rate stabilizes the z′ as well. This is unanticipated, unprecedented and considered to be the primary discovery of this research if it is peer-reviewed and verifed to be true.

kg Figure 3.14: Centaur II - Torque form ˙ = 0.5 s

The gravity gradient torque is oscillatory form ˙ = 0, seen in Figure 3.14; the torque due to the atmospheric drag about the z′ axis is sinusoidal, while it simultaneously oscillates and decreases about the y′ axis. The addition of the mass transfer tem-

52 porarily stabilizes the drag torque about the y′ axis; it then begins to return to it’s oscillatory response pattern. Notice how the gravity gradient torque becomes temporarily bounded with the addition of the mass transfer.

kg Figure 3.15: Centaur II - System Response form ˙ = 10 s

Referring to Figure 3.15, the orientation about both axes is damped almost entirely for a nonzero mass transfer. However, the magnitude of the angular velocities about both axes are on the order of 10−7 and decreasing with time after the fuel transfer

kg is completed. The ability to transfer to transfer mass atm ˙ = 10 s would be ideal as this results in a change in orientation of less than a quarter of a degree. This

53 mass fow rate has not been tested and validated in an orbital refueling scenario.

kg kg Rates of 1 s and 2.5 s are included in the Appendices and have been simulated to kg kg show the transition from 0.5 s to 10 s . Notice that the response for a nonexistent mass transfer is identical, but in diferent time scales for the diferent mass fow rates used. This shows congruency between the simulations as for fxed masses, the system responses exhibit an identical trend in diferent time scales for various mass fow rates. Refer to Figures 7 through 12 in the Appendices for a more distinct trend.

kg Figure 3.16: Centaur II - Diference between System Responses form ˙ = 10 s

The diference between system responses with and without mass transfer for a complete refueling of Centaur II is comparable to that of the scenario involving an orbital refueling of TRMM with low transfer mass, shown by comparing Figure 3.16 with Figure 3.9. Consequently, the extricated control efort will be commensurate for these scenarios. A generalization can be made suggesting that a low value of η with a low mass fow rate yields similar dynamics to a high value of η with a high

54 kg mass fow rate. However, the mass fow rate needs to be high (m ˙ > 1 s ) in order for the damping to fully take efect.

kg Figure 3.17: Centaur II - Torque form ˙ = 10 s

The torques depicted in Figure 3.17 form ˙ = 0 are once again more resolved versions of the same subplots depicted in Figure 3.14. The addition of the mass transfer causes the torque due to the atmospheric drag to decrease about both axes, while it causes the gravity gradient torque to increase about both axes.

55 Chapter 4

Conclusion

The successful orbital refueling missions conducted thus far have transferred an unsubstantial amount of mass during inconsequential time spans, resulting in negligible dynamics during the fuid transfer scenarios. For future missions, specifcally those involving the refueling of upper stages of rockets, the body adversely rotates if the mass fow ratem ˙ is too low due to a longer total transfer time T .

M T = m˙

Reasonable orbital perturbations were considered for various simulations involving a variety of spacecraft shapes, sizes, masses and mass fow rates. The derived model of connected spacecraft sustaining fuel transfer could be made more exemplary by including: a time-varying mass fow rate, an inertia tensor for an authentic spacecraft with and a comprehensive dynamic efect of slosh. The amplitudes of the responses decrease, while the periods increase for the case of a nonzero mass fow rate. This is indicative of a damping efect occurring as

56 a result of the internal mass transfer; implying that the control efort required to stabilize spacecraft sustaining fuel transfer could be less than what is necessary in the absence of an internal mass fow rate. A ratio of η closer to 1,

m η = f , M coupled with a relatively high mass fow rate results in a defned damping efect kg about both the y′ and the z′ axes. The mass fow rate needs to be high, (m ˙ ≥ 1 ) s in order for the damping to fully take efect. The control efort would be extricated for a controller maintaining the attitude of connected spacecraft sustaining an expedited fuel transfer. The value of η needs to be high ( > 0.05 ) in order for notable dynamics from the mass transfer to be present. A large value of η coupled with a relatively low mass fow rate results in an incontestable damping efect about the y′ axis. A larger transfer mass for a given mass fow rate results in better damping. It is possible to damp both responses about the y′ and z′ axes with a fast enough mass transfer rate for any realizable value of η.

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61 Appendices

Figure 1: TRMM - Orientation and Angular Velocity for 85% Fuel Transfer

62 Figure 2: TRMM - Orientation and Angular Velocity - Displacement between Simulations for 85% Fuel Transfer

63 Figure 3: TRMM - Torque due to Atmospheric Drag and Gravity Gradient for 85% Fuel Transfer

64 Figure 4: TRMM - Orientation and Angular Velocity for 70% Fuel Transfer

Figure 5: TRMM - Orientation and Angular Velocity - Displacement between Simulations for 70% Fuel Transfer

65 Figure 6: TRMM - Torque due to Atmospheric Drag and Gravity Gradient for 70% Fuel Transfer

66 kg Figure 7: Centaur II - Orientation and Angular velocity withm ˙ = 1 s

67 Figure 8: Centaur II - Orientation and Angular velocity - Displacement between kg Simulations form ˙ = 1 without Mass Transfer s

68 Figure 9: Centaur II - Torque due to Atmospheric Drag and Gravity Gradient for kg m˙ = 1 s

69 kg Figure 10: Centaur II - Orientation and Angular velocity withm ˙ = 2.5 s

70 Figure 11: Centaur II - Orientation and Angular velocity - Displacement between kg Simulations form ˙ = 2.5 without Mass Transfer s

71 Figure 12: Centaur II - Torque due to Atmospheric Drag and Gravity Gradient kg form ˙ = 2.5 s