Low-Impact and Damped State Feedback Control of a Solar Sail on an Optimal Non- Keplerian Planet-Centered Orbit

Thesis

Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University

Ryan Micah Gero

Graduate Program in Aeronautical and Astronautical Engineering

The Ohio State University

2009

Thesis Committee:

Dr. Richard J. Freuler, Advisor

Dr. Gerald M. Gregorek

Ryan Micah Gero

2009

Abstract

Consider the most fundamental difference between a solar sail and conventional spacecraft: propellant. In order to effect propulsion, solar sails receive a constant supply of massless photons to reflect while conventional spacecraft must carry a limited supply of fuel. At first glance, solar sails are infinitely more efficient than conventional spacecraft simply because of this fact. However, while certainly an advantage to solar sailing, propellant consumption is not the proper metric for spacecraft comparison or the only appealing facet of the solar sail.

It can be shown by a reasonable and straightforward analysis that solar sails have the potential to out-perform conventional spacecraft on the basis of effective specific impulse, a parameter that incorporates launch and payload masses as well as total mission duration via an adaptation of the illustrious rocket equation. Pair an aggressive specific impulse with the orbital possibilities that arise when solar sail performance is at a level capable of producing spacecraft accelerations the same order of magnitude as local solar or planetary gravitational acceleration, and the engineer finds significantly fewer constraints limiting the design of future space missions. Imagine a spacecraft for which a

Lagrange equilibrium point becomes a large surface, rather than a singular location, on which it is able to remain at rest. Picture a space vehicle hovering high above an ecliptic plane or perhaps racing along some other non-Keplerian orbit taking measurements and relaying signals from positions previously untenable. Solar sails can do all of these

ii things, and it is the intent of this body of work to generate a proof of concept for one of the most attainable and pertinent capabilities unique to solar sails mentioned thus far.

In the pages that follow it will be shown that a solar sail is inherently stable for some of the optimal non-Keplerian family of planet-centered orbits, and can be stabilized by

straightforward control schemes for the rest. Beginning from scratch with a radiation

pressure model, gain parameters were developed for low-impact and damped state

feedback control via sail pitch attitude variation. Optimal orbits are attainable, as these

trajectories were designed to minimize the required spacecraft acceleration and thus

lower the solar sail performance requirement. Planet-centered orbits are pertinent, since

a solar sail must inevitably begin its journey by escaping from the planet Earth and most

of NASA’s recent efforts in space are geared towards the exploration of nearby planets

and their moons. Uniqueness stems from the specification of the non-Keplerian family of

orbits, since solar sails are capable of sustaining them whereas modern conventional

spacecraft are not. In today’s day and age, with payload miniaturization and the ability to

manufacture extremely light weight reflective materials, solar sailing has the potential to

become reality within the next five to ten years. The concepts highlighted in this thesis

have a significant probability of being among the first demonstrated capabilities of solar

sail spacecraft once they take flight.

iii Dedication

Dedicated to the memory of my late grandmother Eleanor R. Gero, who has been there for all of my accomplishments up until this one.

iv Acknowledgements

To my advisor, Dr. Richard J. Freuler, I wish to extend the deepest sense of gratitude. You have been with me since the very beginning of my college career, and you looked on as my team’s robot that was built for the annual Freshman Engineering Honors program competition could not decide where it wanted to go or what it wanted to do.

Several years later you watched as I could not decide where I wanted to go academically or what I wanted to do professionally. I think I’m headed in the right direction now, and you have been a significant source of guidance.

To the professor of my first aeronautical and astronautical engineering course, Dr.

Gerald M. Gregorek, know that you are regarded by me as a beacon of lucidity.

Together, you and Dr. Freuler always managed to keep my attention focused on aerospace as I wandered through many different fields of study as an undergraduate.

Know that your signatures of approval concerning my research give me great pride.

To the graduate studies committee chaired by Dr. Mo-How Herman Shen, and the graduate program coordinator Ms. Carol Scott, I thank you for the opportunity to finish my degree after many years of absence. Without your help none of this would have been possible.

Lastly and most importantly, I am fortunate to have always had the support and encouragement of my family and close friends. For this I am eternally indebted, and promise to return the favor whenever and however I can.

v Vita

December 9, 1980 ...... Born - New London, Connecticut, USA

1999 ...... Floyd E. Kellam High School, Virginia

Beach, Virginia

2000 – 2004 ...... Undergraduate Teaching Assistant,

Freshman Engineering Honors Program,

The Ohio State University

2004 ...... B.S. Engineering Physics, B.S. Applied

Mathematics, Astronomy, The Ohio

State University

2004 – 2006 ...... Designated Student Naval Aviator,

United States Navy

2006 – present ...... Designated Naval Aviator,

United States Navy

Fields of Study

Major Field: Aeronautical and Astronautical Engineering

Other Fields: Engineering Physics Applied Mathematics Astronomy

vi Table of Contents

Abstract ...... ii

Dedication ...... iv

Acknowledgements ...... v

Vita ...... vi

List of Tables ...... ix

List of Figures ...... x

Nomenclature ...... xii

Chapters:

1 INTRODUCTION ...... 1 1.1 Hunting Halley ...... 1 1.2 Keeping the Embers Burning ...... 3 1.3 Experiments in Space ...... 4 1.4 Less is More ...... 6 1.5 Motivation for this Thesis ...... 8

2 SOLAR SAIL DESIGNS ...... 11 2.1 Square Sail ...... 11 2.2 Heliogyro ...... 13 2.3 Disc Sail ...... 15

3 ANALYTICAL FOUNDATIONS ...... 17 3.1 A Quantum Portrayal of Radiation Pressure Physics ...... 18 3.1.1 Momentum Transport by a Single Photon ...... 18 3.1.2 Momentum Transport by a Flux of Photons ...... 18 3.2 Solar Sail Force Model ...... 19 3.2.1 The Solar Sail as a Perfect Reflector ...... 20 3.2.2 Effects of Imperfect Reflection on a Solar Sail ...... 22

vii 3.3 Characterizing Solar Sail Performance ...... 23 3.3.1 Solar Sail Buoyancy Ratio ...... 23 3.3.2 Critical Solar Sail Burden ...... 24 3.3.3 A Realistic Performance Estimate ...... 24 3.4 The Solar Sail vs. Conventional Spacecraft ...... 26 3.4.1 Assessing Conventional Rockets ...... 26 3.4.2 Solar Sail Equivalent Specific Impulse ...... 27 3.4.3 Substantiating a Worthwhile Venture ...... 28 3.5 Planet-Centered Non-Keplerian Orbital Dynamics ...... 29 3.5.1 Equations of Motion ...... 30 3.5.2 Orbit Optimization ...... 34 3.5.3 Variational Equations ...... 36

4 STABILITY ...... 40 4.1 The Characteristic Polynomial ...... 40 4.2 Forbidden Regions in the - z Plane ...... 41

5 CONTROLLABILITY ...... 43 5.1 Modifying the Variational Equation ...... 43 5.2 Modeling in State Space ...... 44 5.3 State Controllability ...... 45 5.4 Output Controllability ...... 46

6 OBSERVABILITY ...... 49 6.1 Observability Matrix ...... 49 6.2 Displacement Outputs ...... 50 6.3 Rate Outputs ...... 50

7 STATE FEEDBACK ...... 52 7.1 Gain Matrix Definition ...... 52 7.2 Transfer Function Determination ...... 53 7.3 Pole-Zero Configuration ...... 55 7.4 Pole Placement ...... 60 7.4.1 Gain Determination ...... 61 7.4.2 Low-Impact Control Method ...... 63 7.4.3 Testing for Lowest Impact ...... 64 7.4.4 Damped Control Method ...... 65

8 SUMMARY AND CONCLUSION ...... 70

References ...... 72

viii List of Tables

Table 1.1 Chronological History of Solar Sail Missions to Space ...... 7

Table 3.1 Solar Sail Equivalent Specific Impulses for Selected Mission Durations and

Characteristic Accelerations (√p = 1/5) ...... 28

Table 8.1 Comparison of Low-Impact and Damped Gain Components for the Cosmos 1 and NanoSail-D Spacecraft on Selected Earth-Centered Optimal Non-Keplerian Orbits (0.7 > -ln[] > 0.2 for 0.5 <  < 0.8) ...... 71

ix List of Figures

Figure 1.1 Artist’s Conception of the Comet Halley Square Sail Design ...... 2

Figure 1.2 Artist’s Conception of the Comet Halley Heliogyro Design ...... 3

Figure 1.3 Artist’s Conception of a Solar Sail Race to the Moon ...... 4

Figure 1.4 Artist’s Rendering of a Solar Sail Delivery Vehicle Orbiting Mars ...... 5

Figure 1.5 Deployment Test of the Znamya 2 Spacecraft, 4 Feb 1993 ...... 6

Figure 1.6 Deployment Test of the IAE Reflector, 19 May 1996 ...... 8

Figure 1.7 Model of a Solar Sail on Display at K2001 ...... 9

Figure 1.8 Artist’s Rendering of Cosmos 1 in Orbit ...... 10

Figure 2.1 Mathematica Drawing of a Square Sail Design with Four Control Vanes . . 12

Figure 2.2 Mathematica Drawing of a Heliogyro ...... 14

Figure 2.3 Mathematica Drawing of a Disc Sail Design with Control Boom ...... 15

Figure 3.1 Mathematica Drawing of a Perfectly Reflecting Flat Sail Film ...... 20

Figure 3.2 Planet-Centered Non-Keplerian Orbit Diagram ...... 30

Figure 3.3 Relative Orientations Between the Unshared Basis Vectors of the Inertial and Rotating Reference Frames ...... 31

Figure 4.1 Depiction of the Torus-like Shape Encompassing Bounded Oscillations for a Solar Sail on a Stable Optimal Non-Keplerian Planet-Centered Orbit ...... 41

Figure 5.1 Block Diagram of the Solar Sail Pitch Attitude Control System in State Space ...... 45

x Figure 5.2 Block Diagram of the Solar Sail Pitch Attitude Control System in State Space with an Output ...... 46

Figure 7.1 Block Diagram of the Solar Sail Pitch Attitude Control System in State Space with State Feedback Control ...... 53

Figure 7.2 Control System Diagrams resulting from Consecutive Loop Elimination . . 54

Figure 7.3 Pole-Zero Configuration Continuum ...... 55

Figure 7.4 Unstable Open-Loop Pole-Zero Configuration for (0 <  < 2z) . . . . . 56

Figure 7.5 Unstable Open-Loop Pole-Zero Configuration for ( = 2z) ...... 57

Figure 7.6 Unstable Open-Loop Pole-Zero Configuration for (2z <  < 22z) . 58

Figure 7.7 Stable Open-Loop Pole-Zero Configuration for ( = 22z) ...... 59

Figure 7.8 Stable Open-Loop Pole-Zero Configuration for ( > 22z) ...... 60

Figure 7.9 Stable Damped Closed-Loop Pole-Zero Configuration for ( < 2z) . . . 66

Figure 7.10 Stable Damped Closed-Loop Pole-Zero Configuration for ( = 2z) . . 67

Figure 7.11 Stable Damped Closed-Loop Pole-Zero Configuration for ( > 2z). . . 68

xi Nomenclature

Constants c Speed of Light G Universal Gravitational Constant g Earth Gravity g0 Characteristic Solar Gravity

LS Solar Luminosity

MS Mass of the Sun  Pi

S Critical Solar Sail Burden

Parameters in Order of Appearance

E Total Energy m,M Mass p Momentum  Energy Flux r Distance between Bodies A Sail Film Surface Area t Time P Radiation Pressure n Sail Normal Vector t Sail Tangential Vector r Reflected Photon Vector s Incident Photon Vector  Solar Sail Pitch Angle f Radiation Pressure Force Vector a Radiation Pressure Acceleration Vector  Solar Sail Buoyancy Ratio  Solar Sail Burden  Solar Sail Efficiency √ Mass Fraction

a0 Solar Sail Characteristic Acceleration v Speed

Isp Specific Impulse T Mission Duration

xii x, y, z Cartesian Coordinates , , z Cylindrical Polar Coordinates  Angular Velocity Vector r Position Vector V Gravitational Potential C Compound Potential  Mean Radiation Pressure Acceleration  Keplerian Orbit Angular Speed  Perturbation Vector , ,  Infinitesimal Perturbation Components  Linear Expansion Matrix  Cross Product Matrix  Preferred Variational Matrix c1, c2 Transformed Perturbation Constants  Eigenvalue I Identity Matrix 0 Null Vector P Pitch Control Vector x State Vector u Control Signal A, B, C, D State Space Equation Arguments M Controllability Matrix  Finite Time Interval y Output Vector  Output Controllability Matrix O Observability Matrix K State Feedback Gain Matrix s Laplace Transformation Frequency G(s) Transfer Function Vector Y(s) Laplace Transformed Output Vector U(s) Laplace Transformed Control Signal

pi,i=1,2,… Transfer Function Poles

zi,i=1,2,… Transfer Function Zeros T Transformation Matrix W Characteristic Polynomial Matrix  Characteristic Coefficient Vector

an Open-Loop Characteristic Coefficients

n Closed-Loop Characteristic Coefficients f(s) Characteristic Equation

n Undamped Natural Frequency  Damping Ratio  Pole Rotation Angle

xiii Subscripts

0 Constant, At Rest, or in Equilibrium E Earth f Final i Initial P Planetary p Payload r Reflected S Solar s Incident from the Sun

Superscripts

` Denotes a Unit Vector  d/dt Operator ’ Relative to the Rotating Frame * Relative to the Transformed Perturbation

xiv 1 INTRODUCTION

Solar sailing is not a new concept, and many aspects of its history have been made available thanks to Dr. Colin R. McInnes.1 The idea has actually been around since the

1920s when the Russians first considered using light pressure as a means of propulsion.

Shortly after James Clerk Maxwell made public his discoveries regarding electromagnetic radiation, visionaries like Konstantin Tsiolkovsky and Fridrickh

Arturovich Tsander were dreaming of giant clipper ships in the sky. Back then the solar sail was thought to be a romantic idea, albeit not a very practical one. The concept sat on the back burner for some fifty years until it emerged as a novel idea again in the seventies.

1.1 Hunting Halley

At the forefront of the solar sailing scene in the seventies was Jerome Wright from

Battelle Memorial Institute in Columbus, Ohio.2 He found a way to meet up with

Halley’s Comet in only four years using a solar sail. Before this astonishing discovery, no one imagined that a complex rendezvous mission on such a small time scale could be feasible. A comet Halley rendezvous mission using solar-electric ion propulsion was already on the drawing board, but it would take seven or eight years to make that trip.

Given this information, the National Aeronautics and Space Administration (NASA)

Figure 1.1 Artist’s Conception of the Comet Halley Square Sail Design (NASA/JPL).

became intrigued with solar sails, and a small study began at Jet Propulsion Laboratory

(JPL) in 1975 under director Bruce Murray.

A formal proposal to use solar sail propulsion for the comet Halley rendezvous mission was submitted to NASA management in September 1976, and by November of that same year it had been approved and plans set in motion. Originally hatched as an

800 x 800 m square sail design, in May 1977 this idea was abandoned due to deployment concerns in favor of a spin-stabilized heliogyro configuration with a dozen 7.5 km blades

(see Figures 1.1 and 1.2).

Competing head-to-head for funding, it eventually came down to a decision between the solar-sail and solar-electric propulsion concepts to determine which design NASA would move forward with. In September 1977 the solar-electric system, with a larger

support group both within NASA and in industry, was selected for its smaller perceived 2

Figure 1.2 Artist’s Conception of the Comet Halley Heliogyro Design (NASA/JPL).

risk of failure. While disappointing to its proponents, this verdict by itself did not

severely impede the forward progress of solar sail technology. It wasn’t until later on

down the road, when the entire comet Halley rendezvous mission was scrapped due to

rising cost estimates, that NASA halted research in the solar sailing arena for a time.

1.2 Keeping the Embers Burning

The concept of solar sailing has been a long while in the making, but even now as

this thesis is being written, a true solar sail has yet to be flown. Throughout history

NASA has rejected it three times as a means of propulsion for spacecraft. In defiance, a

number of NASA teams, private research groups, and foreign space agencies remain confident in solar sailing as a realistic technology and have continued research despite

numerous setbacks. The World Space Federation (WSF) was formed in 1979 to ensure

solar sailing didn’t fall by the wayside. The organization was started by JPL engineer 3

Figure 1.3 Artist’s Conception of a Solar Sail Race to the Moon (U3P).

Robert Staehle and several other former members of JPL’s solar sail design team. In

1981 the Union pour la Promotion de la Propulsion Photonique (U3P) proposed a

motivating race to the moon in order to spur on development of solar sail technology (see

Figure 1.3). The Solar Sail Union of Japan (SSUJ) came into being in 1982, and when the quincentenary of the discovery of the New World took place in 1992 the US

Columbus Quincentennial Jubilee commission suggested a solar sail race to Mars (see

Figure 1.4). It is hard to say why, but none of these solar sail races ever materialized.

Even so, they are responsible for a much needed revitalization of solar sailing following the letdown after the comet Halley debacle.

1.3 Experiments in Space

Advancements have been occurring one piece of technology at a time, with some

notable developments during the 1990s. In February 1993 Vladimir Syromiatnikov and

the Russian Space Regatta Consortium successfully deployed a rotating 20 m reflector

with an on-board electric motor for autonomous opening (see Figure 1.5). Known as the

Figure 1.4 Artist’s Rendering of a Solar Sail Delivery Vehicle Orbiting Mars (NASA/JPL).

Znamya experiment (‘banner’ or ‘flag’ in English), this early demonstration of solar sailing technologies actually came secondary to the primary objective of illuminating northern Russian cities during dark winter months in hopes of bolstering economic development.

Another reflector was sent to space hot on the heels of the first Znamya. In May

1996 a 14 m diameter Inflatable Antenna Experiment (IAE) was scheduled aboard the

STS-77 Space Shuttle mission (see Figure 1.6). The deployment sequence was botched up, however, when some of the trapped air in the stowed film vented into space. The experiment was not a complete failure, although the antenna did lack the kind of precision its designers were hoping for. Researchers continue to exhibit confidence in the strong and dependable operation of inflatable structures in space, and many newer solar sail designs have been inspired by these early trials.

Figure 1.5 Deployment Test of the Znamya 2 Spacecraft, 4 Feb 1993 (SRC Energia).

The two experiments mentioned above are not the only tests being done in space to further solar sailing technology, but they do have the distinction of being the first. At the current stage of solar sail development, apart from the occasional launch vehicle failure, the most difficult design aspect encountered is still the deployment process. Solar sails

have proven to be so delicate and finicky that most missions undertaken by NASA and its

Russian and Japanese counterparts focus on testing mechanisms for deployment rather

than demonstrating propulsion capabilities. Reliability first, followed by proof of

concept – with deployment statistics hovering around a thirty percent success rate, solar

sails still have a ways to go (see Table 1.1).

1.4 Less is More

Mankind has only just begun to explore the solar system and the vast expanse of the

universe beyond. Our journeys are limited by the time it takes to travel between planets and other solar systems, and also by the small size of the payloads that conventional rockets can carry. For example, the U.S. space shuttle and the Russian Proton spacecraft 6 Table 1.1 Chronological History of Solar Sail Missions to Space. Date Project Agency Summary 1993 – Feb – 4 Znamya 2 RSA1 Success; 20 m reflector tested from Mir space station 1996 – May – 19 IAE NASA Failure; 14 m Inflatable Antenna Experiment failed to deploy properly 1999 – Feb – 5 Znamya 2.5 RSA Failure ; 25 m reflector de-orbited after getting caught on Mir antenna 2004 – Aug – 9 S-310-34 ISAS2 Success; clover and fan-type solar sail deployment mechanisms tested 2005 – Jun – 21 Cosmos 1 TPS / CS / RAS3 Failure; 600 m2 heliogyro lost when its delivery vehicle shut down prematurely 2006 – Feb – 21 SSP JAXA4 Fa ilure; 15 m solar sail delivered to orbit but failed to open completely 2008 – Aug – 3 NanoSail-D NASA Failure; 9.3 m2 solar sail weighing less than 10 lbs, lost during launch failure

are only able to ferry 5 tons to Mars or 2 tons to Jupiter. In order to take space exploration to the next level, we need to find a means of propelling spacecraft that uses less fuel. This would in turn allow for lar ger payloads.

Solar sails provide the capability to fly in space free of propellant and free of mechanical assistance. This translates to more room for payloads, and together with the advantages inherent in the unique design of the solar sail a complete package is ready to take space exploration to the next level. As payloads miniaturize and the manufacturing of extremely light weight reflective materials matures (see Figure 1.7), solar sailing becomes a more believable technology. Furthermore, as spacecraft miniaturization progresses, solar sails take on a size not only more humble in appearance, but more importantly, easier to deploy. With all of these advances available solar sails are no longer massive, unruly structures fit only for the imagination or for some future day and

1 Russian Federal Space Agency 2 Japanese Institute of Space and Astronautical Science 3 The Planetary Society / Cosmos Studios / Russian Academy of Science 4 Japan Aerospace Exploration Agency 7

Figure 1.6 Deployment Test of the IAE Reflector, 19 May 1996 (NASA/JPL).

age when mankind has the ability to design and build the impossible. Solar sails have the

potential to become reality within the next five to ten years.

1.5 Motivation for this Thesis

Recently an event took place that could have been the compelling factor that solar

sailing has been waiting for. From beneath the Barents Sea a Russian nuclear submarine launched Cosmos 1, the world’s first true solar sail, inside a Volna missile left over from

the Soviet Cold War arsenal (see Figure 1.8). Experienced scientists and engineers from

the United States and Russia worked together under the guidance of The Planetary

Society with funding from Cosmos Studios in an attempt to transform the dreams of

Tsander, Tsiolkovsky and many others into reality. Unfortunately, their efforts were

thwarted by the aforementioned and ever present enemy...launch vehicle failure. Cosmos

1 never made it to earth orbit. Its eight gigantic, reflective blades never had a chance to

unfurl and harness the sun’s energy.

Figure 1.7 Model of a Solar Sail on Display at K2001 (ESA).

The Babakin Space Center and the Space Research Institute in Russia collaborated to

build Cosmos 1. Their hope was to accomplish a controlled flight which would

demonstrate to the world once and for all the feasibility of the solar sail concept. Their

achievement was building and launching a spacecraft with the potential to bring

interstellar travel within reach, but unfortunately they fell just short of displaying its capabilities. In the words of Louis Friedman, the Cosmos 1 project director, “To get as

far as we did was—by itself—a great accomplishment, and The Planetary Society board,

staff, and technical team, together with our partners at Cosmos Studios, are dedicated to

trying again.”3

In the spirit of the Cosmos mission, the intent of this work is to pick up theoretically

where Cosmos 1 never got to go experimentally and investigate the stability of a solar sail

in a planet-centered non-Keplerian orbit. The hope is to further show the controllability

Figure 1.8 Artist’s Rendering of Cosmos 1 in Orbit (Babakin Space Center/The Planetary Society).

of solar sail spacecraft in their most challenging environment – the steep gravity well that exists near a planet or some faraway deep space object. At this moment, solar sailing is the only technology in existence that can take spacecraft far outside the solar system, and hopefully, someday, it will.

10 2 SOLAR SAIL DESIGNS

The prime focus for any solar sail design is to present a large, flat reflective surface

while using the smallest amount of structural support mass possible. Ease of

manufacturing and reliability of deployment are secondary yet very important

considerations. Factors such as these can make or break a successful solar sail design.

Tension forces at the edges of the sail film are what keep the surface flat, and there is

some leeway as to how these forces can be applied. Cantilevered spars are one way to do

it, or the solar sail can also be spun to generate the tensile forces. A combination of both

mechanical and centripetal forces is also feasible. Following are descriptions of three of the most basic solar sail designs.

2.1 Square Sail

Chronologically first in the anthology of solar sail designs, the square sail was the

configuration chosen for the comet Halley rendezvous mission. The best possible

arrangement for this concept utilizes four deployable spars cantilevered from a central

load-bearing hub. The payload is fixed to this hub, as well as the spar deployment

system. In order to increase performance, the spar deployment system should be

jettisoned after setting up the sail. Attitude control for the spacecraft can be provided by

articulated control vanes attached to the tips of the spars, or by an articulated control

Figure 2.1 Mathematica Drawing of a Square Sail Design with Four Control Vanes.

boom that offsets the center-of-mass and center-of-pressure of the sail. The comparatively rudimentary design of the square solar sail bestows it with an innate ruggedness and ease of control (see Figure 2.1). The difficulty is making sure the spacecraft is packed soundly and deploys properly. There are such a large number of steps to be completed in the deployment process, with each item dependent upon the success of its predecessors, that the opportunity for failure is quite substantial. The procedure can be compared to breaking out the fancy lights during the holiday season 12 after a year of storage and having all of the serial strands light up without any

troubleshooting.

2.2 Heliogyro

A sail film under pressure is not a very heavy burden, but the spars used in the

square sail design must be rigid enough to support any bending loads encountered. As a

result, the spars can contribute significantly to the overall mass of the solar sail. Using

spin-induced tension instead of mechanical means to maintain a flat, uniform reflective

surface may be more appealing to some. In the heliogyro design a number of long,

slender blades attached to a central load-bearing hub support the reflective sail film (see

Figure 2.2). The surface is kept flat by slowly spinning the spacecraft. The blades of the

heliogyro can be rotated in a cyclic fashion to induce torques that will precess the spin

axis of the vehicle, much like a helicopter. Even though centripetal forces are used to

provide tension on the sail film instead of mechanical spars, the heliogyro blades must maintain some rigidity in order to handle radial loads and permit cyclic blade rotations.

This may thwart efforts to reduce overall mass. The chief virtues of the heliogyro design are ease of packing and deployment. Each of the blades can be rolled up during manufacture, and then when the structure is spun during deployment the blades will unroll. The solar sail Cosmos 1 designed by The Planetary Society is a heliogyro variant that used triangular-shaped inflatable blades instead of long, slender rectangles. With each of its panels flattened to zero degrees of pitch angle, the outline of the world’s first constructed solar sail appeared disc-like, though in truth it was an octagon. This brings us to the final concept.

Figure 2.2 Mathematica Drawing of a Heliogyro. 14

Figure 2.3 Mathematica Drawing of a Disc Sail Design with Control Boom.

2.3 Disc Sail

The disc solar sail is characterized by either a continuous film, or sections of film, held flat by centripetal force (see Figure 2.3). It can have a reduced structural mass, much like the heliogyro, since it uses spin-induced tension instead of mechanical spars, but the disc sail doesn’t have long, slender blades to be concerned with. The articulated control boom mentioned earlier is the best way to provide attitude control for the disc sail. Although the bending loads are quite small, it will be necessary for the disc sail to maintain some degree of stiffness during precession. One way to accomplish this is to 15 use radial spars, or a hoop structure can be used to provide edge tension. The disc solar sail allows for a very appealing passive deployment method, which would eliminate most of the complications present during the packing and deployment process. If flexible radial spars are wound around the central hub during manufacture, then the elastic energy could be used to unfurl the sail film during deployment. Provided the sail doesn’t get caught on the way out, this one-step process has many significant advantages.

There has only been one solar sail actually constructed to date, The Planetary

Society’s Cosmos 1, and in its final form this spacecraft incorporated features from each of these basic layouts. The square sail, heliogyro, and disc sail are something of an underpinning upon which newer and better solar sail designs can be derived.

16 3 ANALYTICAL FOUNDATIONS

The building blocks of this thesis take shape in the paragraphs to follow, beginning

with a simple model for solar radiation pressure. Any thorough examination, no matter what the degree of complexity, must first begin modestly with fundamental assumptions and approximations about the phenomenon to be studied. Intricacies can be added in later, once a general understanding has been accomplished, or left out completely when they are found to be inconsequential. It may seem that a number of shortcuts are taken early on in the development of this thesis, and for these a very broad explanation is offered. The stability of a solar sail in an irregular orbit against “perturbations” by use of feedback control laws is what is being tested. These “perturbations” can be considered to

be the manifestation of incomplete modeling for any combination of the following

elements: solar radiation pressure; solar sail force; solar, lunar, or planetary gravitational

fields; solar wind; or some other imperfection, known or unknown, that happens to be a

part of the real world. Simplifications will be justified whenever possible, but for all the

other instances it will be chalked up to stability and control privilege. It should be

mentioned that several developments due to Dr. Colin R. McInnes have been adapted or

borrowed in this section1, and mechanical and mathematical methods were referenced frequently throughout its formation.4,5

17 3.1 A Quantum Portrayal of Radiation Pressure Physics

According to quantum mechanics, light is composed of quantum packets of energy

commonly referred to as photons. These energy packets transport momentum, and this is

one way to explain the concept of radiation pressure.

3.1.1 Momentum Transport by a Single Photon

The equivalence principle of special relativity describes the total energy E of a

moving body in terms of its rest energy and its energy of motion by the equation

2 2 4 2 2 E = m0 c + p c (3.1)

8 where m0 is the rest mass of the body, p is its momentum, and c = 2.998 × 10 m / s is the speed of light in free space. Given that a photon has zero rest mass, its energy is due exclusively to the transport of momentum:

E = p c (3.2)

3.1.2 Momentum Transport by a Flux of Photons

The pressure exerted on an extended body is calculated by considering the

momentum transported by a flux of photons. If the Sun is taken to be a single point

source of light that radiates equally in all directions, then the solar energy flux S (energy crossing unit area in unit time) at a distance r from its center can be expressed in terms of

its solar luminosity LS as

2 S = LS / (4r ) (3.3)

26 where LS = 3.839 × 10 W. In actuality, the Sun is sizable enough to appear as a

disc even to the naked eye and furthermore its surface is non-Lambertian. A Lambertian

18 surface is one which appears equally bright to a viewer from any angle, and the Sun’s limbs appear darker than its center when viewed obliquely so it doesn’t quite behave this way. Later on in this thesis the solar radiation field will be taken to be uniform since it falls several orders of magnitude short of matching the variance of a planetary gravitational field on the orbital scale. So, with this information, it makes no sense to strive for painstaking accuracy at this stage. Proceeding on to calculate the energy E incident on a surface of area A normal to the Sun’s rays in time t we use

E = S A t (3.4)

Exploiting Eq (3.2) we find the momentum p transported by a flux of photons to be

p = E / c (3.5)

By definition, the pressure exerted on the surface of an extended body is equal to the momentum transported per unit area per unit time, or

P = p / (A t) (3.6)

By combining Eqs (3.4), (3.5), and (3.6) the surface pressure due to a flux of photons can be cast in the much more concise form

P = S / c (3.7)

3.2 Solar Sail Force Model

Again, intending to focus on the stability and controllability of the solar sail, very little generality is lost in the assumption of perfect reflection. That is why the effects of an imperfect reflector are ignored in the solar sail force model used for this thesis.

However, since it is a simple matter to discuss, these effects will be mentioned briefly after developing the force model for an ideal solar sail.

19 ` ` t n

s` Sail

Figure 3.1 Mathematica Drawing of a Perfectly Reflecting Flat Sail Film.

3.2.1 The Solar Sail as a Perfect Reflector ` ` Let the unit vectors n and t define the orientation of the solar sail normal and tangential, respectively. Additionally, the direction of travel of photons will be given by ` ` the unit vectors r for reflected and s for incident along the Sun-sail line (see Figure 3.1).

The angle  is the angle between the incident photons and the sail normal, also referred to as the solar sail pitch angle. In order to find the total force on an ideal solar sail due to photon reflection, we must first consider the force due to incident photons: ` ` ` fs = P A (s ◊ n) s (3.8)

20 where the solar radiation pressure P has been multiplied by the projection of the solar sail ` ` surface area A (s ◊ n) along the Sun-sail line to give the magnitude of the incident ` force. As these incident photons are reflected along r, an equal and opposite reaction force propels the solar sail according to ` ` ` fr = -P A (s ◊ n) r (3.9)

The net force on the solar sail due to perfect reflection is just the sum of these two

components, or

f = fs + fr (3.10) ` ` Adding Eqs (3.8) and (3.9) yields a term (s - r), which can be simplified by

projecting the two vectors along the normal and tangential components of the solar sail

and evaluating the expression. Here are the vector equations after projection: ` ` ` s = cos[ n + sin[ t (3.11a) ` ` ` r = -cos[ n + sin[ t (3.11b)

Subtracting Eq (3.11b) from Eq (3.11a) yields ` ` ` (s - r) = 2 cos[ n (3.12)

Now, using the definition of the inner product ` ` (s ◊ n) = cos[ (3.13)

and combining it with the developments of Eqs (3.8), (3.9), (3.10), and (3.12) the solar

radiation pressure force due to a perfectly reflecting solar sail can be written as ` ` ` f = 2 P A (s ◊ n)2 n (3.14)

21 3.2.2 Effects of Imperfect Reflection on a Solar Sail

Accounting for the optical behaviors and material properties of a sail film provides

increased accuracy in determining the magnitude and direction of the solar radiation

pressure force. These additional rigors result in the calculation of a solar radiation

pressure force that, relative to the force f in Eq (3.14) has reduced magnitude and an ` orientation slightly off-axis relative to the sail normal vector n.

The optical solar sail force model begins the same way as the ideal model with a

force due to incident photons along the Sun-sail line according to Eq (3.8). A small

fraction of these photons are absorbed by the sail film in the optical model, however, so

that the magnitude of the reflected force given by Eq (3.9) is reduced. This disparity

between incident and reflected force magnitudes provides the most straightforward

explanation for the off-axis orientation and reduced size of the solar radiation pressure ` force. The direction of the resultant force is tilted from the sail normal n toward the Sun- ` sail vector s owing to the stronger influence of incident photons relative to reflected

photons, and the reduction in number of reflected photons due to absorption weakens the

overall pressure force.

To take this scrutiny one step further, the effects of non-specular reflection (uniform

scattering) and re-emission of radiation from the front and back surfaces of the solar sail

combine to further influence the orientation of the solar radiation pressure force and

reduce its magnitude. Still another factor to consider and incorporate into the production of a future solar sail is the billowing effect that a light, thin reflective film may experience when under pressure from solar radiation. Generally speaking, minimizing the impact of all the aforementioned deficiencies will be pivotal to improving the 22 performance of solar sails from one design to the next. However, control laws can be

written to account for a reduced, off-axis radiation pressure force so imperfect sail

reflection will not be included in the model used for this thesis.

3.3 Characterizing Solar Sail Performance

The motion of a solar sail within a planetary system is the result of a delicate balance

between gravity and the solar radiation pressure acting on the spacecraft. The

accelerations from both of these forces follow an inverse square law with distance, so that

it is easy to relate the two when they stem from the same source.

3.3.1 Solar Sail Buoyancy Ratio

For a solar sail in a heliocentric orbit, its acceleration due to solar radiation pressure

is best expressed in terms of the solar gravitational acceleration via

2 ` ` 2 ` a =  (G MS / r ) (s ◊ n) n (3.15)

-11 2 2 where G = 6.674 × 10 N ÿ m / kg is the universal gravitational constant, MS =

1.989 × 1030 kg is the mass of the Sun, and  is the solar sail buoyancy ratio. A solution for  can be obtained for an ideal solar sail using Newton’s 2nd Law (a = f /

m) and Eqs (3.3), (3.7), and (3.14):

 = LS / (2c G MS ) (3.16a)

 = m / A (3.16b)

where  is the mass per unit area, or sail burden, of a solar sail with mass m and reflective area A.

23 3.3.2 Critical Solar Sail Burden

It is now possible to crudely quantify the performance requirements that must be met by a solar sail in order for it to travel along unusual orbits such as the non-Keplerian

variety, or perhaps even “hover” in space when gravity and solar radiation pressure are

balanced. With a buoyancy ratio of unity, the solar radiation pressure on a solar sail ` ` oriented parallel to the Sun-sail line (n = s) exactly balances the solar gravitational

acceleration. Therefore, sustained voyages along high-performance orbits are effectively

unlocked when  = S according to

 = S /  (3.17a)

S = LS / (2c G MS) (3.17b)

2 where S = 1.535 g / m is the critical solar sail burden. In practice, for a high performance solar sail, the critical solar sail burden will be an upper design limit.

3.3.3 A Realistic Performance Estimate

Accounting for the various imperfections discussed in Section 3.2.2, an efficiency 

must be factored into the calculation of realistic solar sail acceleration. The performance

loss from this, shown in Eq (3.18a), must be offset following the logic of Eq (3.18b). The

end result is a reduction of the solar sail burden, consistent with Eq (3.18c), in order to

compensate and thus preserve the solar sail’s extraordinary capabilities:

a Ø  a (3.18a)

 2 \  Ø  ' ||a|| = (G MS / r ) (3.18b)

ﬂ  Ø  S (3.18c)

24 Assuming a solar sail efficiency of  = , a realistic solar sail burden of  =

1.228 g / m2 is required in order for radiation pressure to match the solar gravitational

acceleration.

At this point the feasibility of constructing high performance solar sails using

modern technology and materials is presented. We begin by estimating a generous

payload mass fraction of √p = 1/5, where

√p = mp / m (3.19)

In the equation above, the portion of total spacecraft mass m contributed by the payload is

mp. With the reflective film and support structure for the solar sail each contributing an equal share of the remaining 80 % of the spacecraft mass, a figure can be derived for the target surface density of potential reflective material candidates. Multiplying the realistic

solar sail burden from the paragraph above by 40 % yields a film surface density f =

0.491 g / m2. This is roughly 10 % of the surface density of currently available thin

film materials such as Kapton®, Mylar®, and Lexan® (trademark names of E. I. duPont de

Nemours & Co.). If we assume that in their infancy solar sails will fall short of design

aspirations by a factor of 10 across the board, just as the reflective material component has, then we can initially expect characteristic accelerations approximately 10 % of the solar gravitational pull. Using the mean Earth-Sun distance, or astronomical unit (1 AU

= 1.496 × 1011 m), as the base separation between Sun and sail, it follows that the

characteristic solar gravitational acceleration is

2 g0 = GS / AU (3.20)

2 with a numerical value of 5.931 mm / s . Taking 10 % of g0 and rounding up slightly,

2 the characteristic solar sail acceleration is found to be a0 @ 0.6 mm / s . This number 25 represents the performance that might be expected from the very first solar sails, with the

potential that as technology improves characteristic accelerations as high as 5 or 6 mm /

s2 can be reached.

3.4 The Solar Sail vs. Conventional Spacecraft

Evaluating solar sails in comparison with one another is a straightforward task using

the solar sail buoyancy ratio developed in the last section. This number doesn’t have any

equivalent in conventional spacecraft design, however. What metric, then, does an

engineer use to logically decide between using a solar sail or a conventional propulsion

system to get a payload where it needs to go? If the mission demands sustained orbits

that only solar sails can provide then the choice is obvious. It can be shown, however,

that even without considering their inimitable orbital capabilities, solar sails are often a

more efficient alternative to conventional vehicles for a typical space mission.

3.4.1 Assessing Conventional Rockets

The rocket equation contains the principle metric used among conventional spacecraft for comparison:

mf = mi exp[-v / (g Isp)] (3.21)

In the equation above mf is the final mass of a spacecraft after accelerating through a

speed change v resulting from the firing of its rockets at initial mass mi. The constant g

2 = 9.807 m / s is the acceleration due to gravity at the Earth’s surface, and Isp is the specific impulse of the engine used. With conventional rockets it is desirable to achieve the largest increase in speed per unit mass of propellant consumed, and specific impulse is the metric used to express this ratio. It is typically given in seconds, with larger 26 specific impulses corresponding to more efficient propulsion systems. Introducing the

mass fraction √ = mf / mi, Eq (3.21) may be rearranged to yield

-1 Isp = v / (g ln[√ ]) (3.22)

3.4.2 Solar Sail Equivalent Specific Impulse

At first glance it seems unfair to apply the specific impulse equation above to solar

sails due to the fact that their mass remains constant from one instant to the next. This

yields an infinite specific impulse, leading one to believe that solar sails should be used

for every mission in space from now until there is nothing left to explore. Avoiding such

a misrepresentation, it will now be shown that by taking the speed change v and mass

fraction √ over the span of an entire mission instead of just a short period of time,

conventional spacecraft and solar sails find themselves on equal footing for evaluation

using specific impulse. Furthermore, with this approximation solar sails are found to

have an infinite specific impulse only for infinite mission durations.

If the payload mass of a conventional spacecraft is separated from its total mass,

what remains is almost entirely propellant. Since one of the most useless things to a

conventional space vehicle at the end of a mission is unused propellant, it makes sense to

assume that for any properly planned mission the final mass of a spacecraft is comprised

of just its hollow shell with nothing but payload inside. Over the course of an entire

mission, then, the mass fraction √ reduces to the familiar payload mass fraction √p of Eq

(3.19). Furthermore, the speed change for a solar sail from beginning to end of a mission

(in the vicinity of one astronomical unit from the Sun) can be taken to be v ~ a0 T,

27 Table 3.1 Solar Sail Equivalent Specific Impulses for Selected Mission Durations and Characteristic

Accelerations (√p = 1/5).

2 T (days) a0 (mm / s ) Isp (s) 30 0.6 99 1.5 246 3 493 6 985

180 0.6 591 1.5 1478 3 2956 6 5912

365 0.6 1199 1.5 2997 3 5994 6 11988

where a0 is the characteristic acceleration from Section 3.3 and T is the mission duration.

The equivalent specific impulse of a solar sail, to be used for evaluation against

conventional spacecraft, then becomes

-1 Isp ~ a0 T / (g ln[√p ]) (3.23)

where the linear variation with mission duration gives rise to infinite specific impulses

only as T Ø ¶.

3.4.3 Substantiating a Worthwhile Venture

Solar sails accelerate at a much slower rate than conventional spacecraft, but when

their propellant is all gone the solar sail keeps on going. Will solar sails then only be

practical for missions to extremely remote locations? Table 3.1, showing solar sail

equivalent specific impulses for a variety of performance levels and mission durations,

indicates otherwise. A typical chemical rocket has a specific impulse between 250 and

28 450 seconds, with currently proven designs for nuclear thermal rockets and ion thrusters around 850 and 3000 seconds, respectively. It is clear from the results of Table 3.1 that the earliest solar sail designs will probably only be beneficial as a proof of concept and technology test bed, but there is a slight chance they might be chosen as a delivery vehicle for a mission with duration greater than a few years. Moderate and high- performance solar sails, on the other hand, are able to out-perform conventional propulsion systems for missions of much shorter length. Of course, specific impulse is not the only factor that must be considered when planning a mission in space. The findings in this section simply illustrate that solar sail technology has more than one advantage over modern space vehicles and may someday find its own niche in the space flight continuum.

3.5 Planet-Centered Non-Keplerian Orbital Dynamics

Keplerian orbits are a solution to the classic two-body problem in which only gravitational attraction is considered. Typically taking the form of ellipses, parabolas, and hyperbolas, solutions to the Kepler problem can also be solutions for solar sails in planet-centered orbits. However, accounting for the effects of solar radiation pressure in addition to gravitational attraction necessarily transforms the problem of solar sail orbital dynamics into an inherently non-Keplerian situation. New families of non-Keplerian orbits result from the solar sail’s ability to find equilibrium by pitting solar radiation against gravity.

29 y

Sail x Sun Planet r r

w z

Figure 3.2 Planet-Centered Non-Keplerian Orbit Diagram.

3.5.1 Equations of Motion

Consider a perfectly reflecting solar sail in a rotating frame of reference with origin centered on a planet approximated by a point mass. The inertial frame of reference for this problem will be the x, y, and z-axes, which share a common origin with the rotating frame spanned by the , , and z-axes (see Figures 3.2 and 3.3). The Sun-planet line (z- axis) thus defines the axis of rotation orbited by the solar sail with angular velocity vector

. The position vector r locates the solar sail center of mass relative to the origin in the inertial frame.

In order to construct the equations of motion it is necessary to find an expression for .. the acceleration r of the solar sail in the inertial reference frame. Building up to this one  step at a time, the velocity vector r in the inertial frame is calculated first via coordinate transformation. Using cylindrical polar coordinates, the position vector in the inertial

30 y

cos q

 

sin q ` q   r`

q x -sin q z cos q

    Figure 3.3 Relative Orientations Between the Unshared Basis Vectors of the Inertial and Rotating Reference Frames.

frame can be expressed as

r = { cos[], sin[],z} (3.24) where  is the radial distance of the solar sail from the z-axis, and  is the angle between ` the unit vector  and the x-axis (see Figure 3.3). Solving directly for  r = {(d/dt) rx,(d/dt) ry,(d/dt) rz} (3.25) the unrefined solution is       r = { cos[] -  sin[] , sin[] +  cos[] ,z} (3.26) ` ` By referencing Figure 3.3 the basis vectors  and  can be transformed in the inertial frame to `  = {cos[],sin[],0} (3.27a) `  = {-sin[],cos[],0} (3.27b) and this leads to the simplified expression   `  `  ` r =   + z z +    (3.28) 31 Noting that the position vector in the rotating (primed) reference frame is given by

r’ = {,,z}, (3.29) it follows that its primed time derivative (d’/dt) must be    r’ = {,,z}, (3.30)

and this accounts for the first two terms on the right hand side of Eq (3.28). Now, using

the vector cross product `  â r =    (3.31)  and also substituting  for , the last term on the right hand side of Eq (3.28) is put in

new light and the general relationship between time derivatives of rotating coordinate

systems emerges:   r = r’ +  â r (3.32)

Taking the time derivative (d/dt) with respect to the inertial reference frame of each

side of Eq (3.32), we have ..    r = (d’/dt) r +  â r +  â r (3.33) where the order of derivatives for the first term on the right hand side of the equation  above has been switched so that r from Eq (3.32) may be substituted twice to yield .. ..  r = r’ + 2  â r’ +  â ( â r) (3.34)

Eq (3.34) stems from the Coriolis theorem, where in this case any term involving a time

derivative of  has been thrown out due to conservation of angular momentum for a solar

sail in orbit.

It is now possible to express the non-linear vector equation for non-Keplerian motion

about a planet: ..  r’ + 2  â r’ +  â ( â r) = a - V (3.35)

32 where a is the solar radiation pressure acceleration mentioned in Section 3.3,  is the

gradient operator, and V is the planetary gravitational potential given by

V = -G MP / r (3.36)

Analogous to the effective potential in Newtonian dynamics, which incorporates angular momentum into the potential energy of a system, a compound potential function C can be created that includes both the centripetal and gravitational acceleration terms from Eq

(3.35):

  C = -(  / 2 + G MP / r) (3.37)

Just to verify, the vector cross product `  â ( â r) = -   (3.38)

should give the same solution as the gradient of the first term on the right hand side of Eq

(3.37). In cylindrical polar coordinates, the gradient operator takes the form ` ` `  = (/)  + (1/)(/)  + (/z) z (3.39)

and it is plain to see that using the compound potential is indeed equivalent to the vector

triple product. Eq (3.35) can now be written as ..  r’ + 2  â r’ + C = a (3.40)

Finally, it must not be forgotten that the solar radiation field is approximately uniform

over the distance span of the orbits under consideration. Thus the expression for solar

radiation pressure acceleration greatly simplifies to ` ` ` a =  (s ◊ n)2 n (3.41a)

2  = LS / (2  c  rP ) (3.41b)

where  is the mean radiation pressure acceleration at distance rP equal to the average

separation of the Sun and the orbited planet.

33 3.5.2 Orbit Optimization

Once in orbit, the orientation of a solar sail must be adjusted so as to align the solar

radiation pressure a with the compound potential gradient C. Furthermore, the only

allowable trajectories are those for which ||a|| = ||C||, so that the solar sail reaches

equilibrium in the rotating reference frame and the first two terms of Eq (3.40) disappear.

The solar sail normal vector can then be expressed as ` n = C / ||C|| (3.42)

with ` ` C =  (2 - )  + z 2 z (3.43a)

2 3  = G MP / r (3.43b)

r = r’ = (2 + z2)1/2 (3.43c)

where  is the Keplerian orbit angular speed about the planet with semimajor axis equal

to r. Now that the compound potential gradient has been identified as a scalar multiple ` ` of n, cross products and dot products with s can be used to determine the attitude of the

solar sail by its pitch angle : ` ` tan[] = ||s â C|| / s ◊ C (3.44) ` ` Since s = z, Eq (3.44) evaluates to

tan[] = ( / z) (1 – ( / )2) (3.45)

Thus for a given elevation z and orbit radius , the attitude of a solar sail is shown to be dependent upon its angular velocity . The question to ask now is: for what value of angular velocity is the solar radiation pressure requirement minimized to give an optimized orbit? In order to answer that, the parameter  must be evaluated in terms of

34 ` the angular velocity. This can be done by taking the dot product of Eq (3.40) with n,

again removing the first two terms to satisfy the equilibrium condition, with the results of

Eq (3.41a) used in place of the solar sail acceleration: ` ` `  = C ◊ n / (s ◊ n)2 (3.46)

The outcome above can be further reduced using Eq (3.42) and the trigonometric identity

1 + tan2[] = 1 / cos2[], yielding

 = ||C|| (1 + tan2[]) (3.47)

Finally, using

||C|| = z 2 (1 + ( / z)2 (1 - ( / )2)2)1/2 (3.48)

combined with the result of Eq (3.45), we have a solution for  that is useful:

 = z 2 (1 + ( / z)2 (1 - ( / )2)2)3/2 (3.49)

Now, by solving for / and setting it equal to zero, we find that  is minimized

and consequently an optimized orbit is obtained with

 =  (3.50)

This also means that for an optimized orbit

tan[] = 0 (3.51)

and

 = z 2 (3.52)

The solar sail normal is therefore perpendicular to the orbital plane for this family of

trajectories, aligned with the direction of incoming solar radiation. This is to be

expected, since the absolute minimum solar radiation pressure force required to keep the

solar sail elevated in its non-Keplerian planetary orbit is that which exactly balances the

sunward component of gravity. The transverse component of gravity then acts to keep

35 the motion bounded in the azimuthal plane. Noting the identical angular velocities, the optimal planet-centered non-Keplerian orbit was found to be synchronous with Keplerian orbits having semimajor axis equal to the distance r between planet and sail.

3.5.3 Variational Equations

The premise of this thesis requires the development of a model for solar sail orbit

perturbations. To accomplish this, the infinitesimal vector given by

’ = { } (3.53)

is added to the position vector r0’ located along a stable orbit trajectory (the subscript 0

will be used from this point forward to signify calculations made along equilibrium orbits). Next, the equations of motion developed in Section 3.5.1 are re-evaluated so that

criteria for bound solutions may ultimately be determined. The complete task is to solve

2 2 (d’ /dt )(r0’ + ’) + 2  â (d’/dt)(r0’ + ’) + C(r0’ + ’)

= a(r0’) (3.54)

where the perturbation is shown to have no effect on the solar radiation pressure

acceleration a since it has already been established as a constant for this situation. Now,

evaluating the time derivatives for r0’ and ’ individually, it is found that the terms

involving r0’ drop out since the orbit is an equilibrium solution. This leaves ..  ’ + 2  â ’ + C(r0’ + ’) = a(r0’) (3.55)

To proceed from this point a Taylor expansion of C about r0’ is taken out to first order

in ’, giving a linear approximation for C(r0’ + ’):

C(r0’ + ’)  C(r0’) + (’/r)0 C(r’) ’ + ... (3.56)

36 where the result of the operator (’/r)0 acting on C(r’), henceforth labeled , is

given by the 3 x 3 linear expansion matrix

∑ 1 ∑ ∑ “ Cr “ C r “ Cr ∑r r ∑q ∑ z ∑ 1 ∑ ∑  = “ C q “ C q “ C q (3.57) ∑r r ∑q ∑ z    ∑ 1 ∑ ∑   “Cz “Cz “Cz   ∑r r ∑q ∑ z 0       By stopping at first order in the Taylor series above, the family of solutions to Eq (3.55)   will not provide a complete set of constraints to ensure stability. The goal of proceeding

with these calculations then becomes to quantify absolute conditions for instability and to

show that stable solutions are possible for an optimal planet-centered non-Keplerian orbit

by exception. With the linear approximation, the new equations are ..  ’ +  ◊ ’ + C(r0’) +  ◊ ’ = a(r0’) (3.58)

where another 3 x 3 matrix has been introduced to replace the vector cross product:

0 2 0  = 2 00 (3.59) 000     Noting that the compound potential and solar sail acceleration evaluated along the   equilibrium orbit cancel each other out, the general variational equations may now be

written in vector form as ..  ’ +  ◊ ’ +  ◊ ’ = 0 (3.60) or in component form ..   - 2  0  + 11  + 13  = 0 (3.61a) ..  0  + 2   = 0 (3.61b) ..  + 31  + 33  = 0 (3.61c)

37 where according to Eq (3.43a) both the second row and the second column of  evaluate

to zeros since C = 0 and the other two components of C are azimuthally symmetric

(do not depend on the variable ). As for the components of  that do not evaluate to

zero, what remains is

2 2 2 2 11 =  (1 – 3  / r ) -  (3.62a)

2 2 13 = 31 = -3   z / r (3.62b)

2 2 2 33 =  (1 – 3 z / r ) (3.62c)

Now, taking the azimuthal symmetry advantage one step further, it is evident that by

integrating Eq (3.61b) to obtain  0  + 2  ( - 0) = 0 (3.63)

where 0 represents both constants of integration rolled up into one, the variational

equations may be reduced in number by substituting Eq (3.63) into Eq (3.61a):

.. 2 2  + (4  + 11)  + 13  = 4  0 (3.64a) ..  + 31  + 33  = 0 (3.64b)

It will be useful later on to manipulate this new set of variational equations in vector form, but first the 2 x 2 preferred variational matrix  must be introduced:

 =  11  12 (3.65) 21 22 where   2 2 2 2 2 11 = 4  + 11 = 3  +  (1 – 3  / r ) (3.66a)

12 = 13 (3.66b)

21 = 31 (3.66c)

22 = 33 (3.66d)

38 Next, the non-homogeneity of Eqs (3.64) needs to be remedied. There is a coordinate

transformation of the form

* = {**} (3.67a)

* =  + c1 (3.67b)

* =  + c2 (3.67c)

where c1 and c2 are constants, such that .. * +  ◊ * = 0 (3.68)

The equation above is the preferred vector representation of the variational equations. In

order to find c1 and c2, Eq (3.68) is evaluated while incorporating the results of Eqs

(3.64). This yields the set of equations

2 11 c1 + 12 c2 = -4  0 (3.69a)

21 c1 + 22 c2 = 0 (3.69b) which can be solved to obtain

2 c1 = -4  0 22 / det[] (3.70a)

2 c2 = 4  0 21 / det[] (3.70b)

where det[] is the matrix determinant expressed as 11 22 - 12 21.

39 4 STABILITY

Using a common method also adopted by Dr. Colin R. McInnes, solutions to Eq

(3.68) were investigated using

* = 0 exp[ t] (4.1)

1 where 0 is a constant vector and  represents the eigenvalues of the system. This produced the set of equations

(2 I + ) ◊ * = 0 (4.2) for which non-trivial solutions were assured only by satisfying

det[2 I + ] = 0 (4.3)

4.1 The Characteristic Polynomial

Evaluating Eq (4.3) generated a characteristic polynomial of the form

4 + tr[] 2 + det[] = 0 (4.4)

where tr[] is the trace of the matrix given by 11 + 22. An algebraic equation of fourth order in  meant that there were 4 eigenmodes that had to be kept on the left-hand side of the complex plane in order to guarantee bound oscillations in the variables  and

. The mathematical equivalent to this guarantee was to ensure both tr[] and det[] were positive, providing necessary but not sufficient conditions for stability.

Referring to Eqs (3.62) and (3.66) and substituting  =  for an optimized orbit, their

40 y

Sail x Sun

Planet r

Figure 4.1 Depiction of the Torus-like Shape Encompassing Bounded Oscillations for a Solar Sail on a Stable Optimal Non-Keplerian Planet-Centered Orbit.

values became

tr[] = 2 2 (4.5a)

det[] = 4 (1 – 9 z2 / r2) (4.5b)

4.2 Forbidden Regions in the - z Plane

The requirement for a positive trace was met without any sort of dependency on

orbital trajectory, as plainly seen by Eq (4.5a). The determinant, however, was found to be greater than zero only if

 > 22 z (4.6)

Consequently, an uncontrolled solar sail subject to a perturbation while on a trajectory

that lay within the cone defined by Eq (4.6) would absolutely diverge from its orbit.

However, solar sails traveling along a path outside of this cone might be expected to 41 spiral about the equilibrium trajectory while ultimately remaining within a torus-like boundary with maximum radial distance from the nominal orbit determined by the

amplitude vector 0.

42 5 CONTROLLABILITY

It was resolved to pursue control of the solar sail via adjustments in pitch attitude  due to the simplicity of the design. From Section 3.5.2 the nominal pitch angle for an optimized planet-centered non-Keplerian orbit is zero, so it makes sense to use a method other than changing the pitch to control the solar sail. One option which would minimize departure of the solar sail from its equilibrium orientation was sail area variation. The conceivable control laws and mechanisms used for trims to the sail area resulting in small acceleration changes  versus those for regulating the sail pitch were considered. It was decided that measuring  directly by sensing where the sun is relative to the sail normal and proceeding to effect a new sail orientation was probably much simpler than computing a desired sail area and altering the shape of what could be quite a sizable structure. In addition to being less complicated, the sun sensor with pitch changes was also a logical choice since it was less likely to be bulky and massive. The texts by Dr.

Katsuhiko Ogata and Dr. McInnes were referenced while generating this section.1,6

5.1 Modifying the Variational Equation

With control actions now contributing to the variational motion developed in Section

3.5.3, Eq (3.68) was transformed to yield .. * +  ◊ * = P  (5.1)

43 where P is the pitch control vector given by

P = {(/) a,(/) az} (5.2)

In the reduced two-dimensional arena that resulted from the act of eliminating the azimuthal dimension from the variational equations in Section 3.5.3, the solar sail received the freedom of unchecked tangential speed changes along the nominal orbit.

This compromise was inconsequential for stability and did not give rise to any new

constraints. With the azimuth effectively ignored, however, the radiation pressure

acceleration a could now be limited to only the  and z directions with no adverse

effects: ` ` a =  cos2[](sin[]  + cos[] z) (5.3)

The components of the pitch control vector were then

3 2 P1 =  cos [](1 – 2 tan []) (5.4a)

2 P2 = -3  cos [] sin[] (5.4b)

5.2 Modeling in State Space

Using a state space representation (see Figure 5.1) opened up a variety of tools that

were used to analyze the system. The familiar state variable format is  x = A ◊ x + B u (5.5)

and by choosing the state vector to be   x = {*,*,*,*} (5.6)

Figure 5.1 Block Diagram of the Solar Sail Pitch Attitude Control System in State Space.

the other arguments of Eq (5.5) could be determined in accordance with Eq (5.1):

0 1 0 0   A = 11 0 12 0 (5.7a) 0001  21 0 22 0        B = {0,P ,0,P }   1 2  (5.7b) u =  (5.7c)

5.3 State Controllability

The condition for complete state controllability of the system was that the 4 x 4 state controllability matrix M given by

M = {B|A ◊ B|A2 ◊ B|A3 ◊ B} (5.8) be of rank four, or contain four linearly independent column vectors. The solution for M turned out to be

0P1 0 P1 11  P2 12 M = P 1 0  P 1  11  P 2  12 0 (5.9) 0P2 0 P1 21  P2 22  P2 0 P1 21  P2 22 0            45

Figure 5.2 Block Diagram of the Solar Sail Pitch Attitude Control System in State Space with an Output.

and its column vectors could be characterized as linearly independent if its determinant was found to be nonzero:

2 2 2 det[M] = -(P1 21 + P1 P2 (22 - 11) – P2 12) (5.10)

Evaluating P1 and P2 with the constraint  = 0 for optimal orbits

P1 =  (5.11a)

P2 = 0 (5.11b)

and using the relation z 2 =  from Eq (3.52), Eq (5.10) reduced to

det[M] = -9 6 2 / r4 (5.12) which has a negative, nonzero value. Therefore, the state of the solar sail could be

transferred from an initial state x(t0) to any final state x(t0 + ) in the  - z plane within a finite time interval  by regulating its pitch angle with control signal u(t).

5.4 Output Controllability

It would be impractical to design a controller to adjust all four quantities of the

system state x when only the outputs * and * need to be controlled. Creating the

output vector y, the system (see Figure 5.2) could then be described by the state

equations 46  x = A ◊ x + B u (5.13a)

y = C ◊ x + D u (5.13b)

where

C = 1 0 0 0 (5.14) 0010 and   D = 0 (5.15)

Complete output controllability of a system refers to the ability of that system to generate

a control signal u(t) that will transfer an initial output y(t0) to any final output y(t0

+ ) within a finite interval of time . Complete output controllability does not follow

directly from complete state controllability, nor is complete state controllability necessary

to achieve complete output controllability. There is a separate test for each, and the

condition for complete output controllability of this system was that the 2 x 5 matrix

 = {C ◊ B|C ◊ A ◊ B|C ◊ A2 ◊ B|C ◊ A3 ◊ B|D} (5.16)

be of rank two. Evaluating the above matrix equation yielded

 = 0 P 1 0  P 1  11  P 2  12 0 (5.17) 0P2 0 P1 21  P2 22 0 and in order for the two nonzero column vectors to be linearly independent, the equation  

P1 / P2 = (P1 11 + P2 12) / (P1 21 + P2 22) (5.18) was required to be without a non-trivial solution. Linear independence of the two nonzero column vectors of the output controllability matrix  was verified with the constraints applied for optimal planet-centered non-Keplerian orbits with pitch control,

but the job was found to be much simpler to do by direct substitution of the values for P1

and P2 using Eqs (5.11) rather than solving Eq (5.18) directly:

 = 0  0   11 0 (5.19) 000 21 0 47   Complete output controllability of the solar sail was determined to be possible since the above expression showed the rank of  to be two.

48 6 OBSERVABILITY

Observable systems add flexibility to their design by allowing the engineer to choose

outputs that are easy to measure and still maintain the ability to construct state feedback

control signals using all of the state variables. Observability was tested for a solar sail on

an optimal planet-centered non-Keplerian orbit for two sets of outputs, the displacements  from the nominal orbit * and * as well as their corresponding displacement rates *  and *. While it is conceivable to use a combination displacement and rate output for

control in the  - z plane, it is not practical so these groupings were not considered. This

section includes control analysis techniques explained by Dr. Ogata in his textbook.6

6.1 Observability Matrix

In order for a system to be classified as completely observable, it must be possible to

reconstruct every state x(t0) from observation of the outputs y(t) over a finite interval

of time t0 < t < t0 + . This condition, in tensor form, required the 4 x 8

observability matrix O given by

O = {C*|A* ◊ C*|(A*)2 ◊ C*|(A*)3 ◊ C*} (6.1)

to be of rank four. Note that the matrices A and C came about from the state space representation of the system in Section 5.4, and * denotes the conjugate transpose matrix

operation.

49 6.2 Displacement Outputs

The matrix C given by Eq (5.14) is configured for displacement output, and the observability matrix produced from it was:

1 0 0 0 11 21 0 0 0010 0 0   O = 1 1 21 (6.2) 010012 22 00  0001 0 0 12 22          Noting that the first four column vectors of O were obviously linearly independent, it followed that the remaining column vectors could be constructed from linear combinations of them and the rank of the observability matrix was four. The state of the solar sail was therefore completely observable with displacement outputs.

6.3 Rate Outputs

If the control scheme used for the solar sail involved the measurement of

displacement rates via some sort of gyro, then the output y must be generated from the

state x using

C = 0 1 0 0 , (6.3) 0001 which was found to produce the observability matrix given by   2 0011 21 0011 12 21 11 21 21 22 O = 10 0 0  1 1  2 1 00 (6.2) 2 0012 22 0011 12 12 22 12 21 22    01 0 0 12 22 00       The first two column vectors exhibited linear independence, and could be scaled and   combined to generate the fifth and sixth column vectors. This left the four remaining

column vectors to be tested systematically for dependence on one another. With each 50 check it was discovered that for any pair to be orthogonal and thus increase the rank of the observability matrix from two to four, the following inequality would have to hold true:

det[]   (6.3)

Therefore, using the results for det[] from Eq (4.5b), the state of the solar sail on an optimal planet-centered non-Keplerian orbit was found to be completely observable using displacement rate outputs if and only if

  22 z (6.4)

51 7 STATE FEEDBACK

The solar sail pitch attitude control system developed so far has no input, or rather

has a reference input of zero. The desired output is also zero. Systems such as this that strive to maintain a constant output are called regulator systems. The solar sail pitch attitude control system hasn’t been shown to be very successful at regulating up to this point, however, since stable orbits only exist for certain regions of the  - z plane and even the stable orbits exhibit undamped oscillations about the nominal. For this reason state feedback control was added to the system, so that the control signal u(t) could be generated based on the state of the system x(t). The control theories put to work in this section were researched from the textbook by Dr. Katsuhiko Ogata.6

7.1 Gain Matrix Definition

A block diagram of the new closed-loop system is shown in Figure 7.1. The control

signal was updated to reflect the addition of state feedback control in accordance with

u =  = - K ◊ x (7.1a)

K = {K1,K2,K3,K4} (7.1b) where K is the state feedback gain matrix. Using Eq (7.1a), another way to write Eq (5.5) became apparent:

x = (A – B ◊ K) ◊ x (7.2)

Figure 7.1 Block Diagram of the Solar Sail Pitch Attitude Control System in State Space with State Feedback Control.

The stability and transient response characteristics of the system could now be calculated

from the eigenvalues of the matrix A – B ◊ K.

7.2 Transfer Function Determination

Before setting out to find a suitable gain matrix, the transfer function of the system

was examined to figure out the general behavior of the open loop poles and zeros for

different values of  and z. For a system defined in state space, the transfer function is

calculated from

G(s) = C(s I – A)-1 ◊ B + D (7.3)

where the matrices A, B, C, and D are as defined in Chapter 5 and I is the identity matrix.

Eq (7.3) was officially derived mathematically, but its form could be verified quite easily

by reducing the block diagram of the system. Figure 7.2 shows the result of rearranging

the block diagram by eliminating first the inner loop, then the outer loop, and finally re- ordering the leftover blocks and combining them into one. Two feedforward paths were generated, one from the system state to the control signal, and one from the control signal

Figure 7.2 Control System Diagrams resulting from Consecutive Loop Elimination

to the system output. The transfer function G(s) = Y(s) / U(s) came out naturally.

After performing the matrix multiplication specified by Eq (7.3), the outcome was

2 P1 s  P1 22 P2 12 G(s) = s 4  tr  s 2  det  (7.4) 2 P2 s P2 11 P1 21 4 2  s tr s det          which didn’t simplify much when the optimal  planet-centered    non-Keplerian orbit   constraints of Eqs (3.52), (4.5), and (5.11) were applied:

 s2 2z 1  3 z2 r2 s2  3  1  s2  3  1 r z r z G(s) Ø   (7.5) 3 2  r2    s2  3  1s2  3  1   r z r z         The poles and zeros of the system  were found to be located  at  

p1,2 =  i  (3/r + 1/z) (7.6a)

p3,4 =   (3/r - 1/z) (7.6b)

2 2 z1,2 =  ( z) (3/r - 1/z ) (7.6c) and according to Eq (7.5) the poles apply to both outputs * and * while the zeros only apply to *. 54

Figure 7.3 Pole-Zero Configuration Continuum.

7.3 Pole-Zero Configuration

In control theory, any poles located on the right-hand side of the complex plane give rise to unstable, divergent systems. It was previously shown by Eq (4.6) that the stability of the solar sail on an optimal planet-centered non-Keplerian orbit could not be

guaranteed except for values of  greater than 22z. By investigating the possible pole- zero configurations of the open-loop solar sail pitch attitude control system, more

detailed information concerning the stability of the system was obtained.

It was discovered that two ratios between  and z gave rise to poles or zeros at the

origin of the complex plane. One of them was the ratio given in Eq (4.6). These two

critical ratios were then used to isolate the regions where the pole-zero configuration of

the system was different (see Figure 7.3). Plots of the relative pole-zero configurations

were then generated for these two exact ratios as well as for the regions around them (see

Figures 7.4-7.8).

55 Scaled Pole - Zero Configuration for 0

   4

p1 2 ixA p4 z2 z1 p3

gam 0 Is

p -2 2

-4

-4 -2 0 2 4 Real Axis

Figure 7.4 Unstable Open-Loop Pole-Zero Configuration for (0 <  < 2z).

It was evident that in addition to  > 22z, the open-loop system was also stable at 

= 22z. All other values of the ratio led to instability as seen by the presence of a pole on the positive real axis. It is noteworthy to mention, however, that only one of the poles

caused the system to lose its stability in every case. Pole p3 becomes a positive real

number for  < 22z.

56 Scaled Pole - Zero Configuration for r= 2z

   4

p1 2 ixA p4 z1,z2 p3

gam 0 Is

-2 p2

-4

-4 -2 0 2 4 Real Axis

Figure 7.5 Unstable Open-Loop Pole-Zero configuration for ( = 2z).

57 Scaled Pole - Zero Configuration for 2z

    4

p1 2

z1 ixA p4 p3

gam 0 Is z2

-2 p2

-4

-4 -2 0 2 4 Real Axis

Figure 7.6 Unstable Open-Loop Pole-Zero Configuration for (2z <  < 22z).

58 Scaled Pole - Zero Configuration for r=2 2z

   4

2 z1 ixA p3,p4

gam 0 Is

-2 p2

-4

-4 -2 0 2 4 Real Axis

Figure 7.7 Stable Open-Loop Pole-Zero Configuration for ( = 22z).

59 Scaled Pole - Zero Configuration for r>2 2z

   4

z1 p 2 3 ixA

gam 0 Is

p4 -2 z2

-4

-4 -2 0 2 4 Real Axis

Figure 7.8 Stable Open-Loop Pole-Zero Configuration for ( > 22z).

7.4 Pole Placement

If a system is found to be completely state controllable, as the solar sail pitch attitude control system was in Section 5.3, then a state feedback gain matrix exists which will drive the poles of the system to any desired locations. Making use of this fact, poles were chosen to evoke desired transient and steady-state response behaviors for the solar sail in 60 an optimal, planet-centered non-Keplerian orbit. There is one major constraint inherited

by the engineer opting to design a controller in this fashion, though. All of the state

variables must be measurable, or else a state observer must be inserted into the system.

Some engineering applications may not benefit enough from the pole placement approach

to justify this cost in design limitation. For a spacecraft, however, the measurement of all

position and velocity errors is required not only for control but also for navigation.

Therefore, it was assumed that all four state variables in the system at hand could be

measured and the pole placement approach was then used to calculate gain matrices for

both low-impact and damped control schemes.

7.4.1 Gain Determination

The transformation method was used to determine the components of the state feedback gain matrix K, which called for the use of the transformation matrix T given by

T = M ◊ W (7.7) where M is the state controllability matrix given by Eq (5.9) and W is formed from the coefficients of the characteristic polynomial, or the denominators in G(s) of Eq (7.4):

0 tr  0 1 tr  010 W = 010 0 (7.8)    1000           The typical use for a transformation matrix is to transform the state equation for the system into the controllable canonical form. In this case it was used to generate the state feedback gain matrix K from the coefficients of the open-loop and closed-loop

61 characteristic equations. The general result for T was

P1 22  P2 12 0 P1 0 0P1 22  P2 12 0P1 T = (7.9) P2 11  P1 21 0P2 0  0P2 11  P1 21 0P2          and upon application of the constraints for optimal planet-centered non-Keplerian orbits

it became

2 1  3z 0  0 z r2 0 2 1  3z 0  z r2 T Ø (7.10)   3 2    2 000  r    2    0 3   00  r2            The equation used to obtain K from the open- and closed-loop characteristic equations and the transformation matrix was

K =  ◊ T-1 (7.11a)

 = {0 – a0,1 – a1,…,n-2 – an-2,n-1 – an-1}, (7.11b)

with the open- and closed-loop characteristic equations giving rise to the coefficients an

and n according to

n n-1 fopen(s) = an s + an-1 s + … + a1 s + a0 (7.12a)

n n-1 fclosed(s) = n s + n-1 s + … + 1 s + 0 (7.12b)

Recall that the open-loop coefficients for the solar sail on an optimal planet-centered non-

2 2 2 Keplerian orbit are a4 = 1, a2 = /z, and a0 =  (1/z - 9/r ) with the other

coefficients equal to zero.

62 7.4.2 Low-Impact Control Method

This control scheme involved fixing the only pole which posed a problem for

stability, pole p3, at s = 0. The other poles were placed at their naturally occurring positions from the open-loop system. The low-impact control method was named for its

minimal interference with the open-loop eigenmodes of the system. This approach

yielded a state feedback gain matrix K that could guarantee stability throughout the entire

range of the ratio between  and z. Thus the allowable optimal planet-centered non-

Keplerian orbits for the solar sail became limited only by the ability of its onboard

control equipment to generate the required control signals. Note that with the low-impact

control scheme, there is no damping involved. The solar sail is stable in the sense that it

is constrained to move within a torus-like shape about its chosen nominal orbit, as

discussed in Section 4.1. With the poles placed as specified above, the closed-loop

characteristic equation coefficients became

4 = 1 (7.13a)

3 =  (3/r - 1/z) (7.13b)

2 = (3/r + 1/z) (7.13c)

 1 =  (3/r + 1/z)(3/r - 1/z) (7.13d)

0 = 0 (7.13e)

63 which in turn led to the state feedback gain matrix variables

K1 = 3/r - 1/z (7.14a)

K2 = (1/) (3/r – 1/z) (7.14b)

2 2 K3 = ((r + z) – 2  ) / (r z ) (7.14c)

K4 = ((r + z) / ) (1/) (3/r – 1/z) (7.14d)

7.4.3 Testing for Lowest Impact

The magnitude of the low-impact gain was compared to the magnitude of the gain

required to fix both poles p3 and p4 at s = 0, with poles p1 and p2 undisturbed from

their open-loop locations. The closed-loop characteristic equation coefficients resulting

from the placement of two poles instead of just one were found to be

4 = 1 (7.15a)

3 = 0 (7.15b)

2 = (3/r + 1/z) (7.15c)

1 = 0 (7.15d)

0 = 0 (7.15e)

When the above values were carried through the mathematical process to generate the

state feedback gain matrix K, only half of the gain components were nonzero:

K1 = 3/r - 1/z (7.16a)

K2 = 0 (7.16b)

2 2 K3 = ((r + z) – 2  ) / r z  (7.16c)

K4 = 0 (7.16d)

64 In fact, those nonzero values were found to match the corresponding gain components for the low-impact case. With half the components identical and the other half zero, it was beginning to look like fixing both poles may provide an advantage in the form of a smaller gain required for stability and control. There was only one way to make sure:

2 2 2 2 2 ||Kone|| – ||Ktwo|| = (3 z - r)(r + z) +  ) / ( r z  ) (7.17)

In the equation above, Kone is the low-impact gain and Ktwo is the gain resulting from the

placement of both poles p3 and p4 at s = 0. If ||Kone|| > ||Ktwo||, then the right-hand side

of Eq (7.17) must be greater than zero. This could only occur if

z < r / 3, (7.18)

 > 22 z (7.19)

Since the solar sail is already inherently stable in the region specified by Eq (7.19)

according to the developments of Section 4.2, fixing both poles does not provide any

realizable benefit. Therefore the low-impact control method in which only the

troublesome open-loop pole was placed at the origin of the complex plane withheld its

place as the gain with the lowest impact.

7.4.4 Damped Control Method

Consider a vector drawn from the origin of the complex plane to either one of a pair of complex conjugate poles which lay in its stable, left-hand side. This vector has

magnitude equal to the undamped natural frequency n of these eigenmodes and its dot product with the negative x-axis yields their damping ratio . Applying this knowledge to the solar sail pitch attitude control system, a control scheme was developed that

65 Scaled Pole-Zero Configuration with Damping Gain for r< 2z

   4

2 p1

f ixA z2 p3,p4 z1

gam 0 Is

p2 -2

-4

-4 -2 0 2 4 Real Axis

Figure 7.9 Stable Damped Closed-Loop Pole-Zero Configuration for ( < 2z).

involved placing poles p3,4 at zero and rotating poles p1,2 away from the imaginary axis

by an angle /2 -  such that their natural damping frequency was unchanged, but a damping ratio  given by

 = cos[] (7.20) was introduced to the system (see Figures 7.9-7.11). Placing the poles in this fashion led

66 Scaled Pole-Zero Configuration with Damping Gain for r= 2z

   4

p 2 1

f ixA p3,p4,z1,z2

gam 0 Is

p2 -2

-4

-4 -2 0 2 4 Real Axis

Figure 7.10 Stable Damped Closed-Loop Pole-Zero Configuration for ( = 2z).

67 Scaled Pole-Zero Configuration with Damping Gain for r> 2z

   4

p1 2

f ixA p3,p4

gam 0 Is z2

p2 -2

-4

-4 -2 0 2 4 Real Axis

Figure 7.11 Stable Damped Closed-Loop Pole-Zero Configuration for ( > 2z).

68 to the closed-loop characteristic equation coefficients

4 = 1 (7.21a)

3 = 2   (3/r + 1/z) (7.21b)

2 = (3/r + 1/z) (7.21c)

1 = 0 (7.21d)

0 = 0 (7.21e)

and the state feedback gain matrix variables followed from Eqs (7.11):

K1 = 3/r - 1/z (7.22a)

K2 = 2  (1/) (3/r + 1/z) (7.22b)

2 2 K3 = ((r + z) – 2  ) / (r z ) (7.22c)

2 2 K4 = 2  ((2 z -  ) / (3 z )) (1/) (3/r + 1/z) (7.22d)

The presence of the damping ratio  was observed only in the components of the   gain matrix that were multiplied by the rate error state variables * and * in the

feedback control equation. Also, the gain for the position errors was found to depend

only upon the shape of the desired planet-centered optimal non-Keplerian orbit. This

control system exhibited proportional control of the position state variables to set the

solar sail moving from a displaced location toward its nominal orbit, and then applied rate dampening along the way to arrive on trajectory after a certain number of overshoots.

In the general analysis of systems with dampening, underdamped systems (0 <  < 1)

with values of  between 0.5 and 0.8 get closer to their steady state quicker than critically

damped ( = 1) and overdamped ( > 1) systems.

69 8 SUMMARY AND CONCLUSION

Two control schemes have been developed for a solar sail on an optimal non-

Keplerian planet-centered orbit. These methods provide stability throughout the entire

family of trajectories, with each orbit shape identified by a particular value of the ratio

between  and z. The low-impact and damped pole placements that were developed allow prospective solar sail mission designers the flexibility to choose between a simple, minimalist gain characterized by bounded oscillations about the nominal orbit and a gain

which dampens errant motion to precisely follow the equilibrium trajectory.

8.1 Gain Comparison Among Current Experimental Solar Sails

Table 8.1 presents a practical assessment of the relative gain magnitudes required to

control Cosmos 1 and NanoSail-D while they travel along a small contingent of optimal non-Keplerian orbits about Earth. Examining the results, it was clear that Cosmos 1 required less gain magnitude than NanoSail-D for stability and control along each orbit shape, which was identified by the orbit period value 2/. Therefore, higher- performance solar sails with a smaller solar sail burden  are easier to control.

Furthermore, the gain magnitude required to maintain stability increased as the orbital period decreased for each experimental sail. This just meant that greater control inputs were necessary for trajectories with increased orbital angular velocities. It was also

70 Table 8.1 Comparison of Low-Impact and Damped Gain Components for the Cosmos 1 and NanoSail-D Spacecraft on Selected Earth-Centered Optimal Non-Keplerian Orbits (0.7 > -ln[] > 0.2 for 0.5 <  < 0.8).

Solar Sail  E 2/ Low-Impact Gain (-ln[|Kn|]) Damped Gain (-ln[|Kn|])   days g/m mm/s |K1| |K2| |K3| |K4| |K1| |K2| |K3| |K4| Cosmos 1 167 0.055 90 18.6 4.4 18.5 4.3 18.6 3.4-ln[] 18.5 2.2-ln[] 180 21.0 5.6 20.7 5.3 21.0 3.9-ln[] 20.7 3.8-ln[] 365 21.1 5.7 20.4 4.9 21.1 4.4-ln[] 20.4 6.3-ln[]

NanoSail-D 488 0.019 90 17.3 3.2 17.2 3.2 17.3 2.4-ln[] 17.2 0.1-ln[] 180 18.8 4.0 18.8 3.9 18.8 3.0-ln[] 18.8 1.7-ln[] 365 21.0 5.1 20.8 4.8 21.0 3.6-ln[] 20.8 3.3-ln[]

found that the greatest components of the gain matrix for each control scheme were the ones that multiplied the displacement rates in the state vector. Most of the solar sail’s response is then determined by how quickly it is diverging from the nominal orbit.

Finally, in comparing the low-impact gain to the damped gain, it generally requires more energy to follow a precise trajectory without perpetual bounded oscillations and the amount of this energy increases as the damping ratio  is varied from 0.5 to 0.8 for a preferred underdamped system.

71 References

[1] McInnes, Colin R. Solar Sailing: Technology, Dynamics and Mission Applications. Chichester, UK: Praxis Publishing Ltd, 1999.

[2] Wright, Jerome L. Space Sailing. Langhorne, PA: Gordon and Breach Science Publishers, 1992.

[3] Friedman, Louis. “Solar Sail Update: The End of Cosmos 1; The Beginning of the Next Chapter.” Planetary News: Cosmos 1 - Solar Sail. 2005. The Planetary Society. 23 May 2009 .

[4] McQuarrie, Donald A. Mathematical Methods for Scientists and Engineers. Sausalito, CA: University Science Books, 2003.

[5] Symon, Keith R. Mechanics – 3rd Ed. Reading, MA: Addison-Wessley Publishing Company, Inc., 1971.

[6] Ogata, Katsuhiko. Modern Control Engineering – 4th Ed. Upper Saddle River, NJ: Prentice-Hall, Inc., 2002.