Calhoun: The NPS Institutional Archive Theses and Dissertations Thesis Collection 2000-06-01 Remote nanosatellite formation designs with orbit perturbation corrections and attitude control/propulsion subsystem correlation Tomlin, Stephen D. Monterey, California. Naval Postgraduate School http://hdl.handle.net/10945/7781 OXUBRARV -cT<Ey -<ADUATE SCHOOL 93943-5101 NAVAL POSTGRADUATE SCHOOL Monterey, California THESIS REMOTE NANOSATELLITE FORMATION DESIGNS WITH ORBIT PERTURBATION CORRECTIONS AND ATTITUDE CONTROL/PROPULSION SUBSYSTEM CORRELATION by Stephen D. Tomlin June 21, 2000 Thesis Advisor: Brij Norm N. Second Reader: Approved for public N. Agrawal release; distribution is Sorenson unlimited. 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REPORT TYPE AND DATES COVERED June 2001 Master's Thesis REMOTE NANOSATELLITE FORMATION TITLE AND SUBTITLE: DESIGNS WITH ORBIT PERTURBATION CORRECTIONS AND ATTITUDE CONTROL / PROPULSION SUBSYSTEM CORRELATION 6. AUTHOR(S) Tomlin, Stephen D. 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 4. Naval Postgraduate School 9. Monterey, CA 93943-5000 SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 5. PERFORMING ORGANIZATION REPORT NUMBER SPONSORING / MONITORING AGENCY REPORT NUMBER 10. N/A 11. SUPPLEMENTARY NOTES The views expressed in this thesis are FUNDING NUMBERS those of the author and do not reflect the official policy or position of the Department of Defense or the U.S. Government. 12a. DISTRIBUTION / AVAILABILITY STATEMENT Approved 13. for public release; distribution ABSTRACT (maximum is 12b. DISTRIBUTION CODE unlimited. 200 words) The innovative idea of distributing the functionality of current larger satellites among smaller, cooperative satellites has been sincerely considered for assorted space missions to accomplish goals that are not possible or very difficult to do with a single satellite. Additionally, the maximized within formations and clusters to conduct missions such as interferometry and earth-sensing. This paper presents a methodology to describe, populate and analyze numerous formation designs employing the use of Hill's equations of motion to describe a formation's dynamics. These equations of motion are then programmed into a MATLAB code to produce Cartesian elements for input into a Satellite Tool Kit™ (STK) simulation that demonstrates numerous possible cluster formation designs. These simulations are then used to determine AV requirements for overcoming LEO-type perturbations that were modeled within STK's High Precision Orbit Propagator (HPOP). utilization of smaller Finally, satellites is components from two subsystems [Attitude Determination and Control (ADCS) and Propulsion], using the AV calculations from the simulation analysis and current advances in MicroElectroMechanical systems (MEMs) and nanosatellite technology, are presented based on a mass constraint of 1 0kg for the entire satellite. 14. SUBJECT TERMS Nanosatellite, 15. Orbit Satellite Propulsion, Dynamics, Satellite Formation, Satellite Cluster, STK SECURITY CLASSIFICATION OF CLASSIFICATION OF THIS 19. SECURITY CLASSIFICATION OF REPORT PAGE ABSTRACT 17. Unclassified NSN 7540-01-280-5500 18. SECURITY Unclassified NUMBER OF PAGES 136 16. PRICE CODE 20. LIMITATION OF ABSTRACT UL Unclassified Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. 239-1 THIS PAGE INTENTIONALLY LEFT BLANK 11 Approved for public release; distribution is unlimited. Remote Nanosatellite Formation Designs Corrections and Attitude Control/Propulsion Perturbation With Orbit Subsystem Correlation Stephen D. Tpmlin Lieutenant, United 'States Navy B.S., Rensselaer Polytechnic Institute, Submitted in partial fulfillment 1994 of the requirements for the degree of MASTER OF SCIENCE IN ASTRONAUTICAL ENGINEERING from the NAVAL POSTGRADUATE SCHOOL June 2001 * EY ABSTRACT The innovative idea of among distributing the functionality of current larger satellites smaller, cooperative satellites has been sincerely considered for assorted space missions to accomplish goals that are not possible or very difficult to do with a single Additionally, the utilization of smaller satellites satellite. is maximized within formations and clusters to conduct missions such as interferometry and earth-sensing. This paper presents a methodology to describe, populate and analyze numerous formation designs employing the use of Hill's equations of motion to describe a formation's dynamics. MATLAB code to (STK) simulation These equations of motion are then programmed into a produce Cartesian elements for input into a that demonstrates These simulations are then used numerous possible to determine type perturbations that were modeled within Satellite Tool Kit cluster formation designs. A V requirements for overcoming LEO- STK's High Precision Orbit Propagator (HPOP). Finally, components from two subsystems [Attitude Determination and Control (ADCS) and current Propulsion], using the advances in A V calculations from MicroElectroMechanical systems the simulation analysis and (MEMs) and nanosatellite technology, are presented based on a mass constraint of 1 Okg for the entire satellite. THIS PAGE INTENTIONALLY LEFT BLANK VI TABLE OF CONTENTS I. INTRODUCTION NANOTECHNOLOGY PHENOMENON B. DEFINING THE STARTING LINE CURRENT AND FUTURE EMPHASIS C. D. II. 1 A. 1 1 3 OBSTACLES E. CONCEPTS OF THE AEROSPACE CORPORATION F. THE SCOPE LITERATURE REVIEW: PAST AND CURRENT SYSTEMS A. PAST SYSTEMS SIZE 7 9 9 1. 1958 1 990-1 995 9 10 3. 1 995-2000 2000 - Present 13 11 JA WSA T SNAP-1 Munin a. b. c. 13 15 75 16 STUDENT SATELLITE PROJECTS 1. 2. 3. C. 6 2. 4. B. 5 SSETI - Student Space Exploration University nanosatellite program & Technology Initiative a. Three Corner SAT. b. ION-F c. Emerald 17 18 18 19 d. UW nanosatellite (Dawgstar) 20 20 22 22 24 24 25 Miscellaneous Projects CURRENT SYSTEMS AND INDUSTRY LEADERS 1 Aerospace Corp 2. NASA 3. Foreign Universities NANOSATELLITES FLYING TOGETHER NANOSATELLITE FORMATION DYNAMICS A. INTRODUCTION B. INITIAL EQUATIONS OF MOTION FORMATIONS D. III. 2. In-Plane Formation 3. In-Track Formation Circular Formation Projected Circular Formation REMOTE in the WITH CLUSTERS DISTRIBUTION 27 27 OF SATELLITE 29 32 33 34 35 37 Formation Basic 5. C. Remote Motion 1. 4. 16 CONSTANT APPARENT 39 vii 1 2. 3. Problem Description Linearized Approach Orbital Mechanics a. Apparent Angular Width Correction for Inclination Apparent Vertical Size Phase Separation in Apparent Orbit Geometry of the Orbit b. c. d. e. 4. D. IV. 52 53 Example Populating a Remote Cluster PERTURBATIONS AND A V REQUIREMENTS 1. STK Perturbation Propagators a. Two-Body, b. HPOP. Perturbation Effects 3. Formation Keeping Station Keeping 63 67 67 SUBSYSTEM DESIGN A. ADCS 2. 3. 69 70 Magnetic Control 71 a. All-Magnetic Torquer System Reaction wheel / Magnetic Torquerod System b. Propulsion Option 72 Components 73 74 c Micromechanical gyroscope Magnetometers Reaction Wheels d. Sensors. a. b. B. 75 76 77 79 80 Systems Cold/Hot Gas 81 81 a. b. MEMS. c. Electrical. Performance Propulsion System Comparison (Cold gas 3. EXAM PLE SN API (SSTL) 82 84 85 2. C. V. 72 74 e. DGPS PROPULSION 1. 54 55 61 62 OTHER TOPICS 1 LOW 60 60 60 J2 and J 2. 4. F. 57 POPULATING AND MAINTAINING A CLUSTER IN EARTH ORBIT (LEO) 1. E. 40 43 45 46 48 49 : CONCLUSION A. THE NANOSATELLITE PUSH B. FORMATION DESIGNS C. PERTURBATION UPKEEP viii / fiPPT) 86 88 89 89 89 90 APPENDIX 2.1 93 APPENDIX 2.2 95 APPENDIX 3.1 97 APPENDIX 3.2 99 APPENDIX 3.3 103 APPENDIX 3.4 105 APPENDIX 4.1 107 LIST OF REFERENCES 1 INITIAL DISTRIBUTION LIST 09 113 IX THIS PAGE INTENTIONALLY LEFT BLANK LIST OF FIGURES Figure 1.1 Small Satellite Launch Mass 3 Figure 2.1 Vanguard- Figure 2.2 TUB SAT-N (bottom box) / -Nl Figure 2.3 Figure 3.1 Aerospace tethered picosatellites (artist interpretation) SNAP-1 from SSTL (UK) Inside a CubeSat (10cm per side) Artist Conception of orbiting AEROSPACE Nanosatellite Hill's reference frame for satellite relative motion Figure 3.2 Moving Frame 22 23 30 32 Figure 3.3 Reference, relative, and apparent orbits 41 Figure 3.4 Figure 3.5 Geometry description and definitions Orbital elements of the eccentric orbit Figure 3.6 Apparently circular cluster of satellites Figure 3.7 Figure 4.1 Ring of eight equally spaced satellites Cyclic motion of Subsat Orbit Remote Geometry as viewed along Mothersat Velocity vector BEI GYROCHIP™ Model QRS1 1 Micromachined Angular Rate Sensor 42 43 45 53 55 56 Figure 4.2 Litton Figure 4.3 Model 533: Miniature Figure 4.4 Small reaction wheel developed by Figure 4.5 EDO Figure 4.6 Details of the structure of the micro-mirrors are Figure 4.7 Aerospace MEMS chip compared to Penny Marotta microthruster compared to Dime Full-sized Pulsed Plasma Thrusters from Primex Aerospace Company Basic diagram of a pulsed-plasma thruster Snap-1 Propellant tube Figure 2.4 Figure 2.5 Figure 2.6 Figure 3.8 Figure 3.9 9 G2000 gyroscope with Barnes Model (top plate) 11 14 15 75 75 electronics 3 Axis, Fluxgate Magnetometer 76 77 HIT 13-500 wide-angle miniature solid-state horizon 78 79 sensor Figure 4.8 Figure 4.9 Figure 4.10 Figure 4.1 XI shown 83 84 84 85 ; ....88 THIS PAGE INTENTIONALLY LEFT BLANK XII LIST Table 1.1 OF TABLES 2 Satellite Classifications Table 4.1 Assumptions made Table 4.2 Comparison of uPPT and cold-gas propulsion systems 69 for Satellite Physical Characteristics performance) (single thruster 87 Xlll THIS PAGE INTENTIONALLY LEFT xiv BLANK LIST OF a semimajor axis Cd drag coefficient Ce effective exhaust velocity ff SYMBOLS 8 angle that defines shape and orientation of suborbit plane e eccentricity E eccentric (f> rotary angle h angular rj anomaly momentum dimensionless pattern generator scale factor i inclination Ibit minimum impulse bit Isp specific impulse k elevation angle X lateral M mean anomaly // gravitational parameter {398, 600. 441 5 v true angle km 3/(solar sec) 2 ) anomaly phase angle of satellite {measured clockwise from the cross-track(z) direction) decay q orbit r radius of the formation rate xv r position of remote from Earth (vector) p scale factor for formation creation Re radius of Earth Rp radius of planet R position of mothersat from Earth (vector) t time T thrust u angle from line of nodes to radial vector (w v velocity vector w argument of perigee co mean motion coe rotation rate of the Earth Q right ascension x radial difference v along-track difference z cross-track difference AV velocity increment (0. v) 000 072 921 158 553 of the ascending node / + 'delta-V xvi rad/solar sec) ACKNOWLEDGMENTS Of first thank enough all the help, support and inspiration my wife Kimberly to enable more ways than To and I my who I received writing this paper, I wanted has affectionately encouraged, assisted and loved completion of this project. Without your help me would be stuck I to in one, thank you for the smiles, the laughter and the unwavering love. my three boys, Joshua, Daniel and thank you guys for waiting patiently Adam, at the I am truly blessed to be your father chess table, train table, and changing table respectively! Also who I wanted my sincere to express thanks to all my family members helped through words of prayer and encouragement to show possible, and that a true faith and a loving family especially to Professor Brij Agrawal, Dr. Norm is all me and friends that anything is that is important. Thank you Sorenson, Dave Cook, and the all OCFers. I also wanted read and critique this paper. evaluating to Jeff it with me my to express Thank you afterwards. King who helped with sincere thanks to Jon Strizzi, It was certainly the best review knowledge of I through His blessing and grace, receive my thank my heavenly Father wonderful family, words of knowledge, wisdom and love. vain. xvii my Without got, thanks! And MATLAB. Thank the time spent and hopefully your part of the puzzle in this field will I took the time to for the time spent in reading, proofing and his far superior Finally and most important, who who blessed Him fit I also you for nicely! made possible, and the ability to has life would be building in THIS PAGE INTENTIONALLY LEFT BLANK xvin INTRODUCTION I. Nanotechnology provides the capability to manipulate matter at the atomic level. In the future, we will measure the way we design and build our systems by the atom, not by the pound. Today, we are developing material systems, at the molecular level, that are 100 times stronger than steel at 1/6 the weight. We will also develop sensors and detectors capable of responding to a single photon of light or the stimulus from a single electron. Using nanotechnology, we will build systems on a scale 1000 times smaller than today ~ at true molecular level. They will be based on concepts emerging from biology, quantum mechanics and chemistry, all of which have no current parallel. [Ref 1] A. NANOTECHNOLOGY PHENOMENON Commercially and systems militarily, (MEMS), nanoscale space systems based on MicroElectroMechanical design and materials, low power quantum electronics, and high bandwidth photonics are of special interest, as are the demonstrations of space subsystems based on these technologies. Significant reductions in individual spacecraft mass and cost, large gains in capabilities and robustness, and novel architectures involving large constellations and closely coupled spacecraft are expected with the introduction of these technologies. Novel spacecraft architecture concepts for Earth orbiting missions including communication, navigation, remote earth-sensing, and monitoring of the local space environment that can be created with the introduction of micro/nano-technologies will be outlined briefly below. B. DEFINING THE STARTING LINE First of all, it is worth defining what current small satellite world Cheaper'. Small satellite is we mean by a small encompassed by the slogan projects are compared with the conventional space months. Leading-edge technology is characterized satellite. The spirit of the 'Faster, Better, Smaller, by rapid development when industry, often ranging from six to thirty-six routinely included in order to provide innovative 1 solutions, permitting lighter satellite systems to be designed inside smaller volumes. manned space programs, Frequently, traditional procedures, with roots in the military and can no longer be justified, and low cost solutions are favored to match the reducing space budgets. So in many ways it mass of the the philosophy, and not the size or is satellite that matters. Many terms are used to describe this rediscovered class of satellites, SmallSat, Cheapsat, MicroSat, MiniSat, NanoSat and even PicoSat. Advanced Research Naval Space Projects Command method of adopted (see Table TACSat's (Tactical medium Satellites). Nevertheless, in recent years a mass has been generally 1.1). tradeoffs are typically classes are an indication of made, which Although the satellite', this Defense refers to these as LightSats, the U.S. classifying satellites in terms of deployed The boundaries of these 'wet mass'). US as SPINSat's (Single Purpose Inexpensive Satellite Systems), and the U.S. Air Force as general Agency (DARPA) The including is also why the mass is where launcher or cost defined including fuel satellites in the 5 00- 1000kg are typically causes confusion and until a better term appears it designated as a 'small will be defined here as a sized satellite. B B E Group name Wet Mass Large satellite >1000kg Medium sized satellite 500-1000kg Mini 100-500kg satellite Micro n H Nano Pico satellite satellite 0.1-lkg satellite Femto satellite Table 10-100kg l-10kg 1.1 <100g Satellite Classifications "(i.e. Small Satellites The mass ire no clear mass boundaries, although there '.00kg class. )f distribution for small satellites plotted in Fig. 1.1 illustrates that there launched The emove a general lack of spacecraft in the positive sloping line in the figure depicts a gradual increase in the satellites. vithin the next is few Although years, the this trend will continue as heavier satellites are number of smaller satellites 1 00- mass launched launched should start to this trend. Small Figure 1 . 1 satellite Small mass Satellite Launch Mass CURRENT AND FUTURE EMPHASIS Few modern Nano and aunched, although there echnology is is Pico-Satellites weighing less than considerable interest in this area as advanced microsat applied to miniaturize satellite systems even further. nrcrosats almost fall 10kg have been into this category with weights Some launched of 11-1 4kg, most notably the AMSAT than microsat-series [Ref 2]. These satellites are cubical in shape and measure less 150mm on each side. Nanosatellites are attractive to commonly space, as many available technology educational institutions to get involved in now makes most importantly affordable. Although operational are some time still nanotechnology makes on entire satellites weighing it JAWSAT just recently below). less than launched a Increasingly, 1kg of set micro and possible to fabricate entire satellite sub-systems, and possibly a chip. Considerable effort than less 2A: weighing picosatellites Aerospace Corporation Chapter (see picosatellites off, type of satellite feasible and this with 0.1kg, is being spent on these Femtosatellites applications in remote inspection, distributed measurement and disposable sensors. For Nanosatellites, autonomous operation using a single on-board computer making use of technology developed feasible, minimize mass, active attitude systems. to The downlink data be operated at The key low data palmtop computers. To and orbit control are often ignored, and omni-directional The main antennas are employed. for laptop and. is rate is limited by the rates, or in burst mode. to the successful advanced technology. Each of the by limits are set the downlink and orbit average power generation power generation, and has development of nanosatellites and constellations many is enabling technologies toward nanotechnology represents a breakthrough in performance, capability, or application in a unique way. This technological challenge is formidable, since currently, the smallest "full-service" microspacecraft weighs 100 kg (220 lb) or of five. more - size must be reduced by almost a factor This will require revolutionary advances in microelectronics and spacecraft component technologies. In addition, at present day, almost space missions are flown as single all spacecraft. This is because controlling spacecraft in flight is a very problem of flying several spacecraft as one system is further communication path from a constellation of spacecraft communication to behave stations on the ground. "intelligently" — In order to work autonomously staying 4 complex process. The compounded by in flight the complex high above Earth with properly, the spacecraft will have in constant contact with each other, sharing information, and re-configuring onboard instruments and systems to behave as a single unit. SIZE OBSTACLES D. A fundamental problem in spaceflight is spacecraft size. It is very expensive to place satellites into space. For example, using expendable cost with eight launches a year Space Shuttle, for the it payload into low-Earth as possible. It costs about $13,200 per kilogram ($6,000 per As orbit. pound) to deploy a a result, engineers try to design spacecraft to be as small has been estimated that satellites using nanotechnology could measure inches wide by 2 inches thick and weigh about 2 pounds. These satellites 1 would then require smaller and therefore less expensive launchers. [Ref. 3] Most Earth-observing kg (2,200 spacecraft with science payloads lbs). Microsatellites are much range. Nanosatellite are even smaller — weigh smaller, typically in the 100 in the in excess kg (220 lbs). However, small spacecraft even today capabilities. Typically they lack control, lb) or larger sub-20 kg (44 lbs) range. Small spacecraft are nothing new. In fact, the first satellite launched into space, Sputnik, about 90 kg (180 of 1 ,000 weighed only are very limited in their any propulsion, possess only limited ability for attitude and carry one single-function payload. What service", is needed meaning they is new a era of "smart", miniature spacecraft that will be "full will carry a wide range of spacecraft services including guidance, and control, attitude control, propulsion, high bandwidth and complex navigation communication functions. Nanosatellites not only reduce launch costs, but they also reduce the risks associated with flying missions. Currently, several instruments and payloads are flown on a single, large spacecraft. disable the entire A single instrument or system failure mission. Constellations complementary instruments reduce the instrument fails. risk of numerous of an entire may severely degrade or spacecraft, mission failing if each carrying one system or E. CONCEPTS OF THE AEROSPACE CORPORATION Most of the radical the nanosatellite concept new concepts for building and using spacecraft represented by were developed The Aerospace Corporation and formally at introduced in a paper, "The Concept of 'Nanosatellite' for Revolutionary Systems," presented at Low Cost Space the 44th International Astronautical Federation Congress in Graz, Austria, in 1993. — who Authors included Robinson, Siegfried Janson, Ph.D., and originated the concept [Ref nanosatellite scientist the in microtechnology ~ 3] and center coined the term and Henry Helvajian, Ph.D., a senior of the just-published editor book, "Microengineering Aerospace Systems." A of reports written by Helvajian, Janson, and Robinson and issued series the 1993 conference presented the details on nanosatellites. national and how to design, build, These nanosatellite technologies are international research now organizations after power and maneuver being explored by a number of in addition to The Aerospace Corporation The First International Conference on Integrated Micro/Nanotechnology for Space Applications was hosted jointly by the Aerospace Corporation and Johnson Space Center in Houston, from October 30 conference was to bring th together through November scientists and 3 rd 1995. engineers The purpose of from the fields the of microtechnology, nanoelectronics and space technology to explore the possibilities for applying newly emerging capabilities in microtechnology to space operations. The evolution of microelectronic technology coupled with the growth of 5 years has had a significant impact in the commercial MEMS in the past 4- terrestrial sector. This influence can be evidenced particularly in sensor, optical switching and mass data storage applications that have been inserted into major industries such as transportation, medicine, telecommunications and computers. The focus of this conference was to anticipate and extend the incorporation of nanoelectronics Specific Integrated Microinstruments of space systems. (ASIMs) and MEMS into Application in order to revolutionize the development THE SCOPE F. This paper sets out to introduce, explore and design nanosatellite formation designs with their required orbital dynamics. Chapter two will introduce past and current Numerous systems systems that are based on novel nanosatellite concepts. will be covered in areas ranging from experimental military designs to innovative student-driven ideas and future-looking commercial enterprises, developing realm of 'smaller, faster, all expecting to capture the quickly cheaper' with regards to nanosatellite technology. This paper presents a methodology to describe, populate and analyze numerous formation designs employing the use of Hill's equations of motion to describe a formation's dynamics. These equations of motion are then code to that demonstrates Satellite Tool Kit formation designs. produce Cartesian elements for input into a MATLAB and numerous possible STK to create cluster understand the required delta- V dimensional MATLAB (STK) simulation After utilizing (AV) to (LEO) perturbations will be analyzed maintain the formation within given criteria. Finally, (ADCS) and current into a formation simulations modeled within STK's High Precision Orbit Propagator (HPOP), low-earth orbit to programmed components from two subsystems [Attitude Determination and Control Propulsion], using the advances in AV calculations from the simulation analysis and MicroElectroMechanical systems (MEMs) and technology, are presented based on a mass constraint of 10kg for the entire nanosatellite satellite. THIS PAGE INTENTIONALLY LEFT BLANK LITERATURE REVIEW: PAST AND CURRENT II. SYSTEMS PAST SYSTEMS A. 1. The 1958 first U.S. earth satellite, Explorer Launched by a modified today's standards. February 01, 1958 Army diameter, and with a I, mass of 13.6kg and an nanosatellite first Ballistic Missile orbit by Agency Jupiter-C on 150mm of 347x1,859 km at in 33.2 deg developed by the Jet Propulsion Laboratory, carried the U.S.-IGY (International Geophysical Year) experiment of James A. Van Allen and resulted discovery of the radiation belt around the earth [Ref. follow would weigh the containing the upper stage, measured 2.03m long and it, inclination. Explorer was almost I, more and become bigger 4]. Although most in size, there was one US in the satellites to that could be considered the 'forefather' of today's nanosatellites: Vanguard-1. Although the first official 'nanosatellite' years, the first satellite launched in the 1-1 0kg still orbiting Earth. in July of 1957 it Originally, a simple nose was decided would not appear mass range used instead to exercise the tracking stations (see Fig. 2.1). Vanguard-1 9 ironically the oldest satellite cone was to be carried on Vanguard-1, but that a small 1.47-kg (3.25 Figure 2.1 is for another thirty pound) test satellite would be Launched March orbit, this test satellite aluminum 108 17, 1958 into an would the Vanguard technical and scientific results (656x3866 km), 34.25 deg inclination consist of a simple 16 centimeter (6.4 inch) polished alloy sphere equipped with MHz. Although elliptical is 1 two transmitters operating satellite must be looked Technically impressive. its at frequencies around at as a test satellite purpose was to its and test evaluate solar cells, satellite terminal design and on board instrumentation. All these test objectives were met. The solar six years, (interfering the scientific results 108 cells MHz worked so well proved to were also a success. As Soviet be a very useful tool. were active frequency) well beyond the expected lifetime. scoffed at the diminutive size of Vanguard it that its transmitters 1 and authorities its The and even some in the West lack of sophisticated instrumentation, Analysis of the motion of Vanguard fact that the earth is not spherical but for established the 1 has a bulge, disclosing an unsuspected stress deep within the earth. These measurements indicated that there was large-scale convection taking place inside the Earth, which supported the emerging theories of continental drift and sea floor spreading. Analysis of the drag exerted by the atmosphere on Vanguard proved the atmosphere to be far more extensive and Perturbations in Vanguard l's orbit also led to a 1 variable than previously believed. more refined estimate of the Earths oblateness. [Ref. 5] 2. 1990-1995 On September 03, 1990 China launched two atmospheric balloons (1990- 081B/C), weighing 4kg each to measure the magnetosphere. 775x804 into km orbit at an 833x886 99 degrees and decayed on March 1, QQW1 1991. km orbit at 99 degrees and decayed on July 24, was launched QQW2 into a was launched 1991 Also on two occasions in mid- 1990 (February 03, 1994, February 04, 1995), a series of spherical objects were released from the Orbital DEbris RAdar Calibration Spheres shuttle, typically in pairs. (ODERACS) were a few centimeters in diameter, and were intended to provide calibration for radar echoes. provides the data on these objects. 10 These Appendix 2.1 1995-2000 3. A the MIR commemorative functioning replica of the original Sputnik (SPUTNJX-40 - aka space station was deployed from Sputnik-II, PS-2, RS-17, Sputnik Jr., 1997- 05 8C, 24958) during an extravehicular activity (EVA) by Russian cosmonauts Anatoly Solovyov and Pavel Vinogradov on November 03, 1997. km orbit inclined at 51.6 degrees. rocket on October 9 th . The satellite It It was therefore in a 383x391 originally arrived via a Progress automated cargo was 1/3 scale and weighed only 3kg and was built by French students from the l'Aeroclub of France (radio transmitter), staff from the Russian Aeronautical Federation (structure), industry. It The and funded by various sponsors in the space May 21, stopped transmitting on the December 29, and decayed on the first satellites TUBSAT-N1 launched from a submarine were (1998-042B). The 8kg TUBSAT-N (1998-042 A) and TUBSAT-N and the 3kg TUBSAT-N1 were two nanosatellites launched on July 07, 1998 as a 1998. satellite cluster (see Fig. 2.2) from a Russian nuclear powered submarine with a Shtil-1 converted missile in the Barents Sea, and was reported to have cost on the order of $ 100k (US$1998). orbit via telecommand and were placed Figure 2.2 The each spacecrafts contained three into a TUBSAT-N 400x776 different experimental satellites were separated km orbit inclined at (bottom box) were flat-box shaped with a The / -Nl (top solar panel payloads in 78.9 degrees. plate) on the largest face, and provided by the Technical University of Berlin (TUB): reaction wheel performance, star sensor performance, and •11 and forward communication. The store communication transceivers 1 200 and 2400 baud. in the Two for store payload consisted of four independent and forward communication with a baud transceivers worked 70-cm frequency band with FFSK additional downlink transmitter with latter in the rate 2m-frequency band, the other two (Fast Frequency Shift Keying) modulation. 9600 Baud of GMSK (Gaussian Minimum An Shift Keying) modulation was available. The attitude control of TUBS AT-N consisted of two magnetic coils, a magnetometer, a reaction wheel and a star sensor. All attitude control devices were also developments of the Technical University of Berlin. Electrical power was provided by 9 NiCd-battery cells Ah (SANYO). The of 5 battery cells were connected serially and provided an unregulated bus voltage from 9 to 13 V. The current of the nanosatellites utilization mammals, is being used for tracking medium-sized and large stolen cars and to collect data from autonomous buoys for earth environmental observation. These buoys are located in the northern Atlantic Ocean near the Canary Islands. A MIR second sputnik, Sputnik-41 (aka RS-18, 1998-062C), was launched from the spacestation into a 1998. A 313x318 inclined orbit at 51.7 degrees Progress-M40 cargo rocket delivered it to MIR on the November on the October 25, 1998. It 10, was financed by the Aeroclub de France, and built by French and Russian students. Sputnik41's 200m Russian. from W transmitter broadcasted pre-recorded voice greetings in French, English, and The orbit on the The from the 1999. A spacecraft 1 measured in diameter, spacestation RS-19 (aka Sputnik-Jr. 3rd, It decayed earlier spacecraft in this series, it was supposed was launched "switched off, company (Swatch) were 1 999-01 5C), was launched by Jean-Pierre Haignere during a spacewalk on the 16th April Progress-M41 cargo rocket delivered spacecraft and weighed 4kg. 1th January 1999. third "junior" sputnik, MIR 230mm it to MIR on the 2nd April 1999. Like to transmit simple messages, as advertising messages carried in breach by however the a commercial of International Telecommunication Union (ITU) regulations regarding amateur bands [Ref 6]. 12 2000 -Present 4. JAWSAT a. On January 26, 2000 an Launch Vehicle) carried JAWS AT, Those several OSPSLV Program Space (Orbital Suborbital within microsatellites a payload adapter called developed jointly by the U.S. Air Force and Weber State University in Utah. included FalconSat, an experimental satellites Academy, ASUSat built 1, by students at satellite built by the U.S. Air Force Arizona State University; the Optical Calibration Sphere Experiment, an inflatable 3.5-meter (11.5-foot) balloon built by L'Garde for the Air Force Research Laboratory that served as a target for low-powered ground-based lasers; and Opal, a Stanford University satellite that, in turn, deployed six smaller "picosatellites" built by Santa Clara University, the Aerospace Corporation, and ham radio operators. In addition to those satellites, remained attached provided by altitudes, to NASA's A microsatellite. Satellite Test (PEST), at orbital Control Platform tested a low-cost three-axis satellites are covered below in more detail. DARPA Picosat (OPAL-#l&2), built by Rockwell and funded by DARPA and UCLA weighing just 0.5kg tethered set of Picosatellites, 100x750x250mm each (see Fig. They were deployed on February established 24 hours the pair. The Plasma Experiment State's Attitude The launched the Aerospace Corp. at measuring after launch. included two other payloads that Marshall Space Flight Center, studied plasma found Weber while stabilization system. and it JAWSAT later. Gold strands in the tether The spacecraft performed basic tests on 2.3), 6th, launched and were instrumental MEMS RF OPAL communications were in radar tracking switches. primary batteries (lithium thionyl chloride) ran out by February 10th. 13 from the The spacecraft Aerospace tethered Figure 2.3 A ARTEMIS in a Picosat 5400 hour named picosatellite picosatellites (artist interpretation) after (OPAL-#3&4, Thelma and effort over a period of 1 the Greek Goddess of the moon, the Louise), weighed just 0.5kg and GaAs not attitude stabilized. solar cells. carried a It were deployed on February MASAT, 1 the 1th, It was designed Very built months by an exclusively female team of 7 Santa Clara University students. The spacecraft employed a battery cells and was Low for 68HC1 1 microcontroller, AA one week of operations and was Frequency (VLF) receiver. The spacecraft and were reported not to be operational. [Ref. Miniature Amateur Satellite, and 7] STENSAT, NASA's amateur radio picosatellite built by Goddard Space Flight Center (GSFC), deployed on February 12 picosatellite th from the OPAL weighting just 0.5kg, and Both spacecraft were reported not ASUS AT- was 1 University. when microsatellite. It carried GPS, to MASAT STENSAT (OPAL-#6) (OPAL-#5, JAK) was a weighed in at just 0.25kg. be operational. a 5kg nano-satellite designed and built a camera, and a radio amateur at Arizona State FM repeater that only operated requested on the uplink. The batteries failed to charge and therefore the satellite operated for approximately 15 hours during which telemetry longer operational. 14 was received, and is no b. SNAP-1 SNAP-1 (2000-033C, 26385) Satellite is a 6.5kg nanosatellite developed by Surrey Technology Limited (SSTL) and the Surrey Space Centre (SSC) Fig. 2.4). The spacecraft was launched COSMOS-3M as a piggyback ride orbit inclined at 98.13 degrees. It carries a communicate with Tshinghua-1. nanosatellite to was placed microsatellite into a remote inspection payload, and an carries a The subsystem design. 684x707 km intersatellite Butane propulsion system with a 3m/s be presented on capability. Further information will regards It It UK (see on June 28, 2000 on a launcher from Plesetsk, together with the Tsinghua-1 (China) and the Nadezhda-06 (Russia) primary payload. link to in the this satellite in spacecraft is still Chapter 4 with reported to be operational. -r-'&cp^- J^^*^ - SNAP-1 from SSTL (UK) Figure 2.4 c. Munin Munin (2000-0, 26621 A) is a Swedish 6kg satellite (7.5kg including separation system) to measure the electron and ion distribution in the aurora ovals and was launched into a 698x1800 km orbit. Detector of Ions and Neutral Atoms), The Swedish University Institute is The satellite carries a spectrometer (DINA, cubic in shape and measures 200x200x250mm. of Space Physics (IRF) and the Dept. of Space Physics of (RYP) designed the satellite. Munin was launched from VAFB, 15 Umea together with EO-1 (Earth Observing-1), and SAC-C last contact with Munin was February ('Satelite de Aplicaciones Cientificas'-C). The 12, 2001. STUDENT SATELLITE PROJECTS B. In recent years, an increased effort to design, build, and operate small satellites has taken place in universities and laboratories and nanosatellites provide numerous fraction of the cost of larger all over the world. These microsatellites flight opportunities for science traditional missions. experiments find themselves working progress to continue. Appendix 2.2 known to be involved lists an international environment for space the most current educational establishments in small spacecraft projects. The European Space Agency (ESA) has to involve students into the building is to create a Tomorrow's in SSETI - Student Space Exploration 1. of & Technology Initiative started this ambitious educational project satellites: "...The network of students, educational main objective of this institutions Internet) to facilitate the distributed design, construction initiative and organizations (on the and launch of (micro)-satellites and potentially more complex projects such as a moon-lander [Ref. 8]." The distribution a In addition, there has been an increasing trend towards international cooperation on space projects. engineers will at round of the sub-systems resulted in the following distribution: AOCS: Institute Superior Tecnico, Portugal Communication: UNICAL, Cosenza, Ground Instituto Superior Tecnico, Portugal stations: first Italy Instruments: University of Florence, Italy Lander, Avionics: Escola Politecnica Superior at Universitat de Girona, Spain Mechanism: EPFL(Lausanne), Switzerland Mission Analysis: University ofZaragoza, Spain On-Board Data Handling (OBDH): 16 UK University of Newcastle, • Power: Euroavia-Napoli, Italy • Programmatic: University • Propulsion: University of Stuttgart, • SSETI • Simulation: • Structures/Configuration: of Stuttgart, Germany Infrastructure: Escola Politecnica Superior at Universitat de Girona, Spain TU Vienna, Austria University of the • Basque Country, Spain Structures/Structural calculations: Kingston University, • Germany Manchester University, Thermal: 2. University nanosatellite The Air Force Advanced Research UK program Office of Scientific Research Projects grants of $50k/year over Agency (DARPA) two years UK to design (AFOSR), NASA, and the Defense are jointly funding 10 universities with and assemble 10 nanosatellites (~10kg). The universities will conduct creative low-cost space experiments to explore the military usefulness of nanosatellites in such areas as formation flying, miniature bus technologies, enhanced communications, miniaturized sensors, attitude control, distributed capabilities and maneuvering. The satellites are planned to launch mid-2002. satellite The 10 university nanosatellites provide a broad range of technology demonstrations in the areas of miniature spacecraft subsystem components and formation flying. numerous science measurements and experiments GPS wind, magnetic fields, in such areas as scintillation, solar and upper atmosphere ion density. The Air Force Research Laboratory (AFRL) structure, securing a launch, is developing a deployment and providing such advanced microsatellite hardware as high efficiency solar cells and micropropulsion units. the universities to provide approximately flying technologies as There are also NASA $1.2M funding Goddard has also teamed with to demonstrate such formation advanced crosslink communication and navigation hardware and 17 flight control algorithms. Numerous industry partners are also supporting the universities with hardware and design and testing services. This program has the potential to provide significant payoff for very modest DoD funding by NASA and given the broad university resources being applied and support by industry partners. If these flight demonstrations are successful, it very likely is government sponsorship can be secured for follow-on launches of nanosatellites universities built by and other agencies. Three Corner SAT a. This project University of Colorado Aptly named Three Sat (3 demonstrate and communications, a joint effort between Arizona State University (ASU), Boulder (CU), and Comer will nanosatellites at is innovative New Mexico State University (NMSU). A Sat), the proposed constellation of three identical stereo imaging, command and formation data flying/cellular-phone handling. In addition, each University in the 3 A Sat constellation has the opportunity to fly an individual unique payload should b. it desire. [Ref. 9] ION-F Utah State of University University, Washington, and Virginia Polytechnic Institute are designing and developing a system of three 10-kg spacecraft to investigate satellite coordination ionospheric measurements. The and management three will coordinate on technologies satellite design, and distributed formation flying and management mission development, and science instruments and mission. Advanced hardware for distributed space system to be demonstrated includes micro-pulsed plasma thrusters (uPPT), gimbaling magnetic attitude control, addition, an Internet control that its satellite would and an advanced tether system. In based operations center will be designed to enable each university to from its own remote location. ION-F will focus benefit TechSat 21 and future Air Force and 18 NASA on mission objectives missions. In addition, (SDL, Primex, Honeywell) support has been industrial students, hardware, and design support. [Ref. satellite Spacecraft formation flying and commercial potential EMERALD, a low cost, As and University Stanford is an evolving technology with vast that ranges radical reductions in operations cost. Program, 0] Emerald c. military, 1 identified, including funding for from enhanced mission performance to of the TechSat 2 1 University Nanosatellite part Santa two-satellite scientific, Clara University mission for are jointly validating developing formation-flying technologies. SSDL (Space Systems Development Lab) and Santa Clara (Santa Clara Stanford's University's SCREEM Remote Extreme Environment Mechanisms Laboratory) will work as a unified team to develop, construct, test and eventually operate the EMERALD Stanford's spacecraft. The formation flying experiments will ARL (Aerospace Robotics Lab) The EMERALD from a simple single Mission satellite to two is divided into three distinct stages that progress free flying satellites in a coarse formation. building block experimental strategy, the research payloads isolation. be coordinated through first will Using a be characterized in Then, they will be coordinated and combined to permit simple demonstrations of fundamental formation flying control functions such as relative position determination and position control. At release, the two spacecraft will be stacked together and travel as a single object. This will allow initial checkout, calibration, will and some limited experimentation. During the second stage of operation, the satellites will separate and a simple tether or flexible boom will uncoil, linking the two vehicles. This tethered stage will allow full formation flying experimentation including on-orbit relative position determination and simple closed loop relative position control using the drag panels and microthrusters. During the true final stage two-body formation flying of operation, the tether will be cut for a limited period 19 in order to permit of time. The tether will have a simple sub-satellite at its midpoint. Each will separate. Upon ground command, satellite will retain The University of Washington space. to design, build of UW 1 1] UW nanosatellite (Dawgstar) d. in a cluster sub-satellite half of the tether and half of the sub-satellite, providing very rough gravity gradient stabilization. [Ref. program two halves of the the Nanosatellite and launch the smallest self-propelled flown in 2002 as one of the satellites program first is a student run (15kg) to be used satellite distributed satellite testbeds in has university partners (Utah State and Virginia Tech), each of which designing a allow the "cluster" testbed. The focus on the satellite to UW is program has been on distributed control. The sensing will be done using several technologies relating to GPS, and a cross-link system designed by APL-JHU. The actuation will be a set of eight pulsed plasma thrusters, to be used for both attitude and position control. In addition, there are several other important technology developments in this program, including: development of uPPTs for both and attitude orbit/formation development of horizon and sun sensors using small, lightweight, CMOS control, camera technology and distributed ionospheric science. [Ref. 12] Miscellaneous Projects 3. Project Starshine. small, reflective optically Washington, DC assembled The Student Tracked Atmospheric Research spherical it spacecraft. Satellite is a The Naval Research Laboratory from eleven hundred machined by Utah technology students and shipped sets in kits in of aluminum mirror blanks by project officials to schools around the world where students polished the blanks. The eye. satellite is 48cm Students recorded satellite's orbit decayed (19in) and their was very bright and easily visible observations online while tracking the at a rate proportional to how much heated by solar activity, thus monitoring sunspots. to the naked satellite. the upper atmosphere Starshine was deployed by The was NASA from a Hitchhiker canister on the Space Shuttle Discovery into a highly inclined low 20 on mission STS-96 earth orbit atmosphere on Feb Sunsat. African 1 8, satellite. It cm x 45 cm of 1999, and the satellite re-entered Earth's 2000 Graduate students was launched on February 23, 1999. 45 May in SUNSAT University of Stellenbosch built this South at the as a piggyback payload is x 60 cm, with an a micro satellite, on a Delta from Vandenberg II weighing 64 kg, with dimensions of The polar orbit (620 by 850 km). elliptical satellite holds amateur radio transponders and several other instruments that allow digital store- and-forward capability and a voice The educational demonstrations. 'parrot' repeater that unit has two VHF is being used primarily for and two UHF transmit-receive systems. In addition to amateur radio and school science payloads, Sunsat carries two NASA experiments and an experimental push-broom imager capable of taking pictures of Earth. The high-resolution imager operates computer also can be stored in RAM in real aboard the time on S-band frequencies. Images satellite and then downloaded speeds for retrieval by hams and schools. The participants have made at lower a big effort to use the project to inspire interest in science and engineering in South African high school students. CubeSat. The CubeSat (Stanford's Space System's is a nano-satellite design from Development Laboratory). Bob Twiggs of the SSDL The motivation is to develop a standard, off-the-shelf-satellite small satellite kit that can be cheaply built, easily adapted for different missions, CubeSat is and launched in clusters to lower the per satellite launch cost. The about 10cm per side and weighs a kilogram (see next page - Fig. 2.5), allowing student groups to be able to build and launch them for around $50k each. Eventually, multiple CubeSats will of a single large formed to satellite. together in formation to provide the capabilities The company OSSS (One-Stop-Satellite-Solutions) has been commercialize SSDL's CubeSat. Several college teams are CubeSats for launch CubeSat work at the in 2001: CubeSat at now building Cuesta College Amateur Radio Organization, University of Tokyo, and Cube-sat 13] 21 at Tokyo Institute of Technology. [Ref Inside a CubeSat Figure 2.5 PANSAT. The and designed Petite Amateur Navy by military built officer (10cm per Satellite students, side) (PANSAT) faculty, and a small satellite is staff at Naval the Postgraduate School (NPS). The main objective was to support the Space Systems Engineering and Space Systems Operations curricula by providing a 'hands-on' hardware project where exposure can be experienced. to the PANSAT many facets of a space system development and Shuttle into a low-Earth orbit spacecraft communications using bits C. PANSAT was launched from on the STS-95 Discovery mission as per second and 9 itself direct amateur radio 70 to utilize NPS. part of the the third Extreme Ultraviolet Hitchhiker (IEH-3) experiment. The in the cycle further provides educational training while in orbit through a space-based laboratory for officer students at International life provides store-and-forward (packet sequence spread spectrum modulation. cm band with center frequency at 436.5 radio) digital PANSAT operates MHz, a bit rate of 9842 MB of message storage. Amateur radio ground stations will be able PANSAT via a bulletin-board type user interface. CURRENT SYSTEMS AND INDUSTRY LEADERS Aerospace Corp. 1. In an effort sponsored by Defense Advanced Research Projects Agency (DARPA), Aerospace Corporation scientists 22 and engineers are collaborating with Lockwell Science Center and Stanford University to develop miniature low-cost space latforms to validate microsystems for space applications and advance the development f mass-producible, fully functional nanosatellites (see Fig. 2.6). Aerospace supported reflight activities for an experiment involving two tiny picosatellites to be deployed OPAL rom Stanford University's tie new Air Force OSPSLV. The within the ormed the rrays third in space to MEMS A picosat its launch by mounted on a 50-meter ground antenna MEMS element of a rudimentary constellation. were also tested on /[EMS 2000-present ) after picosats were tethered to emulate formation flying range of low-power radios. The mission he satellite (see section radio-frequency switch this mission. represents one of several programs for systematic testing and use of be designed and implemented by The Aerospace Corporation. One of devices, designed and fabricated at The Aerospace Corporation, comprises an rray of 19 microthrusters that could be used to orient a nanosatellite. Each of the 19 cells epresents a separate thruster like a solid rocket motor on a launch vehicle (see Chapter B for propulsion components). The mission closed out February validate atellites MEMS 0, 2000 and was the operating in constellations. Another picosat mission a third first of a series designed to technology and demonstrate the capabilities of mass-produced miniature iboard the MightySat 2.1 satellite built md 1 more complex was launched July 29, 2000 by the Air Force Research Laboratory (AFRL), "inspector" mission is planned for 2003. i^^^^^^^^H 1 ^j^H ^^^ P^kHt?^ ^H fV': :^>?^^^^BB *H| j*9^z ' fcV*r- BBkL I .* "^'ilSffifipM Figure 2.6 Artist Conception of orbiting 23 AEROSPACE Nanosatellite NASA 2. The Nanosat Constellation Trailblazer mission NASA's New Millennium Program. Known mission will attempt to spacecraft is fly three is Space Technology 5 (ST-5 as about the size of a 'big' birthday cake; spacecraft communication, and payoff from a in for short), the miniature spacecraft high above the Earth. Each of the 42cm (17 in) across and 20 ST5 high^ and weighs about 21.5 kilograms (47 pounds). separate the fourth deep space mission in performing constellation, scientific observations as it was cm (8 in) will attempt to fly three movements, coordinated a single, larger spacecraft. The technology demonstration will enable a series of multi-spacecraft this missions in the future. Large numbers of small spacecraft are planned in "constellation perform in-situ measurements of space weather class" missions in the next century to conditions. Space Technology 5 will about 200x 35,790 a as will single be used system. to test methods The mission technologies in the magnetosphere. will The mission is for operating a constellation also test 3. Flight Center in Greenbelt, innovative eight Maryland and is is new managed by NASA's budgeted at $28 million. Foreign Universities The purpose of the Surrey Nanosatellite Applications Program, by Surrey Technology Ltd (SSTL) located multi-mission of planned for launch in 2003 as a secondary payload on an expendable launch vehicle. The mission Goddard Space orbit at km altitude. The spacecraft spacecraft major fly three nanosatellites in a stable satellite at Surrey University, is to Satellite develop practical, modular, buses within mass ranges of 1-10 kg, and to demonstrate the use of miniature electrical and mechanical COTS technologies in space. It also is to provide vehicles for the education and training of students in spacecraft engineering at the undergraduate and post-graduate level. The first accomplishment of this was by the development of the ultra-low-cost demonstration spacecraft ('SNAP-1' - see above 2000- 24 present ) within a year, by a team comprising undergraduates, Masters and Ph.D. with the supporting expertise from MicroStructure Additionally, Sweden will launch in a the Center for SSTL CSER staff. and Technology (MST) few years the first European Advanced Micro Engineering (AME) at Uppsala University nanosatellite. D. in Researchers within are contributing on microsystems research hoping to increase the European nanosatellite knowledge base. project has been initiated at students, A nanosatellite AME to promote system oriented research work. NANOSATELLITES FLYING TOGETHER One distinctive nanotechnology and concept, which many of the above Many the small satellite designs, next chapter's thrust will be to present numerous formations. demonstrates how to unique is capabilities formation flying. of The use these smaller nanosatellites in concepts have started to surface on what types of formations are better at certain missions, and cluster lifetime orbit analysis simulations are quickly narrowing down the required fuel expense and attitude control needed to maintain the formation dynamics. 25 THIS PAGE INTENTIONALLY LEFT BLANK 26 NANOSATELLITE FORMATION DYNAMICS III. INTRODUCTION A. In recent years, the innovative idea of distributing the functionality of current among larger satellites smaller, cooperative satellites has been sincerely considered for various space missions to achieve goals that are not possible or very difficult to accomplish with a single clusters of imaging. For instance, one possible use satellite. satellites flying together in In this case, groups for nanosatellites is formation for high-resolution, synthetic-aperture of nanosatellites are operated cooperatively sparse aperture with an effective dimension larger than can be achieved larger satellite. The system adds flexibility since the formation and orientation are adjustable on orbit. [Ref. 14] Flying to act as a by a single, and therefore aperture size many satellites in formation presents flexibility to mission designers given that the individual satellites will be capable of repositioning themselves with respect to each other to achieve diverse tasks. By accurately computing the preliminary Keplerian orbit elements, the satellites can realize the desired close separation and cluster orientation desired to operate the necessary missions. In particular, formation flying NASA (MSFF) buzzword denoting and U.S. Air Force have identified multiple spacecraft as an enabling technology for future missions. this division of labor among smaller as Distributed Satellite Systems NASA's (DSS) Mission to Planet Earth and benefits of satellite formation flying in New DoD, satellites is NASA referred to sectors. Millennium programs have acknowledged the and have incorporated an enhanced formation a sophisticated formation for Earth's magnetosphere, the Orion program in formation flying, becoming and the commercial experimentation in the Earth Observing System mission. [Ref. 16] Agency has engineered [Ref 15] Another its flying The European Space Cluster mission to study the intended for a low cost display of GPS uses is and the Laser Interferometer Space Antenna (LISA) mission heliocentric formation-flying mission intended to identify gravity waves. Force Research Laboratory's TechSat 21 program 27 is The US is a Air a technology demonstration of the The TechSat 21 program was reviewed 'virtual satellite' concept. [Ref. 17] in Section II (above). Advances nanosatellites masses of less than 3 a mass of order 10" may have large MEMS in miniaturization using numbers of such 1 kg kg. In some technology leads to estimation of instances, simple satellites-on-a-chip A particular original (picosats). concept is the use of nanosatellites/picosatellites to shape constellations to permit the real-time acquisition of distributed information. For example, a spherical constellation of nanosatellites has been proposed to provide a real-time, three-dimensional magnetosphere. Each nanosatellite can be thought of as a data To supply set. view of the of a three-dimensional 'pixel' high-quality spatial and temporal resolution for such a mission, large numbers of nanosatellites are necessary. Because these ultra-low mass can be satellites passive sensors without active orbit control, environmental effects such as air drag will form the evolution of the constellation. For of order 0.1 kg or less, constellations MEMS fabricated nanosatellites, with a could include several thousand elements. Other concepts have envisioned large numbers (>10 deployed from a dispenser to mass 3 ) of nanosatellites provide a continuous planar ring of to satellites be for Such constellations would be fashioned by dispensing communication purposes. nanosatellites over a range of orbit radii at the identical inclination to induce differential azimuthal motion, therefore forming a homogeneous ring. satellites in basically locations, communication Furthermore, the constellation for on-orbit failures. eventually remove random azimuthal all communications, where the constellation is vehicle. Analysis [Ref. to 1 8] The make up shows that constellation for losses 400 random links large grow A orbits to Formation flying clusters of performance during times of be robust are attractive for military related concept requires clusters and air -95% coverage of the globe. 28 nanosatellites drag removal. satellites also satellite failure. of piggyback launches on any available would require the continual deposition of new failures to short lived because air drag will nanosatellites can provide from on-orbit number of positioned from a single launch vehicle and formed to sustain a dedicated remote operation. nanosatellites to be launched into is Such concepts the nanosatellites. With a provide for graceful degradation of If a single large satellite has a system failure, the entire mission satellites in the cluster The is at risk. may If a single satellite in a cluster fails, the maintain mission objectives be brought back up cluster could then to at a particular lower performance level. mission design specifications or even improved with the addition of another inexpensive replacement The following chapter remaining satellite. investigates several satellite formation-flying designs, with emphasis on the projected circular formation. These cluster formations are described with regard to their basic orbital equations of motion and then are populated through computation from a technique derived from the Aerospace Corporation. [Ref. A MATLAB program was written position and velocity vectors, simulation provides an discussed in AIAA satellite applications. motion for Earth Interferometric imaging are Satellite Constellations Imaging [Ref. 20], and in the following sections four satellite in-track , circular , projected circular) are obtained for a variety The of desired trajectories are carefully designed natural orbits so that the energy costs to fly along these trajectories relative basic formation OF MOTION OF SATELLITE FORMATIONS paper 98-4379, Optimization of Geosynchronous ( in-plane , Satellite STK. This look into the orbital dynamics and a brief look into formations that are optimal for Interferometric Earth formations in turn entered into LEO cluster with the A V required to maintain that cluster. INITIAL EQUATIONS Satellite 9] determined both the mothersat and eight remotes' which were initial perturbations that affect a B. that 1 is minimized. formation flying designs can be derived from the linearized equations of for two objects under the influence of a point mass These equations are commonly known as Hill's equations. 21], a detailed derivation of Hills's equation is From gravitational field. Vallado's text [Ref. presented, which take the following form for unperturbed motion: x-2o)y-3a) 2 x = y + 2cox = z (3.1) + co 2 z = 29 Here, x is the radial difference between the is two objects, the cross-track difference (see Fig. 3.1 below), and y is co the along-track difference, z the is mean motion of the reference object. Relative Orbit Reference Orbit Hill's reference frame for satellite relative motion. Figure 3.1 The unperturbed version of Hill's equations can be solved analytically, with the solution being: x(t) = ^ sm(cot) + (3x + -^-) cos(fi>f ) + (4x + ^_) CO CO y(t) = 2x — 2- cos(a>/) 4v + (6x + -Z±) sm{cot) - (6cox + 3y CO z{t) CO CO = -^sin(ft)/') + z )t 2xG -+y (3.2) CO cos(<y/) co In order to avoid secular, or long-term, growth in the relative motion, Eq. (3.2) needs a constraint on its y = -2x secular term to be set to zero: co (3.3) 30 It can then be shown that this constraint results in a displaced orbit and thus the same semi-major By enforcing the displacements. x(t) = x — axis, as the reference orbit sin(&>0 +x through with the same energy, first order in the small become: constraint, Hill's equations cos(cot) CO y{t) = 2x — - - 2x cos(cot) 2x °- sin(cot) z(t) = — + yQ (3 .4) CO CO sin(<yf ) +z cos(cot) CO where terms with '0' subscripts refer to initial conditions. These equations provide for and along-track (y) formation flying design. In the above equations, components of motion can be seen it are uncoupled that the radial (x) from the cross-track motion. The motion in the radial/along-track plane can be eccentricity (e flying designs = 0.866) with an must project to be an ellipse relative of fixed All unperturbed formation motion into the radial/along-track plane. In the linearized model, the cross-track (z) Combining simple harmonic oscillator. shown arbitrary along-track offset. this elliptical component of the (z) component of the the elliptical motion relative motion is a in the radial/along-track direction with the oscillatory motion in the cross-track direction yields the family of ellipses that describe all formation-flying designs. [Ref. 14] Six conditions must be specified in the solutions to Hill's equations. These initial conditions can be thought of as Cartesian or Keplerian element differences initial between the two satellites on the formation. from the solution One and allow for constraint was of the specified to Hill's equations (see requiring the semimajor axis of the offset six constraints or design parameters to elliptical motion two when the secular terms were Eq. (3.3) above). satellites to of relative motion. be equal. Another constraint in the radial/along-track plane, y . is the This leaves four design initial location in the These also can be thought of as the size of the 31 removed This can be thought of as parameters, which describe the size, eccentricity, orientation, and ellipse be placed ellipse in the radial/along-track plane, the initial location within that ellipse, the amplitude of the and the oscillation in the cross-track plane, initial location in the cross-track oscillation. Basic Remote Motion in the Formation 1. The remote formation, center of the formation a is as mentioned above, satellite flying in which then has a moving frame attached to direction x and y orbit, defined as 'mothersat', directions are shown in Fig. 3.2 pointing upward orthogonal to both the x and y axes. is this coordinate system, the The Moving Frame Figure 3.2 The z derived from Hill's equations. a circular Its it. is dynamic equations of a remote, Under 'Sat2', are: _2-]" /2 2 ..2 x-2coy-co\r +x){l-r J [(r +xy+y-+z } = v. . z y + 2o)x-co y{\-r 3 z + 6T r z[(r + x) Under the 2 +y +z moving frame, xd (?) ~ (*o I yd if) - (2x zd (/) 2 = yJ3(x ]~ Yl } =v (3.5) : = v3 + x cos(wt) cos(cot) 1 co) 2 z the desired trajectory of Sat2 in the circular formation satisfies: <y ) sin(<2tf ) 1 co) +xy+y +z TA z [(r , - 2x sin(cot) sin( wt) (3.6) + v3jc cos(wO 32 The desired path is a nonthrust trajectory for the linearized dynamics of Eq. (3.5). x and x free variables, fixed frame, the path r are the initial value of x and the derivative of x. The In the inertially is: = 2yjxllo) 2 +x] (3.7) All formation designs that follow will be constrained to these equations of As motion of the remotes (Sat2s). will be seen, most parts of these equations can be canceled out, or simplified. In-Plane Formation 2. The in-plane formation of satellites occupying the same equations, setting The solutions are x{t) all initial the simplest is orbital plane of all cluster designs. It consists of a group and separated by mean anomaly. In conditions, except for y , Hill's to zero, represents this formation. then: = y(0=y (38) z(t) = where y represents the amount of in-plane spacing between two related to the mean anomaly satellites. This can be separation by: AM = ^_ (3.9) a where AM is the mean anomaly separation and a is the semimajor axis of the orbits. Again, this formulation being based on Hill's equations assumes that the orbits are circular, are at least nearly so. The primary advantage of simplicity in design, deployment and control. 33 the in-plane formation is its In-Track Formation 3. The in-track cluster design is a special case satellites all share the same ground different orbital planes separated To do track. by of the in-plane formation. Here the so, the satellites right ascension have to occupy slightly of the ascending node (Q), which accounts for the rotation of the earth. The difference in the equations can be seen by the addition of a cross-track oscillation that represents the difference in right ascension of the ascending node. The solutions to Hill's equations are then: x(t) = y(t) =y z(t) = —-sin(i)y (3-10) cos(cot) CO where y represents the amount of in-plane spacing between two motion of the Earth. The orbits, i is the inclination of the orbits, and coe trailing satellite is relate this difference in some time mean the rotation rate of the behind the lead by some difference in mean anomaly; mean anomaly to a difference in time nodal difference AD. such that the second satellite at is satellites, co is the in the future (t + At). satellite is (i.e. At). Then calculate the over the same point as the lead The equations take the following form: AM = ^- co aco (3.11) AQ = co„At = -co ^0 aco z = . zo a sin(z) -sm(i)y co 34 AM where the is mean anomaly separation and attractiveness of the in-track formation the same exact spots on example screenshot from the ground, STK is AQ is the nodal The separation. that each satellite in the formation passes over which is An valuable for Earth sensing missions. for an in-track formation is shown in Appendix 3.1 . Circular Formation 4. The one circular formation is each other. It in which satellites maintain a constant distance from can be derived from Hill's equations analytically or geometrically. analytic approach takes the solutions to Hill's equations and determines relations The between the initial conditions given the constraint: x where 2 2 2 +y +z =r (3.12) r is the radius fact that the relative From 2 of the formation. The geometric approach takes advantage of motion in the radial/along-track planes the (x/y) is fixed in eccentricity. either approach, the following relations are found: 2 . CO y = -2cox (3.13) z = ±V3x i = ±V3;*: where the first constraints show Both two conditions that there are set the along-track offset and drift to zero. two planes in which the circular formation is possible. intersect the cross-track/along-track plane along the along-track axis but inclined 30° to that plane and the other is inclined at -30°. the following solutions to Hill's equations are found: 35 These If the 30° case is one is chosen, x(t) = —sin(a>t) + x cos(cot) CO y(t) = 2x — -cos(cot)-2x sin(cot) (3.14) CO z(t) = >j3—sm(cot) + y/3x cos(cot) CO Note that the formation design again has two free parameters: parameters specify the radius and phasing of one The four other reference satellite. along-track x and x satellite in its circular . These free path about the conditions were constrained for eliminating initial eliminating the along-track offset, and setting the eccentricity and drift, orientation of the ellipse of relative motion. If the initial conditions are formulated in terms of the solutions to Hill's equations presented in Eq. (3.14) of the cluster radius and phasing, the following equations arise: xn = ^3 tan 2 (#) +4 (3.15) 2 rco 2 where r is y]3t<m (0) ^3 tan 2 (0) +4 6 is the radius of the formation, and the phase angle of the satellite measured clockwise from the cross-track(z) direction. From Equation radius and phasing, it (3.14)/(3.15), given a circular-reference orbit is and desired cluster possible to convert the reference elements from Keplerian to Cartesian, after coordinate transformations add the Cartesian differences to get the position and velocity vectors of the satellites in the circular clusters, and then convert the Cartesian elements back to Keplerian for The Keplerian element on the phasing and cluster, there will initial all the satellites in the cluster. differences for a circular formation are highly dependent conditions of the reference orbit. For two arbitrary points in the be differences in inclination 36 (/), right ascension of the ascending node (Q), argument of perigee (w), and will mean anomaly {M). Generally, satellites in a cluster have the same eccentricity except in the case where the reference orbit There are conditions, however, where two satellites in the cluster is not circular. can have the same inclination or right ascension of the ascending node. The circular formation has two properties which make attractive: it 1) the satellites maintain a constant distance from each other, enabling precise measurements for use in missions such as interferometry, and 2) unlike the in-plane and in-track formations, the circular cluster presents a two dimensional array, increasing dimensional resolution for imaging, geolocation potential, and numerous other missions. from STK for a circular formation with four remotes is shown in An example Appendix 3.1 screenshot . Projected Circular Formation 5. The projected circular formation is very close in design to the circular formation. This formation will be covered briefly here in this section, and in more detail with another example of equation setup in Section D (satellite formations with constant apparent distribution - see below). The difference is that the projected circular cluster only maintains a fixed distance in the along-track/cross-track (y/z) plane. formation is to say that track/cross-track plane, when it Another way to describe the projected circular the ellipse of relative motion produces a circle. is projected onto the along- This results in the following constraint: 2 2 2 y +z =r where r is (3.16) the radius of the projected circle. Like the circular cluster, the constraint the solutions of Hill's equations to produce: 37 can be applied to the initial conditions to =—2 x . y o CO y = -2cox (3.17) 2 = ±2* i = ±2x where the first two conditions set the along-track drift planes in which the projected circular formation is and offset to zero. There are two possible. track/along-track plane along the along-track axis but one plane and the other is intersect the cross- inclined 26.565° to that inclined at -26.565° 26.565° case If the is Both is chosen, the following solutions to Hill's equations are found: x(t) = —sin(cot) + x cos(cot) CO y(t) = 2x — -cos{cot)-2x (3.18) s\n(cot) CO z(t) = 2x — ^-sin(6tf) + 2x cos(cot) CO The formation design again has the same two free parameters, which represent the radius and phasing of projected circular path. If the initial conditions of the solutions to Hill's equations in Eq. (3.17) are formulated in terms of the cluster radius and phasing, the following arise: xo =-cos(0) (3-19) x =— sin(^) 38 Like the circular formation design, the Keplerian element differences show up in of the ascending node, argument of perigee, and mean inclination, right ascension anomaly, and the variations are dependent on the reference orbit and phasing. primary advantage of the projected circular cluster over the circular cluster distance separates the satellites when formation the is that The a fixed projected onto the along- is track/cross-track plane. This has applications for ground observing missions. REMOTE CLUSTERS WITH CONSTANT APPARENT DISTRIBUTION C. The work presented one - in this section continues the the problem of creating a cluster of satellites that nearly constant, shape and size to work have a constant apparent would have a satellites (as when viewed from opposed started at the end of the would maintain a the Earth. Such a to physical) distribution. In clear field of view of the surface, and cluster constant, or would seem such a formation, would remain last all in sufficiently close formation to share their information. Such formations are of interest for large distributed-aperture sensing, for example. Another possibility is that of forming clusters from many small, inexpensive satellites, computing power. Remaining in a close formation information and computing ability For such missions, positions. It it each with a particular type of sensor and some among know satellites to share themselves. might be unnecessary might be sufficient to would allow the to maintain extremely precise relative the relative position accurately, and to remain in close enough proximity to allow intersatellite communication. In addition, an effect of constant apparent distribution their ground targets Since it is that the angular dispositions of the satellites relative to and each other would be constant. has been mentioned that one of the desires of nanosatellite designs is to reduce the need for station-keeping thrust, orbits in which the natural motion of the satellites keeps them in a cluster are required. The need for station keeping would then be reduced to eliminating the effects of perturbations on the array (covered below in Section E). These of the effects will be found satellites in the array. in the bulk For missions motion of the array and in in the relative motions which precise knowledge of position 39 is the may be primary goal, some cyclic perturbations acceptable if they are sufficiently small and well understood. Problem Description 1. In what follows, the motion of a moving reference point and as seen satellite will be described with respect both to a by an observer on the surface of a spherical planet. The motion of the reference point can be visualized by thinking of it (possibly hypothetical) reference satellite on a circular orbit as the position of a of radius R, which will be referred to as the reference orbit. This reference point will serve as the origin of a local coordinate system. This reference frame work on the motion of the Let moon was first described by Hill, who derived it in his about the Earth. R be the position of the reference satellite (mothersat), following the reference orbit under idealized two-body motion, and r the position of a nearby point (the remote). Both R and r are expressed Hill's coordinate frame aligned with R. in an inertial moves with The y-axis is frame centered tangent to the reference orbit illustrated earlier in Fig. 3.1. moving reference Hill's the reference point, and z-axis completes the orthogonal set, as Thus, the reference point frame allows the motion of the actual It will be shown create an orbit; this orbit will (passing through the plane at the relative motion as is The at is also the origin of the frame. to the reference point. position the planetary center of mass. the reference point, and rotates such that the x-axis is positive in the direction of the orbital motion. was at it that the be called the some satellite to motion of the relative orbit, as be described with respect satellite in this shown The primary arbitrary angle). frame will in Fig. 3.3 interest below however, is in appears to a fictional observer on the planetary surface, whose always on the ray connecting the center of the planet to the reference point. This observer represents the point on the surface that is the subject of observation by the satellite cluster. The concept of an apparent planetary surface that orbit now moves with time such 40 can be defined. Consider a point on the that it is always between the center of the planet and the origin of Hill's frame. relative to the reference point as seen of sight from the viewer orbit. However, of sight as it it is The apparent from this point. to the satellite, there is helpful to visualize passes through the y - it orbit is the As motion of the this is purely a matter no physical meaning as the trace left satellite of the line to the apparent by the intersection of the z plane in the Hill's coordinate frame, as line shown in Fig. 3.3. Relative Orbit Reference Orbit Figure 3.3 The Reference, relative, and apparent orbits analysis that follows will be primarily concerned with three angles that describe the position of the satellite on the apparent orbit. lateral angle X, and the rotary angle ^are shown, with <f> shown in Fig. 3.4. The elevation angle The angles are all increasing in right-hand rotation about the Hill's-frame x-axis. 41 k, the positive as Viewer Geometry description and Figure 3.4 The maximum value of X the maximum the value R - Rp of the altitude Using total a: orbit, and will denote the radius of the planet, and of the will be referred to as the angular width as the angular height. will be called the altitude Rp of the reference orbit. This is also the mean satellite cluster. this definition, a perfectly circular apparent orbit would result in a constant angle between the line of sight and the line joining the viewpoint to the origin of the The apparent Hill's frame. Rp. value of definitions The orbit can then difference in this formal be calculated by multiplying the angles by method and the R— visualization suggested above is its mean semimajor axis and eccentricity of an orbit. extremely small for clusters in which the cluster radius is small with respect to altitude. The symbols a and The variable for an co is the e will denote the mean motion elliptic orbit is or average angular motion of a body on an orbit, and given by the relation co = yj/j I a 3 where , /j. is the gravitational parameter of the central body (the mass of the planet times the universal gravitational constant). The periapsis, as shown circular orbit, orbit planes. this line true anomaly (v) of a point on an eccentric orbit will be measured from in Fig. 3.5. In this case, the reference plane will be the plane and the line of nodes will be taken to be the The argument of periapsis (w) of the of nodes. The inclination will be denoted 42 line of intersection of the two eccentric orbit will be i and will of the measured from be the angle between the two orbit planes. Finally, the angle u Note vector. = w+v is the angle from the line of nodes to the radial that this is defined for circular as well as eccentric orbits. Periapsis Line of Nodes Reference Orbit Rising Node Orbital elements of the eccentric orbit. Figure 3.5 Linearized Approach 2. The Clossey - Wiltshire (C-W) equations motion of a satellite reference frame as Remember is an accepted choice for describing the near a circular reference orbit [Ref 21]. They are defined in Hill's in Fig. (from Section B ) 3.1, and are generally referred to as Hill's equations. the equations are written: x-2o)y-3co'x = y + 2cox = z It + co 2 (3.1) = - follows that the out-of-plane motion z(t) where Az slightly = A: is cos(a>t the more is given by: + <p_) magnitude and difficult. (3.20) ^ is a constant phase shift. Following usual practice, integrate y to y = -2o)x + k Solving for x and y is only get: (3.21) 43 where & is a constant of integration. Substitute this into the x equation to get: 2 x = -a) x + 2o)k This is directly solved to get: x{t) which (3.22) + cos(cot + 2klcQ x) (f> (3.23) substituted into Eq. (3.1) and integrated to get: is y{t) where c = Ax = -2Ax is sm(cot + )-2>kt (f>x +c (3.24) the constant of integration. Because a purely cyclic motion method, k orbit for this = 0; c is is requires with the same period as the reference simply an offset term and can be given the value zero without loss of generality. Only the orbit as it appears from the surface of the planet upon. In keeping with the assumptions assumption orbit is made = -2AX what is to linearize the equations being focused of motion, the that the variation in r is negligible with respect to the altitude The appearance of the y{t) made is sm(cot + orbit will then be its projection on they - of the z plane. This shows: (/> x) (3.25) z(t) = A, cos(cot Setting <j> x = + <f>, the initial conditions, Setting fa = fa + n <f>z it ) results in a perfectly elliptic projection. is possible to set A z = 2A X and the The statement projection assumes that the viewpoint is on the properly specifying apparent orbit becomes a also results in an elliptic projection, projected motion will be reversed. By however the circle. direction of the that the apparent orbit is the y-z line connecting the center of the planet to the center of the relative orbit. Because the many initial angle fa in Eqn. (3.25) satellites in a circular some constant angle. This is arbitrary, it is possible to place as apparent orbit as desired, each separated from the next by would give a "pinwheel" 44 effect from the planetary surface as the satellites rotate about the center of the cluster. The cluster would spin about its own center once during each orbit of the planet. The radius of the apparent orbit cluster of an almost arbitrary number of reference satellite, as in Fig. 3.6. is also arbitrary. Thus, satellites, The period of it is possible to create a by creating concentric rings about the the apparent orbit reference orbit, so that the entire cluster will maintain its is always that of the shape relative to the center. Thus, separation angles such as fa in Fig. 3.6 should remain constant. z V o Figure 3.6 As the Apparently circular cluster of satellites. C-W equations are linearizations, describe the relative motion of the orbital equations satellite. it is to be expected that they do not Examining the higher-order terms can make an approximation of their This error. is fully in the the subject of the next section. The height of Thus, this shown to be independent of the width. technique can also be used to generate a relative orbit that circular in actuality, 3. the apparent orbit can be and which appears elliptic when viewed from is very nearly the planetary surface. Orbital Mechanics The motion of the satellite described by the C-W differences in the orbital parameters of the satellites. 45 The equations is caused by slight satellite in the elliptic orbit moves more slowly than It the reference when at apoapsis, thus seems to lag behind and then to catch up. and more quickly near When combined periapsis. with inclination, this leads to an apparent orbit about the reference satellite. Investigating the apparent orbit in terms of the orbital elements forward as using the linear equations, but limits this is approach allows an investigation of the of the linearized approach, and an estimate of the corrections due In this section, the eccentricity not as straight and inclination parameters required to to nonlinearity. produce a circular apparent orbit will be investigated. In developing the ideas of this section, attention will be confined to orbits in which periapsis is 90° from the line of nodes. This ensures the symmetry of the apparent orbit about the Hill's-frame z-axis. (Recall that the line of nodes as defined here will be the line of intersection of the orbital plane with the reference plane.) The of the description of the apparent orbit will be expressed in terms of the eccentricity satellite orbit. may be These terms will be on the order of e\ thus, second-order terms in e considered first-order corrections. In the following analysis, estimates for these correction terms will be found. Apparent Angular Width a. The needed eccentricity is first The width particular width. will examined to produce an apparent orbit of a be defined by projecting the line of sight to the satellite onto the reference plane. The angle between this projection and the line of sight to the reference point is the angle X is then the angular separation of the satellite and the reference point (this in Fig. 3.4). The width of the If the inclination orbit is then the of the orbit is maximum zero, the angle is value of this angle. easily computed, taking the reference orbit and the eccentric orbit to have equal semimajor axes and thus equal periods. The motion of eccentric orbit. the reference point The separation at is any point then equivalent to the is then the difference between the true anomaly v and the mean anomaly M. Assuming the 46 mean motion of the inclination is small enough that its of some information. The expansion as given effects are negligible then allows extraction in Vallado [Ref. 21] is: v-M = 2esinM + ±e sm2M + 2 Finding the maximum, (3.26) ... taking the expansion through second order, and taking the derivative with respect to M results in: 4ecosM + 5e 2 cos2M=0 For e — > 0, this leads to (3.27) values of M approaching through by e allows Eq. (3.27) to be rewritten nil. Setting M = nl 2 + 6 and dividing as: -2sin£-5ecos2£ = Again, e 8 is less - gives 5= 0, (3.28) and for very small e, important than the knowledge that 8 Using these results, have 8 is of order Eq. (3.26) can approximately -5 el is 2. The sign of e. now be written: 2 v-7W = 2esin£-{e sin2£ + Because 8 is small, cos 8~\ (3.29) ... and sin2S ~ 2b. Because 8 the right in the above equation is -e 2 and is of order e, on the second term can be ignored. Thus, to second order: v-M = 2e (3.30) Thus, the angular width of the apparent orbit as seen from the center of the central body is 4e. From the surface of the planet, the angular spread is wider. Letting X denote the angle between the lines of sight to the satellite and to the reference, geometry gives: A, = rsin(v-M) tan" (3.31) yrcos(v-M)-R py Expanding the radius of the r elliptic orbit in M and e, again writing M as nl 2 + 8, gives: = R[l-esmS + (e 2 l2)(cos2S-\) + ...] Using the expansions for v - M and r as expanded, the 47 (3.32) first order approximation is: , A= IRcosS R-R p e~ B and 2Re R-R p , , (3.33) n of the second-order term that the coefficient order. This gives a total angular width is of order of the apparent orbit 5, making of: 2A*4e[R/(R-R p )] For a low-altitude orbit, the term of third (3.34) Rp may be nearly as large as R, and thus the angular spread of the apparent orbit as seen from the planetary surface might be several times larger than the value of v - M. Correction for Inclination b. To allow of the apparent orbit must be extended between the projection of the the eccentricity vector (that ' ( rsinv ^ is, , it _,ftanv (3.35) v C0Si the apparent orbit M of the reference is then the difference between this angle orbit. This angle will be maximized near v = nfl argument of periapsis is w = njl , this about this point, the small parameter s to v and the projection of shows: = tan-'P=— The width of Consider the angle the vector from the center of the planet to periapsis). i^rcosvcos/y and the mean anomaly to include the inclination. radial vector onto the reference plane Letting this angle be denoted v v = tan compute the width for orbital inclination, the expansions used to . Since the assumption is the corresponds to the descending node. Expanding = v - njl is defined, and the angle corresponding as: tan£ = =>£ = tan [atan^] rcos£ 48 (3.36) where for brevity use a of cos in place the projection of the radial vector. This angle i. The expectation measured from the is is s » 0, so that line of nodes to that Eq. (3.36) should be valid over the area of interest. The expansion 2 a£+[(a + a s = for £is: +0(s 3 /3]e The angular width is 5 found by introducing the expansion for v into expression. At this point, the assumption correct, and thus that 8 is in fact constant c at least through Knowing third order combined first 5 that in the (3.37) ) made is that the earlier results of about the same size as e and that is two of the order of e allows the disregard to 2 2 2 )S -3cos£-13cos3J](a/12)£> expansion some all terms above 2 ]e 3 (3.38) 3 +... (ce) for a and expanding the trigonometric terms gives the as: £-S = 2e-S 2 2 2 e-(5 + c )Se /2-(c 2 +%)e 3 through third order combined. Through there are for 2 + [\6(\-a )Scos2S-5sm2S](a/4)e +[32(l-a )cos S = cos & ce variables. Thus: 2 i were "nearly" order. £-S = (a-l)S + (l-a )aSy3 + 2acosS[l + (\-a Substituting cos i this first (3.39) order, again the width is 2e, and once more no second-order terms. Apparent Vertical Size c. The apparent between the line of sight vertical size to the satellite of the orbit is determined by the elevation angle and the reference plane. Relative to the center of the planet, this angle can be expressed as: kc = sin "' [sin m sin/'] (3.40) 49 where u the angle from the line of nodes to the radial vector. is As a of the result requirement on the argument of periapsis, the absolute value will be maximized for u—± n/y. maximum kc = to For a circular value of v -1 sin there is maximum requires that the it is value of k c equal the Thus, from Eq. (3.30): . v-M = 2e (3.41) e. The the lateral. =i= (sin/) second order in M orbit, vertical angle as seen However, maximum as the from the surface of the planet vertical spread occurs at the is widened, as is extrema of the radius, a first-order contribution of e to r that must be examined. When the satellite is apparent elevation angle can be at the apses, the written as: ( k= i?(l-f-,?)sinz' ' tan R(l + where s is equal at periapsis. tan magnitude in Expand about s= - f k= (3.42) £)cosi-R p R cos i — R p positive at apoapsis and negative RRsini (R-R p ) +R 2 . is to get: \ . R sin i ' and to the eccentricity, 2 sin £+ 2 (3.43) ... i Recalling Eq. (3.41) and making small angle approximations, the first term reduces K^2R-e/(R-R p ) The second term is assumption is made Rp approaches R, is that the coefficient that the product Re I k. of the correction contains (R - Rp) is The I and Rp. it is An when seen that the correction term (R- Rp). coefficient of the correction term at periapsis R small. This is of concern only as for a low-altitude orbit. In this case, of second order in Re of the angle (3.44) the first-order correction to Note to: and a contraction at apoapsis. 50 is negative, The effect showing an expansion of this could be to raise the apparent orbit slightly with respect to the reference plane. If in fact the apparent orbit is elevated, there should be a corresponding elevation at the points of maximum lateral spread. To orbit at investigate this, a value which the apparent is orbit achieves it required for the position of the satellite on maximum width. Return to Eq. (3.38) in an attempt to find this value. If the derivative of this equation are found that are of first taken, however, order combined. Because the value of s - show information. Numerical investigations is in fact is 6, no terms and thus of i, is second order, first-order terms are necessary to obtain any defined only through width its that the elevation of the point of maximum not linear with e [Ref 21]. Phase Separation d. When point, they will there is in Apparent Orbit more than one satellite in apparent orbit about the reference be separated by some angle, constant equations. In Fig. 3.6, the angle denoted fa is to the accuracy of the C-W such a separation angle. The separation angle between two satellites in the same apparent orbit is a function of the angle between the lines of nodes of their orbits. Nominally, the separation angle equals this angle. The separation will vary as a result of the second-order deviations of the apparent orbit from perfect circularity; the actual amount of variance will be dependent upon the separation. Although angle, the angular rate will sweep same its this of the angle faster as that for the true makes it impractical to define the change in separation satellite in its when it is apparent orbit can be considered. The at periapsis (f> is than at apoapsis; the ratio will be the anomaly rates: *„M=[(l + *)/(l-«)r where satellite (3-45) the phase angle in the apparent orbit and the subscripts refer to periapsis and apoapsis. 51 An estimate of the total variation in fit) from that predicted by the equations can be had by finding the value of the true anomaly equals x/l v= . This is accomplished through the same expansion used M + 2<?sinM + (5e 2 /4)sin2M +...=> v =7u/2 can be expressed approximately — + tan" = R(y-M) Recalling that apparent angle as two i M goes from satellites in orbit mean anomaly earlier: (3.46) orbit, measured from the vertical, as: rsinwsin/^ ' the + 2e through second order. The phase angle in the apparent <f>(M =7r/2) when C-W « *^/2 + sini (3.47) sweeps through Ae 2e, this implies that the satellite n/L to 3i ^A less than during the other half of the orbit. Thus, about the reference, separated by a 180-degree difference in phase, will vary in their relative positions by ±4e radians during the orbit. Geometry of the Orbit e. The apparent orbit described in the previous sections is in a sense the projection of the relative orbit onto a plane normal to the radius of the reference orbit (see The Fig. 3.3/3.4 above). from the actual relative orbit results gained in Section The depth of apoapsis, which by the apparent orbit 4Re, the motion is As from the reference point maximum ^ A first As motion a result of to the satellite will vary satellite crosses the times this amount distance will be (to This follows between periapsis and the first-order approximation of the diameter of this implies that the relative apparent orbital radius (when the approximately elliptical. clearly the difference angled 60° with respect to the reference plane. actual distance not circular, but C above (Linearized Approach). definition is 2Re. is is (at lies this, near a plane that the variation in the from a minimum of the reference plane) to a periapsis and apoapsis). order in e) V5(/? 52 -R )e is maximum Thus the of actual Example 4. An example linearized solution orbital is presented. To tie all the chapters together, the nanosatellite design's parameters are used in this example. As has been mentioned in the preceding section, the appearance of the relative of the angles between the orbit is a matter reference, lines of sight to the satellite and to the from a point on the planetary surface along the vector joining the center of the planet to the reference. precise definition. by the of the correction terms on the to demonstrate numerically the effects altitude For To speak of this case, of the reference The nominal orbit. satellites, such as in Fig. orbital altitude is -1111 km second-order effects, set the radius of the apparent orbit spherical Earth. The cluster consists of a single ring, such that their nominal apparent angular separation is at 3.7, in apparent orbit about a (600 nm). To exaggerate the 25 km. Additionally, assume a with eight satellites equally spaced 45 deg. Nominal Angular Separation Figure 3.7 more simply multiply the angles describing the relative orbit Consider a cluster of Earth central point. the radius of the apparent orbit requires a Ring of eight equally spaced 53 satellites. 45° Using the X = 25/1111 = 0.0225023 From rad. 6378.1363 km, and £=7489.5457 km) 0.0225023 = 2 Re/(R The The have the angular spread of the desired results derived above, Eq. (3.34) the equation works out (with is \0e 0.1 = 1.84 x 10 -4 which when multiplied by the nominal 204.07 m. to the 10" 3 rad the spread angle, so first-order corrections to the vertical spread Ak = The shape of the apparent be Rp = that: -Rp )=* e=\.67 x inclination angle orbit to i = .09563 deg = 2e = of the 3 3.34 x 10" rad = 0.1913 deg. orbit are: = 0.0105 deg altitude of the orbit produces an apparent orbit is so close to circular that it shift of would appear so naked eye. D. POPULATING AND MAINTAINING A CLUSTER IN ORBIT (LEO) The following analysis was used to generate both cartesian LOW EARTH and classical orbital elements (Keplerian) that were entered into Satellite Tool Kit® (STK). Both in-plane and out-of-plane clusters were developed and varying plane angles of the remotes were looked at. some of the Creating a Matlab program (see final results satellites 3.2 ) to calculate the data points, of the different cluster formations can be found For a cluster constellation such as chosen so that (1) each Appendix satellite that in Fig Appendix 3.8, the initial orbit 3.3 , elements are occupies a node in an arbitrary spatial pattern, and (2) the undergo a cyclic motion that allows the formation AV for maneuvering. in Depending on the be occupied by a mother ship or the suborbit plane will maintain may be its application, the center position in the cluster empty. An arbitrary distribution little may of spacecraft in configuration, and the suborbit normal will stay at a fixed angle 8 relative to the mothersat orbit normal. can to persist with relatively occupy the disconnected nanosatellites characteristics of a larger (km-scale) spacecraft. 54 suborbit Large numbers of physically plane and can imitate the Normal Vector Mothersat Orbit Normal Vector Subsat Orbit of Subsat Plane Mothersat Orbit adir Vector Cyclic motion of Subsat Orbit Figure 3.8 Populating a Remote Cluster 1. Each subsat (remote) undergoes a satellite (mothersat). cyclic motion in the reference The remote completes one suborbit cycle and frame of a center returns to original its position relative to the mothersat after one revolution about the Earth, as illustrated in Fig. 3.8. Therefore in the cluster satellites all semimajor axis (and hence the same The cluster can must have the same value orbital period) if perturbing forces are neglected. be populated based on the idea that each remote o'clock position in the suborbit exactly once per revolution. orbital velocity vector is taken to for the is in At the 6 o'clock or 12 this instant the remote be parallel to the mothersat velocity vector, but the two velocity magnitudes are different. No further assumptions concerning the position and velocity of the remote at other points in the orbit are needed to determine the orbital elements. The inclination initial i c, mean anomaly mothersat orbit epoch M c o- specified by the semimajor axis Q a c eccentricity , argument of perigee wc Three parameters that specify the cluster geometry (p, right ascension at is of the ascending node 55 c, , ec, and rj, S) to are illustrated in Fig 3.9. The scale factor assigned the same distance value for pattern generator arrangement rj all remotes is 25 km), while the dimensionless The angle £ specifies and the orientation of the suborbit plane Choosing (i.e. the overall cluster size of and can be different for each remote and determines the geometrical for circular rings, 77=1). (i.e. p determines different values for 8 in the shape of the suborbit relative to the orbit plane of the mothersat. one cluster will produce a swarm having multiple suborbit planes. Remote rs (12 'clock) r|p sin5 Earth rs (6 o'clock) \ Remote Remote Geometry Figure 3.9 The following paragraphs Astrodynamics textbook. [Ref. 21] choosing a set Ec of as viewed along Mothersat Velocity vector utilize values (e.g. 0°, 45°, 90°, etc.) where o'clock position (either 61 =2tan -i + e, when e„ sin Ec orbital is elements begins by the eccentric the corresponding remote is at anomaly of the 6 or 12 populate the cluster). At these 'population points' radius and orbital speed of the mothersat are: (E„ tan 1-e. M=E- may be used to mean anomaly, l and equations from Vallado's Determining the remote the mothersat at the location in the orbit the true anomaly, definitions (3.48) v E„ (3.49) 56 1 (3.50) + e, cos 0, r V c=JV /? c p r. Q cJ (3.51) v where 2 = a f (1 - e: ) population point - vs = are: 8) + (rc ± rjp cos Sy y](rjp sin f 2 jM l> rs l s where the ± sign is (3.52) (3.53) \ it 3.9 the radius and orbital speed of the remote at the 2 rs Here From Fig . in assumed J Eq that (3.52) is negative for the 6 o'clock and positive at 12 o'clock. a s = a c for all remotes, but this choice will be modified Using the above expressions together with the construction velocity vectors in the mothersat perifocal coordinates when later. in Fig. 3.9, the radius a remote is at and the population point are: COS0. r. = r. sin^ (3.54) -s\r\6c v„ rs = = ec +cos# c (re ± rjp cos 5) cos 6C (rc ± rjp cos 8) sin 6C ±rjp sin 8 (3.55) V, = V. 57 The magnitudes of these vectors agree with the scalar values in Eqns. (3.50)- Equation (3.55) expresses the key idea that the remote and mothersat velocity (3.53). vectors are parallel Knowing at the population points. that the radius vector and velocity vector of a remote orbit is sufficient for calculating its orbital elements (a s , es , is to transform is , Q s, vv s , at M one point in s0 ). The first its step remote radius and velocity vectors from mothersat perifocal the coordinates to Earth-centered inertial coordinates as follows: r*=*-r. (3.56) vp =-R-vJ where R is a 3x3 rotation matrix 1 defined by: R = R(Clc )*R(ie )*R(wc ) (3.57) and with the three components identified *(a)= cos(-Q c ) sin(-Q c ) -sin(-Q c ) cos(-Q.) as: 1 R(ic ) = cos(-ic ) sin(-ic ) -sin(-/c ) cos(-*c ) cos(-wf ) *to) = -sin(-w f ) (3.58) sin(-wc ) cos(-wc ) 1 The second step is to apply the set of equations relating the Earth-centered inertial radius and velocity vectors to the classical orbital elements. This procedure yields (a s , es , 1 Matrix found in Matlab program in Appendix 3.2 58 Q w is , s, S [1 =2 tan E. s =E The • S is the true anomaly anomaly and mean anomaly eccentric M 6 ) where 6 s, mothersat s at the -M»fr -e sinE s (3.59) (3.60) s mean anomaly M at s the is M same c0 =0 o . Using Eq. (3.49) point, the initial vv s , M s0 M c at the initial population point and orbital elements. is M The j0 = M -M s c , resulting es , is , are distributed around the mothersat values, with the difference depending on the magnitude of/? and on the location The for and hence the remote mean anomaly which completes the determination of the remote s, The corresponding are: cluster is initialized with the mothersat at perigee, Eq. (3.51)for Q population point. initial on the remote within the cluster. subsat orbital elements can be refined to enhance cluster stability in the presence of the non-spherical geopotential. This is accomplished by calculating the subsat displacements (relative to their original positions) after one orbital period of the mothersat. is due The most to the mean-motion significant perturbation (which will be discussed in a later section) second zonal harmonic vP Jj, which appears in the expressions for the mean and the secular rates of change of Q and w. Propagating the cluster for one revolution in low-Earth orbit under the influence of 1/2 reveals that the remote displacements are primarily in the mothersat along-track coordinate (y-axis), with cross-track and radial displacements being at least 50 times smaller. This suggests that the semimajor axis of each remote can be adjusted slightly to cancel the along-track displacement. the same for all satellites in the cluster, semimajor axis as In the unperturbed solution the that namely a s = a c in Eq. (3.53). compensates for along-track displacements due =ac+ ^- semimajor axis was to J2 The revised is: (3.61) 3/T 59 where AY is one revolution. the along-track displacement after semimajor axis differs from a c by less than 5m 1111 at km Typically the revised altitude. Of course, adjusting a s does not reduce the cross-track and radial displacements caused by Ji. t E. PERTURBATIONS AND A V REQUIREMENTS 1. STK Perturbation For real Propagators geopotential and other perturbing forces analysis the non-spherical conspire to disrupt cluster coherency. Data from propagated the orbits and presented STK's numerous based influences on perturbation modelers two-body, and Ji J a, contributions, solar/lunar gravitational effects, solar radiation pressure, and atmospheric drag for short term (3-10 days) and long term a. Two-Body, A Two-Body, gravity from the Earth, the year) investigation. J2 and J or Keplerian motion, propagator considers only the force of which is modeled as a point mass. The two-body propagator uses same basic technique outlined technique assumes the Earth (1 is in the two-body equation of motion development. This a perfect sphere and the only force acting on a gravity. This propagator doesn't account for satellite is any perturbations. J2 Perturbation (first-order) and J4 Perturbation (second-order) propagators account for secular (long-term) variations in the orbit elements due to Earth oblateness. These propagators don't model atmospheric drag or solar or lunar gravitational J2 and J4 are zonal harmonic coefficients in an forces. infinite series representation of the Earth's gravity field. J2 represents the dominant effects of Earth oblateness. The even zonal harmonic coefficients of the gravity field are the only coefficients that result in secular changes in satellite orbital elements. The includes the first-order secular effects of the J2 coefficient while the and second-order of J2 and the includes the first- coefficient, which produces long period periodic effects 60 J2 propagator J4 first-order effects effects, isn't propagator of J4. The J3 included in either propagator. J4 oblateness. Since the second-order J? and the first-order there is little is difference between the orbits generated b. J4 and is a result of Earth secular effects are very small, by the two propagators. HPOP HPOP the is High Precision Orbit Propagator, and was the main propagator for formation analysis for can handle J2 approximately 1000 times smaller than this thesis. circular, elliptical, parabolic HPOP, included as part of and hyperbolic orbits Moon the surface of the Earth to the orbit of the at distances and beyond. As name its STK/PRO, ranging from implies, it uses a powerful propagation technique to incorporate sophisticated orbit perturbation models. HPOP uses a Runge-Kutta-Fehlberg integration method of order 7-8 to propagate the satellite state in the J2000 reference frame. including the Joint Gravity Earth's oblateness) with a A variety of high-fidelity models are utilized, Model 2 (JGM2; a highly maximum precise model (70 X 70) of the degree/order of 21 The atmospheric density model used the Jacchia-Roberts (similar to Jacchia-1971 but uses analytical methods to improve performance) to model drag effects on the spacecraft. This model takes into account daily variations in the height of the atmosphere due to solar heating among other parameters, and was based on values of Cd = 2.0, of 0.020 daily/average F10.7 of 150, a geomagnetic index of 3.0 and an area/mass ratio m /kg. 2 Solar radiation pressure was modeled using C r =1.0 and the same area/mass ratio as above. Since sunlight produces a small force on any exposed surface. This force varies depending on more how reflective the surface is reflective than a black surface). been updated to The (i.e. solar radiation pressure a mirrored surface model in be consistent with other commonly used propagators such as Finally, third body gravity models (solar/lunar HPOP is has GTDS. gravitational effects) are based on U.S. Naval Observatory data and are accurate to within 0.03 arc seconds. Additionally, third-body gravitational perturbations are also planetary ephemeris from JPL for the Sun and Moon. 61 computed using the DE 405 Perturbation Effects 2. Propagations in the presence of perturbations show the circular formation to be The primary highly unstable. The oblateness or Ji effect. factor disrupting the formation design Jj contributions to the relative motion are magnitude larger than the disturbing accelerations including the earth's is at least an order of tesseral resonance (for short repeat ground track cases), atmospheric drag, solar radiation pressure, and third gravitational effects. Earth oblateness effects are most prevalent in the secular motion of the right ascension of the ascending anomaly node (Q), argument of perigee right ascension of the ascending node. Since the two orbital planes have slightly for each orbit to precess at slightly different rates. drifting apart An of the ascending This results in the orbital planes and a cross-track error growth. additional contribution to the error growth of apsides of the the and mean cross-track error growth can be attributed to the difference in precession rates different inclinations, the secular Jj effect causes the right ascension node (w), (A/). The of the body This orbits. mean anomaly and is is caused by the rotation of the line close to causing the Ji effect to be equal but opposite on the argument of perigee. If the orbit is not circular (i.e. all the remotes' orbit) the orbit line of the apsides begins to rotate and disrupt the formation design. Large variations in eccentricity and argument of perigee that occur in near. circular LEO will disrupt the formation, but they can mothersat into a 'frozen' orbit. be greatly reduced by placing the This choice also simplifies the ephemeris representation, &fc. The NASA/JPL stability and maintenance because here are no secular or long-period variations in ec and TOPEX mission has successfully demonstrated the long-term of this type of orbit. [Ref. 22] ec that is A frozen orbit requires w c determined by the choice of inclination and satellites in the cluster (typically the expected that the total AV = 90° and a altitude. particular value Although only one of the mothersat) can achieve frozen conditions, of formation keeping will be minimized for 62 of it is this choice. Periodic maneuvers must be performed by the mothersat to follow a reference orbit having an altitude and frozen eccentricity. Typical the mothersat will have a higher areato-mass ratio then the remotes, and compensation rate is its altitude will therefore decay more quickly. Drag applied so that the mothersat reference orbit matches the average decay of all the remotes. This minimizes the fuel requirements for each remote. For the long-term behavior (~3 years), solar radiation pressure and atmospheric drag will disturb the frozen orbit conditions, because the eccentricity and argument of perigee are no longer constant. Small oscillations in inclination are common to the mothersat and remotes and are not expected to disrupt the formation. For the short-term behavior (1-10 days), effects of atmospheric drag, solar radiation pressure, and the Jj to J& gravitational harmonics disrupt the cluster geometry. The position deviations indicate that frequent formation-keeping maintain the desired separations position deviations at 1111 among km the remotes. and altitude, mothersat maneuvers. The cross-track deviation Drag this is is is maneuvers are needed to the largest contributor to the compensated efficiently using caused primarily by differential nodal regression: HS ts.Vm where tm orbit < =3^L is (3. 63) the time between maneuvers, decay rate in m/sec, co is the orbit increment of each two-burn transfer. summing 3. 362) L is mean The the in-track tolerance in meters, q motion, in rad/sec, and total AV AVm is is the the velocity for the remotes are obtained by the velocity increments for each maneuver. Formation Keeping The circular and projected circular formations were highly unstable and require formation-keeping maneuvers to account for earth oblateness, atmospheric drag, and tesseral resonance (for short repeat ground track cycle orbits) 63 effects. Both atmospheric drag and The track direction. tesseral resonance affect the formation in the along- along-track error growth induced by these perturbations can be controlled via small adjustments in the semimajor axis of the satellites. Based on the results of the propagation, these semimajor axis adjustments are sub-centimeter for drag effects and on the centimeter level for deep tesseral resonance. From Gauss' variation of parameters plane, the (VOP) equations [Ref 23] for Keplerian elements in the normal-tangential change in semimajor axis due —= at to a disturbing acceleration is: aA (3.64) fj. where a is V is the semimajor axis, constant for the earth, and a d If a velocity , impulse is the magnitude of the velocity, ju is the gravitational the acceleration in the direction of the velocity vector. is assumed, Eq. (3.64) can be rearranged to determine the velocity impulse required to produce a desired change in semimajor axis: AV,=-^-Aa (3.65) 2a V where the changes to the axis in velocity, AV, and semimajor From Eq. nominal values. (3.65), it axis, Aa, are can be shown by 10cm, a velocity impulse of 0.00478 cm/sec near-circular orbit. is assumed small compared that to change the semimajor required for an 1111 maneuvers will altitude Thus, the AV, and therefore propellant requirements, to account for the differential drag and tesseral resonance effects will be quite small. the km depend on the extent of the drag and resonance The frequency of effects, formation keeping error bounds, and several control related issues, such as the accuracy with which these maneuvers are affected. The effects contributions. of Ji were seen In terms in the cross-track direction of Keplerian elements, the J2 secular effects on right ascension of the ascending node, argument of perigee, and [Ref. 23], the change with indirect along-track in right ascension (cross-track) acceleration mean anomaly are of concern. of the ascending node due is: 64 From to an out Battin of plane dQ. rsin(MH-v) dt h sin(z') l where r (3.66) dh w the satellite radius magnitude, is anomaly, h is the angular momentum, is the argument of perigee, v the orbit inclination, and momentum of the angular in the direction i is vector. If a velocity a& is impulse is the true the acceleration is assumed, Eq. (3.66) can be rearranged to determine the velocity impulse required to produce a desired change in right ascension AVh = h sin(z') of the ascending node: AQ (3.67) rsin(w + v) where the changes are assumed small. The differential oblateness effects on right ascension of the ascending node can be derived analytically by taking the partial derivative of the governing equation with respect to the inclination (taken from Vallado [Ref. 21]): 2 3 2 (£] \P ) wcos(z') (3.68) > . \ 3 = — J2 2 dt wsin(/)3/ <p J Combining Eqs. (3.67)-(3.68), required to maintain nodal spacing is it is seen that the amount of velocity impulse proportional to the size of the formation and the length of the mission: A^ = 5QA/ = h sin(i') *K where (3.69) 3 wsin(i)diAt J, rsin(w+\ 2 p j 8/ is the inclination difference in the formation maneuvers. If the near circular assumption AK„ = wW(,') a sin( w + v) 3 is 1-a-A/ and At is the time between made, Eq. (3.69) simplifies to: (3.70) 2 65 The AV total requirement is approximately ~7.2m/sec per kilometer of separation per year for an 1111-kilometer altitude circular formation. maneuvers must be performed It should be noted that the at certain points in the orbit to avoid disturbing the inclination. The second disruptive influence of the earth's oblateness flying is the rotation of the orbit line on satellite formation of apsides. Since the effects of J2 on the argument of perigee and mean anomaly are nearly equal and opposite: 3 '** 4 kp w = —J • co(4-5sin(i)) J (3.71) ( M =-J 4 The effects 2 D N 2 2 \^-\ yll-e G)(3sm (i)-2)' \P) of accelerations in the normal direction are also nearly equal and opposite on these elements for near circular orbits: ~17 at = ~17 2e + - C0S ( V ) adn eV\ a \ — dM = at J (3.72) A — ~ eaV\a -b (r -cos(v) \adn J In Eq. (3.72), only accelerations in the normal direction (mutually perpendicular to h and v are considered since impulses semimajor axis maneuvers in the tangential (velocity) direction in the cross track direction do not affect the would affect mean anomaly. Since the effects of these maneuvers are nearly equal and opposite on the mean anomaly and argument of perigee, the formation keeping analysis can be focused on maintaining either one of the elements with maneuvers in the normal direction. assumes that the other element will be maintained by those maneuvers as well. For now, the argument of perigee If is looked at. impulse maneuvers are assumed, Eq (3.72) can be rewritten in terms of the amount of velocity impulse required AVn This for a given =-^—Aw change in argument of perigee: (3.73) cos(v) 66 where the eccentricity is The required change considered to be small. in argument of perigee can be derived from Eq. (3.71): Aw = wAt = — 6>A/ J., 4 - (3.74) AV = n eV 6 3 —--J cos(v) 4 coAt 2 where the amount of A V required 1111 km year. It is again a function of the length of the mission. For an altitude circular mission, the total Av requirement is approximately 2.1m/sec per should be noted again that the maneuvers must be performed at certain points in the orbit to avoid disturbing the eccentricity. 4. Station Keeping The primary atmospheric drag. orbit's keeping concerns station For an 1 1 1 1 semimajor axis by close drag are dependent on the atmospheric density, which km formation all flying designs altitude satellite, atmospheric drag could to 0.2 kilometers in satellite is for altitude, one year. Of course decay the the effects of area to mass, drag coefficient, and the a function of the solar cycle. A high-density atmosphere near the peak of the solar cycle was used in this analysis so the drag estimates overly conservative. is may be Station keeping a 0.2 km/year decay in semimajor axis could require approximately .097 m/sec of velocity impulse. F. OTHER TOPICS Another topic specifically important to formation problems is reconfiguration. Formations are designed based on the optimization of the various performance metric functions for a given mission. the mission requirements. These performance metric functions are defined meet Various primary mission requirements, such as achieving the best image quality or gaining the highest probability of detecting different to performance metric functions. As a 67 result, moving targets, lead to image or interferometry qualities for example, is closely related to the formation baseline and the distribution of the satellites in the formation, whereas the performance of moving on the number of satellites and the footprint. To meet target indication systems depends multiple mission requirements, it is important for the formation controller to have the capability of easy reconfiguration. Furthermore, if one of the satellite has a malfunction in the middle of a mission, the adjustment satellite distribution to keep the system working, or the replacement of the with the malfunction, requires reconfiguration of the formation. Reconfiguration could include adjustment of the relative distance between satellites, of a formation, the changing of the numbers of satellites in the combination of two formations flying closely. 68 satellite reassignment of the leader formation, and the SUBSYSTEM DESIGN IV. To two incorporate a useable and attainable design for potential nanosatellite clusters, satellite subsystems show up as integral to the proposal: 1) the attitude and control subsystem, and 2) the propulsion subsystem. satellite bus are mutually coupled with all Although all determination subsystems of a other subsystems onboard, these two systems allow the close formation flying needed for the numerous cluster missions so far mentioned. The assumptions component offered in the remaining chapter's selection are derived from given reference payload requirements, the constraint of the previous chapter's calculations of AV required over a course of a year to maintain the cluster formation, and to minimize mass, power and physical Table 4.1 sums up most size. physical assumptions of the overall spacecraft. Value Characteristic Satellite shape and composition Cylindrical and Mass 10 kg 0.21m Radius Ixx, homogenous m 2 m 0.25 m 0.208 m 0.104 m 2 0.16 kg hy 0.22 kg hz Height Moment arm (z-axis) Moment arm (x.v-axis) Propulsion & ADCS 35% (3.5kg) Mass Allowance Table 4. Assumptions made for Satellite Physical Characteristics 69 A. ADCS The requirements of ADCS systems are twofold: Support formation flight • Manage maneuvers • attitude/rates to the extent required to support precision propulsive if available/ required. Maintain pointing within small angles (i.e. ± .5° in and maintain rates to within precise control pitch and roll, ± 1° in yaw), ±0.1° per second) of the (i.e. desired rates about each axis. • Maintain attitude knowledge and spacecraft stability at all times Support ground communication • Point communication antenna when within sight Attitude Control exception. Until is of ground for command uplink and telemetry downlink station required for any satellite system and small satellites are no now most small (i.e. rudimentary attitude control systems. Microelectromechanical Systems and manufacturing techniques. micro-/nanosats) used only very simple and This can (MEMS) and New now change other small consumer electronics devices control systems are nanosatellite formation flying missions with the development of now needed for missions such as and space-based interferometry. Nanosatellites represents a flexible tool to carry out scientific and technological research in space. Nevertheless, obvious limitations in size, mass, onboard available power, and costs impose several constraints on the design of nanosatellite subsystems. The attitude control subsystem, as particularly affected by these one of the more complex subsystems of a constraints, especially require stringent attitude control. Therefore new when satellite, is the considered applications solutions in terms of components and operating logic need to be investigated to reduce costs, volume, and power requirements. 70 The Attitude Determination and Control Subsystem measures and spacecraft's angular orientation (pointing navigation, and control system, both The simplest spacecraft orbit). its direction), or, in the case of guidance, orientation and linear velocity (which affects are either uncontrolled or achieve control methods as spinning or interacting with the Earth's magnetic or gravity may may or employ not use sensors to measure the attitude or position. controllers to process the spacecraft attitude, Because of of particular its in gravity gradient it attitude control be made from those gravity, the Earth's results. is stabilized and procession control. [Ref. 24/25] documented the power fields. These and actuators, torquers, or momentum. low mass and power consumption requirements, magnetic control interest for small satellites, since damping acquisition, attitude by passive Magnetic Control 1. attitude its More complex systems propulsion subsystem thrusters to change attitude, velocity or angular is controls the extensively adopted also for active of microsatellites, and a case for nanosatellites can then field, With regard use of gravity gradient and aerodynamic drag, to control the spacecraft to this, various solutions booms with eddy magnetic damping rods to damp by torques, such as those caused by means of passive and/or semi-passive devices allows savings. attitude Several authors have studied and The use of environmental magnetic despin, satellites,' initial substantial mass and have been proposed based on the current dampers, fluid ring dampers, or soft- the satellite residual attitude motion. Nevertheless, these solutions achieve poor attitude control accuracy (~ 5-10deg). When a finer control is required, as in the case of remote sensing applications (~ 0.1 deg), various configurations of low-mass, low-power momentum/reaction wheels have been proposed. In use of magnetic torquers for momentum dumping, as an alternative to this case the more traditional gas jets [Ref. 26], reduces the control system complexity and mass. Components available for use in these situations will be presented later below). 71 on this section ( Section A3 All-Magnetic Torquer System a. To develop 3-axis attitude control given the very limited power and weight availability on a nanosatellite, an all-magnetic torquer system where permanent magnets on stepper motors could be used instead of traditional torquer coils. The attitude determination would be achieved by a combination of Earth horizon and sun sensors, giving three-axis control to approximately two to three degrees. Although this concept does not provide fine control ability for most remote sensing applications, the idea is to progress the knowledge base by getting these nanosatellite systems into space and start operational testing and evaluation. Reaction wheel / Magnetic b. Torquerod System As a last example, the attitude control system of the Italian Scientific Microsatellite for Advanced Research and Technology (SMART) was reviewed. Their microsatellite attitude control system consisted of three small reaction wheels and three magnetic torquers (torquerods). The wheels were used for three-axes attitude control during station keeping. The reaction wheel design had been performed using a technique that minimized mass and power consumption. As concerned, it was primarily driven by unloading and initial far as the magnetic torquer design was the requirements for onboard wheel attitude acquisition following the separation momentum from the launcher. Nevertheless, the possibility of using the magnetic torquers for attitude control during station keeping was considered. Their technique was attitude control between wheels and magnetic torquers, which minimized the to distribute the torque required for total power consumption. The result of this example [Ref. 27] presented the analytical model of the proposed technique and demonstrated particular, the numerical analysis • effectiveness by numerical simulations. The In shows the following: Wheels and torquerods can be simultaneously used torque with • its to realize a required control minimum power consumption. control torque distribution between torquerods and wheels is governed by the torquerod control efficiency, which strongly depends on the Earth's magnetic 72 field variation (where R t is along the orbit and on the torquerod design parameter When more to the torquerod / scf supply current). efficient torquerods are used, the control torque portion transferred to the torquerods increases so that the total • t the torquerod winding resistance and scf is the ratio of the torquerod magnetic dipole • R power consumption minimized. is The use of torquerods with low values of the design parameter allows power savings with substantial respect to the case of optimal control with reaction wheels only. • As the required torque percentage assigned to the torquerods increases, the attitude control accuracy reduces. Nevertheless, substantial power savings can be realized while retaining attitude control accuracy adequate for remote sensing applications (0.016 deg for a • The attitude control 40% power savings). accuracy could be improved by slightly increasing the numerical values of the control gains • The proposed at the cost of higher power consumption. control technique could be used in addition to unloading and attitude acquisition considerations to design the Finally, the wheel momentum satellite proposed technique does not increase the attitude control system complexity and mass because the torquerods are also used for acquisition and wheel momentum initial attitude unloading. Propulsion Option 2. The attitude control system is very closely coupled with several of the other systems aboard the spacecraft, and influences or spacecraft to interaction torquerods some is is influenced by every system on the degree. In the remote's attitude control configuration, the normally that with the Propulsion system. nanosatellite designers are utilizing consists most notable One concept that many of micro-pulsed-plasma thrusters (jiPPTs) provide control of both attitude and translation. The shared nature of the ADCS to and propulsion actuators also has a profound effect on the formation flight mission, placing 73 limits on the controller design. This design option is explained more in detail under the Propulsion section below ( Section Bl below). Components 3. Appendix 4. shows the overview of Micromachined solid-state gyroscopes use vibrating They have no in this section. modes of a mechanical highest rotation sensitivity is mechanical elements rotating parts that require bearings, so they can miniaturized. All vibration gyroscopes are based vibration components covered Micromechanical gyroscope a. to sense rotation. ADCS structure, obtained on the transfer be easily of energy between two caused by Coriolis acceleration [Ref. 28]. The when the drive and sense modes have the same resonant frequency. Resolution, drift rate, zero-rate output, and scale factor are the most important factors that determine the performance of a gyroscope. inertially static, the output signal is a random function that is the When a gyroscope sum of white is noise and a cyclic noise function of the mechanical resonant frequencies. Based on resolution, drift rate, QRS-11 micromachined angular and zero-rate output, the Systron Dormer rate sensor (see Fig. 4.1) was seen as the leader from seven commercially available micromachined gyroscopes to complement magnetometers for yaw stability determination. This gyroscope has a resolution of (100s at 0.004%, a short-term bias constant temperature) of 0.004°/s, and a zero-rate output Hz) of 0.01°/s. The gyroscope has a mass of 60g [Ref 29]. 74 (DC to 100 Figure BEI 4. A Fig. 4.2). This GYROCHIP™ Model QRS1 close alternative to the component 1 Micromachined Angular Rate Sensor QRS-11, would be the Litton G2000 gyro (see offers a two-axis gyro with a size 0.76 inches wide and weighs 25 grams [Ref. 30]. of only 0.97 inches tall by This gyro system can be combined with a star sensor system to provide attitude position information to provide a pointing accuracy of approximately 0. 1 degree. The system is still in production and testing and has not been space flown. **M Figure 4.2 b. Litton G2000 gyroscope with electronics Magnetometers Magnetometers are simple, lightweight sensors direction and magnitude complex software as compared of the Earth's magnetic for interpretation to horizon, sun, and field. They that are reliable but require and provide relatively coarse star sensors. GPS measure both the attitude determination position measurements are used with a computer model of the Earth's magnetic field to approximate the field direction at the spacecraft's current position. Over the course of an 75 orbit, the Earth's magnetic field direction usually changes computation of the enough to rapidly enough with respect field's magnetometer. The Earth's magnetic star the spacecraft magnetometer is all three Euler angles field also varies using only a three-axis with time and cannot be calculated often used with another sensor such as a sun, horizon or sensor or a gyroscope in order to improve the accuracy. The Applied Physics Systems Model 533 miniature magnetometer (see Fig. 4.3) laboratory environment. This its make to time derivative possible. These field variations are large enable determination of precisely, so a to three-axis fluxgate can provide direction accuracy to better than 0.1° in a model extremely low mass (18g) and is its well suited for use in the nanosatellite because of small size [Ref. 31]. Yaw attitude knowledge is maintained between magnetometer readings by integration of angular rate measurements. Figure 4.3 c. The of many Model 533: Miniature 3 Axis, Fluxgate Reaction HTteels reaction wheel is used as an actuator for the attitude control and in vacuum, is one important key technologies of nanosatellites. Hokkaido Institute of Technology (HIT) has developed a small reaction wheel, which Nms Magnetometer maximum is storable angular is about 150g in weight and 0.015 momentum. The motor, which can be used 30g and has a torque of 28gcm/2W. The wheel mass balance precisely and the vibrational level is restricted 76 as low as possible. is in a adjusted Small reaction wheel developed by HIT Figure 4.4 Sensors d. (1) Horizon sensors Horizon sensors are infrared devices that detect the contrast between the of deep space and the heat of the Earth's atmosphere. Horizon sensors can provide :old )itch and ).1° to roll attitude knowledge for Earth-pointing spacecraft, with an accuracy 0.25°. For the highest accuracy in low Earth orbit (LEO), it is of necessary to correct he data for the Earth oblateness and seasonal changes in the apparent horizon. Two EDO ;ensors (see Fig. 4.5) errors. ill Barnes Model 13-500 wide-angle miniature solid-state horizon can be used to provide pitch and These sensors have been space-proven on lave fields of view is knowledge 77 ± of up 1°, including The sensors to ± 1 1 °, but Each sensor has a mass of 0.113 kg roughly cylindrical with a diameter of about 4.1 Ref. 32]. to six missions to date. sufficient to allow pointing at off-nadir angles >eak performance is limited to angles less than 9°. md roll cm and a height of about 5.6 cm EDO Barnes Model Figure 4.5 13-500 wide-angle miniature solid-state horizon sensor Star Tracker (2) Current pounds, attain 5 state-of-the-art to 10 arc-second commercial accuracy, star sensors typically weigh and use roughly 10 watts of power. Unfortunately, the current state-of-the-art commercial star sensors do not meet NASA's 15 "next-generation" spacecraft and instrument needs. Nor do they many of satisfy DoD's need for micro/nano-satellite systems. The Intelligent Star Tracker [Ref. 33], built by AFRL, presents a low cost, miniature star tracker for nanosatellite attitude determination and navigation. The Intelligent Star Tracker incorporates adaptive optic catadioptric telescopes in a single, compact, robust Silicon Carbide housing. mechanical (MOEMs) The Micro-opto-electro- micro-mirrors (see Fig. 4.6) are used to compensate for various aberrations as well as introduce aberrations such as leveraging off of adaptive optics research, the active pixel position sensors enable imaging of faint and bright stars in a single wide dynamic range and simultaneous image frame. 78 *N- r ]r I i gi ib. & v&m^gj^ f :=5sJ. rz CI) « . 1 >K£ Figure 4.6 Details of the structure of the micro-mirrors are shown. The adaptive optics telescope, using extremely accurate tracking, and the ability based on algebraic coding theory - - MOEMs micro-mirrors, facilitates when coupled with a star-matching scheme enables the capability to track at least 5 stars simultaneously. Moreover, the massively parallel architecture enables the star tracker to operate autonomously without burdening the spacecraft processor and supplement the onboard processor. Because the design utilizes may be used to technologies that inherently integrate well together and lend themselves to batch processing, estimations have that the Intelligent Star Tracker will have a recurring cost less than $100k. In addition to low cost, preliminary analysis indicates that the Intelligent Star Tracker will have a pointing accuracy exceeding 0.20 arc-sec, consumption less than e. NEA better than 0.10 arc-sec, power 2W and a weight of approximately 200 g. DGPS Formation flying will quickly revolutionize the way science, sensing and surveillance missions are performed in space, enabling a whole applications for small satellites. Currently, there are stages involving formation flying of a constellation to truly achieve the goals numerous missions of micro- or remote new range of in the planning nanosatellites. However, of these formation-flying missions, an accurate means of 79 determining ranging, relative time and position communications, and controlling the formation The need of integrated capabilities timing among for measurements, becoming states is future formation flying missions to spacecraft within the constellation attitude sensor with a The system. result critical. [Ref. 34] have technology with communicating, relative ranging, and exchanging precise is fast approaching. AeroAstro Inc. developed a system by integrating a Carrier Phase Differential and inter-satellite GPS (CDGPS) is navigation low power, inexpensive, compact ranging and communications of this integration is a low-cost, robust, secure GPS micro navigation and communication system for micro and nanosatellite constellations called Star Ranger. The ranging accuracy of Star Ranger the ultimate goal is set at 3 mm. Using CDGPS, between spacecraft will be determinable it cm is expected to be is expected that the relative attitude 1 or better, and to 0.5° or better. In addition, the overall relative position of each spacecraft with respect to each other will also be measurable to less than 1 cm, with a goal of 5 B. mm. PROPULSION There spacecraft. is an increasing need for on-board propulsion systems for micro- and nano- These include upper stage engines as very small-scale boosters to launch and new 1 to boost spacecraft into final orbits as well kg class payloads for military, commercial scientific applications. Decreasing payload sizes will increase demand platforms, including the ability to maneuver and change for orbit; smaller, more capable hence the need for small propulsion systems. Such propulsive missions could include: • Remote inspector to rendezvous and • Constellations • De-orbiting In addition to low maneuver around a host spacecraft on the same launch vehicle requiring separation of space junk requiring rendezvous, docking and cost, orbit changing low mass and short delivery some more specific requirements these propulsion systems include: 80 for • Low power consumption • Low, • High propellant • High density controllable thrust The simplest Isp Isp spacecraft do not require thrust and hence have no propulsion But most spacecraft need some controlled equipment. some form of metered propulsion - thrust, so their design includes a propulsion system that can be turned on and off in small increments. The propulsion system has The three objectives. first objective is to provide the remotes with attitude control, which involves rotational disturbance rejection and angular positioning. The second To maintain objective of the propulsion system a formation with the mother center is to satellite, enable formation flying. and other remotes, the propulsion system must have the ability to reject translational disturbances and reposition the remotes the when A V necessary the satellite drifts out of the formation. for the orbital The third objective is to provide maneuvers throughout the mission as determined by any mission plan. These maneuvers could include changing from one formation to another, changing orbital parameters, correct velocity (e.g., drag), errors, maneuver, counter disturbance forces control attitude during thrusting, and control and correct angular The equipment the in propulsion subsystem (propellant, tankage, distribution system, pressurant, includes momentum. a propellant supply and propellant controls) and thruster or engines. Significant sizing parameters for the subsystem are the total impulse and number, orientation, and thrust levels of the 1. thrusters. Systems a. Cold/Hot Gas Cold-gas is a more traditional propulsion system. expanding high-pressure gas through a nozzle. system are tank, tubing, filter, It provides thrust by Some of the components required for the pressure regulator, valves, and thrusters. Currently the 81 GN&C's NASA's Goddard Space Propulsion Branch of the (GSFC) Flight Center is conducting a broad technology development program for propulsion devices that are ideally suited for nanosatellite missions. The goal of the program nanosatellite propulsion systems that can be flight qualified in a is to develop few years and flown in support of nanosatellite missions. The miniature cold gas thruster technology, the first GSFC's propulsion component technology development program, will product from the be flown on the upcoming ST-5 mission in 2003. The ST-5 mission validate various nanosatellite technologies in mission to more ambitious nanosatellite major subsystem all missions such areas. as is It is designed to a precursor Magnetospheric the Constellation mission. [Ref. 35] MEMS b. Small them in place. The satellites flying in clusters require periodic required impulse satellites in rigid formation, is very small cluster. the goal is not to keep the individual but only to keep them in well-defined orbitals with respect to one another. The necessary impulse, therefore, the difference in drag - "stationkeeping" to keep only the amount needed to overcome is between the most-affected and the least-affected satellites in the Estimates are that the differential drag can be overcome by providing ~1 (micro-Newton second) throughout each satellite's mN sec (milli-Newton second) every 10 to 100 seconds mission. propulsion Currently miniaturized systems. ~1 to Most notable technology is MEMS is developing technology. The Micro-thruster Array (see Fig. 4.7) thrusters have very low for ignition (~10 m Watts, highly reliable. [Ref. 36] mNsec A -100 u Joules), and no moving rapidly MEMS towards Mega-pixel power and energy thresholds parts so are expected to be single thruster array contains a quarter of a million separate thrusters. 82 Aerospace Figure 4.7 Marotta Scientific Vticroelectromechanical System the spacecraft; it :ontrol systems. MEMS chip compared to Penny Controls (MEMS) designing is very tiny chip that provides fine attitude adjustments on uses 8.5 times less power and weighs 2 times less than proven attitude [Ref. 37] Marotta is also in the process of developing a unique microthruster (see Fig. 4.8) and electronics driver combination power operation a (<1W peak), response time of <5 msec, which minimum is capable of low pulse rate of <1 Hz, and has a mass of 50g. The resulting low power component provides an order of magnitude reduction in solenoid coil heating when compared to an ordinary continuous duty solenoid valve. Aerospace, Primex, Honeywell and MEMS AFRL are working separately on based thrusters such as micro-hydrazine. These will be flown on numerous commercial and university based nanosatellite systems when the maturity of the technology will allow it. The small modular nozzles would allow many options microthruster size. Although development time will years, the potential for nanosatellites is very high. 83 most likely require as to more than two Marotta microthruster compared to Dime Figure 4.8 Electrical c. Washington to scale Company Aerospace Primex down Thrusters (see Fig. 4.9). The the is working power requirements of UW nanosatellite will fly a with the University their full-sized Pulsed of Plasma propulsion system, and will be either fiPPT's, or a cold gas system. V* Full-sized Pulsed Figure 4.9 A typical Plasma Thrusters from Primex Aerospace Company pulsed plasma thruster consists of two electrodes, a solid Teflon" propellant bar, an igniter (spark plug), a feed spring, a power supply, and a capacitor (shown in Fig. 4.10). The power supply charges the electrodes. When capacitor, which a small plasma puff from the spark plug 84 is connected to the two is released between the electrodes, the puff creates a low-resistance arc path, discharging the energy stored in the capacitor. This arc ablates a small it into plasma. resulting JxB The amount of the Teflon propellant bar and turns part of current flowing through the arc also creates a magnetic field, and the force accelerates the plasma away from Feed the thruster, thus generating thrust. Eleclrock Is»riiler Spring Plasrtu Electrode Capacitor Figure 4. Basic diagram of a pulsed-plasma thruster 1 Performance 2. Three parameters determine the performance of a propulsion system. These three parameters are thrust (7), minimum impulse and specific impulse bit {hit), propulsion system hardware, the thrust can be measured experimentally. (Isp ). From Given these experimental data, the average thrust can be calculated. However, for a theoretical analysis, the thrust is calculated from Eq. (4.1): T = rhC,tf where m is the (4.1) mass flow Ce/f rate at the thruster exit and amount of thrust be delivered by a is the effective exhaust velocity. The hit is the smallest that can thruster. This is given in the units of an impulse (force multiplied by time), such as 70 uNs. Although the hit may be experimental calculated theoretically in data since the some cases, it is usually calculated from the minimal thrusting capability depends highly on the 85 propulsion system hardware. The /#/ is calculated using Eqn. (4.2) from the thrust profile data of an experiment in which a propulsion system is activated for the shortest duration possible. = [T-dt Ibil (4.2) In equation (4.2), respectively. For the f, and tf are the time at the start most spacecraft a low 4„ is and the end of the desired for fine control of the attitude, and especially during dead-band limit cycling. The specific impulse as shown thrusting, is generally defined in equation (4.3): hP = ~ ™-g (4-3) The where g is impulse a measure of the amount of thrust a propulsion system can provide mass flow The Isp diminishes is propellant lifetime. The the acceleration due to gravity at the surface of the Earth. rate. specific impulse is essentially a propulsion system, so a high Isp 3. is for a given most propulsion systems over their measure of the mass efficiency of a desired for the propulsion system. Propulsion System Comparison (Cold gas Two main uPPT and for specific / uPPT) propulsion systems were analyzed and compared. The results of the cold-gas system performance analyses are summarized in Table 4.2 below. 86 Propulsion Total System Mass Type (kg) Propellant T Isp Ibit (s) GiNs) (mN) Mass per AV (g-s/m) A V Time Energy Duration per (s 2 /m) (J-s/m) uPPTf 3.80 500 70 0.07 2 1.43xl0 cold-gas 4.58 65 100 4.5 16 2.22x1 5 3 f The performance of the {iPPT was analyzed assuming a J X The energy AV requirement for per Peak Power (W) AV 17.9xl0 l~5xl0 6 12.5 4 10.1 J Hz firing frequency. a cold-gas thruster depends on the firing mode, pulsed or continuous. Comparison of uPPT and cold-gas propulsion systems Table 4.2 (single thruster performance). Both uPPT and cold-gas maximum translational disturbances, has a lower mass due to and more A V for its lower a given propellant due mostly Ibu uPPT of less compensate for remotes drag. The uPPT system , mass than the cold-gas system. The an impulsive bum, or a short duration per is to the thrust to and higher Isp providing better pointing accuracy required for attitude control can be characterized maneuvers, enough thrusters provide by fine thrust control the long time duration per AV. Thus, AV burn, which is optimal for most orbital importance. The peak power and energy consumption are high for systems. Most importantly, uPPTs do not have a history of complexity and failing due problems suffer from propellant leakage. Cold-gas systems to propellant leakage. inherent to a Compared high-pressure to the cold-gas system's and system the miniaturization capability due to the flow characteristic of gases and liquids, the more attractive. The \iPPT has a simple feed system with minimal moving to higher reliability. Also, size uPPT system mass can be further reduced of the electronics, the most massive component of the characteristics are suited to small satellites future commercial nanosatellites uPPT limited uPPT is parts, leading by decreasing the system. The uPPT and thus will become the stepping stone for that will utilize 87 uPPTs. EXAMPLE: SNAP-I :. SNAP-1 >STL), rid is it is (SSTL) a low-cost nanosatellite built ingle Satellite Technology Ltd. amongst other objectives a technology demonstrator for 3-axis orbit control for a future constellation itellite by Surrey uses a single miniature of small Y-momentum satellites stabilization during formation flying. The wheel, 3-axis magnetic torquers and a butane gas thruster to ensure a nominal nadir-pointing attitude with and ontrol in-track AV full pitch The magnetic torquers do momentum maneuverability. mintenance and nutation damping of the Y-wheel. The primary attitude sensor used, A 1 1 GPS receiver supported by an on-board low cost propulsion system was designed and 7 months from concept to launch site. It utilizes olution. lunch was The site. stored in a spacecraft SNAP-1 was was no 1), spacecraft was used for the valves. The formed titanium tube, rather than a tank, giving a low cost successfully launched The providing 65 propellant cm Figure 4.1 3 1 pipework assembly was stored in 1 . 1 it to Snap-1 Propellant tube is that meter of coiled titanium of storage volume. [Ref. 38] 88. prior to shipping on 28 June 2000. feature of the complete propulsion propellant tank. ibe (see Fig. 4.1 SNAP-1 butane stored as a liquid and operating was loaded with 32.6 grams of butane The most obvious lere orbit estimator. built for the a cold gas mode. Miniature conventional technology ropellant a magnetometer. Precise orbital knowledge was obtained using a liniature 3-axis flux gate mall single antenna is CONCLUSION V. THE NANOSATELLITE PUSH A. In recent years, an increased effort to design, build, and operate small satellites has taken place in universities and laboratories and nanosatellites provide numerous fraction thrust over the world. These microsatellites flight opportunities for science experiments at a of the cost of larger traditional missions. This paper has presented an enormous toward innovative ways, not only of satellite's roles 'some distant in all and abilities in the future', and with built, opportunities to show how commercial and fields years. but a shift in thinking about Nanosatellites are far from being the and indeed are not even the end of the space commercialization. designed coming satellite design, Picosatellites space line for this new movement and femtosatellites are currently being engineering paving the way for many powerful useful and cunning these systems can be to both the to the military. FORMATION DESIGNS B. Imagine satellites numbering in the tens, hundreds or even thousands being launched off surface combatant ships, submarines, mobile Army vehicles or even Air Force cargo planes. The possibility of throwing a large quantity of 'sensors' quickly and efficiently into an engaged theater wherever/whenever removes the dependence on costly, vulnerable national-asset satellites. The preceding chapters looked at the possibility of creating clusters that through their natural motion retain a constant shape when viewed from the planet's surface. shown the shape of the apparent circular orbit can be described in terms of that eccentricity of the orbit, and It is that terms through second order in e are sufficient to describe the motion to a high order of accuracy. Many formation designs have been presented to show that utilizing natural orbits allows formation dynamics to help reduce fuel requirements for formation-, and station-keeping needs. These natural coupled with robust control laws and precise position knowledge and 89 orbits, inter-satellite communication DoD key abilities, are to the NASA, growing need and requirement for future and commercial space missions. PERTURBATION UPKEEP C. This paper has presented the effects of the earth's oblateness on satellite formation flying designs like the circular and projected circular clusters. There are two components of motion that must be accounted for: 1) differential changes in the right ascension of the ascending node, and 2) secular changes in the argument of perigee and mean anomaly. The formation. cost to maintain relative For a circular cluster of 25 km node spacing is dependent on the size of the radius, the cost is approximately 7.2m/sec per year of velocity impulse. The cost to maintain the argument of perigee, not being a function of the cluster size, is roughly 2.1m/sec per year for the circular maneuvers cannot be coupled, a cluster 1 like the cluster. If the one presented here could require close to Om/sec per year of velocity impulse. Based on the orbit propagations and an assumed 10% error bounded on the formation, maneuvers would be required every 40 hours or may Other perturbing effects also require maneuvers but It may be satellite in the all directions in the along-track, cross-track, maneuvers will also vary from and each and radial directions. satellite to satellite formation must be able to thrust The amount of formation keeping within the formation. Station keeping cost for the classes of orbits discussed here Only atmospheric drag decay of the semimajor axis to correct for this effect is is that is is relatively small. a major concern. Velocity impulse on the order of .097m/sec per year using high drag conditions. Another major issue maneuvers frequency and cost should also be pointed out that maneuvers than the oblateness induced maneuvers. required in at far less so. that impacts the frequency and cost of formation keeping of attitude control. The formation keeping maneuvers discussed above require thrusting in the along-track, cross-track and radial directions. dynamics are very sensitive to acceleration in the along-track directions. maneuvers are required in the cross-track has substantial errors when cross-track unwanted acceleration might be applied and radial directions. or radial maneuvers The The satellite largest If the satellite pointing are performed, some in the along-track direction with significant 90 consequences. This will undoubtedly drive the frequency and cost of formation-keeping higher than what has been calculated here. Orbit determination knowledge control. is another factor that will influence formation The STK's simulated propagations show that centimeter level differences in semimajor axis cause significant along-track error growth over time. determine the orbits to this level of precision control. 91 is vital for precise and The ability to efficient formation THIS PAGE INTENTIONALLY LEFT BLANK 92 APPENDIX 2.1 International Mass Launched from (g) Initial orbit @ 51.6 deg Decayed Designator Oderacs A STS60 9Feb94 1994-006B 4200 225x463 km @ 51.6 deg 10Feb94 Oderacs B STS60 9Feb94 1994-006C 4200 239x451 km @ 51.6 deg 10 Apr 94 Oderacs C STS60 9Feb94 1994-006D 500 Oderacs D STS60 9Feb94 1994-006E 500 Oderacs E STS60 9Feb94 1994-006F 5 266x272 km @ 5 Oderacs F STS60 9Feb94 1994-006G 5 266x271 km @ 51.6 deg Oderacs 2 STS63 4Feb95 1995-004C 5000 267x277 km @ 51.6 deg 13 Oderacs 2B STS63 4Feb95 1995-004D 4200 323x349 km @ 51.6 deg 29 Sep 1995 Oderacs 2C STS63 4Feb95 1995-004E 500 Oderacs 2D STS63 4Feb95 1995-004F 1.5 314x320 km @ 51.6 deg 2 Oderacs 2E STS63 4Feb95 1995-004G 1.5 320x334 km @ 51.6 deg 27 Feb 1995 Oderacs 2F STS63 4Feb95 3 Mar 95 6 Feb95 1 .6 deg 3 Mar 95 24 Feb 95 Mar 1995 7 Feb 1995 Mar 1995 Not catalogued by 0.5 NORAD ODERACS Data Table 93 20 Feb 1995 THIS PAGE INTENTIONALLY LEFT BLANK 94 APPENDIX 2.2 Educational institutions involved in small Increasingly, satellites. This satellites to becoming possible it is is largely due to be designed and satellites be involved in small affordable yet sophisticated technology. This then allows for educational institutions to built within the course duration, or via a series of student projects. The following list of educational establishements are those known to be involved in small spacecraft projects. This can be either in experiments, parts of satellites, or entire Where known, specific spacecraft project names apear in brackets, inducing a year when launched (or expected to be launched). For more details on the projects, please satellites. refer to the satellite and future mission lists. All the links on this page will lead you to the individual institutions Europe HUniversidad Complutense, Madrid, Spain BlJniversidad Politecnica de Madrid, Spain BSurrey Space Centre (UPMSAT *95) University of Surrey, Surrey, United Kingdom (UoSAT series '80 onwards) HTechnical University of Helsinki in Finland (HUTS AT) (CATS AT) Germany (TUBS AT HUniversity of Leicester, England HTechnical University of Berlin, series '90 onwards) Buniversity of Bremen, Germany (BREMSAT BRoval Stockholm Sweden. (KTHSat) Institute of Technology Blnstituto Superior Tecnico , in '94, Abrixas '97) . Lisbon Portugal. (PoSAT-1) Germany (Phase 3D) of Umea University (RYP) (Munin) BUnitersitaet Kaiserslautern , EDept. of Space Physics North America In the U.S., NASA encourages participation in small missions via the University Explorer (UNEX) Programme. BUniversity of Illinois Urbana-Champain. BStanford University, U.S. (SAPHIRE and OPAL) BArizona State University. Tempe. Arizona. U.S. (ASUSAT) BUni versify of Arizona Tuscon, U.S., Students Satellite Program (UASAT) Blowa State Univerity, Ames, Iowa, U.S. (ISAT) BUniversidad Nacional Autonoma de Mexico Mexico City, Mexico (UNAMSAT) at , . , 95 (PANSAT) BSan Jose State University, San Jose, California, U.S. ( SPARTNIK ) BSierra Community College Rocklin, California, U.S. ( MINERVA ) BUniversitv of Alabama Huntsville, Alabama, U.S. (SEDSAT) BUtah State University Logan, Utah, U.S. (NUSAT '85, Webersat etc CAST ) BNaval Postgraduate School Monterey, , California, U.S. , , , BJohns Hopkins University Applied Physics Laboratory U.S. , BPenn State University U.S. , (SAC-B CUBIC instrument) BUniversitv of Colorado U.S. BMassachusetts Institute of Technology Centre for Space Research U.S. (HETE) BWeber State Univerity Ogden, Utah, U.S. (NUSAT *85, WEBERSAT '90 and more) , , , BUniversitv of New Hampshire U.S. , (CATSAT) BBoston University U.S. (TERRIERS) BAnahuac University, Mexico (ANISAT , Nano-satellite project) BCarleton University, Canada. (SUA microsatellite) BUniversitv of Toronto (MOST microsatellite) Africa BUniversitv of Stellenbosch South Africa , (SUNSAT) Asia BKorean Advanced Institute of Science /Technology (KAIST), S.Korea (KITSAT BMahanakorn University, Thailand (TMSAT) series) BNational Central University, Taiwan (TUU-Sat) BATSB, Malaysia, (JjungSAT) BTsinghua University, China (Tsinghua-1) South America BUniversidad de La Frontera, Temuco, Chile, Involved with the design and construction of CESAR- 96 APPENDIX 3.1 STK Screen Shots In-Track Formation(2 remotes) 97 y* • • * s _A ISfeta ^ ^ ^</ *^^;~* ^~~"'""" **N» F&^^^Siz'^ ** " «•-*. ' V***- \ WM HTM *9a * _ "Sgj^frVi AmL. -^X-<""-' ., . / "Si; j-^^L ^sJ? 6; 7 ^ / A, ^ / / / / P" / *^-, • ^ / fism / / / / / 1 .^fen 2001 / / / ; / / OOOOOqio Time : f / / * . V / / / ; ' / / / / Circular Formation (4 remotes) 98 I J / \vA xN s^fe. X ^ 2 X APPENDIX 3.2 Matlab Code for STK data input in inertial position/velocity parameters %Cluster Formation clear clc %Need Mother parameters %ac - semimajor axis %ec - eccentricity %ic - inclination %RAANc - right ascension of ascending node argument of perigee %argc - %Mc - mean anomaly at epoch %Ec - eccentric anomaly %Tc - true anomaly %rc - radius %vc - orbital speed ac = 7489.137000; ecdeg ec = ecdeg*pi/180 %deg 63.4; = RAANcdeg RAANc = argcdeg argc = = ; %rad %deg ; RAANcdeg *pi/ 180 ;%rad = %deg ; argcdeg*pi/180; pc = ac* (l-ec = %rad ; icdeg* (pi/180) ic mu %deg 0.00; = icdeg %km A 398600.5; %rad 2) %km^3 s A -2 %Cluster geometry parameters %rho - scale factor; same for all subsats %nu - dimensionless arrangement determines pattern generator; %delta - angle; specifies shape suborbit plane wrt centersat orbit 99 overall of cluster size/ determines suborbit & assigned geometrical orientation of %rs = subsat radial speed %vs = subsat orbital speed %as = subsat semimajor axis rho = deltadeg delta %km 100; = %deg 63.4; deltadeg*pi/180; = %rad %km as = ac; %First choose eccentric anomaly, Ec, of centersat numsats=8 for m=0 :numsats-l Anomaly %Eccentric Ec= (2*pi/numsats*m) spacing circular Equal %Pattern Generator for Desired Pattern nu= . %Circular Spacing, one loop 5 %Calculte Center-Sat parameters Tc=2*atan(sqrt (1+ec) ( Mc=Ec-ec*sin (Ec) rc=pc./ / (1-ec) ) vc=sqrt (mu* (2/rc-l/ac) ) . *tan(Ec/2) ) ; ,- +ec. *cos (Tc) (1 ) %Calculte Sub-Sat parameters (use 12 o'clock insertion position) %Assumed for all subsats as=ac; rs=sqrt (nu*rho*sin (delta) ( vs=sqrt (mu* (2/rs-l/as) A 2+ (rc+nu*rho*cos (delta) A 2) ) %Calculate Vectors Rc=rc* [cos (Tc) sin(Tc) Vc=sqrt (mu/pc) * 0] ' [-sin(Tc) ec+cos(Tc) Rs= (rc+nu*rho*cos (delta) *cos nu*rho*sin (delta) ) [ 0] [rc+nu*rho*cos (delta) (Tc) ] ' Vs=vs/vc. ' *Vc,- 100 ) *sin (Tc) vectors %Transform subsat radius/velocity Earth-centered inertial coord Rxx = 3x3 rotation matrix from RSW to Rraanc [cos(-RAANc) = sin(-RAANc) 0; UK from centersat coord frame -sin(-RAANc) cos(-RAANc) (-sin(-ic)) (cos(-ic))]; 1]; Ric =[10 Rargc R = [cos(-argc) = (sin(-ic)); (cos(-ic)) 0; sin(-argc) 0; -sin(-argc) Rraanc*Ric*Rargc; Rijk=R*Rs Vijk=R*Vs Rsub (m+1, 1 :numsats-l) =Ri jk ' ,- Vsub(m+1, 1 :numsats-l) =Vijk' ; %Calculate Orbital Elements %As should equal Ac for all subsat s (Assumed) end display 'SubSatl ( ' ) 101 cos(-argc) 0; 1] 0; to THIS PAGE INTENTIONALLY LEFT BLANK 102 APPENDIX 3.3 Example orbital Orbital 5 = Parameters MOM 63.4 Position Inclination 1.5 63.403457 deg 3 4.5 Parameters from Matlab to a c = 6978.137 km STK coordinate alt = 600 transformation km Ring radius -25 km Arg RAAN Mean Anomaly 0.003 180 -0.006913 deg 0.003 225.012421 deg 180 134.987579 deg 63.406913 deg 0.003 270.024842 deg 63.403457 deg 0.003 315.012421 deg Eccentricity 63.4 6 63.4 0.003 0.006913 deg 7.5 63.396543 deg 0.003 44.987579 deg 9 63.393087 deg 0.003 89.975158 deg 10.5 63.396543 deg 0.003 134.987579 deg 12 63.4 0.003 180 F. or 5 = 63.4 MOM 63.4 Position Inclination Eccentricity 63.400654 0.00328833 63.06986 0.00325885 63.400654 (all a c = 6978. 137 89.975158 deg 44.987579 deg Bottom 315.012421 deg -0.000001 deg 270.024842 deg 225.012421 deg -0.006913 deg second outt. tr ring: e=006 Orbital Parameters 0.000001 deg Top 180 other parameters stay the same) km alt = 600 km Ring radius -25 km RAAN Mean Anomaly 359.630879 deg 359.953493 deg 270 deg Odeg Odeg 0.00328833 179.788017 deg 0.369121 deg 0.046908 deg 63.730337 0.00326474 90 deg Odeg Odeg 63.400654 0.00328833 359.630879 deg 359.953493 deg Arg 0.21 1983 deg Top 1.5 3 4.5 6 7.5 9 10.5 12 0.21 1983 deg For second outer ring: e=006 103 (all other parameters stay the same) Bottom Orbital Parameters MOM Position 5 = 63.4 a c = 7489.1 37 km alt = 11 11 km Rin9 radius —23 km 63.4 Eccentricity Argft o<°) M(°) 63.400508 0.00300701 180.170765 359.61862 180 63.058986 0.00300701 90 Odeg 180 63.400508 0.00300701 359.829235 0.38138 180 63.741014 0.00300701 270 Odeg 180 63.400508 0.00300701 180.170765 359.61862 180 Inclination (°) Top 1.5 3 4.5 6 7.5 9 10.5 12 For second outer ring: e-.006 104 (all other parameters stay the same) Bottom APPENDIX 3.4 RAAN over 1 year(deg) 350 300 - ORC\ 0) 0) oc\c\ zuu . ^ 150 - i— O) I uu 50 n 2001001 2001051 2001101 2001151 Date 2001051 2001101 2001251 2001301 2001351 (YYYYMMDD.hhmmss) Arg of Perigee over 2001001 2001201 1 2001151 year(deg) 2001201 Date 105 - uncorrected 2001251 2001301 2001351 THIS PAGE INTENTIONALLY LEFT BLANK 106 APPENDIX 4.1 Sensor Performance Power Unit (kg) (W) Price short-term bias and Notes Units Use 0.0047s Micro- Gyroscope Mass 0.060 0.030 $450 between 1 out resolution using regularly the magnetometer. Nominal use Magnetometer ±3° magnetometer measurements. Must be zeroed attitude 0.018 0.030 $2200 1 only. is yaw for Can be used attitude for three-axis attitude if necessary. 8300Reaction Wheel 8400 rpm 0.015 Built by 0.450 2.0 0.113 0.030 $40,000 2 Pitch and roll attitude 0.200 <2.0 $100,000 1 Tracker's processor HIT 3 Nms Horizon sensor Star Tracker ±1° attitude pointing accuracy: 0.20arcsec NEA: better to than O.lOarcsec TOTALS 0.954 4.120 $182,700 8 Comparison of attitude sensors 107 may be used supplement onboard processor THIS PAGE INTENTIONALLY LEFT BLANK 108 LIST OF REFERENCES 1 Daniel Goldm, Administrator of National Aeronautics and Space Administration before S. Subcommittee on VA, the HUD and Independent Agencies Committee on Appropriations House of Representatives, March 15, 2000. 2. Online World Wide 3. Online Web (WWW), http://www.amsat.org/ , March 1 7, 200 1 WWW, http://www.nas.nasa.gov/Pubs/NASnews/95/ll/nanotech.htmL April 09, 2001. 4. 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