<%BANNER%>

Development of Spacecraft Orientation Buoyancy Experimental Kiosk

Permanent Link: http://ufdc.ufl.edu/UFE0022280/00001

Material Information

Title: Development of Spacecraft Orientation Buoyancy Experimental Kiosk
Physical Description: 1 online resource (78 p.)
Language: english
Creator: Leve, Frederick
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: attitude, control, reaction, satellite, spacecraft, testbed
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: DEVELOPMENT OF THE SPACECRAFT ORIENTATION BUOYANCY EXPERIMENTAL KIOSK TESTBED Most satellites are unique and therefore usually take 10 to 15 years to design, fabricate, test, and finally launch. To expedite future advances in space technology, setting standards in terms of ?black-boxed? subsystems and interfaces, drastically reduces the time and long term costs needed to complete these tasks. What is meant by ?black-boxed? is a stand alone subsystem that is adaptable enough to work with any other subsystems without alteration. Government organizations such as the Department of Defense?s (DoD?s) Operationally Responsive Space (ORS) office take these tasks into consideration and look to change this pattern by expediting the processes of design, fabrication, test, evaluation, and launch. To accomplish these tasks, smallsats which range from 1 to 1000 kg in mass, are considered for their cheaper platforms and launch costs. With utilization of smallsats of the pico- (0.1-1 kg), nano- (1 to 10 kg), and micro-classes (10-100 kg) for space missions, obstacles occur in terms of power, mass, and volume constraints. Many of these same smallsats obstacles relate to the attitude control system (ACS). Despite these obstacles for many space missions, small satellites must be held to the same pointing requirements as their larger counterparts to be of use. To compensate for these obstacles while maintaining close pointing requirements, innovations in attitude control actuators and determination sensors that satisfy these constraints need to be developed. An excellent choice for attitude control actuators of smallsats are those of the flywheel sort known as zero-momentum and momentum bias actuators. These actuators have the ability to perform precision attitude maneuvers without use of propellant. There are two classes of flywheel attitude control actuators that are used in satellites, reaction/momentum wheels (RWs/MWs) and control moment gyroscopes (CMGs). Due to the scarcity of CMGs available for small satellites below the micro-class, RWs are presently utilized. An analysis completed in this thesis shows that although the current state-of-the-art CMGs are too massive to fit inside smallsats, the performance relationships in terms of output torque per mass and input power are more favorable than that for RWs. This analysis is an additional motivation for the research in this thesis, which is carried out on the premise that an ACS consisting of single-gimbal control moment gyroscopes (SGCMGs) can meet the performance requirements providing rapid-retargeting and precision-pointing (R2P2) for smallsats while enduring their power, mass, and volume constraints. To further test this premise the topic of this thesis is the development of a ground based SGCMG testbed known as Spacecraft Orientation Buoyancy Experimental Kiosk (SOBEK) was developed to validate the utility of using SGCMGs for smallsat attitude control. The thesis justifies the reasons for using SGCMGs on SOBEK, discusses the spacecraft and actuator dynamics and kinematics associated with an ACS containing CMGs, addresses the geometrical configurations of systems of these actuators and the configuration chosen, and then finally validates the testbed numerically through simulation and experimentation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Frederick Leve.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Fitz-Coy, Norman G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0022280:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022280/00001

Material Information

Title: Development of Spacecraft Orientation Buoyancy Experimental Kiosk
Physical Description: 1 online resource (78 p.)
Language: english
Creator: Leve, Frederick
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: attitude, control, reaction, satellite, spacecraft, testbed
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: DEVELOPMENT OF THE SPACECRAFT ORIENTATION BUOYANCY EXPERIMENTAL KIOSK TESTBED Most satellites are unique and therefore usually take 10 to 15 years to design, fabricate, test, and finally launch. To expedite future advances in space technology, setting standards in terms of ?black-boxed? subsystems and interfaces, drastically reduces the time and long term costs needed to complete these tasks. What is meant by ?black-boxed? is a stand alone subsystem that is adaptable enough to work with any other subsystems without alteration. Government organizations such as the Department of Defense?s (DoD?s) Operationally Responsive Space (ORS) office take these tasks into consideration and look to change this pattern by expediting the processes of design, fabrication, test, evaluation, and launch. To accomplish these tasks, smallsats which range from 1 to 1000 kg in mass, are considered for their cheaper platforms and launch costs. With utilization of smallsats of the pico- (0.1-1 kg), nano- (1 to 10 kg), and micro-classes (10-100 kg) for space missions, obstacles occur in terms of power, mass, and volume constraints. Many of these same smallsats obstacles relate to the attitude control system (ACS). Despite these obstacles for many space missions, small satellites must be held to the same pointing requirements as their larger counterparts to be of use. To compensate for these obstacles while maintaining close pointing requirements, innovations in attitude control actuators and determination sensors that satisfy these constraints need to be developed. An excellent choice for attitude control actuators of smallsats are those of the flywheel sort known as zero-momentum and momentum bias actuators. These actuators have the ability to perform precision attitude maneuvers without use of propellant. There are two classes of flywheel attitude control actuators that are used in satellites, reaction/momentum wheels (RWs/MWs) and control moment gyroscopes (CMGs). Due to the scarcity of CMGs available for small satellites below the micro-class, RWs are presently utilized. An analysis completed in this thesis shows that although the current state-of-the-art CMGs are too massive to fit inside smallsats, the performance relationships in terms of output torque per mass and input power are more favorable than that for RWs. This analysis is an additional motivation for the research in this thesis, which is carried out on the premise that an ACS consisting of single-gimbal control moment gyroscopes (SGCMGs) can meet the performance requirements providing rapid-retargeting and precision-pointing (R2P2) for smallsats while enduring their power, mass, and volume constraints. To further test this premise the topic of this thesis is the development of a ground based SGCMG testbed known as Spacecraft Orientation Buoyancy Experimental Kiosk (SOBEK) was developed to validate the utility of using SGCMGs for smallsat attitude control. The thesis justifies the reasons for using SGCMGs on SOBEK, discusses the spacecraft and actuator dynamics and kinematics associated with an ACS containing CMGs, addresses the geometrical configurations of systems of these actuators and the configuration chosen, and then finally validates the testbed numerically through simulation and experimentation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Frederick Leve.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Fitz-Coy, Norman G.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0022280:00001


This item has the following downloads:


Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID E20101109_AAAADL INGEST_TIME 2010-11-10T03:07:57Z PACKAGE UFE0022280_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
FILE SIZE 5084 DFID F20101109_AACBFU ORIGIN DEPOSITOR PATH leve_f_Page_27thm.jpg GLOBAL false PRESERVATION BIT MESSAGE_DIGEST ALGORITHM MD5
b5fd32aae8c904f5b183f5032899d2cd
SHA-1
439409ba2e7038b8a0ff3e9f8eef7e99099a2c2f
41416 F20101109_AACBAX leve_f_Page_56.pro
33faee5579e3c033733807e78174d723
39f2f58181cb58fefdab6d1b601bfe64ded4cf14
5926 F20101109_AACARB leve_f_Page_05thm.jpg
7001ed78db9736d2ce635d2446ec7dfd
fb319363ce9383d0002cd8bcd4d0e07316fefa6e
25271604 F20101109_AACAYX leve_f_Page_72.tif
99e54b36f5be2f68925090871ff126e7
dda9badd5df729f9b3ce1053e6cae4490b4b4e8a
19385 F20101109_AACBFV leve_f_Page_29.QC.jpg
6be05ac730820b82e5f7328e6315dfa8
1c9286cdde63b2951d23dbc9d0fafce96db328eb
48046 F20101109_AACBAY leve_f_Page_57.pro
b38c27aad1d84ae2cbb064bfc2524085
17b1851984ae8e85250b953f1e7212af584c584f
36269 F20101109_AACARC leve_f_Page_66.jpg
e6f81ed782c193ad84e18c392c7068c4
c37026356f5e8081762c1d88b14c9f91a3b97d11
991995 F20101109_AACAWA leve_f_Page_58.jp2
b60091c3641a261c04451f6885e9b8af
5ca5c1cc82b2d342ea395f844b6633fbe423bf85
F20101109_AACAYY leve_f_Page_73.tif
1016ee5e45f2df57178dc4fd2ec70ea9
90203e1e1e0d744273433969b92eec1c69520113
5686 F20101109_AACBFW leve_f_Page_29thm.jpg
3cc6299be580fba2bfa599b7a635cbe8
fd964d25f4dd71ba08bcef564b994350173a25e0
43814 F20101109_AACBAZ leve_f_Page_58.pro
e85c63cee74dc3ccd248fbbce1a74404
4ec3671a2cf4b70fbddf18f43b02d3ae6645ae2e
85106 F20101109_AACARD leve_f_Page_39.jpg
3c385fffb42d145213d69b771799380d
e922b4524516bb11166bb69f9866aca9e5881fc3
997151 F20101109_AACAWB leve_f_Page_59.jp2
bbec1c83fe13da2d578f14e9c5cdc141
82a25bc7d6db39fc7debc97b0bf5651a838b4391
F20101109_AACAYZ leve_f_Page_74.tif
f9b6a4d054453d3fd9858ffaa4a6e346
bb844ae580354ec3c9ac4680d2ad39f2f0c9e734
23120 F20101109_AACBFX leve_f_Page_30.QC.jpg
71717b6182667b6bfa11116ecca32606
2fd297480658eb6b995647e6b6b017bcf5e356a4
F20101109_AACARE leve_f_Page_68.tif
76f49589db4684bb765e2d80df751d6c
c7f81774582f0c41da36b18662a4d911ef398503
1051948 F20101109_AACAWC leve_f_Page_60.jp2
70b23092a37d99d7f05bb9489266036d
86201a851ba7b05f7875b44cfdeb713ad59cc8b9
6579 F20101109_AACBFY leve_f_Page_30thm.jpg
ee8020467e1717f3ca29daa5ac95e223
06f7131201739a39907c06a2a5a7c668e5b23c82
610 F20101109_AACARF leve_f_Page_22.txt
232b817ae277f0c0f46d00641b1c12fc
c860dfa2a44a749e0ff369bdb7451a034fd5b3b3
224 F20101109_AACBDA leve_f_Page_49.txt
746b82c436381cf12f6fba2a238d151c
8c13151f26f5d8232a9fd23b83a99b7715e32c36
567504 F20101109_AACAWD leve_f_Page_61.jp2
53f8c25d1681c777099baafc4ffdc2d8
37e0127696f47539e2c387f8a910ae2f61f9c4b9
22504 F20101109_AACBFZ leve_f_Page_31.QC.jpg
d9afece991801faf5de20a54b95223ef
e3095028817049d7d118b5c67f04c00563cbdfb3
19085 F20101109_AACARG leve_f_Page_69.jpg
092b41c45399e1936576c7ff6bb63cde
c1c537df5408a3e7ad18d0dfc7d14a575519a762
166 F20101109_AACBDB leve_f_Page_50.txt
40fbd27bdb979148716c7b3d1d6e4676
935e1afc57de09c2acf338e4a4f1cce8866c3385
584580 F20101109_AACAWE leve_f_Page_63.jp2
c2311491b03df92c4cc19f79ebbb5188
e17c0c2a9d88fe534fa8c353dfb9a4cfc113ec80
14342 F20101109_AACARH leve_f_Page_50.QC.jpg
a292f37609b7c7fbff7a1b5ca80889b9
93d00f0c74f04a6f32dfafa237144f61cb0d7d4c
483 F20101109_AACBDC leve_f_Page_51.txt
f13fdcebee2cf361b4891386b4645ba8
48aa1457fa6e8dc4aa6b20151a4f1a232b6e3639
4398 F20101109_AACBIA leve_f_Page_61thm.jpg
e8aa6b507471628846d94a19e2dc4d8d
0955c82558af4bd5a4577490a9af9d78912f0043
3847 F20101109_AACARI leve_f_Page_48thm.jpg
d9650c412219fd04990320b2196c04d2
12005f73b2aedf2a609df6585b75dd8f4c795a41
1569 F20101109_AACBDD leve_f_Page_52.txt
3f8d41000907508aaec64d8b26e74e3f
55b706a20dd61ff6c3687312f8bd90e3900c3da1
498646 F20101109_AACAWF leve_f_Page_64.jp2
e25962f0bc881a4fd711d2b6617f36c8
43be83803989a6afb522e409c4ffa770905af9f5
3550 F20101109_AACBIB leve_f_Page_62thm.jpg
a4a3388e85914e320e87d9b15f8d9f7f
87297ac43a99f9d716b5c36c08ba30a881fc5099
1051966 F20101109_AACARJ leve_f_Page_13.jp2
a43c8e807908fb98f646c449cce8b800
0daaa42fc47f8dc7f37de8c2cf3cc3acf4cdf95e
1692 F20101109_AACBDE leve_f_Page_53.txt
bdf55c5f14fb05b5aa04dc580d09f86f
c64ed86da521917bf70441504516902c5df4a693
602530 F20101109_AACAWG leve_f_Page_66.jp2
57c13846c0b88fb589e923223235658e
e5a45e88220130b759c20bc50e2b35bec106e6fc
4063 F20101109_AACBIC leve_f_Page_63thm.jpg
8f6b9a5fadb18f8cfcdc9806f92c9ae8
12c69d9f34738ae6d8dd6db8476752ec14ecf60c
1502 F20101109_AACBDF leve_f_Page_54.txt
2104c8fe425c902719cf219b9adaff2b
55484f8848203dc1ee0bae2db41e4f7cd4363011
678206 F20101109_AACAWH leve_f_Page_67.jp2
2cb5154d9a70ef0b88366392556f2c6c
8bf16228607e821345f755b87aabc82409e23f9e
22594 F20101109_AACARK leve_f_Page_56.QC.jpg
1a19400746d4b3bd94b6c155480e4bd2
eb54eab8c94cfac494aab5a29871339f4a200d6c
11026 F20101109_AACBID leve_f_Page_64.QC.jpg
f5e0b6ed02e94117d22f0f1afb723161
7a5c5ad38ea529bd0fb5ff4d8ec092074c7eb0a0
1758 F20101109_AACBDG leve_f_Page_56.txt
82345dc1350729512eb5720f476cdf2f
4f4031511bedc25598234f005378fb5daa1049c1
669753 F20101109_AACAWI leve_f_Page_68.jp2
176bec09c4c34819f572bf536811acd3
985138b1616423ae5aa1fd81833492f2b0022fba
663 F20101109_AACARL leve_f_Page_48.txt
e5bae50325603f2d5049ea972ad1998d
5395006d243f02205dc193d25d4d14cd5105a688
3632 F20101109_AACBIE leve_f_Page_64thm.jpg
9efbb5dcfb1b779481630ca0d5f5cdca
e164ca542271d628f11db2e354a7a18247fb2222
1978 F20101109_AACBDH leve_f_Page_57.txt
99968a9005311d6bc51d138d0c0fec33
3c7c643943548da248d685e70bacf5cf206a9f2c
238642 F20101109_AACAWJ leve_f_Page_69.jp2
e8f52c041bd7853ad12bec1be6392b27
fddc27901c057ece12f53b0cdc6878bfc27b9f36
70751 F20101109_AACARM leve_f_Page_58.jpg
ec59711b7eb915d17855595db654fb5e
6bbf6be8d1f81427fa866c623e4604d87a627076
13867 F20101109_AACBIF leve_f_Page_65.QC.jpg
43f6a4b660c2584ee4413c8aee1a58a9
802ac464714cffb874172be5b02f335040d82744
1767 F20101109_AACBDI leve_f_Page_58.txt
bd0c60560e085f90797129946bc0dc7d
1760d1a9aff706bdc8f2fba14b14e41b83ca95d7
681739 F20101109_AACAWK leve_f_Page_70.jp2
02e2004c3fd45690d1fb8cf1eb3cedc8
bf77b1b69f3dfa1845c82d12d2353eab74236428
41634 F20101109_AACARN leve_f_Page_06.jpg
b52dbbfde13e0e323e0b4479bca14f9c
53c5bd9dc06ff19bbbd834f01260e9c2eef2e607
11563 F20101109_AACBIG leve_f_Page_66.QC.jpg
e6d0852cda01ae49d48b4c6511e14245
a176a62c228e1a0eac13aa0273dab2c93bce327c
1830 F20101109_AACBDJ leve_f_Page_59.txt
b3e43ebaf2b8328ab7c21c134ab5150b
cb45b630b10a2d2a6b26ddb9f2168cc11dd65b2b
515416 F20101109_AACAWL leve_f_Page_71.jp2
3bed550c7f119d13310fb4f88b8f8fcf
7ccf27ea6bd0fd9407541699c6815926750edcbe
29564 F20101109_AACARO leve_f_Page_52.pro
fbc43b37d1c20e6bd664af123f020ec6
067d6d1d24bca72446ffe90010a90ddb9b78b648
3897 F20101109_AACBIH leve_f_Page_66thm.jpg
3945fffed93008eaba61aa4d69c1eaca
d1fac6170bf7d6e8012ef201f16186b3fda1a3c4
2129 F20101109_AACBDK leve_f_Page_60.txt
1e6790d2c8aa5b212e0f45e192840c1c
1ec35cde2c9ca45f12adab515fb7af7f40071503
297633 F20101109_AACAWM leve_f_Page_72.jp2
eb8fbd3c154b1fbbc56777888bee6415
934bfd4d8c50eba322d4be314c9d3f14ce068937
F20101109_AACARP leve_f_Page_06.tif
ea9d54b8464e58fe2b97a7f6f30a985e
d88b98e523fd1ce189e2eebc74b1815f95938400
12640 F20101109_AACBII leve_f_Page_67.QC.jpg
10bcec4a4412f4757eef7f0f9578b626
7859c25afe6526ebe3029f36301f81be7b4f2ffa
1084 F20101109_AACBDL leve_f_Page_61.txt
63d82344bfd6a9417b79a4a9b6d8bb3a
88de8416544e1adce68b79747d3af42f58db9651
428769 F20101109_AACAWN leve_f_Page_73.jp2
a1af24fedea97bd7cd9f1465c66345a8
c994c068e6f7be6b039495a5db2d3dc6857d8e1b
1261 F20101109_AACARQ leve_f_Page_78.txt
49a5a64e1e8f1e21eedceb00b79b475c
2e1fe87599b2cfe67d98fb2cabdb4f973a9f1192
4062 F20101109_AACBIJ leve_f_Page_67thm.jpg
1af1ff1a93c8a2c0cd2b4e952804410c
9cd54451083bf5bd5940372d6f6fb09774d63386
379 F20101109_AACBDM leve_f_Page_63.txt
3333f0d8711e03b4d73108f715da1f3c
7b04507bad21a4cdd595f8516a37cbf5cdd54b7a
1051958 F20101109_AACAWO leve_f_Page_74.jp2
fc694c025b90068af7bfd726c07b61a2
adb2d0ba9a053898a3002011f497cccc0dfbfbd7
71531 F20101109_AACARR leve_f_Page_09.jpg
9fc2f20628b17d6fc0da374b7710b5d8
f330f10f957b6c958c5cb2e40681a919511f3635
12210 F20101109_AACBIK leve_f_Page_68.QC.jpg
a0a49e34d9d59c0ebcfa1af8970315e3
9c2ad42b624e5751df36e23d42dc6bd15b3e2cae
1333 F20101109_AACBDN leve_f_Page_64.txt
64d61bc77d7570d6bc308d3ae12be903
29a23529ef4c156d75a19e6fbf82ed0394462880
1051979 F20101109_AACAWP leve_f_Page_75.jp2
b236137c93809fbb23674fe149b1c0ad
4b19bdf76c24221e0ae86682eab7196813b0fb33
875 F20101109_AACARS leve_f_Page_67.txt
2223246005ad01359de38bb0a3a9858b
21263a028ca9bf88318a6a2f2f8b5fcd5cb2128d
3969 F20101109_AACBIL leve_f_Page_68thm.jpg
ef83de79937d8412de171ea7481aa14b
7f09a6a3083c299f793debfe538ed556d41e82be
706 F20101109_AACBDO leve_f_Page_65.txt
b88a24b1c9c2975cb7f3c732000f91ff
b7e35d5603fa30da9e22dd9051aa224d34f9e3fe
1051962 F20101109_AACAWQ leve_f_Page_77.jp2
a3987b06f1f5f97a7849ed510b4b7a52
076e4c42ef0e89184a91494bd0cad1a405832b49
117123 F20101109_AACART UFE0022280_00001.xml FULL
25fdaeb6776da4af01e68c377a33714d
fbce324e1126ce6b4c95223ab06324ba8cbc5cc6
5718 F20101109_AACBIM leve_f_Page_69.QC.jpg
3d8f0c700f0e8ed62b974433b154e7eb
5e8a0e5a88850681d3162b47dc8daa2c6659a6d1
703 F20101109_AACBDP leve_f_Page_66.txt
a9e685776b01fc44cf3a393155dadd73
c252bce10ccff6bed38d477d1e7e33dbd80276b3
713965 F20101109_AACAWR leve_f_Page_78.jp2
52f3a4405c15b90a61602aa1ae1204a3
5e67f4812f1bc91cda6982d61db3cca4826a241f
2162 F20101109_AACBIN leve_f_Page_69thm.jpg
627eb322595d9bc1ce021ad1b1197267
70a35cffad3624ca6fe555f1e74f0b436f4ec898
649 F20101109_AACBDQ leve_f_Page_68.txt
2de7d0191ad4093763470eb72a53d033
8f948035627139bcf996b68c2bd3251a017c0008
F20101109_AACAWS leve_f_Page_01.tif
8847443091038abcec1b81b9834ded35
e398f8148f20fa8771a681156147e5926aa003ab
16867 F20101109_AACBIO leve_f_Page_70.QC.jpg
68775677c33092388cbe14ca4957ebb8
62ef638d56a69cdb83cd0ecdbd82932edc0ce447
F20101109_AACAWT leve_f_Page_03.tif
f03af922f48bf59d64f471563890b2e1
d6396d1314807eae2ebb4cf3cc44a0c4706c8ba9
24544 F20101109_AACARW leve_f_Page_01.jpg
cad0ac9e009f39bd8ca2758a598327de
224d6f3820368f20dde966cb3d3eccae398eef28
4721 F20101109_AACBIP leve_f_Page_70thm.jpg
fb7c7cd1ec119d48cd12f00f2337ff55
b6a6bbec6d6e7de6c326988931ddab753b75c1ee
322 F20101109_AACBDR leve_f_Page_69.txt
c0c2e7191ea83241bfb68fa5b9677516
de779a075e8656f81158160ebdd92efdbc4e7352
F20101109_AACAWU leve_f_Page_04.tif
b66447aff1a3372ce8e63bbd15bdbee2
921104358a296dc685b091231f115f6ff2f39337
9914 F20101109_AACARX leve_f_Page_02.jpg
a58b837f5a85a1660cb5121b2a4fe47c
1a3026634625efc6af0b3532ce009bbfb590722f
12690 F20101109_AACBIQ leve_f_Page_71.QC.jpg
be56378f639b5683dbdd4b1c3b0bd70a
63e58b7ddcaca476f454a074ba5cd2d58fe1034f
1313 F20101109_AACBDS leve_f_Page_70.txt
e7110bdcce0b134958ce8adf8ca8436c
f8577f9326d33c3fc8cda7443ff91d5a765432c9
F20101109_AACAWV leve_f_Page_05.tif
2a2c64fb1c1b3bc6f6c99ba96f0cf3ba
0e75d84f28adafb0b0259e7344e1f0fb5980a197
9759 F20101109_AACARY leve_f_Page_03.jpg
c9c60863bcc27d5775c3cf110f5785e3
3289be6b74f165d6db8c3699e20e4ee0519591d6
4097 F20101109_AACBIR leve_f_Page_71thm.jpg
cebf1ff978cd5b408607d7473d72a227
60a50a8616e112dc99c1a837572a01c6e05a6241
1625 F20101109_AACBDT leve_f_Page_71.txt
8931dc861f61575410ed43db520a5525
4856e1bead6fd77a219a3f95f68cf2ee6cc809d6
F20101109_AACAWW leve_f_Page_07.tif
60e641efd42c980866c7d261a4e318c9
ad0d07e26e9398da8bd7150188b3a34294ee11e6
F20101109_AACAPA leve_f_Page_13.tif
3267aa934b72558819e27ef020c18693
85d67749d38e5ed6601784ef152dd2087841a279
34603 F20101109_AACARZ leve_f_Page_04.jpg
feff1abe2176ce1b648463a3765b59b1
9e942c95ffb2b10e17915c0c8f18798dcc3187cf
8567 F20101109_AACBIS leve_f_Page_72.QC.jpg
0a99c4521f816499364f6215b1abaa88
fa380de6617f03297e67c0891a699e8146c681c1
1276 F20101109_AACBDU leve_f_Page_73.txt
02335855221fa7112a26236b20dc15e4
0e8dff9e80b2d684ec2f7557b1d886fd46b779a3
F20101109_AACAWX leve_f_Page_08.tif
9bd5b06fe97e999a76e127602242f442
381e47559c847af8fbd529394e63c7adf6965c59
933570 F20101109_AACAPB leve_f_Page_56.jp2
71013c89278edfc378654c6ac3207365
e4b56fcb06f3ea15d699da08bb1ce24023ab4ff0
2546 F20101109_AACBIT leve_f_Page_72thm.jpg
dbf3892865c6989ede52aa329ef94c3c
a80ce54c361270b9281a9e8e87a48c2a25b984cf
2340 F20101109_AACBDV leve_f_Page_74.txt
0832bc533346a2b6d6055522bbf36e80
baded77f8b3c4b418066181976ab146667618d29
51796 F20101109_AACAUA leve_f_Page_70.jpg
0d5dc05bfc991e6da0e78993fcfd435d
30b2ed25806ce18e9a4a5f813c14a5466475e1c3
F20101109_AACAWY leve_f_Page_10.tif
1a14188b91e1b127e322d785bb4a1df2
d95589a489c787d3906daeafad3846920cd58f2c
14063 F20101109_AACAPC leve_f_Page_48.pro
2909c7c717e378d362461eb4a26b959e
01d0a1effc6e5cbda16730719af2b19306371c50
11560 F20101109_AACBIU leve_f_Page_73.QC.jpg
45a04d02e5a5060b59afa9c6e78f99ff
1e1a512e7a84095e8ef6287ee8ab02b215482c0f
2518 F20101109_AACBDW leve_f_Page_75.txt
7a14e0fc549ce16a6114ffc8513e3601
8fa17b454ee2a17a0c5b8db380d1f8397eb30152
39221 F20101109_AACAUB leve_f_Page_73.jpg
32d2647f27cd3726ed66e70e4b4d91d4
601b88bfbcfba61dec20e6e6a5da2a520f3945a9
F20101109_AACAWZ leve_f_Page_11.tif
3bc98970a7cb5baab2cf0bed5d4c5deb
dd40a3415866820fde3d9889cdb840e7c55f8d72
1051832 F20101109_AACAPD leve_f_Page_65.jp2
3d55b8ec8da9a9e1e6194b62ec585c5a
090fdad84d9328f25806f077d63577a945a1e98c
3488 F20101109_AACBIV leve_f_Page_73thm.jpg
8690cec03817e96ea42fe887970392ab
a5c8b7481037d3b2106afb27ab00894f49d99666
2737 F20101109_AACBDX leve_f_Page_76.txt
f2d52f0b776e3d5ead4832eb7246b8cf
c59e854f0b9e90619dec0ddbea33b5f729bf9873
95793 F20101109_AACAUC leve_f_Page_74.jpg
0afcec98c28c113af2bbcc7566ab2d25
5b49ea97ff7aad42039c2425fed266097a2f5435
6690 F20101109_AACAPE leve_f_Page_44thm.jpg
826949c0b4af27c9689e2951262c7b25
7d8faf91e764b85f398e9c9ad6a793e3877049ea
1503744 F20101109_AACBDY leve_f.pdf
fea1b44448ac208f2ab57990ce2a662e
a8ffe5ac3ff333d6d9b4476e97555f60294eb521
14159 F20101109_AACAPF leve_f_Page_66.pro
0e5b696f6c8238717a615f3ee2cf692f
528ff33d1fe5d3f22392d8f9cb31bbe349ed6446
F20101109_AACAZA leve_f_Page_76.tif
04a86fbabc7d2ddacc99e8eaa3f87234
ac17bf0bbc69e280d3d7ce79f1daa00478a69e03
53460 F20101109_AACBBA leve_f_Page_60.pro
c92a3844de53af978949d6e2206c6309
1f6f2d54a23a83d353dd2b56205bc14d8bb0b5ce
27233 F20101109_AACBIW leve_f_Page_74.QC.jpg
52a818e14680b3c48760929f81433b0d
5f33e60dde09f67421fc5889e494ab3113be1f17
7200 F20101109_AACBDZ leve_f_Page_01.QC.jpg
57548aee3d26070f5dc17954360ca6bb
7951b901fe12bec1cdf7a68e33e0eea27200a5e8
101873 F20101109_AACAUD leve_f_Page_75.jpg
beb84dcbff22fabdbf4ec272010a0573
9d026a372a881044a54715f60f011118d73821fd
846615 F20101109_AACAPG leve_f_Page_32.jp2
5a10d3005d90fa3d5de4e5a9023602d5
182c76821e137f39694e034a9d7a4016e3d0dd4a
F20101109_AACAZB leve_f_Page_77.tif
7db45394e17e118ae24f08dc98152977
17ac5142231b9fbf267ee3526043a8bb332ce666
24647 F20101109_AACBBB leve_f_Page_61.pro
a683c423e8dc5bb75647a1428af0ca27
2acaa0a3bcb807ddba823238acf8292350635952
7036 F20101109_AACBIX leve_f_Page_74thm.jpg
021e30997cf53af72500186d61339c27
b76960d757d6560c15e495928e146360536f39ae
87387 F20101109_AACAUE leve_f_Page_77.jpg
bdc6045303be35af2d1ed012ce141366
6f2c2da162b46234ba1a1d704e53249db19e754f
F20101109_AACAPH leve_f_Page_48.tif
a3e90fd4f3b2900cdcf7a26a48f2073d
37c4dab3b89f8e2e58edc1426d2627edb1c834c3
F20101109_AACAZC leve_f_Page_78.tif
86c88810ed20243dae109e04c61c1f16
f247c56efa176326fcd1469849db5ee6ff69efbd
16056 F20101109_AACBBC leve_f_Page_62.pro
dc6d04ee30da0e949bd2c9972c2f2f75
3a96f0028500777d80070996f46bab313a314db8
28535 F20101109_AACBIY leve_f_Page_75.QC.jpg
a56f6251c7e30c2c36f7df03b189eea5
ddd5f98efc40f57f985211d71500da1c15a7cfce
6317 F20101109_AACBGA leve_f_Page_31thm.jpg
d6282eaa28b4825b70829bfffea68e44
ef90cc02838792fa49bf8d6679f0a55019ac99bd
53889 F20101109_AACAUF leve_f_Page_78.jpg
25016a16c2277c249e262087670c69a1
3bc46447153f5191fcd04b29e38c2f46db83e251
2828 F20101109_AACAPI leve_f_Page_50.pro
d9cb2d64e544b4f68ceba0cad641c68a
d93c59db0b35590ef98ea9fdbb9953d9dda39da6
7858 F20101109_AACAZD leve_f_Page_01.pro
5a1def7cb94a3db490372a92b173b63e
d944c1ca10561e25ddcfd7a0d4e2c88f93c93cbc
8502 F20101109_AACBBD leve_f_Page_63.pro
92c1fca72ceaef2d6e7a78e1cae1a4be
75aa02ab42f731f7e20a538e3e757d8535ea3d2c
7643 F20101109_AACBIZ leve_f_Page_75thm.jpg
f09c8b96ca8c53d24ea0c433978bea98
bb8d4f9ada0f6f8be2548842f3c0eda3263805c4
21144 F20101109_AACBGB leve_f_Page_32.QC.jpg
dee4d40955ac38a5a14b48a0d7a6bb87
56b83c30420a653826f275ad9b38d2c2e4c2a9e4
19574 F20101109_AACAPJ leve_f_Page_07.pro
368f2c8d1a9ac41134bd3559b6818968
6ca796cbd823df36cfec9e97240e214c947eb266
910 F20101109_AACAZE leve_f_Page_02.pro
81c44776616dc9af7f06df68a478b1bd
93af25935db58db310192339fc91a827056e232d
24829 F20101109_AACBBE leve_f_Page_64.pro
1146189955bc86c90c5f99f7990e6417
f07adf8590a941098bd94cb31fa59f2b06bf2958
237552 F20101109_AACAUG leve_f_Page_01.jp2
fe5f9ad37228362b02a90e90688288e8
1a5e6cf77d732f359905f2fa95cea2257c3af785
5788 F20101109_AACBGC leve_f_Page_32thm.jpg
e5d14a0c743fb1a04435d4bfe0ea992d
d3060ccdebaead22ae0dfcc326a827b41ba89b81
1607 F20101109_AACAPK leve_f_Page_35.txt
80488498a805a9bbd6650c96d3befd60
2d5ecbe69a0f7513694ae3e83d10e4e358ba7f84
882 F20101109_AACAZF leve_f_Page_03.pro
4205c8e6b613968748cb962382872915
116518448ed7654794bb43516597475161b8d027
16255 F20101109_AACBBF leve_f_Page_67.pro
3d35950efa0e97faaef8189468875152
14511642b034af9d7171dd4666ba99f173b72b96
26981 F20101109_AACAUH leve_f_Page_02.jp2
f12548f543f4bb18caee832ce4b1f299
b254e2f6f2fd9716143efca7635477a5952bf986
19232 F20101109_AACBGD leve_f_Page_33.QC.jpg
1edf494272966949df0eb7533c5e73da
9f1312219a6b2eace5021c254a671d1358f54140
2066 F20101109_AACAPL leve_f_Page_47.txt
18d41cf8b0211bfa0e1a68850a5899f0
c2bdaf29f10fe172fc6bf4a852d9fa63e27917a3
17315 F20101109_AACAZG leve_f_Page_04.pro
766a55817f824a9960433b6df1d1fb66
8e68666322aa7d49aa23f99db31e1569aed76af3
13198 F20101109_AACBBG leve_f_Page_68.pro
ab805d382b90fe38712a204f9f5073d6
11c35e5c1191755c59bbfc347b448c8441c95686
24671 F20101109_AACAUI leve_f_Page_03.jp2
01b96ddc0fd44c04a64190223ea077d9
70695fceae3046040199426e6ea742b68dcd2804
6003 F20101109_AACBGE leve_f_Page_33thm.jpg
b4fb6707ef6db530f9d95f8edee88b0b
e6e4b5bc5668836d1ad57e9ee33c1b3cb5d999f1
10409 F20101109_AACAPM leve_f_Page_62.QC.jpg
a89fa8ceb41ce300de764452f30ce2f5
fa0f02398920c1ddf6d6d3e8b03e7c900e694f03
97064 F20101109_AACAZH leve_f_Page_05.pro
c8993df2692f3fc238044328c077fe85
230d8edccdf8e6b5d792a88552ca5b18e7c0a171
3566 F20101109_AACBBH leve_f_Page_69.pro
9a56935d06fc9d84c76656ea33346b37
4b8e8590f751bb8263461c059e43d7120539f7f4
408175 F20101109_AACAUJ leve_f_Page_04.jp2
d49d958a14043c446c22be9e321bf101
26c40c4be1f58f7cf9ec956eb01106a418f99c4a
22918 F20101109_AACBGF leve_f_Page_34.QC.jpg
51b8207a3a23b7cb6459ec5cb0d0d534
52a63188d10022929194ff25522f45eb9c5da4da
F20101109_AACAPN leve_f_Page_41.tif
10d2285ae8aff7decb2726505dbc8d81
592d971a8597ac6ab17b0a17ed57129f4e0a4ddb
29162 F20101109_AACBBI leve_f_Page_70.pro
7273d4e410ccb93d3d0552c4db350c19
fb76683d6ab572ad7184b3b6373eba14266de618
1051937 F20101109_AACAUK leve_f_Page_05.jp2
71185f6090f80f53329c4cbfea4dbb43
9e7f3968dc4a6b85df4d30b6d469b6a428af3537
17109 F20101109_AACBGG leve_f_Page_35.QC.jpg
808e98a5ac5cc9b8763d3b5a61836f6a
7767c4eda75693ebaddfb6896d7a922b215902d7
2134 F20101109_AACAPO leve_f_Page_77.txt
22bbe7742f173caa7429c5903190ecd1
fc37159ddd3feaa0b111f3e970202e7b3d7ee3a7
39054 F20101109_AACAZI leve_f_Page_06.pro
d8cbb8bf7f6e3cae8646d3d71c8916b8
ab0047e6e56fe9b22a134f381f1973452f766ca1
26282 F20101109_AACBBJ leve_f_Page_71.pro
85f828d3cb91d3d6a4ac205d13c2133e
108bcd9d9b028257f24c2456b219bc46444f07a0
1051959 F20101109_AACAUL leve_f_Page_06.jp2
9e7352fb140a9347266e2067bef2a824
b2a6c58344a95057cd77047f5a3bcda2b1dda9e2
5178 F20101109_AACBGH leve_f_Page_35thm.jpg
18304f560f3e0ab8aaecf6acde0e0b70
ed6ca8ef9b1070e73ff706a6299944a5099a7b8f
7231 F20101109_AACAPP leve_f_Page_60thm.jpg
b42015d8affd35dafa515e8f41c47443
6af2f8aec81999680db62b46293b253ddc02ec2d
73168 F20101109_AACAZJ leve_f_Page_08.pro
9094b6d5615be1c91622955d7ac63f73
c6dd37a842ff92db79e4f46230e93b4cc2a42cee
12032 F20101109_AACBBK leve_f_Page_72.pro
fe73905a704d1d69dba9c6027c143d7f
4ff43ebe2ac93ac84d4b32fc35c6c8cf40a20e13
610546 F20101109_AACAUM leve_f_Page_07.jp2
63e5d0b73bb65c93b57a8412f48d67e3
fc505460b9a8d99ab082f5ddfe95c18d3d6a1f18
20054 F20101109_AACBGI leve_f_Page_36.QC.jpg
360a51c3b698b21289b9763928547ac3
8e0077aa5aeaae3203dbd97fec2e54ceaf9f6faf
7364 F20101109_AACAPQ leve_f_Page_14thm.jpg
0820294628fe06c384da318119d03b84
97b7aa8dfdeb7fef4d0213068bec8855d23ef08c
49684 F20101109_AACAZK leve_f_Page_09.pro
199a12356a277a545295abc41fb88620
5c4aa494339cb3e466eda215fc1f5e618b902e0e
25264 F20101109_AACBBL leve_f_Page_73.pro
40106597490cb43f724cb8650c165d92
c871befc8723e5ffe84832ae2740efa24e9066ba
1051974 F20101109_AACAUN leve_f_Page_08.jp2
53845d258ba4fbb8b787bfc44fa96fc5
a156f193ea39996105ce6cf628130e5f02f91f6f
5995 F20101109_AACBGJ leve_f_Page_36thm.jpg
ea196d1d1d2612acf4938c506711ad19
a2d9758e6225eb20b6a5970ed0c6eb160b9ef250
F20101109_AACAPR leve_f_Page_62.tif
8c4511094e115081f7010a41d5cc93e2
c85bfa5b68c03fd092a80bfe5e7ca9fc2de45763
48488 F20101109_AACAZL leve_f_Page_11.pro
a5eb2a8266065edf78cb3f93e1c129ee
268ec9983c2376c08b62c5a3a08179415797e46a
57912 F20101109_AACBBM leve_f_Page_74.pro
276c6eb5312bd73fe431fc8a3d33de41
74cb38c80808a701f4ae43098ba72d5e561d6095
1051968 F20101109_AACAUO leve_f_Page_09.jp2
ca74333e8fdaf786ce85cf1e60057fa4
b6fcb5b8bc3dd9178bf0bcf9d9bae3e33af81671
19980 F20101109_AACBGK leve_f_Page_37.QC.jpg
984d58d0702c0e6e173a9d7cf7c5513e
2cbdf20db49d408b40e9edb129dea451724c1912
4691 F20101109_AACAPS leve_f_Page_26thm.jpg
c0a391673d599df376f0fa4887b741eb
04ca57582ab99f310c388df95469b2da8f77bff6
45887 F20101109_AACAZM leve_f_Page_12.pro
072328db082314dd8f9566cde5c91e5b
e157e0447be911714f06ae37e5844e33231b64a0
62863 F20101109_AACBBN leve_f_Page_75.pro
a025903b3dfe9ce21ed32d714cf80701
4f56fc56c5192a3c6b47e0fe08b1ff5b67b7c2ac
517428 F20101109_AACAUP leve_f_Page_10.jp2
afb885f341f6449558ddaf9914876f28
9a6709ec9dfab1c53e2941affe40dff0985f37de
5998 F20101109_AACBGL leve_f_Page_37thm.jpg
bb0b5bee616d1566cc8c227a69111d44
52e741bd94c88ecb6cb5c794709e8cc443011cd5
1051986 F20101109_AACAPT leve_f_Page_39.jp2
5e455c253edc412593c70c0a0892c982
e74d6a7a4569fbff0ccaadb55fcac572c74f8032
54151 F20101109_AACAZN leve_f_Page_13.pro
6a2c646c0b654c69670a74d4acb410bc
b33a276658f6e0ccf1d0d434eca607f84a5d1053
30551 F20101109_AACBBO leve_f_Page_78.pro
d3b02da9ea7f6197b25363989b5b4044
211b66f43af107621a92d0cd643eb9914e7c6f11
1051936 F20101109_AACAUQ leve_f_Page_12.jp2
6c541c6f6e0ed07f6df156d40a76dc9c
6933194491c3dbfc09d8d2f0b8ba5dfb17403460
27245 F20101109_AACBGM leve_f_Page_38.QC.jpg
eeb65c560908f94cdc0fdb0dffbbcf17
fbada1554e1bfcd9ae198e530d443c08a949aff3
176 F20101109_AACAPU leve_f_Page_23.txt
74de9fa45ca44e88f37a6ebbe4904591
f12a139bf6b328a132cc8b959054bf76b9207716
53146 F20101109_AACAZO leve_f_Page_14.pro
38960ca6c51b38c88c3bccb5cfdfd707
f6628edf0c32feab6581ed9655257e12e1630d25
1051932 F20101109_AACAUR leve_f_Page_14.jp2
e96154c477bc3a8f59cf998b836d6f22
80dbc1d7dc19cf29c821f6c0e7ceeb170332478e
7548 F20101109_AACBGN leve_f_Page_38thm.jpg
b198c61be2dd98e0cb7ca660e8eb76b4
d3ace51704c828fa7d9f8df268063a66f2089df2
922021 F20101109_AACAPV leve_f_Page_28.jp2
8e5f77d07b21b08298174cd08fe5cfce
a56e1bd1e285e56e0f376d215ec8914e85fbf641
51831 F20101109_AACAZP leve_f_Page_17.pro
a3c7908706211837d828f0df8f34239d
67ed091fdef29494ea0b0093f266714416c6eb9d
456 F20101109_AACBBP leve_f_Page_01.txt
a10949ef01edb7ab0c5c0b53d83428a9
8d814055ddf7cbb5c8aba7f2a8243ee67ff7f5e7
884369 F20101109_AACAUS leve_f_Page_15.jp2
834b92952eb9e1e958ba9e9d1abe4d21
cf1493149ac1ae8e4ad4fb8538376a32293d7b47
26935 F20101109_AACBGO leve_f_Page_39.QC.jpg
83334e891785306845fdaf051fe3e4e5
4d0d39d55829d454f048ee760152a3ed90f51d19
F20101109_AACAPW leve_f_Page_24.tif
226b8bcc06e9d9cedf11248d66d5509a
b0cc74babe90e3f39b8240a25b8b9ac1e0bf475f
51794 F20101109_AACAZQ leve_f_Page_18.pro
19c910b770f1dd2be2c64568ace1c11f
ae1c7e05067617ddadcc7f3ff9857ac8fd1469d1
89 F20101109_AACBBQ leve_f_Page_02.txt
3d9252c1d2683781c9637b73961bb362
a8608885c856d7ebccc40d61ed39a64e225754a6
1049520 F20101109_AACAUT leve_f_Page_16.jp2
a91ea573a9b5acecec8d16816968d951
569b3e03039e14281726683d41a0550b67c3a0f2
7480 F20101109_AACBGP leve_f_Page_39thm.jpg
9868a2c61e86fd6e46032879ba3d66da
f44a32e7565a57f0531a361b2fc7d814608dda9a
F20101109_AACAPX leve_f_Page_39.tif
719465c2b7b6f8a13c5575f3c9bd4eae
eece6b20f0dcd2848893f1e26488b48db7591953
44375 F20101109_AACAZR leve_f_Page_19.pro
69776fb10e8acacd87b0d920472a400d
32c77d3674563b3ce1ab2e793b0699db75f5f4f9
86 F20101109_AACBBR leve_f_Page_03.txt
43cf51c4e432e397b64981585b15f313
9edccfbfcbeae39b9ec6a948fb9bc3c5c17fbaa0
1051969 F20101109_AACAUU leve_f_Page_17.jp2
be85e01f43b783151a8230917c317d90
83dbe397bb49dd7be16924ad9d41dce4509e6e92
20659 F20101109_AACBGQ leve_f_Page_40.QC.jpg
42ab8abd0fb78ac5ece0a96f6058f3c5
389576a91096a3c4355053cebeddf0e15e6b5dd3
52300 F20101109_AACAPY leve_f_Page_77.pro
ccae453e830641473d9096833f15175f
6450acbf8ee98ad006de5044432b2fc511ba0d63
53058 F20101109_AACAZS leve_f_Page_20.pro
db411bae4c91013d81369645281801bb
2f430f62e9e10dfde9f7f02f6e7d7809000011bb
725 F20101109_AACBBS leve_f_Page_04.txt
e66a4790dd29534830f8670d5081b779
45e5d5f356477508162bd5ef439a3b7b5ff17db1
1051965 F20101109_AACAUV leve_f_Page_18.jp2
bfa224c6969b30f37b1212d737136f01
ec257bfca3d480c2ab3802d262a578b53dba8d11
6152 F20101109_AACBGR leve_f_Page_40thm.jpg
139da20602d0196b3ddbe16cc9c1f51b
e701043baca0b10a1b18f223f4de3cda2ec28613
F20101109_AACAPZ leve_f_Page_32.tif
b23c4ff8a9ec0789c6221c73568a447f
f6f66efa76ba4b0a10609a520bc2c78caa3e83e5
41005 F20101109_AACAZT leve_f_Page_21.pro
23323729b6dafe0cb438448403a819be
43144173336e870f0a0ac3b8839c0d495fc698c6
3996 F20101109_AACBBT leve_f_Page_05.txt
00d7f05f76e8a6d466c7cd055dc45db8
f8e6c35dd661f5dc90187e0a0019b6813fc663e0
1025985 F20101109_AACAUW leve_f_Page_19.jp2
de08cdceda86c8792e73ce6f9f063e84
da3c5291093868ce1e696daccf8433c215b30142
11579 F20101109_AACBGS leve_f_Page_41.QC.jpg
75fc6bfd6b6fa7de76497a694cf999c3
6f95ddbb2b01e897ccfdfb4cc85c6a25900099ee
12156 F20101109_AACAZU leve_f_Page_22.pro
64e93cb8487fac594d380fea67400960
248130f8d5d84b3efdc8e6ada36d1d769bbb8d91
1666 F20101109_AACBBU leve_f_Page_06.txt
ad1e747ccecb8fe07a66d1721e8c96eb
b507dc9fea1f5a8c7709c62d1dde00e5aec92992
F20101109_AACAUX leve_f_Page_20.jp2
f67c5c6e7541db9c6d5977f610596118
0548ebcf638c2b5c156684ad1a71f8d3bc505e65
3965 F20101109_AACBGT leve_f_Page_41thm.jpg
246ddd4d49f513b29a7178433950b7fd
b353b26a7189aa145f7e0a7a8749ddea1721881e
2981 F20101109_AACAZV leve_f_Page_23.pro
3037f9c2e63a695c584a2d2395e7082b
171ae0219acb807d667ed39d13b791b63d951a6c
816 F20101109_AACBBV leve_f_Page_07.txt
28c40c3175b160b3126d2cbcc9ed7ed3
46e515ee26c0db1156e1424cc41ee2d630a3dfe9
84017 F20101109_AACASA leve_f_Page_05.jpg
b7e154d410682829fd1de4eaae1504cf
07e60a9db5e8c5f7a5232fd16a4b783370d8accf
1009580 F20101109_AACAUY leve_f_Page_21.jp2
3bf8cf6ab81511372f0f107c7daa7651
7073457d1487e54930098a9ae81c67914dcc5be0
39693 F20101109_AACAZW leve_f_Page_24.pro
c8d4c7e6bfbba2758cf88452b22ee443
f68210e983bb83921d6b7064c95bbed384bbaba0
2918 F20101109_AACBBW leve_f_Page_08.txt
2920a31ed4cf7db5950931db22166336
834af2f458b52eed07c19f31551089b0343393a9
595054 F20101109_AACAUZ leve_f_Page_23.jp2
d5088f08e4594edd5c84fd6d6722aa54
e8180d8cff9c49bb3f5ab64c4bea893ae208a6e6
7030 F20101109_AACBGU leve_f_Page_42.QC.jpg
df280673568dfdbee036f44d23bbc997
7d7307fec2bee2d21e42dcba74cc85d61c84443f
31844 F20101109_AACAZX leve_f_Page_25.pro
38c3f861687c7741441c92ab9a7ea35c
abf54f94b782c5f1d9f672186d9de0be1e87e4f6
2031 F20101109_AACBBX leve_f_Page_09.txt
f42b5ddb4894a89937a50744de714714
d8010840c826bcf6d6d70985606f94a09062d3a5
28002 F20101109_AACASB leve_f_Page_07.jpg
8d33d8dc8c91d5f3103f883cfd7bef9e
5499e9f97115e753d65020dea15b9c0c4cd53866
2485 F20101109_AACBGV leve_f_Page_42thm.jpg
a3b7977e42d0aaee5b1db47070db0a62
fab41d704350ced0158b1c56e874716692e5518f
F20101109_AACAXA leve_f_Page_12.tif
26151d05b7065b12d6f457e8a7ca388f
0189057d4b725c3e04d4ff59d8ca07c6395db8bd
27307 F20101109_AACAZY leve_f_Page_26.pro
e2d13aeb49a2c0f3406cfe5f52010297
318ffc5e8f239c46d931b7da1e4a2d9a11b57ab3
1142 F20101109_AACBBY leve_f_Page_10.txt
f605ae819ca88b19dd40e765fc56f3bd
2b03bd89f96d19045bcdf1f19e72f1756dedb5fb
75361 F20101109_AACASC leve_f_Page_08.jpg
a066423e0299517760146d6631de6841
7cda5bf309d5799d2a4c7f3a6ff5d30529f8963e
7147 F20101109_AACBGW leve_f_Page_43thm.jpg
7e88d138615449bd5cc45e0d10ece62c
e0b30f8b9680d71eb31013ca1d329a833b3e9597
F20101109_AACAXB leve_f_Page_14.tif
478126dfead1ba47790283cbf0aa3700
ce211955a25ee1643e16133167fec6138f346d14
28801 F20101109_AACAZZ leve_f_Page_27.pro
141eadf1ceed82a0fe9ec4c605e25607
d7a8b8495c0df7eb24e0d472ce62087dd0d3fdf7
1826 F20101109_AACBBZ leve_f_Page_12.txt
f7819021bec8c836c5ebe33e9bbb3025
370b5d5126f24d1e4fb3eb3cf729d60e3f094228
41878 F20101109_AACASD leve_f_Page_10.jpg
3c221d4ee10da0ed34fa6a503d753993
f9dc1b475ecdf2ec3c46911d81649d502aeaa838
24166 F20101109_AACBGX leve_f_Page_44.QC.jpg
8ae2c7c9d7ffb687e62efcddad49cd6f
4847905a9b1025ac0b41b3778b02894d39886ab7
F20101109_AACAXC leve_f_Page_15.tif
43551ee6a000b81e3a99542e2437ffd4
cdc3cef10d6287f084d96f24766b967b0fb2fade
83131 F20101109_AACASE leve_f_Page_11.jpg
c59303127b2ccc84be6f4180f443e59b
51c4941c27b9a0218178cb870d8ff831af51e254
26692 F20101109_AACBGY leve_f_Page_45.QC.jpg
31e71b9016beeebb624a6be86741ea9b
ff07d5dd88d9bfce67be766f2715d9e8f0f24757
2513 F20101109_AACBEA leve_f_Page_01thm.jpg
51a593686eccca397ef1de616510f9e3
3a50427476c93bedddaf8e6e677bed3ae926d468
76798 F20101109_AACASF leve_f_Page_12.jpg
5310e1c26411bc86c73faefadc155958
8bce85bb3aff1f41b25e78ee64524c3e7c4c241b
F20101109_AACAXD leve_f_Page_16.tif
ec1c51357ab052010cd97913f7591bbb
3e09f98973e17410ad12eece8a5ff2c5ad46cbf0
7284 F20101109_AACBGZ leve_f_Page_45thm.jpg
6db2ca0bfd32f48de58132eb5e00d06f
2c3051df745470aed15e14121c286c54804c4dd4
1351 F20101109_AACBEB leve_f_Page_02thm.jpg
163f6c21ec78a79e54e0cfd6d00574cb
043ab14a8a762f6e51e09b15a5243f808847a4a4
86334 F20101109_AACASG leve_f_Page_13.jpg
3a1ffcf9da9d6ce7a94b23c5cd1285a8
45b1ec61b5090cbb9fd819fa9d37d01a52669809
F20101109_AACAXE leve_f_Page_17.tif
a33b40637aff4b587fbc4398655cee46
57e27e7cfb4cbd6ad3d26c3d83c9be08856d81fd
3110 F20101109_AACBEC leve_f_Page_03.QC.jpg
2b8f073376eb3c1d17035d14e4368f60
118621348bd1e576c3acf4af688d7030d8a20759
81825 F20101109_AACASH leve_f_Page_14.jpg
d671ea0b065bf2cc4b5e4ea52611a76c
d94561126d1d26479278b34d4b3eb9fd8fa75b50
F20101109_AACAXF leve_f_Page_18.tif
3c9f5d195799e98b439e6a6cde2b7c18
a3475c8c0439c9d47aa0dc14619cada6d41e7ac5
29675 F20101109_AACBJA leve_f_Page_76.QC.jpg
61d017edb34890de8a697662a303876d
f9c94f75a3cd7dd587d3025eeba7da572457aa9b
1341 F20101109_AACBED leve_f_Page_03thm.jpg
e7a64ae01f71aa0c77406fc2ae70d731
41595a4c188354e53b6df3b5f47d7e6d2e31acd6
46505 F20101109_AACASI leve_f_Page_15.jpg
16833388f00a286bfb8d0bdbbfa6a4e1
19d355985f872aeb014199138f0a95487c7d8236
7580 F20101109_AACBJB leve_f_Page_76thm.jpg
0472a67d39bec1f7134c92f54afb9363
1b321cd6a0aac258d83eb7e21941454e0d85f350
10565 F20101109_AACBEE leve_f_Page_04.QC.jpg
6286f387b6b492cc0e0df3af8031570b
603bbd53ec7e719823cb14fa23715168e89d76a4
79603 F20101109_AACASJ leve_f_Page_17.jpg
6d97de4c84370610dea86e1f16d8d691
21c8fffabc3f9d8f88d453e314b3815c16f97b1f
F20101109_AACAXG leve_f_Page_20.tif
5f8d5590a0e75a17e967ec06a2e40338
06ba22eb05e5312172807922e9b12151c11fbf42
24447 F20101109_AACBJC leve_f_Page_77.QC.jpg
8bcadb361698297c622df714eb266fb2
1d2b91e4229fe50e54578f26a5f8f6a36a8debee
3308 F20101109_AACBEF leve_f_Page_04thm.jpg
3b22d5dd5732aef0baa422f1fcf025ed
f3cfcab640e1b1f8bf37d131e240b1db2ea4dc67
82406 F20101109_AACASK leve_f_Page_18.jpg
29bf0d473f6a9e3a5673c1f48da91057
deb258518910536a5548513696856e3f59e6e2bd
F20101109_AACAXH leve_f_Page_21.tif
8d045058ce60dee3f06d99959c199e36
2bed893686a42481cf090e5dcb0d74e76b7a238c
6641 F20101109_AACBJD leve_f_Page_77thm.jpg
6c7af66cab08951f22d80fe29471af43
24ad3149e2a509aea4f3be73628eecd3d571b712
23408 F20101109_AACBEG leve_f_Page_05.QC.jpg
3387f05c032e1d1b5b3644550e4524ec
f6ad55fd5218d904dd43af471b529bd95de84be4
73805 F20101109_AACASL leve_f_Page_19.jpg
cf0e724bb9c90b90ae881b8610b212fa
b522ed5dd373d6d0264cc4254fd62752f0857388
F20101109_AACAXI leve_f_Page_22.tif
3802657182659b028d7ee95b1f7e2131
6f1369a837b0b9741b8b7c44ebace3ada0a89110
16987 F20101109_AACBJE leve_f_Page_78.QC.jpg
c6f82153d6a8fc44c008a09c91d5ea00
4f64d7a8bd253b90be8f6d74f61eb53ac5930049
11108 F20101109_AACBEH leve_f_Page_06.QC.jpg
a9c5591ef16b5598109ccf58e448508b
bcf1adecaa3215bf1c6ffe61d8c0cd343e2324d2
69862 F20101109_AACASM leve_f_Page_21.jpg
27bafab5353e127c62536fcc781dccbc
804bdcaa2f79abc0b608c770aeabc55d3bd2fac0
F20101109_AACAXJ leve_f_Page_23.tif
73487601c5a82dcf036824db0d0f45da
86570b3ff4d7d6213b9c3608e1f4037b24d17a13
4886 F20101109_AACBJF leve_f_Page_78thm.jpg
f477594039ffc959c8a435236c1853bc
43f8401a54ed3618a85843faf7fce89d50090d95
3274 F20101109_AACBEI leve_f_Page_06thm.jpg
9fabeb9b1968a44afad1b09f2250e9a9
461e0ef6e98d64d5ffb9771985e76d79e1167803
39563 F20101109_AACASN leve_f_Page_22.jpg
6426da7a5aaff0811e0cbf27523e673d
a9d68f698026aabd1345ec2435ce1c704befe394
F20101109_AACAXK leve_f_Page_25.tif
69126090f24d83fda55aaabd2bfde19e
e5ad6f7e7c9073d47f964fcdfe1e3341175b252a
90614 F20101109_AACBJG UFE0022280_00001.mets
6114e3faff81719e6973a306de2ab332
515cc9112a7e3b8c646375f9d9c9e2039583c9da
8690 F20101109_AACBEJ leve_f_Page_07.QC.jpg
0e309934d5b9b4407785041835507f8a
83f7e23111cb523882dfff962ac6efce12c95705
30048 F20101109_AACASO leve_f_Page_23.jpg
24752b943cb8b2b7c873ccded6edd89c
b42453ef946a6f68cd33f0d8b2f05c57f1fa20df
F20101109_AACAXL leve_f_Page_26.tif
625e8145152b45eb0284464fd22c29ea
4a49dda36df8e686f8593685f8ceebbb676762e3
2612 F20101109_AACBEK leve_f_Page_07thm.jpg
f1eeb5a651c2b316e3f4c75d472f3c36
6fb31978452cbafb22150251c38e793cf73e889f
69182 F20101109_AACASP leve_f_Page_24.jpg
5b7c07dd92d262e941339ccc95b949ec
34c2d2ee34ac06043f99f72862c3bb1d5cb9eae6
F20101109_AACAXM leve_f_Page_27.tif
c4a0fc087d4fc160d54be04c1aed0b84
f4ccc0280aec7edee202882e156f592a6e2ca236
22997 F20101109_AACBEL leve_f_Page_08.QC.jpg
a33fb56df6985a90780d90d24de5596d
98b8e4404b17a248386fb667bcf8f6725568d426
51149 F20101109_AACASQ leve_f_Page_25.jpg
8927b4e7a228f6a799dfa6e75566f0ee
d9e8c0e94bc974c548a557315711e2103d27c6a9
F20101109_AACAXN leve_f_Page_28.tif
eb4f7e54677591687f34b797b9a0cf37
d7fcbfdeec16e70e7256c9c308e80bcfa519e2d6
5566 F20101109_AACBEM leve_f_Page_08thm.jpg
84d3c6eb3aff27b0a859a05ceb387f00
7d68ee04ec491d7983cb0d7e69a6a477e1bb80ce
46811 F20101109_AACASR leve_f_Page_26.jpg
286c292df21d1fb6a171b45111397a64
bc486fbd0718beac0d75fe3a7d576be3f6f99d7e
F20101109_AACAXO leve_f_Page_29.tif
2e431e2aea6bfd8d4db2c60e97571701
e13d4d0d1a3b449c4681a73aeeb048692b607803
18397 F20101109_AACBEN leve_f_Page_09.QC.jpg
1478ae1b0bed4beb42ac03300ebf3e2e
9ff47d5541525cbe32cd4060062db3d41ff9ced3
52141 F20101109_AACASS leve_f_Page_27.jpg
18cd758baaf97693991085da9e6a4a9a
ea1f20ec4b36b004fc1168e25375ca04fcf5eeb6
F20101109_AACAXP leve_f_Page_30.tif
49836f788ff1989496e82311347c23f8
7fab7207752fb456f4fe5fc9905cceacbf2bd1b5
5064 F20101109_AACBEO leve_f_Page_09thm.jpg
4b3ff2ca744e642d5709cfb632d5b313
1ed8e29f8cf977db9e4d3527f74ecafaebb5dba2
67427 F20101109_AACAST leve_f_Page_28.jpg
1541151c3377f0a1ac49c25acb5f65b9
d8374825992d6d4b8d0746b73bcc9ef7984d3619
F20101109_AACAXQ leve_f_Page_31.tif
fc020b4bf0c6cb30a2808a200ccb7300
0bef594f2b4ce35f319c87859c9207c660782af7
13807 F20101109_AACBEP leve_f_Page_10.QC.jpg
cb9f7af41fde274b823d5dcb64cfc6db
f2147a7a59257acfea2fa1480cbbfb3596e5c508
60633 F20101109_AACASU leve_f_Page_29.jpg
de8befa8686354b1aab6460c583c9f7f
e86118c13aa6f5b0631c553b47bbdbf6815a1dce
F20101109_AACAXR leve_f_Page_33.tif
08621550ff791a704784ce6179eb88e1
e4e493b56705770cc3b1b44289266fffdb4b6206
4130 F20101109_AACBEQ leve_f_Page_10thm.jpg
8607eb7bfcce28a3ea82ea0fca256345
055942c99a0bac8ba99c55c6faa2deea987001b5
75215 F20101109_AACASV leve_f_Page_30.jpg
7dffbd434f53d3e7aa29fc9414626d0a
ce257a5ad5e7d057b2e3a87af75a37b0b75805bb
F20101109_AACAXS leve_f_Page_34.tif
e5873f45ebf83520ba3e3b9d08ecff0c
51c121d3209b51702128431186b3008e79455d48
25755 F20101109_AACBER leve_f_Page_11.QC.jpg
e40bc95ec9001444f1020307f888f63f
0471c3162aebde53031588a92119eaacdd187fe5
67032 F20101109_AACASW leve_f_Page_31.jpg
514526a1b331c52b687009c3866b46d6
1d118a0eb49d9a03234735c3720cd5c30fc40325
F20101109_AACAXT leve_f_Page_35.tif
d0ea0c81e6abf2699d6782361a09cf92
e5a954022950b9c8ae2ce424de54d012ffb8ee15
63016 F20101109_AACASX leve_f_Page_33.jpg
6d9b71dcc9ea6f59911a023171aadb57
42320130e6acf892122c41d5d4e14b37b000baaa
F20101109_AACAXU leve_f_Page_36.tif
ed821d19e3c6b11202d856b1f9185bc0
160652dfea1dc45a59aa4c57239ce92f828dcb69
7246 F20101109_AACBES leve_f_Page_11thm.jpg
2ab6515e1b0ea10e5c8d8d0099c35178
0de1bc0a5a545b8596401add11a33ba4da6a48a3
76953 F20101109_AACASY leve_f_Page_34.jpg
b9a095d4fd2e05473553ddce23fc64fb
4ce22955f646af9a5e518e15a207e6cacb1ff174
F20101109_AACAXV leve_f_Page_37.tif
c907c429976eb86b94cc48a104bbc952
9bf0d2723aaedd87d41773176c3346301ae0cbee
25034 F20101109_AACBET leve_f_Page_12.QC.jpg
97f9391cc7d4496b6e57268a1be29f46
eb61c3309e5d8bcb1e7df0c05ba0b4f96ea11138
52605 F20101109_AACASZ leve_f_Page_35.jpg
c71747b9be749b49311a1e1bef5fedb0
24b210012e4d5b354117e01f580e9b81d3f022ae
F20101109_AACAXW leve_f_Page_38.tif
89cc55b8646848e7b1c61dba81025186
2199ddfebc171b1cd71957dbf2607cc9cfa8950e
16373 F20101109_AACAQA leve_f_Page_27.QC.jpg
d8601a6ad05f4f266a69317e7540dbe3
23ef9c69082d218a75f438ad6f654234b0d60511
6730 F20101109_AACBEU leve_f_Page_12thm.jpg
2429afe300ae3cc6ef6fdd4fd1f71687
1f08cba16a7fa87af3fa8db90a59fcc085b95965
F20101109_AACAXX leve_f_Page_40.tif
4303559d1d893bfbc845654b8ef0e542
c3ab3e837e520e47ab1dadfe3e89b61cd7e9dc9d
11824 F20101109_AACAQB leve_f_Page_65.pro
7c6c77993b826a00ae2e8398cbd92786
c191e5884606da25a6079b7bb16be28128a48606
F20101109_AACBEV leve_f_Page_13.QC.jpg
79c5e3205731d3f3be0a0ca1312953b3
0c68dc76437d23d3fe0eccd911cb8ec549d8dc49
936252 F20101109_AACAVA leve_f_Page_24.jp2
dc3e25bcf7728657ed5e1935ee6d2aa4
fb3f76b027c0565a678fe513621a19682988f195
F20101109_AACAXY leve_f_Page_42.tif
d8f82f5b18b9fcca1e4e011b7e443ede
e535fff433841ce71bd46cfe0b18177f816a1ed7
14776 F20101109_AACAQC leve_f_Page_15.pro
b1532cc7d63d63eeec562aebd66d54cd
183b530f78471624083e45ffc4bc3d8958bb8951
7350 F20101109_AACBEW leve_f_Page_13thm.jpg
0e675f382b83a3ff0d99a4621623eeb9
e423052f23d07147dcc090bccdf5413b2d2f6a80
681079 F20101109_AACAVB leve_f_Page_25.jp2
684ef1ea7214fc4f04465acfd56aca90
a3d7296bc4f756b01266f81fc35af763202b37e4
F20101109_AACAXZ leve_f_Page_43.tif
369ba2c6725a0585ad84c4ba2f34da15
4751a8ed8613149c67052b68c9477d360e8a4f8e
1762 F20101109_AACAQD leve_f_Page_33.txt
7ec0131e0f73138e200534d46ede65a5
a5a0edc0fefc7e11aa137f94521ce0754803ca6b
26947 F20101109_AACBEX leve_f_Page_14.QC.jpg
2ecfc8cbaf79a51d0bc0e8697e8889db
4e4376c15d81c8fd79f433e150a021ae056e0d39
595298 F20101109_AACAVC leve_f_Page_26.jp2
3b6223988d46241cad2caeb74bc033bd
584f3e569ce44c40ce08abd974acc08c583f70c3
25469 F20101109_AACAQE leve_f_Page_17.QC.jpg
a1c7e82a2ced77e1a9a54441532c8944
e293aa4b61c5b430d821cfe0c5d40fa5e4ba90cc
15439 F20101109_AACBEY leve_f_Page_15.QC.jpg
8a2b539c773e6da6e6a99622bb5d73e9
8baaa9c3f3cd616d8c45d574e9c9eb71bf1df095
2291 F20101109_AACBCA leve_f_Page_13.txt
216756183a4b18cf2423ad109a1f5d7f
f3a12c409ea15998e1a926f6571c97915283c2b5
653226 F20101109_AACAVD leve_f_Page_27.jp2
204506fcd6788473e97512bb864b115f
4cce44cf3bfeaadb23ef88258e86a318548751d8
16687 F20101109_AACAQF leve_f_Page_52.QC.jpg
3258562f8e3cb3c7faa6c8b0298a3b8e
f29f98009d7775b575bd5a96f09f86c75dcb4ac6
F20101109_AACBEZ leve_f_Page_15thm.jpg
c1de5ebaf97c5541df26e2129fc235bf
469568f23b7e23befb828923f2166305476866fe
2181 F20101109_AACBCB leve_f_Page_14.txt
afd6d6da01b488367611a51d0ba8324c
2b39a06e4470e96ddf23d8cdbb012c7eaec291b9
F20101109_AACAQG leve_f_Page_49.tif
d197be8e6a884abb32242fe100841488
791178a93e93d15baff4fe14bd92822596f7a6d8
640 F20101109_AACBCC leve_f_Page_15.txt
3b39c913f8dc09ab034919d1032bf3e9
701ff747499beee20285bec1fb61b5cb72ef9de6
803690 F20101109_AACAVE leve_f_Page_29.jp2
ce2569ed63e46a5d9688f6f6baa06eeb
edca2b46ba0ce580402d8d8201552b79fb000b52
83893 F20101109_AACAQH leve_f_Page_47.jpg
c633c481a0c85ec319aaf7e5e9320a07
0d2115dd1d6d9bea0f08bb48fa0da33001483640
28214 F20101109_AACBHA leve_f_Page_46.QC.jpg
75479781f8eee293df0f5bdf78229f1a
ac6efb64e9f8a2b2258183992fcfe79de1069e94
2088 F20101109_AACBCD leve_f_Page_16.txt
c6dba0e154aa61f294acbfae8eab0aa9
739582ee513dc21f5da399c9f0f060f0947942bd
1018300 F20101109_AACAVF leve_f_Page_30.jp2
186fe73e72595cafd774100c172e1f96
cb2444cf606698ed7146efb5163244f4ea6e827b
86704 F20101109_AACAQI leve_f_Page_20.jpg
45497aecf6ff31376fe7daf9b2e0e7b5
35c1418c6b85a0b485576a92bc05c7f8ed257a37
7455 F20101109_AACBHB leve_f_Page_46thm.jpg
54eeed3aa7c0dd96dc2f568cd5e5de1c
98634076e7b46f399ac5f30fc546618c201e0c90
2090 F20101109_AACBCE leve_f_Page_17.txt
61f961bb12fe82e233106a4d464beb51
ba5095e6ccca80a3dae71631cc43736b37431c87
923058 F20101109_AACAVG leve_f_Page_31.jp2
f46d098ee628e0183b51c45729d05f7b
6a5ae8d1d55eeea7280238fae1e4d7c38d088d96
23885 F20101109_AACAQJ leve_f_Page_10.pro
c28dc34526550ab4b8e46b3b36059fff
40f130348bf51a633528e4099e74292c1dd1ae30
26093 F20101109_AACBHC leve_f_Page_47.QC.jpg
cde97f9a2df72d8e47a53e2e0392fb8f
22c03cfd6f802fa7ad1525e1d923c1ceb166d081
2050 F20101109_AACBCF leve_f_Page_18.txt
61371ad017fd31da3f643694e1763bb5
90213b63917cdbf6a0a23b0f76efaa99284ee868
876932 F20101109_AACAVH leve_f_Page_33.jp2
212e55a2a9f4727e627506968c52931e
33a97bf04aa3271798fdfd62a2154ce245acb44f
F20101109_AACAQK leve_f_Page_76.jp2
b447646333faf2824ce11b30962454aa
d47945a51e6ee84d9034bc22d0b976dd397aa81a
7164 F20101109_AACBHD leve_f_Page_47thm.jpg
8267a0a02e1f89920c644bd8e5bf8e37
be6309cf09c5467daf307afc9e7520245723ce58
1041075 F20101109_AACAVI leve_f_Page_34.jp2
5792ff8e9042057aa9c0107e9ebbda8b
69c952e448fc28663df5524cea5ae819d2155478
831821 F20101109_AACAQL leve_f_Page_55.jp2
fb26ea5dadd8b1644728710174d6137b
11de1bc310a66a51d1bde198425d99687e626aa0
1805 F20101109_AACBCG leve_f_Page_19.txt
d585e483d7343d8e6f89f11a986df4d6
61bc0fbb6fa02314791ccf1e0bbe3cb9c319dd8c
11953 F20101109_AACBHE leve_f_Page_48.QC.jpg
0e5383b28bb8cf26624ed9ce06249945
854930ce784ad1d1cc2082fb62bd13051f39f470
675460 F20101109_AACAVJ leve_f_Page_35.jp2
1e60fed1fd2a79f493bb64438ec81628
cffc2c544729818af06ef29a813c23762b14dd79
F20101109_AACAQM leve_f_Page_64.tif
a7716dc7d6e25ad16c781e49fbcd8a15
5d0a2bd6f1604e3a1e4e74983422f0734b1a6911
2083 F20101109_AACBCH leve_f_Page_20.txt
add97ee089884c36a7c404d8d9bf3b24
0440076a59f9b0f80156e0b1067d0e10f60f59e1
8361 F20101109_AACBHF leve_f_Page_49.QC.jpg
ec577bc50484dd27093c6360e0a44aaf
daeb7df364950cd2a8e31a338f3a0959a93d3538
860021 F20101109_AACAVK leve_f_Page_36.jp2
f1e105e6c616adb75f5d6da7a140bfc8
a497cd4863262c21726b67d0dedb24dbaf9dec86
41437 F20101109_AACAQN leve_f_Page_71.jpg
d9000e90e8d1f6bd829f4226f9aa3f98
33326108e7c80ede03da9962cec77f315b3077fe
1861 F20101109_AACBCI leve_f_Page_21.txt
3ad7e3ae21f0b11e444fe3e7c4bdde17
6118bb70dabbd4d8c3313925c30892616ffe4324
2764 F20101109_AACBHG leve_f_Page_49thm.jpg
60f3bf661900fa3259b87bb0cdcc6c50
01e1915b0d4847948bb36be97ec4cac6cc58c876
895903 F20101109_AACAVL leve_f_Page_37.jp2
c0a4bd93947d06eaa638e3f0866acb8e
20b7509598b7454d0c52b41f5b7bba9fbf999a07
63989 F20101109_AACAQO leve_f_Page_36.jpg
32588d6657af730e68677e37dce6ac19
1f2bc4267330cb5ff865838d0ff5f22aad9d1330
1928 F20101109_AACBCJ leve_f_Page_24.txt
06f083fa72aa830bf4ed789759703adc
a7dfe5d1942ca7b16b189ec84898e8a2db842a92
5216 F20101109_AACBHH leve_f_Page_50thm.jpg
46ed6aac9ac9d63a8807f1bb07adbacd
9d150dca01317b87d57c3b417be7bac3c2fc215f
1051980 F20101109_AACAVM leve_f_Page_38.jp2
c06952ada7978ae5194739f282890df7
14e53f4efebb6dd3dae59c13729a1fc74c2b9f8b
35823 F20101109_AACAQP leve_f_Page_29.pro
2e53495c8f80908d7826f32aef1e7a7d
6f9dc701bed5e52bb28ecf6f3f7706d21d635192
1734 F20101109_AACBCK leve_f_Page_25.txt
d12f67e0ea30c4e6c881351403e38db0
9b6eb1867ecd90e51b6cb7b1cc1c5132dbb70fd8
19836 F20101109_AACBHI leve_f_Page_51.QC.jpg
4a3b5c171e5e6eb8037a9c04bbf40e70
4f3dc1ddbe5ce6e0affb4d1d61e9a2f15d7cb8dc
1024772 F20101109_AACAVN leve_f_Page_40.jp2
ffb9474a0516fa3f2aa3bb46817f2d07
7ada5deef18d851b856777a6e9568d4f735c5ba3
68811 F20101109_AACAQQ leve_f_Page_56.jpg
31e876d3cf13e04172c4c1f7fed8f3b8
5f1d870a1e05545b8e85cc5d830e0762ef7e8af6
1760 F20101109_AACBCL leve_f_Page_26.txt
6e3ec63c7f78f23974b0e18eecb80e2b
7af62a57801c157fb11408a5859e277588d09f68
6129 F20101109_AACBHJ leve_f_Page_51thm.jpg
e9b1e36386dea2da2a5805aad70e07f6
c0e88066a603c325c4dc16fe07787a82f2875046
943189 F20101109_AACAVO leve_f_Page_41.jp2
57118ee222a3a8056f503e9c6f565ecb
dd15dafccfb0ea27d824de22b725ef57605b466e
4559 F20101109_AACAQR leve_f_Page_65thm.jpg
d7075bdf6a1acda4bb748343957966cd
9cd4ccace142385f51ecb8fed8f5f4de24c0af8f
1328 F20101109_AACBCM leve_f_Page_27.txt
ca686f413ea70de4f828e895e7a57f11
0be190136304b699feebc3f89eae4c52b10d566f
5016 F20101109_AACBHK leve_f_Page_52thm.jpg
af0dca0a6e4688aa5a44f2bdfbd984a7
d20d4432633a96492d1a870a7f5af3bce46600c6
348107 F20101109_AACAVP leve_f_Page_42.jp2
554f1cfaed67ab27958f691e982805cb
27ad5d20d2f98568addfbfc24a65a146d86e55ef
1043334 F20101109_AACAQS leve_f_Page_57.jp2
15c1aac90eaef6b598afc94131a32d1a
387aa8e41fc041d2cbbd30a8a6c4a9df2e97a3a6
1799 F20101109_AACBCN leve_f_Page_29.txt
4a53fcc804e07ee1c1bd6ec38510b86a
f01247bb1734ce6ce1e200a2f98ccc5f71e3fde2
18299 F20101109_AACBHL leve_f_Page_53.QC.jpg
9ad384900da8020e8baadc69c747eb63
9efadd8b842f239924699cf4ade4855fc3db2412
1031692 F20101109_AACAVQ leve_f_Page_44.jp2
e18e8b5cf7f52ba6b84ebe01fa8bf31b
7d482f55cd9d4120a4d30f0623d7adb64e24970c
6596 F20101109_AACAQT leve_f_Page_28thm.jpg
c0bf51576fe473fef720eb06a9cf2f4c
faa52e34402eef128c45906846fab1897e74b9c1
1713 F20101109_AACBCO leve_f_Page_31.txt
b03e42595bdcda07e15eab599f689f0e
8b5e6ab4bacc873a56ebb6cd99a909ac13e1202c
5578 F20101109_AACBHM leve_f_Page_53thm.jpg
e6f687ee595fe8c8f93329c062d90dab
5fb9e782bb9b99b8bbf3fe86f187fde94ca4c55c
1051978 F20101109_AACAVR leve_f_Page_45.jp2
8620e9d93c3b31fc6b1fd9094866b43a
82282bba8743dfa1ff7dcf531f867973b91095a3
60155 F20101109_AACAQU leve_f_Page_53.jpg
2daa660c45177c05978a8f92be3bff13
fd27864513075c127caaecf10fb4e7ad466bd732
1508 F20101109_AACBCP leve_f_Page_32.txt
5ea9a3c8e289eb05f9f87b9826f526c1
a7742f956f552c62a2c8650b2e318e97b62d9017
18083 F20101109_AACBHN leve_f_Page_54.QC.jpg
f5765da0d6cc608497292b3b634989f4
6f6a7ae94d4f944d791f99176e9b644c0027c748
1051952 F20101109_AACAVS leve_f_Page_46.jp2
0a298e5dbd4895077362b7b68388afde
e8f9b655aa85b49b85be2ac81ec897ee58004c47
442613 F20101109_AACAQV leve_f_Page_62.jp2
b2018e40f5c684d58e623e32e7efb1d3
60c4b2222047658d15fc1abcb3446687b628c51c
5287 F20101109_AACBHO leve_f_Page_54thm.jpg
c06a8178ebab4d0e5f96d41b9989dcd1
e025582a0d76ddc13c587f67a680da6581e3407c
1051972 F20101109_AACAVT leve_f_Page_47.jp2
49db09de46be6489477ae263a98fe02a
984b713922aed086c4b26def5c82a968d913bc39
107597 F20101109_AACAQW leve_f_Page_76.jpg
19c1e4a97fb794f47c84dfa18356714b
ba62a135c4f8cb4ae70f89377ef3475da8fd5ee5
1961 F20101109_AACBCQ leve_f_Page_34.txt
6fac5705551ce11249af226f181a910b
40c2c3992b85340d166b54eef4661d07efce4e72
19959 F20101109_AACBHP leve_f_Page_55.QC.jpg
5512c3ad6b10e04a9f9e75c98af03fcf
2bb3ce585e9768592651f129160670a4cc051492
621472 F20101109_AACAVU leve_f_Page_48.jp2
71e0a29eb50ae8556dca7fd917e46a59
4cd8bf92d39be619c8715fbdef6ca680c61f4fe4
6728 F20101109_AACAQX leve_f_Page_34thm.jpg
937f3b2c7f3f474fbc906cedca9bbd2b
a7a5d0b72a382fb6c8e1144942204734c50bafb6
1686 F20101109_AACBCR leve_f_Page_36.txt
e98bf18aa8448e4448197a3eda52da1d
a3551a0a214aa6958d6a55b1c6fa15adfe51011e
5731 F20101109_AACBHQ leve_f_Page_55thm.jpg
2e7ef2c29269aed90bce36d2f51188c3
dbb95b8bf8756791fe7099051cf3f73aba70a664
603061 F20101109_AACAVV leve_f_Page_49.jp2
fda3995aa93101df64acfa902b592836
4da44da56452f1a639795eb6e9f5346a328519d2
824 F20101109_AACAQY leve_f_Page_62.txt
2ce1a740ab611081ef35e621d2992f07
165f35844dd890d8422893e92f8505f60e3c7934
1925 F20101109_AACBCS leve_f_Page_37.txt
739e4d42625d3258538eb2d133e6172c
4bb52bdb11a148c086a757b5873b69b70b6e6a7a
6267 F20101109_AACBHR leve_f_Page_56thm.jpg
01e304e8a0a968bce44b8572aabf4b37
609a0ecbc3d4151eec0271babe4a302a1895a6b9
1051985 F20101109_AACAVW leve_f_Page_51.jp2
253eff1d56303d0a6225c965c4746c66
c738fe434655ef118e7e952801ae0ac8b2893327
3123 F20101109_AACAQZ leve_f_Page_02.QC.jpg
0858df0ac82c023fceb0a07058d8d815
a61538f13b3139a1108d272c6f66ecb572ab2334
2095 F20101109_AACBCT leve_f_Page_39.txt
3990069114f6fd3054a75fcdb7c9240f
6e019889dbcb434f4b9455951863100c72f6f0f4
24118 F20101109_AACBHS leve_f_Page_57.QC.jpg
263066836ca175713522ec248c8e7435
8db789ff759eb62cfefa03ec5021d77316398dba
647501 F20101109_AACAVX leve_f_Page_52.jp2
418203d6ec3a25a105326e62b53aa131
e7379543171254c38c9e2efae7a30f03e761d587
1309 F20101109_AACBCU leve_f_Page_40.txt
65ab8d0cde1010c46b43973b8fbcb011
0b411b0ebe6b2b4eb77288a8571f6c554f22464a
6744 F20101109_AACBHT leve_f_Page_57thm.jpg
d64e4003e81c37e67001c9afa383878f
9ff20f7542a1747f5626ee375aae89691415872b
799201 F20101109_AACAVY leve_f_Page_53.jp2
4b01fed177e41a080a14ed8c06312cc2
1aac4ff3c10204846ff5bd197c929edffaf0e567
63129 F20101109_AACAOC leve_f_Page_32.jpg
e76c00646adb50d33cbda1e7e19cb4f4
68000464fae77b1eb7edd1bc60ec4fbb2a33d35b
354 F20101109_AACBCV leve_f_Page_42.txt
c3849e0fb27966107fac3662144c6de0
bbfe872ae717a610a36cf7d143583ea9b1a5b19e
67231 F20101109_AACATA leve_f_Page_37.jpg
cfb255f9ba1d6feedbf6d122168d2561
fc5ca1df4bb9608f689f6cb64bf6278087caaad0
22200 F20101109_AACBHU leve_f_Page_58.QC.jpg
d18bb3276e81ab6353f12e2947a56e9e
8d7b961d597c6fdc30b1db95ff111b600e798021
763125 F20101109_AACAVZ leve_f_Page_54.jp2
083726a414c091c475d23eee3111ec50
ddba48e282ff116f97802ad4ef83103aca3d94b3
75825 F20101109_AACAOD leve_f_Page_16.jpg
7090a0673cee62546fb109a985278a78
1d467c44e1c75541f9950918834d127faec8dd4d
2218 F20101109_AACBCW leve_f_Page_43.txt
eed93f7f967c790bac503ce145e6d253
7c6346dc30f4feac8890ada2651200880467535c
86275 F20101109_AACATB leve_f_Page_38.jpg
6bd3b68ddf645e9928164ae01abe385a
4560cb0172d2eef480a6f0f74f2bdcdf0bf010a8
45516 F20101109_AACAOE leve_f_Page_59.pro
6ab9492e300a9d690d6fa74fe618c95f
fda17989f39b4278c9073c6d49ac28a74ee4a8f2
1877 F20101109_AACBCX leve_f_Page_44.txt
00e4e8ebf1a6e9f91045a2bdcfebd94b
8e10544aa6939d6c23faa56d0c1886f2a440a83e
6505 F20101109_AACBHV leve_f_Page_58thm.jpg
fae50671fd75bab66efb8d76857e2564
a3dbe50d8f2c1b2567f2fdaa8fad82827ee641bf
558980 F20101109_AACAOF leve_f_Page_22.jp2
51a9b474a5c2b88e2116f06a4628684c
9fa98942a09d2dea563cc342824685a5591f8343
F20101109_AACAYA leve_f_Page_44.tif
aebf8b25d037e86dedcc162f4b38c290
6868d15f51d0cb3831590b2607c95af031ab1061
39801 F20101109_AACBAA leve_f_Page_28.pro
4a4afced31c69f5b4bc5cc4968c605d4
cfeeeaa0baf26bfd7e4d67719047e50d29e52145
2057 F20101109_AACBCY leve_f_Page_45.txt
ed766af5b914294cf9d0a3963fcea46f
f11d05faf9cebd69951b998af51a14b43cebcd60
64429 F20101109_AACATC leve_f_Page_40.jpg
9511623584b9abe8ec1e2890badcc826
a231ece184a668913bdaf475ff76031ca9c3683d
23247 F20101109_AACBHW leve_f_Page_59.QC.jpg
559c023ec2752b62463c655c0a93bad0
823a8aff8fe7fc6999497ecf728465e38c26720a
1051981 F20101109_AACAOG leve_f_Page_43.jp2
40865e4d4789fb43198180567519dcc5
b305c25597e33721caf55aefbd2393f124be6077
F20101109_AACAYB leve_f_Page_45.tif
c7540c16a72367a66d566319f1de8a19
c7362522bbf9cf4cf6029753adfc7372d6884f2d
45457 F20101109_AACBAB leve_f_Page_30.pro
807ec54c74263487609d82f3f5e47d4a
ce515bb4d935570babc795a545b1583cf62d8725
2203 F20101109_AACBCZ leve_f_Page_46.txt
418d5d9bce90d3f483945395cd7c6a5c
803575b38ecd2476d92b5a3fdb7a13a847e396e2
39443 F20101109_AACATD leve_f_Page_41.jpg
07580494e491eacf5db05bcba13da1ed
fc8dec85f34002a5be5e75167df1a141ca1e1a78
6535 F20101109_AACBHX leve_f_Page_59thm.jpg
9cb3de712f9cf9740c835d1f795d45e9
14109ac929ac867784edbc182a3611ec73a301c8
1884 F20101109_AACAOH leve_f_Page_55.txt
d19d9b6b1e90e80e3b10768dc753a38e
acabb368b1dbc6006ff945a48be06aa047718667
F20101109_AACAYC leve_f_Page_46.tif
36e0e330fb271851f1bc29fe335f7682
fdecfe59b39f03570976b9be037a393cc76a1dd5
41082 F20101109_AACBAC leve_f_Page_31.pro
26e490e60e115c4987f9691cbd60b109
00d00ff1efe782c7ae126a04543b0670d4b98469
22823 F20101109_AACATE leve_f_Page_42.jpg
0163c3de810fac0db3a1fecba02988be
9b1a5baca560f6a1bd80c92ce04a6d1d39193538
26176 F20101109_AACBHY leve_f_Page_60.QC.jpg
963e35e6883a82c4e08c8728eb2fb39b
df612e2b1caabf126f281b252ad954242625ccdf
25075 F20101109_AACBFA leve_f_Page_16.QC.jpg
80238f66097ad5876290d43c5b8f2f2f
8dc44b26d81eee41e5dca15f3a50b1d3b0fc45de
12120 F20101109_AACAOI leve_f_Page_63.QC.jpg
964a0dddd7ee7e3ef038ac0b54c34a79
2deeec276555cad102b69baa5982a4e3c228429f
F20101109_AACAYD leve_f_Page_47.tif
dfdfb03e659de5998ae636ec54f9f5ca
1660253f928e98df7c0ae34d6518451e8c1c410b
36299 F20101109_AACBAD leve_f_Page_32.pro
db1979610e1cdee578325e188281cd4b
0c9eaf426f017304c4c931d8e968107b937b6f8b
82282 F20101109_AACATF leve_f_Page_43.jpg
d8423360d02fa1f69be74f2c35a8e811
c36503e4689743d010be44c72faee5b8d03f8f4e
15095 F20101109_AACBHZ leve_f_Page_61.QC.jpg
96783486a8e183ec751b6620ace53c63
b51dfd1150d9257f049992e42942d80d2f936861
6778 F20101109_AACBFB leve_f_Page_16thm.jpg
2c9e15977d8c1a9313faf7ee88957f34
b32d382d091543b58e0890962c90c2eeb3269799
1788 F20101109_AACAOJ leve_f_Page_28.txt
0850532982d613e6da1a9d6cba8976c4
81f83dd33deaff1e2d1e533280fc51cf867b8ab8
F20101109_AACAYE leve_f_Page_50.tif
913095d99fa367870f27ba3d7c3692bf
8a1bcccf735515196c42cf3e095f086c302b2cc8
39085 F20101109_AACBAE leve_f_Page_33.pro
84207f8965bfedbfa21ad224b763f568
7e7568d9e240920bd339a7ed93d6c9d9971479dc
75549 F20101109_AACATG leve_f_Page_44.jpg
a60a324f290d71b5a6c66c6be0447c5d
86f1a15d010523d42d1528958449e38ef32d1ce3
7206 F20101109_AACBFC leve_f_Page_17thm.jpg
134f40c4cf74bcf0c9856f8a7c2254fb
30fe530efb612e119d1975f09fbed5061caae98f
437 F20101109_AACAOK leve_f_Page_41.txt
d235730923d5c02d0e4a76b5401d8115
3ba14b2e99ac142bc5fecd8e00be24a885b54a61
F20101109_AACAYF leve_f_Page_51.tif
036620d0fa557f1c8da9d14915a943b4
0290f180ddf7f6bc8901522e1cb4015d9ac6031a
46338 F20101109_AACBAF leve_f_Page_34.pro
b058601221c84c66f6586b7b01b8f7db
1d2657e0a9ed576f59ccc7bfc09e356137247480
84100 F20101109_AACATH leve_f_Page_45.jpg
c17b5090a66bd6ff3bb1ab42133b4154
a08497963314f9b93a1860420e88c11f3ecd2f00
26507 F20101109_AACBFD leve_f_Page_18.QC.jpg
2a068059ebcfeb2803ecf69c134bc986
81ad71b4066cab8e2f055a2d83145e5c46efd6af
F20101109_AACAOL leve_f_Page_09.tif
dab19582f8f7677ae2785f228c3956f5
4a5e56387f01210af9ec05002e277e3843d5c5a1
F20101109_AACAYG leve_f_Page_52.tif
cc7f7deac75c198d875fb8910245589e
5d6220505079cdb80d6c217707fb286191163a1a
30834 F20101109_AACBAG leve_f_Page_35.pro
9526cde2a8311e8483d06a55ecb3c4e9
9bb6a80723d5bd5f23e215cdd950f22edc67509c
88247 F20101109_AACATI leve_f_Page_46.jpg
b4e09e1bf28208d45d0f77ab0d9968ac
00a6598ec9f972e961ac979dbf7046c7f99decc9
7248 F20101109_AACBFE leve_f_Page_18thm.jpg
463d13bc7671037d8152a329afdb70fb
f9e8f5498ab91ab06235e79eaa69f271d9f4c56d
F20101109_AACAOM leve_f_Page_75.tif
f6af458a1ec541916f95f2587ba9e702
ffb3c5673b8ecaf4067e6a31562cb375db6998dd
38029 F20101109_AACBAH leve_f_Page_36.pro
8f424fbf054ced95c8019689f5fe5e64
60ad6e37aa0cf99c8332fa104c9debde408acf39
37604 F20101109_AACATJ leve_f_Page_48.jpg
d4dfd89d87f3c735a7854bc5a30e3c1b
fe7975c59eb4380dae2e059156a48706a5d3b162
23777 F20101109_AACBFF leve_f_Page_19.QC.jpg
e986d1e0aafb55ae794b0e334aae9c05
da568d20345631ef63cfbaa69b8d2ed988c8d6a2
27946 F20101109_AACAON leve_f_Page_72.jpg
53311eb41a97343dda842e983e81998f
b882e7d56b3e73691655fb81e436860a44b29ed8
F20101109_AACAYH leve_f_Page_53.tif
c6af815170e2e78e8b539bc8b8735f87
0202d5a19d3e7bcabd501e3a15ff76abb3816d1c
40642 F20101109_AACBAI leve_f_Page_37.pro
8b41f7781add7645a84da17670eb59cf
1303dd5feb7dc1ebcbe5f2bf1d58144ef53179ff
27159 F20101109_AACATK leve_f_Page_49.jpg
08f527e3e3a6472b691b1785875157f9
87a73f481384fad93e6031e92b7b54b755e9c118
6750 F20101109_AACBFG leve_f_Page_19thm.jpg
677d3d1ed6e5afdc49fd70c5986c96a8
476c47551fa2ea545bc6653cf3094bf635a14dce
F20101109_AACAYI leve_f_Page_54.tif
b2c5fba8595ef3f0f100fd91c8da8c39
45068c7ac66ea6fa378845d3b3fb0fb6a7b6fb50
51776 F20101109_AACBAJ leve_f_Page_38.pro
cb3dfc2b1d698c00a2f6ca511da55619
4f08483dcb54138afbe8aa66fdceee7136fc1b5d
40339 F20101109_AACATL leve_f_Page_50.jpg
6665a58179096d2411acfef0bf9d8ccd
cd39d54dba7218a7192fc081f6b0c5144370d470
F20101109_AACAOO leve_f_Page_02.tif
a2a8f1bdbd5542f1fcf7e1a829786dfe
14fa7e54aca5a7567ce82e33d0782640e8e33d29
27611 F20101109_AACBFH leve_f_Page_20.QC.jpg
d005d9113415ccf77d36f771f283c1d5
56ccc19ad24cd235354f57b98e98778d2d9bbeb6
F20101109_AACAYJ leve_f_Page_55.tif
ab010fbc400881e6f6c8d822ffbdda53
2abd5ba289682b1ad980c4230518f2221ad5d5ce
53428 F20101109_AACBAK leve_f_Page_39.pro
3252499b9e34087f02aa82178aa81802
faa1728e89839154078a85d14de841e5c740c401
60984 F20101109_AACATM leve_f_Page_51.jpg
ac014b95afc16d7ab95419eb1d5ad0b0
1dc7ef8528d8fc20633ecacacbfac69ce8698f4f
2116 F20101109_AACAOP leve_f_Page_11.txt
a3e14b93d78f47bee6a5aa69dbe4678d
406c0f30a370078ea904e62e4f7c2f536a0818d7
7429 F20101109_AACBFI leve_f_Page_20thm.jpg
2871f5f7aa738297ec4a01e45e65d626
ddfe9a640f16e998be54a4b863cf3104cae5cf0e
F20101109_AACAYK leve_f_Page_56.tif
fd3043c2804a597bd64383cb8dfe13ff
20d2f703112ae8aa7181c19feb1a32e53bf39942
31238 F20101109_AACBAL leve_f_Page_40.pro
1107d36cde4cbae4cba52d110ecd20ab
50c538ae4a05e1374798bb28bacf41326573d0fd
49163 F20101109_AACATN leve_f_Page_52.jpg
d49629dc796a2f8ccaf363c6e41c6b23
098546899927dc27bb3e14f9b5844069ca7919af
F20101109_AACAOQ leve_f_Page_19.tif
ecde1f0a18d8e9cee584b1d252e644ef
05a75cc1a320e6ce3dff5ae984a6fb1e1abc7ba2
22055 F20101109_AACBFJ leve_f_Page_21.QC.jpg
c41e8d8fde801d3e57187f1c8c3546d1
b0e7ec83909039b2855fdd4a0c627e27eeec888a
F20101109_AACAYL leve_f_Page_57.tif
cdecd833e2f074067eda3031ed0b233f
c11706222a1561c9633c7c814c2cf46f6f105346
7403 F20101109_AACBAM leve_f_Page_41.pro
20386ea245e9b84072e8d048dd62814c
244646dd09e4e840903fc9ef8e16b8bb2f6fddcf
56699 F20101109_AACATO leve_f_Page_54.jpg
ba946f51de1e135feb14c1c3cdf236d5
77e103cf7c619a2cd792b6b457a86c56e88d19c0
2092 F20101109_AACAOR leve_f_Page_38.txt
2366f5947488780ab6659627360d6ad4
cd11449c5783ebe48fba5697a182019c2f6634b6
6429 F20101109_AACBFK leve_f_Page_21thm.jpg
4ec0deabdf6a612ee0b70450fc46348d
027845a64d3b6c9f1191a07a1bd7e73351fae071
F20101109_AACAYM leve_f_Page_58.tif
9950a1d6dc03e5900d2c51532729d55f
247485dac2b61961869fb036ec303f6876669c69
5885 F20101109_AACBAN leve_f_Page_42.pro
c9c16e2c4e18ac41852494569e920c46
f63c866d6da50d67156c07a59f8a9113b91a5431
60882 F20101109_AACATP leve_f_Page_55.jpg
9a800636ae6c8f459a5bd369e45058c8
f2f78a42ce7118dc36b4d6179c85b60349d1c45d
55770 F20101109_AACAOS leve_f_Page_46.pro
0487e3510f671e309cb811c227d56198
ceaa655c4f4b94247ca613c5566907d604ff7020
13696 F20101109_AACBFL leve_f_Page_22.QC.jpg
82c8f5e3e28737de20a5205fe08471b1
9f64443ea37357f385121921462e9821cb549234
F20101109_AACAYN leve_f_Page_59.tif
3ee239aa2f4b4001966efd3d0c911bdd
2ad30cbf284eab016f5980f4bfb112e91cc6bd99
76051 F20101109_AACATQ leve_f_Page_57.jpg
9e1c32207a342fa53cd69252a24a8453
720de410cca96b479ba7e7036fb3930eca9a92a6
25448 F20101109_AACAOT leve_f_Page_43.QC.jpg
275205dda1353c63103d5226142a4152
399712e219a11ba7753579a25c2d53871ab1d626
4975 F20101109_AACBFM leve_f_Page_22thm.jpg
9f4a5539bd84396dd671eccc2620b86f
047e0f817de2c1623fd20ea04e599fa366280f41
F20101109_AACAYO leve_f_Page_60.tif
8cd16cdf03adc15b38ed85886e5302cc
54bcb47faca3e90870dd96e7923e74e847b9437b
50573 F20101109_AACBAO leve_f_Page_43.pro
52104183e7cd0b5ca4efabdcd4526120
271ed021686aeb26e2eff812d50f39a7b96e13de
73485 F20101109_AACATR leve_f_Page_59.jpg
7b5d1edebef4dae386eed05e0e4c9ead
bda74f7ed61ff1de783eb51dc7866fc71bd07ab5
22303 F20101109_AACAOU leve_f_Page_28.QC.jpg
41b86da47db1620a49de378dd7f97021
19392d4acd51b647ff69678913558c167ba74c8a
9952 F20101109_AACBFN leve_f_Page_23.QC.jpg
46b92ce2b9bdb224b2f7fa6ef87dcc5e
b35826215f4a7d1988eb1c2a3e2b6a9bec00f0ea
F20101109_AACAYP leve_f_Page_61.tif
7737770d22670f913596611ae36e3a9e
70eefbe098cd2c49e8ffd206b687e7a77928a2ff
44076 F20101109_AACBAP leve_f_Page_44.pro
b029616e45eaeae3251745015f5e84de
2b5926d7b503d39feb6474e45ca024f37fff8a00
84304 F20101109_AACATS leve_f_Page_60.jpg
492c2666379bd5d96940b25554369f23
db775347fd2604f4801558c2b7a61af13e9f500e
47265 F20101109_AACAOV leve_f_Page_16.pro
8f64483f49132928dd4772d07198e785
6b7d5e17307fa39aa9d3bc8a03ef50fe9dcdcd78
3278 F20101109_AACBFO leve_f_Page_23thm.jpg
36f24119c75bf699883bd5de5e9c7740
945b005abb62fc11f4f852aacd609cdcc57d700a
F20101109_AACAYQ leve_f_Page_63.tif
3ed6297a13f0072dd1dac5480457765f
8cb2cd6bd9d0442c11f9b1bfdc26c47c5b600b9c
51921 F20101109_AACBAQ leve_f_Page_45.pro
cfc23ee9cfa5f0f5f9524a9a2afbe4ad
469de931a0f0ebfa43ff980d70d7fcf940fcb406
44206 F20101109_AACATT leve_f_Page_61.jpg
1f3bf6b6b3d1e69c59361715f7748d99
dda0c15205765cd1b6e9302938d2b8ddb75672af
1908 F20101109_AACAOW leve_f_Page_30.txt
15765046bea4a83d12915c9eccbb2d12
9aab7057f3b73244d95c274c951c0252034cd2b5
22524 F20101109_AACBFP leve_f_Page_24.QC.jpg
c94332721e1a0ff522c564abb192709d
7b107531952d188e01de14db38f31a3fd3bfe0f5
F20101109_AACAYR leve_f_Page_65.tif
9b67d703268a4a1065f3540ac207349c
a809fba60fe51a5fcb10c3d4ddbc036eba475f1b
52184 F20101109_AACBAR leve_f_Page_47.pro
17c679e5911b0db9436cbc0879ebda47
45ccf1d236e92c1be661cbb2a51676320a0152af
33996 F20101109_AACATU leve_f_Page_62.jpg
ebf4e3e6611bff7be6d0600fd2f94180
05c4f38789178760b6031e0ceb8f88aeec3cda81
529 F20101109_AACAOX leve_f_Page_72.txt
5cb105858f6664746500982f8d4c4925
d2363cb370685e6de36ceaa3c37095feefb59e5e
6365 F20101109_AACBFQ leve_f_Page_24thm.jpg
17ab527e6ffa2a2bc2907cde29c84bcf
cd441e427e6c92deac7f3101874477ae5b6a59bc
F20101109_AACAYS leve_f_Page_66.tif
c90cf802a70c224816dad4249bfe4b5e
8aa9485c4284365b9153b5d19cfd854d58cd5955
3386 F20101109_AACBAS leve_f_Page_49.pro
189ad54b0ef8c9c4778c3a054be51189
0b6467064c48e2b5357a1d2dfc809e4e60829479
38066 F20101109_AACATV leve_f_Page_63.jpg
f3dca9566801976e5cd6b5a2f755a32c
bc13e5ed32533f0b305644381b23171af57e7d56
798968 F20101109_AACAOY leve_f_Page_50.jp2
7a459c3eccd0c21dcd6025ce2ecb6b3f
0627e4d5e061d23dea156afb7bb27fef24b48ae1
16294 F20101109_AACBFR leve_f_Page_25.QC.jpg
693286994529a32ad951a727c1ac8cf2
98d30c4f1651e84e3d2224044c7315e1d46a9409
F20101109_AACAYT leve_f_Page_67.tif
d266be0637fda7172e8d4a0a2ff4a877
5c54e7a69dd7c813c4ed62163e74473ef31b9f1e
9278 F20101109_AACBAT leve_f_Page_51.pro
f28954f46de557f032441497fb8d7c39
6c3045206bbec1e2624a63908daf60b18f7fafbb
35299 F20101109_AACATW leve_f_Page_64.jpg
456ae2464fc4f4b10f1487a9b3268d73
5158e1a753a42bf25fb83a9d099c61b364534217
1051960 F20101109_AACAOZ leve_f_Page_11.jp2
3d8345d2c1df6d98fc6edd6c2739bf7c
250819d61582d76c742e27af909c95683864080d
5093 F20101109_AACBFS leve_f_Page_25thm.jpg
752ed86c63a985fad87ffae4db43348a
71874e898bc8ecd75dc296b81c1433eb3875c777
F20101109_AACAYU leve_f_Page_69.tif
9edec5aa725f6ce42ed6adda978cc4d7
9c8fa47e07b44d15b70e0b44f721966008827f51
35577 F20101109_AACBAU leve_f_Page_53.pro
595c5302a747813c9ffb3e95b4ec48e2
e982622a4afb8f459e6834681c526a6a034333ba
44393 F20101109_AACATX leve_f_Page_65.jpg
908577cf668fccfdaafe6d1ac782dfba
ef22458a6f443ac936e44b6bd1a4a696c216d123
F20101109_AACAYV leve_f_Page_70.tif
01f4b1a125ffb8e484561550faedf507
307f7fd2ea53cd058e0bd9467b5c56abbabfd21c
34097 F20101109_AACBAV leve_f_Page_54.pro
37cb3827332be2bab94870c042447199
3a22d32ddadafe9d7ecca4d21c890d31f37573ad
40894 F20101109_AACATY leve_f_Page_67.jpg
b8babf47450d49b4980335db5c04658b
95786b4f4759e743c5121db04e5ccffcb39ca8bb
14494 F20101109_AACBFT leve_f_Page_26.QC.jpg
322ca506a54c1b9b97a699cf6bfef3f0
5c260494f2c66633ec5e9f7b9880a64c0a5fe126
F20101109_AACAYW leve_f_Page_71.tif
4501bd32905a1b9a2ffa8d252350d328
c04ab1745355c8528eee4acf1cea61b10794cfc0
40297 F20101109_AACBAW leve_f_Page_55.pro
9ecbf258a5824a3f742c23fa37477e27
18b065fa9bd9db1e738ed01776fcbb68e1a2092d
68268 F20101109_AACARA leve_f_Page_76.pro
f51370246f5426e8388037b013b5d273
1a4aae068c54e2f7f95537fadd9888c1c4bd1cfb
40054 F20101109_AACATZ leve_f_Page_68.jpg
cbb1997c7ec676fe3a939985fd7c0cfd
a32d9762eb3f8d27dc91fdc80382869a34e5bbfc








DEVELOPMENT OF THE SPACECRAFT ORIENTATION BUOYANCY
EXPERIMENTAL KIOSK

















By

FREDERICK AARON LEVE


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2008
































2008 Frederick A. Leve
































To my mother and father









ACKNOWLEDGMENTS

I thank my supervisory committee chair (Dr. Norman Fitz-Coy) for giving me the chance

to embark on this innovative research. I thank him for his patience and advisement. Without his

help I would have certainly been lost. I would like to thank my fellow colleagues in my research

group were of great assistance to me; Dr. Andrew Tatsch, Shawn Allgeier, Andrew Waldrum,

Jaime Bestard, Sharan Asundi, Dante Buckley, Vivek Nagabhushan, Josue Munoz, Nick

Martinson, and Will Mackunis. Lastly I would also like to thank my supervisory committee (Dr.

Warren Dixon and Dr. Gloria Wiens) for their assistance in validating my master's research.









TABLE OF CONTENTS

page

A CK N O W LED G M EN T S ................................................................. ........... ............. .....

LIST OF TABLES ................... .....................................................

LIST OF FIGURES. .................................................................... 8

L IST O F A B B R E V IA T IO N S ....................... ........................................................... 10

CHAPTER

1 INTRODUCTION ............... .......................................................... 13

R research M motivation .......... ........... ................................ .. .. .... ........ .......... ..... 13
M issio n E x am p le s ............................................................................................................. 14
R e se a rc h F o cu s ..........................................................................................................14

2 ATTITUDE CONTROL ACTUATORS................................................... ..................16

Types of A attitude A ctuators........................................................................... ....................16
T ypes of C M G s .................. .... .............. ...................................................... 19
Single Gimbal Control Momentum Gyroscopes .................................. ...............20
Double Gimbal Control Momentum Gyroscopes................................. ...............20
Variable Speed Control Momentum Gyroscopes...................... ..................21

3 SGCMG DYNAMICS AND PERFORMANCE CHARACTERIZATION ..........................24

D y n a m ic s .......................................................................................................................... 2 4
Torque Am plification ................................... .. ........... .. ............28
S in g u laritie s ............................................................3 0
External Singularities .................................. .. .. ........ .. ............30
Intern al S in gu larities ................. ............................................ ........ .......... .. .... 1
Singular Surfaces................................................ 31
S te e rin g L o g ic s ................................................................................................................. 3 3
M om entum M anagem ent....................................................... .................. ...............38
External Angular M momentum Offloading ............................................ ............... 39
Internal Angular M momentum Offloading ............................................. ............... 39

4 SGCMG ACS CONFIGURATIONS ..............................................................................43

R ooftop C configuration ........ ........................................................................ ..................43
Box Configuration .................................. ... .. .... .... ................. 43
Pyramid Configuration ....................... ................... ...................43
SOBEK Pyram id Configuration ......................................................... .............. 44
SO B EK H ardw are ........................ .. ........................ .. .... ........ ........ 45
SOBEK M mechanical D design .................................................. ........ .... ............... 46
SOBEK Mechanical, Power and Output Specifications.................. .................47
5









SOBEK Attitude Determination...................... ...... .............................. 47

5 R E SU L T S .............. ... ................................................................52

S im u latio n M o d el ...................................................................................................................5 2
Lyapunov Stability Analysis of EMK Attitude Controller.................................. 53
Lyapunov Stability Analysis of Actual Attitude Controller with State Uncertainty .......55
Sim ulation R results and D discussion .............................................................. .....................56
Experim ental R results and D iscussion.......................................................... ............... 58

6 CONCLUSION AND FUTURE RESEARCH ................................ ........................ 70

C o n clu sio n ................... ...................7...................0..........
F utu re R research ................................................................70

APPENDIX

Off-The-Shelf Rw, CMG, And Magnet Torquer Data Specifications................ ..................71

L IST O F R E F E R E N C E S .............................................................................. ...........................74

B IO G R A PH IC A L SK E T C H .............................................................................. .....................78































6









LIST OF TABLES


Table page

2-1 Attitude control actuators and their specifications.................................. ...............21

4-1 Mechanical and Power Specifications of the SOBEK ACS....................... ...........48

5-1 Sim ulation param eter values ...................... .. .. ............. ..............................................6 1

A-i Off-the-shelf performance specifications for RWs .........................................................71

A-2 Off-the-shelf performance specifications for CMGs .....................................................72

A-3 Off-the-shelf performance specifications for Magnet Torquers .............. ...................73









LIST OF FIGURES.


Fig. page

1-1. On-orbit assembly ................................................................... 15

1-2. E arth m monitoring in L E O .............. ................................................... ......... ......... 15

1-3 P ost-docking stabilization ........................................................................ ...................15

2-1. Magnet coil from Tokyo Institute of Technology Cute-1.7 CubeSat.............................21

2-2 D ynacon m miniature R W .......................................................................... ....................22

2-3. Semi-log plot of input power versus output torque for off-the-shelf RWs and CMGs ....22

2-4. Semi-log plot of mass versus output torque for off-the-shelf RWs and CMGs................22

2-5. SSTL microsatellite SGCM G flown on Bilsat-1 ........................................ ...................23

2-6. DGCMG developed by L3 Communications ........................................ ............... 23

3-1. The SGCMG with gimbal coordinate axes.................. .................................... 40

3-2. A 3-2-3 rotation sequence through angles 0,, .................................................. 41

3-3. Intersecting planes of the spanning of SGCMG angular momentum ..............................41

3-4. Angular momentum envelope for 0 = 35.26 and e = [1,1,1,1] .................. ...............41

3-5. Internal singular surface 0 = 35.26 and = [1,1,1,-1] .............................................42

3-6. Z ero-m om entum configuration .............................................................. .....................42

4-1. Honeywell roof-top configuration ..............................................................................48

4-2. H oneyw ell box configuration ........................................ .............................................48

4-3. External singular surface for box type SGCMG ACS .................................................49

4-3. H oneyw ell pyram id configuration .......................... ........................................ ............49

4-4. Pyram idal SG C M G A C S ......................................................................... ...................50

4-5. R edition of SOBEK tested ......................................... .......... ....... ............... .50

4-6. A C S w ith hardw are com ponents ............................................. ............................. 51

4-7. PhaseSpace attitude determination system ............................................ ............... 51

8









5-1. ACS simulation plots for k = 0.08 and c = 2.5k: A) Quaternion error vector elements
versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D)
Singularity param eter versus tim e. ............................................ ............................ 62

5-2. ACS simulation plots for k = 0.08 and c = 2.5k: A) Quaternion error vector elements
versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D)
Singularity m measure versus tim e. .............................................. ............................. 63

5-3. ACS simulation plots for k = 4 and c = 3k: A) Quaternion error vector elements
versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D)
Singularity m measure versus tim e. .............................................. ............................. 64

5-4. SOBEK tested GUI .............. ....................................... ............... 65

5-5. ACS experimental plots for k = 0.8 and c = 2.5k: A) Quatemion error vector
elements versus time, B) Gimbal rates versus time, C) Angular rates versus Time,
and D) Singularity measure versus time. ........................................ ....................... 66

5-6. ACS experimental plots for k = 0.24 and c = 2.0k: A) Quatemion error vector
elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and
D) Singularity m measure versus tim e ...................................................... ............... 67

5-7. ACS experimental plots for k = 0.32 and c = 3.0k: A) Quatemion error vector
elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, D)
Singularity m measure versus tim e. .............................................. ............................. 68

5-8. ACS experiment quaternion error of Moore Penrose solution ..................................69









LIST OF ABBREVIATIONS

ACS Attitude control system

API Application program interface

CMG Control moment gyroscope

CM Center of mass

DCM Direction cosine matrix

DoD Department of Defense

DOF Degree of freedom

EADS European Aeronautic Defense Space Company

EMK Exact model knowledge

GUI Graphical user interface

FACETS Flywheel Attitude Control and Energy Transmission System

IPACS Integrated Power and Attitude Control System

LEO Low Earth Orbit

MW Momentum Wheel

ORS Operationally Responsive Space

RW Reaction Wheel

R2P2 Rapid Retargeting and Precision Pointing

SGCMG Single-Gimbal Control Moment Gyroscope

SOBEK Spacecraft Orientation Buoyancy Experimental Kiosk

SSTL Surrey Space Technology Lab

UUB Uniformly Ultimately Bounded









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

DEVELOPMENT OF THE SPACECRAFT ORIENTATION BUOYANCY EXPERIMENTAL
KIOSK TESTBED
By

Frederick Aaron Leve

May 2009

Chair: Norman Fitz-Coy
Major: Aerospace Engineering

Most satellites are unique and therefore usually take 10 to 15 years to design, fabricate,

test, and finally launch. To expedite future advances in space technology, setting standards in

terms of "black-boxed" subsystems and interfaces, drastically reduces the time and long term

costs needed to complete these tasks. What is meant by "black-boxed" is a stand alone

subsystem that is adaptable enough to work with any other subsystems without alteration.

Government organizations such as the Department of Defense's (DoD's) Operationally

Responsive Space (ORS) office take these tasks into consideration and look to change this

pattern by expediting the processes of design, fabrication, test, evaluation, and launch. To

accomplish these tasks, smallsats which range from 1 to 1000 kg in mass, are considered for their

cheaper platforms and launch costs. With utilization of smallsats of the pico- (0.1-1 kg), nano-

(1 to 10 kg), and micro-classes (10-100 kg) for space missions, obstacles occur in terms of

power, mass, and volume constraints. Many of these same smallsats obstacles relate to the

attitude control system (ACS). Despite these obstacles for many space missions, small satellites

must be held to the same pointing requirements as their larger counterparts to be of use. To

compensate for these obstacles while maintaining close pointing requirements, innovations in

attitude control actuators and determination sensors that satisfy these constraints need to be

developed.









An excellent choice for attitude control actuators of smallsats are those of the flywheel

sort known as zero-momentum and momentum bias actuators. These actuators have the ability

to perform precision attitude maneuvers without use of propellant. There are two classes of

flywheel attitude control actuators that are used in satellites, reaction/momentum wheels

(RWs/MWs) and control moment gyroscopes (CMGs). Due to the scarcity of CMGs available

for small satellites below the micro-class, RWs are presently utilized. An analysis completed in

this thesis shows that although the current state-of-the-art CMGs are too massive to fit inside

smallsats, the performance relationships in terms of output torque per mass and input power are

more favorable than that for RWs. This analysis is an additional motivation for the research in

this thesis, which is carried out on the premise that an ACS consisting of single-gimbal control

moment gyroscopes (SGCMGs) can meet the performance requirements providing rapid-

retargeting and precision-pointing (R2P2) for smallsats while enduring their power, mass, and

volume constraints.

To further test this premise, the topic of this thesis is the development of a ground based

SGCMG testbed known as Spacecraft Orientation Buoyancy Experimental Kiosk (SOBEK) was

developed to validate the utility of using SGCMGs for smallsat attitude control. The thesis

justifies the reasons for using SGCMGs on SOBEK, discusses the spacecraft and actuator

dynamics and kinematics associated with an ACS containing CMGs, addresses the geometrical

configurations of systems of these actuators and the configuration chosen, and then finally

validates the testbed numerically through simulation and experimentation.









CHAPTER 1
INTRODUCTION

Research Motivation

The Department of Defense's (DOD's) Operationally Responsive Space (ORS) office [1]

seeks to better the capabilities of development, fabrication, testing, evaluation and launch of

space assets. To accomplish the goals of the ORS office, standards such as "black-boxed"

systems are considered useful to aid these tasks and reduce the uniqueness factor of every

satellite project. To accomplish some of these tasks, it is believed that switching from larger

more massive satellites to smallsats of the pico- (0.1-1 kg), nano- (1-10 kg), and micro-classes

(10-100 kg) opens up the doors for a wide variety of missions and provides a quicker and less

costly method (i.e., small satellites are cheaper and faster in terms of integration and launch) to

develop these black-boxed technologies. The budget of developing smallsats combined with

components-off-the-shelf (COTS) makes it possible for some universities and smaller aerospace

companies to contribute to the knowledge base and evolution of space technologies. Numerous

university satellite projects consisting of missions involving CubeSat and nano-satellite projects

have utilized COTS and been flight tested successfully [2-4].

Standardizations for smallsats in the academic sector are being led by California

Polytechnic Institute and Stanford University. These endeavors include CubeSat Kit CubeSat

bus architecture (Pumpkin Inc.) and P-Pod launch vehicle integration [5-7].

Work is also being done on standardization of satellite bus architecture in the military

sector such as the ANGELS, XSS-11, and the tactical satellite programs at AFRL [8] and [9].

Although smallsats are a viable choice for testing of new space technologies, they have

limited attitude control and propulsive capabilities which leaves some in the aerospace

community to regard smallsats as toys or space debris with no significant functionality. This

perception of smallsats can be challenged with the development of low power and mass attitude

13









control systems (ACSs) which are near the same pointing and slew requirements of those for

larger satellites and hence are still able to accomplish an assortment of many space missions for

smallsats where propulsive capabilities can be limited or non-existent.

Mission Examples

Some space missions that could possibly utilize smallsats are laser communication, space

science, on-orbit assembly, satellite servicing, formation flight, blue force tracking, and Earth

monitoring. Smallsats that have propulsive capabilities would most likely have a minimum

amount of fixed thrusters, therefore their attitude and translation would be coupled. Taking this

into account, for a smallsat distribution used for such missions as on-orbit assembly shown in

Fig. 1-1, precision in terms of position and orientation need to be maintained for collision

avoidance and attitude control where redundant thrusters is most likely not possible.

Earth sensing and blue force tracking which have satellite systems in Low Earth Orbit

(LEO) require tracking of a specific point on the surface of earth. This becomes nontrivial due to

smaller orbital periods (i.e., faster orbital rates). To acquire a longer time of coverage, higher

slew rates while maintaining precision are needed. An example of this is shown in Fig. 1-2.

When dealing with missions such as satellite servicing, there may be a situation when the

satellite being serviced is non-cooperative and is more massive than the satellite which is

providing the servicing. In this situation the ACS on board the servicing satellite must be able to

provide large amounts of torque responsively to stabilize the system. An illustration of this is

shown in Fig. 1-3.

Research Focus

The research focus of this thesis is in the development of a ground based testbed to test and

evaluate propellantless ACS methodology for smallsats which can endure the volume, mass, and

power constraints while still maintaining performance in terms of high torque, slew rates, and









precision. The ACS methodology presented in this thesis attempts to validate single-gimbal

control moment gyroscopes (SGCMGs) as a viable option of attitude control actuators for

smallsats which can contribute high torque, slew rates, and attitude precision while meeting their

volume, mass, and power constraints. Examples of other attitude actuators including SGCMGs

are discussed in the next chapter for comparison.




0 Z I ,/I


Fig. 1-1.









Fig. 1-2.


On-orbit assembly


Earth monitoring in LEO


Post-docking stabilization


Fig. 1-3.









CHAPTER 2
ATTITUDE CONTROL ACTUATORS

Types of Attitude Actuators

Attitude control actuators for satellites can be passive or active. Passive attitude control

actuators such as gravity gradient booms use the gravitational field of the Earth to provide a

bounded attitude between 0, = +5 in the orbital plane. Because the gravitational field is

conservative, gravity gradient torques ,_, do not dissipate the satellites kinetic energy or dump

its angular momentum. The torque is dependent on the distance R between the spacecraft's

center of mass (CM) and Earth's CM, the spacecraft principle moments of inertias J,, and the

small angles of deviation between the spacecraft's principle axes and the Local vertical-Local

horizon coordinate frame 0,. For small angular displacements, the torque is expressed as,

(J2 J3 )81
rg = 3 -'3( J3 J1 02


Another type of passive attitude control utilizes hysteresis rods or permanent magnets.

Typically permanent magnets carry accuracy on the same order of magnitude as gravity gradient

booms but cause a satellite to flip at the poles. These actuators like gravity booms are

conservative and do not dissipate energy or dump angular momentum. Hysteresis rods on the

other hand are passive actuators that may dissipate energy and dump angular momentum. These

actuators can be used to null the angular rates of satellites with residual angular momentum after

launch [10].

There are three main categories of active attitude control actuators. These categories

include magnet torquers, reaction control devices, and flywheel actuators.

Magnet torquers are the most common attitude control actuators utilized in smallsats (e.g.,

magnet coils and torque rods). These actuators are cheaper and lighter than other attitude control
16









actuators used onboard smallsats. Figure 2-1 shows an example of a magnet coil for attitude

control.

Magnetic torquers produce a torque zg as a reaction to the change of the local magnetic

field of the satellite to that of the Earth. This torque is modeled as an interaction (cross product)

between, the magnetic moment of the spacecraft / and Earth's magnetic field direction B as,


r- =,uxB

Since the torque is a function of the cross product between the satellite's magnetic moment

and Earth's magnetic field, magnet torquers suffer from a singularity which limits their actuation

where no components of torque are available along / and B Due to the change of the local

magnetic field of the satellite, sensors for attitude determination such as magnetometers are

unable to take readings during use of these actuators until the residual magnetic field of the

satellite has decayed. These actuators loose performance as a function of distance squared hence

are less useful for satellites at orbits with higher altitudes than LEO.

Reaction control devices such as thrusters require propellant which is at this time not

available for the majority of nano- and pico-satellite missions. These devices produce a torque

on the satellite via propulsion through jets of fluid, heated or cooled plasma and gas, or charged

ions with use of a magnetic field. Thrusters that utilize the chemical energy within the propellant

(e.g., hydrazine) produce greater magnitude of output thrust but require large radiators to dump

the expended heat created. Thrusters of this sort are not realizable for smallsats which do not

have the extra mass and volume to contain large radiators for the thermal protection system.

The last type of attitude control actuators known as flywheel actuators can be broken down

into two types, reaction and momentum wheels (RWs/MWs) and control moment gyroscopes

(CMGs). These are the most accurate sources of attitude control.









Reaction and momentum wheels work off the same principle by producing a reaction

torque opposite in direction to the acceleration of a flywheel. The difference between MWs and

RWs is that, RWs have a zero nominal operating speed (i.e., the speed that the wheels are spun

back down to after the maneuver is completed). When spinning RWs/MWs back down to their

operating speed, the satellite will revert to its original orientation due to the conservation of

angular momentum. For this reason both MWs and RWs must have other actuators onboard

such as thrusters or magnet torquers to dump the excess angular momentum, or these actuators

will suffer angular momentum saturation of their flywheels. Three RWs/MWs are needed for

full three-axis attitude control of a satellite. An example of a RW used for CubeSat and nano-

satellite missions is shown below in Fig. 2-2.

The second type of flywheel attitude control actuator known as CMGs provide an

instantaneous gyroscopic torque along a torque axis defined as the axis perpendicular to the

flywheel and gimbal axes. These actuators rely on shifting the direction of satellites body

angular momentum to apply a torque. Three CMGs are needed for three-axis control in general,

although a fourth is typically added for singularity avoidance and/or redundancy.

Performance characteristics for typical attitude control actuators were investigated and are

shown in Tables 2-1 and Tables A-i through A-3 in the appendix. Table 2-1 is taken from Space

Mission Analysis and Design by Wertz [11]. The tables in Appendix A are for currently

available RWs, CMGs, and magnet torquers and supplement that which is shown in Table 2-1

[12-23].

In Table 2-1, the only attitude control actuators that can meet the performance demands of

high precision while maintaining the low power requirement for small satellite systems are

flywheel actuators consisting of RWs, MWs, and CMGs. Magnet torquers/coils do not produce









sufficient output torque for a given input power and are less accurate due to the uncertainty of

Earth's dynamic magnetic field model.

A survey was conducted of existing RWs and CMGs. The results of the survey are

shown in Figs. 2-3 and Fig. 2-4. The supporting data for these figures is provided by Tables A-i

and A-2

Figures 2-3 and 2-4 show that CMGs including that which is developed in the Spacecraft

Orientation Buoyancy Experimental Kiosk (SOBEK) have a higher torque output for a given

mass and input power than RWs. These figures also illustrate that CMGs are currently

nonexistent for classes of satellites equal to or smaller than micro-satellites. The scarcity of

these actuators is due to their complexity in design and use. If this complexity was removed and

they were designed in a "black-boxed" manner (i.e., accepting only an input torque or attitude

and producing it), then they could potentially find more use in smaller satellites due to higher

torque per mass and power advantage over RWs.

There are three CMGs that have been developed for micro-satellites by companies such as

EADS/Astrium, Honeywell, and Surrey Space Technology Laboratory (SSTL) [24-28] Fig. 2-5

shows one of these three, the SSTL single-gimbal control momentum gyroscope (SGCMG)

which flew on the Turkish micro-satellite Bilsat-1 [29].

Types of CMGs

There are two varieties of CMGs: ones with a singe controllable degree of freedom (DOF)

known as SGCMGs and those with multiple controllable DOF known as double-gimbal control

momentum gyroscopes (DGCMGs) and variable-speed control moment gyroscopes (VSCMGs).

The benefits and drawbacks of each type of CMG is discussed next.









Single Gimbal Control Momentum Gyroscopes

SGCMGs such as the ones shown in Fig. 2-5 are the least mechanically complex form of

CMGs. They utilize a single gimbal axis to produce a gyroscopic torque by rotating a constant

speed flywheel about a gimbaled axis perpendicular to the spin-axis. A minimum of three

SGCMGs are needed to obtain three-axis attitude control with these actuators. However an ACS

using SGCMGs in some cases utilizes four of these actuators to avoid performance inefficiencies

known as internal singularities. The benefit of these actuators among other CMGs lies in the

property of torque amplification which is the ratio of output torque from the ACS containing

SGCMGs to input torque from the gimbal motors. Discussion of singularities as well as

performance characterization of SGCMGs in terms of momentum management and torque

amplification is discussed in chapter 3.

Double Gimbal Control Momentum Gyroscopes

Double-gimbal control momentum gyroscopes (DGCMG) are the most mechanically

complex form of CMG actuators. These actuators produce torque by the same method as

SGCMGs but possess a redundant mechanical controllable DOF in terms of an extra gimbal axis.

This redundancy can be useful when a set of two actuators is used for underactuated attitude

control of a satellite but can be harmful when gimbal lock occurs. Gimbal lock occurs when the

spin axis of a DGCMG aligns with its axis of freedom and in consequence deprives the DGCMG

of one of its controllable DOF thereby eliminating its useful properties. There are however

acceptable methods of avoiding these singularities which in turn leaves DGCMGs the benefit of

multiple degrees of freedom. Although with the addition of an extra gimbal to these actuators,

they are heavier and more mechanically complex than other forms of CMGs which excludes

them from being a viable option for smallsats. A figure of a DGCMG developed by L3

Communications for the International Space Station shown in Fig. 2-6.









Variable Speed Control Momentum Gyroscopes

VSCMGs are used for their extra DOF added through their reaction wheel mode where

flywheel accelerations are nonzero. This extra DOF has its benefits in that it does not require

steering logic and is able to produce the required torque when a singularity is encountered [30-

32]. Although when in reaction wheel mode the flywheel accelerations require shaft power and

therefore input torque thereby increasing their required power and reducing their available torque

amplification. Another effect of the reaction wheel modes of VSCMGs is that it becomes near

impossible to isolate vibrations from the varying flywheel speeds. In addition to the larger

power requirement, less torque amplification, and induced vibration, the motor driver circuitry of

VSCMGs is more complicated, and they require more computational and electrical power than

standard SGCMGs to control.

SGCMGs are considered here as the most viable option for the study of smallsat attitude

control and hence were chosen for the SOBEK testbed.

Table 2-1. Attitude control actuators and their specifications
Actuator Output Accuracy Mass (Kg) Power (W)
(deg)
Magnetic Torquers 1 to 4,000 Am2 1 to 5 0.4 to 50 0.6 to 16
Hot Gas Thrusters 0.5 to 9,000 Nm t 5 v
1 to 5 variable N/A
Cold Gas Thrusters <5Nm

RW & MW 0.005 to 1 Nm 0.001 to 1 1 to 20 10 to 110
CMG 10 to 500 Nm 0.001 to 1 >1 5 to 150












Fig. 2-1. Magnet coil from Tokyo Institute of Technology Cute-1.7 CubeSat

21





















Dynacon miniature RW

40tA .n,


120 4
100
80
60 .
40
20
0 ^ -


* vV
MVV
r, RA_
SC:-E-El/


logio(torque) mNm

Semi-log plot of input power versus output torque for off-the-shelf RWs and
CMGs


140
120
100
80
60
40
20
-- 9 0


U


U
U

U
U


* RV
* CMc


loglo(torque) mNm

Semi-log plot of mass versus output torque for off-the-shelf RWs and CMGs


Fig. 2-2.


Fig. 2-3.


Fig. 2-4.





















Fig. 2-5. SSTL micro-satellite SGCMG flown on Bilsat-1














Fig. 2-6. DGCMG developed by L3 Communications









CHAPTER 3
SGCMG DYNAMICS AND PERFORMANCE CHARACTERIZATION

Dynamics

CMGs produce a gyroscopic torque about a torque axis ( orthogonal to both the gimbal

and spin axes g, and ,, respectively. An illustration of the CMG gimbal frame 3 and how it

relates to the spacecraft body frame 3, is shown in Fig. 3-1 with gimbal angles, velocities and

accelerations, ,, and flywheel velocities and accelerations, Q_, .

To analyze the torque generated by CMGs, we first develop the angular momentum

expression and coordinatize it in the spacecraft body fixed frame. Application of Euler's

equation assuming the external torque is zero yields the governing equation for the CMG system.

This development is as follows. The total spacecraft-centroidal angular momentum of the

system coordinatized in the body frame is


BH-C = B c B ( + "hC (3-1)

with the spacecraft centroidal inertia tensor including the CMG components Bj the spacecraft


body fixed angular velocity BW)B, and the total CMG angular momentum Bhe about the

spacecraft center of mass (CM) coordinatized in the body frame.

The spacecraft centroidal inertia BJc in Eq. (3-2) contains a fixed inertia BJ time

n
varying inertia from the combined gimbal-wheel system CB G T B, and the parallel axis
1=1

components of inertia associated with each CMG m, (&RcT R1 B B~) Rc, with mass m,


and position from the spacecraft CM to that of the flywheel, BR .



Bc Bj + L[CB C +m, i( ,T& 1B BRc T) (3-2)

24=1
24









Assuming that BRc is held fairly constant and therefore its time derivative is zero, we


differentiate BJC BCB with respect to time as,

d(BC BOB) aBJ BB d, (
= + J_ GC = A, + -JC G (3-3)
dt d3 dt

This CMG contribution of angular momentum B h coordinatized in the body frame is

expressed as,

4
Bh =Y (3-4)
h = -BG, G h (3-4)
i=1

and is found in by summing the contributions of each individual CMG angular moment 'h ,

in S which is transformed to the body frame by the DCM CBG Note that for SGCMGs, the

DCM CBG varies only with the gimbal angles of the ith CMG S,.


The angular momentum of the ith CMG G'h in G is,



G,h 0L
G hr,= 0


where 0, and I, are the ith flywheel's angular speed and centroidal inertia about its spin axis ,

and t, and Ig, are the ith gimbal rate and the centroidal inertia of the combined wheel-gimbal

system about its gimbal axis g,.


Taking the time derivative of Bhc and observing that "~h = B h (3,, ,), we get,


dBhC O8h dt8 9Bh d8 98h dB M
= +- + (3-5)
dt 98 dt 85 dt MQ dt
with the Jacobian coefficient matrices of the CMG states are defined as,









a"h
= A2 (3-7)
98 h
a dB

=B (3-8)

ahC
= C (3-9)
aQ =
Combining these terms we get,


d B h
d B A2 +B3+CO
dt =2- =-
If we also combine A, with A2 which are both multiplied by the gimbal rates d, we have a

more compact representation of the output torque from the CMGs in Eq. (3-10) where the

complete Jacobian matrix is D = [(A + A2) B C.



com+ (3e
S=A,- + B =(A+A) B C S =D( 5,,)X (3-10)
dt dt -

The form of Eq. (3-10) is similar to Schaub and Junkins [33] with exception of the different

notation.

The gyroscopic components of torque are derived from the product of the skew symmetric

matrix I B, ] with the total centroidal angular momentum of the system 5 cH as,


[B B]X BHc (BB)X Bj BWB +[BB]X Bh +h

(3-11)
[B] Cx + B Rc B&rI B B& T)-Rc B RTB
i=l
The complete rotational equation of motion for this system in Eq. (3-12) is found through

the sum of Eqs. (3-3), (3-10), and (3-11) giving,









dBH
d B C B o B + B +B 0) B ] J o B)B + [ BB] BD + +
dt BBB-B B[ B

ZlCBG, CG, + m, (BRC, Bk Bkc Bkc, )] + (3-12)

S J 2 G, BG _C _c
=1

coxI B-B CBG G CTG 1 (BkT Bk 1_Bk BkT)J BB

If it is assumed that,

BJC BJ

* Ig << I < ,

* 1 < I,, and Ig, <<,I, ,

* Q, 0

and there are no external torques applied to the spacecraft the reduced equation of motion for this

system is

diH -JE Bs B + ) B]x BJs BcoB hoAd+ [coB x B = 0 (3-13)
dt I -
with h0 = I,, and B = h0Ad .

A 3-2-3 DCM through Euler angles ,, 0,, 8, where and 8, are the constant spacing

and inclination angles, and 8, is the time varying gimbal angle transforms 3, to 3, for the

four-CMG pyramid configuration in Fig. 3-1. This DCM is expanded as,


~c(8,)c()s(O,)-s(6,)s(O) -c(9,)s(O)-s(9,)c(O)s(0,) -c( )c(O,)
CBG,= S(S,)C()+C(S,)S ,)S( ,) c(S,)cg ()-sS(,)s( )s(O,) -s )c(o,)
s(8)+c(c,)c(O,) c(8,)-s(8)c(O) c (0,)
where c(*),s(*) = cos(*),sin ()
The 3-2-3 Euler sequence can be visualized in Fig. 3-2.









Now that we have reduced the general equations of motion to an easily useable form we

can address the performance characterization of SGCMGs. There are three areas that describe

the performance characteristics of SGCMGs: torque amplification, singularities, and momentum

management. Each one of these areas is discussed next.

Torque Amplification

Torque amplification for a SGCMG is defined as a higher amount of output torque

produced than a given gimbal motor input torque required, assuming negligible torque needed by

the flywheel motors. This is a reasonable assumption when considering SGCMGs because their

flywheels require minimum torque to spin at a constant speed. In the literature (e.g., [34]),

torque amplification is defined for a single actuator as,



Torque Amplification 2- 2 (3-15)
2 BB ]X Bh 0)


If it is assumed that there are no disturbance torques on the system, the equation for

internal torque r can be found by decomposing Eq. (3-13) of a single SGCMG actuator into two

separate elements with equal magnitudes and opposite signs as,

Bj BO) B _B]X x B)B B


whereBL =[X BhC


The internal torque can be decomposed to into an output torque t = [Bh ]X x from the


CMG and input torque = BB ]X B ~ which is the gyroscopic contribution from the

satellite's angular velocity in Eq. (3-13). If the two-norm of newly defined r, is divided by that

of L, we get the same form of the torque amplification equation shown in Eq. (3-15).

28









When considering pico- or nano-class satellites where the ACS might take up a majority of

the mass and volume of the overall satellite and the gimbal-wheel system mass may make up a

majority of the mass and volume of the ACS, the contributions to torque, and angular momentum

from the gimbal-wheel system and the gimbal motor friction inefficiencies may not be assumed

negligible. Therefore, Eq. (3-15) is an upper bound for the torque amplification equation.

Useful information is gathered from Eq. (3-15) such as the insight that the spacecraft angular

velocity has to be less than the gimbal rate to have a torque amplification greater then one. To

understand torque amplification of a single SGCMG for use on smallsats a lower bound was

developed [35] and [36]. This lower bound is


Irot 211 h0,
JTt2-- (3-16)
S2 [B w +h + Fi +F, sgn()

with gimbal input torques I and motor friction inefficiencies with dynamic and static

coefficients F andF,.

If the motor friction inefficiencies are of the main concern we get a reduced form of Eq.

(3-16) as,



2 B 1 +F sgnO) +
BO + h
n2 B -


and more compactly as,


,(3-17)
^l sgn( ) Fd w
O+) + d 1
h h

where torque from static friction is assumed to be negligibly small when compared to that for

dynamic friction.









Equation (3-17), which considers the mechanical inefficiencies of the gimbal motors, is a

lower bound in the presence of these motor friction inefficiencies on the torque amplification of

an SGCMG. From this equation, it can be seen that choosing the correct gimbal motor to make

h
the optimization parameter f = as high as possible, will reduce the motor friction effects on
Fd

the SGCMG torque amplification. It should be noted that increasing the flywheel speed as much

as possible will give the largest torque amplification for small satellites where mass, volume and

power are limited but will also increase their jitter. The motor inefficiencies of the DC motors

used to rotate the flywheels are overlooked for SGCMGs that are not variable speed, due to the

fact that they have a very small dynamic friction coefficient and their static friction and residual

torques are largest on startup. Now that torque amplification has been discussed, singularities

associated with SGCMGs are addressed.

Singularities

SGCMGs experience two classes of singularities. These singularities are internal and

external singularities each of which are instances where the required control torque cannot be

produced.

External Singularities

External singularities occur when the addition of angular momentum from an external

source (e.g., disturbances) saturates the SGCMGs of the ACS. Of these disturbance torques,

aerodynamic torques are considered to be the most harmful when in LEO due to their size and

constant nature in comparison with other disturbance torques. External singularities are usually

addressed in mission planning and considered the most harmful at the end of a satellites life

cycle when de-orbiting.









Internal Singularities

Internal singularities are encountered on the fly when the Jacobian matrix of the SGCMG

ACS becomes rank deficient, whereby the torque vectors lie in a plane and no torque can be

produced normal to this plane. Unlike external singularities they must be handled online and

cannot simply be designed for a priori. The two types of internal singularities are known as

elliptic which are unavoidable/inescapable through null motion (i.e., null motion is motion of the

gimbals that produces no net torque on the satellite) and hyperbolic which are avoidable through

null motion. Singularity avoidance strategies are discussed later in this chapter

Singular Surfaces

The CMG torque vector directions f are the columns of the Jacobian matrix. Therefore,

singularity occurs when all i lie in the same plane. There are 2n singular configurations (i.e.,

singular sets of gimbal angles, 3) for any singular direction u normal to this plane. All possible

for each CMG span a plane and an example of these planes for a four-CMG pyramid

configuration can be visualized clearly in Fig. 3-3. The CMG torque vector direction is defined

as,


= torque vector =gk x


With a given singular direction u not parallel to a gimbal axis, there are two possibilities

for singularity to occur along each singular direction. These possibilities are shown in shown

below.


u_. =O, u-j >Ooru =0, u_ <0

It should be noted that the occurrence of singularity when two gimbal axes are aligned has

already addressed as gimbal lock and occurs in DGCMGs.









Using the notation found in reference [35] and [32], we can define E, = sign (u ) With

these definitions, the spin and torque axis vectors at singular states can be defined as,


--= (g X")X ,--, U g-

and




Therefore the total normalized angular momentum Bh of the CMGs at singular states

corresponding to u is,



Rh =E= --, X )x-- u g


When E, = +1 (or = -1 due to symmetry), the external singular surface known as the

angular momentum envelope shown in Fig. 3-4 is formed as the locus of the maximum

projections of the angular momentum of each CMG at the singular directions. External

singularities occur on the surface of this envelope when the total internal SGCMG angular

momentum is less than that which is external to the spacecraft, and the SGCMGs are then

saturated.

The internal singular surface is formed in the same way as that of the external singular

surface with exception to one E, = -1. An examples of the internal singular surfaces for an ACS

containing four SGCMGs in a pyramidal arrangement at an inclination angle 0 = 35.26 is

shown in Figs. 3-5. The inclination angle should not be confused with the skew angle / used to

characterize the angular momentum envelope [37] and [38]. The skew angle/? is the minimum

angle of the SGCMG planes shown in Fig 3-3 with that of the horizontal. For a spherical angular









momentum envelope, / = 90' 6 = 54.74'. Units of Fig. 3-4 and Fig. 3-5 are in terms of the

nominal angular momentum magnitude h0 (i.e. diameter of angular momentum envelope

involving four SGCMGs is less than or equal to four h0). In Fig. 3-4, locations that have white

circles or holes on the momentum envelope are shown where the gimbal axes are located and

there is no angular momentum available.

The knowledge of where these singularities occur is known, and there have been methods

developed that steer away or escape from these singularities known as steering logics. These

methods are discussed in the next section.

Steering Logics

An ACS utilizing SGCMGs requires an addition of a fourth actuator for minimum

redundancy to avoid singularities through null motion (i.e., motion is motion of the gimbals

whereby no net torque is produced). This fourth actuator renders the system's Jacobian matrix to

be non-square which in turn requires a pseudo-inverse to map the output torque B, c R3 onto

the gimbal rates 3 e R4 Recall, that there are 2n singular configurations for each singular

direction of n SGCMGs. Therefore, it is important to note that the addition of actuators does not

eliminate the problem of singularities but provides a null space where singularity avoidance may

be provided.

The Moore-Penrose pseudo-inverse solution to the gimbal rates is represented as

-1
3 =-A BA C (3-18)
h,-
where the Moore-Penrose pseudo-inverse is,

A+ =AT(AAT









The Moore-Penrose pseudo-inverse solution for the gimbal rates in Eq. (3-18) fails when AA

becomes singular. To avoid/escape singularities when mapping the output torque onto the

gimbal rates, a steering logic is applied.

Steering logics can be broken up into two main groups: those which provide null motion

for singularity avoidance such as local gradient methods and global avoidance algorithms, and

pseudo-inverse solutions which escape singularities through loading of the Jacobian matrix

singular values. Common drawbacks to these steering logics are computational complexity,

inability to avoid all internal singularities, or loss accuracy by adding torque errors to keep the

Jacobian non-singular with loading of its singular values.

Local gradient methods were of the earliest methods used to steer away from internal

SGCMG singularities by Margulies and Auburn [39]. This local gradient method, like most

steering logics, requires a fourth SGCMG added to provide a 1-dimensional null space where

null motion can be applied to steer the gimbals away from singularities. The Jacobian matrix

null space provides multiple solutions to the gimbal state equation to steer the gimbals away

from singularity while maneuvering. An example of local gradient methods is,


k1
S= 1-(A+ B~ [ -A +AAd) (3-19)


where B A is the SGCMG output torque and h, is the nominal magnitude of angular

momentum. This equation is similar to Eq. (3-18) with the addition of a null motion vector d

which is mapped to the Jacobian matrix null space by the projection matrix II A+A. The

contribution of null motion is scaled by the singularity parameter


S= y0 exp(-/ m2) (3-20)

which is an explicit function of the singularity measure m,










m= det(AA) (3-21)

and design constants p and yo.



When using local gradient methods, the null motion vector d can be chosen as the

gradient,

T
d: {8f 8f Of /fY
d=\- -,--
as- 0a2 3 'a 4

of a objective function

f=
m
This null vector d produces null motion in the direction that maximizes m or the distance away

from singularity.

It is important to note that steering logics utilizing only null motion are unable to avoid

elliptic internal singularities [40] and [41]. Local gradient methods shown here are

computationally intensive and cannot avoid all internal singularities such as elliptic internal

singularities where null motion is unsuccessful.

Global avoidance methods shown in Eq. (3-22) such as those developed by (Paradiso and

Kuhns [42] and [43]) are similar to local gradient methods and differ in the fact that the null

motion vector is produced by the difference of the gimbal angle positions from a set of alternate

gimbal angle configurations known as preferred trajectories. This difference is shown in Eq. (3-

22), where 5* are the preferred trajectories.


= (A+ B +7[- A+A](-*)) (3-22)
ho0









These preferred trajectories are usually calculated offline. This method is computationally

intensive and is not suitable for online use.

The most popular pseudo inverse solution is the SR inverse developed first by

(Nakamura and Hanafusa [44]), modified later by (Wie [45], and Ford [46]). The SR inverse


A" =A (A AA + y) (3-23)

produces the solution to the gimbal rates


3= A#Bhc (3-24)

Because the A# always exists, the maneuver can start out near singularity where local

gradient and global avoidance methods are unable.

This SR-inverse solution works by adding a positive definite matrix of torque errors y!

to the positive semi-definite matrix AAT to leave the matrix(AAT + y7) positive definite and

hence non-singular. Rather than avoiding internal singularities as discussed previously, this

method which is developed from the Moore-Penrose pseudo-inverse, approaches singularity and

makes a rapid transition to escape. At escape there is a jump in gimbal rates due to the added

torque error. As a consequence of these added torque errors, this method is not the preferred

method for precision pointing. Under certain circumstances (i.e., when det AA ) = 0


and B e null(A#)), this method can become locked into a singularity, so the generalized SR-

inverse (GSR) was developed by Wie [45] .

The GSR-inverse


A# = A (AA yE)1 (3-25)

is made up a positive definite symmetric matrix









1 0o sin(03t+ 3) 0 sin (2t +2)
E_= o sin (3t +3 ) 1 0 sin (0t + ) >0 (3-26)
S~ sin (2t+2) 2 0 sin(w1lt+ 0) 1
composed of time varying modular parameters E, and scaled by the singularity parameter y in

Eq. (3-20). These modulating parameters ensure that B will not stay locked in the null space

of the pseudo-inverse, which is not explicitly dependent on time.

The GSR-inverse has been shown to avoid all internal singularities but adds torque error to

all directions. Therefore, the original SR-inverse was modified again to minimize the amount of

torque error added to the gimbals when avoiding internal singularities. This method developed

by Ford is known as Singular Direction Avoidance (SDA) with the pseudo-inverse

ADA = V StAU and from which the gimbal rates are


3= DA B (3-43)
ho=-

where V and U are unitary matrices and SDA is,


1 1 3
diag -,-, --
SIDA 1 02 3
OT

The benefit of using this method over the SR-inverse methods is that it only adds torque error to

the smallest singular value, which in turn requires less null motion and has smoother gimbal rate

trajectories than the previous SR and GSR-inverse methods. A drawback of this method is that it

is required that a SVD be calculated forA every time step, making the steering logic

computationally intensive. Also it is developed directly from the original SR-inverse and

therefore can become locked in singularity when the control torque is in the singular direction of

the pseudo-inverse.









The discussed methods of avoiding or escaping singularities are of the most popular of

steering logics developed for SGCMGs [47]. Additional methods have been developed to avoid

internal singularities associated with SGCMGs [42-44]. An example of one such method

includes game theory [48]. Other methods for singularity avoidance have been developed with

the addition of an extra degree of freedom while using VSCMGs instead of SGCMGs.

In the case of VSCMG, steering logics associated with gimbal velocity and acceleration

have been developed by Schaub and Junkins [31] and [49]. These methods use the extra degree

of freedom through flywheel acceleration to give an ACS using these actuators the ability to

always produce the required output torque needed.

Recall it was stated that VSCMGs do not require steering logics due to their reaction

wheels modes. Therefore the addition of a fourth VSCMG instead may provide power tracking

for systems such as the Flywheel Attitude Control and Energy Transmission System (FACETS)

and the Integrated Power and Attitude Control System (IPACS) [50]. These systems are able to

use the null space instead to monitor power transmission while converting the extra kinetic

energy gathered from spinning down the flywheels after a maneuver to electricity charging the

power system on board. Systems such as FACETS require ceramic wheels to make use of the

kinetic energy provided from spinning down the flywheels and were deemed unsafe due to the

high flywheel speeds of forty-thousand rpm and have henceforth not been flight tested.

The next section discusses momentum management which deals with the dumping of

external and internal angular momentum in order to maintain the performance of an ACS using

SGCMGs.

Momentum Management

There are two situations where the angular momentum of the system must be managed in

order to maintain the required ACS performance. These situations occur from external angular









momentum added into the system from torque disturbances or internal components of angular

momentum remaining post maneuver from gimbal angle saturation.

External Angular Momentum Offloading

External angular momentum accumulated by the spacecraft from non-zero disturbance

torques is considered a priori for each specific mission. If angular momentum is added into the

system from the disturbance torque, and it is greater than that from which is allotted from the

CMGs, the system encounters an external singularity as discussed in section 3.2. Assuming that

a spacecraft is in LEO orbit then the addition of aerodynamic torque is the only source that is

considered to saturate the system. This is so because the magnetic disturbance torque is

exceptionally small and the disturbance torque from the gravity gradient does not saturate the

system with angular momentum due to its periodic nature. Also disturbance torques from solar

pressure effects are neglected due to the small amount of solar cell surface area on small

satellites. When considering small satellites, aerodynamic torques add a negligible rate of

angular momentum into the system in most standard LEO orbits [36]. The addition of angular

momentum from disturbance torques is a slower process than that for the internal angular

momentum buildup from the SGCMGs. For this reason there is greater concern for offloading of

internal angular momentum.

Internal Angular Momentum Offloading

Internal angular momentum accumulates when components of angular momentum from

the SGCMG cannot be cancelled out due to gimbal angle saturation. As previously mentioned

these gimbal angle constraints can be reduced by using magnetic bearings and/or slip rings which

allow for full range of gimbal rotation. Two cases arise when internal angular momentum needs

to be offloaded. The first case occurs when excess components of internal angular momentum is

left over from maneuvers due to gimbal angle saturation where unwanted components of angular









momentum are unable to be cancelled out. The second case is at startup of the flywheels, where

needs to be angular momentum dumped to stop the satellite from tumbling due to the offset in its

angular momentum direction. A typical startup configuration of an ACS containing SGCMGs is

at zero-momentum. This configuration for a four-pyramid SGCMG cluster has all of the angular

momentum vectors of each SGCMG in the body x-y plane at a 3 = 0 leaving a zero net

momentum for the ACS. An example of this configuration is shown in Fig. 3-6. In this figure,

all of the angular momentum vectors are in the plane although it is not a singular configuration

because all of the torque vectors are not in the same plane. For smallsats the offloading of the

excess angular momentum on startup as well as that due to gimbal angle saturation can be

accomplished by use of magnetic actuators.

Now that mathematical and physical aspects of ACSs containing SGCMGs have been

discussed, the types of ACS configurations containing these actuators as well is the development

and choice of the SOBEK configuration is addressed in the next chapter.









Fh








Fig. 3-1. The SGCMG with gimbal coordinate axes


































A 3-2-3 rotation sequence through angles 0, ,, 6,


&4~ir~'l~ \I


Fig. 3-3.


Intersecting planes spanning of SGCMG torque directions


3-

2-

1 -

0i~


-2-

.3
.5
5 -4


Angular momentum envelope for 0


Fig. 3-2.


Fig. 3-4.


35.26 and E


[1,1,1,1]














5 i .

-1.5-T-2-
-2
2 = 35. 2 2
Internal singular surface 0 = 35.26" and _s= [1,1,1, 1]


t .
ft2


II L W

It Q34


Zero-momentum configuration


Fig. 3-5.


Fig. 3-6.


JA


lA-.









CHAPTER 4
SGCMG ACS CONFIGURATIONS

Rooftop Configuration

The rooftop design uses two groups of SGCMGs, each with parallel gimbal axes slanted at

an angle to each other shown in Fig. 4-1. A CMG system in this configuration will never have

elliptic internal singularities [40]. This means that they will always have continuous momentum

trajectories and therefore have continuous gimbal trajectories. This configuration suffers from

situations where its Jacobian may approach rank 1. The singular direction where this occurs is on

the intersection of the two rows of SGCMGs shown as the red arrow in Fig. 4-1.

Box Configuration

The box configuration is a variation of the rooftop configuration containing four SGCMGs

with an inclination angle of 0 = 90. This configuration is shown in Fig. 4-2 where the red

arrow indicates the same rank 1 direction where the torque vectors lie.

The internal singular surface associated with this configuration covers a large surface of

the angular momentum envelope and is seen as the large empty area present inside the external

singular surface in Fig. 4-3. The singularity represented by this singular surface shown at Ih is

avoided in the box configuration by utilizing only three SGCMG and keeping the third as a spare

while staying within the Ih constraint. Only three of the four SGCMG will be used at any given

time so the Jacobian is square and there is no need for a pseudo-inverse to map the gimbal rates

onto the torque. The major drawbacks of this design are that it is not as compact as other designs

and its performance is constrained to Ih of angular momentum, causing the ACS to require

larger SGCMGs to meet the missions angular momentum requirements.

Pyramid Configuration

Common pyramid configurations have numbers of SGCMGs in groups of four or six.

ACSs containing these amounts of SGCMGs and have both elliptic and hyperbolic internal

43








singularities. Although an ACS utilizing SGCMGs in a pyramid configuration is susceptible to

the elliptic internal singularities, there are methods of escaping these singularities that were

discussed in the previous chapter. This configuration has the benefit of giving a somewhat

spherical angular momentum envelope for the right choice of inclination angle of the SGCMGs.

The SOBEK testbed will utilize a minimal redundancy (i.e., four SGCMGs) pyramid

configuration for three-axis control while applying the GSR-inverse steering logic method

discussed in chapter 3.

SOBEK Pyramid Configuration
The SOBEK testbed's ACS is shown in Fig. 4-4 with motion capture for attitude tracking

provided by Phase Space and its visual interpretation of the mapped wireless LEDs in the upper

right hand corer. SOBEK's ACS uses four SGCMGs in a pyramid configuration. The testbed

was originally designed to float on an air bearing table with multiple ACSs where the footprint of

the ACS was limited while trying to minimize weight. For this reason, the inclination was

chosen as angle 0 = 54.74'. It should also be mentioned that the formulations in literature

consider a zero reference gimbal angle position shifted 90 from where the SOBEK zero

reference angle is. The new formulated SGCMG angular momentum vector

S s0S()c(81)~ s(2) 2 s(/ c(,3) S(c 4)
h,= hk -s(5i) + -s(0)c(52) + s(53) + s(0)c(54) (4-1)
S-cC (0) c() -cL C(0c5(2) -c(9)c(53) L c(9)c(54),
and Jacobian matrix


Ls(0)s(8) c(2) -s(0)s((3) -c(4)
A=- -c (() s (0)s (2) c (3) -s(0)s () (4-2)
Sc()s(gl) c(0)s(2) c (0)s(,3) c(0)s(34)
takes these changes into account.









Other testbeds such as those at Virginia Polytechnical Institute, Lawrence Livermore

Laboratory, SSTL, and Honeywell's Line of Site (LOS) testbed are of the few testbeds that

comprise the known heritage for this research [51-53]. This testbed differs from the ones

previously referenced in that it is physically smaller and does not utilize an air bearing but a

pivot point to mitigate the forces of gravity. When making the testbed near marginally stable at a

point, the only forces and torques considered are the small amount of friction from that point as

well as the residual moment from gravity. An illustration of SOBEK that contains the ACS in

and stand which holds the ACS on a pivot point is shown in Fig. 4-5. This illustration was made

in ProEngineer, where the SOBEK inertia and mass properties were also estimated.

The next section discusses the hardware chosen for the SOBEK and its design shown in

Fig. 4-6 and [54].

SOBEK Hardware

The basic electrical hardware for preliminary setup of the SOBEK ACS consists of four

Arsape AM2224 two-phase Stepper motors, four Faulhaber 1525 BRC brushless DC motors with

integrated electronics and four State Electronics 600 Series digital optical encoders. The

electrical hardware for the current setup only contains a wireless Phasespace LED controller, a

Roving Networks RS232 bluetooth module, and the EZHR17EN stepper driver boards with an

RS232 to RS485 adapter. The entire ACS is powered by two lithium polymer batteries. This is

possible because there are voltage regulators present to step down the voltage from 14 to 12 volts

and 5 volts to meet the specific voltages of the mechanical hardware.

Currently the setup has no onboard processing and runs code written in C++ from a

Panasonic Toughbook with a Linux operating system. The future setup of this testbed will have

everything coded on a single processor board containing the wireless Bluetooth module and

motor driver circuitry.









There are significant reasons for the choice of the hardware for the testbed. One of these

reasons deals with the digital optical encoders. These encoders happen to be more accurate then

potentiometers and are dual quadrature which are directly compatible with the Allmotion driver

boards chosen. The Allmotion EZHR17EN stepper driver boards have the required output

current and voltage necessary for the given Arsape AM2224 stepper motors and so were chosen

for this reason. In addition to having the correct current and voltages, these boards also have the

ability to daisy chain to other boards. That is for a given set of four motor driver boards

controlling a total of four stepper motors two daisy chain boards are needed to send commands to

all four motors through the serial port. It is assumed that the gimbal rate commands are sent fast

enough serially that they arrive at all four motors at the same time from the daisy chain boards.

The nominal speed of the stepper motors needed to be geared down from 134:1. There was a

significant loss in efficiency by doing this so the optical rotary encoders are added on the other

end of the shaft to reduce the effect of this uncertainty by feedback.

The Faulhaber 1525 BRC motors were chosen because they have a high enough stall

torque to spin up the flywheels as well as integrated electronics including a frequency output to

measure the speed of the motors. This is valuable for use in the testbed, because knowledge of

the flywheel angular speed is an integral part of the SGCMG dynamics and control. If the

flywheels were spinning at different speeds the angular momentum vector of each SGCMG

would have different magnitudes and this would add attitude control error.

SOBEK Mechanical Design

There are some unique design considerations that were carried out in the design of the

SOBEK testbed. One of these design considerations dealt with how the ACS would rotate.

To approximate the mass and inertia properties, ProEngineer was used. The inertia

properties were measured about the ACS CM. If the CM was made to be below the point of









rotation, then the system would be stable (i.e. a pendulum equilibrium point). It should be

mentioned that the distance between the point of rotation and the CM cannot be too great or the

reaction momentum from gravity will saturate the SGCMGs. For this reason the CM was chosen

to be just below the point of rotation to make the system close to neutral stability.

To account for the body frame x and y deviation of the CM from the point of rotation, the

ACS is connected to a platform that contains a system of weight ballasts. These ballasts allow

two-dimensional adjustment of the CM. The motor driver and daisy chain boards as well as the

batteries and the power terminal are mounted on this platform to conserve space. The z-axis

deviation of the CM was accounted for by the vertical adjustment of the ACS center.

SOBEK Mechanical, Power and Output Specifications

Mechanical specifications of the SOBEK ACS are shown in Table 4-1 with a total output

torque of 267 mNm, mass of 5.489 kg, and power consumption of 8.36 W.

SOBEK Attitude Determination

The SOBEK ACS is fitted with LEDs that are distinguished by different blinking

frequencies where they are read by a system of eight cameras. Shown in Fig. 4-7 is the testbed

setup including the Phasespace attitude determination system. The Phasespace motion capture

system is chosen because it is a simple and compatible off-the-shelf way to measure attitude

accurately. This is possible because Phasespace works on the principle of taking position

measurements from LEDs located on the SOBEK robot thereby calculating the attitude of the

tracked object. This completes the attitude determination system for SOBEK.

The next chapter discusses the simulation and experimental results of this testbed. The

simulation results utilize the mass and inertia properties estimated in ProEngineer as the basis for

its model. Also discussed in this chapter will be the limitations of the current design iteration of









the SOBEK testbed as well as an introduction of the application package interface software


(API) that runs it.


Table 4-1 Mechanical and Power Specifications of the SOBEK ACS
Current Voltage Weight
Micro ACS Power (W) (A) (V) (kg) Output Torque (mNm)
Stepper motors 6.000 0.500 12.000 267
DC motors 1.776 0.148 12.000
Encoders 0.600 0.120 5.000
Total 8.376 5.489


6 -


Fig. 4-1.


Honeywell roof-top configuration


Honeywell box configuration


Fig. 4-2.

























o 1
2 -2


External singular surface for box type SGCMG ACS


Honeywell pyramid configuration


Fig. 4-3.


Fig. 4-3.































Pyramidal SGCMG ACS


-.


t ,..w.- .


Rendition of SOBEK testbed


Fig. 4-4.


Fig. 4-5.































ACS with hardware components


Cameras collect rangi
measurements from
LEDs



Motor driver boards
send command to the
stepper motors
T


control calculated and
sent via bluetooth to
ACS


q

Quaternions
sent to
command PC


PhaseSpace attitude determination system


Range
data
processed


Fig. 4-6.


Fig. 4-7.









CHAPTER 5
RESULTS

Simulation Model

The error quaternion state equation is


r L[B"B]' e+le 1 4[e
e 2 2 1 G eo) 1Q e
e4 B B T 2- e4 2 J= e4
2 -

where, e and e4, are the vector and scalar elements of the error quaternion and B0B is the

spacecraft angular velocity coordinatized in the body frame.

The equation for the spacecraft model is


S= B B B +B [ BB] Bj B)B

where, BjC is the spacecraft centroidal inertia mentioned in Eq. (3-1) and r is the internal control

torque from the SGCMGs mentioned in Eq. (3-14), both of which are coordinatized in the body

frame.

The equation for the commanded SGCMG output torque is,
B=-_Z- [BB]j Bh

The output torque is mapped onto the gimbal rates with the GSR-inverse discussed in

chapter 3 giving the gimbal rate solution of



-o At L A" (5-1)
ho=

where, (ct is the actual column matrix of commanded gimbal rates used to generate the given

output torque considering the torque error added from the GSR-inverse.

The internal SGCMG torque r is found in through a nonlinear rest to rest control logic









r = -Ke CBB + [B B B BB (5-2)

with the symmetric positive-definite controller stiffness and damping gain matrices K = 2kBJ

and C = cBJ The non-linear exact model knowledge (EMK) controller shown in has been

proven to be asymptotically stable [45] and a Lyapunov stability analysis was performed on this

controller to understand its behavior when uncertainty is present in the system.

Lyapunov Stability Analysis of EMK Attitude Controller

A positive definite Lyapunov function is chosen as,


V= e'e+(l-e4)2l[ ] i BBK L>0 (5-3)
2 53
with time derivative equal to

V = 2e +2(1-e4 )(,4 )+[B w B K [ B]=

2e ]Xe+- B e4 +2(1-e4) B B]Te (5-4)

[B B]) ] -K1 [ BB ]X BjBC9B

To obtain asymptotic stability, Eq. (5-4) must be made negative definite. To achieve

asymptotic stability the control torque (i.e., internal torque of the SGCMGs) is chosen as Eq. (5-

2). Substituting r from Eq. (5-1) into V that of Eq. (5-4), we end up with


S C [B B]TBB
V, = c It)]T c < 0
2k -
The final result of this substitution is that the time derivative of the Lyapunov function is

negative semi-definite. To prove that the controller in Eq. (5-2) gives a global asymptotically

stable result, we must note that the spacecraft angular velocity goes to zero asymptotically for the

following reasons:









* Bo) e Lee L z-e L, ~ B6B L


* BoB E L2

* B )B is uniformly continuous

If we apply Barbalat's lemma with these conditions, the spacecraft angular velocity goes to

zero asymptotically. If we revisit the control law in Eq. (5-2), we can see now that it is not

possible for the angular velocity to go to zero asymptotically unless the error quaternion vector

elements also go to zero, therefore this controller is asymptotically stable.

This controller is EMK which assumes full state feedback, therefore the global asymptotic

stability does not necessarily hold if accurate full state feedback is not available.

The current iteration of SOBEK does not contain gyros for angular rate calculation and

encoders for gimbal rate position. This iteration also has no online method of calculating the

flywheel speeds, therefore the gimbal angles and angular rates are estimated with uncertainty

present and the flywheel speeds are calculated before experimentation. The gimbal angles are

estimated through an Euler forward integration of the solution to Eq. (5-1) as,


= + At
where Atk is the time step of the control loop.

The equation for the calculation of the ACS angular rates is


B 0)c B [AS Ae e ,+l e
S= L= A +=



where, Q\ is always invertible.
e4k









Due to the uncertainty present in this iteration, the best result that the controller can

produce on this testbed are shown to be uniformly ultimately bounded (UUB). To account for

the many sources of uncertainty in this iteration and obtain the UUB result, another stability

analysis of this controller is carried out with uncertainty considered. The control torque is

dependent on both the estimated angular rates and gimbal angles. The gimbal angles are

integrated from the gimbal rates which are mapped from the control torque. Recall also that the

torque utilizes minimal torque errors for singularity avoidance when using the GSR-inverse and

therefore these torque errors must be considered in addition to the uncertainty.

Lyapunov Stability Analysis of Actual Attitude Controller with State Uncertainty

For the stability analysis, the actual control torque and angular velocities of the system are

defined as,


7ct= + 6

Tact =0 +
with the uncertainties and .

If we substitute the actual control torque and angular velocities into Eq. (5-3), we are left

with


v2 =e +(1-e4)2 +I TK-B
V2e e+(-e4 2 oact _- B ) act > 0
2
and its time derivative


V )= _K_1 1" -c _B_ + 0) .._e (5-5)
V2 actK L -0 act cacte (5-5)

Assuming that the uncertainties and & have negligible time derivatives, the control

torque from Eq. (5-2) give the result

V= TK-1 K-1eC I B e B + O) + a + e
2 cih a simifid r actsct

with a simplified results










BCB ]' (,I+ B] -1 ( B B 1_ T BB)
V, = (5-6)


Examining Eq. (5-6) while assuming that the uncertainty is less than the actual control

torque and angular velocity, the best result possible is UUB on the order of the same magnitude

as that of the uncertainty for this control law.

Now that the stability analyses have been carried out, the results from simulation and

experiment is discussed.

Simulation Results and Discussion

The simulations were carried out in a Matlab environment with parameters shown in Table

5-1.

Shown in Fig. 5-1 (A) is the error quaternion vector elements for the first simulation

where, el, e2, and e3 are the quaternion error vector elements about the roll, pitch, and yaw

directions.

This result at first glance seems to have great performance. Recall that this simulation was run

assuming exact model knowledge in the absence of uncertainty. The next plot shown in Fig. 5-1

B) shows the commanded gimbal rates throughout the simulation.

In this figure it is clear that the gimbal rates are large in the beginning of the simulation.

This is due to the fact that the simulation starts off with the SGCMG gimbals in a singular

configuration for a torque needed about the axis. Figure 5-1 (B) also shows the trend of the

gimbal rates going null after the maneuver is completed which is not the case when disturbance

torques are added to the system. In this situation, the gimbal rates will need to compensate for

the addition of angular momentum and will continue to be nonzero.









Figure 5-1 (C) shows the spacecraft angular rates of the ACS. In this figure, the angular

rates are nullified after the maneuver is completed. This is the desired affect for a rest to rest

maneuver. The final plot of this simulation is that of the singularity measure m in Fig. 5-1 (D).

Two things can be attributed to Fig. 5-1 (D). The first is that the singularity measure is zero at

the beginning of the simulation (i.e., Jacobian is rank deficient when all gimbal angles are

zeroed) and it ramps up quickly to transit from singularity within the first two seconds.

Although the ACS is initially at a singular configuration, use of this steering logic enables it to

transit away from singularity in a timely manner. The second conclusion for this figure is that m

becomes constant after the maneuver is completed. This should be obvious because it is an

explicit function of the gimbal angles.

Before an experiment was carried out, another simulation was run with the same k and c

values from the previous simulation including the addition of a random error signal added to the

angular velocity measurement and gimbal angle integration. This random error was added in an

attempt to model the uncertainty of the gimbal angles and angular rates as


Bck+~ = B cB +O.03rand)

1 = +0.1rand(.)


It is clear from the results of this simulation are shown in Figs. 5-2, while considering an

EMK control logic when the controller has uncertainty within its states, the performance is

heavily degraded to the point of instability.

In addition to this simulation another one including a higher choice of gains k and c was

run to compare results and validate the uniformly ultimately bounded result of the stability

analysis in Eq. (5-6). The results in Figs. 5-3 prove the opposite end of the stability analysis









which is with higher choices of a k and c, the steady state error of the UUB solution can be

reduced.

Experimental Results and Discussion

The flywheels onboard the SGCMGs of the ACS are not balanced and are unstable at

certain wheel speeds. For safety reasons the wheels are left to spin at a maximum speed of

around 6000 rpm to keep them within a stable vibration frequency. This speed does not produce

enough angular momentum, and hence, enough torque to overcome the moment from gravity to

do large off-axis A or b-axis maneuvers and therefore the experimental results are for a yaw

maneuvers about the b axis.

The GUI associated with this testbed is shown below in Fig. 5-4. This GUI in addition to

rest of the API software was developed by Andrew Waldrum at the University of Florida

specifically for SOBEK.

The flywheel angular speeds were calculated offline through the output frequency

feedback of the DC motors and found to be of different magnitude among the four SGCMGs. A

variation of Eq. (5-1)




is used to account for the variation in nominal angular moment for the individual SGCMGs

where Ho is the diagonal matrix of nominal angular moment.

In this GUI the gains k and c are increased at intervals c = 2.5k to where k varies from .08

and 0.2 in the first experiment. There is no torque to steer away from singularity at startup

because the gains are zeroed out initially. All experiments were carried out with the same

steering logic parameters and initial conditions as those in the first simulation and only differed

by choice of control gain. An initial experiment was run with the controller given in Eq. (5-2).









It is shown in Fig. 5-5 (A) that at around forty seconds the quaternion error diverges,

which can be caused by either the controller's instability, propagation uncertainty from the

estimated states, or both. The reason for this error propagation is due to the error rates produced

from the uncertainty of the estimates. The maneuvers in these experiments start out with a high

initial error about the 3 axis and attempt to align the ACS to a quaternion of q = [0 0 0 1].

The next three plots shown in Figs. 5-5 (B)-(D) show the gimbal rates, angular rates, and

singularity measure associated with this maneuver.

The next experiment took into consideration the result of the stability analyses and the

last two simulations. This experiment was carried out with higher values of the control gains and

its results are shown in Figs. 5-6. In these experiments k was chosen greater than that for the

previous experiment keeping the same c/k ratio.

The quaternion error vector elements in Fig. 5-6 (A) contain better results than the

previous experiment although they also posses low frequency oscillations. The steady state error

in this figure is measured as,


b = min[2sin ( e 2,2r 2sin 1( e) = 0.0826 rad= 4.7441V

where, the angle e, represents the minimum angle needed to be traced out to align the two

frames 3B and 3G about an eigen-axis.

This metric is valid because the error quaternion is still a unit quaternion whereby the 2-

norm of the vector elements is equal to sin (Oe/2). Noted in red on the plots of Fig. 5-6 (B) and

(D), are the large jumps in gimbal rates when transiting a singularity. This was previously said

to occur when the SR or GSR-inverse method is applied for singularity escape.









The result is still oscillating about zero after sixty seconds and the steady state error

S= 0.0826 rad = 4.7441 is still quite large. It is believed through a larger choice of gains,

preferably ones with a higher ratio of c/k will reduce the oscillations and steady state error.

The next experiment shown in Figs. 5-7 was carried out with a larger value of control gains with

a higher c/k ratio to improve the results from the previous experiment. These values were k =

0.32 and c = 3.0k. The error quaternion vector elements for this experiment converged with a

much lower steady state error of = 0.0376 rad = 1.0779 and a smaller amount of steady-state

oscillation due to the higher c control gain.

The behavior of the system for the chosen control gains is more stable as well as more

efficient in terms of less wasted control effort than that for the previous experiment. This is due

to the larger ratio of c/k and the fact that the maneuver did not encounter an additional internal

singularity within the time of the experiment. With this larger c/k ratio, the steady state error

oscillations are reduced by a much larger c gain. The next plot shown is the angular rates versus

time for this experiment. Angular rates shown in Fig. 5-7 (C) show a more desirable behavior

than that of Fig. 5-6 (C) in terms of high slew rates with less oscillation.

The last plot of this experiment is that of the singularity measure. It has already been

discussed that behavior of this system is more desirable than the previous experiments with the

choice of the current gains, and now it also seen that at no time did this system become singular.

A final experiment was carried out were the Moore-Penrose pseudo-inverse which

possessed no steering logic (i.e. A = 0). The quaternion error of this experiment is shown in

Fig. 5-8.

The SOBEK ACS for all experiments is initially at singularity, therefore without a steering

logic large gimbal rates are commanded and the system tumbles out of control as shown in Fig.









5-8. From this figure, it is clear that without use of steering logics while utilizing SGCMGs,

drastic consequences may arise.

In the future iterations of the SOBEK testbed, if encoders are provided are for gimbal angle

measurement, tri-axial gyros provided for angular rate measurements and use of onboard

feedback for the flywheel angular speeds will produce more accurate attitude maneuvers with

elimination of state uncertainty.

Table 5-1. Simulation parameter values
Parameters Values Units


0.04

0.8


J= diag {J1, J22, J33}

qc = [q, q2c 3c q4c
0

q(0)= [q q2 q3 4]T

S(0) = [0, 02 )3]T

E1 = Eorand(1)

g2 = orand(2)
E3 = Orand(3)

A = A0 exp(- /m2)

3(0)= [g, g2 3 64


diag {0.0668,0.0756,0.0815}

[0 0 0 1]f

54.74

[0.9380 0 0 0.3466]T

[0 0 Of
0.0lrand(l)

0.0 lrand(2)

0.0lrand(3)

0.01exp(-20m2)

[0 0 0 O]


Nms

none

none

Nm2

none

degrees

none

rad/s

none

none

none

none

rad















0

-0.05

-0.1

o -0.15

-0.2

-0.25

-0.3

-0.35


40 50


002


-






CO -0 08
-0


-dal/dt
-da2/dt
-da3/dt
da4/dt


-012


10 20 30
time (s)


40 50 60


0.25

0.2

0.15

0.1

. 0.05

S 0
3
-0.05

-0.1

-0.15

202 10 20


30 40 50 60
time (s)


E
004


002


0 10 20 30 40 50 60
time (s)


ACS simulation plots for k = 0.08 and c = 2.5k: A)
elements versus time, B) Gimbal rates versus time,
and D) Singularity parameter versus time.


Quaternion error vector
C) Angular rates versus time,


10 20 30
time (s)


-e

-e2
-e3


Fig. 5-1.











-d5/dt
-d&Idt


10 23 30
tirm s (f


tie (s)


ACS simulation plots for k = 0.08 and c = 2.5k: A) Quaternion error vector
elements versus time, B) Gimbal rates versus time, C) Angular rates versus time,
and D) Singularity measure versus time.


times)


-u) 0.05




-0.1

-0.10


C

Fig. 5-2.


time(s)















-.1 -O i:
-0.1 2 -001 --d.
0) -e3 -.i
.2 .-d6dt

-0 -003


40 10 0 30 40 0 6 40 10 2 3 40 5 60
time (s) time (s)

A B


Q3, 1.6

0.25 -0 1.4
Q2-- 1.2

1 1
01 E0.
0
a0.6
0.05 0.4
0 \ 0.2
0.2

0 10 23 30 40 50 6 70 00 10 20 30 40 50 60 70
time (s) time (s)

C D

Fig. 5-3. ACS simulation plots for k = 4 and c = 3k: A) Quaternion error vector elements
versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D)
Singularity measure versus time.









Euler angles


M e


U. UU O

I w-3 I I I
I i. uunE
I yl 7Lj I 1

Seal wiSTOP
Senal wire


~'U fu llla m rIaiumlls p: 1
Y: I0

r" Radians
Degrees
Kn
Kr
Kz
Ka O7

Draw Range Stop Gimbals
Loop is Closed Si re Data
Status: BAD Singularity BA --
Controller NULL
K: J S StepNumber |P 1
C: I Commands Sent 77
Rate of Operation | L| I


SOBEK testbed GUI


1 i :j -a


Fig. 5-4.














0.5


0 0
U,


0


-d61/dt
-d62/dt
-d63/dt
d /dt
4


.3

.2



S11 r,/ ,fr


-0.1
0 10 20 30 40 50 60 70 80
time (s)


1.4

S1.2

0 1

0.8
E
0.6

0.4

0.2


0 10 20 30 40 50 60 70 80
time (s)


Fig. 5-5.


0 10 20 30 40 50 60 70 80
time (s)


ACS experimental plots for k = 0.8 and c = 2.5k: A) Quaternion error vector
elements versus time, B) Gimbal rates versus time, C) Angular rates versus Time,
and D) Singularity measure versus time.


time (s)


C I i~v











1.2
1

0.8

0.6

0.4

0.2

0

-o0 1 0 20 30 40 50 60 70
time (s)


. & I


, I .I.


-d61/dt
-da /dt
-ds3/dt
... d /dt





" a'w''."."


-1
0 10 20 30 40 50 60 70
time (s)


B


-1
--2
--3


W- ------ I -


0 10 20 30 40 50 60 70 10 11 40 50 60 70
time (s) time (s)


C D


Fig. 5-6. ACS experimental plots for k = 0.24 and c = 2.0k: A) Quaternion error vector
elements versus time, B) Gimbal rates versus time, C) Angular rates versus time,
and D) Singularity measure versus time.














-e2
-e2
-e,


A


P 10 20 30 40 50 60 70 80
time (s)


0.5o



-0.5



-1.5


--1 2/dt

-.J 3/dt
4J ,/dt


0 10 20 30 40 50 60 70 80
time (s)


B


1.5,


10 20 30 40 50 60 70
time (s)


D


ACS experimental plots for k = 0.32 and c = 3.0k: A) Quaternion error vector
elements versus time, B) Gimbal rates versus time, C) Angular rates versus time,
D) Singularity measure versus time.


S0

-0.1
s


Fig. 5-7.


time (s)


"Awcss


--


























-0.1


-0.
S 2 4 6 8 10 12 14
time (s)


Fig. 5-8. ACS experiment quaternion error of Moore Penrose solution


-e

-e









CHAPTER 6
CONCLUSION AND FUTURE RESEARCH

Conclusion

This thesis discusses the development and testing of an attitude control testbed utilizing

four SGCMGs in an orthogonal pyramid configuration. SOBEK is of great utility for evaluating

attitude control algorithms and has provided insights into the understanding and application of

control moment gyro-based actuators. In its current form, SOBEK has only attitude feedback and

does not have onboard processing or internal feedback (i.e., operates in an open-loop manner).

With the addition of onboard processing and sensors for state feedback, it is expected that the

testbed will prove to be invaluable for testing control algorithms and steering logics while

showing precision in terms of attitude accuracy from SGCMGs.

Future Research

Future research for this attitude control system will include the changes to the new

iteration of SOBEK and testing of different control algorithms and steering logics. A pico-

satellite CMG-based ACS will be developed using SOBEK as the alpha-model to analyze

software that will fly on this miniaturized actuator.









APPENDIX A
OFF-THE-SHELF RW, CMG, AND MAGNET TORQUER DATA SPECIFICATIONS

Table A-1. Off-the-shelf performance specifications for RWs
RWA Torque (mNm) Mass (kg) Power (W)
Bradford Engineering
W05 100 3.2 73
W18 200 4.95 63
W45 300 6.95 64

Honeywell
HR 0610 55 3.6 80
Dynacon
MicroWheel 200 30 0.93 2
MicroWheel 1000 30 1.3 9

Vectronic Aerospace
RW1 20 1.8 25
SunSpace
SunSpace RW 50 1.98 35

TELDIX
RSI 01-5/15 5 0.6 4
RSI 01-5/28 5 0.7 4

Goodrich (Ithaco)
TW26E300 300 13.9

L3 Space Comm.
MARS
RWA 15
MWA-50 160 10.5 100

Orbital Sciences
LEO Star Wheel 140 3.6 55

SSTL
MicroWheel-5S 5 0.5 8
MicroWheel-3S 3 0.75 3
MiniWheel-20S 20 3.2 14
MiniWheel-20S-X 20 2.6 14
NanoWheel-m500S 0.5 0.08 0.5
MicroWheel-10SP-S 10 1.1 5









Table A-2. Off-the-shelf performance specifications for CMGs


CMG
Honeywell
M50
M95
M160
M225
M325
M325D
M715
M600
M1400
M1300


EADS/ASTRIUM
CMG 15-45S

SSTL
Bilsat-1 CMG

SSG
SOBEK ACS


Torque (mNm) Mass (kg) Power (W)


74570
128803
216931
305059
440641
440641
969410
813491
2E+06
2E+06


45000


95000


66.7


33.1
38.6
44
54
61.2
61.2
89.8
81.6
132
125


2.2


1.38


75
129
217
305
441
441
949
814
1899
1716


12


2.09











Table A-3.


Strauss Space
Micro Magnetic Torque Rods
Nano Magnetic Torque Rods
Magnetic Torque Coils

Vectronic Aerospace
MTR-5


0.6
0.2
0.04


0.75


3.5
0.36
0.1


6.0


Off-the-shelf performance specifications for Magnet Torquers
Magnet Torquer Output (Am2) Mass (kg) Power (W)
Microcrosm
MT2-1 2.5 0.2 0.5
MT5-2-M 6.0 0.3 0.77
MT6-1 7.0 0.23 0.25
MT6-2 8.0 0.3 0.5
MT10-2-H 12 0.35 1.0
MT15-1-M 20 0.43 1.11
MT30-2-CGS 40 1.050 3.6
MT30-2-GRC 35 1.4 1.5
MT70-1 75 2.6 3.8
MT70-2 75 2.2 2.6
MT80-1 100 4.12 3.0
MT80-2-M 90 2.3 4.7
MT110-2 120 3.8 2.9
MT140-2 170 5.3 1.9
MT250-2 300 5.5 4.8
MT400-2-L 500 7.8 9.0
MT400-2 550 11.0 11.4
MT400-1 550 9.2 7.7









LIST OF REFERENCES


[1] Cebrowski, A. and Raymond, J., Operationally Responsive Space: A New Defense
Business Model, Parameters, 2005, pp. 67-77.

[2] Alminde, L., Bisgaard, M., Vinther, D., Viscor, T., and ostergard, K., The AAU CubeSat
Student Satellite Project: Architectural Overview and Lessons Learned, 16th IFAC
Symposium on Automatic Control in Aerospace, (Russia), 2004.

[3] Waydo, S., Henry, D., and Campbell, M., CubeSat Design for LEO-Based Earth Science
Missions, IEEE Aerospace Conference Proceedings, Vol. 1, 2002.

[4] Long, M., Lorenz, A., Rodgers, G., Tapio, E., Tran, G., Jackson, K., Twiggs, R., and
Bleier, T., A CubeSat Derived Design for A Unique Academic Research Mission in
Earthquake Signature Detection, AIAA Small Satellite Conference, 2003.

[5] Vladimirova, T., Wu, X., Jallad, A., and Bridges, C., Distributed Computing in
Reconfigurable Picosatellite Networks, Proceedings of the Second NASA/ESA Conference
on Adaptive Hardware and Systems, IEEE Computer Society Washington, DC, USA,
2007, pp. 682-692.

[6] Puig-Suari, J., Turner, C., and Twiggs, R., CubeSat: The Development and Launch
Support Infrastructure for Eighteen Different Satellite Customers on One Launch, 15 th
Annual AIAA/USU Conference on Small Satellites, Logan, Utah, 2001.

[7] Twiggs, B. and Puig-Suari, J., CUBESAT Design Specifications Document, 2003.

[8] Straight, S. and Davis, T., Tactical Satellite 3: Requirements Development for Responsive
Space Missions, Air Force Research Lab, Georgia Institute of Technology, 2005.

[9] Wegner, P. and Kiziah, C., Pulling the Pieces Together at AFRL

[10] Kim, K., Analysis of Hysteresis for Attitude Control of a Microsatellite, San Jose State
University, http://www.engr.sjsu.edu/spartnik/adac.html.

[11] Larson, W. and Wertz, J., Space Mission Analysis and Design, Microcosm, Inc., Torrance,
CA (US), 1992.

[12] http://www.sstl.co.uk/Products/Subsystems/Available_Subsystems, Surrey Space Centre,
Tech. rep., [Accessed 07/15/2008].

[13] http://www.smad.com/ie/ieframessr2.html, Microcrosm,, Tech. rep., [Accessed
12/06/2008].

[14] http://www.stras-space.com/strasspace products, Stras Space, Tech. rep., [Accessed
12/06/2008].









[15] http://www.bradford space.com/pdf/be_datasheet rwu_sep2006.pdf, Bradford
Engineering, Tech. rep., [Accessed 07/15/2008].

[16] https://commerce.honeywell.com/webapp/wcs/stores/servlet/ECategoryDisplay?catalog
Id=10106&storeld=10651 &categoryld=13832&cacheld=1000000000000001&langld=-1,
Honeywell Engineering, Tech. rep., [Accessed 07/15/2008].

[17] http://microsat.sm.bmstu.ru/e library/Algorithms/Hardware/wheels/productpdf 6.pdf,
Dynacon Enterprises Limited, Tech. rep., [Accessed 07/15/2008].

[18] http://www.vectronic aerospace.com/html/magnet torquer.html, Vectronic Aerospace,
Tech. rep., [Accessed 12/06/2008].

[19] http://www.sunspace.co.za/products/index.htm, Sunspace, Tech. rep., [Accessed
07/15/2008].

[20] http://microsat.sm.bmstu.ru/e library/Algorithms/Hardware/wheels/RSIO1.pdf, Teldix,
Tech. rep., [Accessed 07/15/2008].

[21] http://www.oss.goodrich.com/ReactionWheels.html, Goodrich, Tech. rep., [Accessed
07/15/2008].

[22] http://www.l3com.com/products services/productservice.aspx?id=310&type=b, L3
Communications, Tech. rep., [Accessed 07/15/2008].

[23] http://microsat.sm.bmstu.ru/e library/Algorithms/Hardware/wheels/ssci5.pdf, Orbital
Sciences, Tech. rep., [Accessed 07/15/2008].

[24] Defendini, A., Morand, J., Faucheux, P., Guay, P., Rabejac, C., Bangertt, K., and Heimel,
H., Control Moment Gyroscope(CMG) Solutions For Small, Agile Satellites, Advances in
the Astronautical Sciences, Vol. 121, 2005, pp. 51-67.

[25] Abbott, F., Hamilton, B., Kreider, T., Di Leonardo, P., and Smith, D., MCS Revolution,
Advances in the Astronautical Sciences, Vol. 125, 2006, pp. 99.

[26] Lappas, V., Steyn, W., and Underwood, C., Design and Testing of a Control Moment
Gyroscope Cluster for Small Satellites, Journal of Spacecraft andRockets, Vol. 42, No. 4,
2005, pp. 729.

[27] Davis, P., Momentum System Concepts and Trades for the New Class of Smaller Lower
Cost Satellites, Advances in the Astronautical Sciences, Vol. 125, 2006, pp. 13.

[28] Hamilton, B. and Underhill, B., Modern Momentum Systems for Spacecraft Attitude
Control, Advances in the Astronautical Sciences, Vol. 125, 2006, pp. 57.

[29] Lappas, V. and Underwood, C., Experimental Testing of a CMG Cluster for Agile
Microsatellites, 54th International Astronautical Congress of the International
Astronautical Federation (IAF), (Bremen, Germany), Sep.-Oct, 2003.









[30] Wie, B., Bailey, D., and Heiberg, C., Singularity Robust Steering Logic for Redundant
Single-Gimbal Control Moment Gyros, Journal of Guidance, Control, and Dynamics,
Vol. 24, No. 5, 2001, pp. 865-872.

[31] Schaub, H. and Junkins, J., CMG Singularity Avoidance Using VSCMG Null Motion
(Variable Speed Control Moment Gyroscope), AIAA/AASAstrodynamics Specialist
Conference and Exhibit, Boston, MA, 1998, pp. 213-220.

[32] Yoon, H. and Tsiotras, P., Singularity Analysis of Variable-Speed Control Moment Gyros,
Journal of Guidance, Control, and Dynamics, Vol. 27, No. 3, 2004, pp. 374-386.

[33] Schaub, H., Vadali, S., and Junkins, J., Feedback Control Law for Variable Speed Control
Moment Gyros, Journal of the Astronautical Sciences, Vol. 46, No. 3, 1998, pp. 307-328.

[34] Lappas, V., Steyn, W., and Underwood, C., Torque Amplification of Control Moment
Gyros, Electronics Letters, Vol. 38, No. 15, 2002, pp. 837-839.

[35] Leve, F., Tatsch, A., and Fitz-Coy, N., A Scalable Control Moment Gyro Design for
Attitude Control of Micro-, Nano-, and Pico-Class Satellites, Advances in the
Astronautical Sciences, Vol. 128, Published for the American Astronautical Society by
Univelt; 1999, 2007, p. 235.

[36] Leve, F. and Tatsch, A., Fitz.-Coy. N., Three-axis attitude control design for on-orbit
robotics, 2007.

[37] Lappas, V., Steyn, W., and Underwood, C., Practical Results on the Development of a
Control Moment Gyro Based Attitude Control System for Agile Small Satellites, 16th
Annual AIAA/USU Conference on Small Satellites, (Logan, UT), Utah State University,
Aug, 2002.

[38] Wie, B., Space Vehicle Dynamics and Control, AIAA, 1998.

[39] Margulies, G. and Aubrun, J., Geometric Theory of Single-Gimbal Control Moment Gyro
Systems, Journal of the Astronautical Sciences, Vol. 26, No. 2, 1978, pp. 159-191.

[40] Bedrossian, N., Paradiso, J., Bergmann, E., and Rowell, D., Steering Law Design for
Redundant Single-Gimbal Control Moment Gyroscopes, Journal of Guidance, Control,
and Dynamics, Vol. 13, No. 6, 1990, pp. 1083-1089.

[41] Kurokawa, H., Survey of Theory and Steering Laws of Single-Gimbal Control Moment
Gyros, Journal of Guidance Control and Dynamics, Vol. 30, No. 5, 2007, pp. 1331.

[42] Joseph, A., Global Steering of Single Gimballed Control Moment Gyroscopes Using a
Directed Search, Journal OF Guidance, Control, and Dynamics, Vol. 15, No. 5, 1992.

[43] Kuhns, M. and Rodriguez, A., A Preferred Trajectory Tracking Steering Law for
Spacecraft with Redundant CMGs, Proceedings of the American Control Conference,
1995., Vol. 5, 1995.









[44] Nakamura, Y. and Hanafusa, H., Inverse Kinematic Solutions with Singularity Robustness
for Robot Manipulator Control, ASME, Transactions, Journal ofDynamic Systems,
Measurement, and Control, Vol. 108, 1986, pp. 163-171.

[45] Wie, B., Bailey, D., and Heiberg, C., Rapid Multitarget Acquisition and Pointing Control
of Agile Spacecraft, Journal of Guidance Control and Dynamics, Vol. 25, No. 1, 2002,
pp. 96-104.

[46] Ford, K. and Hall, C., Singular Direction Avoidance Steering for Control-Moment Gyros,
Journal of Guidance Control and Dynamics, Vol. 23, No. 4, 2000, pp. 648-656.

[47] Jung, D. and Tsiotras, P., An Experimental Comparison of CMG Steering Control Laws,
Proceedings of the AIAA Astrodynamics Specialist Conference, 2004.

[48] Lee, J., Bang, H., and Lee, H., Singularity Avoidance by Game Theory for Control
Moment Gyros, AIAA Guidance, Navigation, and Control Conference and Exhibit, 2005,
pp. 1-20.

[49] Schaub, H. and Junkins, J., Singularity Avoidance Using Null Motion and Variable-Speed
Control Moment Gyros, Journal of Guidance, Control, and Dynamics, Vol. 23, No. 1,
2000, pp. 11-16.

[50] Fausz, J. and Richie, D., Flywheel Simultaneous Attitude Control and Energy Storage
Using a VSCMG Configuration, Proceedings of the 2000 IEEE International Conference
on Control Applications, 2000., 2000, pp. 991-995.

[51] Ledebuhr, A., Kordas, J., Ng, L., Jones, M., Whitehead, J., Breitfeller, E., Gaughan, R.,
Dittman, M., and Wilson, B., Autonomous Agile Micro Satellites and Supporting
Technologies, AIAA Space Technology Conference and Exposition, 1999, pp. 99-4537.

[52] Underhill, B. and Hamilton, B., Momentum Control System and Line-of-Sight Testbed,
Advances in the Astronautical Sciences, Vol. 125, 2006, pp. 543.

[53] Jung, D. and Tsiotras, P., A 3-DoF Experimental Test-Bed for Integrated Attitude
Dynamics and Control Research, 2003.

[54] Leve, F., Design of a 3DOF Testbed For Microsatellite Autonomous Operations, AIAA
Aerospace Sciences Meeting, 2006.









BIOGRAPHICAL SKETCH

Frederick Aaron Leve was born in Hollywood, Florida, in 1981. In August 2000 he was

accepted into the University of Florida's Department of Aerospace Engineering in the College of

Engineering where he pursued his bachelor's degrees in Mechanical and Aerospace Engineering.

After completing his bachelor's degrees in May 2005, he was accepted into the master's

program in aerospace engineering at the University of Florida. While in the master's program,

he received two awards in academia. In January 2007 he received the American Institute of

Aeronautics and Astronautics Abe Zarem Award for Distinguished Achievement in Astronautics.

For this award he was invited to Valencia, Spain, where he competed in the International

Astronautical Federations's International Astronautical Congress Student Competition. Here he

received the silver Herman Oberth medal in the graduate category. In May 2006, he was

accepted to the Air Force Research Lab Space Scholars Program, where spent his summer

conducting space research.

His interests include satellite attitude control, satellite pursuit evasion, astrodynamics, and

orbit relative motion.





PAGE 1

1 DEVELOPEMENT OF THE SPACECRAFT ORIENTATION BUOYANCY EXPERIMENTAL KIOSK By FREDERICK AARON LEVE A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008

PAGE 2

2 2008 Frederick A. Leve

PAGE 3

3 To my mother and father

PAGE 4

4 ACKNOWLEDGMENTS I thank m y supervisory committee chair (Dr. No rman Fitz-Coy) for giving me the chance to embark on this innovative research. I thank him for his patience and ad visement. Without his help I would have certainly been lost. I would like to thank my fellow colleagues in my research group were of great assistance to me; Dr. Andr ew Tatsch, Shawn Allgeier, Andrew Waldrum, Jaime Bestard, Sharan Asundi, Dante Buck ley, Vivek Nagabhushan, Josue Munoz, Nick Martinson, and Will Mackunis. Lastly I would al so like to thank my supervisory committee (Dr. Warren Dixon and Dr. Gloria Wiens) for their assi stance in validating my masters research.

PAGE 5

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 LIST OF ABBREVIATIONS........................................................................................................ 10 CHAP TER 1 INTRODUCTION..................................................................................................................13 Research Motivation............................................................................................................ ...13 Mission Examples...................................................................................................................14 Research Focus.......................................................................................................................14 2 ATTITUDE CONTROL ACTUATORS................................................................................16 Types of Attitude Actuators.................................................................................................... 16 Types of CMGs.......................................................................................................................19 Single Gimbal Control Momentum Gyroscopes............................................................. 20 Double Gimbal Control Momentum Gyroscopes............................................................ 20 Variable Speed Control Momentum Gyroscopes............................................................ 21 3 SGCMG DYNAMICS AND PERFORMA NCE CHARACTERIZATION.......................... 24 Dynamics................................................................................................................................24 Torque Amplification........................................................................................................... ..28 Singularities............................................................................................................................30 External Singularities......................................................................................................30 Internal Singularities.......................................................................................................31 Singular Surfaces.............................................................................................................31 Steering Logics................................................................................................................ .......33 Momentum Ma nagem ent........................................................................................................38 External Angular Momentum Offloading....................................................................... 39 Internal Angular Momentum Offloading........................................................................ 39 4 SGCMG ACS CONFIGURATIONS..................................................................................... 43 Rooftop Configuration.......................................................................................................... ..43 Box Configuration..................................................................................................................43 Pyramid Configuration.......................................................................................................... .43 SOBEK Pyramid Configuration......................................................................................44 SOBEK Hardware...........................................................................................................45 SOBEK Mechanical Design............................................................................................46 SOBEK Mechanical, Power and Output Specifications .................................................. 47

PAGE 6

6 SOBEK Attitude Determination......................................................................................47 5 RESULTS...............................................................................................................................52 Simulation Model............................................................................................................... ....52 Lyapunov Stability Analysis of EMK Attitude Controller..............................................53 Lyapunov Stability Analysis of Actual Att itude Controller w ith State Uncertainty....... 55 Simulation Results and Discussion......................................................................................... 56 Experimental Results and Discussion..................................................................................... 58 6 CONCLUSION AND FUTURE RESEARCH...................................................................... 70 Conclusion..............................................................................................................................70 Future Research......................................................................................................................70 APPENDIX Off-The-Shelf Rw, CMG, And Magne t Torquer Data Specifications ........................................... 71 LIST OF REFERENCES...............................................................................................................74 BIOGRAPHICAL SKETCH.........................................................................................................78

PAGE 7

7 LIST OF TABLES Table page 2-1 Attitude control actuators and their specifications............................................................. 21 4-1 Mechanical and Power Speci fications of the SOBEK ACS .............................................. 48 5-1 Simulation parameter values.............................................................................................. 61 A-1 Off-the-shelf performance specifications for RWs............................................................71 A-2 Off-the-shelf performance specifications for CMGs......................................................... 72 A-3 Off-the-shelf performance spec ifications for Magnet Torquers ........................................ 73

PAGE 8

8 LIST OF FIGURES. Fig. page 1-1. On-orbit assem bly............................................................................................................. 15 1-2. Earth m onitoring in LEO..................................................................................................15 1-3. Post-docking stabilization .................................................................................................15 2-1. Magnet coil from Tokyo Institut e of Technology Cute-1.7 CubeSat............................... 21 2-2. Dynacon miniature RW ....................................................................................................22 2-3. Semi-log plot of input power versus output torque for off-the-shelf RWs and CMGs....22 2-4. Semi-log plot of mass versus output to rque for off-the-shelf RWs and CMGs................22 2-5. SSTL m icrosatellite SGCMG flown on Bilsat-1..............................................................23 2-6. DGCM G developed by L3 Communications.................................................................... 23 3-1. The SGCM G with gimbal coordinate axes.......................................................................40 3-2. A 3-2-3 rotation sequence through angles iii ........................................................41 3-3. Intersecting planes of the spa nning of SGCMG a ngular momentum............................... 41 3-4. Angular m omentum envelope for 35.26 and 1,1,1,1.......................................41 3-5. Intern al singular surface 35.26 and 1,1,1,1 ...................................................42 3-6. Zero-m omentum configuration......................................................................................... 42 4-1. Honeywell roof-top configuration .................................................................................... 48 4-2. Honeywell box configuration ........................................................................................... 48 4-3. External singular surfac e for box type SGCMG ACS ...................................................... 49 4-3. Honeywell pyram id configuration.................................................................................... 49 4-4. Pyram idal SGCMG ACS..................................................................................................50 4-5. Rendition of SOBEK te stbed............................................................................................50 4-6. ACS with hardware components ......................................................................................51 4-7. PhaseSpace attitude determ ination system....................................................................... 51

PAGE 9

9 5-1. ACS si mulation plots for k = 0.08 and c = 2.5k : A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity parameter versus time.....................................................................................62 5-2. ACS si mulation plots for k = 0.08 and c = 2.5k : A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity measure versus time........................................................................................ 63 5-3. ACS si mulation plots for k = 4 and c = 3k : A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity measure versus time........................................................................................ 64 5-4. SOBEK testbed GUI .........................................................................................................65 5-5. ACS experim ental plots for k = 0.8 and c = 2.5k : A) Quaternion error vector elements versus time, B) Gimbal rates ve rsus time, C) Angular rates versus Time, and D) Singularity measure versus time............................................................................ 66 5-6. ACS experim ental plots for k = 0.24 and c = 2.0 k : A) Quaternion error vector elements versus time, B) Gimbal rates vers us time, C) Angular ra tes versus time, and D) Singularity measure versus time................................................................................... 67 5-7. ACS experim ental plots for k = 0.32 and c = 3.0 k : A) Quaternion error vector elements versus time, B) Gimbal rates vers us time, C) Angular ra tes versus time, D) Singularity measure versus time........................................................................................ 68 5-8. ACS experim ent quaternion error of Moore Penrose solution......................................... 69

PAGE 10

10 LIST OF ABBREVIATIONS ACS Attitude control system API Application program interface CMG Control moment gyroscope CM Center of mass DCM Direction cosine matrix DoD Department of Defense DOF Degree of freedom EADS European Aeronautic Defense Space Company EMK Exact model knowledge GUI Graphical user interface FACETS Flywheel Attitude Control and Energy Transmission System IPACS Integrated Power and Attitude Control System LEO Low Earth Orbit MW Momentum Wheel ORS Operationally Responsive Space RW Reaction Wheel R2P2 Rapid Retargeting and Precision Pointing SGCMG Single-Gimbal Control Moment Gyroscope SOBEK Spacecraft Orientation B uoyancy Experimental Kiosk SSTL Surrey Space Technology Lab UUB Uniformly Ultimately Bounded

PAGE 11

11 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science DEVELOPMENT OF THE SPACECRAFT ORIENTATION BUOYANCY EXPERIMENTAL KIOSK TESTBED By Frederick Aaron Leve May 2009 Chair: Norman Fitz-Coy Major: Aerospace Engineering Most satellites are unique and therefore usuall y take 10 to 15 years to design, fabricate, test, and finally launch. To expedite future advances in space technology, setting standards in terms of black-boxed subsystems and interfac es, drastically reduces the time and long term costs needed to complete these tasks. What is meant by black-boxe d is a stand alone subsystem that is adaptable enough to work w ith any other subsystems without alteration. Government organizations such as the Depa rtment of Defenses (DoDs) Operationally Responsive Space (ORS) office take these tasks in to consideration and look to change this pattern by expediting the proce sses of design, fabric ation, test, evaluation, and launch. To accomplish these tasks, smallsats which range from 1 to 1000 kg in mass, are considered for their cheaper platforms and launch costs. With utiliza tion of smallsats of the pico(0.1-1 kg), nano(1 to 10 kg), and micro-classes (10-100 kg) fo r space missions, obstacles occur in terms of power, mass, and volume constraints. Many of these same smallsats obstacles relate to the attitude control system (ACS). Despite these obstacles for many space missions, small satellites must be held to the same pointi ng requirements as their larger counterparts to be of use. To compensate for these obstacles while maintaining close pointing requirements, innovations in attitude control actuators and determination sensors that satisfy these constraints need to be developed.

PAGE 12

12 An excellent choice for attitude control actua tors of smallsats are those of the flywheel sort known as zero-momentum and momentum bias actuators. Thes e actuators have the ability to perform precision attitude maneuvers without use of propellant. There are two classes of flywheel attitude control actuators that are used in satellites, reaction/momentum wheels (RWs/MWs) and control moment gyroscopes (CMGs) Due to the scarcity of CMGs available for small satellites below the micro-class, RWs are presently utilized. An analysis completed in this thesis shows that although the current stateof-the-art CMGs are too massive to fit inside smallsats, the performance relati onships in terms of output tor que per mass and input power are more favorable than that for RWs. This analysis is an additional motivation for the research in this thesis, which is carried out on the premise that an ACS consisting of single-gimbal control moment gyroscopes (SGCMGs) can meet th e performance requirements providing rapidretargeting and precisio n-pointing (R2P2) for smallsats while enduring their power, mass, and volume constraints. To further test this premise, the topic of th is thesis is the development of a ground based SGCMG testbed known as Spacecraft Orientation Buoyancy Experimental Kiosk (SOBEK) was developed to validate the utility of using SGCMGs for smallsat attitude control. The thesis justifies the reasons for using SGCMGs on SO BEK, discusses the spacecraft and actuator dynamics and kinematics associated with an AC S containing CMGs, addresses the geometrical configurations of systems of these actuators and the configur ation chosen, and then finally validates the testbed numerically th rough simulation and experimentation.

PAGE 13

13 CHAPTER 1 INTRODUCTION Research Motivation The Departm ent of Defenses (DODs) Operationally Responsive Space (ORS) office [ 1] seeks to better the capab ilities of developm ent, fabrication, testing, evaluati on and launch of space assets. To accomplish the goals of the ORS office, standards such as black-boxed systems are considered useful to aid these ta sks and reduce the uniquene ss factor of every satellite project. To accomplish some of these task s, it is believed that switching from larger more massive satellites to smalls ats of the pico(0.1-1 kg), nano(1-10 kg), and micro-classes (10-100 kg) opens up the doors for a wide variety of missions a nd provides a quicker and less costly method (i.e., small satellites are cheaper a nd faster in terms of integration and launch) to develop these black-boxed techno logies. The budget of developing smallsats combined with components-off-the-shelf (COTS) makes it possi ble for some universitie s and smaller aerospace companies to contribute to the knowledge base and evolution of space te chnologies. Numerous university satellite projec ts consisting of missions involving CubeSat and nano-satellite projects have utilized COTS and been f light tested successfully [ 24]. Standard izations for smallsats in the acad emic sector are being led by California Polytechnic Institute and Stanford University. These endeavors include CubeSat Kit CubeSat bus architecture (Pumpkin Inc.) and P-Pod launch vehicle integration [ 57]. Work is also being done on standardization of satellite bus architecture in the military sector such as the ANGELS, XSS-11, and the tactical sa tellite programs at AFRL [ 8] and [ 9]. Although sm allsats are a viable choice for tes ting of new space technologies, they have limited attitude control and propulsive capabilit ies which leaves some in the aerospace community to regard smallsats as toys or space debris with no significant functionality. This perception of smallsats can be challenged with the development of low power and mass attitude

PAGE 14

14 control systems (ACSs) which are near the same pointing and slew requirements of those for larger satellites and hence are still able to accomplish an assortment of many space missions for smallsats where propulsiv e capabilities can be li mited or non-existent. Mission Examples Som e space missions that could possibly utilize smallsats are laser communication, space science, on-orbit assembly, satellite servicing, formation flight blue force tracking, and Earth monitoring. Smallsats that have propulsive capab ilities would most likely have a minimum amount of fixed thrusters, theref ore their attitude and translati on would be coupled. Taking this into account, for a smallsat distribution used fo r such missions as on-orbit assembly shown in Fig. 1-1, precision in terms of position and orient ation need to be maintained for collision avoidance and attitude control where redundant thrusters is most likely not possible. Earth sensing and blue force tracking which have satellite systems in Low Earth Orbit (LEO) require tracking of a specific point on the su rface of earth. This becomes nontrivial due to smaller orbital periods (i.e., faster orbital rates) To acquire a longer time of coverage, higher slew rates while maintaining precision are neede d. An example of this is shown in Fig. 1-2. When dealing with missions such as satellite servicing, there may be a situation when the satellite being serv iced is non-cooperative and is more ma ssive than the satellite which is providing the servicing. In this si tuation the ACS on board the servic ing satellite must be able to provide large amounts of torque re sponsively to stabilize the system An illustration of this is shown in Fig. 1-3. Research Focus The research focus of this thesis is in the developm ent of a ground base d testbed to test and evaluate propellantless ACS me thodology for smallsats which can endure the volume, mass, and power constraints while still maintaining performa nce in terms of high torque, slew rates, and

PAGE 15

15 precision. The ACS methodology presented in this thesis attempts to validate single-gimbal control moment gyroscopes (SGCMGs) as a viable option of attitude control actuators for smallsats which can contribute high torque, slew rates, and attitude precision while meeting their volume, mass, and power constraints. Examples of other attitude actuators including SGCMGs are discussed in the next chapter for comparison. Fig. 1-1. On-orbit assembly Fig. 1-2. Earth monitoring in LEO Fig. 1-3. Post-docking stabilization

PAGE 16

16 CHAPTER 2 ATTITUDE CONTROL ACTUATORS Types of Attitude Actuators Attitud e control actuators for satellites can be passive or active. Passive attitude control actuators such as gravity gradie nt booms use the gravitational fi eld of the Earth to provide a bounded attitude between 5i in the orbital plane. Because the gravitational field is conservative, gravity gradient torques g do not dissipate the satellit es kinetic energy or dump its angular momentum. The tor que is dependent on the distance R between the spacecrafts center of mass (CM) and Eart hs CM, the spacecraft prin ciple moments of inertias iJ, and the small angles of deviation between the spacecrafts principle axes and the Local vertical-Local horizon coordinate frame i For small angular displacements the torque is expressed as, 231 312 33 0gJJ JJ R Another type of passive attitude control util izes hysteresis rods or permanent magnets. Typically permanent magnets carry accuracy on the same order of magnitude as gravity gradient booms but cause a satellite to flip at the poles. These actuators like gravity booms are conservative and do not dissipate energy or dump angular moment um. Hysteresis rods on the other hand are passive actuators that may dissipate energy and du mp angular momentum. These actuators can be used to null the angular rates of satellites with residual angular momentum after launch [ 10]. There are th ree main categories of active at titude control actuators. These categories include magnet torquers, reaction control devices, and flywheel actuators. Magnet torquers are the most co mmon attitude control actuators utilized in smallsats (e.g., magnet coils and torque rods). Th ese actuators are cheaper and lighter than other attitude control

PAGE 17

17 actuators used onboard smallsats. Figure 2-1 show s an example of a magnet coil for attitude control. Magnetic torquers produce a torque B as a reaction to the change of the local magnetic field of the satellite to that of the Earth. This torque is modele d as an interaction (cross product) between, the magnetic moment of the spacecraft and Earths magnetic field direction B as, B B Since the torque is a function of the cross product between the satellites magnetic moment and Earths magnetic field, magnet torquers suffer from a singularity which limits their actuation where no components of torque are available along and B Due to the change of the local magnetic field of the satellite, sensors for att itude determination such as magnetometers are unable to take readings during use of these actuators until the residual magnetic field of the satellite has decayed. These actu ators loose performance as a function of distance squared hence are less useful for satellites at or bits with higher altitudes than LEO. Reaction control devices such as thrusters re quire propellant which is at this time not available for the majority of nanoand pico-sat ellite missions. These devices produce a torque on the satellite via propulsion through jets of fluid, heated or cooled plasma and gas, or charged ions with use of a magnetic field. Thrusters that utilize the chemical energy within the propellant (e.g., hydrazine) produce greater magnitude of output thrust but require la rge radiators to dump the expended heat created. Thrusters of this so rt are not realizable fo r smallsats which do not have the extra mass and volume to contain larg e radiators for the thermal protection system. The last type of attitude c ontrol actuators known as flywheel actuators can be broken down into two types, reaction and momentum wheel s (RWs/MWs) and control moment gyroscopes (CMGs). These are the most accurate sources of attitude control.

PAGE 18

18 Reaction and momentum wheels work off th e same principle by producing a reaction torque opposite in direction to the acceleration of a flywheel. The difference between MWs and RWs is that, RWs have a zero nominal operating sp eed (i.e., the speed that the wheels are spun back down to after the maneuver is completed) When spinning RWs/MWs back down to their operating speed, the satellite will revert to its original orientation due to the conservation of angular momentum. For this reason both MW s and RWs must have other actuators onboard such as thrusters or magnet torquers to dump the excess angular momentum, or these actuators will suffer angular momentum saturation of thei r flywheels. Three RWs/MWs are needed for full three-axis attitude control of a satellite. An example of a RW used for CubeSat and nanosatellite missions is shown below in Fig. 2-2. The second type of flywheel attitude c ontrol actuator known as CMGs provide an instantaneous gyroscopic torque al ong a torque axis defined as the axis perpendicular to the flywheel and gimbal axes. These actuators rely on shifting the dire ction of satellites body angular momentum to apply a torque. Three CMGs are needed for three-axis control in general, although a fourth is typically added for si ngularity avoidance a nd/or redundancy. Performance characteristics for typical attitude control actuators were investigated and are shown in Tables 2-1 and Tables A-1 through A-3 in the appendix. Table 2-1 is taken from Space Mission Analysis and Design by Wertz [ 11]. The tables in Appendix A are for currently available R Ws, CMGs, and magnet torquers and supplement that which is shown in Table 2-1 [ 1223]. In Table 2-1, the only attitude control actuato rs th at can meet the performance demands of high precision while maintaining the low power requirement for small satellite systems are flywheel actuators consisting of RWs, MWs, and CMGs. Magnet torquers/coils do not produce

PAGE 19

19 sufficient output torque for a gi ven input power and are less accura te due to the uncertainty of Earths dynamic magnetic field model. A survey was conducted of existing RWs a nd CMGs. The results of the survey are shown in Figs. 2-3 and Fig. 2-4. The supporting data for these figures is provided by Tables A-1 and A-2 Figures 2-3 and 2-4 show that CMGs includi ng that which is developed in the Spacecraft Orientation Buoyancy Experiment al Kiosk (SOBEK) have a highe r torque output for a given mass and input power than RWs. These figures also illustrate that CMGs are currently nonexistent for classes of satellite s equal to or smaller than micr o-satellites. The scarcity of these actuators is due to their complexity in de sign and use. If this complexity was removed and they were designed in a black-boxed manner (i .e., accepting only an input torque or attitude and producing it), then they could potentially find more use in smaller satellites due to higher torque per mass and power advantage over RWs. There are three CMGs that have been developed for micro-satellites by companies such as EADS/Astrium, Honeywell, and Surrey Space Technology Laboratory (SSTL) [ 2428] Fig. 2-5 shows one of these three, the SSTL single-gi m bal control momentum gyroscope (SGCMG) which flew on the Turkish micro-satellite Bilsat-1 [ 29]. Types of CMGs There are two varieties of CMGs: ones with a singe controllable degree of freedom (DOF) known as SGCMGs and those with multiple cont rollable DOF known as double-gimbal control momentum gyroscopes (DGCMGs) and variable-s peed control moment gyroscopes (VSCMGs). The benefits and drawbacks of each type of CMG is discussed next.

PAGE 20

20 Single Gimbal Control Momentum Gyroscopes SGCMGs such as the ones shown in Fig. 2-5 are the least mechanica lly complex form of CMGs. They utilize a single gimbal axis to pr oduce a gyroscopic torque by rotating a constant speed flywheel about a gimbaled axis perpendi cular to the spin-axis. A minimum of three SGCMGs are needed to obtain thre e-axis attitude control with th ese actuators. However an ACS using SGCMGs in some cases utilizes four of th ese actuators to avoid performance inefficiencies known as internal singularities. The benefit of these actuators among other CMGs lies in the property of torque amplification which is the ratio of output torque from the ACS containing SGCMGs to input torque from the gimbal moto rs. Discussion of singularities as well as performance characterization of SGCMGs in terms of momentum management and torque amplification is discussed in chapter 3. Double Gimbal Control Momentum Gyroscopes Double-gimbal control momentum gyroscope s (DGCMG) are the most mechanically complex form of CMG actuators. These actu ators produce torque by the same method as SGCMGs but possess a redundant mechan ical controllable DOF in term s of an extra gimbal axis. This redundancy can be useful when a set of two actuators is used for underactuated attitude control of a satellite but can be harmful when gimbal lock occurs. Gimbal lock occurs when the spin axis of a DGCMG aligns with its axis of freedom and in consequence deprives the DGCMG of one of its controllable DOF thereby elimina ting its useful propertie s. There are however acceptable methods of avoiding these singularities which in turn leaves DGCMGs the benefit of multiple degrees of freedom. Although with the addi tion of an extra gimbal to these actuators, they are heavier and more mechanically comp lex than other forms of CMGs which excludes them from being a viable option for smalls ats. A figure of a DGCMG developed by L3 Communications for the International Space Station shown in Fig. 2-6.

PAGE 21

21 Variable Speed Control Momentum Gyroscopes VSCMGs are used for their extra DOF adde d through their reacti on wheel mode where flywheel accelerations are nonzero. This extra DOF has its benefits in that it does not require steering logic and is able to produce the required torque when a singularity is encountered [ 3032]. Although when in reaction wheel mode the fl ywheel acceleration s require shaft power and therefore input torque thereby increasing their re quired power and reducing their available torque amplification. Another effect of the reaction wheel modes of VSCMGs is that it becomes near impossible to isolate vibrations from the varyi ng flywheel speeds. In addition to the larger power requirement, less torque amplification, and i nduced vibration, the moto r driver circuitry of VSCMGs is more complicated, and they require more computational and electrical power than standard SGCMGs to control. SGCMGs are considered here as the most viable option for th e study of smallsat attitude control and hence were chos en for the SOBEK testbed. Table 2-1. Attitude control actuators and their specifications Actuator Output Accuracy (deg) Mass (Kg) Power (W) Magnetic Torquers 1 to 4,000 Am2 1 to 5 0.4 to 50 0.6 to 16 Hot Gas Thrusters Cold Gas Thrusters 0.5 to 9,000 Nm <5Nm 1 to 5 variable N/A RW & MW 0.005 to 1 Nm 0.001 to 1 1 to 20 10 to 110 CMG 10 to 500 Nm 0.001 to 1 >1 5 to 150 Fig. 2-1. Magnet coil from Tokyo Inst itute of Technology Cute-1.7 CubeSat

PAGE 22

22 Fig. 2-2. Dynacon miniature RW 0 20 40 60 80 100 120 140 -202468 log10(torque) mNmPower W RW CMG SOBEK Fig. 2-3. Semi-log plot of input power versus output torque for off-the-shelf RWs and CMGs 0 20 40 60 80 100 120 140 -202468 log10(torque) mNmMass kg RW CMG SOBEK Fig. 2-4. Semi-log plot of mass versus out put torque for off-the-shelf RWs and CMGs

PAGE 23

23 Fig. 2-5. SSTL micro-satell ite SGCMG flown on Bilsat-1 Fig. 2-6. DGCMG developed by L3 Communications

PAGE 24

24 CHAPTER 3 SGCMG DYNAMICS AND PERFORMA NCE CHARACTERIZATION Dynamics CMGs produce a gyroscopic tor que about a torque axis it orthogonal to both the gimbal and spin axes ig and is, respectively. An illustration of the CMG gimbal frame iG and how it relates to the spacecraft body frame B is shown in Fig. 3-1 with gimbal angles, velocities and accelerations, iii and flywheel velociti es and accelerations, ii To analyze the torque generated by CMGs, we first develop the angular momentum expression and coordinatize it in the spacecraf t body fixed frame. Application of Eulers equation assuming the external torq ue is zero yields the governing equation for the CMG system. This development is as follows. The total spacecraft-centroidal angul ar momentum of the system coordinatized in the body frame is BBBBB CCCHJh (3-1) with the spacecraft centro idal inertia tensor incl uding the CMG components B CJ, the spacecraft body fixed angular velocity BB and the total CMG angular momentum B Ch about the spacecraft center of mass (CM) c oordinatized in the body frame. The spacecraft centroidal inertia B CJ in Eq. (3-2) contai ns a fixed inertia B CJ, time varying inertia from the comb ined gimbal-wheel system 1iin GT BGgwBG iCIC, and the parallel axis components of inertia associated with each CMG 1iiiiBTBBBT iCCCCmRRRR with mass im and position from the spacecraft CM to that of the flywheel, iB C R 11iiiiiin BBTBTBBBT CCBGgwBGiCCCC iJJCICmRRRR (3-2)

PAGE 25

25 Assuming that iB C R is held fairly constant and ther efore its time derivative is zero, we differentiate BBB CJ with respect to time as, 1 BBB BBB C C BBBBBB CCdJ J d JAJ dtdt (3-3) This CMG contribution of angular momentum B Ch coordinatized in the body frame is expressed as, 4 1i iG B CBG ihCh (3-4) and is found in by summing the contributions of each individual CMG angular momenta iGh iniG which is transformed to the body frame by the DCM iBGC. Note that for SGCMGs, the DCM iBGC varies only with the gimbal angles of the ith CMG i The angular momentum of the ith CMG iGh in G is, 0iwi G g wiI h I where i and w I are the ith flywheels angular speed and centroidal inertia about its spin axis is andi and g w I are the ith gimbal rate and the centroidal inertia of the combined wheel-gimbal system about its gimbal axis ig Taking the time derivative of B Ch and observing that ,,BB CChh we get, BBBB CCCCdhhhh ddd dtdtdtdt (3-5) with the Jacobian coefficient matrices of the CMG states are defined as,

PAGE 26

26 2B Ch A (3-7) B Ch B (3-8) B Ch C (3-9) Combining these terms we get, 2 B Cdh ABC dt If we also combine 1A with 2A which are both multiplied by the gimbal rates we have a more compact representation of the output torque from the CMGs in Eq. (3-10) where the complete Jacobian matrix is 12||DAABC 11 2|| ,,B B C Cdh d hAAABCDX dtdt (3-10) The form of Eq. (3-10) is similar to Schaub and Junkins [ 33] with exception of the different notation. The gyrosco pic components of torque are deri ved from the product of the skew symmetric matrix BB with the total centroidal angular momentum of the system B CH as, 11iiiiiiBBBBBBBBBBB CSC n BBTBTBBBTBB BGgwBGiCCCC iHJh CICmRRRR (3-11) The complete rotational equati on of motion for this system in Eq. (3-12) is found through the sum of Eqs. (3-3), (3-10), and (3-11) giving,

PAGE 27

27 1 11 1iiiiii iiiiiiB BBBBBBBBBBB C SSC n TBTBBBTBB BGgwBGiCCCC i n BBGTBTBBBTBB BGgwBGiCCCC idH JJDXh dt CICmRRRR CICmRRRR (3-12) If it is assumed that, BB CSJJ g wiwiII and g wiwigwiwiiIIII 0i and there are no external torques applied to the spacecraft the reduced equation of motion for this system is 00B BBBBBBBBBBB C SSCdH JJhAh dt (3-13) with 0 whI and 0 B ChhA A 3-2-3 DCM through Euler angles iii where i and i are the constant spacing and inclination angles, and i is the time varying gimbal angle transforms iG to B for the four-CMG pyramid configuration in Fi g. 3-1. This DCM is expanded as, ,cos,siniiiiiiiiiiiii BGiiiiiiiiiiii iiiiiiiccssscsscscc Csccssccssssc scccscc cs where (3.14) The 3-2-3 Euler sequence can be visualized in Fig. 3-2.

PAGE 28

28 Now that we have reduced the general equati ons of motion to an easily useable form we can address the performance char acterization of SGCMGs. There are three areas that describe the performance characteristics of SGCMGs: torque amplification, singularities, and momentum management. Each one of these areas is discussed next. Torque Amplification Torque amplification for a SGCMG is defi ned as a higher amount of output torque produced than a given gimbal moto r input torque required, assuming negligible torque needed by the flywheel motors. This is a reasonable a ssumption when considering SGCMGs because their flywheels require minimum torque to spin at a constant speed. In the literature (e.g., [ 34]), torque am plification is defi ned for a single actuator as, 22 2 2Torque Amplification = B Ci out i BBB in Ch h (3-15) If it is assumed that there are no disturba nce torques on the system, the equation for internal torque can be found by decomposing Eq. (3-13) of a single SGCMG actuator into two separate elements with equal ma gnitudes and opposite signs as, BBBBBBBBBBBB SSCCJJhh where BB CiChh The internal torque can be decom posed to into an output torque B outCih from the CMG and input torque BBB inCh which is the gyroscopi c contribution from the satellites angular velocity in Eq. (3-1 3). If the two-norm of newly defined out is divided by that of in we get the same form of the torque am plification equation shown in Eq. (3-15).

PAGE 29

29 When considering picoor nano-class satellite s where the ACS might take up a majority of the mass and volume of the overall satellite and the gimbal-wheel system mass may make up a majority of the mass and volume of the ACS, the contributions to torque and angular momentum from the gimbal-wheel system and the gimbal mo tor friction inefficiencies may not be assumed negligible. Therefore, Eq. (3-15) is an uppe r bound for the torque am plification equation. Useful information is gathered from Eq. (3-15) such as the insight that the spacecraft angular velocity has to be less than the gimbal rate to have a torque amplification greater then one. To understand torque amplification of a single SGCMG for use on smallsats a lower bound was developed [ 35] and [ 36]. This lower bound is 0 2 2 2sgnout i BBB in Cgwidisih hIFF (3-16) with gimbal input torques g wi I and motor friction inefficiencies with dynamic and static coefficients dFand s F. If the motor friction inefficiencies are of th e main concern we get a reduced form of Eq. (3-16) as, 2 2 2 2 2sgnout ii in disi BB B CFF h h and more compactly as, 1 sgnii si d i iF F hh (3-17) where torque from static friction is assumed to be negligibly small when compared to that for dynamic friction.

PAGE 30

30 Equation (3-17), which considers the mechanical inefficiencies of the gimbal motors, is a lower bound in the presence of these motor friction inefficiencies on the torque amplification of an SGCMG. From this equation, it can be seen that choosing the correct gimbal motor to make the optimization parameter dh F as high as possible, will reduce the motor friction effects on the SGCMG torque amplification. It should be no ted that increasing the flywheel speed as much as possible will give the largest torque amplification for small satellites where mass, volume and power are limited but will also increase their jitt er. The motor inefficiencies of the DC motors used to rotate the flywheels are overlooked for SG CMGs that are not variable speed, due to the fact that they have a very small dynamic friction coefficient and their static friction and residual torques are largest on startup. Now that torque amplification has been discussed, singularities associated with SGCMGs are addressed. Singularities SGCMGs experience two classes of singularities. These singularities are internal and external singularities ea ch of which are instances where th e required control torque cannot be produced. External Singularities External singularities occur when the addition of angular momentum from an external source (e.g., disturbances) saturates the SGCMGs of the ACS. Of these disturbance torques, aerodynamic torques are considered to be the most harmful when in LEO due to their size and constant nature in comparison wi th other disturbance torques. External singularities are usually addressed in mission planning and considered the most harmful at the end of a satellites life cycle when de-orbiting.

PAGE 31

31 Internal Singularities Internal singularities are encountered on the fly when th e Jacobian matrix of the SGCMG ACS becomes rank deficient, whereby the torque vectors lie in a plane and no torque can be produced normal to this plane. Unlike external singularities they must be handled online and cannot simply be designed for a priori. The tw o types of internal si ngularities are known as elliptic which are unavoidable/inescapable thr ough null motion (i.e., null motion is motion of the gimbals that produces no net torque on the sate llite) and hyperbolic wh ich are avoidable through null motion. Singularity avoidance strategi es are discussed later in this chapter Singular Surfaces The CMG torque vector directions it are the columns of the Jacobian matrix. Therefore, singularity occurs when all it lie in the same plane. There are 2n singular configurations (i.e., singular sets of gimbal angles, ) for any singular direction u normal to this plane. All possible it for each CMG span a plane and an example of these planes for a four-CMG pyramid configuration can be visualized clearly in Fig. 33. The CMG torque vector direction is defined as, torque vector =ii itg s With a given singular direction u not parallel to a gimbal axis, there are two possibilities for singularity to occur along each singular direct ion. These possibilitie s are shown in shown below. 0, 0 or 0, 0iiiiutusutus It should be noted that the occurrence of si ngularity when two gimbal axes are aligned has already addressed as gimbal lock and occurs in DGCMGs.

PAGE 32

32 Using the notation found in reference [ 35] and [ 32], we can define iisignus. With these definitions, the spin and torque axis ve ctors at singular states can be defined as, 2 i iiii ig sguug gu and 2 i iii igu tug gu Therefore the total norma lized angular momentum B Ch of the CMGs at singular states corresponding to u is, 2 i B Ciiii ii ig hsguug gu When1i (or 1i due to symmetry), the external singular surface known as the angular momentum envelope shown in Fig. 34 is formed as the locus of the maximum projections of the angular mo mentum of each CMG at the singular directions. External singularities occur on the surface of this envelo pe when the total internal SGCMG angular momentum is less than that which is external to the spacecraft, and the SGCMGs are then saturated. The internal singular surface is formed in the same way as that of the external singular surface with exception to one i = -1. An examples of the inte rnal singular surfaces for an ACS containing four SGCMGs in a pyramidal arrangement at an inclination angle 35.26 is shown in Figs. 3-5. The inclination angle s hould not be confused with the skew angle used to characterize the angular momentum envelope [ 37] and [ 38]. The skew angle is the minimum angle of the SGCMG planes shown in Fig 3-3 with that of the horizontal. For a spherical angular

PAGE 33

33 momentum envelope,9054.74. Units of Fig. 3-4 and Fi g. 3-5 are in terms of the nominal angular momentum magnitude 0h (i.e. diameter of angular momentum envelope involving four SGCMGs is less than or equal to four 0h). In Fig. 3-4, locations that have white circles or holes on the momentum envelope are shown where the gimbal axes are located and there is no angular momentum available. The knowledge of where these singularities oc cur is known, and there have been methods developed that steer away or escape from th ese singularities known as steering logics. These methods are discussed in the next section. Steering Logics An ACS utilizing SGCMGs requires an addi tion of a fourth actuator for minimum redundancy to avoid singularities through null motion (i.e., moti on is motion of the gimbals whereby no net torque is produced). This fourth actuator renders the systems Jacobian matrix to be non-square which in turn requires a pseudo-inverse to map the output torque 3B Ch onto the gimbal rates 4 Recall, that there are 2n singular configurations for each singular direction of n SGCMGs. Therefore, it is important to note that the addition of actuators does not eliminate the problem of singularities but provide s a null space where singularity avoidance may be provided. The Moore-Penrose pseudo-inverse solution to the gimbal rates is represented as 01B CAh h (3-18) where the Moore-Penrose pseudo-inverse is, 1TTAAAA

PAGE 34

34 The Moore-Penrose pseudo-inverse solution for th e gimbal rates in Eq. (3-18) fails when TAA becomes singular. To avoid/escape singularit ies when mapping the output torque onto the gimbal rates, a steering logic is applied. Steering logics can be broken up into two main groups: those which provide null motion for singularity avoidance such as local gradient methods and global avoidance algorithms, and pseudo-inverse solutions which escape singularities through load ing of the Jacobian matrix singular values. Common drawbacks to these st eering logics are computational complexity, inability to avoid all internal singularities, or loss accuracy by adding torque errors to keep the Jacobian non-singular with lo ading of its singular values. Local gradient methods were of the earliest methods used to steer away from internal SGCMG singularities by Margulies and Auburn [ 39]. This local gradient method, like most steering logics, requires a fourth SGCMG added to provide a 1-dim ensional null space where null motion can be applied to steer the gimbals away from singularities. The Jacobian matrix null space provides multiple soluti ons to the gimbal state equation to steer the gimbals away from singularity while maneuvering. An ex ample of local gradient methods is, 01B CAhIAAd h (3-19) where B Ch is the SGCMG output torque and 0h is the nominal magnitude of angular momentum. This equation is similar to Eq. (3-18) with the addition of a null motion vector d which is mapped to the Jacobian matr ix null space by the projection matrix I AA The contribution of null motion is scal ed by the singularity parameter 2 0exp m (3-20) which is an explicit functi on of the singularity measure m

PAGE 35

35 1 det 2TmAA (3-21) and design constants and0 When using local gradient methods, the null motion vector dcan be chosen as the gradient, 1234,,,T f fff d of a objective function 1f m This null vector d produces null motion in the direction that maximizes m or the distance away from singularity. It is important to note that steering logics utilizing only null motion are unable to avoid elliptic internal singularities [40] and [ 41]. Local gradient methods shown here are com putationally intensive and ca nnot avoid all internal singulari ties such as elliptic internal singularities where null motion is unsuccessful. Global avoidance methods shown in Eq. (3-22) such as those developed by (Paradiso and Kuhns [ 42] and [ 43]) are similar to local gradient methods and differ in the fact that the null motion vector is produced by the dif ference of the gimbal angle positions from a set of alternate gimbal angle configurations known as preferred trajectories. This difference is shown in Eq. (322), where are the preferred trajectories. 01B CAhIAA h (3-22)

PAGE 36

36 These preferred trajectories ar e usually calculated offline. This method is computationally intensive and is not suitable for online use. The most popular pseudo inverse soluti on is the SR inverse developed first by (444546 1 # TTAAAAI (3-23) # 01B CAh h (3-24) #A This SR-inverse solution works by adding a positive definite matrix of torque errors I to the positive semi-definite matrix TAA to leave the matrix TAAI positive definite and hence non-singular. Rather than avoiding internal singularities as discussed previously, this method which is developed from the Moore-Penros e pseudo-inverse, approaches singularity and makes a rapid transition to escape. At escape ther e is a jump in gimbal rates due to the added torque error. As a consequen ce of these added torque errors, this method is not the preferred method for precision pointing. Under certain circumstances (i.e., when det0TAA and # B ChnullA), this method can become locked into a singularity, so the generalized SRinverse (GSR) was developed by Wie [ 45] The GSR-inverse 1 # TTAAAAE (3-25) is made up a positive definite symmetric matrix

PAGE 37

37 033022 033011 0220111sinsin sin1sin0 sinsin1tt Ett tt (3-26) composed of time varying modular parametersi and scaled by the singularity parameter in Eq. (3-20). These modulati ng parameters ensure that B Ch will not stay locked in the null space of the pseudo-inverse, which is not explicitly dependent on time. The GSR-inverse has been shown to avoid all internal singularities but adds torque error to all directions. Therefore, the original SR-inve rse was modified again to minimize the amount of torque error added to the gimbal s when avoiding internal singularities. This method developed by Ford is known as Singular Direction Avoidance (SDA) with the pseudo-inverse T SDASDAAVSU and from which the gimbal rates are 01B SDACAh h (3-43) where V and U are unitary matrices and SDAS is, 3 2 12 311 ,, 0SDA Tdiag S The benefit of using this method ove r the SR-inverse methods is that it only adds to rque error to the smallest singular value, which in turn requ ires less null motion and has smoother gimbal rate trajectories than the previous SR and GSR-inverse methods. A drawback of this method is that it is required that a SVD be calculated for A every time step, making the steering logic computationally intensive. Also it is developed directly from the original SR-inverse and therefore can become locked in si ngularity when the cont rol torque is in the singular direction of the pseudo-inverse.

PAGE 38

38 The discussed methods of avoiding or escap ing singularities are of the most popular of steering logics developed for SGCMGs [ 47]. Additional methods have been developed to avoid internal singularities associat ed with SGCMGs [42-44]. An exam ple of one such method includes game theory [ 48]. Other methods for singularity av oidance have been developed with the addition of an extra d egree of freedo m while using VSCMGs instead of SGCMGs. In the case of VSCMG, steering logics asso ciated with gimbal velocity and acceleration have been developed by Schaub and Junkins [ 31] and [ 49]. These methods use the extra degree of freedom through flywheel accel eration to give an ACS using these actuators the ability to always produce the required output torque needed. Recall it was stated that VSCMGs do not re quire steering logics due to their reaction wheels modes. Therefore the addition of a fourth VSCMG instead may provide power tracking for systems such as the Flywheel Attitude Control and Energy Transmission System (FACETS) and the Integrated Power and Att itude Control System (IPACS) [ 50]. These systems are able to use the nu ll space instead to monitor power tr ansmission while converting the extra kinetic energy gathered from spinning down the flywheel s after a maneuver to electricity charging the power system on board. Systems such as FACETS require ceramic wheels to make use of the kinetic energy provided from spinning down the flywheels and were deemed unsafe due to the high flywheel speeds of forty-thousand rpm and have henceforth not been flight tested. The next section discusses momentum ma nagement which deals with the dumping of external and internal angular momentum in order to maintain the performance of an ACS using SGCMGs. Momentum Management There are two situations where the angular mo mentum of the system must be managed in order to maintain the required ACS performance. These situations occur from external angular

PAGE 39

39 momentum added into the system from torque di sturbances or internal components of angular momentum remaining post maneuver from gimbal angle saturation. External Angular Momentum Offloading External angular momentum accumulated by the spacecraft from non-zero disturbance torques is considered a priori for each specific mi ssion. If angular momentum is added into the system from the disturbance torque and it is greater than that fr om which is allotted from the CMGs, the system encounters an external singularity as discussed in section 3.2. Assuming that a spacecraft is in LEO orbit then the addition of aerodynamic torque is the only source that is considered to saturate the system. This is so because the magnetic disturbance torque is exceptionally small and the disturbance torque from the gravity gradient does not saturate the system with angular momentum due to its period ic nature. Also disturbance torques from solar pressure effects are neglected due to the small amount of solar cell surface area on small satellites. When considering small satellites, aerodynamic tor ques add a negligible rate of angular momentum into the system in most standard LEO orbits [ 36]. The addition of angular mom entum from disturbance tor ques is a slower process than that for the internal angular momentum buildup from the SGCMGs. For this re ason there is greater concern for offloading of internal angular momentum. Internal Angular Momentum Offloading Internal angular momentum accumulates when components of angular momentum from the SGCMG cannot be cancelled out due to gimbal angle saturation. As previously mentioned these gimbal angle constraints can be reduced by using magnetic bearings a nd/or slip rings which allow for full range of gimbal rotation. Two cases arise when internal angular momentum needs to be offloaded. The first case occurs when ex cess components of internal angular momentum is left over from maneuvers due to gimbal angle saturation where unwanted components of angular

PAGE 40

40 momentum are unable to be cancelled out. The s econd case is at startup of the flywheels, where needs to be angular momentum dumped to stop the satellite from tumbling due to the offset in its angular momentum direction. A typical startup confi guration of an ACS c ontaining SGCMGs is at zero-momentum. This configuration for a f our-pyramid SGCMG cluster has all of the angular momentum vectors of each SG CMG in the body x-y plane at a 0 leaving a zero net momentum for the ACS. An example of this confi guration is shown in Fig. 3-6. In this figure, all of the angular momentum vectors are in the plane although it is not a singular configuration because all of the torque vectors are not in the same plane. Fo r smallsats the offloading of the excess angular momentum on startup as well as that due to gimbal angle saturation can be accomplished by use of magnetic actuators. Now that mathematical and physical aspect s of ACSs containing SGCMGs have been discussed, the types of ACS configurations containing these actuat ors as well is the development and choice of the SOBEK configuration is addressed in the next chapter. Fig. 3-1. The SGCMG with gimbal coordinate axes i 3 b 1 b 2 b i G B ,,iii ,ii

PAGE 41

41 Fig. 3-2. A 3-2-3 rotation sequence through angles iii Fig. 3-3. Intersecting planes sp anning of SGCMG torque directions Fig. 3-4. Angular momentum envelope for 35.26 and 1,1,1,1 it 1 it 2 it i g is it 1 b 2 b 3 bi i i i i

PAGE 42

42 Fig. 3-5. Internal singular surface 35.26 and 1,1,1,1 Fig. 3-6. Zero-momentum configuration 1h 2h 3h 4h

PAGE 43

43 CHAPTER 4 SGCMG ACS CONFIGURATIONS Rooftop Configuration The roof top design uses two groups of SGCMGs, each with parallel gimbal axes slanted at an angle to each other shown in Fig. 4-1. A CMG system in this configuration will never have elliptic internal singularities [ 40]. This means that they will always have continuous mom entum trajectories and therefore have c ontinuous gimbal trajectories. Th is configuration suffers from situations where its Jacobian may approach rank 1. The singular direction where this occurs is on the intersection of the two rows of SGCMGs shown as the red arrow in Fig. 4-1. Box Configuration The box configuration is a variation of the r ooftop configuration containing four SGCMGs with an inclination angle of 90. This configuration is shown in Fig. 4-2 where the red arrow indicates the same rank 1 dire ction where the torque vectors lie. The internal singular surface associated with this configuration cove rs a large surface of the angular momentum envelope and is seen as the large empty area presen t inside the external singular surface in Fig. 4-3. The singularity represented by this singular surface shown at 1h is avoided in the box configuration by utilizing only three SGCMG a nd keeping the third as a spare while staying within the 1h constraint. Only thre e of the four SGCMG will be used at any given time so the Jacobian is square and there is no need for a pseudoinverse to map the gimbal rates onto the torque. The major drawback s of this design are that it is not as compact as other designs and its performance is constrained to 1h of angular momentum, causing the ACS to require larger SGCMGs to meet the missions angular momentum requirements. Pyramid Configuration Common pyramid configurations have numbers of SGCMGs in groups of four or six. ACSs containing these amounts of SGCMGs and have both elliptic and hyperbolic internal

PAGE 44

44 singularities. Although an ACS utilizing SGCMGs in a pyramid configuration is susceptible to the elliptic internal sing ularities, there are methods of escap ing these singularities that were discussed in the previous chapte r. This configuration has the benefit of giving a somewhat spherical angular momentum envelope for the right choice of inclination angle of the SGCMGs. The SOBEK testbed will utilize a minimal redundancy (i.e., four SGCMGs) pyramid configuration for three-axis control while applying the GS R-inverse steering logic method discussed in chapter 3. SOBEK Pyramid Configuration The SOBEK testbeds ACS is shown in Fig. 4-4 with motion capture for attitude tracking provided by Phase Space and its visual interpretati on of the mapped wireless LEDs in the upper right hand corner. SOBEKs ACS uses four SGCM Gs in a pyramid configuration. The testbed was originally designed to float on an air beari ng table with multiple ACSs where the footprint of the ACS was limited while trying to minimize weight. For this reason, the inclination was chosen as angle 54.74. It should also be mentioned that the formulations in literature consider a zero reference gimbal angle position shifted 90 from where the SOBEK zero reference angle is. The new formulat ed SGCMG angular momentum vector 1234 01234 1234 B Cscsscs hhsscssc cccccccc (4-1) and Jacobian matrix 1234 1234 1234 B Csscssc h Acsscss cscscscs (4-2) takes these changes into account.

PAGE 45

45 Other testbeds such as those at Virginia Polytechnical Institute, Lawrence Livermore Laboratory, SSTL, and Honeywell s Line of Site (LOS) testbed are of the few testbeds that comprise the known heritage for this research [ 5153]. This testbed differs from the ones previous ly referenced in that it is physically smaller and does not utilize an air bearing but a pivot point to mitigate the forces of gravity. Wh en making the testbed near marginally stable at a point, the only forces and torque s considered are the small amount of friction from that point as well as the residual moment from gravity. An illustration of SOBEK that contains the ACS in and stand which holds the ACS on a pivot point is shown in Fig. 45. This illustration was made in ProEngineer, where the SOBEK inertia a nd mass properties were also estimated. The next section discusses the hardware c hosen for the SOBEK and its design shown in Fig. 4-6 and [ 54]. SOBEK Hardware The basic electrical hardware for preliminary setup of the SOBEK ACS consists of four Arsape AM2224 two-phase Stepper motors four Faulhaber 1525 BRC brushless DC motors with integ rated electronics and four State Electr onics 600 Series digital optical encoders. The electrical hardware for the current setup only contains a wireless Phasespace LED controller, a Roving Networks RS232 bluetooth module and the EZHR17EN stepper driver boards with an RS232 to RS485 adapter. The entire ACS is power ed by two lith ium polymer batteries. This is possible because there are voltage regulators presen t to step down the voltage from 14 to 12 volts and 5 volts to meet the specific volta ges of the mechanical hardware. Currently the setup has no onboard processing and runs code written in C++ from a Panasonic Toughbook with a Linux operating system. Th e future setup of th is testbed will have everything coded on a single processor board co ntaining the wireless Bluetooth module and motor driver circuitry.

PAGE 46

46 There are significant reasons for the choice of th e hardware for the testbed. One of these reasons deals with the digital opt ical encoders. These encoders happen to be more accurate then potentiometers and are dual quadrat ure which are directly compatible with the Allmotion driver boards chosen. The Allmotion EZHR17EN stepper driver boards have the required output current and voltage necessary for the given Arsa pe AM2224 stepper motors and so were chosen for this reason. In addition to having the correct current and volta ges, these boards also have the ability to daisy chain to other boards. That is for a given set of four motor driver boards controlling a total of four steppe r motors two daisy chain boards are needed to send commands to all four motors through the serial port. It is assumed that the gi mbal rate commands are sent fast enough serially that th ey arrive at all four motors at the same time from the daisy chain boards. The nominal speed of the stepper motors needed to be geared down from 134:1. There was a significant loss in efficiency by doing this so th e optical rotary encoders are added on the other end of the shaft to reduce the effect of this uncertainty by feedback. The Faulhaber 1525 BRC motors were chosen because they have a high enough stall torque to spin up the flywheels as well as integrated electronics includi ng a frequency output to measure the speed of the motors. This is valuable for use in th e testbed, because knowledge of the flywheel angular speed is an integral part of the SGCMG dynamics and control. If the flywheels were spinning at different speeds the angular momentum vector of each SGCMG would have different magnitudes and this would add attitude control error. SOBEK Mechanical Design There are some unique design c onsiderations that were carri ed out in the design of the SOBEK testbed. One of these design consider ations dealt with how the ACS would rotate. To approximate the mass and inertia proper ties, ProEngineer was used. The inertia properties were measured about the ACS CM. If the CM was made to be below the point of

PAGE 47

47 rotation, then the system would be stable (i.e a pendulum equilibrium point). It should be mentioned that the distance between the point of rotation and the CM cannot be too great or the reaction momentum from gravity will saturate th e SGCMGs. For this reason the CM was chosen to be just below the point of rotation to make the system close to neutral stability. To account for the body frame x and y deviation of the CM from the point of rotation, the ACS is connected to a platform that contains a sy stem of weight ballasts. These ballasts allow two-dimensional adjustment of th e CM. The motor driver and daisy chain boards as well as the batteries and the power terminal are mounted on th is platform to conserve space. The z-axis deviation of the CM was accounted for by the vertical adjustment of the ACS center. SOBEK Mechanical, Power and Output Specifications Mechanical specification s of the SOBEK ACS are shown in Ta ble 4-1 with a total output torque of 267 mNm, mass of 5.489 kg, and power consumption of 8.36 W. SOBEK Attitude Determination The SOBEK ACS is fitted with LEDs that are distinguished by different blinking frequencies where they are read by a system of ei ght cameras. Shown in Fig. 4-7 is the testbed setup including the Phasespace attitude determination system. The Phasespace motion capture system is chosen because it is a simple and co mpatible off-the-shelf way to measure attitude accurately. This is possible because Phasesp ace works on the principle of taking position measurements from LEDs located on the SOBEK r obot thereby calculating the attitude of the tracked object. This completes the attitude determination system for SOBEK. The next chapter discusses the simulation and e xperimental results of this testbed. The simulation results utilize the mass and inertia properties estimated in ProEngineer as the basis for its model. Also discussed in this chapter will be the limitations of the current design iteration of

PAGE 48

48 the SOBEK testbed as well as an introduction of the application package interface software (API) that runs it. Table 4-1 Mechanical and Power Specifications of the SOBEK ACS Micro ACS Power (W) Current (A) Voltage (V) Weight (kg) Output Torque (mNm) Stepper motors 6.000 0.500 12.000 267 DC motors 1.776 0.148 12.000 Encoders 0.600 0.120 5.000 Total 8.376 5.489 Fig. 4-1. Honeywell roof-top configuration Fig. 4-2. Honeywell box configuration

PAGE 49

49 Fig. 4-3. External singular surface for box type SGCMG ACS Fig. 4-3. Honeywell pyramid configuration

PAGE 50

50 Fig. 4-4. Pyramidal SGCMG ACS Fig. 4-5. Rendition of SOBEK testbed LEDs Phase Space Graphical User Interface

PAGE 51

51 Fig. 4-6. ACS with hardware components q c Fig. 4-7. PhaseSpace att itude determination system Phase Space Wireless LED controller Digital Optical Encoders Bluetooth Module Allmotion EZHR17EN Stepper Driver Boards Allmotion Daisy Chain Boards Arsape AM2224 Stepper Motors Faulhaber 1525 BRC DC motors Lithium Polymer Battery

PAGE 52

52 CHAPTER 5 RESULTS Simulation Model The error quaternion state equation is 4 44411 11 22 1 22 2BB BB T BBee eee GQ eee e where, e and 4e are the vector and scalar elements of the error quaternion and BB is the spacecraft angular velocity coor dinatized in the body frame. The equation for the spacecraft model is BBBBBBBB CCJJ where, B CJis the spacecraft centroidal inertia mentioned in Eq. (3-1) and is the internal control torque from the SGCMGs mentioned in Eq. (3-1 4), both of which are coordinatized in the body frame. The equation for the commanded SGCMG output torque is, BB B B CChh The output torque is mapped onto the gimbal rates with the GSR-inverse discussed in chapter 3 giving the gimb al rate solution of # 01B actCAh h (5-1) where, act is the actual column matrix of commanded gimbal rates used to generate the given output torque considering the torque error added from the GSR-inverse. The internal SGCMG torque is found in through a nonlinear rest to rest control logic

PAGE 53

53 BBBBBBB CKeC J (5-2) with the symmetric positive-definite controller stiffness and damping gain matrices 2B CKkJ and B CCcJ. The non-linear exact model knowledge (E MK) controller shown in has been proven to be asymptotically stable [ 45] and a Lyapunov stability analysis was performed on this controller to understand its be havior when uncertainty is present in the system Lyapunov Stability Analysis of EMK Attitude Controller A positive definite Lyapunov function is chosen as, 2 1 141 10 2T TBBBBB CVeeeKJ (5-3) with time derivative equal to 1 144 44 1221 111 221 222T TBBBBB C T TBBBBBB T BBBBBBB CVeeeeKJ eeeee KJ (5-4) To obtain asymptotic stability, Eq. (5-4) must be made negative definite. To achieve asymptotic stability the control to rque (i.e., internal torque of the SGCMGs) is chosen as Eq. (52). Substituting from Eq. (5-1) into1V that of Eq. (5-4), we end up with 10 2T BBBBc V k The final result of this substitution is that the time derivative of the Lyapunov function is negative semi-definite. To prove that the contro ller in Eq. (5-2) gives a global asymptotically stable result, we must note that the spacecraft angu lar velocity goes to zero asymptotically for the following reasons:

PAGE 54

54 ,BB BBLeLLL 2BB L BB is uniformly continuous If we apply Barbalats lemma with these conditio ns, the spacecraft angular velocity goes to zero asymptotically. If we revisit the control la w in Eq. (5-2), we can see now that it is not possible for the angular velocity to go to zero asymptotically unless the error quaternion vector elements also go to zero, therefore this controller is asym ptotically stable. This controller is EMK which assumes full st ate feedback, therefore the global asymptotic stability does not necessarily hold if accurate full state f eedback is not available. The current iteration of SOBEK does not contai n gyros for angular ra te calculation and encoders for gimbal rate position. This iter ation also has no online method of calculating the flywheel speeds, therefore the gimbal angles a nd angular rates are estim ated with uncertainty present and the flywheel speeds are calculated be fore experimentation. The gimbal angles are estimated through an Euler forward integra tion of the solution to Eq. (5-1) as, 1 kkkkt where kt is the time step of the control loop. The equation for the calculation of the ACS angular rates is 11 1 1 1 4442, kk kkk k BB k kk kee e e Qe ee e t where, 4kke Q e is always invertible.

PAGE 55

55 Due to the uncertainty present in this iterati on, the best result that the controller can produce on this testbed are shown to be uniformly ultimately bounded (UUB). To account for the many sources of uncertainty in this itera tion and obtain the UUB re sult, another stability analysis of this controller is carried out with uncertainty considered. The control torque is dependent on both the estimated angular rates and gimbal angles. The gimbal angles are integrated from the gimbal rates which are mapped from the control torque. Recall also that the torque utilizes minimal torque errors for singularity avoidance when using the GSR-inverse and therefore these torque errors must be cons idered in addition to the uncertainty. Lyapunov Stability Analysis of Actual At titude Controller with State Uncertainty For the stability analysis, the ac tual control torque and angular velocities of the system are defined as, act BB act with the uncertainties and If we substitute the actual contro l torque and angular velocities into Eq. (5-3), we are left with 2 1 241 10 2TT B act CactVeeeKJ and its time derivative 1 2TBT actactactCactactVKJe (5-5) Assuming that the uncertainties and have negligible time derivatives, the control torque from Eq. (5-2) give the result 11 2TB B B B B B BBT act C actCactactVKKeCJ Je with a simplified results

PAGE 56

56 1 2 111 2T BBBBBBTBBB CCC TT BBBBBBB CCCcIJJ J V k Jc I JJ (5-6) Examining Eq. (5-6) while assuming that the uncertainty is less than the actual control torque and angular veloc ity, the best result possible is UUB on the order of the same magnitude as that of the uncertainty for this control law. Now that the stability analyses have been carried out, the results from simulation and experiment is discussed. Simulation Results and Discussion The simulations were carried out in a Matlab environment with parameters shown in Table 5-1. Shown in Fig. 5-1 (A) is th e error quaternion vector elements for the first simulation where, 1e 2e and 3e are the quaternion error vector elemen ts about the roll, pitch, and yaw directions. This result at first glance seems to have great performance. Recall that this simulation was run assuming exact model knowledge in the absence of un certainty. The next plot shown in Fig. 5-1 B) shows the commanded gimbal rates throughout the simulation. In this figure it is clear that the gimbal ra tes are large in the beginning of the simulation. This is due to the fact that the simulation st arts off with the SGCMG gimbals in a singular configuration for a torque needed about the 3 b axis. Figure 5-1 (B) also shows the trend of the gimbal rates going null after the maneuver is comp leted which is not the case when disturbance torques are added to the system. In this situati on, the gimbal rates will need to compensate for the addition of angular momentum an d will continue to be nonzero.

PAGE 57

57 Figure 5-1 (C) shows the spacecraft angular rates of the ACS. In this figure, the angular rates are nullified after the maneuver is completed. This is the desired affect for a rest to rest maneuver. The final plot of this simula tion is that of the singularity measure m in Fig. 5-1 (D) Two things can be attributed to Fig. 5-1 (D). The first is that the singularity measure is zero at the beginning of the simulation (i.e., Jacobian is rank deficient when all gimbal angles are zeroed) and it ramps up quickly to transit from singularity within the first two seconds. Although the ACS is initially at a singular configuration, use of th is steering logic enables it to transit away from singularity in a timely manner. The second conclusion for this figure is that m becomes constant after the maneuver is completed. This should be obvious because it is an explicit function of the gimbal angles. Before an experiment was carried out, another simulation was run with the same k and c values from the previous simulation including th e addition of a random error signal added to the angular velocity measurement and gimbal angle in tegration. This random error was added in an attempt to model the uncertainty of the gimbal angles and angular rates as 10.03BBBB kkrand 10.1kkrand It is clear from the results of this simulation are shown in Figs. 5-2, while considering an EMK control logic when the controller has uncerta inty within its states, the performance is heavily degraded to the point of instability. In addition to this simulation another one including a higher choice of gains k and c was run to compare results and validate the uniformly ultimately bounded result of the stability analysis in Eq. (5-6). The results in Figs. 53 prove the opposite end of the stability analysis

PAGE 58

58 which is with higher choices of a k and c, the steady state error of the UUB solution can be reduced. Experimental Results and Discussion The flywheels onboard the SGCMGs of the AC S are not balanced and are unstable at certain wheel speeds. For safety reasons the wh eels are left to spin at a maximum speed of around 6000 rpm to keep them within a stable vibration frequency. This speed does not produce enough angular momentum, and hence, enough torque to overcome the moment from gravity to do large off-axis 1 b or 2 b-axis maneuvers and therefore the e xperimental results are for a yaw maneuvers about the 3 b axis. The GUI associated with this testbed is shown below in Fig. 5-4. This GUI in addition to rest of the API software was developed by Andrew Waldrum at the University of Florida specifically for SOBEK. The flywheel angular speeds were calcul ated offline through the output frequency feedback of the DC motors and found to be of different magnitude among the four SGCMGs. A variation of Eq. (5-1) 1# 0B actCHAh is used to account for the va riation in nominal angular momenta for the individual SGCMGs where 0H is the diagonal matrix of nominal angular momenta. In this GUI the gains k and c are increased at intervals c = 2.5 k to where k varies from .08 and 0.2 in the first experiment. There is no to rque to steer away from singularity at startup because the gains are zeroed out initially. All experiments we re carried out with the same steering logic parameters and initial conditions as those in the first simulation and only differed by choice of control gain. An initial experiment was run with the controller given in Eq. (5-2).

PAGE 59

59 It is shown in Fig. 5-5 (A) that at around forty seconds the quaternion error diverges, which can be caused by either the controllers instability, propagation uncertainty from the estimated states, or both. The reason for this error propagation is due to the error rates produced from the uncertainty of the estimates. The mane uvers in these experiment s start out with a high initial error about the 3 b axis and attempt to align the ACS to a quaternion of 0001T cq The next three plots shown in Figs. 5-5 (B)(D ) show the gimbal rate s, angular rates, and singularity measure associated with this maneuver. The next experiment took into consideration the result of the stability analyses and the last two simulations. This experi ment was carried out with higher values of the c ontrol gains and its results are shown in Figs. 5-6. In these experiments k was chosen greater than that for the previous experiment keeping the same c/k ratio. The quaternion error vector elements in Fi g. 5-6 (A) contain be tter results than the previous experiment although they also posses low frequency oscillations. The steady state error in this figure is measured as, 11 22min2sin,22sin0.0826 4.7441eeer a d where, the angle e represents the minimum angle needed to be traced out to align the two frames B and G about an eigen-axis. This metric is valid because the error quaternion is still a unit quaternion whereby the 2norm of the vector elements is equal to sin2e. Noted in red on the plots of Fig. 5-6 (B) and (D), are the large jumps in gimbal rates when tr ansiting a singularity. This was previously said to occur when the SR or GSR-inverse method is applied for singularity escape.

PAGE 60

60 The result is still oscillating about zero after sixty seconds and the steady state error 0.0826 4.7441erad is still quite large. It is believed through a la rger choice of gains, preferably ones with a higher ratio of ck will reduce the oscillations and steady state error. The next experiment shown in Figs. 5-7 was carried out with a larger value of control gains with a higher c/k ratio to improve the results from the pr evious experiment. These values were k = 0.32 and c = 3.0 k. The error quaternion vect or elements for this expe riment converged with a much lower steady state error of 0.0376 1.0779erad and a smaller amount of steady-state oscillation due to the higher c control gain. The behavior of the system for the chosen control gains is more stable as well as more efficient in terms of less wasted control effort than that for the previous experiment. This is due to the larger ratio of c/k and the fact that the maneuver did not encounter an additional internal singularity within the ti me of the experiment. With this larger ck ratio, the steady state error oscillations are reduced by a much larger c gain. The next plot shown is the angular rates versus time for this experiment. Angular rates shown in Fig. 5-7 (C) show a more desirable behavior than that of Fig. 5-6 (C) in terms of high slew rates with less oscillation. The last plot of this experiment is that of the singularity measure. It has already been discussed that behavior of this system is more desirable than the previous experiments with the choice of the current gains, and now it also seen that at no time did this system become singular. A final experiment was carried out were the Moore-Penrose pseudo-inverse which possessed no steering logic (i.e.00 ). The quaternion error of th is experiment is shown in Fig. 5-8. The SOBEK ACS for all experiments is initially at singularity, theref ore without a steering logic large gimbal rates are commanded and the system tumbles out of control as shown in Fig.

PAGE 61

61 5-8. From this figure, it is clear that without use of steeri ng logics while utilizing SGCMGs, drastic consequences may arise. In the future iterations of the SOBEK testbed, if encoders are provided are for gimbal angle measurement, tri-axial gyros provided for a ngular rate measurements and use of onboard feedback for the flywheel angular speeds will produce more accurate attitude maneuvers with elimination of state uncertainty. Table 5-1. Simulation parameter values Parameters Values Units 0h 0.04 Nms k 0.8 none c 2 none 112233,,JdiagJJJ 0.0668,0.0756,0.0815diag Nm2 1234CCCCT Cqqqqq 0001T none 54.74 degrees 12340Tqqqqq 0.9380000.3466T none 1230T 000T rad/s 10(1)rand 0.01(1)rand none 202 rand 0.01(2)rand none 30(3)rand 0.01(3)rand none 2 0expm 20.01exp20m none 12340T 0000T rad

PAGE 62

62 A B C D Fig. 5-1. ACS simulation plots for k = 0.08 and c = 2.5k: A) Quaternion error vector elements versus time, B) Gimbal rates ve rsus time, C) Angular rates versus time, and D) Singularity parameter versus time. 0 10 20 30 40 50 60 70 0 0.02 0.04 0.06 0.08 0.1 time (s)m 0 10 20 30 40 50 60 70 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 time (s) (rad/s) 1 2 3 0 10 20 30 40 50 60 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 time (s)d /dt(rad/s) d 1/dt d 2/dt d 3/dt d 4/dt 0 10 20 30 40 50 60 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 time (s)e e1 e2 e3

PAGE 63

63 A B C D Fig. 5-2. ACS simulation plots for k = 0.08 and c = 2.5k: A) Quaternion error vector elements versus time, B) Gimbal rates ve rsus time, C) Angular rates versus time, and D) Singularity measure versus time. 0 10 20 30 40 50 60 70 0 0.5 1 1.5 2 time (s)m 0 10 20 30 40 50 60 70 -0.1 -0.05 0 0.05 0.1 0.15 time (s) (rad/s) 1 2 3 0 10 20 30 40 50 60 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 time(s) d /dt(rad/s) d 1/dt d 2/dt d 3/dt d 4/dt 0 10 20 30 40 50 60 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 time (s)e e1 e2 e3

PAGE 64

64 A B C D Fig. 5-3. ACS simulation plots for k = 4 and c = 3k: A) Quaternion error vector elements versus time, B) Gimbal rates versus time, C) Angular rates versus time, and D) Singularity measure versus time. 0 10 20 30 40 50 60 70 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 time (s)m 0 10 20 30 40 50 60 70 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 time (s) (rad/s) 1 2 3 0 10 20 30 40 50 60 -0.04 -0.03 -0.02 -0.01 0 0.01 time (s)d /dt(rad/s) d 1/dt d 2/dt d 3/dt d 4/dt 0 10 20 30 40 50 60 -0.4 -0.3 -0.2 -0.1 0 0.1 time (s)e e1 e2 e3

PAGE 65

65 e e Fig. 5-4. SOBEK testbed GUI

PAGE 66

66 A B C D Fig. 5-5. ACS expe rimental plots for k = 0.8 and c = 2.5k: A) Quaternion error vector elements versus time, B) Gimbal rates ve rsus time, C) Angular rates versus Time, and D) Singularity measure versus time. 0 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time (s)m 0 10 20 30 40 50 60 70 80 -1 0 1 2 3 4 time ( s ) (rad/s) 1 2 3 0 10 20 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 time (s)d /dt(rad/s) d 1/dt d 2/dt d 3/dt d 4/dt 0 10 20 30 40 50 60 70 80 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 time (s)e e1 e2 e3

PAGE 67

67 A B C D Fig. 5-6. ACS expe rimental plots for k = 0.24 and c = 2.0k: A) Quaternion error vector elements versus time, B) Gimbal rates ve rsus time, C) Angular rates versus time, and D) Singularity measure versus time. 0 10 20 30 40 50 60 70 0 0.2 0.4 0.6 0.8 1 1.2 1.4 time (s)m 0 10 20 30 40 50 60 70 -3 -2 -1 0 1 2 3 time (s) (rad/s) 1 2 3 0 10 20 30 40 50 60 70 -1 -0.5 0 0.5 1 1.5 2 2.5 3 time (s)d /dt(rad/s) d 1/dt d 2/dt d 3/dt d 4/dt 0 10 20 30 40 50 60 70 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 time (s)e e1 e2 e3

PAGE 68

68 A B C D Fig. 5-7. ACS expe rimental plots for k = 0.32 and c = 3.0k: A) Quaternion error vector elements versus time, B) Gimbal rates ve rsus time, C) Angular rates versus time, D) Singularity measure versus time. 0 10 20 30 40 50 60 70 80 0 0.5 1 1.5 time (s)m 0 10 20 30 40 50 60 70 80 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 time (s) (rad/s) 1 2 3 0 10 20 30 40 50 60 70 80 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 time (s)d /dt(rad/s) d 1/dt d 2/dt d 3/dt d 4/dt 0 10 20 30 40 50 60 70 80 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 time (s)e e1 e2 e3

PAGE 69

69 Fig. 5-8. ACS experiment quaterni on error of Moore Penrose solution 0 2 4 6 8 10 12 14 -0.2 -0.1 0 0.1 0.2 0.3 time (s)e e1 e2 e3

PAGE 70

70 CHAPTER 6 CONCLUSION AND FUTURE RESEARCH Conclusion This thesis discusses the development and tes ting of an attitude control testbed utilizing four SGCMGs in an orthogonal pyramid configurat ion. SOBEK is of great utility for evaluating attitude control algorithms and has provided insights into the unde rstanding and application of control moment gyro-based actuators. In its current form, SOBEK has only attitude feedback and does not have onboard processing or internal feed back (i.e., operates in an open-loop manner). With the addition of onboard processing and sensor s for state feedback, it is expected that the testbed will prove to be invaluable for testi ng control algorithms and steering logics while showing precision in terms of attitude accuracy from SGCMGs. Future Research Future research for this attitude control system will include the changes to the new iteration of SOBEK and testing of different cont rol algorithms and steering logics. A picosatellite CMG-based ACS will be developed using SOBEK as the alpha-model to analyze software that will fly on this miniaturized actuator.

PAGE 71

71 APPENDIX A OFF-THE-SHELF RW, CMG, AND MAGNET TORQUER DATA SPECIFICATIONS Table A-1. Off-the-sh elf perfor mance specifications for RWs RWA Torque (mNm)Mass (kg)Power (W) Bradford Engineering W05 100 3.2 73 W18 200 4.95 63 W45 300 6.95 64 Honeywell HR 0610 55 3.6 80 Dynacon MicroWheel 200 30 0.93 2 MicroWheel 1000 30 1.3 9 Vectronic Aerospace RW1 20 1.8 25 SunSpace SunSpace RW 50 1.98 35 TELDIX RSI 01-5/15 5 0.6 4 RSI 01-5/28 5 0.7 4 Goodrich (Ithaco) TW26E300 300 13.9 L3 Space Comm. MARS RWA 15 MWA-50 160 10.5 100 Orbital Sciences LEO Star Wheel 140 3.6 55 SSTL MicroWheel-5S 5 0.5 8 MicroWheel-3S 3 0.75 3 MiniWheel-20S 20 3.2 14 MiniWheel-20S-X 20 2.6 14 NanoWheel-m500S 0.5 0.08 0.5 MicroWheel-10SP-S 10 1.1 5

PAGE 72

72 Table A-2. Off-the-sh elf performance specifications for CMGs CMG Torque (mNm)Mass (kg)Power (W) Honeywell M50 74570 33.1 75 M95 128803 38.6 129 M160 216931 44 217 M225 305059 54 305 M325 440641 61.2 441 M325D 440641 61.2 441 M715 969410 89.8 949 M600 813491 81.6 814 M1400 2E+06 132 1899 M1300 2E+06 125 1716 EADS/ASTRIUM CMG 15-45S 45000 15 SSTL Bilsat-1 CMG 95000 2.2 12 SSG SOBEK ACS 66.7 1.38 2.09

PAGE 73

73 Table A-3. Off-the-shelf performance specifications for Magnet Torquers Magnet Torquer Output (Am2) Mass (kg)Power (W) Microcrosm MT2-1 2.5 0.2 0.5 MT5-2-M 6.0 0.3 0.77 MT6-1 7.0 0.23 0.25 MT6-2 8.0 0.3 0.5 MT10-2-H 12 0.35 1.0 MT15-1-M 20 0.43 1.11 MT30-2-CGS 40 1.050 3.6 MT30-2-GRC 35 1.4 1.5 MT70-1 75 2.6 3.8 MT70-2 75 2.2 2.6 MT80-1 100 4.12 3.0 MT80-2-M 90 2.3 4.7 MT110-2 120 3.8 2.9 MT140-2 170 5.3 1.9 MT250-2 300 5.5 4.8 MT400-2-L 500 7.8 9.0 MT400-2 550 11.0 11.4 MT400-1 550 9.2 7.7 Strauss Space Micro Magnetic Torque Rods15 0.6 3.5 Nano Magnetic Torque Rods 1.5 0.2 0.36 Magnetic Torque Coils 0.1 0.04 0.1 Vectronic Aerospace MTR-5 100 0.75 6.0

PAGE 74

74 LIST OF REFERENCES [1] Cebrowski, A. and Raymond, J., Oper ationally Responsive Space: A New Defense Business Model, Parameters, 2005, pp. 67. [2] Alminde, L., Bisgaard, M., Vinther, D., Viscor, T., and stergard, K., The AAU CubeSat Student Satellite Project: Architectur al Overview and Lessons Learned, 16th IFAC Symposium on Automatic Control in Aerospace, (Russia), 2004. [3] Waydo, S., Henry, D., and Campbell, M., CubeSat Design for LEO-Based Earth Science Missions, IEEE Aerospace Conference Proceedings, Vol. 1, 2002. [4] Long, M., Lorenz, A., Rodgers, G., Tapio, E., Tran, G., Jackson, K., Twiggs, R., and Bleier, T., A CubeSat Derived Design for A Unique Academic Research Mission in Earthquake Signature Detection, AIAA Small Satellite Conference, 2003. [5] Vladimirova, T., Wu, X., Jallad, A., a nd Bridges, C., Distri buted Computing in Reconfigurable Picosatellite Networks, Proceedings of the Second NASA/ESA Conference on Adaptive Hardware and Systems, IEEE Computer Society Washington, DC, USA, 2007, pp. 682. [6] Puig-Suari, J., Turner, C., and Twiggs R., CubeSat: The Development and Launch Support Infrastructure for Eighteen Differe nt Satellite Customers on One Launch, 15 th Annual AIAA/USU Conference on Small Satellites, Logan, Utah, 2001. [7] Twiggs, B. and Puig-Suari, J., CUBE SAT Design Specifications Document, 2003. [8] Straight, S. and Davis, T ., Tactical Satellite 3: Require ments Development for Responsive Space Missions, Air Force Research Lab, Georgia Institute of Technology, 2005. [9] Wegner, P. and Kiziah, C., Pulling the Pieces Together at AFRL [10] Kim, K., Analysis of Hysteresis for Attitude Control of a Microsatellite, San Jose State University, http://www.engr.sjsu.edu/spartnik/adac.html. [11] Larson, W. and Wertz, J., Space Mission Analysis and Design, Microcosm, Inc., Torrance, CA (US), 1992. [12] http://www.sstl.co.uk/Products/Subsystems/Available_Subsystems Surrey Space Centre, Tech. rep., [Accessed 07/15/2008]. [13] http://www.smad.com/ie/ieframessr2.html Micro crosm,, Tech. rep., [Accessed 12/06/2008]. [14] http://www.stras-space.co m/strasspace products Stras Space, Tech. rep., [Accessed 12/06/2008].

PAGE 75

75 [15] http://www.bradford space.com/ pdf/ be_datasheet_rwu_sep2006.pdf Bradford Engineering, Tech. rep., [Accessed 07/15/2008]. [16] https://commerce.honeywell.com/webapp/wcs/ stores/servlet/ECate goryDisplay? catalog Id=10106&storeId=10651&categoryId=13832&cacheId=1000000000000001&langId=-1 Honeywell Engineering, T ech. rep., [Accessed 07/15/2008]. [17] http://microsat.sm.bmstu.ru/e library/A lgorithm s/Hardware/wheels/productpdf_6.pdf Dynacon Enterprises Limited, Tech. rep., [Accessed 07/15/2008]. [18] http://www.vectronic aerospace. com/htm l/magnet_torquer.html Vectronic Aerospace, Tech. rep., [Accessed 12/06/2008]. [19] http://www.sunspace.co.za/products/index.htm Sunspace, Tech. rep., [Accessed 07/15/2008]. [20] http://microsat.sm.bmstu.ru/e library/Algorithms/Hardware/wheels/RSI01.pdf Teldix, Tech. rep., [Accessed 07/15/2008]. [21] http://www.oss.goodrich.com/ReactionWheels.html Goodrich, Tech rep., [Accessed 07/15/2008]. [22] http://www.l3com.com/products servi ces/productservice.aspx?id=310&type=b L3 Communications, Tech. re p., [Accessed 07/15/2008]. [23] http://microsat.sm.bmstu.ru/e library/Algorithms/Hardware/wheels/ssci5.pdf O rbital Sciences, Tech. rep., [Accessed 07/15/2008]. [24] Defendini, A., Morand, J., Faucheux, P., Gu ay, P., Rabejac, C., Bangertt, K., and Heimel, H., Control Moment Gyroscope(CMG) Solutions For Small, Agile Satellites, Advances in the Astronautical Sciences, Vol. 121, 2005, pp. 51. [25] Abbott, F., Hamilton, B., Kreider, T., Di Leonardo, P., and Smith, D., MCS Revolution, Advances in the Astronautical Sciences, Vol. 125, 2006, pp. 99. [26] Lappas, V., Steyn, W., and Underwood, C., Design and Testing of a Control Moment Gyroscope Cluster for Small Satellites, Journal of Spacecraft and Rockets, Vol. 42, No. 4, 2005, pp. 729. [27] Davis, P., Momentum Syst em Concepts and Trades for the New Class of Smaller Lower Cost Satellites, Advances in the Astronautical Sciences, Vol. 125, 2006, pp. 13. [28] Hamilton, B. and Underhill, B., Modern Momentum Systems for Spacecraft Attitude Control, Advances in the As tronautical Sciences, Vol. 125, 2006, pp. 57. [29] Lappas, V. and Underwood, C., Experime ntal Testing of a CMG Cluster for Agile Microsatellites, 54th International Astronautical Congress of the International Astronautical Federation (IAF),(Bremen, Germany), Sep.-Oct, 2003.

PAGE 76

76 [30] Wie, B., Bailey, D., and Heiberg, C ., Singularity Robust Stee ring Logic for Redundant Single-Gimbal Contro l Moment Gyros, Journal of Guidance, Control, and Dynamics, Vol. 24, No. 5, 2001, pp. 865. [31] Schaub, H. and Junkins, J., CMG Singul arity Avoidance Using VSCMG Null Motion (Variable Speed Control Moment Gyroscope), AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Boston, MA, 1998, pp. 213. [32] Yoon, H. and Tsiotras, P., Singularity Analys is of Variable-Speed Control Moment Gyros, Journal of Guidance, Control, and Dynamics, Vol. 27, No. 3, 2004, pp. 374. [33] Schaub, H., Vadali, S., and Junkins, J., F eedback Control Law for Variable Speed Control Moment Gyros, Journal of the Astronautical Sciences, Vol. 46, No. 3, 1998, pp. 307. [34] Lappas, V., Steyn, W., and Underwood, C ., Torque Amplification of Control Moment Gyros, Electronics Letters, Vol. 38, No. 15, 2002, pp. 837. [35] Leve, F., Tatsch, A., and Fitz-Coy, N., A Scalable Control Moment Gyro Design for Attitude Control of Micro-, Nano-, and Pico-Class Satellites, Advances in the Astronautical Sciences, Vol. 128, Published for the American Astronautical Society by Univelt; 1999, 2007, p. 235. [36] Leve, F. and Tatsch, A., Fitz.-Coy. N., Three-axis attitude co ntrol design for on-orbit robotics, 2007. [37] Lappas, V., Steyn, W., and Underwood, C ., Practical Results on the Development of a Control Moment Gyro Based Attitude Control System for Agile Small Satellites, 16th Annual AIAA/USU Conference on Small Satell ites,(Logan, UT), Utah State University, Aug, 2002. [38] Wie, B., Space Vehicle Dynamics and Control, AIAA, 1998. [39] Margulies, G. and Aubrun, J., Geometric Theory of Single-Gimbal Control Moment Gyro Systems, Journal of the Astronautical Sciences, Vol. 26, No. 2, 1978, pp. 159. [40] Bedrossian, N., Paradiso, J., Bergmann, E., and Rowell, D., Steering Law Design for Redundant Single-Gimbal Cont rol Moment Gyroscopes, Journal of Guidance, Control, and Dynamics, Vol. 13, No. 6, 1990, pp. 1083. [41] Kurokawa, H., Survey of Theory and St eering Laws of Single-Gimbal Control Moment Gyros, Journal of Guidance Control and Dynamics, Vol. 30, No. 5, 2007, pp. 1331. [42] Joseph, A., Global Steering of Single Gi mballed Control Moment Gyroscopes Using a Directed Search, Journal OF Guidance, Control, and Dynamics, Vol. 15, No. 5, 1992. [43] Kuhns, M. and Rodriguez, A., A Preferred Trajectory Tracki ng Steering Law for Spacecraft with Redundant CMGs, Proceedings of the American Control Conference, 1995., Vol. 5, 1995.

PAGE 77

77 [44] Nakamura, Y. and Hanafusa, H., Inverse Kinematic Solutions with Singularity Robustness for Robot Manipulator Control, ASME, Transactions, Journal of Dynamic Systems, Measurement, and Control, Vol. 108, 1986, pp. 163. [45] Wie, B., Bailey, D., and Heiberg, C., Rapid Multitarget Acquisition and Pointing Control of Agile Spacecraft, Journal of Guidance Control and Dynamics, Vol. 25, No. 1, 2002, pp. 96. [46] Ford, K. and Hall, C., Singular Direction Avoidance Steering for Control-Moment Gyros, Journal of Guidance Control and Dynamics, Vol. 23, No. 4, 2000, pp. 648. [47] Jung, D. and Tsiotras, P., An Experime ntal Comparison of CMG Steering Control Laws, Proceedings of the AIAA Astrodynamics Specialist Conference, 2004. [48] Lee, J., Bang, H., and Lee, H., Singul arity Avoidance by Game Theory for Control Moment Gyros, AIAA Guidance, Navigation, and Control Conference and Exhibit, 2005, pp. 1. [49] Schaub, H. and Junkins, J., Singularity Avoidance Using Null Motion and Variable-Speed Control Moment Gyros, Journal of Guidance, Control, and Dynamics, Vol. 23, No. 1, 2000, pp. 11. [50] Fausz, J. and Richie, D., Flywheel Simultaneous Attitude Control and Energy Storage Using a VSCMG Configuration, Proceedings of the 2000 IEEE International Conference on Control Applications, 2000., 2000, pp. 991. [51] Ledebuhr, A., Kordas, J., Ng, L., Jones, M., Whitehead, J., Breitfeller, E., Gaughan, R., Dittman, M., and Wilson, B., Autonomous Agile Micro Satellites and Supporting Technologies, AIAA Space Technology Conference and Exposition, 1999, pp. 99. [52] Underhill, B. and Hamilton, B., Momentum Control System and Line-of-Sight Testbed, Advances in the Astronautical Sciences, Vol. 125, 2006, pp. 543. [53] Jung, D. and Tsiotras, P., A 3-DoF Expe rimental Test-Bed for Integrated Attitude Dynamics and Control Research, 2003. [54] Leve, F., Design of a 3DOF Testbed For Microsatellite Autonomous Operations, AIAA Aerospace Sciences Meeting, 2006.

PAGE 78

78 BIOGRAPHICAL SKETCH Frederick Aaron Leve w as born in Hollywood, Florida, in 1981. In August 2000 he was accepted into the University of Floridas Departme nt of Aerospace Engineering in the College of Engineering where he pursued his bachelors de grees in Mechanical an d Aerospace Engineering. After completing his bachelors degrees in May 2005, he was accepted into the masters program in aerospace engineering at the University of Florida. While in the masters program, he received two awards in academia. In Janua ry 2007 he received the American Institute of Aeronautics and Astronautics Abe Zarem Award for Distinguished Achievement in Astronautics. For this award he was invited to Valencia, Spain, where he competed in the International Astronautical Federationss International Astronautical Congress Student Competition. Here he received the silver Herman Oberth medal in the graduate category. In May 2006, he was accepted to the Air Force Research Lab Space Scholars Program, where spent his summer conducting space research. His interests include satellite attitude control, satellite pur suit evasion, astrodynamics, and orbit relative motion.