SPACECRAFT ATTITUDE CONTROL: Modelling and Controller

Transcription

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+DF=B; HIN - Prosjektrapport
Master i teknologi
Postboks 385, Postboks 385
8505 NARVIK
Telefon: 76 96 60 00
Telefax: 76 96 68 10
Tittel
Dato
SPACECRAFT ATTITUDE CONTROL:
Modelling and Controller Design considering
Actuator Dynamics
07. juni 2006
Forfatter
Antall sider
David Hagen
56
Gradering
Åpen
Vedlegg
3
Institutt for
Studieretning
Data-, elektro- og romteknologi
Romteknologi
Veileder
Associate Professor Dr.ing. Per Johan
Nicklasson
Oppdragsgiver
Oppdragsgivers kontaktperson
HiN
PhD Student Raymond
Kristiansen
Sammendrag
Denne hovedoppgaven omhandler attitydekontroll av romfartøy med fokus på
aktuatordynamikk. En generell satellittmodell beskrevet med Euler parameter sammen
med generell modellering av aktuatorer er utledet. Aktuatordynamikk er undersøkt og to
generelle modeller er utledet. Tre kontrollere er utledet, hvor to tar hensyn til
aktuatordynamikken. En studie av SSETI ESMO satellitten er gjennomført og brukt som
simuleringsobjekt.
Abstract
This thesis concerns the attitude control of spacecrafts with special considerations on
actuator dynamics. A general model of a spacecraft described with Euler parameters
along with general models of actuators are deduced. Actuator dynamics are investigated
and there are deduced two general mathematical models. Three controllers are derived in
which two concern actuator dynamics. A case study of the SSETI European Student
Moon Orbiter is conducted and used for simulations.
Norske stikkord
Keywords
Attityde-kontrollsystemer
Modellering
Aktuatordynamikk
Regulatordesign
Attitude Control System
Modelling
Actuator Dynamics
Controller design
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3% 3& " 3$ 3$ 3& 3% "
3$ 3$ 5B8 =BG9FH=B; H<9 H<FIGH9F D@579A9BH :FCA ;=J9G
" " " "
,<9 HCH5@ HCFEI9 :FCA H<9 :CIF F957H=CB H<FIGH9FG 75B BCK 69 9LDF9GG98 5G
" *CH5H=CB 5FCIB8 5BM 5L=G C: H<9 G5H9@@=H9 =G A589 DCGG=6@9 6M OF=B; H<9 H<FIGH9FG =B D5=FG EI5H=CB =AD@=9G H<9 D5=FG DF9G9BH98 =B ,56@9 :CF >C=BH OF=B;
,56@9 ,<FIGH9F OF=B; H56@9
$ $ % % ,<9 H<FIGH :FCA 957< H<FIGH9F 6M DF=@ " =G
&
& & K<=7< ;=J9G H<9 HCFEI9 =B &A KCF?=B; CB H<9 G5H9@@=H9 5G
$
%
& % & $ !B CF89F HC ;=J9 5 EI5G=@=B95F 7CBB97H=CB 69HK99B H<9 7CBHFC@@9F =BDIH 5B8 57HI5HCF CIHDIH 5 DI@G9 K=8H<
AC8I@5HCF K=@@ 69 IG98 =B H<9 G=AI@5H=CBG ,<9 (/% K=@@ <5J9 5 8958 NCB9 G75@98 HC ;=J9 5 @CK9GH
DCGG=6@9 CIHDIH DI@G9 8IF5H=CB C: G97CB8 ,<=G ;=J9G 5 89G=F98 %= C:
&G
,<9 A5H<9A5H=75@ AC89@ C: H<9 F957H=CB H<FIGH9F 8MB5A=7G :CF H<9 +%' H<FIGH9FG 5F9 5GGIA98 75B 69
89G7F=698 6M H<9 G97CB8 CF89F ;9B9F5@ 57HI5HCF AC89@ =B K<9F9 H<9 H=A9 7CBGH5BHG 5F9 HIB98 HC
,<=G @95J9G H<9 AC89@ :CF H<9 89G=F98 57HI5HCF =BDIH 5G
%$
# #
)
=;IF9 G<CK 5 7CAD5F=GCB C: H<9 89G=F98 %= :FCA H<9 7CBHFC@@9F 5B8 H<9 F95@ %= :FCA H<9 H<FIGH9FG
0.12
w MiB
uMiB
0.1
0.08
0.06
0.04
0.02
0
0
0.05
0.1
0.15
0.2
=;IF9 CAD5F=GCB C: %% = 5B8 #% = !BJ9GH=;5H=CB C: #% = 5B8 BIA9F=75@ =BH9;F5H=CB G<CK H<5H
%% = #% =
5:H9F G97CB8 K<=7< =G H<9 <=;<9GH DCGG=6@9 DI@G9 F5H9 :FCA H<9 (/%
+'):/54 %.++29
,<9 F957H=CB K<99@ GMGH9A CB H<9 +%' =G F9GDCBG=6@9 :CF 5HH=HI89 A5BC9IJF9G CB H<9 G5H9@@=H9 5G K9@@
5G GH56=@=N5H=CB CF H<9 G=AI@5H=CBG =B H<=G H<9G=G 5 H9HF5<98FCB 7CBO;IF5H=CB C: H<9 F957H=CB K<99@G 5G
GI;;9GH98 =B 3$IB8 4 K=@@ 69 IG98 ,<=G =G 5 7CBO;IF5H=CB 7CBG=GH=B; C: :CIF K<99@G K<=7< 7C=B7=89
K=H< H<9 J97HCFG 89OB98 6M H<9 ;9CA9HF=75@ C6>97H H9HF5<98FCB ,<=G C6>97H 7CBG=GHG C: :CIF 9EI5@ HF=5B;@9G
5GG9A6@=B; 5 DMF5A=8 ,<9 J97HCF GHF9H7<9G :FCA H<9 J9FH9L9G 5B8 D9FD9B8=7I@5F HC =HG 5DDIFH9B5BH
6CHHCA D@5B9 ,<=G 7CBO;IF5H=CB =G 89G=;B98 :CF 5JC=8=B; 7C@8 F98IB85B7M C: F957H=CB K<99@G 5G K9@@ 5G
:I@O@@=B; F98IB85B7M GD97=O75H=CBG ,<=G =G A589 DCGG=6@9 5G 5 F9GI@H C: H<9 H9HF5<98FCB 7CBO;IF5H=CB
0.6
1
Z-axis
0.4
0.2
2
0
-0.2
3
4
-0.4
0.5
0
-0.5
Y-axis
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
X-axis
=;IF9 *957H=CB K<99@ D@579A9BH
K<=7< 5@@CKG CD9F5H=CBG 5:H9F 5 6F95?8CKB C: 5 F5B8CA F957H=CB K<99@ ,<9 D@579A9BH C: H<9 F957H=CB
K<99@ J97HCFG =B H<9 G5H9@@=H9 =G 89D=7H98 =B =;IF9 ,CFEI9 :FCA H<9 H9HF5<98FCB F957H=CB K<99@
7CBO;IF5H=CB 75B 69 KF=HH9B 5G 3$IB8 4
#
K=H<
# K<9F9 =G H<9 K9=;<H=B; A5HF=L 5B8 =G H<9 F957H=CB K<99@ 5@@C75H=CB A5HF=L CF H<9 7CBO;IF5H=CB
CB +%' 5@@ :CIF K<99@G DFCJ=89 HCFEI9 K<=7< =AD@M H<5H 5B8 # 697CA9G
# K<9F9 =G H<9 DG9I8C =BJ9FG9 C: ,<9 5@@C75H=CB A5HF=L 89OB9G H<9 J97HCFG H<9 K<99@G 5F9 5@=;B98
K=H< 5G
5$ 5$ 5$ 5
$
" " " "
5% 5% 5% 5
% 5& 5& 5& 5
&
M IG=B; H<9 ?BCKB 5B;@9 69HK99B 5@@ H<9 7C@IAB J97HCFG =B 5 H9HF5<98FCB K<=7< =G 5B8
:I@O@@=B; H<9 HKC 7F=H9F=5
5$ 5$ 5$ 5
$ 5% 5% 5% 5
% 5& 5& 5& 5
& 5 5% 5& $
5 5% 5& $
5 5% 5& $
5
$ 5
% 5
& 5G K9@@ 5G H<9 J97HCF DFCD9FH=9G
'$ ($ '% (% '& (& ;=J9G H<9 7CAD@9H9 5@@C75H=CB A5HF=L 5G 3$IB8 4
,<9 8=A9BG=CB=B; C: H<9 F957H=CB K<99@G :CF +%' =G BCH M9H 89H9FA=B98 5B8 :CF H<9 G=AI@5H=CBG =B H<=G
H<9G=G HKC D5F5A9H9FG 5F9 G9H ,<9 ACA9BH C: =B9FH=5 C: H<9 F957H=CB K<99@G # 5F9 7<CG9B 5G # ?;A 5B8 H<9 A5L=AIA K<99@ GD998 =G G9H HC FDA ,<9 HCH5@ 5B;I@5F ACA9BHIA 957< K<99@ 75B
GHCF9 697CA9G
-# # &AG
,<9 A5H<9A5H=75@ AC89@ C: H<9 F957H=CB K<99@ 8MB5A=7G :CF H<9 +%' K<99@G 5F9 5GGIA98 75B 69
89G7F=698 6M H<9 G97CB8 CF89F ;9B9F5@ 57HI5HCF AC89@ =B K<9B H<9 H=A9 7CBGH5BHG 5F9 HIB98 HC
,<=G @95J9G H<9 AC89@ :CF H<9 89G=F98 57HI5HCF =BDIH 5G
%$
# #
)
=;IF9 G<CK 5 7CAD5F=GCB C: H<9 89G=F98 HCFEI9 :FCA H<9 7CBHFC@@9F 5B8 H<9 F95@ HCFEI9 F95@=N98 6M
H<9 F957H=CB K<99@G
0.5
0.4
w
u
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
0.8
0.9
=;IF9 CAD5F=GCB C: % 5B8 # :CF H<9 +%' F957H=CB K<99@G
1
.'6:+8 !/3;2':/549
!B H<=G 7<5DH9F H<9 G=AI@5H=CB F9GI@HG C: H<9 H<F99 7CBHFC@@9FG :FCA ,56@9 =B <5DH9F K=@@ 69 DF9G9BH98
,<9 H<F99 7CBHFC@@9FG 5F9 G=AI@5H98 =B HKC AC89@G C: H<9 +%' G5H9@@=H9 K=H< H<9 H97<B=75@ GD97=O75H=CBG
DF9G9BH98 =B <5DH9F ,<9 OFGH G=AI@5H=CBG IG9 H<9 F957H=CB H<FIGH9F GMGH9A 5G 57HI5HCFG 5B8 H<9
CH<9F IH=@=N9 H<9 F957H=CB K<99@ GMGH9A +=B79 H<9 9AD<5G=G =B H<=G H<9G=G =G CB H<9 8MB5A=7G C: H<9
57HI5HCFG GCA9 G=AD@=O75H=CBG 577CF8=B; HC H<9 A=GG=CB GD97=O75H=CBG 5B8 9BJ=FCBA9BH =G 8CB9 ,<9
G=AI@5H=CBG 5F9 7CB8I7H98 =B 5 ?A 7=F7I@5F @IB5F CF6=H K<9F9 5@@ D9FHIF6=B; :CF79G 5F9 5GGIA98
HC 69 N9FC !BH9F57H=CBG 69HK99B H<9 HKC 57HI5HCFG :CF 9L5AD@9 ACA9BHIA 8IAD=B; 5B8 HF5BG:9F C:
GHCF98 ACA9BHIA =B H<9 F957H=CB K<99@G 9H7 K=@@ BCH 69 HF95H98 !BGH958 H<9 G=AI@5H=CBG K=@@ 7CB79FB
H<9 69<5J=CIF 5B8 D9F:CFA5B79 C: H<9 H<F99 7CBHFC@@9FG CB H<9 HKC 57HI5HCF GMGH9AG =GC@5H98 =B HKC
=B89D9B89BH G=AI@5H=CB 75G9G CF 957< C: H<9 HKC AC89@G C: +%' H<9F9 K=@@ 69 G=AI@5H98 HKC 5HH=HI89
A5BC9IJF9G CB9 GA5@@ 5B;@9 5B8 CB9 @5F;9 5B;@9 A5BC9IJF9 ,<9 GA5@@ 5B;@9 A5BC9IJF9 K=@@ 69 :FCA
H<9 =B=H=5@ 5HH=HI89
K<=7< =B I@9F 5B;@9 F9DF9G9BH5H=CB 7CFF9GDCB8 HC H<9 5HH=HI89 A5BC9IJF9 =B FC@@ D=H7< 5B8 M5K :FCA
,<9 @5F;9 5HH=HI89 A5BC9IJF9 K=@@ 69
K<=7< =B I@9F 5B;@9 F9DF9G9BH5H=CB 7CFF9GDCB8 HC
,<9 5B;I@5F J9@C7=HM 69HK99B H<9 =B=H=5@ 5B8 H<9 CF6=H :F5A9 =B H<9 % 5L=G :CF H<9 7=F7I@5F @IB5F
CF6=H 697CA9G
3
K<9F9 =G H<9 ;F5J=H5H=CB5@ 7CBGH5BH 5B8 =G H<9 A5GG C: H<9 ACCB
,<9 DFC79GG C: HIB=B; H<9 7CBHFC@@9FG GH5FHG K=H< 7CBHFC@@9F CB H<9 7CAD@9H9 G5H9@@=H9 AC89@ =B7@I8=B; H<9
57HI5HCF 8MB5A=7G !B=H=5@@M GI=H56@9 J5@I9G C: 5B8 K=H< F9GD97H HC G9HH@=B; H=A9 C: H<9 5HH=HI89 5B8
5B;I@5F J9@C7=H=9G 5F9 :CIB8 +I779GG=J9@M 7CBHFC@@9F =G HIB98 K=H< ?99D=B; 5B8 IB7<5B;98 :CF
:IFH<9F =ADFCJ9A9BH C: H<9 G9HH@=B; H=A9 =B5@@M =B 7CBHFC@@9F =G HIB98 :CF DCGG=6@M 69HH9F G9HH@=B;
H=A9 H<5B H<9 HKC OFGH 7CBHFC@@9FG FCA 957< G=AI@5H=CB H<9 I@9F D5F5A9H9F 9FFCF J97HCF ( 5B8 H<9
5B;I@5F J9@C7=H=9G =G D@CHH98 =B H<9 OFGH FCK C: H<9 D@CHG !B H<9 G97CB8 FCK H<9 89G=F98 HCFEI9 :FCA
H<9 7CBHFC@@9F =G D@CHH98 :C@@CK98 6M H<9 CIHDIH :FCA H<9 57HI5HCF 8MB5A=7G AC89@ !H =G BCH9KCFH<M H<5H
! =B ( =G D@CHH98 5G ! :CF 69HH9F J=G=6=@=HM =B H<9 D@CHG !B 5@@ D@CHG H<9 F9GI@H :FCA 7CBHFC@@9F H=@@ =G
D@CHH98 OFGH 5B8 OB5@@M 5 D@CH C: 7CBHFC@@9F K=H<CIH 57HI5HCF 8MB5A=7G ,<=G D@CH =G D9F:CFA98 CB H<9
G5H9@@=H9 K=H< CB@M H<9 CIHDIH :FCA H<9 (/% :CF H<9 H<FIGH9FG 5B8 A5L=AIA GD=B GHCF5;9 :CF F957H=CB
K<99@G 5G @=A=H5H=CBG G A9BH=CB98 =B <5DH9F H<9 7CBHFC@@9FG IG9 H<9 89F=J5H=J9 C: H<9 GH5H9G K<=7<
=G BCH F97CAA9B898 6IH =G 8CB9 =B H<9 G=AI@5H=CBG ,<9 A5L=AIA GH9D G=N9 =B H<9 GC@J9F =G G9H HC G97CB8 =B CF89F HC A=B=A=N9 H<9 DCH9BH=5@ 9FFCF ,<=G GI6>97H K=@@ 69 8=G7IGG98 @5H9F =B H<=G 7<5DH9F
!/3;2':/54 5, ! =/:.
+'):/54 ".8;9:+89
CF H<9G9 G=AI@5H=CBG H<9 7CBHFC@@9FG 5F9 HIB98 K=H< H<9 D5F5A9H9FG G<CKB =B ,56@9 /<9F9 5B8
=G 7CAACB :CF 7CBHFC@@9F =G 7CAACB :CF 7CBHFC@@9F 5B8 =G CB@M :CF 7CBHFC@@9F !B
H<9 HKC :C@@CK=B; GI6G97H=CB H<9 G=AI@5H=CB F9GI@HG C: H<9 H<F99 7CBHFC@@9FG D9F:CFA98 CB H<9 GA5@@ 5B;@9
A5BC9IJF9 5B8 GI6G9EI9BH@M CB H<9 @5F;9 5B;@9 A5BC9IJF9 5F9 DF9G9BH98 &C D5F5A9H9FG 69G=89G H<9
=B=H=5@ 5HH=HI89 5F9 AC8=O98 =B 69HK99B H<9 HKC G=AI@5H98 A5BC9IJF9G
,56@9 CBHFC@@9F HIB=B; D5F5A9H9FG
(5F5A9H9F
9G7F=DH=CB
(FCDCFH=CB5@ 7CBGH5BH
9F=J5H=J9 7CBGH5BH
=FGH 57HI5HCF 7CBGH5BH
+97CB8 57HI5HCF 7CBGH5BH
!3'22 4-2+ '45+;<8+
Euler Parameter Error Vector
Angular Velocities [deg/s]
0.06
0.4
1- η
ε
0.04
0.2
0.02
2
ε
3
Angle [deg]
1
ε
0
-0.02
-0.04
0
-0.2
-0.4
0
20
40
60
80
-0.6
100
0
20
40
Time [s]
60
ω
1
ω
2
ω
3
80
Desired Torque
Real Torque
0.12
0.12
0.1
0.1
0.08
0.08
Torque [Nm]
Torque [Nm]
Thruster
0.06
Thruster
0.04
Thruster
Thruster
0.02
Thruster
0
0
20
40
100
Time [s]
60
80
1
2
3
Thruster
Thruster
Thruster
1
2
3
4
0.06
0.04
0.02
4
0
100
Time [s]
0
20
40
60
Time [s]
=;IF9 CBHFC@@9F K=H< F957H=CB H<FIGH9FG CB GA5@@ 5B;@9 A5BC9IJF9
80
100
0.06
0.4
1- η
ε
0.04
0.2
0.02
2
ε
3
Angle [deg]
1
ε
0
-0.02
-0.04
0
-0.2
-0.4
0
20
40
60
80
-0.6
100
0
20
40
Time [s]
60
ω
1
ω
2
ω
3
80
100
Time [s]
Desired Torque
Real Torque
0.12
0.12
0.1
0.1
0.08
0.08
Torque [Nm]
Torque [Nm]
Thruster
0.06
Thruster
0.04
Thruster
Thruster
0.02
Thruster
0
0
20
40
60
1
2
3
Thruster
Thruster
Thruster
1
2
3
4
0.06
0.04
0.02
4
80
0
100
0
20
40
Time [s]
60
80
100
Time [s]
0.06
0.4
1- η
ε
0.04
0.2
0.02
2
ε
3
Angle [deg]
1
ε
0
-0.02
-0.04
0
-0.2
-0.4
0
20
40
60
80
-0.6
100
0
20
40
Time [s]
60
ω
1
ω
2
ω
3
80
Desired Torque
Real Torque
0.12
0.12
0.1
0.1
0.08
0.08
Torque [Nm]
Torque [Nm]
Thruster
0.06
Thruster
0.04
Thruster
Thruster
0.02
Thruster
0
0
20
40
100
Time [s]
60
80
1
2
3
Thruster
Thruster
Thruster
1
2
3
4
0.06
0.04
0.02
4
0
100
Time [s]
0
20
40
60
Time [s]
80
100
!H 75B 69 GIAA98 ID H<5H 5@@ H<F99 7CBHFC@@9FG 5F9 GH56@9 K<=7< =G =B 5;F99A9BH K=H< H<9 DFCJ9B
+-+ :CF H<9 7CBHFC@@9FG =B <5DH9F !B 588=H=CB H<9 7CBHFC@@9FG <5J9 F95GCB56@M ;CC8 D9F
:CFA5B79 CBHFC@@9F =B =;IF9 =G C6G9FJ98 HC G9HH@9 H<9 5B;I@5F J9@C7=H=9G 5:H9F 56CIH G97CB8G
6IH @95J9 5 A=BCF GH5H=7 9FFCF =B ( !H 75B 5@GC 69 G99B H<5H H<9 57HI5H=CBG :FCA H<9 H<FIGH9FG 5F9 89
75M=B; F5D=8@M 5:H9F H<=G H=A9 K<=7< =G F95GCB56@9 CK9J9F =H =G C6G9FJ98 H<5H H<9 H<FIGH9FG <5J9 GCA9
57H=J5H=CBG 5:H9F H<9 G9HH@=B; C: H<=G 75B 69 9LD@5=B98 6M 7CFF97H=CBG D9F:CFA98 =B H<9 6CIB85F=9G
5FCIB8 H<9 8958 NCB9 C: H<9 (/%
CBHFC@@9F =B =;IF9 =B7@I89G H<9 OFGH CF89F 57HI5HCF 8MB5A=7G 5B8 =G 5 A=BCF =ADFCJ9A9BH C:
7CBHFC@@9F K=H< F9GD97H HC G9HH@=B; C: ,<9 GH5H=7 9FFCF C: ( =G 5@GC C6G9FJ98 HC 69 @9GG H<9B :FCA
7CBHFC@@9F =;IF9 G<CKG H<9 F9GI@HG :FCA 7CBHFC@@9F K<=7< =B7@I89 H<9 G97CB8 CF89F 57HI5HCF
8MB5A=7G ,<9 F9GI@H =G EI=H9 G=A=@5F HC H<9 F9GI@H C: 7CBHFC@@9F 6IH GCA9 @9GG GH5H=7 9FFCF 75B 69
C6G9FJ98 5G K9@@ 5G :5GH9F 8975M=B; C: H<9 89G=F98 HCFEI9 :FCA H<9 7CBHFC@@9F ,<=G =G F95GCB56@9 G=B79 H<9
7CBHFC@@9F H5?9 6CH< 89@5MG :FCA H<9 57HI5HCF 8MB5A=7G =BHC 7CBG=89F5H=CB
'B9 9LD97H98 :95HIF9 K<=7< =G C6G9FJ98 =B 7CAD5F=GCB C: H<9 89G=F98 5B8 F95@ HCFEI9 =B 5@@ H<F99 O;IF9G =G
H<5H H<9 @9J9@ C: H<9 %= =B H<9 89G=F98 HCFEI9 =G <=;<9F H<5B H<9 7CFF9GDCB8=B; @9J9@ =B H<9 CIHDIH HCFEI9
,<=G =G 75IG98 6M H<9 H=A9 89@5MG =B H<9 57HI5HCF 8MB5A=7G K<=7< <5J9 GHF9H7<98 H<9 89G=F98 HCFEI9 CJ9F
5 @CB;9F D9F=C8 C: H=A9 ,<=G =G DFC656@M H<9 9LD@5B5H=CB C: H<9 :5GH9F G9HH@=B; C: =B 7CBHFC@@9F 5B8
J9FGIG 7CBHFC@@9F CF :IFH<9F 7CAD5F=GCB 5 G=AI@5H=CB C: 7CBHFC@@9F HIB98 K=H< G5A9 5B8 CB H<9 +%' K=H< =895@ 57HI5HCFG 75B 69 :CIB8 =B =;IF9 &CH9 H<5H 89G=F98 HCFEI9 =G 9EI5@ HC F95@
HCFEI9
0.06
0.4
1- η
ε
0.04
0.2
0.02
2
ε
3
Angle [deg]
1
ε
0
-0.02
-0.04
0
-0.2
-0.4
0
20
40
60
80
-0.6
100
0
20
40
Time [s]
0.1
0.1
0.08
0.08
0.06
Thruster
0.04
Thruster
Thruster
0.02
Thruster
20
40
2
ω
3
80
100
Real Torque
0.12
Torque [Nm]
Torque [Nm]
Desired Torque
0
1
ω
Time [s]
0.12
0
60
ω
60
80
1
2
3
0.06
Thruster
0.04
Thruster
Thruster
0.02
Thruster
4
0
100
Time [s]
0
20
40
60
80
1
2
3
4
100
Time [s]
=;IF9 CBHFC@@9F CB H<FIGH9FG K=H<CIH 8MB5A=7G GA5@@ 5B;@9 A5BC9IJF9
'8-+ 4-2+ '45+;<8+
!B H<=G G97H=CB H<9 H<F99 G=AI@5H=CBG C: H<9 @5F;9 5B;@9 A5BC9IJF9 K=H< 7CBHFC@@9F 5F9 DF9G9BH98 =B
=;IF9 CJ9F H<9 HKC :C@@CK=B; D5;9G
0.3
1
1- η
ε
0.2
0.5
0.1
2
ε
3
Angle [deg]
1
ε
0
-0.1
-0.2
0
-0.5
-1
-1.5
0
20
40
60
80
-2
100
0
20
40
Time [s]
60
ω
1
ω
2
ω
3
80
100
Time [s]
Desired Torque
Real Torque
0.12
0.12
0.1
0.1
0.08
0.08
Torque [Nm]
Torque [Nm]
Thruster
0.06
Thruster
0.04
Thruster
Thruster
0.02
Thruster
0
0
20
40
60
1
2
3
Thruster
Thruster
Thruster
1
2
3
4
0.06
0.04
0.02
4
80
0
100
0
20
40
Time [s]
60
80
100
Time [s]
=;IF9 CBHFC@@9F K=H< F957H=CB H<FIGH9FG CB @5F;9 5B;@9 A5BC9IJF9
0.3
1
1- η
ε
0.2
0.5
0.1
2
ε
3
Angle [deg]
1
ε
0
-0.1
-0.2
0
-0.5
-1
-1.5
0
20
40
60
80
-2
100
0
20
40
Time [s]
60
ω
1
ω
2
ω
3
80
Desired Torque
Real Torque
0.12
0.12
0.1
0.1
0.08
0.08
Torque [Nm]
Torque [Nm]
Thruster
0.06
Thruster
0.04
Thruster
Thruster
0.02
Thruster
0
0
20
40
100
Time [s]
60
80
1
2
3
Thruster
Thruster
Thruster
1
2
3
4
0.06
0.04
0.02
4
0
100
Time [s]
0
20
40
60
Time [s]
80
100
0.3
1
1- η
ε
0.2
0.5
0.1
2
ε
3
Angle [deg]
1
ε
0
-0.1
-0.2
0
-0.5
-1
-1.5
0
20
40
60
80
-2
100
0
20
40
Time [s]
60
ω
1
ω
2
ω
3
80
Desired Torque
Real Torque
0.12
0.12
0.1
0.1
0.08
0.08
Torque [Nm]
Torque [Nm]
Thruster
0.06
Thruster
0.04
Thruster
Thruster
0.02
Thruster
0
0
20
40
100
Time [s]
60
80
1
2
3
Thruster
Thruster
Thruster
1
2
3
4
0.06
0.04
0.02
4
0
100
Time [s]
0
20
40
60
80
100
Time [s]
@GC G=AI@5H=CBG C: H<9 @5F;9 5B;@9 A5BC9IJF9 G<CK H<5H 5@@ H<F99 7CBHFC@@9FG 5F9 GH56@9 K<=7< =G =B
577CF85B79 K=H< H<9 A5H<9A5H=75@@M DFCJ9B +-+ G :CF H<9 GA5@@ 5B;@9 A5BC9IJF9 H<9 G5A9
:95HIF9G K=H< F9GD97H HC H<9 G9HH@=B; C: ( 5B8 75B 69 G99B =B H<9 G=AI@5H=CB F9GI@HG CBHFC@@9F =B
=;IF9 =G A5F;=B5@ :5GH9F H<9B 7CBHFC@@9F =B =;IF9 K<=7< =G A5F;=B5@ :5GH9F H<5B 7CBHFC@@9F =B =;IF9 IH H<9 ACGH =BH9F9GH=B; :95HIF9 75B 69 G99B =B H<9 G9HH@=B; C: 9GD97=5@@M 5:H9F H<9
CJ9FG<CCH ,<9 8=GHIF65B79 K<=7< H<9 57HI5HCF 8MB5A=7G 5F9 =BHFC8I7=B; =G 7CAD9BG5H98 :CF =B 7CBHFC@@9F
5B8 K=H< ACF9 DF97=G9 85AD9B=B; C: BCH<9F =B8=75H=CB C: 7CAD9BG5H=CBG C: 8MB5A=7G 75B 69
G99B =B H<9 :F9EI9B7M C: 57H=J5H=CBG C: H<9 H<FIGH9FG :FCA 56CIH G97CB8G =BHC H<9 G=AI@5H=CB @95F@M
7CBHFC@@9F <5G :9J9F 57H=J5H=CBG C: H<9 H<FIGH9FG H<5B H<9 HKC CH<9F 7CBHFC@@9FG K<=7< =G 5B =B8=75H=CB
C: :5GH9F G9HH@=B; C: H<9 5B;I@5F J9@C7=H=9G
CF :IFH<9F 7CAD5F=GCB 5 G=AI@5H=CB C: 7CBHFC@@9F HIB98 K=H< G5A9 5B8 CB H<9 +%' K=H< =895@
57HI5HCFG 75B 69 :CIB8 =B =;IF9 CB H<9 B9LH D5;9 &CH9 H<5H 89G=F98 HCFEI9 =G 9EI5@ HC F95@ HCFEI9
0.3
1
1- η
ε
0.2
0.5
0.1
2
ε
3
Angle [deg]
1
ε
0
-0.1
-0.2
0
-0.5
-1
-1.5
0
20
40
60
80
-2
100
0
20
40
Time [s]
0.1
0.1
0.08
0.08
0.06
Thruster
0.04
Thruster
Thruster
0.02
Thruster
20
40
2
ω
3
80
100
Real Torque
0.12
Torque [Nm]
Torque [Nm]
Desired Torque
0
1
ω
Time [s]
0.12
0
60
ω
60
80
1
2
3
0.06
Thruster
0.04
Thruster
Thruster
0.02
Thruster
4
0
100
Time [s]
0
20
40
60
80
Time [s]
=;IF9 CBHFC@@9F CB H<FIGH9FG K=H<CIH 8MB5A=7G @5F;9 5B;@9 A5BC9IJF9
1
2
3
4
100
!/3;2':/54 5, ! =/:.
+'):/54 %.++29
,<9 HIB=B; C: H<9 7CBHFC@@9FG K=H< F957H=CB K<99@G 5G 57HI5HCFG =G 7CB8I7H98 577CF8=B; HC H<9 G5A9
DFC798IF9 5G :CF F957H=CB H<FIGH9FG =FGH 5B8 =G HIB98 CB 7CBHFC@@9F 5B8 H<9B 5B8 =G
HIB98 CB 7CBHFC@@9F 5B8 F9GD97H=J9@M ,<9 D5F5A9H9FG :FCA H<9 HIB=B; 5F9 DF9G9BH98 =B ,56@9 ,<9G9 J5@I9G 5F9 IB7<5B;98 :CF 5@@ G=AI@5H=CBG K=H< F957H=CB K<99@G 5G 57HI5HCFG
,56@9 CBHFC@@9F HIB=B; D5F5A9H9FG
(5F5A9H9F
9G7F=DH=CB
(FCDCFH=CB5@ 7CBGH5BH
9F=J5H=J9 7CBGH5BH
=FGH 57HI5HCF 7CBGH5BH
+97CB8 57HI5HCF 7CBGH5BH
!3'22 4-2+ '45+;<8+
!B H<9 :C@@CK=B; H<9 H<F99 G=AI@5H=CBG C: H<9 @5F;9 5B;@9 A5BC9IJF9 K=H< 7CBHFC@@9F 5F9 DF9G9BH98 =B
=;IF9 Euler Parameter Error Vector
Angular Velocities
0.06
0.4
1- η
ε
0.04
0.2
0.02
2
ε
3
Angle [deg/s]
1
ε
0
0
-0.2
-0.4
-0.6
-0.02
-0.8
-0.04
0
20
40
60
80
-1
100
0
20
40
Time [s]
1
ω
2
ω
3
80
100
Time [s]
Desired Wheel Speed
Real Wheel Speed
10000
8000
Wheel
Wheel
5000
Wheel
Wheel
1
6000
2
Wheel speed [rpm]
Wheel
Wheel speed [rpm]
60
ω
3
4
0
Wheel
Wheel
4000
Wheel
1
2
3
4
2000
0
-2000
-5000
0
20
40
60
80
-4000
100
Time [s]
0
20
40
60
Time [s]
=;IF9 CBHFC@@9F K=H< F957H=CB K<99@G CB GA5@@ 5B;@9 A5BC9IJF9
80
100
Angular Velocities
0.06
0.6
0.04
1- η
ε
0.4
ε
2
0.2
ε
3
0.02
Angle [deg/s]
1
0
0
-0.2
-0.4
-0.02
-0.6
-0.04
0
20
40
60
80
-0.8
100
0
20
40
Time [s]
Desired Wheel Speed
2
ω
3
80
100
Real Wheel Speed
6000
6000
Wheel
Wheel
4000
Wheel
Wheel
1
Wheel
4000
2
Wheel speed [rpm]
Wheel
Wheel speed [rpm]
1
ω
Time [s]
8000
3
4
2000
0
Wheel
Wheel
2000
1
2
3
4
0
-2000
-2000
-4000
60
ω
0
20
40
60
80
-4000
100
0
20
40
Time [s]
60
80
100
Time [s]
Angular Velocities
0.06
0.2
1- η
ε
0.04
1
2
ε
3
0
Angle [deg/s]
0.02
ε
0
-0.4
-0.02
-0.04
-0.2
0
20
40
60
80
-0.6
100
0
20
40
Time [s]
Desired Wheel Speed
2
ω
3
80
100
Real Wheel Speed
4000
Wheel
3000
Wheel
2000
Wheel
Wheel
1
3000
2
Wheel speed [rpm]
Wheel
4000
Wheel speed [rpm]
1
ω
Time [s]
5000
3
4
1000
0
-1000
Wheel
Wheel
2000
Wheel
1
2
3
4
1000
0
-1000
-2000
-3000
60
ω
0
20
40
60
80
-2000
100
Time [s]
0
20
40
60
Time [s]
80
100
,<9 G=AI@5H=CB F9GI@HG :CF 7CBHFC@@9F 89D=7H98 =B =;IF9 7@95F@M G<CK H<9 9T97HG C: H<9 57HI5HCF
8MB5A=7G !B H<9 D@CH C: H<9 5B;I@5F J9@C7=H=9G =H 75B 69 G99B H<5H H<9 57HI5HCF 8MB5A=7G 5F9 =BHFC8I7=B;
CG7=@@5H=CBG 9GD97=5@ =B=H=5@@M ,<9 G5H9@@=H9 =G 7CAD@9H9@M G9HH@98 5:H9F 56CIH G97CB8G :CF 5@@ H<F99
7CBHFC@@9FG K=H< A5F;=B5@@M 69HH9F D9F:CFA5B79 C: 7CBHFC@@9F =;IF9 5B8 7CBHFC@@9F =;IF9
K=H< F9GD97H HC G9HH@=B; C: !B 5@@ G=AI@5H=CBG =H 75B 69 G99B H<5H H<9 F957H=CB K<99@G =G BCH
F957<=B; =HG A5L=AIA GD=B C: FDA =B H<9 GA5@@ 5B;@9 A5BC9IJF9
FCA H<9 G=AI@5H=CBG K=H< F957H=CB H<FIGH9FG H<9 57HI5HCF 8MB5A=7G =G KCF?=B; CB H<9 89G=F98 HCFEI9
5:H9F H<9 (/% <9B79 CB 957< EI5BH=N98 89G=F98 DI@G9 !B H<9 75G9 C: H<9 F957H=CB K<99@G H<9 57HI5HCF
8MB5A=7G =G KCF?=B; 8=F97H@M CB H<9 57HI5HCF =BDIH K=H< CB@M H<9 588=H=CB5@ @=A=H5H=CB C: A5L=AIA GD=B
GHCF5;9 ,<=G F9GI@H =B DCGG=6@M @5F;9F 9T97H C: H<9 57HI5HCF 8MB5A=7G 9GD97=5@@M K=H< @5F;9 F5H9 7<5B;9G
=B H<9 89G=F98 HCFEI9 G=;B5@ :FCA H<9 7CBHFC@@9F
,<9 ACGH =BH9F9GH=B; :95HIF9 :FCA H<9G9 G=AI@5H=CBG =G H<9 85AD9B=B; C: H<9 CG7=@@5H=CBG =BHFC8I798 6M
H<9 57HI5HCF 8MB5A=7G CBHFC@@9F =G F9ACJ=B; ACGH CG7=@@5H=CBG 8I9 HC =HG :998657? :FCA H<9 57HI5HCF
AC89@ !B 588=H=CB 7CBHFC@@9F =BHFC8I79 5G 9LD97H98 9J9B ACF9 85AD9B=B; G=B79 5@@ 57HI5HCF 8MB5A=7G
=G H5?9B =BHC 7CBG=89F5H=CB ,<=G K=@@ 69 8=G7IGG98 ACF9 H<CFCI;<@M =B H<9 GIAA5FM C: H<9 @5F;9 5B;@9
A5BC9IJF9 K<9F9 5@@ 9T97HG C: 57HI5HCF 8MB5A=7G 5F9 ACF9 9J=89BH CF :IFH<9F 7CAD5F=GCB 5G :FCA H<9
H<FIGH9F G=AI@5H=CBG 7CBHFC@@9F HIB98 K=H< G5A9 5B8 =G G=AI@5H98 K=H< =895@ 57HI5HCFG ,<9
F9GI@HG 75B 69 :CIB8 =B =;IF9 69@CK
Angular Velocities
0.06
0.2
1- η
ε
0.04
0.1
0.02
ε
2
ε
3
Angle [deg/s]
1
0
0
-0.1
-0.2
-0.3
-0.02
-0.4
-0.04
0
20
40
60
80
-0.5
100
0
20
40
Time [s]
Desired Wheel Speed
2
ω
3
80
100
Real Wheel Speed
4000
Wheel
Wheel
2000
Wheel
Wheel
1
3000
2
Wheel speed [rpm]
Wheel
3000
Wheel speed [rpm]
1
ω
Time [s]
4000
3
4
1000
0
-1000
-2000
60
ω
Wheel
Wheel
2000
Wheel
1
2
3
4
1000
0
-1000
0
20
40
60
80
-2000
100
Time [s]
0
20
40
60
80
100
Time [s]
=;IF9 CBHFC@@9F CB F957H=CB K<99@G K=H<CIH 8MB5A=7G GA5@@ 5B;@9 A5BC9IJF9
'8-+ 4-2+ '45+;<8+
!B H<=G G97H=CB H<9 H<F99 G=AI@5H=CB F9GI@HG C: H<9 @5F;9 5B;@9 A5BC9IJF9 K=H< 7CBHFC@@9F 75B 69 :CIB8
=B =;IF9 K<=7< 5F9 DF9G9BH98 CB H<9 HKC B9LH D5;9G
Angular Velocities
1.5
1- η
ε
0.1
ε
2
ε
3
Angle [deg/s]
1
0.2
0
ω
1
1
ω
2
0.5
ω
3
0
-0.5
-1
-0.1
0
20
40
60
80
-1.5
100
0
20
40
Time [s]
x 10
6
Desired Wheel Speed
Wheel
1
Wheel speed [rpm]
80
100
Real Wheel Speed
Wheel
Wheel
0.5
Wheel
1
Wheel
8000
1
Wheel 2
2
Wheel speed [rpm]
1.5
60
Time [s]
3
4
0
-0.5
Wheel
4000
Wheel
3
4
0
-4 000
-1
-1.5
-8 000
0
20
40
60
80
100
0
20
40
Time [s]
60
80
100
Time [s]
=;IF9 CBHFC@@9F K=H< F957H=CB K<99@G CB @5F;9 5B;@9 A5BC9IJF9
Angular Velocities
1.5
1- η
ε
0.2
0.1
ε
2
ε
3
Angle [deg/s]
1
0
ω
1
1
ω
2
0.5
ω
3
0
-0.5
-1
-0.1
0
20
40
60
80
-1.5
100
0
20
40
Time [s]
x 10
6
Desired Wheel Speed
x 10
Wheel
Wheel
1
Wheel speed [rpm]
80
100
Wheel
0.5
Wheel
4
Real Wheel Speed
Wheel
1
0.8
Wheel
2
Wheel speed [rpm]
1.5
60
Time [s]
3
4
0
-0.5
Wheel
0.4
Wheel
1
2
3
4
0
-0.4
-1
-0.8
-1.5
0
20
40
60
80
100
0
Time [s]
20
40
60
Time [s]
80
100
Angular Velocities
1.5
1- η
ε
ω
1
1
ω
2
0.5
ω
3
0.1
ε
2
ε
3
Angle [deg/s]
1
0.2
0
0
-0.5
-1
-0.1
0
20
40
60
80
-1.5
100
0
20
40
Time [s]
x 10
7
Desired Wheel Speed
Wheel speed [rpm]
Wheel
2
Wheel
Wheel
100
Wheel
8 000
1
Wheel
2
3
4
0
-2
-4
80
Real Wheel Speed
Wheel
Wheel speed [rpm]
4
60
Time [s]
Wheel
4 000
Wheel
1
2
3
4
0
-4 000
-8 000
0
20
40
60
80
100
0
Time [s]
20
40
60
80
100
Time [s]
B5@C;CIG HC H<9 G=AI@5H=CBG K=H< F957H=CB H<FIGH9FG =H 75B 69 G99B H<5H H<9 G=AI@5H=CBG K=H< F957H=CB
K<99@G 5F9 5@GC GH56@9 =B 577CF85B79 K=H< H<9 A5H<9A5H=75@@M DFCJ9B +-+ =B H<9 7CBHFC@@9F
89G=;B 7<5DH9F !H =G C6G9FJ98 H<5H 5@@ H<F99 7CBHFC@@9FG G9HH@9 H<9 5B;I@5F J9@C7=H=9G 5B8 H<9 5HH=HI89 5:H9F
5DDFCL=A5H98 G97CB8G FCA H<9 G=AI@5H=CB F9GI@HG C: 7CBHFC@@9F =B =;IF9 =H 75B 69 G99B H<5H
H<9 57HI5HCF 8MB5A=7G =BHFC8I79 CG7=@@5H=CBG =B H<9 K<99@ GD998 K<=7< =B HIFB ;9B9F5H9 CG7=@@5H=CBG =B
!B H<=G @5F;9 5B;@9 A5BC9IJF9 =H =G C6G9FJ98 H<5H H<9 F957H=CB K<99@G 5F9 8F=J9B GHF5=;<H HC G5HIF5H=CB
5H FDA K<9F9 H<9 5B;I@5F J9@C7=HM 5FCIB8 H<9 $ 5L=G F957<9G =HG <=;<9GH DCGG=6@9 J5@I9 C: 5D
DFCL=A5H98 G97CB8 ,<=G 5B;I@5F J9@C7=HM =G 7CBGH5BH IBH=@ H<9 G5H9@@=H9 =G 7@CG=B; =B HC =HG 89G=F98
5HH=HI89 K<9F9 H<9 7CBHFC@@9F 7CAA5B8G H<9 F957H=CB K<99@G HC 8FCD =HG GD=B :H9F 5DDFCL=A5H98 G97CB8G H<9 GD=B C: 5@@ F957H=CB K<99@G 5F9 N9FC 5B8 H<9 G5H9@@=H9 =G 5H F9GH K=H< H<9 89G=F98 5HH=HI89
CAD5F=B; H<9 G9HH@=B; C: H<9 5B;I@5F J9@C7=H=9G :FCA H<9 H<F99 7CBHFC@@9FG =H 75B 69 G99B H<5H 7CBHFC@@9F
K<=7< 7CAD9BG5H9 :CF GCA9 57HI5HCF 8MB5A=7G =G ACF9 DF97=G9 5B8 577IF5H9 H<5B 7CBHFC@@9F B8
K<9B 5@@ 57HI5HCF 8MB5A=7G 5F9 7CAD9BG5H98 :CF =B 7CBHFC@@9F =H 75B 69 G99B H<5H H<=G 7CBHFC@@9F =G ACF9
DF97=G9 5B8 577IF5H9 H<5B 7CBHFC@@9F B =BH9F9GH=B; F9GI@H 75B 69 :CIB8 K<9B 7CBHFC@@9F =;IF9 =G 7CAD5F98 K=H< 7CBHFC@@9F K=H<CIH
57HI5HCF 8MB5A=7G =;IF9 B9LH D5;9 +9HH@=B; C: H<9 5B;I@5F J9@C7=H=9G :FCA 7CBHFC@@9F =G EI=H9
G=A=@5F HC H<9 F9GI@H C: 7CBHFC@@9F G=AI@5H98 K=H< =895@ 57HI5HCFG ,<=G =AD@=9G H<5H 7CBHFC@@9F =G
7CAD9BG5H=B; :CF H<9 8=GHIF65B79G =BHFC8I798 6M H<9 57HI5HCF 8MB5A=7G
Angular Velocities
1.5
1- η
ε
ω
1
1
ω
2
0.5
ω
3
0.1
ε
2
ε
3
Angle [deg/s]
1
0.2
0
0
-0.5
-1
-0.1
0
20
40
60
80
-1.5
100
0
20
40
Time [s]
x 10
6
Desired Wheel Speed
100
Wheel
8000
1
Wheel 2
1
Wheel speed [rpm]
80
Real Wheel Speed
Wheel
Wheel
0.5
Wheel
Wheel
Wheel speed [rpm]
1.5
60
Time [s]
3
4
0
-0.5
Wheel
4000
Wheel
1
2
3
4
0
-4000
-1
-1.5
-8000
0
20
40
60
80
100
0
20
40
Time [s]
60
80
100
Time [s]
=;IF9 CBHFC@@9F CB K<99@G K=H<CIH 8MB5A=7G @5F;9 5B;@9 A5BC9IJF9
536'8/954 5,
+9;2:9 ,853
+'):/54 %.++29
+=B79 H<9F9 =B GCA9 C: H<9 G=AI@5H=CB F9GI@HG 75B 69 8=S7I@H HC G99 8=T9F9B79G =B H<9 D@CHG 5 :IFH<9F 5B8
ACF9 H<CFCI;< =BJ9GH=;5H=CB C: H<9 D9F:CFA5B79 C: H<9 H<F99 G=AI@5H98 7CBHFC@@9FG =G 7CB8I7H98 =B H<=G
G97H=CB ,<9 G=AI@5H=CB 75G9 C: H<9 @5F;9 5B;@9 A5BC9IJF9 C: H<9 +%' G5H9@@=H9 K=H< F957H=CB K<99@G
5G 57HI5HCFG =G 7<CG9B :CF :IFH<9F =BJ9GH=;5H=CB ,<=G =G 8CB9 6975IG9 H<9 =AD57H C: 57HI5HCF 8MB5A=7G
=G ACGH 9J=89BH =B H<5H G=AI@5H=CB 75G9 G 89G7F=698 =B H<9 DF9J=CIG G97H=CB H<9 F9GI@HG C: 7CBHFC@@9F =G EI=H9 G=A=@5F HC H<9 F9GI@HG C: 7CBHFC@@9F CB H<9 G5H9@@=H9 K=H< =895@ 57HI5HCFG !B CF89F HC =@@IGHF5H9
H<=G H<9 8=T9F9B79 =B 5HH=HI89 5B8 5B;I@5F J9@C7=HM 69HK99B 5@@ H<F99 7CBHFC@@9FG =G DF9G9BH98 69@CK 5B8
CB H<9 B9LH D5;9
x 10
-3
Difference in Euler Parameter Error Vector
Difference in Ang ular Velocities
2
1- η
ε1
4
2
ε
2
ε
3
1
Angle [deg/s]
6
0
0
-1
ω
-2
0
20
40
60
80
-2
100
Time [s]
0
20
ω
1
40
60
ω
2
80
Time [s]
=;IF9 CBHFC@@9F K=H< =895@ 57HI5HCFG J9FGIG 7CBHFC@@9F K=H< 57HI5HCF 8MB5A=7G
3
100
x 10
-3
2
1- η
ε1
6
4
ε
2
ε
3
1.5
Angle [deg/s]
8
2
0
-2
1
ω
1
ω
2
ω
3
0.5
0
-0.5
0
20
40
60
80
-1
100
0
20
40
Time [s]
60
80
100
Time [s]
x 10
-3
2
1- η
5
1.5
ε
1
ε
2
ε
3
Angle [deg/s]
10
0
1
ω
1
ω
2
ω
3
0.5
0
-0.5
-5
0
20
40
60
80
-1
100
0
20
Time [s]
40
60
80
100
Time [s]
!B =;IF9 H<9 8=T9F9B79 69HK99B 7CBHFC@@9F K=H< =895@ 57HI5HCFG A=BIG 7CBHFC@@9F K=H< 57HI5HCF
8MB5A=7G =G D@CHH98 !H 75B 69 G99B H<5H H<9 9T97H C: H<9 57HI5HCF 8MB5A=7G =G 5G 9LD97H98 ACGH 9J=89BH =B
H<9 HF5BG=H=CBG C: H<9 5B;I@5F J9@C7=H=9G BCH<9F :95HIF9 C: =B H<9 7CAD5F=GCB C: H<9 H<F99 7CBHFC@@9FG
75B 69 G99B =B H<9 85AD9B=B; C: H<9 CJ9FG<CCH 5G 89G7F=698 DF9J=CIG@M
G 5 F9GI@H C: H<9 85AD9B=B; C: H<9 CG7=@@5H=CBG =B 7CBHFC@@9F 5B8 =H 75B 69 G99B H<5H H<9G9 HKC
7CBHFC@@9FG =;IF9 <5J9 5 GACCH<9F 69<5J=CIF H<5B 7CBHFC@@9F !;33'8? '4* /9);99/54 5, !/3;2':/54
+9;2:9
!B H<=G 7<5DH9F G=AI@5H=CBG C: HKC 7CBO;IF5H=CBG C: H<9 +%' G5H9@@=H9 CB9 K=H< F957H=CB H<FIGH9FG 5B8
CB9 K=H< F957H=CB K<99@G 5G 57HI5HCFG 5F9 DF9G9BH98 FCA 957< 7CBO;IF5H=CB =H =G G=AI@5H98 CB9 GA5@@
5B8 CB9 @5F;9 5B;@9 A5BC9IJF9 ,<9 7CBHFC@@9FG :CF 6CH< 7CBO;IF5H=CBG K9F9 9EI5@@M =AD@9A9BH98 K=H<
CB@M 8=T9F9B79G =B H<9 HIB=B; ,<9 F957H=CB K<99@G <58 5 (/% K=H< @=A=H5H=CBG =B A5L=AIA CIHDIH
HCFEI9 69:CF9 G9BH 5G =BDIH HC H<9 57HI5HCF 8MB5A=7G AC89@ !B H<9 75G9 C: F957H=CB K<99@G H<9 7CBHFC@@9F
CIHDIH =G @=A=H98 6M H<9 A5L=AIA 5@@CK98 GD=B GHCF5;9 69:CF9 G9BH 5G =BDIH HC H<9 57HI5HCF 8MB5A=7G
AC89@ +CA9 8=T9F9B79G K=H< F9GD97H HC G9HH@=B; C: H<9 5HH=HI89 5B8 5B;I@5F J9@C7=H=9G 69HK99B H<9 HKC
7CBO;IF5H=CBG K9F9 F9J95@98 :FCA H<9 G=AI@5H=CBG ,<=G 75B 69 9LD@5=B98 6M H<9 8958 NCB9 C: H<9 F957H=CB
H<FIGH9FG K<9F9 H<9 (/% ;=J9G N9FC 5G CIHDIH 5G @CB; 5G H<9 89G=F98 HCFEI9 =G @CK9F H<5B H<9 8958 NCB9
@=A=H ,<=G ;=J9G GCA9 8F=:H =BG=89 H<9 8958 NCB9 6IH =G 85AD9B98 9J9FM H=A9 H<9 8F=:H 9L7998G H<9 8958
NCB9 @=A=HG *957H=CB K<99@G CB H<9 CH<9F <5B8 <5J9 BC 8958 NCB9 5B8 5F9 75D56@9 C: DFCJ=8=B; HCFEI9
CB 9J9FM G=;B5@ :FCA H<9 7CBHFC@@9F 9L79DH =B H<9 75G9 C: G5HIF5H=CB C: H<9 K<99@G
,<9 ACGH =ADCFH5BH F9GI@H :FCA H<9 G=AI@5H=CBG =G H<5H 5@@ 7CBHFC@@9FG 5F9 GH56@9 ,<=G =B 577CF85B79 K=H<
H<9 DFCJ9B -+ :FCA H<9 7CBHFC@@9F 89G=;B =B <5DH9F 88=H=CB5@@M H<9 G=AI@5H=CBG <5J9 G<CKB H<5H
H<9 7CBHFC@@9FG H5?=B; H<9 57HI5HCF 8MB5A=7G =BHC 7CBG=89F5H=CBG 5F9 GCA9K<5H ACF9 DF97=G9 5B8 577IF5H9
H<5B H<9 7CBHFC@@9F K=H<CIH 57HI5HCF 8MB5A=7G 7CBG=89F5H=CBG
!H 75B 69 8=G7IGG98 K<9H<9F H<9 G=AI@5H=CB CIH7CA9 C: H<9 F957H=CB K<99@G K=H< 8MB5A=7G =G @5F;9F H<5B =B
F95@ @=:9 9LD9F=A9BHG !BHI=H=J9@M =H K5G 9LD97H98 H<5H H<9 G=AI@5H=CB F9GI@H C: H<9 F957H=CB H<FIGH9FG K=H<
8MB5A=7G G<CI@8 69 ACF9 9J=89BH H<9B H<9 G=AI@5H=CB F9GI@HG :FCA H<9 F957H=CB K<99@G K=H< 8MB5A=7G
G=B79 H<FIGH9FG =G ?BCKB HC <5J9 ACF9 8MB5A=7 69<5J=CIF H<9B F957H=CB K<99@G ,<=G 75B DCGG=6@M 69
75IG98 6M DCCF 8=A9BG=CB=B; C: H<9 57HI5HCF AC89@G 7CAD5F98 HC F95@ @=:9 J5@I9G ,<=G =G 5 F9GI@H C: A=GG=B;
H97<B=75@ GD97=O75H=CBG :FCA H<9 MCIB; +%' A=GG=CB ,<=G K=@@ 69 8=G7IGG98 :IFH<9F =B H<9 7CB7@I8=B;
F9A5F?G (CCF 8=A9BG=CB=B; =G 5 EI9GH=CB C: HIB=B; C: H<9 57HI5HCF AC89@G K<=7< K=H< 8=T9F9BH J5@I9G
75B 75IG9 CH<9F CG7=@@5H=CBG 5B8 7<5B;9 =B H<9 G9HH@=B; H=A9 CK9J9F H<9 DFCJ9B +-+ =G GH=@@ J5@=8
577CF8=B; HC H<9 7CBHFC@@9F 89G=;B =B <5DH9F G A9BH=CB98 =B=H=5@@M =B H<=G 7<5DH9F 5B8 =B H<9 =AD@9A9BH5H=CB G97H=CB +97H=CB H<9 7CBHFC@@9FG
IH=@=N9 BIA9F=75@ 89F=J5H=CB C: H<9 GH5H9G ,<9 F95GCB :CF 5JC=8=B; H<=G =B DF57H=75@ =AD@9A9BH5H=CBG
F9GI@H :FCA BC=G9 =B G9BGCFG K<=7< DFCJ=89 H<9 5HH=HI89 ,<=G 75B 75IG9 @5F;9 5@H9FB5H=CBG =B H<9 G=;B5@
5B8 F9GI@H =B @5F;9 89J=5H=CB 69HK99B H<9 F95@ J5@I9 5B8 H<9 89F=J5H=J9 ,<9 BIA9F=75@ 89F=J5H=CB 75B
GCA9K<5H 69 >IGH=O98 6M 5 @CK A5L=AIA GH9D G=N9 =B H<9 GC@J9F 5B8 H<9 :57H H<5H H<9F9 =G BC BC=G9 =B
H<9 G=AI@5H98 AC89@ CK9J9F H<=G 75B =BHFC8I79 9FFCFG =B H<9 G=AI@5H=CBG 9GD97=5@@M =B 75G9G C: @5F;9
F5H9 7<5B;9G 'B9 9LD@=7=H 75G9 K<9F9 H<=G DFC6@9A =G GIGD97H98 HC 5DD95F 75B 69 G99B =B =;IF9 !BGD97H=CBG C: H<9 5B;I@5F J9@C7=H=9G =B H<9 D@CH F9J95@ 5 HF5BG=9BH =B H<9 HF5BG=H=CB C: H<9 J9@C7=HM 5:H9F
56CIH G97CB8G ,<=G 75B 69 75IG98 6M H<9 BIA9F=75@ 89F=J5H=CB ,<=G GI6>97H =G 5 D5FH C: H<9 :IHIF9
KCF? C: H<=G H<9G=G
.'6:+8 54)2;*/4-
+3'819
,<=G 7<5DH9F <C@8G H<9 7CB7@I8=B; F9A5F?G 8F5KB :FCA H<9 KCF? 7CB8I7H98 =B H<=G H<9G=G ,<9 7<5DH9F =G
8=J=898 =BHC H<F99 G97H=CBG KCF? 7CAD@9H98 F9GI@HG C6H5=B98 =B H<9 H<9G=G 5B8 OB5@@M F97CAA9B85H=CBG
:CF :IHIF9 KCF?
%581 5362+:+*
!B H<=G H<9G=G 5 G5H9@@=H9 AC89@ 89G7F=698 K=H< I@9F D5F5A9H9FG <5J9 699B 898I798 ,<F99 HMD9G C: 7CBHFC@
HCFEI9G CB H<9 G5H9@@=H9 <5J9 699B =BHFC8I798 =BJ9GH=;5H98 5B8 89F=J98 AC89@ :CF ,<9 7CBHFC@ HCFEI9G
5F9 5@@ DF9G9BH98 CB 5 ;9B9F5@ :CFA =B CF89F HC =BHFC8I79 5 ;9B9F5@ 57HI5HCF 8MB5A=7G AC89@ K<=7< 75B
DF9GIA56@M 89G7F=69 ACGH 57HI5HCFG ,<9 9AD<5G=G =B H<=G H<9G=G <5G 699B CB 57HI5HCF 8MB5A=7G 5B8 HKC
;9B9F5@ AC89@G <5J9 699B DF9G9BH98 ,<9G9 HKC AC89@G KCI@8 ;=J9 5 69HH9F 5DDFCL=A5H=CB C: H<9 57HI5HCF
F9GDCBG9 5B8 ;=J9 5 ACF9 FC6IGH 5B8 DF97=G9 7CBHFC@@9F ,<=G K5G IG98 5G 5 65G=G :CF H<9 7CBHFC@@9F 89G=;B
5G98 CB H<9 AC89@@=B; C: H<9 G5H9@@=H9 57HI5HCFG 5B8 57HI5HCF 8MB5A=7G H<F99 7CBHFC@@9FG K9F9 89F=J98
5B8 DFCJ9B +-+ 'B9 65G98 CB H<9 G5H9@@=H9 AC89@ 5GGIA=B; =895@ 57HI5HCFG 5B8 HKC 7CBHFC@@9FG
65G98 CB H<9 OFGH 5B8 G97CB8 CF89F 57HI5HCF 8MB5A=7 AC89@G DF9G9BH98 =B <5DH9F ,<9 7CBHFC@@9FG
5GGIA9 H<5H H<9 57HI5HCFG 5B8 57HI5HCF 8MB5A=7G 75B 69 9LDF9GG98 CB H<9 ;9B9F5@ :CFAG DF9G9BH98 =B
<5DH9F 88=H=CB5@@M =G =H 5GGIA98 :CF H<9 7CBHFC@@9F 89G=;B H<5H H<9 G5H9@@=H9 =G =B 5 7=F7I@5F CF6=H 5B8
H<9 DG9I8C =BJ9FG9 C: H<9 57HI5HCF A5HF=L 9L=GHG ,<9 7CBHFC@@9F 89G=;B K5G 5 $M5DIBCJ 65G98 !BH9;F5H=B;
57?GH9DD9F K=H< 5 HCH5@ C: :CIF GH9DG =B H<9 75G9 C: 7CBHFC@@9F ,<9 H<F99 7CBHFC@@9FG 89F=J98 5F9 J5@=8
=B 5@@ 75G9G K<9F9 H<9 A9BH=CB98 5GGIADH=CBG 5F9 :I@O@@98
75G9 GHI8M C: H<9 ++,! +%' G5H9@@=H9 <5G 699B 7CB8I7H98 ,<9 ++,! =B=H=5H=J9 <5G 699B =BHFC8I798
5B8 =HG A=GG=CB <5G 699B 6F=9PM DF9G9BH98 ,<9 +%' @IB5F G5H9@@=H9 =G =BJ9GH=;5H98 5B8 5@@ B979GG5FM
H97<B=75@ GD97=O75H=CBG :CF 5 G=AI@5H=CB 75G9 <5J9 699B DF9G9BH98 ,<9 +%' A=GG=CB =G GH=@@ =B 5B 95F@M
D<5G9 5B8 BCH 5@@ D5F5A9H9FG 5F9 M9H 89H9FA=B98 +CA9 D5F5A9H9FG BCH 89H9FA=B98 <5J9 699B G9H =B
7CCD9F5H=CB K=H< H<9 58J=GCFG =B CF89F HC DFCJ=89 5@@ 85H5 :CF G=AI@5H=CBG ,KC AC89@G C: H<9 +%'
5F9 DF9G9BH98 CB9 K=H< F957H=CB H<FIGH9FG 5G 57HI5HCFG 5B8 CB9 7CBO;IF5H=CB K=H< F957H=CB K<99@G 5G
57HI5HCFG
+=B79 H<9 9AD<5G=G C: H<=G H<9G=G <5J9 F9;5F898 H<9 57HI5HCF 8MB5A=7G GCA9 G=AD@=O75H=CBG K=H< F9GD97H
HC H<9 A=GG=CB K<9F9 8CB9 :CF H<9 G=AI@5H=CBG ,<=G =AD@=9G H<5H =BH9F57H=CBG 69HK99B H<9 HKC 57HI5HCF
7CBO;IF5H=CBG 5F9 8=GF9;5F898 5B8 HKC G5H9@@=H9 AC89@G K=H< H<9 57HI5HCF 7CBO;IF5H=CBG =GC@5H98 5F9
G=AI@5H98 =B5@@M =B H<9 H<9G=G H<9 7CBHFC@@9F CB H<9 DF9G9BH98 AC89@G C: +%' K9F9 G=AI@5H98 ?A @IB5F CF6=H =G 7<CG9B K<9F9 5@@ D9FHIF6=B; :CF79G 5F9 5GGIA98 HC 69 N9FC CF 957< C: H<9 HKC
G5H9@@=H9 AC89@G HKC 5HH=HI89 A5BC9IJF9G K9F9 G=AI@5H98 'B9 GA5@@ 5B;@9 A5BC9IJF9 5B8 CB9 @5F;9
5B;@9 A5BC9IJF9
+9;2:9
*9GI@HG 9A9F;98 =B H<=G H<9G9G 5F9 DF9G9BH98 =B H<9 :C@@CK=B;
5G98 CB H<9 OFGH CF89F @=B95F 57HI5HCF AC89@ =B 3CGG9B 4 5B8 3CGG9B 5B8 >9@@GH58 4 5B
9LD5B898 G97CB8 CF89F AC89@ =G DF9G9BH98 =B H<=G H<9G=G ,<=G 57HI5HCF AC89@ =G 89G=;B98 5G 5B 5@H9FB5H=J9
:CF 89G7F=6=B; 57HI5HCFG K=H< H=A9 89@5M =B =HG F9GDCBG9 'B9 9L5AD@9 C: H<=G =G F957H=CB H<FIGH9FG K<=7<
<5J9 GCA9 89@5M 75IG98 6M CD9B=B; C: J5@J9G 5B8 ;5G PCK =B H<9 GMGH9A 69:CF9 H<9 H<FIGH GH5FHG HC 6I=@8
ID
,<F99 7CBHFC@@9FG K9F9 89F=J98 5B8 DFCJ9B +-+ 577CF8=B; HC $M5DIBCJQG 8=F97H A9H<C8 5B8
$5+5@@9
1CG<=N5K5QG H<9CF9A CBHFC@@9F =G 65G98 IDCB 5B 5IHCBCACIG GMGH9A 5B8 <5G CB@M 7CBG=89F5
H=CBG CB H<9 ?=B9A5H=7G 5B8 8MB5A=7G C: H<9 G5H9@@=H9 CBHFC@@9F =G 89F=J98 65G98 CB H<9 A5=B 7CB79FB
C: 7CFF97H=B; H<9 9FFCF =BHFC8I798 6M H<9 OFGH CF89F 57HI5HCF 8MB5A=7G 5B8 7CBHFC@@9F =G 9LD5B898 HC
=B7@I89 H<9 9FFCF =BHFC8I798 6M H<9 G97CB8 CF89F 8MB5A=7G 5G K9@@ CF 6CH< 7CBHFC@@9F 5B8 =H =G H<9
57HI5HCF A5HF=L K<=7< 89H9FA=B9G K<9H<9F H<9 GMGH9A =G 5IHCBCACIG CF BCB
5IHCBCACIG 6975IG9 H<9
CB@M H=A9 89D9B89BH 9@9A9BH =B H<9 GMGH9A 75B CF=;=B5H9 :FCA H<9 57HI5HCF AC89@ CF H<9 H<F99 HMD9G C:
57HI5HCFG 89G7F=698 =B H<=G H<9G=G CB@M H<9 A5;B9H=7 HCFEI9FG <5J9 57HI5HCF AC89@G K<=7< 5F9 89D9B89BH
CB H<9 DCG=H=CB =B H<9 CF6=H 5B8 <9B79 H<9 H=A9
FCA H<9 G=AI@5H=CBG 7<5DH9F H<9 G=AI@5H=CBG 7CBOFA H<9 A5H<9A5H=75@@M DFCJ9B GH56=@=HM :CF 5@@ H<F99
7CBHFC@@9FG K=H< 6CH< F957H=CB K<99@G 5B8 F957H=CB H<FIGH9FG ,<9 D9F:CFA5B79 C: H<9 7CBHFC@@9FG F9;5F8=B;
H<9 =BHFC8I798 8=GHIF65B79G 6M H<9 57HI5HCF 8MB5A=7 AC89@G K5G 9J=89BH GD97=5@@M =B H<9 75G9 C:
F957H=CB K<99@G 7CI@8 H<9 7CAD9BG5H=CB :CF H<9 57HI5HCF 8MB5A=7G 69 G99B 7CB8I7H98 6M 7CBHFC@@9F 5B8 ,<9 D9F:CFA5B79 C: H<9G9 HKC 7CBHFC@@9FG K5G 5@GC C6G9FJ98 HC 69 A5F;=B5@@M 69HH9F 7CBG=89F=B;
G9HH@=B; H=A9 C: H<9 5HH=HI89 5B8 5B;I@5F J9@C7=H=9G
;8:.+8 %581
,<9 7CBHFC@@9F 89G=;B =B H<=G H<9G=G K5G 65G98 IDCB H<9 5GGIADH=CBG C: 5 7CBGH5BH 5B;I@5F J9@C7=HM K<=7< =AD@=9G 5 7=F7I@5F CF6=H CF :IHIF9 KCF? =H =G F97CAA9B898 HC =BJ9GH=;5H9 H<9 9T97HG CB H<9
7CBHFC@@9F 89G=;B K<9B 5 J5FM=B; =G =BHFC8I798 ,<=G G<CI@8 69 8CB9 6975IG9 ACGH G5H9@@=H9 CF6=HG 5F9
9@@=DH=75@ 5B8 H<9 +%' K=@@ 5@ACGH 79FH5=B@M =B=H=5@@M <5J9 5B 9@@=DH=75@ CF6=H
CF H<9 G=AI@5H=CBG 7CB8I7H98 =B H<=G H<9G=G H<9 9LDF9GG=CBG :CF 5B8 5F9 :CIB8 :FCA BIA9F=75@
89F=J5H=CB C: 5B8 ,<=G 75B 5G 9LD@5=B98 DF9J=CIG@M =BHFC8I79 DFC6@9AG =B H<9 G=AI@5H=CBG 9GD97=5@@M
=B 75G9G C: @5F;9 F5H9 7<5B;9G LDF9GG=CBG :CF 5B8 75B 69 :CIB8 K=H<CIH IG=B; H<9 89F=J5H=J9 C: H<9
GH5H9G 6IH 8I9 HC H=;<H H=A9 G7<98I@9 H<=G K5G BCH 7CB8I7H98 =B H<=G H<9G=G ,<9F9:CF9 =H =G F97CAA9B898
5G :IHIF9 KCF? (FC656@M K=@@ H<9 9T97HG C: H<=G 69 GA5@@ G=B79 H<=G DFC6@9A K5G 7CAD9BG5H98 :CF =B H<9
G=AI@5H=CBG K=H< 5 GA5@@ A5L=AIA H=A9 GH9D =B H<9 GC@J9F 5G K9@@ 5G H<9F9 K9F9 BC =BHFC8I798 BC=G9 =B
H<9 G=AI@5H=CBG K<=7< 7CI@8 =B7F95G9 F5H9 7<5B;9G CK9J9F H<=G 75G9 G<CI@8 69 =BJ9GH=;5H98
+=B79 H<=G H<9G=G <5G 5 ;9B9F5@ H<9CF9H=75@ 5DDFC57< CB =BJ9GH=;5H=CB 5B8 AC89@@=B; C: 57HI5HCF 8MB5A=7G
H<9 J5@I9 C: KCF? 8=F97H@M :CF H<9 +%' A=GG=CB 7CI@8 <5J9 699B @5F;9F 'H<9F G=AI@5H=CBG K=H< 9AD<5G=G
CB A=GG=CB A5BC9IJF9G G=AI@5H98 =B 5B CF6=H 89H9FA=B98 6M H<9 %!+ H95A G<CI@8 69 :95G=6@9 @GC
=B7@IG=CB C: 8=GHIF65B79G K<=7< 9L=GH =B H<5H GD97=O7 CF6=H 7CI@8 <5J9 699B =BJ9GH=;5H98 5B8 G=AI@5H98
,<9 57HI5HCF 8MB5A=7G AC89@G 5F9 GH5H98 HC 69 5 69HH9F 5DDFCL=A5H=CB C: H<9 F95@ @=:9 F9GDCBG9 H<5B 5
G=AD@9F AC89@ !B H<9 75G9 C: H<FIGH9FG =H =G B95F@M =ADCGG=6@M HC AC89@ H<9 H<FIGH DFCO@9 A5H<9A5H=75@@M
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SPACECRAFT ATTITUDE CONTROL: Modelling and Controller

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