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Searches related to ile rodrigues alizoa voyages filetype:pdf

Euler angle representation the Rodrigues parameters and the modified Rodrigues parameters It is widely known that 'all three-dimensional parameterizations have singularities (e g the Rodrigues parameters are singular for 180 degree rotations [4]) The choice of parameters depends on a number of

JohnL.Crassidis

AssistantProfessor

CatholicUniversityofAmerica

DepartmentofMechanicalEngineering

Washington,D.C.20064,r

r_....Lf_

F.LandisMarkley

StaffEngineer

GoddardSpaceFlightCenter

GuidanceandControlBranch,Code712

Greenbelt,MD20771

Abstract

developedfor

Introduction

norm. 71
physicalrepresentationinsight,etc. spacecraft.

AttitudeKinematics

with

Iql(O)q13=q2=_hsin

q3 q4=cos(O)(1) (2a) (2b) givenby 72
[---t.0T!0 q413×3+[q13X]]] where13×3isa3x3identitymatrix. (4b) -a2a10(5) qTq=qT3ql3+q2=1(6) q13__titan(O)(7)p-l+q4 angularvelocity(m),givenby[4] =m__-2m_xp+2(m_-p)p(8)

11Tx]+eer}mel,,x,+[e(9)

BB=A(p_)BI(10)

andA(_p)isgivenby 73

8[e×]2a(e)(l+e e)2

KaimanFilterDevelopment

A_x_f=.A_(12)

Theerrorquaternionisdefinedas

8q=q®O-1(13)

rotation,Equation(13)canbeapproximatedby

8q=[8.1'.3.](14)

gyrobiascorrections(_b),givenby _x=[_](15)

¢.0=_-b-rl1(16)

vector,whichismodeledby -b=_-2(17) 74
w-r ,.1-0-2_1

E{_w(t)}=0

E{w(t)wr(t')}=QS(t-t')

where(18b) (18c):f(p,b,_,t)+g(p,rll,t)(19a) b=r12(19b) 11 (20b) givenby whereAk_=FAx+_Gw_(21) F=5-_ of°"LG21G22J (22b) (23a)

G12=G21=03x3(23b)

G22=13x3(23c)

Theestimatedstate-errorequationisgivenby

___=F1x:_-___(24a) 75

AS-x-_

whereOfx:211+/3/3T} z_k=h_(x_)+v_k where hk(x_)=A(e_)-8,,

Thesensitivitymatrixcanbewrittenas

Hk=[L!03x3]

+__h__A-Xkhk(x_)=hk(__)-PIx_=__ (25a) (25b) (26) (27) (28) (29a) (29b) (30)

Theadditiveapproach

(32) 76
p=8p®/3(33) p=(34) -1+8_2/32_28p./3 (35) From

A(p)=A(Sp)A(_)(36)

itfollowsthat:I-P-(37) andusingthefactthatforsmall8p

A(SP)=13×3-418P×](38)

-4[A(_)BIX]{(1-[_T_)I3x3-2[_X]+2_T}(39) 77
k-_f(j_,t) p=p+prp+dOdr(40a) (4Oh) (40c) (40d) (40e) whered,,!d_...:.-. [03×3!13x3J

SpacecraftSimulationandResults(41)

78

FigureITRMMSpacecraft

SunAvailability

1.5 _e m ca1 0r- II 0 _=-0.s m m

¢-•-,0

T'- 0.5 -10o.. i: i. 2345

Time(Hr)

Figure2PlotoftheSunAvailability!I

678
79

EstimatedModified-RodriguesParameters

012345678

21!,:,!!;/

--Ii!I

012345678

:L.......t

012345678

Time(Hr)

Figure3PlotofKalmanFilterMRPEstimates

KalmanFilterGyro-BiasEstimates

!!III|

IIIIII

1234567

o.1IIIIIIIIl -c-LA"'i"iiii.....

Time(Hr)

80

KalmanFilterModified-RodriguesCovariance

10-15iiiiiI|

012345678

__o...........!

10-_SliiiiiilI

012345678

10"s,,,,

nm10-1°"::

10-_5/iiiiiiI

012345678

Time(Hr)

Figure5PlotofAttitudeCovariances

100KalmanFilterGyro-BiasCovariance

12345678

2345678

IIII 23456

Time(Hr)

Figure6PlotoftheGyro-BiasCovarlances

81
3OO 25O
_200 150
100

5O0TrueAngleofRotation

2345

Time(Hr)

Figure7PlotofTrueRotationAngle8

KalmanFilterAttitudeErrors

0.1 oii -0.1ie01234!! 5678
-0.1

01234567°1Lioil--_........i

o,r_02345678

Time(Hr)

Figure8PlotofAttitudeErrorTrajectories

82

Conclusions

Acknowledgments

References

Aug.1986,AIAAPaper#86-1987,pp.88-96.

Vol.41,No.4,Oct.-Dec.1993,pp.439-517.

Dynamics,inreview.

Austin,TX,Feb.'1996,AASPaper#96-172.

83
quotesdbs_dbs21.pdfusesText_27
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