(GLAS) PRECISION ATTITUDE DETERMINATION - Center for ...

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Rigorous satisfaction of a set of linear dierential equations. No requirement for the evaluation of trigonometric functionsThe lack of trigonometric functions in the computation of quaternions is clearly an advantagein time-critical real time operations. Extensive use of trigonometric function inEuler angles will signicantly increase the computation time, especially with modest performancecomputers such as those used in on-board applications. The quaternions aredened based on Euler's rotation theorem [11] :The most general displacement of a rigid body with one point xed isequivalentto a single rotation about some axis through that point.For some axis, ^e, and a single rotation angle, , the quaternions are dened byq 1 = e x sin( 2 )q 2 = e y sin( 2 )q 3 = e z sin( 2 ) (2.1)q 4 = cos( 2 )where e x , e y and e z are components of rotation axes in terms of the OBF before therotation. Since there are only three degrees of freedom for the rotational motion, thefollowing constraint exists in the quaternion representation :q 2 1+ q 2 2+ q 2 3+ q 2 4=1 (2.2)The single constraint to be observed is a minor disadvantage associated with the fourquaternions. The detailed properties of quaternions and relevant equations are summarizedin Appendix A.The quaternion errors q are frequently represented by another quaternion rotation, whichmust be composed with the estimated quaternions ^q in order to obtain the true quaternionsq true asq true = q ^q (2.3)where the quaternion composition, , is dened in Equation A.9. A benet of this errorrepresentation can be seen by applying the small angle approximations to Equation 2.1 :q 1 = x2q 2 = y2q 3 = z2q 4 = 1(2.4)13
where x , y , and z were dened in the previous section as yaw, roll, and pitch angles,respectively. Only the vector components of quaternions (see Equation A.1) correspond toangle errors and the scalar component becomes insignicant to the rst order. By applyinginverse quaternions (Equation A.2) to Equation 2.3, the error quaternions are expressedasq = q true ^q ;1 : (2.5)Even though the Euler angles are not convenient for numerical computations, their geometricalsignicance in illustrating the rotational motion is more apparent than quaternions.Therefore, they are often used for computer input/output and for analysis. In thisresearch, simulated attitude data were created by Euler angles and, then, converted tothe quaternions using Equation A.13. Euler angles can be recalculated from estimatedquaternions by Equation A.14.2.3 Kinematic Equations of Spacecraft AttitudeIf the quaternion composition (Equation A.9) is performed successively in time, the timeevolution of quaternions during the time interval t is given byq(t +t) =q(t) q(t): (2.6)Let ! x , ! y and ! z be the components of angular velocity vector (~!) at time t, j!j bethe magnitude of the angular velocity, and be a rotation angle during t. From thedenition of quaternions, we can deriveq(t +t) =[ cos()2I 44 + sin()2(~!)]q(t) (2.7)j!jwhere (~!) is(~!) 2640 ! z ;! y ! x;! z 0 ! x ! y! y ;! x 0 ! z;! x ;! y ;! z 0375 : (2.8)This equation predicts the attitude at the future time based on the knowledge of thecurrent attitude if the axis of rotation is invariant over the time interval t. If the averageor instantaneous angular velocity of a spacecraft is known during t, the rotation angle is = j!jt (2.9)about the rotation axis. Assuming t is small enough, the small angle approximationscos 2 1 sin2 1 !t (2.10)214
Page 3: GLOSSARYATBDBDCCDCRFCRSDMTEKFFOAMFO
Page 11: Chapter 2FUNDAMENTALS OFATTITUDE DE
Page 16 and 17: lead to the attitude dierential equ
Page 18 and 19: To take advantage of multiple unit
Page 20 and 21: ppstarz SCF(boresightdirection)66op
Page 22 and 23: Table 3.1: Characteristics of the c
Page 25 and 26: Start?ReadStar Catalog?Input Initia
Page 27 and 28: Table 3.3: Some noise characteristi
Page 29 and 30: 3.2.1 Gyro ModelFarrenkopf's gyro m
Page 31 and 32: Table 3.4: Probability of the numbe
Page 33 and 34: the Sun, ~v, by [52], = j~vjcsin (
Page 35 and 36: 4.1 Pattern Matching Algorithm4.1.1
Page 37 and 38: 0 17(9)5(7)1(11)180 . . .-CC CW25(
Page 39 and 40: . If these conditions are satised,
Page 41 and 42: ....* 30 *400base lineAAAAAAAAAAU*
Page 43 and 44: third star is considered in a posit
Page 45 and 46: Star Tracker FOVat t i+1at t i+1obs
Page 47 and 48: the unique identication. If no star
Page 49 and 50: Declination(a)α 0-∆αα 0+∆αR
Page 51 and 52: The B-matrix is very important sinc
Page 53 and 54: The square roots of the diagonal co
Page 55 and 56: The benet of the FOAM over the SVD
Page 57 and 58: subject to(t k;1 t k;1 ) = I 33 (5.
Page 59 and 60: where ~v k is a discrete white Gaus
Page 61 and 62: Figure 5.1: QUEST angle errors in a
Page 63 and 64: Table 5.2: SFAD computation time ob
Page 65 and 66:
Figure 5.5: Angle errors from batch
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Figure 5.8: True errors versus rms
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Star Field6LRCLaser Reference Camer
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Star TrackerAAAAAAAA Astar A Ab AAA
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Figure 6.5: Pointing determination
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LRSLPAFigure 6.7: Projection of LPA
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A(t), needs to be known at a 40 Hz
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starspacecraftV = V earth+ V s/cV =
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simulation can be applied for the e
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Chapter 7IMPLEMENTATIONCONSIDERATIO
Page 85 and 86:
with the real time determination. T
Page 87 and 88:
Bibliography[1] European Space Agen
Page 89 and 90:
[31] B.E. Schutz. Spaceborne Laser
Page 91 and 92:
ACKNOWLEDGMENTSThe authors would li
Page 93 and 94:
where C ij is a matrix componentini
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(GLAS) PRECISION ATTITUDE DETERMINATION - Center for ...

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