The Cosmic Background Explorer (COBE) spacecraft will exhibit complex attitude motion consisting of a spin rate of approximately -0.8 revolution per minute (rpm) about the x-axis and simultaneous precession of the spin axis at a rate of one revolution per orbit (rpo) about the nearly perpendicular spacecraft-to-Sun vector. The effect of the combined spinning and precession is to make accurate attitude propagation difficult and the 1-degree (3 sigma) solution accuracy goal problematic. To improve this situation, an intermediate reference frame is introduced, and the angular velocity divided into two parts. The nonspinning part is that which would be observed if there were no rotation about the X-axis. The spinning part is simply the X-axis component of the angular velocity. The two are propagated independently and combined whenever the complete attitude is needed. This approach is better than the usual one-step method because each of the two angular velocities look nearly constant in their respective reference frames. Since the angular velocities are almost constant, the approximations made in discrete time propagation are more nearly true. To demonstrate the advantages of this nonspinning method, attitude is propagated as outlined above and is then compared with the results of the one-step method. Over the 100-minute COBE orbit, the one-step error grows to several degrees while the nonspinning error remains negligible

Chu, D.

English

NASA Technical Reports Server

N }0-13418
COBE NONSPINNING ATTITUDE PROPAGATION*
D. Chu
Computer Sciences Corporation (CSC)
ABSTRACT
The Cosmic Background Explorer (COBE) spacecraft will exhibit complex
attitude motion consisting of a spin rate of approximately -0.8 revolution
per minute (rpm) about the x-axis and simultaneous precession of the
spin axis at a rate of one revolution per orbit (rpo) about the nearly per-
pendicular spacecraft-to-Sun vector. The effect of the combined spinning
and precession is to make accurate attitude propagation difficult and
the 1-degree (3 tF) solution accuracy goal problematic.
To improve this situation, an intermediate reference frame is introduced,
and the angular velocity divided into two parts. The "nonspinning" part is
that which would be observed if there were no rotation about the x-axis.
The "spinning" part is simply the x-axis component of the angular veloc-
ity. The two are propagated independently and combined whenever the
complete attitude is needed. This approach is better than the usual "one-
step" method because each of the two angular velocities look nearly con-
stant in their respective reference frames. Since the angular velocities are
almost constant, the approximations made in discrete time propagation
are more nearly true.
To demonstrate the advantages of this "nonspinning" method, attitude is
propagated as outlined above and is then compared with the results of the
one-step method. Over the 100-minute COBE orbit, the one-step error
grows to several degrees while the nonspinning error remains negligible.
" This work was supported by the National Aeronautics and Space Administration (NASA)/Goddard
Space Flight Center (GSFC), Greenbelt, Maryland, under Contract NAS 5-31500.
PRECEDING PAGE BLAP;i( NGT FILMED 65
https://ntrs.nasa.gov/search.jsp?R=19900004102 2020-03-19T23:42:33+00:00Z
COBE ATTITUDE AND THE NEED FOR MORE ACCURATE
PROPAGATION
The attitude of the Cosmic Background Explorer (COBE) is three-axis stabilized with the
minus x-axis maintained 94 degrees from the Sun line (Figure 1). The spacecraft pitches
about that line at a rate of 1 revolution per 100-minute orbit period. It also spins about
the minus x-axis once every 75 seconds. COBE attitude can be expressed as an
Euler 3-2-1 (pitch-roll-yaw) rotation sequence with respect to a rotating Earth-Sun coordi-
nate frame with the z-axis pointing toward the Sun and the y-axis pointing along the cross
product of the Sun and Earth vectors. Nominal pitch and roll are then 0 and -4 degrees,
respectively.
SUN ==
SUN ANGLE94° =SI
P,TCHRA,E=
-O,06°tsec
=
X
Figure 1. COBE Attitude Profile
Propagating the attitude from samples of the angular velocity assumes that the angular
velocity remains constant in the body frame over the sample interval. The COBE gyro
sampling intervals are 0.5, 1, 2, or 4 seconds, depending on the telemetry format and
data rate. For COBE, which spins as much as 19.2 degrees per gyro sample, the pitch
component of the angular velocity can change direction significantly. Under these circum-
stances, the usual method of propagation, which handles the total incremental rotation at
once, introduces errors that accumulate over time and become unacceptably large.
This paper describes a variation on the usual one-step propagation that reduces this error
by introducing an intermediate nonspinning coordinate frame. The advantage of the
nonspinning frame is that the angular velocities used for its propagation and for the
subsequent transformation to the body frame vary much less between gyro samples. The
associated equations for gyro calibration follow along with numerical estimates of the
improvement in COBE propagation accuracy.
66
NOTATION AND THE USUAL ONE-STEP ATTITUDE PROPAGATION
Attitude is represented here by the orthogonal inertial-to-body coordinate transformation
matrix, AB/I, and the kinematic equation for its propagation (Reference 1, p. 512) is
'_B/I = - _B AB/I (1)
The dot (') above ABtI indicates differentiation with respect to time. Since the angular
velocity, tOa, is not constant, Equation (1) is solved numerically. Still, a formal solution
may be written as
Aw, (t) = OB/I (t, to) AB/, (to) (2)
Oa/_ is the attitude propagation matrix satisfying the differential equation:
_B/, = -- WB _B/, (3)
and the initial condition:
(I)B/I (to, to) = I (4)
where I is the identity matrix.
The subscripts I and B refer here to the inertial and body coordinates in which a vector or
matrix is expressed. The vector WB, for example, is the angular velocity in body coordi-
nates, and wa is the antisymmetric matrix derived from it.
I 0 - O)B3 (/)B2 1
O)S -- (OB3 0 --WBl
- O)B2 O)BI 0
(5)
The slash (/) indicates a transformation from the frame on the right to that on the left.
Thus, AB/I is the inertial-to-body coordinate transformation matrix.
NONSPINNING INTERMEDIATE FRAME AND TWO-STEP PROPAGATION
In body coordinates, the spin component of the COBE angular velocity, w_, is constant,
while the pitch component, to_, varies with time:
t tt
WB (t) = WB (t) + w. (6)
If the spin axis is denoted by _ B, these two portions of the angular velocity can be com-
puted as follows:
wh = (I - _B (_B) T) wB (7)
67
ro; = _B (_B) T O')B (8)
An intermediate, nonspinning coordinate system denoted by the subscript N can be intro-
duced that is defined by the propagation equation
,_N/I = - _N AN/I (9)
where
' (A ) ' (10)(-ON _ B/N T roB
Since the magnitude of w_ is less than that of rOB, rO_ does not change as much as rOb
does between gyro samples. Thus, all other things being the same, the error in propagat-
ing AN/_ should be less than the error in propagating AB/_. If the propagation from the
nonspinning frame to the body frame can be done perfectly, as is plausible since the spin
rate is constant, the total propagation error for this two-step method sho'uld also be less
than that for the usual one-step method•
To complete the propagation of AB/_, it remains to compute AB/N, which transforms
from the nonspinning to the body frame:
AB/N = AB/I (AN/I) T (11)
By the product rule for differentiation, the AB/_ is
,_B/N = '_B/I (AN/I) T + AB/I (,_N/I) T (12)
Substituting for ,_u/l and ,_N/I from Equations (1), (6), and (9) and combining attitude
transformations yields
• t
AB/N = -(O)B + _B)AB/N + AB/N (-_)N (13)
Noting that WN iS the similarity transformation of _h,
t s
_N = (AB/N) T _B AB/N (14)
gives the differential equation for the propagation of AB/N:
• H
AB/N = - _B AB/N (15)
With equations for both AB/N and AN/I, the complete attitude, AB/I, can be propagated.
GYRO CALIBRATION
In addition to reducing the propagation error due to the finite gyro sampling interval, it is
usually necessary to calibrate the gyros to reduce systematic errors in the sensed angular
68
velocity. Becausecalibration involves the sametype of computations as propagation, it is
also done more accurately with a two-step method.
With one-steppropagation, gyro calibration errors, Aa, can be found using the solution
to the following error propagation equation (Reference 2, p. 4-13):
0B = -WB 0B + AWB (16)
where 0B is the attitude error expressed in axis and angle form, and the angular velocity
error, AwB, is related to the gyro calibration errors by the matrix G(coB).
AWB = G(OB) Aa (17)
The solution to the error equation has the form
0B (t) = FBa (t, to) Aa + @B/I (t, to) 0B (to) (18)
where the variational matrix, 1-'B/i, transforms the gyro calibration errors into contribu-
tions to the attitude error
FB/I (t, to) = ft' (I)B/I (t, r) G (WB) dr (19)
O
The matrix FB/_ then serves as the partial derivative of the propagated attitude error with
respect to the gyro parameter errors:
00B
= rBa (t, to) (20)
OAa
If one knows 0B at times t and to, Aa can be found from Equation (18).
A corresponding variational matrix is needed for calibration with the nonspinning propa-
gation method. Because the two methods are different, there is no reason to expect the
variational matrices to be the same. Both steps of the nonspinning propagation, however,
follow the same kind of propagation equation as does the one-step method, and the corre-
sponding error equations can be applied to each step separately:
"' = _ ' ' (21)ON Aw_ WN ON
• tt tp tP tt
0B = AWB - rOB 0B (22)
Here, 0_ is the error in the nonspinning propagation expressed in the nonspinning frame.
0_ is the corresponding error in the spinning propagation expressed in body coordinates.
The angular velocity errors, A¢o_ and Aa)_, are defined as follows:
A(.O N _ AN/B [I -- B (_B) T] G ((.OB) aa (23)
69
AwE = _B (_B) TG (O)B) Aa (24)
0_ and 0H can be solved for as in the one-step propagation to give solutions of the form
0_ = FN/I Aa + _N/I 0_ (to) (25)
0H = FB/N Aa + _B/N 0H (to) (26)
where the variational matrices FN/_ and FB/N are computed as follows:
£' ,FN/I = _N/I (t, r) AWN dr (27)
O
_t ,,FwN = _B/N (t, r) AOB dr (28)
O
The total propagation error, 0B, then equals
0B = AB/N 0_ + 0H (29)
This gives the partial derivative of the attitude error with respect to the gyro calibration
errors as
FB/I = FB/N + _B/N FN/I (30)
which can be used in Equation (18) to solve for Aa.
NUMERICAL SIMULATION
Although the large COBE gyro sampling interval can be expected to degrade one-step
propagation accuracy, it is useful to know how much of an effect it actually has. The
nonspinning method must justify its additional computation with significantly better accu-
racy.
To compute the propagation error for each method, the pitch rate, _, roll rate, 0, and
spin rate, _P, measured with respect to the inertial frame are assumed to be constant.
= - 360°/6000 sec -- -0.06*/sec (31)
b = 0 (32)
= -0.8rpm = -4.8°/sec (33)
70
The exact attitude is then found by first performing the pitch rotation, qS, followed by the
roll and spin rotations, 0 and _P.
= (t- to) _ + cpo (34)
0 = -4 ° (35)
_p -- (t- to) _ + IPo (36)
The one-step attitude can be propagated from the following formula for the angular veloc-
ity (Reference 1, p. 765):
COB(t) T = [_0 - sin (0) _, cos (0) sin 0P) _, cos (0) cos 0P) _] (37)
The two angular velocities for the nonspinning propagation are as follows:
CO_4(0 T ---- [0, COS (COrot) COB2 - sin (COB1t) COB3, sin (COBlt) ¢0B2 + COS (COrot) COB3] (38)
COBT = [£0 - sin (0) _, 0, 01 (39)
The one-step and nonspinning attitudes are then compared to the exact attitude, and the
angular differences are computed. A plot of the one-step propagation error for 0.5-second
sampling intervals is shown in Figure 2 for a timespan of one orbit. While the one-step
error grows to a maximum value of 2.4 degrees, the nonspinning error remains less than
0.003 degree.
The results show that, even for the smallest gyro sampling interval, which is 0.5 second,
the one-step propagation errors are quite large. This is counterintuitive. Since a constant
spin can be propagated without error, it would be expected that adding a much smaller
constant pitch rate would have a negligible effect. The flaw in that argument is that,
although the spin is constant in the body frame, the pitch angular velocity is not. It is
constant in the inertial frame. The assumption about the size of the error is also slightly
misplaced. Small pitch rates do produce slowly growing propagation errors, but because
the orbit is correspondingly longer, they have more time to grow.
Even more surprising than the size of the error is its oscillation. Rather than grow without
limit, the propagation error peaks at the middle of the orbit. The reason is that while the
actual pitch angular velocity moves continuously in the body frame, the sampled pitch
angular velocity is fixed in the direction it has at the start of the interval. Thus, the
sampled value lags the true value and introduces a roll component of angular velocity
(Figure 3). Over an orbit period, the roll direction changes by 360 degrees, and the
propagation error that builds up in the first half of the orbit decreases over the second
half.
71
F,,
O
er
ft.
2.0
1,0
VAL
l I l I l I l I I I
0 20 40 60 80 TIME (min)
Figure 2. One-Step Propagation Error Time Dependence
-7,
r
Figure 3. Sampling Introduces a Roll Angular Velocity
SUMMARY
The nonspinning propagation method described here is a means of trading computation
for accuracy when the body angular velocity changes direction between gyro samples.
This method is currently implemented for batch attitude determination in the COBE flight
dynamics support system (Reference 3, p. 3.1.2.42-1), where it is needed to meet the
attitude determination accuracy goal of 1 degree (3o'). Further investigation should still
be done on understanding the effects of nonconstant roll, interpolating the angular
72
velocity to the midpoint of the sampling interval, and using higher order numerical inte-
gration methods.
Whether nonspinning propagation is worth the extra work for other missions depends on
the magnitude and form of the angular velocity, the gyro sampling rate, and the accuracy
requirements. The unexpectedly large errors that would have been observed for COBE,
however, argue for consideration of this effect whenever a three-axis stabilized spacecraft
undergoes a compound rotational motion, such as spinning and pitching.
ACKNOWLEDGMENTS
Nonspinning propagation was proposed for COBE by B. T. Fang, and in presenting it
here, the author seeks only to make it available to a wider audience. For their help in that
effort, the author wishes to acknowledge M. Nicholson and Y. Sheu of Computer Sci-
ences Corporation and F. L. Markley of Goddard Space Flight Center.
.
.
.
REFERENCES
J. R. Wertz (ed.), Spacecraft Attitude Determination and Control. Dordrecht, Holland:
D. Reidel Publishing Company, 1980
Computer Sciences Corporation, CSC/TM-88/6001, Attitude Determination Error
Analysis System (/iDEAS) Mathematical Spec_ications Document, M. Nicholson,
October 1988
--, CSC/TR-87/6004, Cosmic Background Explorer (COBE) Flight Dynamics Support
System (FDSS) Specifications I1. Functional Specifications, K. Hall, June 1987
73



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