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Results of the Attitude Control of "Helios" A
RESULTS OF THE ATTITUDE CONTROL OF "HEllOS" A
L. Schmieder OFVLR -
Oberpfaffenhofen
Institut fur Oynamik der Flugsysteme
Summary The attitude control of "Helios" during the mission is done using measurements of the sUn-sensor alone .
The observabi I ity
of the system is investigated with the result, is observable,
that the system
if the disturbing torque is only caused by the
light-pressure.
Apart from an irregularity between day 26 and
day 44 after launch, which could not be analysed, as the system is unobservable,
if additional disturbating torques are present,
al I observations were in very good agreement with the theory. The fourth attitude maneuver brought the vehicle into the stat ion a r y po i n t .
Sin c e t his t i met her 0 I I - and pit ·c h an 9 I ew e re
nearly constant and remained within the specified limits ± 10 . A fifth maneuver was not necessary until
today (25th January 1976)
241
Introd uction The aim of the attitud e-cont rol of Helios A during the missio n was to keep the spin-a xis of the vehicl e vertic al to the orbitplane within a tolaran ce of one degree . This should be done by means of the sun-se nsor alone, which measur ed the negati ve rollangle, whi le the pitch-a ngle remain ed unobse rved. One year before launch a detaile d study was made in our institu te with the result , that both angles should be known with an accura cy not worse than 0.1 0 [1]. In this study an abunda nt use was made from Kalman 's filteri ng theory [2]. The varian~es were invest igated , and the whole system was tested by means of a numeri cal simuI a t i on .
The equatio ns of motion Figure
shows the orbit of Helios , the inertia l coordi nates Xl'
X2 and the coordi nates Xl' x ' wh i ch turn with Helios on 2
its It is useful , to derive the equatio ns of motion as in the inertia l -- as in the movabl e coordi nate -system [3] . In the first system the vect o r-equa tion
orbit .
(1)
wi th time ,
angular momentum,
disturb ing torque,
is va lid. As in the actual Hel ios measur ements no change of the s pin was observ ed, we may assume
242
and therefore
(2)
const
For the remaining two equations
(3)
we assume,
that the disturbing moment is effected only by the
scalar light-pressure, the resultant of which does not coincide with the center of mass .
Thus a moment is generated, which is
always perpendicular to the radius-vector sun-spacecraft (v true anomaly, see fig.
I), and which is represented by the for-
mulas -Tsinv
Tl
(4)
T2 = T cos v T has the form
(5)
T
wi th distance sun-earth
A
instantaneous distance sun-spacecraft
r =
1+Ecosv a
=
great half-axis of the Helios orbit
E
=
eccentricity of the Hel ios orbit
T o
2
*
96.8
*
10 9 m
0 . 5218
10- 6 Nm was calculated by MBB (Messerschmitt-B6Ikow-Blohm)
243
from the configuration of Hel ios and the light-pressure 2 * solar-constant veloc i ty of light
p
2·1400 =37
~
HI' H2 are very small against H3 (ca . 10
-2
). thus we introduce
(6)
into eq.
and with regard to (4) we get
(3)
T
dX 1
-dt- -- - -H3 sin v
(7) dX2
T
dt
H3
-- = -
cos v .
These equations ar e cons i derably simpl ified by introducing the angular velocity
(~
= Newton's constant
dv
mass of the sun)
(8)
dt
dX
*
1
Tv
= - C sin v
(9) dX
2
dV
= C cos v
wi th 2400 m2 kg/sec 1. 32462804
*
10
20
2 m3 /sec
the constant C accepts the value
244
6.2
C
The solution of (9)
10 -3 radian
*
is the circle
C cos v + x 10 - C
Xl
(10) X
2
=C
sin v + x20
which could be observed by two starsensors on Helios.
For the
actual case (measurements wi th respect to the sun) the variables Xl' x 2 must be replaced by
-c~svl [XT]
v
Xl]
=
[ x2
[-Sin
-s I nv
COSY
( 11)
XI
which leads to the equations
(12)
wi th Cl
C from eq . (10)
C 2
0
XT =
pitch-angle
XI
= negativ roll-angle
It is useful
decl ination of the sun with respect to Hel ios
to introduce the complex functions X-
2
(13) c(v)
Cl +
245
into (lZ), which leads to the equation dz dv + j z
=
c
The homogeneous equation (c _ 0)
is solved by
z=ye- jv
(14a)
This is also the solution of the complete equation (lZ*),
if the
complex constant y becomes a function of v: v y (v)
y
o
+
J
By means of (13) the solution can be written
(14b)
in real variables
( t. v = v - v ) : o
xI = - xTo sin t. v + xlo cos t. v - J l sin v + JZ cos v wi th v
J
l
J Vo
(C l cosw-CZsinw)dw
v
JZ
J v
(C 1 sin w + Cz cos w) dw
o
In the actual case of Hel ios (Cl = C = const, C2 is reduced to
0)
the inte-
grat i ons can be e x ecuted and (15)
(16a) xI = - xTo sin t. v + (xIO + C) cos t. v - C
246
(16b)
The function
(16b), which describes a sine with the amplitude
and the means (1])
XI
= -
C
is observed from Helios.
The observabi 1 i ty of the system The observabi I i ty of the only
I inear system (12)
for
the case,
that
xl" is measured and that Cl' C2 are constants, can be inve-
stigated Extend
in the following way
the system (12)
[4] :
by the two equations
(12**)
The complete system then gets the form
(18) y
H
x =
fXT
x=
observed variable
wi th
l~
F
0
0
-1
0
0
Cl
0
0
0
0
C 2
0
0
0
0
H
[0
0
0]
247
Now form the matrix 0= [H',
F'
(H', F' , (F )n-l
H',
are the transposed matrrces) H'], and
has the dimension four of F, actual
-1
0
0
In the
0
-1
0
0
-1
0
0
-1
the same procedure
equation
(12**)
differential
result
the system (12) with Cl C2
*0
*
0,
is no~ ob~~vable.
is repeated with C = 0, then the second 2 and the system reduces to three
is cancelled,
equations.
~
0
Thus
which has the rank three.
0
0
This
rank of this matrix
case we get the matrix
0
If
if the
the system is observable.
-~J
-1 0
is very
which has the rank three . system (12) with Cl
* 0,
Thus the C2
*0
is
ob~~vable.
-1
0
is not observable,
The matrix 0 then gets the form
important .
It
states,
that the Helios-system
if the disturbing moments have not the form
ofeq.(4). The method, gives, 1 ity
in
employed above,
is
restricted on
this simple form,
no
information about the observabi-
if Cl'
C are functions of v. 2 very general method, which yields a
Ther ~ fore ~' "
e
Thi " JT, c" .hod
velopment of the solution of a
system
dX
1
. . . .. .. . . , .. dx
n
dV =
248
x
f 1 (xl'
n'
another
insight
into
is based on the de-
v) (19)
, . ,.
f n (xl' .. . ,
we give
physical
the probl e m of observability .
dV =
linear systems and
X
n'
v)
into Taylor-series, which have the form (20)
The coefficients xi(O) are given as the initial conditions . coefficients of the second term are the functions (19) at v
The O.
=
The coefficients of the third term are given by the rule (summarize over the index R,)
a {-av
+ f
dx.
a I
-R, axR,
I
J
dV (21)
3 X3i ddV = {
etc .
The operator { } is called, according to Grobner, the Lie-operator [51.
For the equations (12)
the Lie-operator has the form (22)
The function xI(v)
is uniquely determined by the following ex-
pressions, taken at the point v
dXi
C2
dV 2
v
o
:
xI(v o )
XI
d
=
-
xT (23)
xi
--:-T dt
=
3 d x2" ~=
C' 2
- Cl
COl - ClI 2
- XI
-
C2 + xT
ect
The functions Cl (v), C2 (v) are in (23) represented by Taylorseries. The left sides of (23) are known from the diagram xI(v),
249
in which al I measurements are entered.
The observable quanti-
ties are therefore
C~ and,
- Cl
if C2 " 0, the quantities xl"' xT' Cl are observable.
An earth satell ite with disturbing moments caused by the atmosphere, the gravitation, the magnetic field etc . , is not ob~en vab.f.e by an earth-sensor alone, but it is well observable with an additional sun-sensor.
We give the short investigation.
For
the equations (9) with general disturbing torques,
(9*) dX
2 dv
-= 0
2
the lie-operator has the form
,
L}
(24)
and the measured variables are xl
Xl"
= -
(by the sun-sensor) and
xl cos v - x2 sin v
(25)
(by the earth-sensor). From xl and xl" the variable x 2 is known besides in the point = o. From the equations
sin v
(26)
etc.
250
we know further the complete funct ion DI (v).
By means of (24)
we calculate dXI (hi
= xI
sin v - x 2 cos v - DI cos v - D2 sin v (2])
2
d x2
~ dv
=
xI cos v + x 2 sin v + 2 DI sin v - D; cos v - 2 D2 cos v - D~ sin v etc.
In the first equation (Xl) al I quantities are known besides D2 sin v, thus D2 is known,
if sin v
*
O.
Then D' can be deter-
mined in the same way from the second equation etc. sys tem (9*)
Thus the
is observab le.
Now let us return to the discussion of the system of Hel ios . The second equation of (23)
(Cl = constant, C2 = 0) (23*)
permits the calculation of the pitch angle xT from the slope of the curve xI(v), which is given by the observations.
From the
third equation (23) we get in the same way the constant C, but not very accurate. (16b)
The accuracy is improved,
if the function
is fi I led to many observations by varying the quantities
xTo' xIo and C.
This may be done by an averaging calculation
according to Gauss or, in an iterative way by Kalman's filtering theory. In the actual case of Helios the direct evaluation of the observatIons according to (23*) was sufficient.
251
Comparison of the theory with the real observations Figure 2 gives the course of the observed decl ination of the sun (= negat ive roll-angle) .
26 days after the start the decl ina-
tion decreased within five days about 0.4 0 and after that it increased about the same amount.
It is very difficult to find
an explanation for this anomaly.
If there was an additional
disturbing
t~rque,
the system was not observable, and if there
was a fai 1 ing in the sensor, we cannot find it, because there was only one sensor on board (coarse and fine sensor were mounted at the same frame and their measurements were in agreement) . The anomaly caused a conside r able uncertainty, which is demonstrated by the three attitude maneuvers on day nr. 41, 64 and 80 (taken from January I, 1975).
After the thi rd maneuver the
institute was inserted to analyse the situation, which was critical, as the beginning of the blackout (in this case the sun between Helios and earth prevented any communication) was expected within three weeks. formula
The investigation was done according to
(16b), by fitting the parameters to the observations .
We took into consideration the variations of the pitch-angle caused by the well known attitude-maneuvers.
The result was a
very good agreement between theory and observations from day nr. 30 to day nr.
lOO, whi le the anomaly between the days nr. 5
and 30 became evident by a comparison of the observations with the extracted 1 ine, which is a backward calculation from day nr. 41.
Our results were confirmed by independent calculations of
the pitch-angle on certain days, which are based on the measurements of experiment nr. 9 (Dr.
Leinert, Heidelberg) and which
were in agreement with our calculations within 0 . 10 values of v in figure 2).
(see the
Thus we could recommand to execute
a fourth attitude-maneuver to bring the system to the stationary point, which is characterized by
o (28)
- c
252
With these inertial conditions the equations (16) go over in the stationary solution xT :: 0 (29) x2 -= - C The course of the observed decl ination after the fourth attitudemaneuver demonstrates, that the stationary point was reached within an accuracy of 0.19 0
.
The function xI(v)
very good approximation a sine.
in fig . 3 is in
The first maximum is - 0.59°,
the minimum - 0.97°, thus the mean value is Cobserved
Until
=
- 0.78°
and the amplitude
today (January 25,1976) a fifth attitude-maneuver was not
necessary .
It must be noticed, that the second maximum of x (v)
is - 0.54°,
that is 0.05 0 higher than the first.
caused by the fact,
2
This may be
that C is not exactly constant, because the
high-gain antenna of Hel ios is always directed to the earth and therefore must have a varying area with respect to the sun.
But
if there are small additional perturbating torques of statistical nature, we suppose,
in analogy to the Brownian motion of particles
in liquids, that the amplitude of the function
(16b)
creases, because the differential equations (12) solution .
slowly in-
have no stable
We know this either from the solution (16)
itself,
which contents no damping terms, or from the matrix A
of the system (12), which has the eigenvalues ±
Only the observations over many orbits may indicate , if such an increase of the amplitude really takes place.
253
Literature [ 1]
Hofmann, W.,
"The attitude Control of Helios during
Schmieder,
the Mission."
L.
Preliminary Report,
[2]
Jazwinski, A.H.
DFVLR-Oberpfaffen-
hofen,
Institut fur Dynamik der Flugsy-
steme,
Dec .
1973 .
"Stochastic Processes and Filtering Theory". Academic Press, New York and London, 1970 .
[ 3]
Schmieder,
L.
"The Attitude-Control of the Solar-Probe He I i os" . XXVI-th
International Astronautical
Congress, [4]
Bucy,
R.,
Joseph,
P.
"Fi ltering for Stochastic · Processes with Applications to Guidance" . John Wiley
[ 5]
Wanner,
G.
Lissabon 1975, paper no . 180.
&
Sons,
1968 .
"Integration gewohnlicher Di f ferentialgl e ichungen". Hochschultaschenbucher Nr . 831/831a, Bibliographisch e s 1969 .
254
Institut Mannheim,
Perh.l : 15. 3. 75
21.9.75
60
Start 10.d.c 1974
Fig . 1
Orbit of He-lios
0 .5218
£ •
a • 96.8 million5 of km U • 180.3 days
r •
0(1-£2)
1+£casv
6
J
rmax = a ( 1 • £ ) = 148 It 10 km r . = 0(1-£)= 46.4.10 km mon
x, x
T v
I,
x
fixed coordinate system 2 xI rotating coordinate system true
anomaly
255
:=
..
,.
.., z
~
~
> IC:
g.
SI R
~
~
IC:
,. 2
256
5
o
" o
"
~
;
t
i¥
0
ii ~
,
C
.
i :! ,;.
0;:
5:;:
E
257
0 0
o.c ....
?;>
"
"E c ." 5
'Q ~
0
~t 0 0 0
,.; 0
0
~
0 0
~.
8 '" 0o
o
<>
'"
0-
I
o
.,
o
"'-
8N
I
C>
o
'"
N-
I
o
0_
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..
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~
c
:>
0_
....
£
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~~~.::,-~~~~'~. 1;::_:
o
258
.:.-
o
L. Schmieder OFVLR -
Oberpfaffenhofen
Institut fur Oynamik der Flugsysteme
Summary The attitude control of "Helios" during the mission is done using measurements of the sUn-sensor alone .
The observabi I ity
of the system is investigated with the result, is observable,
that the system
if the disturbing torque is only caused by the
light-pressure.
Apart from an irregularity between day 26 and
day 44 after launch, which could not be analysed, as the system is unobservable,
if additional disturbating torques are present,
al I observations were in very good agreement with the theory. The fourth attitude maneuver brought the vehicle into the stat ion a r y po i n t .
Sin c e t his t i met her 0 I I - and pit ·c h an 9 I ew e re
nearly constant and remained within the specified limits ± 10 . A fifth maneuver was not necessary until
today (25th January 1976)
241
Introd uction The aim of the attitud e-cont rol of Helios A during the missio n was to keep the spin-a xis of the vehicl e vertic al to the orbitplane within a tolaran ce of one degree . This should be done by means of the sun-se nsor alone, which measur ed the negati ve rollangle, whi le the pitch-a ngle remain ed unobse rved. One year before launch a detaile d study was made in our institu te with the result , that both angles should be known with an accura cy not worse than 0.1 0 [1]. In this study an abunda nt use was made from Kalman 's filteri ng theory [2]. The varian~es were invest igated , and the whole system was tested by means of a numeri cal simuI a t i on .
The equatio ns of motion Figure
shows the orbit of Helios , the inertia l coordi nates Xl'
X2 and the coordi nates Xl' x ' wh i ch turn with Helios on 2
its It is useful , to derive the equatio ns of motion as in the inertia l -- as in the movabl e coordi nate -system [3] . In the first system the vect o r-equa tion
orbit .
(1)
wi th time ,
angular momentum,
disturb ing torque,
is va lid. As in the actual Hel ios measur ements no change of the s pin was observ ed, we may assume
242
and therefore
(2)
const
For the remaining two equations
(3)
we assume,
that the disturbing moment is effected only by the
scalar light-pressure, the resultant of which does not coincide with the center of mass .
Thus a moment is generated, which is
always perpendicular to the radius-vector sun-spacecraft (v true anomaly, see fig.
I), and which is represented by the for-
mulas -Tsinv
Tl
(4)
T2 = T cos v T has the form
(5)
T
wi th distance sun-earth
A
instantaneous distance sun-spacecraft
r =
1+Ecosv a
=
great half-axis of the Helios orbit
E
=
eccentricity of the Hel ios orbit
T o
2
*
96.8
*
10 9 m
0 . 5218
10- 6 Nm was calculated by MBB (Messerschmitt-B6Ikow-Blohm)
243
from the configuration of Hel ios and the light-pressure 2 * solar-constant veloc i ty of light
p
2·1400 =37
~
HI' H2 are very small against H3 (ca . 10
-2
). thus we introduce
(6)
into eq.
and with regard to (4) we get
(3)
T
dX 1
-dt- -- - -H3 sin v
(7) dX2
T
dt
H3
-- = -
cos v .
These equations ar e cons i derably simpl ified by introducing the angular velocity
(~
= Newton's constant
dv
mass of the sun)
(8)
dt
dX
*
1
Tv
= - C sin v
(9) dX
2
dV
= C cos v
wi th 2400 m2 kg/sec 1. 32462804
*
10
20
2 m3 /sec
the constant C accepts the value
244
6.2
C
The solution of (9)
10 -3 radian
*
is the circle
C cos v + x 10 - C
Xl
(10) X
2
=C
sin v + x20
which could be observed by two starsensors on Helios.
For the
actual case (measurements wi th respect to the sun) the variables Xl' x 2 must be replaced by
-c~svl [XT]
v
Xl]
=
[ x2
[-Sin
-s I nv
COSY
( 11)
XI
which leads to the equations
(12)
wi th Cl
C from eq . (10)
C 2
0
XT =
pitch-angle
XI
= negativ roll-angle
It is useful
decl ination of the sun with respect to Hel ios
to introduce the complex functions X-
2
(13) c(v)
Cl +
245
into (lZ), which leads to the equation dz dv + j z
=
c
The homogeneous equation (c _ 0)
is solved by
z=ye- jv
(14a)
This is also the solution of the complete equation (lZ*),
if the
complex constant y becomes a function of v: v y (v)
y
o
+
J
By means of (13) the solution can be written
(14b)
in real variables
( t. v = v - v ) : o
xI = - xTo sin t. v + xlo cos t. v - J l sin v + JZ cos v wi th v
J
l
J Vo
(C l cosw-CZsinw)dw
v
JZ
J v
(C 1 sin w + Cz cos w) dw
o
In the actual case of Hel ios (Cl = C = const, C2 is reduced to
0)
the inte-
grat i ons can be e x ecuted and (15)
(16a) xI = - xTo sin t. v + (xIO + C) cos t. v - C
246
(16b)
The function
(16b), which describes a sine with the amplitude
and the means (1])
XI
= -
C
is observed from Helios.
The observabi 1 i ty of the system The observabi I i ty of the only
I inear system (12)
for
the case,
that
xl" is measured and that Cl' C2 are constants, can be inve-
stigated Extend
in the following way
the system (12)
[4] :
by the two equations
(12**)
The complete system then gets the form
(18) y
H
x =
fXT
x=
observed variable
wi th
l~
F
0
0
-1
0
0
Cl
0
0
0
0
C 2
0
0
0
0
H
[0
0
0]
247
Now form the matrix 0= [H',
F'
(H', F' , (F )n-l
H',
are the transposed matrrces) H'], and
has the dimension four of F, actual
-1
0
0
In the
0
-1
0
0
-1
0
0
-1
the same procedure
equation
(12**)
differential
result
the system (12) with Cl C2
*0
*
0,
is no~ ob~~vable.
is repeated with C = 0, then the second 2 and the system reduces to three
is cancelled,
equations.
~
0
Thus
which has the rank three.
0
0
This
rank of this matrix
case we get the matrix
0
If
if the
the system is observable.
-~J
-1 0
is very
which has the rank three . system (12) with Cl
* 0,
Thus the C2
*0
is
ob~~vable.
-1
0
is not observable,
The matrix 0 then gets the form
important .
It
states,
that the Helios-system
if the disturbing moments have not the form
ofeq.(4). The method, gives, 1 ity
in
employed above,
is
restricted on
this simple form,
no
information about the observabi-
if Cl'
C are functions of v. 2 very general method, which yields a
Ther ~ fore ~' "
e
Thi " JT, c" .hod
velopment of the solution of a
system
dX
1
. . . .. .. . . , .. dx
n
dV =
248
x
f 1 (xl'
n'
another
insight
into
is based on the de-
v) (19)
, . ,.
f n (xl' .. . ,
we give
physical
the probl e m of observability .
dV =
linear systems and
X
n'
v)
into Taylor-series, which have the form (20)
The coefficients xi(O) are given as the initial conditions . coefficients of the second term are the functions (19) at v
The O.
=
The coefficients of the third term are given by the rule (summarize over the index R,)
a {-av
+ f
dx.
a I
-R, axR,
I
J
dV (21)
3 X3i ddV = {
etc .
The operator { } is called, according to Grobner, the Lie-operator [51.
For the equations (12)
the Lie-operator has the form (22)
The function xI(v)
is uniquely determined by the following ex-
pressions, taken at the point v
dXi
C2
dV 2
v
o
:
xI(v o )
XI
d
=
-
xT (23)
xi
--:-T dt
=
3 d x2" ~=
C' 2
- Cl
COl - ClI 2
- XI
-
C2 + xT
ect
The functions Cl (v), C2 (v) are in (23) represented by Taylorseries. The left sides of (23) are known from the diagram xI(v),
249
in which al I measurements are entered.
The observable quanti-
ties are therefore
C~ and,
- Cl
if C2 " 0, the quantities xl"' xT' Cl are observable.
An earth satell ite with disturbing moments caused by the atmosphere, the gravitation, the magnetic field etc . , is not ob~en vab.f.e by an earth-sensor alone, but it is well observable with an additional sun-sensor.
We give the short investigation.
For
the equations (9) with general disturbing torques,
(9*) dX
2 dv
-= 0
2
the lie-operator has the form
,
L}
(24)
and the measured variables are xl
Xl"
= -
(by the sun-sensor) and
xl cos v - x2 sin v
(25)
(by the earth-sensor). From xl and xl" the variable x 2 is known besides in the point = o. From the equations
sin v
(26)
etc.
250
we know further the complete funct ion DI (v).
By means of (24)
we calculate dXI (hi
= xI
sin v - x 2 cos v - DI cos v - D2 sin v (2])
2
d x2
~ dv
=
xI cos v + x 2 sin v + 2 DI sin v - D; cos v - 2 D2 cos v - D~ sin v etc.
In the first equation (Xl) al I quantities are known besides D2 sin v, thus D2 is known,
if sin v
*
O.
Then D' can be deter-
mined in the same way from the second equation etc. sys tem (9*)
Thus the
is observab le.
Now let us return to the discussion of the system of Hel ios . The second equation of (23)
(Cl = constant, C2 = 0) (23*)
permits the calculation of the pitch angle xT from the slope of the curve xI(v), which is given by the observations.
From the
third equation (23) we get in the same way the constant C, but not very accurate. (16b)
The accuracy is improved,
if the function
is fi I led to many observations by varying the quantities
xTo' xIo and C.
This may be done by an averaging calculation
according to Gauss or, in an iterative way by Kalman's filtering theory. In the actual case of Helios the direct evaluation of the observatIons according to (23*) was sufficient.
251
Comparison of the theory with the real observations Figure 2 gives the course of the observed decl ination of the sun (= negat ive roll-angle) .
26 days after the start the decl ina-
tion decreased within five days about 0.4 0 and after that it increased about the same amount.
It is very difficult to find
an explanation for this anomaly.
If there was an additional
disturbing
t~rque,
the system was not observable, and if there
was a fai 1 ing in the sensor, we cannot find it, because there was only one sensor on board (coarse and fine sensor were mounted at the same frame and their measurements were in agreement) . The anomaly caused a conside r able uncertainty, which is demonstrated by the three attitude maneuvers on day nr. 41, 64 and 80 (taken from January I, 1975).
After the thi rd maneuver the
institute was inserted to analyse the situation, which was critical, as the beginning of the blackout (in this case the sun between Helios and earth prevented any communication) was expected within three weeks. formula
The investigation was done according to
(16b), by fitting the parameters to the observations .
We took into consideration the variations of the pitch-angle caused by the well known attitude-maneuvers.
The result was a
very good agreement between theory and observations from day nr. 30 to day nr.
lOO, whi le the anomaly between the days nr. 5
and 30 became evident by a comparison of the observations with the extracted 1 ine, which is a backward calculation from day nr. 41.
Our results were confirmed by independent calculations of
the pitch-angle on certain days, which are based on the measurements of experiment nr. 9 (Dr.
Leinert, Heidelberg) and which
were in agreement with our calculations within 0 . 10 values of v in figure 2).
(see the
Thus we could recommand to execute
a fourth attitude-maneuver to bring the system to the stationary point, which is characterized by
o (28)
- c
252
With these inertial conditions the equations (16) go over in the stationary solution xT :: 0 (29) x2 -= - C The course of the observed decl ination after the fourth attitudemaneuver demonstrates, that the stationary point was reached within an accuracy of 0.19 0
.
The function xI(v)
very good approximation a sine.
in fig . 3 is in
The first maximum is - 0.59°,
the minimum - 0.97°, thus the mean value is Cobserved
Until
=
- 0.78°
and the amplitude
today (January 25,1976) a fifth attitude-maneuver was not
necessary .
It must be noticed, that the second maximum of x (v)
is - 0.54°,
that is 0.05 0 higher than the first.
caused by the fact,
2
This may be
that C is not exactly constant, because the
high-gain antenna of Hel ios is always directed to the earth and therefore must have a varying area with respect to the sun.
But
if there are small additional perturbating torques of statistical nature, we suppose,
in analogy to the Brownian motion of particles
in liquids, that the amplitude of the function
(16b)
creases, because the differential equations (12) solution .
slowly in-
have no stable
We know this either from the solution (16)
itself,
which contents no damping terms, or from the matrix A
of the system (12), which has the eigenvalues ±
Only the observations over many orbits may indicate , if such an increase of the amplitude really takes place.
253
Literature [ 1]
Hofmann, W.,
"The attitude Control of Helios during
Schmieder,
the Mission."
L.
Preliminary Report,
[2]
Jazwinski, A.H.
DFVLR-Oberpfaffen-
hofen,
Institut fur Dynamik der Flugsy-
steme,
Dec .
1973 .
"Stochastic Processes and Filtering Theory". Academic Press, New York and London, 1970 .
[ 3]
Schmieder,
L.
"The Attitude-Control of the Solar-Probe He I i os" . XXVI-th
International Astronautical
Congress, [4]
Bucy,
R.,
Joseph,
P.
"Fi ltering for Stochastic · Processes with Applications to Guidance" . John Wiley
[ 5]
Wanner,
G.
Lissabon 1975, paper no . 180.
&
Sons,
1968 .
"Integration gewohnlicher Di f ferentialgl e ichungen". Hochschultaschenbucher Nr . 831/831a, Bibliographisch e s 1969 .
254
Institut Mannheim,
Perh.l : 15. 3. 75
21.9.75
60
Start 10.d.c 1974
Fig . 1
Orbit of He-lios
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255
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