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A Low Noise Control System for a High Pointing Accuracy Satellite
A LOW NOISE CONTROL SYSTEM FOR A HIGH POINTING ACCURACY SATELLITE P. Montanari and M. T. Ravazzotti Capra AERITALIA - Space Sector, Corso Marche 41, Torino, Italy
Abstract. This work deals with the design of an attitude control system for a scientific mission which performs astrometric measures by the use of an optical strapdown telescope. This optical system requires a very low vibration level and a high pointing accuracy. This task is performed by a control loop which can filter the noise and assure a smoothed Qotion to the siC in order to follow any path in the sky without loosing star signals. A further improvement is obtained by optimising the observation procedure and sampling time. By the analytical treatment and the simulations performed it results that the jitter level of the system is lower than the allowed one and the dynamical motion of the sic is satisfactory. Keywords. Attitude control; closed loop systems; filtering; gyroscopes; inertial navigation; PID control; satellites, artificial. INTRODUCTION The satellites which carry optical experiments for stars observation belong to a class of scientific satellites which requires special unusual performances in order to fulfil the mission; their optical and detection systems impose the following constraints on the spacecraft: 1) Low vibration level to succeed in detecting very low photon sources. 2) High cleanliness of the space environment in contact with optics to avoid their deterioration.
torques are quite weak. The required operation flexibility and accuracy and the cleanliness level drive our study to the analysis of a mo mentum storage system configured with four r~ action wheels, one of which in skewed arrange ment and spinning at constant rate. This sOlu tion avoids speed reversal and the consequent lack of control. At the same time the global angular momentum of the spacecraft is null, so it is very easy to move the spacecraft without nutation problems and minimizing wheels friction problems. The sensor system includes a star mapper and a 3-axis rate integrat£nggz ro package strapped down to the bus. Many authors discussed the problems of RIG u~ dating, but few studies are available on the vibrations induced by RIG's on spacecraft and payload through attitude control. If we consi der a maximum jitter of 0.025 arcsec. and anattitude reproducibility of 10 arcmin, which are the requirements of the future advanced optical telescopes mounted on satellites to detect very dim stars, it needs to use high accuracy RIG'S, very sensitive actuators, and to adopt a smoothed control.
3) Very smoothed motion to avoid the loss of the observed source. 4) Capability to avoid too bright objects (e. g. sun or earth) which may damage optics. The system analysed in this report is able to satisfy all the above points and, more, it succeeds in pointing anywhere in the sky with high accuracy or to return to po int an object (e.g. a star) after one scan revolution with an accuracy of the order of few arcminutes or, more, to walk through the sky with its optical axis following a predetermined stars path without loo sing the optical references. -
For high pointing accuracy and reproducibility we have chosen a double integrator control loop as shown in Fig. 1. The integrator has a very good effect on wheels friction problems as shown in Fig. 2 and 3 . The dotted line on integrator of Fig. 1 means that the limiter acts on the integrator too in order to prevent divergence of it.
5) Capability of attitude determination with out referring to sun or earth because the se references have not a constant aspectwith respect to the spacecraft. CONTROL SYSTEM
GYRO MODEL
The control system hereafter envisaged is us eful in a geostationary orbit where external
The Gyro model used in the following section
167
168
P. Montanari and M. T. Ravazzotti Capra
is based on various references, maiQly Farren kopf (1914) and.Bendett and others (1975).
comes (5)
Fig. 4 shows a general block diagram of themo del. The gyro outputs are angular rate and pOsition, both affected by errors due to four 00 ise sources introduced into different posi- tions of gyro loop. These noises are supposed uncorrelated. The following noise sources are assumed white: 1) Electronic noise 0e which exists at the s~ gnal generator input or, equivalently, its output. As 0e is difficult to be measured, it is u seful to define the corresponding output standard deviation on angular measure 0E:
where loop.
0E
=
t
is the time constant of the gyro
0e (t /2)1/2 g
(1)
g
°:
various gyro mechanical im 2) Torque noise plementations give rise to this noise whi~ ch acts directly upon the gyro float. On the rate output this noise has a standard deviation 0v
due to the same reason as above. Obviously, the block lis represents the integrating line of the gyro, while KI/S counterbalances the wheels friction and improve static accuracy, and Kp and KR are gains, relati ve to position and velocity signals, respectT vely. The loop output is the e angle. The open loop transfer function is: KR 2 Kp + KI KR GH = - (S + ---:1<-.,.---- S + IS3 -~ If we introduce: i) the phase crossover frequency ww , defined as the value at which arg GH = -rr, in our case Kp KI 1/2 W (--) (7) 1T KR ii) the pseudodamping ~ , such that: Kp+KrKR
n
(2)
The most significant error source is the gyro drift rate bias ago This bias has a random walk variation ° around a bias b g • b can be properly comp~nsated with usual egtimation techniques even if it is assumed a prior unknown at time t = O. The Farrenkopf model (19 7 4) fits the experimental data in the following polinomium to obtain output angular standard deviation
W
(8)
rr =
KR
iii) the gain margin g, defined as: 1 W I rr g = ----
!GH(jw,)
I
(9 )
2~~
we have: GH
+ W 2) rr
(10)
iv) the gain cr0ssoverfrequenry w ' defined as l the value at which IGH I = 1, which, for large values of ~, is:
and for rate (4)
Experimental data give 0u = 2.0 x 10-5 arc sec/(sec)3/2 which is a very low valuewith respect to the other following standard de viations for a short period of time:
w
or,
~n
w rr
(11)
gD7
l
Hz, Wrr
0.0156 arcsec/(sec) 0.04 arcsec
f
1/2
l
(12)
2 1T gl/2
let's consider the two inputs, in A and B, separately.
;10\.',
NOISE ANALYTICAL TREATMENT
A) The closed loop transfer function (TF) is In order to study the system response to the gyro noises, we may treat the gyro outputs, as given in the preceeding paragraph, as inputs to t.h e simplified loop, which is d,awn in Fig. 1. The gyro is represented by the dashed zone and gives two inputs, as a consequence of the superposition principle: - in A the rate noise is applied which reduces to ~3 = o~ through the consideration that 0u is an infinitesimal with respect to o~ and the time between updating is not high - in B the angular noise is applied, which be
S2
Sv (S) nV(S)
S3+ ~S2+ 2g~
wrr 2
g
j
S +wrr -2<:;g (13)
The root mean square on the output angle (8) is:
j
1 ja> 2 2 = --l~ V . . V ITFI dsl 21TJ -Ja> s
G
JW
A low noise control system
169
gative real part if and only if: Sv (S) Sv (-S)dSI nS(S)
~(-S)
(14)
where:
NV
. S=Jw
o~
~
(15)
is the input power spectral density, which in the case of a white noise filtered by a gyro with given bandwith, f , assumes the above expression. g Solving the integral by the residual method, we get
<
s3 + a
3
6.
1
6.
2
6.
a
2 s
2
+ a
s + a
l
O
0
3
equ~
(24) (25)
a a - a a > 0 2 l O 3 2 a a a - a a > 0 2 l O O 3
(26) (27)
N CH (CH = - - ) D CH
(28)
is given by:
1
1 + CH S+W 2 11 (17)
(18)
DCH
fl 112 2 2 1; l/2(1/g-1) (4 f 1 +Ki) 4f K g g I (19)
Now, by applying the superposition principle, (20)
C
C
R
1 + CH
if
(29)
0
NCH
.:!:.
(30)
In our case:
In the same way as above, we get the RMS of the output angle, that is:
o~
0
or identically,
where:
11
(23)
> 0
2
B) The closed loop transfer function is now:
S2 E
n)
The characteristic equation for a system whose open loop transfer function is CH
(16)
that impl ies g
....
(i = 1, 2
where, for a third degree characteristic tion a
= li g
>0
6..
CH= -
~ I
S2+ 1; W1I S + wrr 2 S3
(31)
from which
DCH - NCH
=
S3+
~(S2+2 I
2
1; W11 S +w 11 )
(32) The conditions (25) , (26) , (27) , (28) become now
~ >0
is the total RMS we are interested in opti mizing by varying the values of the parame ters in the equations. Such parameters are the gains K , KR, ~; it is actually easier to consider fl' g, 1;: the relations between these two sets of parameters are the following:
always true
I
(~)2 I
~3
411 3 I f 1 g 3/2
I
(21)
~ 2 (2 1;w )- T w >0 1I 1I
>21;
~ >w 11 I
~2
~ 4 3 (2 1; w ) - - - Wrr >0 =>21;- > W.". 11 I I
But we know that 1/2 W = g w 11 l
(22) while K is given by the (18) I
211 g
1/2
f
l
so
CONTROL SYSTEM STABILITY I
In order to establish the stability, we can use the Routh-Hurwitz criterion, which is a necessary and sufficient condition: all the roots of the characteristic equation have ne-
Thus our system stability condition is g< 1, as we had already found in the preceeding paragraph.
170
P. Montanari and M. T. Ravazzotti Capra SIMULATION RESULTS
Instead of solving the problem by an analytical way, it is more convenient to use a grafic representation to find the values of the parameters which minimize the RMS. First, by comparison of the obtained plots,we got that it is better to consider the sum sign in the expression of KI; it is understood that we must impose ~~ 1.
the input noise, which is now comprehensive of the gyro noise as well as of the wheels 0 ne. The control noise is almost a white noise wi th a constant power spectral density, while the actuators noise gives a contribution only for w
where Figures 5,6 show the behaviour of RMS when and g vary, as a function of fl:
~
a) an increase in ~ shifts fl corresponding to the minimum of RMS towards higher values. b) the smaller is g the lower is the level of the incertitude Oil 6. The values of 6RMS, ancl the parameters for sane cases are reported in table 1; we assumed that: I
0.0128 sec 60 kg m2
Iz
145 kg m2 170 kg m2
L
g
x Iy
SPECIAL OBSERVATION TECHNIQUES The low levels of jitter, shown at la in Table 1 for some cases, may be still reduced by using an appropriate stars observation techni que, as the scientific requirements do not con cerne the actual jitter level but the diffe-rence in jitter between two following observa tions. It is possible to show that the sampl~ ing time is the most critical parameter with regard to this problem. If T is the observation time, the jitter which affects the ohser vat ion of a star i is roughly an averaged va lue like 1 T (33) 6 i " T )0 6(t) dt so the uncertainty in changing star is £iven by
and Now
0(~6l2)
12
~ 6
- 62
1
(34)
is the error we are interested to.
J
2
OO
"
1 ---
2"
2 Ww 2
375 rads/sec.
The integral has been solved numerically and an example is reported in Fig. 10 for the third case of Table 1. Let us note that A is the amplitude of the sinusoidal motion of the wheels; the noise due to the wheels may be truncated or enhanced by the transfer function according to the actual value of wW: ho~ ever we considered always the worst case, that is the Ww corresponding to the greater contri bution to jitter. CONCLUSION
The response to a unit step command of the corn plete control system of Fig. 1 is shown in Fig 7,8,9 for the different gains of Table 1. which minimize the system pointing error due to the input noise. The choice of a set of pa rameters is submitted to the requirements ofdamping, time constant, on board computer loads and other subsystems interfaces.
66
188
-00
M I
12(W) 1
6 Cl)
0
(w) dw
(35)
where 0(w) is the power spectral density of
The designed control system is able to meet all the requirements listed in the introduction of the present paper. As a matter of fact it succeeds in filtering the noise giving, as an output, an error on the attitude much less than the 0.025 arcsec (30) which is required in order to perform astronomical measures with a high accuracy level. Furthermore, it is possible to still reduce the jitter levels, which are shown for some cases in Table 1 at 10 , by introducing an opportune sampling time, after which the observing device turns to ward a different star. REFERENCES Bendett, R.M., Kleine, H.D., Weinstein, S.P. and Zomick, D.A. (1975). Test evaluation of a high performance pulse rebalanced gy ro for strapdown IMU applications. Proceedings on National .\erospace "leeting of the Institute of Navigation. D'Azzo, J.J. and Houpis, C.H. (1960). Feedback Control System Analysis and Synthesis. Mc Graw-Hill, New York. Farrenkopf, R.L. (1974). Generalized Results for Precision Attitude Reference Systems using Gyros.Proceedings on AlAA Mechanics and Control of Flight Conference. Radix, J.C. (1970). Introduction au Filtrage Numerique •. Eyrolles, Paris. Sandhu, G.S. (1974). Rigid Body Mode Pointing Accuracy and Stability Criteria for an Or biting Spacecraft. J. Spacecraft. Engine~ ering notes, l!, pp. 599-601.
171
A low noise control system
TABLE 1
Minimum Jitter Levels and Gains
8 RHS (arcsec)
1;
0.0032
3
0.0041 0.0030
f1
g
(Hz)
K /1
KR/I
K =K = K Ix Iz Iy
P
1/10
0.3
0.113
1. 257
4.394
10
1/4
1.2
0.032
1.031
78.343
10
1/10
1.2
0.010
1. 257
50.139
r----------, I
I
I
I
140--+-------.......1
t-........-r,.-----+---I
I
1 I
SiC
1-- DYNAMICS
Fig. 1.
I JI
Control Loop Simplified Scheme
DEG
o t------------~'=_-- Time 20
Fig. 2.
Control Accuracy with K I
(sec)
0
01-----------.;;:::::0----'--- Ti me (sec) Fig. 3.
Control Accuracy with K 1 0 I
P. Montanari and M. T. Ravazzotti Capra
172 w
b g I I
FLOAT DYNAMICS
SIGNAL GENERATOR
I.-;"';"';~~~~KF
TORQUE REBALANCE
r--,
I
1 em L ~--s1--L_..J
PULSE COUNTER USED IN RIG APPLICATION
DEFINITIONS: H
= Gyro Rotor Momentum
Typically for Strapdown, Rebalanced Gyros:
= Gyro Damping Constant c
K F
Gain
Fixed Gyro Drift Rate
E
9 rG
= CD/KK F
ne
= Broadband Signal Generator = White Float Torque Noise/H
nu Fig. 4.
Ini
E
= Gyro Output = Integral of
nv
«
1
Signal Generator Gain
= Torque Rebalance = Gyro Input Rate c
T
)n i
(t)/
=
(t,)nj (t 2 )
0
I
o
Gyro Output
i, j
= e,
v, u
i
I a
6(t "':'t ):: 1 2
(J = i)
( j
~
i)
Noise
White Torque Derivative Noise/H Gyro Model
e
RMS (orcsec)
Tg
= 0.0128 sec, 9 = 1/10
10
0.001L--------.....L...---------l.-----------J 0.01 0.1
Frequency
f I (Hz)
Fig. 5
A low noise control system
173
aRMS
(orcsec)
TO =
0.0128 sec,
S = 10
0.01
0.001 L -
........
---''--
.....
0.1
0.01
Frequency
f l (Hz) Fig. 6
Command 1.0 1; •
3
g • 1/10
0.0
TIME (SEC)
200
Fig. 7
-1.0
2.0
Conunand
o. 0
1;
10
g
1/4
-I---..I!....r----,----r----r--~-~r__-
200
-2.0 Fig. 8
TIME (SEC)
174
P. Montanari and M. T. Ravazzotti Capra
1.0
Conunand
r
ol~~-----r----J 200
l;
g
10 1/10
i
TIME (SEC)
400
-1 .0
Fig. 9
Error (3
0.0090 arcsec
3 ac
IJ)
(arcsec)
188 Aw
~
tt,
~
375 rad/sec
= 0.0205 arcsec
0.02
0.01
o
IL-'---'----"_"---~_ ___'__~~---...---..... 0.2
Fig. 10.
Jitter Vs Sampling Time
0.4
0.6
0.8
T
(sec)
Abstract. This work deals with the design of an attitude control system for a scientific mission which performs astrometric measures by the use of an optical strapdown telescope. This optical system requires a very low vibration level and a high pointing accuracy. This task is performed by a control loop which can filter the noise and assure a smoothed Qotion to the siC in order to follow any path in the sky without loosing star signals. A further improvement is obtained by optimising the observation procedure and sampling time. By the analytical treatment and the simulations performed it results that the jitter level of the system is lower than the allowed one and the dynamical motion of the sic is satisfactory. Keywords. Attitude control; closed loop systems; filtering; gyroscopes; inertial navigation; PID control; satellites, artificial. INTRODUCTION The satellites which carry optical experiments for stars observation belong to a class of scientific satellites which requires special unusual performances in order to fulfil the mission; their optical and detection systems impose the following constraints on the spacecraft: 1) Low vibration level to succeed in detecting very low photon sources. 2) High cleanliness of the space environment in contact with optics to avoid their deterioration.
torques are quite weak. The required operation flexibility and accuracy and the cleanliness level drive our study to the analysis of a mo mentum storage system configured with four r~ action wheels, one of which in skewed arrange ment and spinning at constant rate. This sOlu tion avoids speed reversal and the consequent lack of control. At the same time the global angular momentum of the spacecraft is null, so it is very easy to move the spacecraft without nutation problems and minimizing wheels friction problems. The sensor system includes a star mapper and a 3-axis rate integrat£nggz ro package strapped down to the bus. Many authors discussed the problems of RIG u~ dating, but few studies are available on the vibrations induced by RIG's on spacecraft and payload through attitude control. If we consi der a maximum jitter of 0.025 arcsec. and anattitude reproducibility of 10 arcmin, which are the requirements of the future advanced optical telescopes mounted on satellites to detect very dim stars, it needs to use high accuracy RIG'S, very sensitive actuators, and to adopt a smoothed control.
3) Very smoothed motion to avoid the loss of the observed source. 4) Capability to avoid too bright objects (e. g. sun or earth) which may damage optics. The system analysed in this report is able to satisfy all the above points and, more, it succeeds in pointing anywhere in the sky with high accuracy or to return to po int an object (e.g. a star) after one scan revolution with an accuracy of the order of few arcminutes or, more, to walk through the sky with its optical axis following a predetermined stars path without loo sing the optical references. -
For high pointing accuracy and reproducibility we have chosen a double integrator control loop as shown in Fig. 1. The integrator has a very good effect on wheels friction problems as shown in Fig. 2 and 3 . The dotted line on integrator of Fig. 1 means that the limiter acts on the integrator too in order to prevent divergence of it.
5) Capability of attitude determination with out referring to sun or earth because the se references have not a constant aspectwith respect to the spacecraft. CONTROL SYSTEM
GYRO MODEL
The control system hereafter envisaged is us eful in a geostationary orbit where external
The Gyro model used in the following section
167
168
P. Montanari and M. T. Ravazzotti Capra
is based on various references, maiQly Farren kopf (1914) and.Bendett and others (1975).
comes (5)
Fig. 4 shows a general block diagram of themo del. The gyro outputs are angular rate and pOsition, both affected by errors due to four 00 ise sources introduced into different posi- tions of gyro loop. These noises are supposed uncorrelated. The following noise sources are assumed white: 1) Electronic noise 0e which exists at the s~ gnal generator input or, equivalently, its output. As 0e is difficult to be measured, it is u seful to define the corresponding output standard deviation on angular measure 0E:
where loop.
0E
=
t
is the time constant of the gyro
0e (t /2)1/2 g
(1)
g
°:
various gyro mechanical im 2) Torque noise plementations give rise to this noise whi~ ch acts directly upon the gyro float. On the rate output this noise has a standard deviation 0v
due to the same reason as above. Obviously, the block lis represents the integrating line of the gyro, while KI/S counterbalances the wheels friction and improve static accuracy, and Kp and KR are gains, relati ve to position and velocity signals, respectT vely. The loop output is the e angle. The open loop transfer function is: KR 2 Kp + KI KR GH = - (S + ---:1<-.,.---- S + IS3 -~ If we introduce: i) the phase crossover frequency ww , defined as the value at which arg GH = -rr, in our case Kp KI 1/2 W (--) (7) 1T KR ii) the pseudodamping ~ , such that: Kp+KrKR
n
(2)
The most significant error source is the gyro drift rate bias ago This bias has a random walk variation ° around a bias b g • b can be properly comp~nsated with usual egtimation techniques even if it is assumed a prior unknown at time t = O. The Farrenkopf model (19 7 4) fits the experimental data in the following polinomium to obtain output angular standard deviation
W
(8)
rr =
KR
iii) the gain margin g, defined as: 1 W I rr g = ----
!GH(jw,)
I
(9 )
2~~
we have: GH
+ W 2) rr
(10)
iv) the gain cr0ssoverfrequenry w ' defined as l the value at which IGH I = 1, which, for large values of ~, is:
and for rate (4)
Experimental data give 0u = 2.0 x 10-5 arc sec/(sec)3/2 which is a very low valuewith respect to the other following standard de viations for a short period of time:
w
or,
~n
w rr
(11)
gD7
l
Hz, Wrr
0.0156 arcsec/(sec) 0.04 arcsec
f
1/2
l
(12)
2 1T gl/2
let's consider the two inputs, in A and B, separately.
;10\.',
NOISE ANALYTICAL TREATMENT
A) The closed loop transfer function (TF) is In order to study the system response to the gyro noises, we may treat the gyro outputs, as given in the preceeding paragraph, as inputs to t.h e simplified loop, which is d,awn in Fig. 1. The gyro is represented by the dashed zone and gives two inputs, as a consequence of the superposition principle: - in A the rate noise is applied which reduces to ~3 = o~ through the consideration that 0u is an infinitesimal with respect to o~ and the time between updating is not high - in B the angular noise is applied, which be
S2
Sv (S) nV(S)
S3+ ~S2+ 2g~
wrr 2
g
j
S +wrr -2<:;g (13)
The root mean square on the output angle (8) is:
j
1 ja> 2 2 = --l~ V . . V ITFI dsl 21TJ -Ja> s
G
JW
A low noise control system
169
gative real part if and only if: Sv (S) Sv (-S)dSI nS(S)
~(-S)
(14)
where:
NV
. S=Jw
o~
~
(15)
is the input power spectral density, which in the case of a white noise filtered by a gyro with given bandwith, f , assumes the above expression. g Solving the integral by the residual method, we get
<
s3 + a
3
6.
1
6.
2
6.
a
2 s
2
+ a
s + a
l
O
0
3
equ~
(24) (25)
a a - a a > 0 2 l O 3 2 a a a - a a > 0 2 l O O 3
(26) (27)
N CH (CH = - - ) D CH
(28)
is given by:
1
1 + CH S+W 2 11 (17)
(18)
DCH
fl 112 2 2 1; l/2(1/g-1) (4 f 1 +Ki) 4f K g g I (19)
Now, by applying the superposition principle, (20)
C
C
R
1 + CH
if
(29)
0
NCH
.:!:.
(30)
In our case:
In the same way as above, we get the RMS of the output angle, that is:
o~
0
or identically,
where:
11
(23)
> 0
2
B) The closed loop transfer function is now:
S2 E
n)
The characteristic equation for a system whose open loop transfer function is CH
(16)
that impl ies g
....
(i = 1, 2
where, for a third degree characteristic tion a
= li g
>0
6..
CH= -
~ I
S2+ 1; W1I S + wrr 2 S3
(31)
from which
DCH - NCH
=
S3+
~(S2+2 I
2
1; W11 S +w 11 )
(32) The conditions (25) , (26) , (27) , (28) become now
~ >0
is the total RMS we are interested in opti mizing by varying the values of the parame ters in the equations. Such parameters are the gains K , KR, ~; it is actually easier to consider fl' g, 1;: the relations between these two sets of parameters are the following:
always true
I
(~)2 I
~3
411 3 I f 1 g 3/2
I
(21)
~ 2 (2 1;w )- T w >0 1I 1I
>21;
~ >w 11 I
~2
~ 4 3 (2 1; w ) - - - Wrr >0 =>21;- > W.". 11 I I
But we know that 1/2 W = g w 11 l
(22) while K is given by the (18) I
211 g
1/2
f
l
so
CONTROL SYSTEM STABILITY I
In order to establish the stability, we can use the Routh-Hurwitz criterion, which is a necessary and sufficient condition: all the roots of the characteristic equation have ne-
Thus our system stability condition is g< 1, as we had already found in the preceeding paragraph.
170
P. Montanari and M. T. Ravazzotti Capra SIMULATION RESULTS
Instead of solving the problem by an analytical way, it is more convenient to use a grafic representation to find the values of the parameters which minimize the RMS. First, by comparison of the obtained plots,we got that it is better to consider the sum sign in the expression of KI; it is understood that we must impose ~~ 1.
the input noise, which is now comprehensive of the gyro noise as well as of the wheels 0 ne. The control noise is almost a white noise wi th a constant power spectral density, while the actuators noise gives a contribution only for w
where Figures 5,6 show the behaviour of RMS when and g vary, as a function of fl:
~
a) an increase in ~ shifts fl corresponding to the minimum of RMS towards higher values. b) the smaller is g the lower is the level of the incertitude Oil 6. The values of 6RMS, ancl the parameters for sane cases are reported in table 1; we assumed that: I
0.0128 sec 60 kg m2
Iz
145 kg m2 170 kg m2
L
g
x Iy
SPECIAL OBSERVATION TECHNIQUES The low levels of jitter, shown at la in Table 1 for some cases, may be still reduced by using an appropriate stars observation techni que, as the scientific requirements do not con cerne the actual jitter level but the diffe-rence in jitter between two following observa tions. It is possible to show that the sampl~ ing time is the most critical parameter with regard to this problem. If T is the observation time, the jitter which affects the ohser vat ion of a star i is roughly an averaged va lue like 1 T (33) 6 i " T )0 6(t) dt so the uncertainty in changing star is £iven by
and Now
0(~6l2)
12
~ 6
- 62
1
(34)
is the error we are interested to.
J
2
OO
"
1 ---
2"
2 Ww 2
375 rads/sec.
The integral has been solved numerically and an example is reported in Fig. 10 for the third case of Table 1. Let us note that A is the amplitude of the sinusoidal motion of the wheels; the noise due to the wheels may be truncated or enhanced by the transfer function according to the actual value of wW: ho~ ever we considered always the worst case, that is the Ww corresponding to the greater contri bution to jitter. CONCLUSION
The response to a unit step command of the corn plete control system of Fig. 1 is shown in Fig 7,8,9 for the different gains of Table 1. which minimize the system pointing error due to the input noise. The choice of a set of pa rameters is submitted to the requirements ofdamping, time constant, on board computer loads and other subsystems interfaces.
66
188
-00
M I
12(W) 1
6 Cl)
0
(w) dw
(35)
where 0(w) is the power spectral density of
The designed control system is able to meet all the requirements listed in the introduction of the present paper. As a matter of fact it succeeds in filtering the noise giving, as an output, an error on the attitude much less than the 0.025 arcsec (30) which is required in order to perform astronomical measures with a high accuracy level. Furthermore, it is possible to still reduce the jitter levels, which are shown for some cases in Table 1 at 10 , by introducing an opportune sampling time, after which the observing device turns to ward a different star. REFERENCES Bendett, R.M., Kleine, H.D., Weinstein, S.P. and Zomick, D.A. (1975). Test evaluation of a high performance pulse rebalanced gy ro for strapdown IMU applications. Proceedings on National .\erospace "leeting of the Institute of Navigation. D'Azzo, J.J. and Houpis, C.H. (1960). Feedback Control System Analysis and Synthesis. Mc Graw-Hill, New York. Farrenkopf, R.L. (1974). Generalized Results for Precision Attitude Reference Systems using Gyros.Proceedings on AlAA Mechanics and Control of Flight Conference. Radix, J.C. (1970). Introduction au Filtrage Numerique •. Eyrolles, Paris. Sandhu, G.S. (1974). Rigid Body Mode Pointing Accuracy and Stability Criteria for an Or biting Spacecraft. J. Spacecraft. Engine~ ering notes, l!, pp. 599-601.
171
A low noise control system
TABLE 1
Minimum Jitter Levels and Gains
8 RHS (arcsec)
1;
0.0032
3
0.0041 0.0030
f1
g
(Hz)
K /1
KR/I
K =K = K Ix Iz Iy
P
1/10
0.3
0.113
1. 257
4.394
10
1/4
1.2
0.032
1.031
78.343
10
1/10
1.2
0.010
1. 257
50.139
r----------, I
I
I
I
140--+-------.......1
t-........-r,.-----+---I
I
1 I
SiC
1-- DYNAMICS
Fig. 1.
I JI
Control Loop Simplified Scheme
DEG
o t------------~'=_-- Time 20
Fig. 2.
Control Accuracy with K I
(sec)
0
01-----------.;;:::::0----'--- Ti me (sec) Fig. 3.
Control Accuracy with K 1 0 I
P. Montanari and M. T. Ravazzotti Capra
172 w
b g I I
FLOAT DYNAMICS
SIGNAL GENERATOR
I.-;"';"';~~~~KF
TORQUE REBALANCE
r--,
I
1 em L ~--s1--L_..J
PULSE COUNTER USED IN RIG APPLICATION
DEFINITIONS: H
= Gyro Rotor Momentum
Typically for Strapdown, Rebalanced Gyros:
= Gyro Damping Constant c
K F
Gain
Fixed Gyro Drift Rate
E
9 rG
= CD/KK F
ne
= Broadband Signal Generator = White Float Torque Noise/H
nu Fig. 4.
Ini
E
= Gyro Output = Integral of
nv
«
1
Signal Generator Gain
= Torque Rebalance = Gyro Input Rate c
T
)n i
(t)/
=
(t,)nj (t 2 )
0
I
o
Gyro Output
i, j
= e,
v, u
i
I a
6(t "':'t ):: 1 2
(J = i)
( j
~
i)
Noise
White Torque Derivative Noise/H Gyro Model
e
RMS (orcsec)
Tg
= 0.0128 sec, 9 = 1/10
10
0.001L--------.....L...---------l.-----------J 0.01 0.1
Frequency
f I (Hz)
Fig. 5
A low noise control system
173
aRMS
(orcsec)
TO =
0.0128 sec,
S = 10
0.01
0.001 L -
........
---''--
.....
0.1
0.01
Frequency
f l (Hz) Fig. 6
Command 1.0 1; •
3
g • 1/10
0.0
TIME (SEC)
200
Fig. 7
-1.0
2.0
Conunand
o. 0
1;
10
g
1/4
-I---..I!....r----,----r----r--~-~r__-
200
-2.0 Fig. 8
TIME (SEC)
174
P. Montanari and M. T. Ravazzotti Capra
1.0
Conunand
r
ol~~-----r----J 200
l;
g
10 1/10
i
TIME (SEC)
400
-1 .0
Fig. 9
Error (3
0.0090 arcsec
3 ac
IJ)
(arcsec)
188 Aw
~
tt,
~
375 rad/sec
= 0.0205 arcsec
0.02
0.01
o
IL-'---'----"_"---~_ ___'__~~---...---..... 0.2
Fig. 10.
Jitter Vs Sampling Time
0.4
0.6
0.8
T
(sec)