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Attitude-Translation Coupling in Drag-Free Satellites
ATTITUDE-TRANSLATION COUPLING IN DRAG-FREE SATELLITES
Alan W. Fleming Research Engineer Daniel B. DeBra Senior Research Associate and Lecturer STANFORD UNIVERSITY Department of Aerona utics and Astronautics Stanford, California, U.S.A. a nd Marcelo Crespo da Silva National Research Council Post-Doctoral Resident Research Associate at NASA-AMES Research Cent er Moffett, Field, California, U.S . A .
Abstract The orbit of a drag-free satellite is determined entirely by gravity bec a use it follows an unsupported proof mass which is shielded from all external surface forces by the s a tellite. A control system in the satellite senses the relative position of the satellite with respect to the proof mass and actuates reaction jets forcing the satellite to follow the proof mass. A drag-fre e control system may be added to any type of satellite, and in principle is independent of the satellite's attitude control syst em. Coupling, however, can o ccur because both the pickoff, which senses the relative position of the s a tellite, and the proof ma ss may be located a way from the satellite mass c enter a nd the thrusters may be misalign e d with r e spect to the mass center. In passive attitude control s y stems, misalign e d translational control thrusters ma y produc e significant disturbances to the attitude. Thes e at titude motions can in turn be coupled back into t he tra nsl a tion control system through an offset of the pickoff with respect to the vehicle mass center. This paper describes the mech a nism of this type of c oupling a nd provides design guid e s for gravity gra dient satellites which are made dra g free. The discussion begins with a simplified model to illustrate the nature of the couplin g a nd develops the full six-degree-of-freedom equ a tions to examine t horoughl y all paths of coupling. It is shown that a n impor ta nt dyna mic pa th for this t y pe of coupl ing is through the interaction of along-tr ac k and radi a l t ranslations (conventional orbital mechaniCS perturbation theory). Thrust which is produced intrack due to relative displa c ement a long the orbit produces radial displacement which in turn activ a t e s thrusters in the radial direction. The v e rtical thrust, if misaligned, produces torques which c hange the vehicle ' s ori e ntation and thereby produ c e a n error sign a l in t he horizontal direction if th e sensor is not lo cat ed at the satellite mass cen t er . This error signal in turn requests addi-
411
tional horizontal thrust, and may thereby close the loop unstably. Stability boundaries as a function of thruster misalignment, pickoff offset, and attitude-motion damping ratio, are established and the alignment requirements are discussed. The analytical discussion is substantiated with results from a simulation which includes the nonlinear characteristics of thrusters and their modulators. A.
Introduction
A drag-free satellite cont a ins an unsupported proof mass which is shi e lded by the satellite from external non-gravit a tional forces. Since only gravitational forces act on the proof mass, it follows a purely gravitational orbit. A control system in the satellite senses motion of the satellite relative to the proof mass and activates thrusters, forcing the sat e llite to follow the proof mass without touching it (Fi g . 1). The satellite therefore also follows a purely gravitational orbit, free from the effec ts of external surface forces. A drag-free orbit is useful in satellite geodesy, for n a vigation satellites where accurate ephemeris predi c tion is needed, and for determining the drag 011 a satellite by measuring the equal and opposite compensating thrust, as Lange(l) has proposed. A drag-free satellite may use any type of attitude control and, in principl e , there need not be any orientation-translation coupling. In practice, coupling exists and two e ffects are discussed here. First, thrusters not acting through the satellite mass center produce torques which disturb the attitude motion, and secondly, if the proof-mass sensor null is not at the satellite mass center, its output c ontains contributions from orientation as well as translation. (In spinning vehicles, velocity information must be carefull y inferred from the proofmass sensor which measures in a rotating reference.) This paper discusses the effect of coupling on the
PROOF MASS HOLLOW CAVITY TRANSLAT IONAL CONTROL THRUSTERS--"" RELATIVE POSITION SENSORS ~
PROOF MASS SATELLITE BODY
POSITION SENSOR NULL POINT VEHICLE MASS CENTER
FIGURE 1 .
PLANNAR SCHEMATIC OF A DRAG- FREE SATELLITE
SATELLITE VELOCITY
-
-
~
SENSOR NULL RADIAL OFFSET (E I)
MOMENT ARM OF IN - TRACK TRANSLATION THRUSTER ( Lt)
f:
IN-TRACK CONTROL FORCES
..".
.....
N
~ ENSOR NULL (5)
VEHICLE MASS CENTER (cm)
EARTH
PROOF MASS (pm)
F'j TRANSLATION CONTROL JETS (j=I , 2 , "' 6)
FIGURE 3.
FIGURE 2.
SCHEMATIC OF DRAG-FREE SATELLITE GEQ\1ETRY
SIMPLE TWO - DEGREE-OF-FREEDOM DUMBBELL MODEL TO ILLUSTRATE THRUSTER COUPLING WITJ-DUT ORBITAL COUPLING
C.
stability of gravity-stabilized, drag-free satel-
Case 1 Without Orbital Coupling
l i tes.
B.
Ignore the coupling of radial and in-track motion in orbit so that
Equations of Motion
(5)
Three models of the vehicle-orbit dynamics will be used in the discussion that follows. Two of these are linear and can be represented by a common set of equations differing only in the coefficient matrix. The third model is discussed subsequently.
and take
K
= (k v s
of k
p
(6)
)U
e
For motion in the orbit plane, has just the ib (pitch) component, and Land E lie in the orbit plane. Then K and (A+K)-l may be trated as scalars, and using L x (E X e ) E(L·e) - e(L'E) -e(L'E), the characteristic equation is
Making the necessary assumptions to obtain a set of linear equations, taking the Laplace transform and ignoring disturbances, there are four relations . needed:
=
=
Translation dynamics A or
o .
=F
Orientation dynamics
B
e
e =L X F
Because is assumed parallel to the 3b axis only the 33 element of this matrix has signfficance and it may be equated to zero directly to get the characteristic equation. The gravity gradient dynamics about the pitch axis are represented by B33 = I 3 (s2 + 2~~gs + ~).
(2 )
Sensor equation
e = - or -
eX
(3)
E
Control law
F
=K
(8)
(1)
In a typical control system the translation control is very much faster (say by a factor of 100) than the att itude libration frequency and, for all practical purposes, the sensor is held at null during the libration. That is, the vehicle is forced to librate about the sensor null instead of its mass center. The force required to make this happen is therefore nearly in-phase (or 180 0 out of phase) with the libration and primarily effects the natural frequency, rather than the damping, of the system mode which, in the uncoupled dynamics, is the gravity gradient attitude behavior. The effect is greatest when the sensor offset and the force moment arm are parallel as indicated by the dot product (i.e., the attitude motion produces a position sensor signal for the axis in which the force has the greatest moment arm). Figure 3 schematically illustrates the geometry of such a case with a simple dumbbell-shaped satellite .
(4)
e
where (Fig. 2) , A, B, and K are 3 X 3 coefficient matrices to be discussed in the next sections or
is motion of the satellite mass center from its equilibrium position of r = -E (3 X 1 matrix)
F
is the control force assuming that all thrusters pass through a common point (3 X 1 matrix)
L
is the location at which the control force acts with respect to the satellite mass center (3 X 1 matrix). For linearity it is assumed that all L ' s are equal.
e
is the attitude of th e vehicle with resp ec t to a local tangent reference frame (3 X 1 matrix)
E
is the location of th e position sensor null with respect to vehicle mas s center (3 X 1 matrix)
e
is the output of the position sensor (3 X 1 matrix)
The locus of characteristic equation roots, as a function of L'E, which starts from the gravity gradient poles moves slowly toward the right half plane and becomes unstable for (9)
where
L X is used to denote the matrix equivalent of the vector cross product operation ( 3 X 3 ma trix) .
and
(B + L X K[-U + (A+K)
KJE
x }e = 0
a
is the a ngle between
Land
E.
For IEI and R equal to 1 cm* and 3 cm respectively, this mechanism should not produce instability as long as ILl ,< 900 meters, which is easily achieved.
The characteristic equation of this system of equations can be derived by direct substitution of 3 and 4 into 1 a nd 2, solving for o r and substituting into 2 to get a single matrix equation in e: -1
R is the vehicle radius of gyration for the mass center,
*Nonzero
(5)
where U is the 3 X 3 unit matrix. The determinant of the coefficient matrix set equal to zero (for a non-trivi a l solution) is the characteristic equation.
413
values of El require a continuous thruster force to hold the satellite up (or down) in the same orbit as the proof mass,which shortens the drag-free life. An El 1 cm is a reasonable upper limit for satellites at altitudes above the atmosphere because the required thrust is about one third of the radiation pressure force that must also be overcome.
=
1RADIAL
PROOF MASS POSITION SENSOR NULL POINT
"
10 MOMENT ARM OF RADIAL TRANSLATION THRUSTER( L Z )
SENSOR NULL RADIAL OFFSET (E,) SATELLITE VELOCITY
"
Ib
VEHICLE MASS - - - - - - - CENTER
~ RADIAL
MOMENT ARM OF IN -TRACK TRANSLATION THRUSTER
II
CONTROL FORCES
"
8,obout 10
IN-TRACK
~::I~ . "
2b' 8 3
( L,)
k
",
"
20
>IN-TRACK CONTROL FORCES
8Z obout 2b
2b
..,. >-' ..,.
"
30
8
" " " " 8s obout 3b
{IO. 20. 30} (0)
>{Ib. 2b. 3b} (bl
,8
"RMAL TO ORBIT PLANE
EARTH
~
"
FIGURE S. FIGURE 4.
SIMPLE THREE-DEGREE-OF-FREEOOM ruMBBELL MODEL TO ILLUSTRATE THRUSTER COUPLING WITH ORBITAL COUPLING
DEFINITION OF BODY AND ORBITAL REFERENCE FRAMES
D.
Case 2 with Orbital Coupling
four sets of equations for all six degrees-of-freedom are given. The results of both a linear analysis and a nonlinear simulation are compared in Section F.
The orbital perturbation equations due to Hill are well known(l). The in-plane motion is coupled and A becomes
1. -21lUl S o 2 ms
A
Referring to Fig. 5, the "orbital frame," (0), is defined by a unit vector 10 along the orbital radius pointing away from the earth, a unit vector 20 in the in-track direction of orbital motion, and a third unit vector ~o normal to the orbit plane, completing the right-handed set; The second frame (b), defined by the three unit vectors lb, 21, and 3b, is fixed in the satellite and aligned with the principal axes of inertia for the vehicle mass center. The orientation of (b) with respect to (0) is specified by a set of three Euler angles as shown in the figure. All quantities, such as forces and moment arms, are fixed in the body (see Fig. 6) except the translation, r, which is most naturally expressed in the orbital reference frame.
(10)
° where Wo is the orbital angular velocity. Using the same control law and previous assumptions the characteristic equation becomes 1 (s 3
2
+ 2sw s + g
2
+
00 )
g
+ (kv S + k p )(M12IL X E l - MllL'E) where
MU
= _ms 2 /(ms 2
+ k s + k ) v p
and
M12
= -21lUl0 s (k v s
+ k ) / (ms
2
P
°
Reference Frame Definitions
(11)
(12)
2.
+k s + k )2 v P
2
The assumption Wo « kp/m has been made to simplify the expression and emphasize the role of IL X El and L·E. For IL X El 0, the behavior is essentially unchanged from the case just discussed. For L'E 0, the coupling is due to IL X El. That is, it is greatest when Land E are perpendicular to each other, which involves a more subtle mechanism of coupling than the first case.
Translation Dynamics
Equation (1) is unchanged except that F must be defined as the sum of the effects of the six thrusters and in general a net disturbance,
=
6
=
Figure 4 is an illustration of a three-degree-offreedom dumbbell model with Land E at an arbitrary relative orientation. If an attitude perturbation occurs, then an in-track error is sensed and the in-track thruster is activated to null the error. The integrated (and therefore phase shifted) intrack force gives a slight change in the orbital speed of the satellite and thus a change in the centripetal acceleration is required to keep the sensor nulled in the radial direction. This requires radi a l thrust whi c h produces a torque due to the moment arm perpendicular to the sensor offset. The resulting attitude motion is then coupled back through the horizontal component of the position sensor . This coupling is important because there is a 90° phase shift introduced by the orbital dynamics. The phase shift causes the IL X El term to effect principally the damping (as opposed to frequency) of the roots of the characteristic equation whose locus starts at the gravity gradient poles. The product of ILIIE I therefore has a much greater effect on stability when Land E are perpendicular than when they a re parallel. E.
Equation of Motion for Simulation
In deriving the equations of motion for a simulation of a gravity stabilized drag-free satellite one can ignore the proof mass perturbat10ns which may be 10-11g or smaller(l) For the s1mulat10n of this system we have included nonlinearities in the translational control system. Each thruster has been assumed to have its own line of action, and the thrust is controlled with a derived rate modulator. First the coordinate frames used are defined, then the
F =
L
F
j
(13)
+ Fd .
j=l For small oscillation the translational motions are the same in orbit and body axes so ro ~ rb' All other vectors are expressed in the body reference frame. 3.
Orientation Dynamics
I f the a ttitude behavior is lightly damped (Sg « l), nonlinearities in the damping mechanism do not have a great effect on the fine structure of the resulting motion and linear damping may be assumed with adequate ac curacy for this study. Assuming a gravity stabilized system with yaw stability augmented with a constant speed pitch wheel, B in Equation (2) becomes(2,3),
(14)
B
where
BU
2 2 lIS +D s+0 -1 )00 +hw
B12
[(-l -1 +1 )w +h]s l 2 3 o
B2l
-B
l
3
2
0
0
B22
12 2 2 12 s +D2s+4(I3-Il) wo+hwo
B33
2 2 13s +D s+3(I -I )w o 3 2 l
and 11, 12, and 13 are principal inertias of the vehicle for the mass center, h is the magnitude of the angular momentum of a roll-yaw coupling momentum
415
--I[II*lsin a>O ----1[11*1 sin a<0
/
ASSUMED PARAMETERS jlOO
m = 86.5 kg 13 = 675 kg. m 2 =0.0033
tg
/~IN'TRACK
CONTROL FORCES
-
PROOF MASS
VEH ICLE MASS CENTER
\
~
FIGURE 6.
/
I
/
GEa-IETRY OF COUPLED PLANAR
-Ikp/m =100 Wg kp/kv =70 .7 Wg
,
I -+-+-t +- + +.. 1- 0 \ - 100
\
RADIAL CONTROL FORCES
/
/''-
\.
", .....
/
~!OTION
FIGURE 7.
I
-j50 ---- ........
I
-+.J __ -1.0 \
"
,
"
"" -)100 "\
-jl .O
ROOTLOCUSFORCOIJPLEDPLANAR~IONASAFUNCfIONOF IL II<1 5ina
t- 2000 •• c
--1
_TIME
(t - i
'0 )
(b)
1[11. \ sin a' +0.05
(COl
(C)
\[1\.1
(a)
NO
COUPLING
0 .10
iE
a , '0)
-=-0.05 ~
STABLE
°0~~---L~9~0~~--L-'I~B~ 0~---L--i360 ANGLE BETWEEN" AND [
FIGURE 8.
cos a-+0.05 (.ina-O)
(a) (dog)
STABJI.ITY RELATIONSHIP FOR L A'W < AS A FUNCfION OF PHYSICAL PARA'lETERS FIGURE 9.
416
REPRESENTATIVE ANALOG CCNPlITER TRACES OF PITCH ANGLE (93 ) WITH NONLINEAR DERIVED RATE TRANSLATIONAL CONTROL
f or
la l =90'
nz,
wheel with its spin axis along lb, and Dl , and D3 are damping coefficients. The torques are due to the six thrusters, and, in general, a disturbance torque. Thus LX F is replaced with:
of Fig. 7.
6
T
=
L
L
j
X F
j
+ Td .
(15)
j=l 4.
=
Sensor Equation
The sensor equation is the same as Eq. (3). A capacitive pick-off used in a laboratory breadboard'exhibits an additional cubic term, but it is negligible for this coupling study. (However, it cannot be neglected in acquisition and transient response studies.) 5.
Control Law
A derived rate modulator(4) was assumed to operate each thruster. This modulator switch with hysteresis is activated by a signal, a j , which is the difference between the error signal, e j , and a lagged feedback signal, fj, from the thruster: a
j
-rf j
= e
j
-
kf
+ fj = F
j
j
=
=
=
G.
Conclusion
A criterion has been obtained which accurately defines a stability boundary for attitude-translational coupled motion, in the plane of the orbit, for a gravity stabilized drag-free satellite. The criterion specifies allowable translational control sensor null offsets and translational thruster moment arms as a function of vehicle physical parameters.
(16) (17)
The question of coupling in roll-yaw and translation normal to the orbit plane has not yet been investigated fully. Tolerances on sensor null offset and thruster moment arm specifications are not expected to be any more stringent than for the in-plane case.
The damping is inherent in the lagged feedback. Fj is full on or off but its average value is approximately linear in ej when averaged over times comparable to the gravity gradient oscillation period. F.
To verify these results, a series of analog computer runs were made using the planar equations and nonlinear derived rate discussed in Section E. Figures 9a, 9b, and 9c show typical traces for 93 as a function of time, starting with an initial pitch angle of 10 degrees. Figure 9a has no coupling and exhibits the damping characteristic associated with Sg 0.0033. Setting IL'EI 0 and ILIIEI sin 0: +0.05 meters 2 , predicted by Eq. (18) to be slightly outside the stable region, results in a slightly divergent pitch motion as shown in Fig. 9b. Notice that the primary change is in damping, not frequency, for this case. Figure 9c is an example of the pitch behavior for the much less critical case IL X El = 0 with ILIIEI cos 0: +0.05 meters2. As predicted previously, there is a marked reduction in frequency with only a slight reduction in damping due to the thruster coupling.
Discussion of Results References
The approximate charact e ristic equa tion (11) is convenient for visualizing the relative contributions of the I L' EI and I L X El terms. Using this equation as a guide, but calculating roots directly from the full equations of planar motion (i.e., wi thout th e assumption that j kp/ mt » wo), the root lo c us can be obtained as a function of I L II E I sin 0: for cos 0: = O. This is shown in Fig. 7. The important part of the figure is the enlarged view of the region near the origin shown in the lower right hand corner. In this locus wi th cos 0: = 0, the principal eff ect of I L II E I is on the damping of the gravity gradient mode causing the poles to cross the imaginary axis for relatively small values of I LI IEI sin 0: > o. To obt a in an analytical relationship among the parame ters L, E, 0: , and Sg for system stability, the Routhian array may be constructed from the approximate characteristic equation (11), and conditions for imaginary axis crossover deduced from this. This yields the criterion,
(18)
for instability. That is, i f Eq. (18) is satisfied, the system is predicted to be unstable. Th i s relationship, represented graphically in Fig. 8, governs the imaginary axis crossover for the loci
417
(1)
Lange, B. 0., "The Dr a g-Free Sa telli te," AIAA Journal, Vol. 2, No. 9, Sept. 1964, pp. 15901606.
(2)
Pisacane, V. L., "Three-Axis Stabilization of a Dumbbell Satellite b y Sma ll Constant Speed Rotors," Applied Physics Laboratory, Johns Hopkins University, APL Report TG-855, Oct. 1966.
(3)
Crespo da Silva, M.R.M., "Attitude Stability and Motions of a Gravity-Stabilized-GyrostatSate11i te in a Circular Orbi t," Stanford Ele c tronics Laboratories, Sta nford University, Scientific Rept. No. 32, De c. 1968.
(4)
Scott, E. D., "Pseudo-Rate Sawtooth-Pulse Res e t Control System Analysis and Design , " 1966 AIAA Guidance and Control Conference, Seattle, Washington, Aug. 1966.
DISCUSSION
Q.
Is there any possibility of the proof mass touching the hollow cavity?
A. There is a possibility of this occurring, although through careful design of the translational control system we sized the thrusters in such a way as to overcome any magnitude of exterior forces that we could forsee. The translational thrusting system is a non-linear pulse modulated system, normally with a good dynamic range. By using either very short or very long pulses it can accomodate quite a wide range of perturbation. In normal operation the proof mass should not touch the cavity.
Q. When you fire the engine on the satellite there is a great perturbation moment. Have you some sort of active system which offsets this perturbation? A. There is indeed a perturbation when the thruster is fired. The order of magnitude of the thrust is very small, something in the order of tenths of a micropound, and the moment arms are quite large; the torque exerted by firing the thrusters is very small compared with the gravity restoring system of the satellite. Thus some sort of resonance phenomenon is needed in order to build up the proper energy into the attitude motion of the satellite.
418
Alan W. Fleming Research Engineer Daniel B. DeBra Senior Research Associate and Lecturer STANFORD UNIVERSITY Department of Aerona utics and Astronautics Stanford, California, U.S.A. a nd Marcelo Crespo da Silva National Research Council Post-Doctoral Resident Research Associate at NASA-AMES Research Cent er Moffett, Field, California, U.S . A .
Abstract The orbit of a drag-free satellite is determined entirely by gravity bec a use it follows an unsupported proof mass which is shielded from all external surface forces by the s a tellite. A control system in the satellite senses the relative position of the satellite with respect to the proof mass and actuates reaction jets forcing the satellite to follow the proof mass. A drag-fre e control system may be added to any type of satellite, and in principle is independent of the satellite's attitude control syst em. Coupling, however, can o ccur because both the pickoff, which senses the relative position of the s a tellite, and the proof ma ss may be located a way from the satellite mass c enter a nd the thrusters may be misalign e d with r e spect to the mass center. In passive attitude control s y stems, misalign e d translational control thrusters ma y produc e significant disturbances to the attitude. Thes e at titude motions can in turn be coupled back into t he tra nsl a tion control system through an offset of the pickoff with respect to the vehicle mass center. This paper describes the mech a nism of this type of c oupling a nd provides design guid e s for gravity gra dient satellites which are made dra g free. The discussion begins with a simplified model to illustrate the nature of the couplin g a nd develops the full six-degree-of-freedom equ a tions to examine t horoughl y all paths of coupling. It is shown that a n impor ta nt dyna mic pa th for this t y pe of coupl ing is through the interaction of along-tr ac k and radi a l t ranslations (conventional orbital mechaniCS perturbation theory). Thrust which is produced intrack due to relative displa c ement a long the orbit produces radial displacement which in turn activ a t e s thrusters in the radial direction. The v e rtical thrust, if misaligned, produces torques which c hange the vehicle ' s ori e ntation and thereby produ c e a n error sign a l in t he horizontal direction if th e sensor is not lo cat ed at the satellite mass cen t er . This error signal in turn requests addi-
411
tional horizontal thrust, and may thereby close the loop unstably. Stability boundaries as a function of thruster misalignment, pickoff offset, and attitude-motion damping ratio, are established and the alignment requirements are discussed. The analytical discussion is substantiated with results from a simulation which includes the nonlinear characteristics of thrusters and their modulators. A.
Introduction
A drag-free satellite cont a ins an unsupported proof mass which is shi e lded by the satellite from external non-gravit a tional forces. Since only gravitational forces act on the proof mass, it follows a purely gravitational orbit. A control system in the satellite senses motion of the satellite relative to the proof mass and activates thrusters, forcing the sat e llite to follow the proof mass without touching it (Fi g . 1). The satellite therefore also follows a purely gravitational orbit, free from the effec ts of external surface forces. A drag-free orbit is useful in satellite geodesy, for n a vigation satellites where accurate ephemeris predi c tion is needed, and for determining the drag 011 a satellite by measuring the equal and opposite compensating thrust, as Lange(l) has proposed. A drag-free satellite may use any type of attitude control and, in principl e , there need not be any orientation-translation coupling. In practice, coupling exists and two e ffects are discussed here. First, thrusters not acting through the satellite mass center produce torques which disturb the attitude motion, and secondly, if the proof-mass sensor null is not at the satellite mass center, its output c ontains contributions from orientation as well as translation. (In spinning vehicles, velocity information must be carefull y inferred from the proofmass sensor which measures in a rotating reference.) This paper discusses the effect of coupling on the
PROOF MASS HOLLOW CAVITY TRANSLAT IONAL CONTROL THRUSTERS--"" RELATIVE POSITION SENSORS ~
PROOF MASS SATELLITE BODY
POSITION SENSOR NULL POINT VEHICLE MASS CENTER
FIGURE 1 .
PLANNAR SCHEMATIC OF A DRAG- FREE SATELLITE
SATELLITE VELOCITY
-
-
~
SENSOR NULL RADIAL OFFSET (E I)
MOMENT ARM OF IN - TRACK TRANSLATION THRUSTER ( Lt)
f:
IN-TRACK CONTROL FORCES
..".
.....
N
~ ENSOR NULL (5)
VEHICLE MASS CENTER (cm)
EARTH
PROOF MASS (pm)
F'j TRANSLATION CONTROL JETS (j=I , 2 , "' 6)
FIGURE 3.
FIGURE 2.
SCHEMATIC OF DRAG-FREE SATELLITE GEQ\1ETRY
SIMPLE TWO - DEGREE-OF-FREEDOM DUMBBELL MODEL TO ILLUSTRATE THRUSTER COUPLING WITJ-DUT ORBITAL COUPLING
C.
stability of gravity-stabilized, drag-free satel-
Case 1 Without Orbital Coupling
l i tes.
B.
Ignore the coupling of radial and in-track motion in orbit so that
Equations of Motion
(5)
Three models of the vehicle-orbit dynamics will be used in the discussion that follows. Two of these are linear and can be represented by a common set of equations differing only in the coefficient matrix. The third model is discussed subsequently.
and take
K
= (k v s
of k
p
(6)
)U
e
For motion in the orbit plane, has just the ib (pitch) component, and Land E lie in the orbit plane. Then K and (A+K)-l may be trated as scalars, and using L x (E X e ) E(L·e) - e(L'E) -e(L'E), the characteristic equation is
Making the necessary assumptions to obtain a set of linear equations, taking the Laplace transform and ignoring disturbances, there are four relations . needed:
=
=
Translation dynamics A or
o .
=F
Orientation dynamics
B
e
e =L X F
Because is assumed parallel to the 3b axis only the 33 element of this matrix has signfficance and it may be equated to zero directly to get the characteristic equation. The gravity gradient dynamics about the pitch axis are represented by B33 = I 3 (s2 + 2~~gs + ~).
(2 )
Sensor equation
e = - or -
eX
(3)
E
Control law
F
=K
(8)
(1)
In a typical control system the translation control is very much faster (say by a factor of 100) than the att itude libration frequency and, for all practical purposes, the sensor is held at null during the libration. That is, the vehicle is forced to librate about the sensor null instead of its mass center. The force required to make this happen is therefore nearly in-phase (or 180 0 out of phase) with the libration and primarily effects the natural frequency, rather than the damping, of the system mode which, in the uncoupled dynamics, is the gravity gradient attitude behavior. The effect is greatest when the sensor offset and the force moment arm are parallel as indicated by the dot product (i.e., the attitude motion produces a position sensor signal for the axis in which the force has the greatest moment arm). Figure 3 schematically illustrates the geometry of such a case with a simple dumbbell-shaped satellite .
(4)
e
where (Fig. 2) , A, B, and K are 3 X 3 coefficient matrices to be discussed in the next sections or
is motion of the satellite mass center from its equilibrium position of r = -E (3 X 1 matrix)
F
is the control force assuming that all thrusters pass through a common point (3 X 1 matrix)
L
is the location at which the control force acts with respect to the satellite mass center (3 X 1 matrix). For linearity it is assumed that all L ' s are equal.
e
is the attitude of th e vehicle with resp ec t to a local tangent reference frame (3 X 1 matrix)
E
is the location of th e position sensor null with respect to vehicle mas s center (3 X 1 matrix)
e
is the output of the position sensor (3 X 1 matrix)
The locus of characteristic equation roots, as a function of L'E, which starts from the gravity gradient poles moves slowly toward the right half plane and becomes unstable for (9)
where
L X is used to denote the matrix equivalent of the vector cross product operation ( 3 X 3 ma trix) .
and
(B + L X K[-U + (A+K)
KJE
x }e = 0
a
is the a ngle between
Land
E.
For IEI and R equal to 1 cm* and 3 cm respectively, this mechanism should not produce instability as long as ILl ,< 900 meters, which is easily achieved.
The characteristic equation of this system of equations can be derived by direct substitution of 3 and 4 into 1 a nd 2, solving for o r and substituting into 2 to get a single matrix equation in e: -1
R is the vehicle radius of gyration for the mass center,
*Nonzero
(5)
where U is the 3 X 3 unit matrix. The determinant of the coefficient matrix set equal to zero (for a non-trivi a l solution) is the characteristic equation.
413
values of El require a continuous thruster force to hold the satellite up (or down) in the same orbit as the proof mass,which shortens the drag-free life. An El 1 cm is a reasonable upper limit for satellites at altitudes above the atmosphere because the required thrust is about one third of the radiation pressure force that must also be overcome.
=
1RADIAL
PROOF MASS POSITION SENSOR NULL POINT
"
10 MOMENT ARM OF RADIAL TRANSLATION THRUSTER( L Z )
SENSOR NULL RADIAL OFFSET (E,) SATELLITE VELOCITY
"
Ib
VEHICLE MASS - - - - - - - CENTER
~ RADIAL
MOMENT ARM OF IN -TRACK TRANSLATION THRUSTER
II
CONTROL FORCES
"
8,obout 10
IN-TRACK
~::I~ . "
2b' 8 3
( L,)
k
",
"
20
>IN-TRACK CONTROL FORCES
8Z obout 2b
2b
..,. >-' ..,.
"
30
8
" " " " 8s obout 3b
{IO. 20. 30} (0)
>{Ib. 2b. 3b} (bl
,8
"RMAL TO ORBIT PLANE
EARTH
~
"
FIGURE S. FIGURE 4.
SIMPLE THREE-DEGREE-OF-FREEOOM ruMBBELL MODEL TO ILLUSTRATE THRUSTER COUPLING WITH ORBITAL COUPLING
DEFINITION OF BODY AND ORBITAL REFERENCE FRAMES
D.
Case 2 with Orbital Coupling
four sets of equations for all six degrees-of-freedom are given. The results of both a linear analysis and a nonlinear simulation are compared in Section F.
The orbital perturbation equations due to Hill are well known(l). The in-plane motion is coupled and A becomes
1. -21lUl S o 2 ms
A
Referring to Fig. 5, the "orbital frame," (0), is defined by a unit vector 10 along the orbital radius pointing away from the earth, a unit vector 20 in the in-track direction of orbital motion, and a third unit vector ~o normal to the orbit plane, completing the right-handed set; The second frame (b), defined by the three unit vectors lb, 21, and 3b, is fixed in the satellite and aligned with the principal axes of inertia for the vehicle mass center. The orientation of (b) with respect to (0) is specified by a set of three Euler angles as shown in the figure. All quantities, such as forces and moment arms, are fixed in the body (see Fig. 6) except the translation, r, which is most naturally expressed in the orbital reference frame.
(10)
° where Wo is the orbital angular velocity. Using the same control law and previous assumptions the characteristic equation becomes 1 (s 3
2
+ 2sw s + g
2
+
00 )
g
+ (kv S + k p )(M12IL X E l - MllL'E) where
MU
= _ms 2 /(ms 2
+ k s + k ) v p
and
M12
= -21lUl0 s (k v s
+ k ) / (ms
2
P
°
Reference Frame Definitions
(11)
(12)
2.
+k s + k )2 v P
2
The assumption Wo « kp/m has been made to simplify the expression and emphasize the role of IL X El and L·E. For IL X El 0, the behavior is essentially unchanged from the case just discussed. For L'E 0, the coupling is due to IL X El. That is, it is greatest when Land E are perpendicular to each other, which involves a more subtle mechanism of coupling than the first case.
Translation Dynamics
Equation (1) is unchanged except that F must be defined as the sum of the effects of the six thrusters and in general a net disturbance,
=
6
=
Figure 4 is an illustration of a three-degree-offreedom dumbbell model with Land E at an arbitrary relative orientation. If an attitude perturbation occurs, then an in-track error is sensed and the in-track thruster is activated to null the error. The integrated (and therefore phase shifted) intrack force gives a slight change in the orbital speed of the satellite and thus a change in the centripetal acceleration is required to keep the sensor nulled in the radial direction. This requires radi a l thrust whi c h produces a torque due to the moment arm perpendicular to the sensor offset. The resulting attitude motion is then coupled back through the horizontal component of the position sensor . This coupling is important because there is a 90° phase shift introduced by the orbital dynamics. The phase shift causes the IL X El term to effect principally the damping (as opposed to frequency) of the roots of the characteristic equation whose locus starts at the gravity gradient poles. The product of ILIIE I therefore has a much greater effect on stability when Land E are perpendicular than when they a re parallel. E.
Equation of Motion for Simulation
In deriving the equations of motion for a simulation of a gravity stabilized drag-free satellite one can ignore the proof mass perturbat10ns which may be 10-11g or smaller(l) For the s1mulat10n of this system we have included nonlinearities in the translational control system. Each thruster has been assumed to have its own line of action, and the thrust is controlled with a derived rate modulator. First the coordinate frames used are defined, then the
F =
L
F
j
(13)
+ Fd .
j=l For small oscillation the translational motions are the same in orbit and body axes so ro ~ rb' All other vectors are expressed in the body reference frame. 3.
Orientation Dynamics
I f the a ttitude behavior is lightly damped (Sg « l), nonlinearities in the damping mechanism do not have a great effect on the fine structure of the resulting motion and linear damping may be assumed with adequate ac curacy for this study. Assuming a gravity stabilized system with yaw stability augmented with a constant speed pitch wheel, B in Equation (2) becomes(2,3),
(14)
B
where
BU
2 2 lIS +D s+0 -1 )00 +hw
B12
[(-l -1 +1 )w +h]s l 2 3 o
B2l
-B
l
3
2
0
0
B22
12 2 2 12 s +D2s+4(I3-Il) wo+hwo
B33
2 2 13s +D s+3(I -I )w o 3 2 l
and 11, 12, and 13 are principal inertias of the vehicle for the mass center, h is the magnitude of the angular momentum of a roll-yaw coupling momentum
415
--I[II*lsin a>O ----1[11*1 sin a<0
/
ASSUMED PARAMETERS jlOO
m = 86.5 kg 13 = 675 kg. m 2 =0.0033
tg
/~IN'TRACK
CONTROL FORCES
-
PROOF MASS
VEH ICLE MASS CENTER
\
~
FIGURE 6.
/
I
/
GEa-IETRY OF COUPLED PLANAR
-Ikp/m =100 Wg kp/kv =70 .7 Wg
,
I -+-+-t +- + +.. 1- 0 \ - 100
\
RADIAL CONTROL FORCES
/
/''-
\.
", .....
/
~!OTION
FIGURE 7.
I
-j50 ---- ........
I
-+.J __ -1.0 \
"
,
"
"" -)100 "\
-jl .O
ROOTLOCUSFORCOIJPLEDPLANAR~IONASAFUNCfIONOF IL II<1 5ina
t- 2000 •• c
--1
_TIME
(t - i
'0 )
(b)
1[11. \ sin a' +0.05
(COl
(C)
\[1\.1
(a)
NO
COUPLING
0 .10
iE
a , '0)
-=-0.05 ~
STABLE
°0~~---L~9~0~~--L-'I~B~ 0~---L--i360 ANGLE BETWEEN" AND [
FIGURE 8.
cos a-+0.05 (.ina-O)
(a) (dog)
STABJI.ITY RELATIONSHIP FOR L A'W < AS A FUNCfION OF PHYSICAL PARA'lETERS FIGURE 9.
416
REPRESENTATIVE ANALOG CCNPlITER TRACES OF PITCH ANGLE (93 ) WITH NONLINEAR DERIVED RATE TRANSLATIONAL CONTROL
f or
la l =90'
nz,
wheel with its spin axis along lb, and Dl , and D3 are damping coefficients. The torques are due to the six thrusters, and, in general, a disturbance torque. Thus LX F is replaced with:
of Fig. 7.
6
T
=
L
L
j
X F
j
+ Td .
(15)
j=l 4.
=
Sensor Equation
The sensor equation is the same as Eq. (3). A capacitive pick-off used in a laboratory breadboard'exhibits an additional cubic term, but it is negligible for this coupling study. (However, it cannot be neglected in acquisition and transient response studies.) 5.
Control Law
A derived rate modulator(4) was assumed to operate each thruster. This modulator switch with hysteresis is activated by a signal, a j , which is the difference between the error signal, e j , and a lagged feedback signal, fj, from the thruster: a
j
-rf j
= e
j
-
kf
+ fj = F
j
j
=
=
=
G.
Conclusion
A criterion has been obtained which accurately defines a stability boundary for attitude-translational coupled motion, in the plane of the orbit, for a gravity stabilized drag-free satellite. The criterion specifies allowable translational control sensor null offsets and translational thruster moment arms as a function of vehicle physical parameters.
(16) (17)
The question of coupling in roll-yaw and translation normal to the orbit plane has not yet been investigated fully. Tolerances on sensor null offset and thruster moment arm specifications are not expected to be any more stringent than for the in-plane case.
The damping is inherent in the lagged feedback. Fj is full on or off but its average value is approximately linear in ej when averaged over times comparable to the gravity gradient oscillation period. F.
To verify these results, a series of analog computer runs were made using the planar equations and nonlinear derived rate discussed in Section E. Figures 9a, 9b, and 9c show typical traces for 93 as a function of time, starting with an initial pitch angle of 10 degrees. Figure 9a has no coupling and exhibits the damping characteristic associated with Sg 0.0033. Setting IL'EI 0 and ILIIEI sin 0: +0.05 meters 2 , predicted by Eq. (18) to be slightly outside the stable region, results in a slightly divergent pitch motion as shown in Fig. 9b. Notice that the primary change is in damping, not frequency, for this case. Figure 9c is an example of the pitch behavior for the much less critical case IL X El = 0 with ILIIEI cos 0: +0.05 meters2. As predicted previously, there is a marked reduction in frequency with only a slight reduction in damping due to the thruster coupling.
Discussion of Results References
The approximate charact e ristic equa tion (11) is convenient for visualizing the relative contributions of the I L' EI and I L X El terms. Using this equation as a guide, but calculating roots directly from the full equations of planar motion (i.e., wi thout th e assumption that j kp/ mt » wo), the root lo c us can be obtained as a function of I L II E I sin 0: for cos 0: = O. This is shown in Fig. 7. The important part of the figure is the enlarged view of the region near the origin shown in the lower right hand corner. In this locus wi th cos 0: = 0, the principal eff ect of I L II E I is on the damping of the gravity gradient mode causing the poles to cross the imaginary axis for relatively small values of I LI IEI sin 0: > o. To obt a in an analytical relationship among the parame ters L, E, 0: , and Sg for system stability, the Routhian array may be constructed from the approximate characteristic equation (11), and conditions for imaginary axis crossover deduced from this. This yields the criterion,
(18)
for instability. That is, i f Eq. (18) is satisfied, the system is predicted to be unstable. Th i s relationship, represented graphically in Fig. 8, governs the imaginary axis crossover for the loci
417
(1)
Lange, B. 0., "The Dr a g-Free Sa telli te," AIAA Journal, Vol. 2, No. 9, Sept. 1964, pp. 15901606.
(2)
Pisacane, V. L., "Three-Axis Stabilization of a Dumbbell Satellite b y Sma ll Constant Speed Rotors," Applied Physics Laboratory, Johns Hopkins University, APL Report TG-855, Oct. 1966.
(3)
Crespo da Silva, M.R.M., "Attitude Stability and Motions of a Gravity-Stabilized-GyrostatSate11i te in a Circular Orbi t," Stanford Ele c tronics Laboratories, Sta nford University, Scientific Rept. No. 32, De c. 1968.
(4)
Scott, E. D., "Pseudo-Rate Sawtooth-Pulse Res e t Control System Analysis and Design , " 1966 AIAA Guidance and Control Conference, Seattle, Washington, Aug. 1966.
DISCUSSION
Q.
Is there any possibility of the proof mass touching the hollow cavity?
A. There is a possibility of this occurring, although through careful design of the translational control system we sized the thrusters in such a way as to overcome any magnitude of exterior forces that we could forsee. The translational thrusting system is a non-linear pulse modulated system, normally with a good dynamic range. By using either very short or very long pulses it can accomodate quite a wide range of perturbation. In normal operation the proof mass should not touch the cavity.
Q. When you fire the engine on the satellite there is a great perturbation moment. Have you some sort of active system which offsets this perturbation? A. There is indeed a perturbation when the thruster is fired. The order of magnitude of the thrust is very small, something in the order of tenths of a micropound, and the moment arms are quite large; the torque exerted by firing the thrusters is very small compared with the gravity restoring system of the satellite. Thus some sort of resonance phenomenon is needed in order to build up the proper energy into the attitude motion of the satellite.
418