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An Active Nutation Control System for Spin Stabilised Satellites
AN ACTIVE NUTATION CONTROL SYSTEM FOR SPIN STABILISED SATELLITES
W.J. Devey Systems Control Department British Aircraft Corporation Six Hills Way Stevenage, Herts England
C.F. Field Head of Control & Optimisation Department The Hatfield Polytechnic Hatfield, Herts England
L. Flook Dynamics Department Hawker Siddeley Dynamics Gunnels Wood Road Stevenage, Herts England
ABSTRACT A familiar problem associated with the design of spin stabilised satellites is the requirement to constrain the satellite to spin about an axis of greatest moment of inertia. This paper is concerned with the design of a nutation control system using a nutation sensor linked to a gas jet system which removes this constraint. The choice of nutation sensor is discussed and the effects of sensor misalignment are investigated. Stability criteria are examined which derive from the sensor/thruster system, energy dissipation and incorrect estimates of the nutation frequency. The results from theoretical analysis are supported by analogue simulation.
author is listed in the List of References. The design of most nutation control systems reported hitherto requires careful location of the nutation sensor to obtain correct phasing of the jet pulses. The design in this paper ensures correct phasing by appropriate electronic delays. It is considered that this approach eases the installation of the system into the satellite, and allows for updates of the nutation frequency late into satellite manufacture. This paper also considers in some depth the effect of misalignment of both the nutation sensor and the principal axis away from the geometric axis, and discusses a filtering technique to overcome the undesirable effects.
1. INTRODUCTION
2. NOTATION Spin stabilised satellites are required to spin about an axis of greatest moment of inertia for stability. To ensure that the spin axis is the axis of greatest moment of inertia requires careful mechanical design and it can place significant constraints upon the overall design. The ability to relax such constraints is especially significant for missions where the satellite is spinning during onlv the first few weeks of a mission lasting several years. The problem may be eased by introducing an active control system which senses the nutation and automatically activates a control actuator to reduce the nutation to a predetermined level. The configuration constraints placed upon a dual spin satellite to ensure passive stability are less severe than those required for a single spinning body. The analysis presented is applicable to such satellites provided that the nutation motion for the total system is considered and that the sensors and actuators are located on the spinning body. The approach adopted is based upon an understanding of the dynamics of the satellite. A criterion for control torque application is derived in a relatively simple manner for an asymmetric satellite. The use of gas jets to provide control torque is examined and the effect of products of inertia and attitude drift is considered. Rate and acceleration devices are considered as control sensing elements. Literature which has been found most useful to the
A,B,C F,G,H Ix, Iy, Iz ~ ~,
Ni
Moments of inertia about the satellite geometric axes Ox Oy Oz respectively. Products of inertia about the geometric axes Principal moments of inertia Inertia Tensor. A general torque Jet pulse length.
M
A generalised complex acceleration.
q
An auxilliary complex variable defined by equation 19.
rwnl /
k
w
n2
Acceleration of P. Pn
Nutation Period.
P
Value of parameter p at t
o
=0
K
Control gain defined by equation 20.
n,N
Integer variable. Laplace transform variable. Coordinates of accelerometer.
345
Lateral misalignment of accelerometer. Time delay. (i) Misalignment of gyro/ accelerometer (ii) Coefficient of W in destabilising torque. (Hi)
8 A \.1
\.1 1 '
\.1 2
n
LIt
Angular misalignment of gyro/accelerometer. Accelerometer sensitivity parameter. Auxiliary variables defined by equation 36. Nutation angle.
n $, 9,
1jJ
Satellite Euler Angles. Complex variable defined by equation (5) Filter damping coefficient.
~
I;
W n W n
CI= W
Nutation frequency. (B - C) ws / A
If a complex variable ~ is defined by I; WJ + ikw2 where k = .; wnl / wn2 where W = (B-C) W nl -A- s
(5) (6) (7)
W = (A-C) W (8) s n2 -Bthen it follows from equations (2) and (3) that (9) ~ + i w I; = M n
and
(10)
where
M
where
\
Nl/A
(11)
and
M2
NzlB
(12)
and
W n
.; W n
\
+ iM2
l
(13)
W n
2 From the definitions of the nutation angle (n), in Appendix I, and 1;, in equation (5) it follows that n ., 0 <=> I; = 0 (14) The basic control problem as posed by equation (9) therfore, is to find a control torque M to control I; The solution of equation (9) when M is zero is -iw (t-to) I; ~o e n (15) where 1;0 I; (t = 0) = wlO + ikw20 (16)
l
= 0)
(A - C) ws/B
where
W s
Spin rate
p
Modulus of
9
Argument of
T
Time constant associated with the Energy Dissipation. Phase error induced by sensor misalignment.
It follows from (15) that both wl and w2 are sinusoidal of frequency w ' n This frequency is termed the nutation frequency. Equation (15) implies that the transverse angular velocity vector and the transverse angular momentum vector rotate at frequency wn' in a
W n
wl O= Wl(t
= 0)
and w20= w2 (t
(17)
2
£
~
,I ~ I ~
W g
Gyro output.
Wgf
Filter output.
3. BASIC THEORY AND SELECTION OF CONTROL LAWS The system equations and control laws are derived initially for the case of a rigid satellite spinning about a principal inertia axis. Subsequently the effect of misalignment of the principal axis from the nominal spin axis is examined. Assuming that the geometric axes Oxyz are aligned with the principal axes, referred to Oxyz Euler's equations of motion are:Aw - (B-C) w w (1) NI 1 2 3
E
(C-A) w w = N2 3 l
(2)
(A-B) w w = N3 (3) l 2 If the satellite is spinning about its Z axis, assuming the spin to be about an axis of minimum moment of inertia, it may be assumed without loss of generality that A> B >C. If wl and w2 the transverse rates, are assumed to be small compared to the spin rate w3' then to first order if N3 = o it follows that (4)
negative sense, with respect to satellite fixed axes. The type of control envisaged is to pulse the control jets at precisely that time when the angular momentum is aligned with the torque vector provided by the gas jets. At this time the change in the magnitude of the transverse angular momentum vector as a result of a control torque will be a maximum. Equation (9) is used to determine when and for how long the gas jets should fire. The solution of equation (9), assuming M,i.e. the gas jet torque to be constant is -iw (t - to) en) iw n
(18)
(19) which can be written I; = 1;' + q where ~' is independent of M. The problem is now reduced to finding the optimum q to reduce I~I. As Iq is most sensitive to the magnitude of q when q is parallel with 1;' we require a control such that (20) q = -K ~ ; for some K If K = 1, q = ~' and I; (t) the nutation, is reduced to zero by the action of q. As the nutation angle will grow slowly as a result of any internal energy dissipation the divergence of the nutation is slow compared with the time constant of the control system. The control stategy can therefore allow the nutation angle or transverse angular momentum to grow to a preassigned value at which point the jets may be
346
fired according to equation (20) to reduce the nutation angle. A stability criterion is derived in Section 5 which formulates the maximum rate of energy dissipation as a function of the control system parameters. In order to solve for the required control M, q is written 2Msina/ q = -2a/ 2 (21) e 2 W
n
a=
where
W (t - to) =w 6t n
n
and 6 t is the jet pulse length. Solving equations (21) and (22) for M M = KPow n ~ (TI +8 - a /2) 2sin~72 e
(22)
(23)
where P and 8 are the real and imaginary parts of 1:;0 r~specti~ely i.e. i 8 E; = P e 0 (24) o 0 It is assumed initially, for simplicity, that one thruster only is available, so that the control torque is limited to one axis of the satellite. M is then written (25) M = Nl/A implying that, w~thout loss of generality, a torque about the x axis is assumed. Equation (25) implies therefore that:Kp
W
o n
2 sina/ and
(26) 2
sin (8 +1f-a/ ) o
2
=0
remembering that I cos (TI +8 - a /)I = 1 the control impulse is definedOrat~er than the torque. If K = 1 in equation (26) the control impulse will be sufficient to reduce the nutation angle to zero in one pul se. Wi th K < 1 then some nutation will remain after the pulse. This iR quite acceptable because a second pulse or even a stream of pulses can be applied, so long as the timing is correct, until the nutation angle is sufficiently small. If K > 1 there will also be some nutation remaining after the pulse. This is clearly inefficient since the control impulse is greater than that required to reduce the nutation to zero, and also if K is sufficiently large the remaining nutation will be greater than the initial nutation, implying instability. It is therefore, required that K ~ l,i.e
(27)
which define the magnitude and phase (timing) of the required control torque respectively. Equation (27) implies that the pulse must begin when E; has an argument 6 defined by:8= a /2-(n-l)'Ir ; n=O, il, ;1;2, ... (28) If time is measured from when w is zero, it 2 follows that the pulse must beg~n at t' seconds where, W t' = -a/ +(n-l)1I', n=O, 1,2, ... (29) In the c~se of the2 jet providing a positive torque about the x axis from equation (27) Pocos (TI+8"'l/2) must be positive. ) is positive, this implies lO n = 0, 2, 4, ... (30) W t' = -a/ +(n-l)~ n 2 Equation (30) implies that the positive x axis jet should fire Gt -a/ 2) /wn seconds after w2 If
The above argument can be repeated similarly for a negative x axis jet, and a y axis jet with the result that each jet should fire when the component of the transverse rate along the jet torque axis is at a peak and opposite in sign to the jet torque. This is equivalent to requiring that each jet should fire when the angular momentum vector is aligned, and opposite in direction to, the torque vector provided by the jet. Equation (26) defines the required magnitude of the jet torque. The control impulse is N 6t, 1 where 6 t is the pulse length (a =w 6 t), so n writing equation (26) as 2NI sin a/ 2 = A K wlOwn (31)
P o (=w
is zero, or on any subsequent nutation period later. The delay term (TI'- a / ) w contains a 2 n 0 a/w term which corresponds cv a 180 phase J1 sh~tt. Thus apart from the a/ term which 2 ensures equal spacing of the pulse either side of the optimum time, the delay causes the jet to fire when w is zero and when w is at a negative I 2 peak. If w is negative there is a further IO 180 0 phase change in the firing delay which still; implies that the firing should occur at a negative peak of w . Thus to summarise, the time for the positive l axis jet to fire is optimum when the pulse is centered on a negative peak of w • I
2N sin a/
2
<
Aw
(32)
lO
Wn or, dividing by 2N sin a/
Cw s 2
<
Nutation
threshold (33)
Cw W s n This in fact sets a lower limit on system accuracy for a given impulse. Effects of Products of Inertia If the principal axis OXp Yp zp are not parallel with the geometric axes Oxvz then the inertia tensor expressed in geometric axes will contain some non-zero off diagnal terms. For simplicity a rotation about the x axis of the principal axe s relative to the geometric axis is considered, in which case the inertia tensor in geometric axes can be written.
r ~ ~ -~ 1
o -F C (34) where (A, B, C) are the moments of inertia, and F is the product of intertia Emyz It can be shown that under torque free conditions the three satellite angular rates about the geometric axis, with zero initial values onw and w are l 2 given by:sin w 't wI 1: n Wn (35) \.I (l-cos w't) _ n \.12
347
(1-
where
cos w' t) n
(35)
BF !J
F
2
+C(A-C) (36)
\.. = A-C 1 B-F w' n
B
w30"
W n
l
B-C-2FBw A
2
30 "'wn 2
!JJll"Wn 2
It follows from equations (35) and (36) that, provided the products of inertia are small, the rates oscillate at approximately the nutation £requency,w · n The most significant result arising from the above equations follows from the form of the solution for w • In general when the response 2 of w is a combination of the effect of products 2 of inertia and a non-zero initial condition, the time variation of w will not be centred on 2 zero, but will have a mean value. (37)
The significance of this is that there is not 0 a simple 90 phase relationship between the zero and the maximum values of w ' so that the 2 maximum value can no longer be predicted from a knowledge of a zero crossing point. There are two possible solutions:(1) to utilise a peak detector network, or (2) pass the measurement w signal through 2 a DC blocking filter to remove the bias. The second approach is adopted as it is shown later, that such a filter can fulfil a dual role and remove the effect of a sensor misalignment, which also adds an unwanted bias to the signal. The filter should remove the bias and preserve the gain and phase at the nutation frequency. A band pass filter of the form 2sw s n
which is the same result obtained from a passive damper. The active system could constrain the satellite to spin about its geometric, non principal axis, if the bandpass filter is not used, but relatively large control torques would be required as they would have to continuously oppose cross coupling torques of the form F w ~. Generally it is suggested that it is sufficient to allow the system to converge to a state of spin about the principal axis. The control scheme as derived in the preceeding paragraphs is summarised in Figure 1. Here N = 1 if the jet has a sign which is opposite to that of the transverse rate at the time of the threshhold crossing. When the signs are the same, N = 3. Choice of Jet Thrust and Pulse Length In most applications the nutation control system will form a part only of an overall attitude control syst em and the nutation control system will make use of the control jets which are provided for the attitude control system as a whole. Equation (33) which can be written w ~t Cw W x Nutation 2N sin _.::n,--_ s n < 2 Threshold (40) defines a m~n~mum nutation threshold for a given jet impulse bit. It is therefore essential to ensure that the thruster system adopted is capable of providing the required system accuracy as defined in equation (40). Assuming that a jet system is available which is compatible with the accuracy requirement, there remains the choice of selecting the pulse length, assuming that equation (40) does in fact allow a degree of choice. In order to allow for system errors it is suggested that ~ t be chosen less than approximately 50% of the value which would imply equality of equation (40) i.e. ~t < 1
sin- l CWs wn x Nutation Threshold
wn
(41)
2N
(38)
is chosen because it is simple, adequate and will also attenuate any high frequency noise associated with the nutation sensor. The choice of damping coefficient s is not critical. Simulation results shows that a value of s = 0.2 represents a reasonable compromise between rejection of frequencies other than w ' and n the filter response time. Use of such a filter implies that the DC term remains uncontrolled. It can be shown that equation (37), which defined this term can be written !J F w = AW (39) 30 30 W = 2 j;"2 (B-C) where A is the angle between the geometric and principal axes. If the motion is reduced to zero apart from this term, the satellite will be in a state of pure spin about its principal axis. It is interesting to note that, apart from the deadzone equal to the nutation threshold, use of a bandpass filter will result in the motion converging to steady spin about a principal axis,
The exact choice is not critical. If ~ t is very small the jet puls e is effic i ent because all the impulse is delivered close to its optimum time; efficiency in fact is equal to sin et let.where et=wn t/2 On the other hand a small pulse could r e sult ~n jitter around the threshold value, although this could be overcome by introducing a two level threshold system and switching to a lower level when a higher level is exceeded. A small pulse length will also result in a longer acquisition time although usually this consideration is not expected to be particularly important. Overall it is considered best to select a small rather than a large value of ~t, i.e. to trade time for efficiency. Attitude Drift From Jet Pulsing Each gas jet firing changes the direction of the system angular momentum vector. Consequently the satellite spin axis direction will vary with time because the nutation control system will always attempt to align the spin axis with the angular momentum ve c tor. A means of minimising this effect is to fire two pulses eql1ally spacF!d half
348
x (or y ) not being exactly zero then the a8celeromgter would respond to the large centrifugal force associated with the spin rate. Choosing x (or y ) as zero, a provides a direct measu~e of 11~ z
a spin period from the optimum jet firing time. This is only possible providing the nutation period is long compared with the spin pe riod.
4. SENSING THE NUTATION Rate gyros and accelerometers are considered to be the most suitable type of sensor as they provide a direct measure of nutation which can easily be used to drive the jet control logic discussed earlier. Sensors which detect satellite motion relative to exterior references, e.g. sun or earth sensors, could be used but these require more complex on-boards processing to derive the nutation angle. Rate Gyros If t is measured from when is zero, under torque fre e conditions it follows t~at
Aw lO
11 (t=o) = 11 0
(42)
Cw
SI
and
11 (t) =11
CCA-B) 2 Wt 1 + A(B-C) sin n
oJ
(43)
J
It follows that by setting a threshold limit on one of the transverse rates, the nutation angle is contained within known bounds. The sensitivity of gyro output (transverse rate) per degree nutation angl e is illustrated in Figure 2 for some typical satellite parameters. Accelerometers Under torque-free conditions the satellite rates are given by:W = a constant (44) s
W cos W t n 10
(45)
sin W t
(46)
n
where t is measured from when w is zero. 2 It can be shown that a general point p , fixed in the satellite has an acceleration a (a ,a ,a ) -
where a
a
y
p
- wlOwnz o cos Wn t - w s2 x0 +wIO wszo cos Wn t
x
x
y
z
(47)
- wlOwnxo sin W t - W 2Yo-wlOws z sin W t n s 0 n -k-
(48) / W
a
z
-y
W
o 10
(
S
\k
+
W)sin W t n
C A
where
W
fw _w!!)
srs
x
k
(SI) 0
Thus a can be used in exactly the same way as a rateZgyro measurement of w or w but with l 2 a different scale factor A The se~sitivity of the acc e lerometer output (a in m/s ) per de gree nutation angle is shownzin Figure 2 for some typical satellite parameters. The effects of sensor misalignment are now examined. Gyro Misalignment If the gyro is misaligned so that its sensitive (input) axis is rotated away from the nominal Ox axis by a rotation of a about the z axis and B about the y axis, the output of the gyro, w
g
nutation cycle, and varies during a nutation pe riod from 11 up to a maximum of o B(A-C 110 A(B-C
k
(SO)
is
The nutation angle is equal to 110 twice per
W10 /
With
n
(4~)
+ xoW10Gs - :n)cos wn t It can be seen that apart ftom the terms in W in equations (47) and (48) which can be avoid~d by suitable choice of sensor location (x or o y = 0) that all three components of ~ are p~oportional to wlO(hence 11 ), vary at the nutation frequency, and the~efore are potential nutation sensors. The z axis component appears most suitable, since, if in the case of a (or a ) x y
(52)
provided that small angle approximations for a and Bare valid. Using equations (44) to (46) which describe the torque-free angular rate response it follows that the gyro output is w = 11 g
Cws
0
A
cos (w t+E) n
- Bw
where the initial nutation angle 11 0
(53)
s
c;;;s
and
(54)
E=a·/k
The effect of a , the angular misalignment about the spin axis, is to cause a phase shift of E= a / , and the effect of B ,the angular error about the k transverse axis is to cause a bias signal Bws or in terms of nutation 11 = BA (SS) b C Assuming an inertia ratio close to unity, equation (SS) implies approximately one degree of nutation bias angle per degree gyro offset. The bias error reflects itself directly as a nutation angle error, whereas the effect of the phase shift is to cause an error in the prediction of the peak of the transverse rate which is requir ed in the control logic discussed earlier. The phase error and bias are plotted as a function of the misalignment in Figure 3. Both of these problems can be overcome by filtering the gyro output with the band pass filter, which is designed earlier when the effects of products of inertia are considered. The filter is only useful of course provided that the bias signal does not saturate the gyro. Accelerometer Misalignment The effects of both lateral and angular alignment errors are important here. It is assumed that the accelerometer is mounted away from its nominal location P = (x , 0, z ) to P where by (ox,n ay ,o a z) 0 o
0
0
z ) o
349
(56)
and additionally it is assumed that there is an angular misalignment of accelerometer sensitive axis away from 0 by an angle a about 0 and B z x about 0 , where a and B are small angles. The output 6f the acceleromet er in t erms of the components of the acceleration of P, a (a,a, a) i s therefore p x y z a = a - a a + Ba (57) sz z y x Using equations (47) and (49) with P defined by equation (56) it follows that , a
where 6 ,
11+ n
sz
ACOS
0
=-
a bias error
(
W
s
W
n
t
e:)
-
2 x B
(58) (59)
0
If the change in nutation angle per jet pulse is greater than the threshold then the system is inefficient, as the transverse rate will have been r e duced from its threshold through zero to a rate of the opposite sign. If the change in nutation angle is greater than twi ce the threshold, then the system is unstable because the nutation following the pulse is greater than the initial nutation . Therefore, to ensure stability it is necessary that 6n < 2 x Nutation Threshold (65) but to retain a reasonable stability margin and to obtain an efficient system it is necessary that
x
and
A
where it assumed that oz
(kWs - wn )
o
Cs -:n)
Cw s
(61)
is neglected compared
o
to z and terms like ao y are neglected compared withOt erms like x B . ~rom equation (60) it is appar ent that theOphase error can be reduced by mounting the accelerometer in the Oxy plane; then z ~ 0, and we have o
- oy o Xo
(w
s + kW) n
kw s
n
sin
Equation (67) sets a limit on the jet impulse bit ( N. 6t ) for a given accuracy and system parameters wn and ws' Error in the Nutation Frequency A prior knowledge of the nutation frequency (w ) n is r equired for the jet control logic. The logic r e lies on predicting a peak by counting either ! or ~ of a nutation period from a zero crossing of the filtered gyro output. The delay time counted from the zero crossing is Ip 4
or
5. STABILITY CONSIDERATIONS There are four main aspects to be considered regarding the system stability, viz:(i) The size of the control i mpulse compared with threshold set by the ac cura cy r equirements. (ii) An error in the estimate of the nutation frequency. (iii) The rate of energy dissipation compared with the size and number of control thrusters. (iv) The effect of delays associate d with the sensors and actuators. The Control Impulse It is shown earlier that the change in the transverse rate per jet pulse ( 6w ) is given by 2N
=-
Aw
n
2N Cw W s n
(62)
W
Th e bias error which is caused by the angular misalignment and the phase error, which is caused by the lateral misalignment, ar e plotted in Figure 4 for typical satellit e parame ters. The e ff e cts of the bias error can be overcome in the s ame way as with the gyro bias , by using a band pass filter centered on the nutation frequency. The effect of the phase error is likely to be insignificant; for the satellite parameters assumed in Figure 4, lcm. offset caus e s a ne J ligible phase shift of approximately 1 degree.
6w 1
(66) Nutation Threshold W 6t < Nutation Threshold n -2(67)
<
6n i. e.
(63) 2
where the torque is applied about the x-axis. The corresponding change in nutation angle is A6w l W 6t 2Nl n CW-w sin 6n = s s n
cw-
lp 4
n
-! 6 t
(68)
This time (t ) can be in error by up to ! Pn d actually ~ncreases . . before the jet t h e nutat~on angle and the system becomes unstable, although the system will become inefficient for errors in t considerably less than this. To obtain r easgnable efficiency ( ~ 90%) it is necessary that the error is less than approximately 1/8 Pn. This requires in the case for tdl the error in the estimate of Pn must be < 100% of Pn for td the error in the estimate of Pn must be < 30% or Pn. A slight modification to th e control logic could reduce the effect of large errors so as to cause loss of efficiency only instead of instability. The sign of the control torque is decided by the sign of the transverse rate when it crosses the threshold. If instead of this the choice of po s itive or negative jet is decided by the sign of the transverse rate at the start of the jet pulse then the jet pulse will always fire so as to r e duce the transverse rate. The above criterion will still be necessary however to ensure reasonable efficiency. Minimum Acceptable Time Constant Associated with the Energy 0issipation There is a necessary criterion for stability arising from the requirement that the increase of nutation angle occurring between jet firings due to energy dissipation must be less than reduction in nutation arising from each jet firing. Suppose the nutation angle in between jet firings follows the law:I) e
n (64)
n
tiT
o
(69)
where T is the time constant associated with energy dissipation. It follows that
2
6n = n-I) = n (e o
350
0
tiT
-1)
(70)
where bn is the change in n in a general interval (O,t). Now suppose the jet impulse bit J corresponds to an angular momentum increment bh where bh = J R, (71) where R, is the jet moment arm. The change in nutation angle per jet firing is then given by bn
I Cw
JR,
(72) s
where Cw is the spin angular momentum, inertia x spin r~te. The number of times the jets can fire in a nutation cycle depends on the thruster configuration. Each jet can fire once per nutation period, therefore the time between jet firings is given by t
= 2IT/
Nw
(73) n
where N is the number of possible jet firings per nutation period. Therefore, in order for the change in n from one jet firing to the next to be less than the change from the jets we require 2'IT/Nw T ' \ (74) no ( e n -~
T
>
2'IT
(1 +bnl
(75) no
where bn is the change in the nutation angle from the pulse, i.e. the minimum T depends upon how often the jets can be fired, the jet impulse bit, the nutation frequency, and n the largest expected initial condition. 0 Sensor and Thruster Delays The effect of these is similar to an error in the knowledge of w n ,i.e. a delay whether within the signal or introduced by the jet itself has the effect of causing the thruster to fire off peak. The delays introduced by sensors and thrusters are expected to be very small compared to the nutation period « 1%), and are likely to be swamped by the errors arising from inaccurate estimates of the moments of inertia. They are unlikely therefore to cause stability problems or loss of efficiency. In theory the same criteria holds as in the previous section, i.e. the time delays should be less than approximately 1/8 of the nutation period. 6.ANALOGUE SIMULATION RESULTS In order to confirm the basic system design, and to complement the theoretical analysis, an analogue simulation of the system was set up on an EAL 680 analogue computor. The simulation (which models a 650 kgrn satellite spinning at 60 rpm) includes 3-axis dynamics, kinematics (see Appendix I for equations) a negative damping term to represent the effect of energy dissipation and a pair of gas jets to supply positive and negative y-axis torques. Two examples of the simulation results, which represent two inertia cases; (A,~,C) = (205, 195, 185) and (220, 200, 160) kgrn , are illustrated in Figures 5 and 7 respectively. Figure 6 illustrates the kinematic response from an initial condition, the system parameters are as in Figure 7 except a 1.0 pulse is used.
7. CONCLUSIONS The design of nutation control systems using gas jets as control actuators and a rate gyro as a nutation sensor is derived for a spin stabilised satellite spinning about its minimum moment of inertia axis. The control law arranges for the jets to fire a control pulse at a peak of the transverse rate so as to reduce the rate and hence the nutation whenever the nutation exceeds a preset threshold. The design is verified with a detailed analogue simulation, which shows that with the chosen laws the system performs satisfactorily. The use of both rate gyros and accelerometers are examined as potential nutation sensors, and both are found to be suitable. The control system is found to be insensitive to misalignment errors of both types of sensor. Apart from small lags which occur and which are shown to be insignificant, the main effect results from the response of sensors to a component of the spin rate. This causes a bias signal on the output of each type of sensor which, provided it is not so large to saturate the instrument, can be filtered out by a band pass filter. The response of the system is analysed when the principal satellite axis is not coincident with the nominal spin axis of the satellite. It is shown that using the same filter which is required to remove the sensor bias induced by misalignment, that the system can converge, to within a dead zone, to a state of steady spin about the principal axis. If it is required to force the satellite to spin about the z axis and not the principal axis then the band pass filter should not be used, and large continous control torques are required. An expression which determines the maximum rate of energy dissipation which is acceptable to the control system is developed. For the parameters assumed in this paper tile minimum time constant is 141 seconds which is very short compared to the value expected in prr.~ tice. There are two other main stability criteria which are developed in Section 5. The first sets an upper limit on the minimum gas jet impulse bit acceptable for a given nutation accuracy. For the parameters assumed in this paper if it is 0 required to keep the nutation to within 0.2 , the ffi1n1mum jet impulse bit must be less than 8.1 Ns. In fact, this impulse bit is quite large for gas jet systems indicating that higher accuracies should be easily achievable. The second criterion requires that the nutation frequellLY should be known to within a given accuracy. The exact requirement depends on the actual system parameters and the precise control strategy adopted, i.e. the number of jets available and the delay time between the zero detection and the start of the jet pulse. 8. REFERENCES GRASSHOFF L.H. "An On-Board Closed Loop Nutation Control System for a Spin Stabilised Spacecraft". Journal of Spacecraft and Rockets, Vol 5, No 5. May 1968 pp. 530-53fi. ~ARNER
351
H.D. and REID H.J.E. Jr.
"Flight Test of a
Pitch Control for a Spinning Vehicle". July 1963 N.A.S.A.
TN D-1922 ZERO DETECT OR
PULSE GENERATOR
DELAY !:!~ - ~6t
6t
4
NEER J.T. "Intelsat IV Nutation Dynamics AIAA Paper 72-537" 4th Communications Satellite Systems Conference April 24-26 1972. DEVEY W.J. "An Active Nutation Control System for Spin Stabilised Satellites". M.Sc Thesis, The Hatfield Pol y technic November, 1973. APPENDIX I Fg.l
BlOCK DIAGRAM
SYSTEM
DEFINITION OF EULER ANGLES AND NUTATION ANGLE Euler Angles In order to descLibe the motion of the satellite spin axis pointing direction it is found most us eful to define a set of Euler angles which define the orientation of an axis system Xl' Y , Z, which is fixed in the satellite excep! fot the spinning motion. The angular velocity of this axes system ( n ) is given by
n=W-
•
where
'\
//
E 02,O
(1)
sinw3t +w2 cos w3 t
1,2
f
0,8 0, 4
j;
0'
V>
DEG/SEC
'1-' 1).
u,. =
//
;j/~ /
G'iRO OUTPUT -
liil
I
I
/
1,6
III
ACCELEROMETER - - - /
l iil
/'
m/si
OUTPUT -
~ ~. ('"
Q,.
/
-
't)%.1).
~
195;
x . = 1m.
= 2s 'Is
'->s
CaS£' ( j)
JP /
A
~
205 , B
C
~
185
Kgm'
220; B ~ 200 ; C ~ 18)
K9 m'
Cose lii)
A
0.1 0.3 02 NUTATION ANGlE 1DEG61.
~
F1Q2. SENSOR SENSITIVITY.
(3)
(wl ' w2 ' w3 ) is the satellite angular velocity vecr'r expressed in satellite fixed axes. W
Nutation n (t) =
2,4
~t3
(~ .~)~
where W is the angular velocity of the satellite fixed axes, and k is a unit vector along the spin axis directIon. Such an axes s ystem responds only to the satellite transverse rates i.e nutation. The Euler angles ~ ,8,W define the orientation of this axes system with respect to inertial space, and are defined by requiring that the transformation from inertial space into this axes system is an ordered set of rotations, w ,8,~ about the Z, Y and X axes respectively. For small angles, ~ and 8 they define the r o tations of the spin axis away from its ~ominal direction, about the two inertially fixed X and Y transverse axes respectively. It follows that w cosw3t - w sin w3 t (2) ~ 2 l 8
lil
'~ u :-~
Hi
9
III
le n is defined by + B2 w 2 2
(4)
W
/
/
0.6
J !f !/
0.4
O. 2
/
Ii
PHASE ERROR v MISAUGNMENT OF GYRO ABOUT SPI N AXIS
BIAS ERROR v MISALlGN/oENT ABOUT A TRANSVERSE AXIS
/
/
z
nl)
l- V/I"I
1,0
0.8
It is therefore the angle between the satellit e principal Z axis, 0 and the angular momentum vector. It can be ~hown (Ref.l) that under torque-free conditions this is equal to the angle be tween the mean direction of the 0 axis and the current direction of the 0 ~xis.
(i)
L'L
1,2
/
/
/
~// /
J
Case III A
A
0.2
~
205
B , 195
C
~
185
K9m'
220
B , 200
c
"60
K9 m'
Case Ill)
O~ 0.6 0.8 1.0 MI SA LI GNMENT E~ I DEGSJ
~
SPIN RATE
Fig.3 ERRORS RESULTING FROM GY RO MISAlIGNMENT.
352
8) RPM
NUTATION SIGNA L PHASE ERR OR
v LATE RAL
MIS-ALl GN MENT -
-
N U TATl ON AN GLE BIAS ERROR
1.2
V ANGULAR MISALl GNMENT - - - -
l il
1j)
--;-/ OB
V
~;:; 0,4
-
~~
l...- f" V
0.2
~
~V
w
~
~
~
V
CA SE ( i J A · 20S 8 · 19 5
iI)
W
M
U
CA S-EI; i) A 220 B 200
C .. 185 Kg rn 2 C 16 0 Kgm 2 SPIN RATE - 6 0 RPM
...--- V
U
~
~( .... 1,'i1
~
LATERAL M I SAU GN~ N T ( ems) ANGUlAR MISALIGNMEN T [EGS . 11f'
Fig 4, ERRORS RESULTI NG FROM ACCELEROMETER MISALlGNME NT,
' OS"'l ;~~T • mA~EBSE RATE -0 5 r / tJ
"
(W
r'
I
1
'0 5
, I,
1
-0 5
f
-0 5 r / a
! ~"'fV
) -0.5 r / .
t;-~Ii;-i'"'~ -'4i I
i r-l't
I'! ~
,
f ":
~
'0 , , / .
i
~
I
I
1 I
,E CONal
l -- - .
~ TRANf'VERSE RATE (W I )
-0 5
!
-,
,_n
_-.....-..:...~_ ' __
rl e
_
I
L_ _ __
'0"/'1~ :.;Tj0
I
!A
"
TRANWE_J: RATE (1612)
~ ~=ERSE ~~~~)____.. __
-0 Sri .
'O'3"'·1 ~ ··- '-.'; ,- -.-- :' .. i '
t
•
,
i
~ F~;E~ ~TPUT IWsfl
:~-,
-<1.33,/ .
i~---....._
_L'
I
I!i
.Ll.LJ .J J_l J....LL.L
LJ
! i
1
I1
GYRO OUTPUT ( W ) II
G YRO OUTPUT (01 ,)
- 0. S r l s
1 '
Ill'
1 ~~ ...:....:_LJJ-'-.Li .l.1-,-1.. ..c. ...l. _.
. a. S r/ .
...
FrLTER OUTPtrT ( C&I&f1 _.: ~ _ _ _L _
'O.33r/s -0, 5 r / e
!I
: ..
.
~
i'"""!1 1
··..'. I "0. 25 rl . t
11l
-0 . 5 r / e
...,. f
- 0001"
J
vivvvI*J.~ VI' " : ' , .. _ _ ---'
i :
iJi 1,I,l'l
' 0, 5 ' / ' }
FIT.TEll OUTPUT (W.,) "
i
+
AA ' 1fo "'... T.",. T,-+ '· ... '
2IJi)!M' , E C0ND8
I!
iT;!; . . ill iT ,in*
-0.33 Ti e
!
r/e
. TRA !oIrVElIB E RA TE (""2'
---+ •• ' .
I : f+~'IAi!o; W'
TRAMJVE RSE RATE (Wl)
---,---,--L-...l..i--,-~~
- 0 . 33 r/ .
I
-t ' -4 :'-1l 'ijlAJl ":'ib\ ;;'1\I\1
)
'O" " f ~ --' ! ~ - OS T/ a
t
f
.
_
~ _ .. --- - .---t-----.- G YRO ~ :.PtrT (w ,)
_~l _'~ __ ~
l_.__ '_
~;;:-;:;;;':~;;:;~~~ :
'/"1 ~ ' I4-i+
. . . . ... , / " - 1. 7"
~UUR ACC'~EIlATION
("'31 ..
IJWfITJ
11; ,. , . It , I l1. __ L~_ -'
O~~ _1 , . ,
.1.- 1
~GL~~~ I -~--, !
"1 7~
I
~
I ,
,
J
~
,
j -,~~~~+4~~~~~+-H t
.
:
. 1 7
-_~'::!~~~~1JJ...:lJJ...lJ..l
--
- 17 Nm
NOMINAL CASE
WITH
GYRO MISALIGNMENT
WITH PRODUCTS OF INERTIA
(iil
(i)
ACQUISITION
353
(iii)
FIG.S
PROJE CTION OF TfP OF SATELLITE Z ."\XIS Ol\TO S PACE-FIXED TRA?\'SVERSE PLA!-.'"E
RESPO?\"SE AFTER
F mST Pt;l.SE
1, ,,
-x
TWO LONG JET Pl' LSES ("SED a t - 1.0 SEC
e
FIG .6
RESPONSE TO AN INITIAL TRAr-8VERSE RATE
~·~f ~ ' ~~~~
~,~,/. r .O ,hl·l -0. 1
r/ •
.""/'1,-
. . "·'I ~~~~ - 12 5 r / •
.... i5
r/.'
"' ' ' ''1 ~~~~ ~O , .
r/.'
~' -"'/] -ttiiTttttti*t+-t+++tt-H++++++~~
I
~•."'.r/" -, ,i'
l
,
t , _, ,,,,
-LLL--'-..1.-'-'--'--LL-.I....L
-LL.LL.L..Ll--L.l--'--LL.LL..Ll--L.l--'--LLLLL.L...LliJLLLLLL...Ll-"--.:~~
-""I ~+++++++t+++--t1~~ . 1f . .
eeNfROl WITH ENERGY
OISSIRtJ~ PRESENT
354
FlG.1
W.J. Devey Systems Control Department British Aircraft Corporation Six Hills Way Stevenage, Herts England
C.F. Field Head of Control & Optimisation Department The Hatfield Polytechnic Hatfield, Herts England
L. Flook Dynamics Department Hawker Siddeley Dynamics Gunnels Wood Road Stevenage, Herts England
ABSTRACT A familiar problem associated with the design of spin stabilised satellites is the requirement to constrain the satellite to spin about an axis of greatest moment of inertia. This paper is concerned with the design of a nutation control system using a nutation sensor linked to a gas jet system which removes this constraint. The choice of nutation sensor is discussed and the effects of sensor misalignment are investigated. Stability criteria are examined which derive from the sensor/thruster system, energy dissipation and incorrect estimates of the nutation frequency. The results from theoretical analysis are supported by analogue simulation.
author is listed in the List of References. The design of most nutation control systems reported hitherto requires careful location of the nutation sensor to obtain correct phasing of the jet pulses. The design in this paper ensures correct phasing by appropriate electronic delays. It is considered that this approach eases the installation of the system into the satellite, and allows for updates of the nutation frequency late into satellite manufacture. This paper also considers in some depth the effect of misalignment of both the nutation sensor and the principal axis away from the geometric axis, and discusses a filtering technique to overcome the undesirable effects.
1. INTRODUCTION
2. NOTATION Spin stabilised satellites are required to spin about an axis of greatest moment of inertia for stability. To ensure that the spin axis is the axis of greatest moment of inertia requires careful mechanical design and it can place significant constraints upon the overall design. The ability to relax such constraints is especially significant for missions where the satellite is spinning during onlv the first few weeks of a mission lasting several years. The problem may be eased by introducing an active control system which senses the nutation and automatically activates a control actuator to reduce the nutation to a predetermined level. The configuration constraints placed upon a dual spin satellite to ensure passive stability are less severe than those required for a single spinning body. The analysis presented is applicable to such satellites provided that the nutation motion for the total system is considered and that the sensors and actuators are located on the spinning body. The approach adopted is based upon an understanding of the dynamics of the satellite. A criterion for control torque application is derived in a relatively simple manner for an asymmetric satellite. The use of gas jets to provide control torque is examined and the effect of products of inertia and attitude drift is considered. Rate and acceleration devices are considered as control sensing elements. Literature which has been found most useful to the
A,B,C F,G,H Ix, Iy, Iz ~ ~,
Ni
Moments of inertia about the satellite geometric axes Ox Oy Oz respectively. Products of inertia about the geometric axes Principal moments of inertia Inertia Tensor. A general torque Jet pulse length.
M
A generalised complex acceleration.
q
An auxilliary complex variable defined by equation 19.
rwnl /
k
w
n2
Acceleration of P. Pn
Nutation Period.
P
Value of parameter p at t
o
=0
K
Control gain defined by equation 20.
n,N
Integer variable. Laplace transform variable. Coordinates of accelerometer.
345
Lateral misalignment of accelerometer. Time delay. (i) Misalignment of gyro/ accelerometer (ii) Coefficient of W in destabilising torque. (Hi)
8 A \.1
\.1 1 '
\.1 2
n
LIt
Angular misalignment of gyro/accelerometer. Accelerometer sensitivity parameter. Auxiliary variables defined by equation 36. Nutation angle.
n $, 9,
1jJ
Satellite Euler Angles. Complex variable defined by equation (5) Filter damping coefficient.
~
I;
W n W n
CI= W
Nutation frequency. (B - C) ws / A
If a complex variable ~ is defined by I; WJ + ikw2 where k = .; wnl / wn2 where W = (B-C) W nl -A- s
(5) (6) (7)
W = (A-C) W (8) s n2 -Bthen it follows from equations (2) and (3) that (9) ~ + i w I; = M n
and
(10)
where
M
where
\
Nl/A
(11)
and
M2
NzlB
(12)
and
W n
.; W n
\
+ iM2
l
(13)
W n
2 From the definitions of the nutation angle (n), in Appendix I, and 1;, in equation (5) it follows that n ., 0 <=> I; = 0 (14) The basic control problem as posed by equation (9) therfore, is to find a control torque M to control I; The solution of equation (9) when M is zero is -iw (t-to) I; ~o e n (15) where 1;0 I; (t = 0) = wlO + ikw20 (16)
l
= 0)
(A - C) ws/B
where
W s
Spin rate
p
Modulus of
9
Argument of
T
Time constant associated with the Energy Dissipation. Phase error induced by sensor misalignment.
It follows from (15) that both wl and w2 are sinusoidal of frequency w ' n This frequency is termed the nutation frequency. Equation (15) implies that the transverse angular velocity vector and the transverse angular momentum vector rotate at frequency wn' in a
W n
wl O= Wl(t
= 0)
and w20= w2 (t
(17)
2
£
~
,I ~ I ~
W g
Gyro output.
Wgf
Filter output.
3. BASIC THEORY AND SELECTION OF CONTROL LAWS The system equations and control laws are derived initially for the case of a rigid satellite spinning about a principal inertia axis. Subsequently the effect of misalignment of the principal axis from the nominal spin axis is examined. Assuming that the geometric axes Oxyz are aligned with the principal axes, referred to Oxyz Euler's equations of motion are:Aw - (B-C) w w (1) NI 1 2 3
E
(C-A) w w = N2 3 l
(2)
(A-B) w w = N3 (3) l 2 If the satellite is spinning about its Z axis, assuming the spin to be about an axis of minimum moment of inertia, it may be assumed without loss of generality that A> B >C. If wl and w2 the transverse rates, are assumed to be small compared to the spin rate w3' then to first order if N3 = o it follows that (4)
negative sense, with respect to satellite fixed axes. The type of control envisaged is to pulse the control jets at precisely that time when the angular momentum is aligned with the torque vector provided by the gas jets. At this time the change in the magnitude of the transverse angular momentum vector as a result of a control torque will be a maximum. Equation (9) is used to determine when and for how long the gas jets should fire. The solution of equation (9), assuming M,i.e. the gas jet torque to be constant is -iw (t - to) en) iw n
(18)
(19) which can be written I; = 1;' + q where ~' is independent of M. The problem is now reduced to finding the optimum q to reduce I~I. As Iq is most sensitive to the magnitude of q when q is parallel with 1;' we require a control such that (20) q = -K ~ ; for some K If K = 1, q = ~' and I; (t) the nutation, is reduced to zero by the action of q. As the nutation angle will grow slowly as a result of any internal energy dissipation the divergence of the nutation is slow compared with the time constant of the control system. The control stategy can therefore allow the nutation angle or transverse angular momentum to grow to a preassigned value at which point the jets may be
346
fired according to equation (20) to reduce the nutation angle. A stability criterion is derived in Section 5 which formulates the maximum rate of energy dissipation as a function of the control system parameters. In order to solve for the required control M, q is written 2Msina/ q = -2a/ 2 (21) e 2 W
n
a=
where
W (t - to) =w 6t n
n
and 6 t is the jet pulse length. Solving equations (21) and (22) for M M = KPow n ~ (TI +8 - a /2) 2sin~72 e
(22)
(23)
where P and 8 are the real and imaginary parts of 1:;0 r~specti~ely i.e. i 8 E; = P e 0 (24) o 0 It is assumed initially, for simplicity, that one thruster only is available, so that the control torque is limited to one axis of the satellite. M is then written (25) M = Nl/A implying that, w~thout loss of generality, a torque about the x axis is assumed. Equation (25) implies therefore that:Kp
W
o n
2 sina/ and
(26) 2
sin (8 +1f-a/ ) o
2
=0
remembering that I cos (TI +8 - a /)I = 1 the control impulse is definedOrat~er than the torque. If K = 1 in equation (26) the control impulse will be sufficient to reduce the nutation angle to zero in one pul se. Wi th K < 1 then some nutation will remain after the pulse. This iR quite acceptable because a second pulse or even a stream of pulses can be applied, so long as the timing is correct, until the nutation angle is sufficiently small. If K > 1 there will also be some nutation remaining after the pulse. This is clearly inefficient since the control impulse is greater than that required to reduce the nutation to zero, and also if K is sufficiently large the remaining nutation will be greater than the initial nutation, implying instability. It is therefore, required that K ~ l,i.e
(27)
which define the magnitude and phase (timing) of the required control torque respectively. Equation (27) implies that the pulse must begin when E; has an argument 6 defined by:8= a /2-(n-l)'Ir ; n=O, il, ;1;2, ... (28) If time is measured from when w is zero, it 2 follows that the pulse must beg~n at t' seconds where, W t' = -a/ +(n-l)1I', n=O, 1,2, ... (29) In the c~se of the2 jet providing a positive torque about the x axis from equation (27) Pocos (TI+8"'l/2) must be positive. ) is positive, this implies lO n = 0, 2, 4, ... (30) W t' = -a/ +(n-l)~ n 2 Equation (30) implies that the positive x axis jet should fire Gt -a/ 2) /wn seconds after w2 If
The above argument can be repeated similarly for a negative x axis jet, and a y axis jet with the result that each jet should fire when the component of the transverse rate along the jet torque axis is at a peak and opposite in sign to the jet torque. This is equivalent to requiring that each jet should fire when the angular momentum vector is aligned, and opposite in direction to, the torque vector provided by the jet. Equation (26) defines the required magnitude of the jet torque. The control impulse is N 6t, 1 where 6 t is the pulse length (a =w 6 t), so n writing equation (26) as 2NI sin a/ 2 = A K wlOwn (31)
P o (=w
is zero, or on any subsequent nutation period later. The delay term (TI'- a / ) w contains a 2 n 0 a/w term which corresponds cv a 180 phase J1 sh~tt. Thus apart from the a/ term which 2 ensures equal spacing of the pulse either side of the optimum time, the delay causes the jet to fire when w is zero and when w is at a negative I 2 peak. If w is negative there is a further IO 180 0 phase change in the firing delay which still; implies that the firing should occur at a negative peak of w . Thus to summarise, the time for the positive l axis jet to fire is optimum when the pulse is centered on a negative peak of w • I
2N sin a/
2
<
Aw
(32)
lO
Wn or, dividing by 2N sin a/
Cw s 2
<
Nutation
threshold (33)
Cw W s n This in fact sets a lower limit on system accuracy for a given impulse. Effects of Products of Inertia If the principal axis OXp Yp zp are not parallel with the geometric axes Oxvz then the inertia tensor expressed in geometric axes will contain some non-zero off diagnal terms. For simplicity a rotation about the x axis of the principal axe s relative to the geometric axis is considered, in which case the inertia tensor in geometric axes can be written.
r ~ ~ -~ 1
o -F C (34) where (A, B, C) are the moments of inertia, and F is the product of intertia Emyz It can be shown that under torque free conditions the three satellite angular rates about the geometric axis, with zero initial values onw and w are l 2 given by:sin w 't wI 1: n Wn (35) \.I (l-cos w't) _ n \.12
347
(1-
where
cos w' t) n
(35)
BF !J
F
2
+C(A-C) (36)
\.. = A-C 1 B-F w' n
B
w30"
W n
l
B-C-2FBw A
2
30 "'wn 2
!JJll"Wn 2
It follows from equations (35) and (36) that, provided the products of inertia are small, the rates oscillate at approximately the nutation £requency,w · n The most significant result arising from the above equations follows from the form of the solution for w • In general when the response 2 of w is a combination of the effect of products 2 of inertia and a non-zero initial condition, the time variation of w will not be centred on 2 zero, but will have a mean value. (37)
The significance of this is that there is not 0 a simple 90 phase relationship between the zero and the maximum values of w ' so that the 2 maximum value can no longer be predicted from a knowledge of a zero crossing point. There are two possible solutions:(1) to utilise a peak detector network, or (2) pass the measurement w signal through 2 a DC blocking filter to remove the bias. The second approach is adopted as it is shown later, that such a filter can fulfil a dual role and remove the effect of a sensor misalignment, which also adds an unwanted bias to the signal. The filter should remove the bias and preserve the gain and phase at the nutation frequency. A band pass filter of the form 2sw s n
which is the same result obtained from a passive damper. The active system could constrain the satellite to spin about its geometric, non principal axis, if the bandpass filter is not used, but relatively large control torques would be required as they would have to continuously oppose cross coupling torques of the form F w ~. Generally it is suggested that it is sufficient to allow the system to converge to a state of spin about the principal axis. The control scheme as derived in the preceeding paragraphs is summarised in Figure 1. Here N = 1 if the jet has a sign which is opposite to that of the transverse rate at the time of the threshhold crossing. When the signs are the same, N = 3. Choice of Jet Thrust and Pulse Length In most applications the nutation control system will form a part only of an overall attitude control syst em and the nutation control system will make use of the control jets which are provided for the attitude control system as a whole. Equation (33) which can be written w ~t Cw W x Nutation 2N sin _.::n,--_ s n < 2 Threshold (40) defines a m~n~mum nutation threshold for a given jet impulse bit. It is therefore essential to ensure that the thruster system adopted is capable of providing the required system accuracy as defined in equation (40). Assuming that a jet system is available which is compatible with the accuracy requirement, there remains the choice of selecting the pulse length, assuming that equation (40) does in fact allow a degree of choice. In order to allow for system errors it is suggested that ~ t be chosen less than approximately 50% of the value which would imply equality of equation (40) i.e. ~t < 1
sin- l CWs wn x Nutation Threshold
wn
(41)
2N
(38)
is chosen because it is simple, adequate and will also attenuate any high frequency noise associated with the nutation sensor. The choice of damping coefficient s is not critical. Simulation results shows that a value of s = 0.2 represents a reasonable compromise between rejection of frequencies other than w ' and n the filter response time. Use of such a filter implies that the DC term remains uncontrolled. It can be shown that equation (37), which defined this term can be written !J F w = AW (39) 30 30 W = 2 j;"2 (B-C) where A is the angle between the geometric and principal axes. If the motion is reduced to zero apart from this term, the satellite will be in a state of pure spin about its principal axis. It is interesting to note that, apart from the deadzone equal to the nutation threshold, use of a bandpass filter will result in the motion converging to steady spin about a principal axis,
The exact choice is not critical. If ~ t is very small the jet puls e is effic i ent because all the impulse is delivered close to its optimum time; efficiency in fact is equal to sin et let.where et=wn t/2 On the other hand a small pulse could r e sult ~n jitter around the threshold value, although this could be overcome by introducing a two level threshold system and switching to a lower level when a higher level is exceeded. A small pulse length will also result in a longer acquisition time although usually this consideration is not expected to be particularly important. Overall it is considered best to select a small rather than a large value of ~t, i.e. to trade time for efficiency. Attitude Drift From Jet Pulsing Each gas jet firing changes the direction of the system angular momentum vector. Consequently the satellite spin axis direction will vary with time because the nutation control system will always attempt to align the spin axis with the angular momentum ve c tor. A means of minimising this effect is to fire two pulses eql1ally spacF!d half
348
x (or y ) not being exactly zero then the a8celeromgter would respond to the large centrifugal force associated with the spin rate. Choosing x (or y ) as zero, a provides a direct measu~e of 11~ z
a spin period from the optimum jet firing time. This is only possible providing the nutation period is long compared with the spin pe riod.
4. SENSING THE NUTATION Rate gyros and accelerometers are considered to be the most suitable type of sensor as they provide a direct measure of nutation which can easily be used to drive the jet control logic discussed earlier. Sensors which detect satellite motion relative to exterior references, e.g. sun or earth sensors, could be used but these require more complex on-boards processing to derive the nutation angle. Rate Gyros If t is measured from when is zero, under torque fre e conditions it follows t~at
Aw lO
11 (t=o) = 11 0
(42)
Cw
SI
and
11 (t) =11
CCA-B) 2 Wt 1 + A(B-C) sin n
oJ
(43)
J
It follows that by setting a threshold limit on one of the transverse rates, the nutation angle is contained within known bounds. The sensitivity of gyro output (transverse rate) per degree nutation angl e is illustrated in Figure 2 for some typical satellite parameters. Accelerometers Under torque-free conditions the satellite rates are given by:W = a constant (44) s
W cos W t n 10
(45)
sin W t
(46)
n
where t is measured from when w is zero. 2 It can be shown that a general point p , fixed in the satellite has an acceleration a (a ,a ,a ) -
where a
a
y
p
- wlOwnz o cos Wn t - w s2 x0 +wIO wszo cos Wn t
x
x
y
z
(47)
- wlOwnxo sin W t - W 2Yo-wlOws z sin W t n s 0 n -k-
(48) / W
a
z
-y
W
o 10
(
S
\k
+
W)sin W t n
C A
where
W
fw _w!!)
srs
x
k
(SI) 0
Thus a can be used in exactly the same way as a rateZgyro measurement of w or w but with l 2 a different scale factor A The se~sitivity of the acc e lerometer output (a in m/s ) per de gree nutation angle is shownzin Figure 2 for some typical satellite parameters. The effects of sensor misalignment are now examined. Gyro Misalignment If the gyro is misaligned so that its sensitive (input) axis is rotated away from the nominal Ox axis by a rotation of a about the z axis and B about the y axis, the output of the gyro, w
g
nutation cycle, and varies during a nutation pe riod from 11 up to a maximum of o B(A-C 110 A(B-C
k
(SO)
is
The nutation angle is equal to 110 twice per
W10 /
With
n
(4~)
+ xoW10Gs - :n)cos wn t It can be seen that apart ftom the terms in W in equations (47) and (48) which can be avoid~d by suitable choice of sensor location (x or o y = 0) that all three components of ~ are p~oportional to wlO(hence 11 ), vary at the nutation frequency, and the~efore are potential nutation sensors. The z axis component appears most suitable, since, if in the case of a (or a ) x y
(52)
provided that small angle approximations for a and Bare valid. Using equations (44) to (46) which describe the torque-free angular rate response it follows that the gyro output is w = 11 g
Cws
0
A
cos (w t+E) n
- Bw
where the initial nutation angle 11 0
(53)
s
c;;;s
and
(54)
E=a·/k
The effect of a , the angular misalignment about the spin axis, is to cause a phase shift of E= a / , and the effect of B ,the angular error about the k transverse axis is to cause a bias signal Bws or in terms of nutation 11 = BA (SS) b C Assuming an inertia ratio close to unity, equation (SS) implies approximately one degree of nutation bias angle per degree gyro offset. The bias error reflects itself directly as a nutation angle error, whereas the effect of the phase shift is to cause an error in the prediction of the peak of the transverse rate which is requir ed in the control logic discussed earlier. The phase error and bias are plotted as a function of the misalignment in Figure 3. Both of these problems can be overcome by filtering the gyro output with the band pass filter, which is designed earlier when the effects of products of inertia are considered. The filter is only useful of course provided that the bias signal does not saturate the gyro. Accelerometer Misalignment The effects of both lateral and angular alignment errors are important here. It is assumed that the accelerometer is mounted away from its nominal location P = (x , 0, z ) to P where by (ox,n ay ,o a z) 0 o
0
0
z ) o
349
(56)
and additionally it is assumed that there is an angular misalignment of accelerometer sensitive axis away from 0 by an angle a about 0 and B z x about 0 , where a and B are small angles. The output 6f the acceleromet er in t erms of the components of the acceleration of P, a (a,a, a) i s therefore p x y z a = a - a a + Ba (57) sz z y x Using equations (47) and (49) with P defined by equation (56) it follows that , a
where 6 ,
11+ n
sz
ACOS
0
=-
a bias error
(
W
s
W
n
t
e:)
-
2 x B
(58) (59)
0
If the change in nutation angle per jet pulse is greater than the threshold then the system is inefficient, as the transverse rate will have been r e duced from its threshold through zero to a rate of the opposite sign. If the change in nutation angle is greater than twi ce the threshold, then the system is unstable because the nutation following the pulse is greater than the initial nutation . Therefore, to ensure stability it is necessary that 6n < 2 x Nutation Threshold (65) but to retain a reasonable stability margin and to obtain an efficient system it is necessary that
x
and
A
where it assumed that oz
(kWs - wn )
o
Cs -:n)
Cw s
(61)
is neglected compared
o
to z and terms like ao y are neglected compared withOt erms like x B . ~rom equation (60) it is appar ent that theOphase error can be reduced by mounting the accelerometer in the Oxy plane; then z ~ 0, and we have o
- oy o Xo
(w
s + kW) n
kw s
n
sin
Equation (67) sets a limit on the jet impulse bit ( N. 6t ) for a given accuracy and system parameters wn and ws' Error in the Nutation Frequency A prior knowledge of the nutation frequency (w ) n is r equired for the jet control logic. The logic r e lies on predicting a peak by counting either ! or ~ of a nutation period from a zero crossing of the filtered gyro output. The delay time counted from the zero crossing is Ip 4
or
5. STABILITY CONSIDERATIONS There are four main aspects to be considered regarding the system stability, viz:(i) The size of the control i mpulse compared with threshold set by the ac cura cy r equirements. (ii) An error in the estimate of the nutation frequency. (iii) The rate of energy dissipation compared with the size and number of control thrusters. (iv) The effect of delays associate d with the sensors and actuators. The Control Impulse It is shown earlier that the change in the transverse rate per jet pulse ( 6w ) is given by 2N
=-
Aw
n
2N Cw W s n
(62)
W
Th e bias error which is caused by the angular misalignment and the phase error, which is caused by the lateral misalignment, ar e plotted in Figure 4 for typical satellit e parame ters. The e ff e cts of the bias error can be overcome in the s ame way as with the gyro bias , by using a band pass filter centered on the nutation frequency. The effect of the phase error is likely to be insignificant; for the satellite parameters assumed in Figure 4, lcm. offset caus e s a ne J ligible phase shift of approximately 1 degree.
6w 1
(66) Nutation Threshold W 6t < Nutation Threshold n -2(67)
<
6n i. e.
(63) 2
where the torque is applied about the x-axis. The corresponding change in nutation angle is A6w l W 6t 2Nl n CW-w sin 6n = s s n
cw-
lp 4
n
-! 6 t
(68)
This time (t ) can be in error by up to ! Pn d actually ~ncreases . . before the jet t h e nutat~on angle and the system becomes unstable, although the system will become inefficient for errors in t considerably less than this. To obtain r easgnable efficiency ( ~ 90%) it is necessary that the error is less than approximately 1/8 Pn. This requires in the case for tdl the error in the estimate of Pn must be < 100% of Pn for td the error in the estimate of Pn must be < 30% or Pn. A slight modification to th e control logic could reduce the effect of large errors so as to cause loss of efficiency only instead of instability. The sign of the control torque is decided by the sign of the transverse rate when it crosses the threshold. If instead of this the choice of po s itive or negative jet is decided by the sign of the transverse rate at the start of the jet pulse then the jet pulse will always fire so as to r e duce the transverse rate. The above criterion will still be necessary however to ensure reasonable efficiency. Minimum Acceptable Time Constant Associated with the Energy 0issipation There is a necessary criterion for stability arising from the requirement that the increase of nutation angle occurring between jet firings due to energy dissipation must be less than reduction in nutation arising from each jet firing. Suppose the nutation angle in between jet firings follows the law:I) e
n (64)
n
tiT
o
(69)
where T is the time constant associated with energy dissipation. It follows that
2
6n = n-I) = n (e o
350
0
tiT
-1)
(70)
where bn is the change in n in a general interval (O,t). Now suppose the jet impulse bit J corresponds to an angular momentum increment bh where bh = J R, (71) where R, is the jet moment arm. The change in nutation angle per jet firing is then given by bn
I Cw
JR,
(72) s
where Cw is the spin angular momentum, inertia x spin r~te. The number of times the jets can fire in a nutation cycle depends on the thruster configuration. Each jet can fire once per nutation period, therefore the time between jet firings is given by t
= 2IT/
Nw
(73) n
where N is the number of possible jet firings per nutation period. Therefore, in order for the change in n from one jet firing to the next to be less than the change from the jets we require 2'IT/Nw T ' \ (74) no ( e n -~
T
>
2'IT
(1 +bnl
(75) no
where bn is the change in the nutation angle from the pulse, i.e. the minimum T depends upon how often the jets can be fired, the jet impulse bit, the nutation frequency, and n the largest expected initial condition. 0 Sensor and Thruster Delays The effect of these is similar to an error in the knowledge of w n ,i.e. a delay whether within the signal or introduced by the jet itself has the effect of causing the thruster to fire off peak. The delays introduced by sensors and thrusters are expected to be very small compared to the nutation period « 1%), and are likely to be swamped by the errors arising from inaccurate estimates of the moments of inertia. They are unlikely therefore to cause stability problems or loss of efficiency. In theory the same criteria holds as in the previous section, i.e. the time delays should be less than approximately 1/8 of the nutation period. 6.ANALOGUE SIMULATION RESULTS In order to confirm the basic system design, and to complement the theoretical analysis, an analogue simulation of the system was set up on an EAL 680 analogue computor. The simulation (which models a 650 kgrn satellite spinning at 60 rpm) includes 3-axis dynamics, kinematics (see Appendix I for equations) a negative damping term to represent the effect of energy dissipation and a pair of gas jets to supply positive and negative y-axis torques. Two examples of the simulation results, which represent two inertia cases; (A,~,C) = (205, 195, 185) and (220, 200, 160) kgrn , are illustrated in Figures 5 and 7 respectively. Figure 6 illustrates the kinematic response from an initial condition, the system parameters are as in Figure 7 except a 1.0 pulse is used.
7. CONCLUSIONS The design of nutation control systems using gas jets as control actuators and a rate gyro as a nutation sensor is derived for a spin stabilised satellite spinning about its minimum moment of inertia axis. The control law arranges for the jets to fire a control pulse at a peak of the transverse rate so as to reduce the rate and hence the nutation whenever the nutation exceeds a preset threshold. The design is verified with a detailed analogue simulation, which shows that with the chosen laws the system performs satisfactorily. The use of both rate gyros and accelerometers are examined as potential nutation sensors, and both are found to be suitable. The control system is found to be insensitive to misalignment errors of both types of sensor. Apart from small lags which occur and which are shown to be insignificant, the main effect results from the response of sensors to a component of the spin rate. This causes a bias signal on the output of each type of sensor which, provided it is not so large to saturate the instrument, can be filtered out by a band pass filter. The response of the system is analysed when the principal satellite axis is not coincident with the nominal spin axis of the satellite. It is shown that using the same filter which is required to remove the sensor bias induced by misalignment, that the system can converge, to within a dead zone, to a state of steady spin about the principal axis. If it is required to force the satellite to spin about the z axis and not the principal axis then the band pass filter should not be used, and large continous control torques are required. An expression which determines the maximum rate of energy dissipation which is acceptable to the control system is developed. For the parameters assumed in this paper tile minimum time constant is 141 seconds which is very short compared to the value expected in prr.~ tice. There are two other main stability criteria which are developed in Section 5. The first sets an upper limit on the minimum gas jet impulse bit acceptable for a given nutation accuracy. For the parameters assumed in this paper if it is 0 required to keep the nutation to within 0.2 , the ffi1n1mum jet impulse bit must be less than 8.1 Ns. In fact, this impulse bit is quite large for gas jet systems indicating that higher accuracies should be easily achievable. The second criterion requires that the nutation frequellLY should be known to within a given accuracy. The exact requirement depends on the actual system parameters and the precise control strategy adopted, i.e. the number of jets available and the delay time between the zero detection and the start of the jet pulse. 8. REFERENCES GRASSHOFF L.H. "An On-Board Closed Loop Nutation Control System for a Spin Stabilised Spacecraft". Journal of Spacecraft and Rockets, Vol 5, No 5. May 1968 pp. 530-53fi. ~ARNER
351
H.D. and REID H.J.E. Jr.
"Flight Test of a
Pitch Control for a Spinning Vehicle". July 1963 N.A.S.A.
TN D-1922 ZERO DETECT OR
PULSE GENERATOR
DELAY !:!~ - ~6t
6t
4
NEER J.T. "Intelsat IV Nutation Dynamics AIAA Paper 72-537" 4th Communications Satellite Systems Conference April 24-26 1972. DEVEY W.J. "An Active Nutation Control System for Spin Stabilised Satellites". M.Sc Thesis, The Hatfield Pol y technic November, 1973. APPENDIX I Fg.l
BlOCK DIAGRAM
SYSTEM
DEFINITION OF EULER ANGLES AND NUTATION ANGLE Euler Angles In order to descLibe the motion of the satellite spin axis pointing direction it is found most us eful to define a set of Euler angles which define the orientation of an axis system Xl' Y , Z, which is fixed in the satellite excep! fot the spinning motion. The angular velocity of this axes system ( n ) is given by
n=W-
•
where
'\
//
E 02,O
(1)
sinw3t +w2 cos w3 t
1,2
f
0,8 0, 4
j;
0'
V>
DEG/SEC
'1-' 1).
u,. =
//
;j/~ /
G'iRO OUTPUT -
liil
I
I
/
1,6
III
ACCELEROMETER - - - /
l iil
/'
m/si
OUTPUT -
~ ~. ('"
Q,.
/
-
't)%.1).
~
195;
x . = 1m.
= 2s 'Is
'->s
CaS£' ( j)
JP /
A
~
205 , B
C
~
185
Kgm'
220; B ~ 200 ; C ~ 18)
K9 m'
Cose lii)
A
0.1 0.3 02 NUTATION ANGlE 1DEG61.
~
F1Q2. SENSOR SENSITIVITY.
(3)
(wl ' w2 ' w3 ) is the satellite angular velocity vecr'r expressed in satellite fixed axes. W
Nutation n (t) =
2,4
~t3
(~ .~)~
where W is the angular velocity of the satellite fixed axes, and k is a unit vector along the spin axis directIon. Such an axes s ystem responds only to the satellite transverse rates i.e nutation. The Euler angles ~ ,8,W define the orientation of this axes system with respect to inertial space, and are defined by requiring that the transformation from inertial space into this axes system is an ordered set of rotations, w ,8,~ about the Z, Y and X axes respectively. For small angles, ~ and 8 they define the r o tations of the spin axis away from its ~ominal direction, about the two inertially fixed X and Y transverse axes respectively. It follows that w cosw3t - w sin w3 t (2) ~ 2 l 8
lil
'~ u :-~
Hi
9
III
le n is defined by + B2 w 2 2
(4)
W
/
/
0.6
J !f !/
0.4
O. 2
/
Ii
PHASE ERROR v MISAUGNMENT OF GYRO ABOUT SPI N AXIS
BIAS ERROR v MISALlGN/oENT ABOUT A TRANSVERSE AXIS
/
/
z
nl)
l- V/I"I
1,0
0.8
It is therefore the angle between the satellit e principal Z axis, 0 and the angular momentum vector. It can be ~hown (Ref.l) that under torque-free conditions this is equal to the angle be tween the mean direction of the 0 axis and the current direction of the 0 ~xis.
(i)
L'L
1,2
/
/
/
~// /
J
Case III A
A
0.2
~
205
B , 195
C
~
185
K9m'
220
B , 200
c
"60
K9 m'
Case Ill)
O~ 0.6 0.8 1.0 MI SA LI GNMENT E~ I DEGSJ
~
SPIN RATE
Fig.3 ERRORS RESULTING FROM GY RO MISAlIGNMENT.
352
8) RPM
NUTATION SIGNA L PHASE ERR OR
v LATE RAL
MIS-ALl GN MENT -
-
N U TATl ON AN GLE BIAS ERROR
1.2
V ANGULAR MISALl GNMENT - - - -
l il
1j)
--;-/ OB
V
~;:; 0,4
-
~~
l...- f" V
0.2
~
~V
w
~
~
~
V
CA SE ( i J A · 20S 8 · 19 5
iI)
W
M
U
CA S-EI; i) A 220 B 200
C .. 185 Kg rn 2 C 16 0 Kgm 2 SPIN RATE - 6 0 RPM
...--- V
U
~
~( .... 1,'i1
~
LATERAL M I SAU GN~ N T ( ems) ANGUlAR MISALIGNMEN T [EGS . 11f'
Fig 4, ERRORS RESULTI NG FROM ACCELEROMETER MISALlGNME NT,
' OS"'l ;~~T • mA~EBSE RATE -0 5 r / tJ
"
(W
r'
I
1
'0 5
, I,
1
-0 5
f
-0 5 r / a
! ~"'fV
) -0.5 r / .
t;-~Ii;-i'"'~ -'4i I
i r-l't
I'! ~
,
f ":
~
'0 , , / .
i
~
I
I
1 I
,E CONal
l -- - .
~ TRANf'VERSE RATE (W I )
-0 5
!
-,
,_n
_-.....-..:...~_ ' __
rl e
_
I
L_ _ __
'0"/'1~ :.;Tj0
I
!A
"
TRANWE_J: RATE (1612)
~ ~=ERSE ~~~~)____.. __
-0 Sri .
'O'3"'·1 ~ ··- '-.'; ,- -.-- :' .. i '
t
•
,
i
~ F~;E~ ~TPUT IWsfl
:~-,
-<1.33,/ .
i~---....._
_L'
I
I!i
.Ll.LJ .J J_l J....LL.L
LJ
! i
1
I1
GYRO OUTPUT ( W ) II
G YRO OUTPUT (01 ,)
- 0. S r l s
1 '
Ill'
1 ~~ ...:....:_LJJ-'-.Li .l.1-,-1.. ..c. ...l. _.
. a. S r/ .
...
FrLTER OUTPtrT ( C&I&f1 _.: ~ _ _ _L _
'O.33r/s -0, 5 r / e
!I
: ..
.
~
i'"""!1 1
··..'. I "0. 25 rl . t
11l
-0 . 5 r / e
...,. f
- 0001"
J
vivvvI*J.~ VI' " : ' , .. _ _ ---'
i :
iJi 1,I,l'l
' 0, 5 ' / ' }
FIT.TEll OUTPUT (W.,) "
i
+
AA ' 1fo "'... T.",. T,-+ '· ... '
2IJi)!M' , E C0ND8
I!
iT;!; . . ill iT ,in*
-0.33 Ti e
!
r/e
. TRA !oIrVElIB E RA TE (""2'
---+ •• ' .
I : f+~'IAi!o; W'
TRAMJVE RSE RATE (Wl)
---,---,--L-...l..i--,-~~
- 0 . 33 r/ .
I
-t ' -4 :'-1l 'ijlAJl ":'ib\ ;;'1\I\1
)
'O" " f ~ --' ! ~ - OS T/ a
t
f
.
_
~ _ .. --- - .---t-----.- G YRO ~ :.PtrT (w ,)
_~l _'~ __ ~
l_.__ '_
~;;:-;:;;;':~;;:;~~~ :
'/"1 ~ ' I4-i+
. . . . ... , / " - 1. 7"
~UUR ACC'~EIlATION
("'31 ..
IJWfITJ
11; ,. , . It , I l1. __ L~_ -'
O~~ _1 , . ,
.1.- 1
~GL~~~ I -~--, !
"1 7~
I
~
I ,
,
J
~
,
j -,~~~~+4~~~~~+-H t
.
:
. 1 7
-_~'::!~~~~1JJ...:lJJ...lJ..l
--
- 17 Nm
NOMINAL CASE
WITH
GYRO MISALIGNMENT
WITH PRODUCTS OF INERTIA
(iil
(i)
ACQUISITION
353
(iii)
FIG.S
PROJE CTION OF TfP OF SATELLITE Z ."\XIS Ol\TO S PACE-FIXED TRA?\'SVERSE PLA!-.'"E
RESPO?\"SE AFTER
F mST Pt;l.SE
1, ,,
-x
TWO LONG JET Pl' LSES ("SED a t - 1.0 SEC
e
FIG .6
RESPONSE TO AN INITIAL TRAr-8VERSE RATE
~·~f ~ ' ~~~~
~,~,/. r .O ,hl·l -0. 1
r/ •
.""/'1,-
. . "·'I ~~~~ - 12 5 r / •
.... i5
r/.'
"' ' ' ''1 ~~~~ ~O , .
r/.'
~' -"'/] -ttiiTttttti*t+-t+++tt-H++++++~~
I
~•."'.r/" -, ,i'
l
,
t , _, ,,,,
-LLL--'-..1.-'-'--'--LL-.I....L
-LL.LL.L..Ll--L.l--'--LL.LL..Ll--L.l--'--LLLLL.L...LliJLLLLLL...Ll-"--.:~~
-""I ~+++++++t+++--t1~~ . 1f . .
eeNfROl WITH ENERGY
OISSIRtJ~ PRESENT
354
FlG.1