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Direct Digital Attitude Control with a Double-Gimballed Momentum Wheel
DIRECT DIGITAL ATT ITUDE CONTROL WITH A DOUBLE-GIMBALLEP MOMENTUM WHEEL G. Hirzinger, Dr,- lng. Th. Lange, Dipl. ~ l ng, Deutsche Forschung s- und Versuchsanstalt fuer Luft- und Raumfahrt e.V. Institut fuer Dynamik der Flugsysteme 0-8031 Oberpfaffenhofen I Post Wessling F.R. Germany SUMMARY A direct digital control scheme is proposed for double-gimballed-momentum-wheel (DGMW) attitude control systems with stepwise gimbal actuation. It is shown that t he use of stepper motors leads to some scheme of pulsefrequency modulat ion (PFM). Presumi,ng that an on-board computer is available for rea l izing state estimation and control, a general design concept for compu ter-controlled PFM-systems is developed and appl ied to DGMW-systems . The theoretical scheme providing high precision control is verified by a physical simulation including an air-bearing simulator, a DGMW with stepw ise gimbal actuation and a process computer as a controller.
175
G. Hi rzinger Th. Lange 1.
INTRODUCTION
Large telecommunication satell ites to be built in the future wi 11 be complex systems with several antennas and large flexible solar panels in order to supply the considerable amount of electric energy needed for an increased communication capability. Required overall beam accuracies may be less than O. I degree. Clearly a sophisticated three-axis stabi 1 ization system wi 11 be used. In a geostationary orbit the disturbing torques are mainly due to solar pressure acting on the panels and thus are composed of secular and cycl ic (zero mean in inertial coordinates) terms. Using a reaction jet control system would require continuous mass expulsion to maintain precise attitude control and so for long time missions of ten years and more mass required would become excessive. A momentum exchange system. however. simply exchanges internal and external angular momentum to compensate the cycl ic disturbances. and only secular disturbances need be counteracted by other torques. for example gas jets which may be used for station keeping anyhow . This is one of the main reasons why double-gimballed momentum wheels seem to be the most promising kind of stabilization system for complex communication satell ites with high pointing accuracies. Though they may consume more power than mass expulsion systems because of their continuous rather than intermittent operation. this electri'c power will be only a fraction of that needed by the communication system and so does not represent a severe shortcoming . As is well known. a double-gimballed moment wheel allows to exert control torques in 3 axes by varying the wheel speed (pitch control). and by torquing the gimbals to rotate the direction of the momentum vector (roll-yaw control). It is assumed in this paper that an on-board computer is available for direct digital control. It is furthermore assumed that stepping motors are used as gimbal actuators as they a)
al low to perform small gimbal steps precisely and rei iably.
b)
ideally fi t into a di rect digi tal control scheme.
This kind of gimbal acutation. however. leads to some scheme of pulsefrequency modulation. Though pulse-frequency-modulated systems are very important especially in space appl ications. no general design method is given for them in the literature. So the main aspect of this paper is to show how modern control theory can help to develop general design concepts for computer-controlled pulse-frequency-modulated systems. Consequently the technique proposed is not restricted to the design of a DGMW system with stepwise gimbal motion. but is likewise applicable to other PFM-systems. Attitude estimation is assumed to be provided by a Kalman-Filter that makes use of any measurement available to estimate the state of the complete dynamical system involved. Surely infrared sensors will be employed for providing roll and pitch information. Due to the momentum bias strong coupling exists between roll and yaw and allows to estimate yaw from roll measurements. Nevertheless direct yaw information from star trackers or radio-frequency sensors would be advantageous. Clearly pointing accuracy is 1 imited by the estimation errors. but in this paper emphasis is layed upon show.ing how optimal use is made of the estimated state under the supposition of computer-controlled stepwise gimbal actuation. 176
G. Hi rzinger Th. Lange 2.
SYSTEM DESCRIPTION
Let us briefly derive the equations of motion for a rigid satellite with DGMW in a geostationary orbit (see Fig. 1). In body fixed coordinates the total angular momentum H writes (including the wheel bias but neglecting the gimbal and wheel moments of inertia).
H
-[
x + cos Yx siny z
w
"~
y - cos Yx cosYz HB
w
z - sin Yx HB
w
""
where Ix' Iy' I z w x' wy' Yx' Yz HB
r
x + yz
w
Iy wy - HB I
"J
(1)
z wz - Yx HB
ro 11, pitch, yaw moments of inertia respectively roll, pitch, yaw body rates respectively
W
z
-
roll, yaw gimbal angle respectively wheel momentum bias
n.J\T~
YAN GIUE.AL
~OLL
GU;SAi,--
VELOCITY
y 'I.
t
i> YAW AXIS
GEOCEtlTER
flg~
1
Reference Coordinate F'ramc
The second part of (1) holds for small gimbal angles and wi 11 be used in the following. From the kinematic relations the angular velocity w refers to the Euler angles as follows (assuming again small angles and angu 1a r rates) :
(2)
where 177
G. Hi rzinger Th . Lange
roll Euler angle
~
o
=
~
=
pitch Euler angle yaw Euler angle orbit rate (= 7,28 . 10- 5 rad/sec)
W o
The spacecraft's motion is determined by Euler's equation
where T is the external torque vector, d/dt( } denotes the time derivative with respect to the inertial space, while (') denotes the time derivative with respect to the spacecraft coordinate system. Substituting (1) and (2) into (3}, 1 inearizing the nonl inear set of equations and transferring the gravity gradient torques to the right hand side Ill, we arr i ve at Tx
Ix ii + a
T y
I
Tz
I z iJj + c
y
o+
~
+ b ~ + Yz Ha + W o Ha Yx
3 Wo2 (Ix - I z) o - Ha ~
+ d
~
(4)
- Ha Yx + W0 Ha Yz
where a
=4
b c d
=
2 W0 (I
Y
- I z) + Wo Ha
Ha - (I x - I y + I z) Wo W Ha + w02 (I y - Ix) 0 - Ha - Wo (I - Ix Y
-
I z)
Apparently the pitch motion is decoupled from roll-yaw and approximately represents a simple double integrator to be controlled via the wheel acceleration (contained i n Ha) . So we fully concentrate on the roll-yaw dynami c s which write in state space form:
o -a
~
o
o
o
o
-b
o
q>
T;
o 0 o -.!!,e
o
Ix
o
o
o
o
-d
r;
-c
T;
o
o
o
o
o
o
o
o
o
o o
o
o
o
o - I --
z
~
~
Yz
: [::l
o
:J
o
o
(5) 178
G. Hi rzinger Th. Lange Obviously the gimbal angle rates are the control variables . Note that this linear model would indicate complete controllabi lity, though from physical considerations the nonlinear system cannot be completely controllable. The gimbal angles may therefore be considered as uncontrollable state variables . Remember that we assume an on-board computer to steer the gimbal step motors as specified by a digital control law. 5tepwise gimbal motion implies that the gimbal rates and so the control variables occur in form of pulses. The gimbal step size and so the pulse areas are fixed, but the instants fo their occurrence are arbitrary apart from a certain dead time 6t between two pulses . So we have to handle a typical pulsefrequency modulated (PFM) system. In state space notation a 1 inear time-invariant single-input system
(6) is transferred from an initial state ing the transition equation
~(T) - efT ~(O)
+
T
f
~(O)
into a final state x(T) follow-
F(T-t} !;.S. u(t} dt
(71
o
Assume that u(t) is a pulse of width
+0
u(t}
T
T
for T1 - "2" t
occurring at t <
T 1 (0 < T 1 < T):
T
T1 +"2 (8)
o Then for
T
~(T) =
else
very small compared with the system time constants we get FT
F(T-Tl)
e- ~(O) + e-
where
S. T
h
f
(9)
T
"2 h
f
u(v) dv T
- "2 As mentioned in the introduction, PFM-systems are nonl inear systems and no general design techniques have been developed for them despite their practical importance. Well known techniques as integral-pulse-frequencymodulation (IPFM) rely on the assumption, that a pulse of area h occurring in the middle of an interval (Tl = T/2) causes the same final state as would do 11 constant input of same area u(t} "hiT. It is easy to show that apart from special cases (e.g. double integrating plant) this assumption is seriously hurt for greater T. So let us search for a more general design technique which in addition fits into a computer control scheme .
179
G. Hirzinger Th. Lange 3.
ON A GENERAL DESIGN CONCEPT FOR COMPUTER-CONTROLLED PFM-SYSTEMS
Computer control of a PFM-system implies that any T seconds -- where T is a fixed sampl ing interval -- measurements of the system outputs are processed (e.g. in a Kalman filter) to compute a sequence of control pulses for the next interval. If we presume that the computer should be employed with control tasks as little as possible, the questions arising are how large T can be made and how the pulses should be arranged. To ask these questions we go through the following considerations (for simpl icity in the single-input case): Apparen~ly we cannot assume that we might vary technique to be discussed controlled system show up pulse of arbitrary area h terval? A simple way to control; that is to say, a performance index
"~ ~. xT 1 k=O --...+
change the pulsed character of control. Yet the pulse area (height) arbitrarily by some later. Up to what sampling time T does the good performance, provided that one control may occur in the middle of each sampl ing inanswer this question is by means of optimal for different sampling times T we minimize
+ -2 uk r
g --... x'-+l
subject to the discrete system equation (see eqn (9) with Tl ~(O) = ~, ~(T) = ~+1)
(10) T/2,
(11) where A
b
--0
=
FT 12
f';
.ll.
T
Note that the discrete control variable uk has been chosen as hk/T to ensure that for doubled sampl ing interval a double pulse area is admitted. The prefactor T in eqn. (10) is due to the fact that for doubled T the number of contributions in the sum is halfed. Q and r are chosen preferably so that the maximal allowable deviations in the interesting state variables (e.g. roll and yaw angle) cause the same costs as do maximal control ampl itudes (e.g. gimbal angle rates). law uk =:;,.T ~ and in order to be independent of some special initial state -- computes as
J 1 is minimized by a linear control
J 1 = trace(~)
where the cost matrix
(12) ~
is easily computed by an iterative procedure 13/.
Clearly Jl increases for increasing T, and by starting with a smallest sampling time T = T, we practically compare with the ideal istic case that pulses may occur at any time (as the dead time 6t is impl ied for T > 6t only). However recal I that we assumed we might realize any pulse area hk (of 180
G. Hlrzing er Th. Lange Though this course only within a certain control amplitu de constra int). red by conside T each for ion assumpt the h approac we ble, impossi is ntal gimarrangin g thi element ary pulses of given area h* (e.g. increme area hk of pulse desired a Now . 2 fig. in shown as bal steps 6yx, 6y;) pulses which is approxim ated by an integer multipl e j of the element ary are set at times ti
+ i . 6t
= ;
where i =
- lJ..:..!l 2
t ' ... ,
i = j
=
.L:.!.
-2, -1, 0, 1, 2,
-2, -I, I, 2, ... , hk integer ( h* )
I I
t
I I
I I
I I
even
for
sampling period T I I I At At I At
... .....
At
for j odd
2
,
.., I I
At
lA!. I
2
I I
T
times of eventual unipolar pulse occurrence fig.
2
Propos ed arrange ment of pulses (j
4 here)
the compute r T should be an integer multipl e N of the spaeing 6t so that pulses) of number desired the to ng (accordi r re9iste a" set only "has to which is sensed via an externa l clock. The actual input vector
~(j) computes as
.L:.!. 2 1:
i=1
-1
b. + -I
1:
'-1
b.)/j
for
odd
(13)
-I
i=-(.Lf!-) 181
G. Hi rzinger Th. Lange
1
-1
2
L: b.
L: ..~)/j
even
(13)
where b. denotes the input vector for a pulse occurring at t to the-~elation F(I- ot ) I( r ot) - 2 FT/2
Due
i=1
-I
+
i=
2
~+e
e
for
_.L
~~2e-.~
holding for small o t, the actual input vectors b(j) may be expected to be close to~ . Of course the degree of misalignment would increase with increasing j. Therefore -- assuming that all j ~re equally probable -- we define a new representative input vector ~ replacing £0 :
b
~
~
b{j) . j)/ j j=1 j=1
=
(14)
So we have replaced the original, unattainable assumption of arbitrary pulse areas by the practically implementable scheme just described. But thereby we have introduced stochastic errors. Presuming again that all admissible control ampl itudes are equally probable, these errors are composed of a)
a quantization error ~lk (in general this is the main error) with zero mean and variance matrix
v
-1
b)
= _1_ (b h*) T (b h*) Jl2' - T - T
a input-misalignment error ~k with zero mean (due to the def i nition of eqn (14) and variance matrix 1
Y..2
=
N
N+T .Ll J=
A measure for the influence of these errors ~lk and ~~ (which occur on the right side of eqn. (11» on system performance I S given by the expected value J 2 (see /3/) : (15) Eqn . (15) presumes that the vectors ~Ik and ~k are uncorrelated. the same linear control law i s optima for Jl and J 2 .
Then
Though we must not add Jl and J2 due to their different defin i tions, a plot of J 1 and J 2 versus sampling t ime T allows to decide on a sampl ing time To for which good system performance can be expected without employing the on-board computer unnecessarily often . Fig. 3 shows a typical plot of this kind. The increase of J2 for small sampling times should not surpr i se as for small T the assumption of uncorrelated errors does no longe r hold and so a simple 1 inear controller would not be opt ima 1.
182
G. Hi rz i nge r Th. Lange
IJ)Ssib~
choicf of sampling period T
fig. 3
Typical plot of J 1 and J 2 versus sampling time
The results of this section ·are immediately appl icable to multi-input PFH-systems (adding the variances of the additional input erros in eqn. (IS)). So we summarize and generalize: It has been shown, how by proper arrangement of pulses a pulsefrequency modulated system can be transformed into a I inear discrete-time system
~+I
=
~~ +!~
(16)
and how an appropriate sampl ing time To can be found. The main difference to a normal discrete system may be seen in the quantization error 6~ due to the fact that only an integer number of pulses is possible. Ways to overcome this shortcoming are shown in the next section .
4.
CONTROL PRINCIPLES FOR STEPWISE ROLL-YAW GIHBAL ACTUATION
In the real appl ication the basic roll-yaw dynamics as developped in eqn. (S) have to be augmented by controllable subsystems (motions of flexible arrays) and uncontrollable subsystems (disturbance torque models, e.g. the mainly sinusoidal solar pressure). A complete discrete system description implying stepwise gimbal motion in a scheme as discussed in section 3 may be written in the form
183
G. Hirzinger Th. Lange
where ~
denotes the set of controllable state variables (~, ~, generalized coordinates of the flexible parts)
~, ~,
y
denotes the set of uncontrollable state variables (gimbal angles, modelled disturbance torques)
u denotes the control pulse areas per sampl ing time, that is here
" •
T~ [:::J
(18)
The computed ideal values of 6yx and 6l z are approximated by an integer number of incremental steps 6y= and 6yz arranged as shown in fig. 2. Those torques Tx , Tz which are known, but cannot be dynamically modelled, as are desaturation or station keeping thruster pulses, antenna steps etc., are introduced with their individual kind of actuation expressed by the input matriees Lx and Lvk' Let us discuss two control principles : a) Assume that
~
~ = .£.(~
is the desired set point.
-
A I inear control law
~s)
(19)
may be determined for the controllable part of system eqn. (17) by applying well-known design techniques for ensuring favourable closed-loop dynamics (e.g. decoupling of roll-yaw motion). A correction term 6~ should be added in order to counterbalance 1. the influence of the uncontrollable state variables and external torques which would cause a deviation (20)
2. the influence of the Quantizat ion error 6~_1 which was made in the preceding interval. After ~_I ha, been computed and the appropriate pulses set, we know the quantization error 6~_1 and so the state error
6~
= :[
6~_1
This state error propagates into a deviation
6~+1
where (21)
Though these adverse influences in general cannot be completely cancel led, they can be counterbalanced in the sense of minimizing an 184
G. Hi rzinger Th. Lange eucl idean norm of 6~+1 by a correcting control o~:
Of course, the matrix products occurring in eqn. (22) should be precomputed. Note that desaturation torques are seen as kind of disturbance torques which are known and so may be anticipated by proper gimbal motion. This implies that the thruster pulses, if possible, should occur in the middle of the sampling interval and theIr pulsewidth should correspond to an integer number of gimbal steps. b) The correction of a 1 inear control law as just discussed is not necessary, if one immediately refers to a reference state. Then for the k-th sampl ing interval the control ~ is computed thus that a norm L of the error between projected state ~+I and reference state 3k+1 is minimized: (23) Expressing
~ yields
~+I
~inL
by the right side of eqn. (17) and minimizing Lover ",T
- -I -T
I.! ~!)
! ~ (~~
+ ~ ~ +
-
.4 lK - ~+ I)
(24)
Eqn. (24) holds only if all saturation constraints -- as given by maximal gimbal rates -- are met; otherwise additional computations would be necessary. However for systems with fairly well-known disturbances it is no problem to keep the actual state close enough to the reference state. In the additude hold mode the reference state equals the set point state~. Yet assume that a set point change -- due to a new commanded roll angle -- occurs. All state variables should remain in rest apart from roll angle ~ and rate ~ which should follow a desired second order transient. In a) this transient is immediately determined by the feedback law . Here it is prescribed by the first two reference state v8riabl ~ s , which represent a second order model with chosen dynamics. This model is implemented in terms of a discrete algorithm, too, and it is driven by the roll angle commands. To ensure that full use . is made of the avai lable gimbal rates without exceeding the existing rate 1 imits, i t is proposed to choose the model eigenvalues thus that for a rather small "basic roll comrnand" 6 ~0 the gimbal rates approach but do not exceed their maximum values. For larger roll angle commands the maneuver may start wi th the "bas i c command" 6 ~0 . When the roll rate -- which nearly equals the required roll gimbal rate due to the coupling of roll rate into yaw acceleration (see eqn. (5)) -- attains its maximum value it may be "frozen in" by increasing the roll command input just with this maximum rate until it gets up to the actually commanded value. By the way a similar technique is possible for the 1 inear control law discussed in a). What concerns the matrix
~
it is proposed to set
185
G. Hi rzinger Th. Lange where ~ is the controllabil ity index. Then L = c represents that boundary of the state space that can be brought to the origin with a certain amount c of control energy (~ squared gimbal rates here) in a minimal number of sampling periods.
5. 5.1
DESIGN AND TEST OF A DGMW-CONTROL SYSTEM USING AN AIR-BEARING SIMULATOR Test set-up and disturbances
A DGMW-control system was designed and tested on an air-bearing simulator following the principles derived in this paper. The DGMW was fabricated under contract by Teldix Co., Heidelberg. It consists of a ballbearing momentum wheel mounted on two gimbals which are suspended by flex-pivots. The gimbals are driven by self-locking disc-type stepper motors. Three modes of operation were of particular interest: normal fine attitude control, maneuvers in roll and pitch, and momentum unloading by cold gas thruster firing. A process computer EAI PACER 100 was used for direct digital control . As shown in fig. 4, the air bearing platform contains the DGMW plus interface electronics, a power battery supply, four gas tanks connected through a common I ine, a FM/FM command receiver, an FM telemetry transmitter and attitude sensors . The latter ones comprise an infrared sensor for roll and pitch sensing and a sunsensor for yaw information. As an additional high precision optical reference, two 2-al!;is colI imators looking a> .. :r4atform fixed mi rrors are installed on the ground. Before closing the DGMW control loop, aA electromagnetic control system has to keep the platform in nul I position. The restoring torque, which is produced by electromagnetic coils equals the disturbance torques. So monitoring the current in the coils serves for an accurate bdlancing as wel I as for measuring the remaining disturbance torques. r-'I ~cEj F.:T, ;::: T ROi-;::;:Cf. I
fig. 4 186
Test setup
G. Hirzing er Th. Lange d in the Unfortu nately the remainin g torque level which can be achieve in space. ng laborato ry is orders of magnitu de greater than that prevaili error sounce This fact is mai·nly due to the presenc e of gravity . One it was caused of this kind is the imbalan ce drift. In our specifi c case actuato rs. gimbal the ing energiz after gimbals the of shift mass a by constan t with The effect was an exponen tial transie nt of one hour time i vely small t compara by 3 followed was It . Nm 10. 2 of an amplitu de in the labo~ changes with time (~ 10-4 Nm) due to tempera ture var i ations energiz e the ratory . So for precise measure ments it was necessa ry to gimbal actuato rs several hours before test. of mass of Another imbalan ce effect was caused by a shift of the center the flexibl e the DGMW when changin g the gimbal angle. This is due to radially fixed pivot gimbal bearing s. where the axis cannot be exactly shift along mass a was result The s. bearing ball with case as is the (see fig. 5) eristic charact ar nonline a up showing axis the nominal spin d. balance counter be not and accordi ngly could
'\ '\
7
if
v
-
.........
o
~
-
\
0
2°
-
I
./
4
4
1\ 4°
1---
o
2° --4. inner gimba l angle
ou t er gimbal an g le
fig. 5
V
~
J
l7
Gimbal pendu losity of the DGMW
Obvious ly the disturb i ng torque level torque due to ea r th rotation (~ 2.5 laborato ry the precess ion effect was dulos i ty of the DGMW and thus of the cession frequenc y wp computes as
is much higher than the precess ion . 10-5 Nm/O) . That is why in the increase d by a predeter mined penplatform . The correspo nding pre-
about 100 where p is to denote the pendulo sity torque. wp was chosen the nominal times earth rate precess ion frequenc y . As a consequ ence. 187
G. Hi rzinger Th. Lange pitch axis had to be aligned to the local vertical instead of the earth rotation axis, since here the precession torque is mainly d~e to the gravity field of the earth. The corresponding reference coordinate frame is dep,cted in fig. 6. As the gravity vector and the earth rotation vector are not coinci'dent, a constant precession torque Wo HB cos>. ( A m geographical latitude angle) due to earth rotation is present even in the null position; this torque had to be balanced.
NORTH
~:/~~r.l:"
I
.~ fig. 6
5.2
x,
ROLL AXIS
Y, PI'i'cn AXIS (::'UCAL VERTICAl.)
Reference coordinate frame
Controller specification and performance
By osci llation tests the main moments of inertia of the platform including DGMW were estimated as Ix - 33.6 Nms Iy
=
39.3 Nms 24.0 Nms
2 2 2
Nonzero products of inertia, especially Iyz, do exist but were neglected in order to test the parameter sensitivity of the chosen controller. The wheel nominal momentum amounts to 20 Nm. The step sizes ~Y ~ and ~y; of the gimbal stepper motors are 10 arcsec with a minimum dead time ~t = 50 msec. From a comparison of sampling periods as proposed in section 3, a sampl ing interval To = 0,5 sec turned out to be favourable for roll-yaw control and, of course, was used for pitch control, too: For a (rigid) s;'ltell ite with Ix = 300 Nms 2 , I z = 500 Nms 2 a sampl ing period To = 5 sec might suffice. Because of the high disturbance torques discussed above which cause large yaw estimation errors if yaw is not measured, the optical system mentioned in section 5.1 was used for measuring ~, 0 , 0/ in the first stage of tests which are reported here. Every To = 0,5 seconds the measurements are sampled, a scalar difference algorithm makes use of the pitch angle information to calculate the pitch control signal, while the roll and yaw information are processed in a Kalman filter algorithm. The estimate of the state of the rollyaw dynamics serves for determining the , optimal number ot gimbal steps.
188
G. Hirzinger Th. Lange The complete control algorithm takes about 10 ms computation time, follow\ng the concept as der(ved in section 3 and 4.
Pitch axis-control (vi'a decelerating and accelerating the wheel) is performed by a scalar difference equation (/5/) that real izes a discrete controller with transfer function C(z)
(25)
The fi rst factor (l)n the right side of eqn. (25) starids for a lead compensator, whose parameters were chosen thus that the (three) closed-loop poles are located at zl/2 = 0,5, z3 = O. By setting a2 = 0,9, b2 = 0,999 an additional lag compensator is real ized which reduces the static errors by a factor f = lOO . In case of a pitch angle cOOllland a timeoptimal controller is applied for avoiding that the linear controller might violate the saturation constraint and for making full use of , the available control torque. As the plant essentially represents a double integrator, the time-optimal switching curve is qiven analytically and so is implementable by a few multipli~ations. Computed control torques are transferred into actual torques by pulse width modulating the wheel motor drive. Fig. 7 shows the system's response to a commanded pitch angle of 10.
Roll-Yaw-Axis Control The dynamical system used in the control algorithm is of tenth order due to a disturbance model of fourth order (see section and 5. I). A correlation analysis was carried out after measuring the disturbance torques for a period of several hours. As a result of this analysis both in roll and yaw axis two independent first order systems with time constants TI ~ I hour and T7 ~ 0,5 min were modelled. They were assumed to be driven by white Gausslan noise processes, the variances of which were chosen thus that the stationary torque variances (without estimation) correspond to the autocorrelation characteristic values /3/. Another source of system noise to be accounted for in the Kalman filter is introduced by the gimbal angle stepsize, which may vary from 8 to 12 arcsec. Measurement noise is inferred by the optical sensors and amounts to 30 arcsec rms value. The gimbal angles are not measured but estimated. Both kinds of controllers as discussed in section 4 were implemented and yielded excel lent results. Fig. 8 shows the system's response to a roll an~~e command for controller b) of section 4 (minimizing a quadratic norm of the difference vector between actual and reference state). A second order reference model with (continuous-time) eigenvalues sl/2 = - 0,25 was chosen for roll maneuvers. Fig. 9 shows a section of the pulse pattern driving the roll gimbal stepper motor . Attitude errors during the transient as well as in the steady-state case amount to 30 arcsec and thus are just in the range of measurement noise.
189
G. Hirzinger
Th. Lange
1500
--
sec
....... 4 sec
J
I ~
/' / - --
0
L
L '-=t-
ROLL -- -- f c--
PITCH
-= c::- r:--f-
-+-- -
,- -
o - - r- , \
SeC
-t--- i- :-iL-- '-- ~
-~i---~ ~ _..- w
-- f- ~
T
--
- --
YAW
- - -
l -
-
t--- --
...
0
- -
___-L
--
,.........
150 sec
- - -+-
c-
-1500
-t--I--I-I i~ - - t- -~-- I....J
-:-r ~
PITCH 150
sec
I + --+--+-+-+---+---+-t----
~-r--,-.-r-r-.-, -
-I--+-+-I--!--!--I '-- --- 1- - -
f -f----1I-=-1~__1--f--f-+-f_-!--.
--
.
- t--+-+--I--+-+--t---t--I--f-- o
....
~~ 11!'2
--I- I----'f--i
.....
-
~.: I- .,
':"'f'f--
--=i-::; : ' --t-_+---1f-~-,.:.+--+-I--+--+- - - f - -1 50 se-H--+-+--i
1- - ~~~~~~c ~ ~_~ b~~-~~:~+-+-+-+-+-~-- ~ ~ ==
YAW
t:~~
fig. 7
Response to a pitch command of 1500 arcsec
fig.
II
•
8
-='F:= --
-re
c
- --
Response to a Roll Command of 1500 arcsec
=-
II ~==
fig.
190
9
A section of the pulse pattern driving the roll gimbal stepper motor
G. Hirzinger Th. Lange
6.
CONCLUSION
It is shown that an attitude control system using a double-gimballed momentum wheel with stepper motors as gimbal actuators turns out to be a pulse-frequency modulated system . Presuming that an on-board computer is available for direct digital control, a general design concept is developed for computer-controlled pulse-frequency modulated systems . The results are specialized to DGMW control and verified by a physical simulation indicating that the proposed concept provides very high pointing accuracies which in practice are I imited only by attitude measurement errors.
7.
ACKNOWLEDGEMENT
The authors gratefully acknowledge the valuable assistance of Mr. Humbert in the process computer implementation of the control schemes . Thanks also to the technical staff of the Institute for their assistance in various aspects of system simulation .
REFERENCES /1/
Lyons, M.G . ; Lebsock, K.L . ; Scott, E. D.: Double gimballed reaction wheel attitude control system for high altitude communications satel lites AIAA Guidance, Control and Fl ight Mechanics Conference 1971.
/2/
Mork, H.L .: Synthesis and design of a gimballed reaction wheel attitude stabilization package AIAA Guidance, Control and Fl ight Mechanics Conference 1971 .
/3/
Kwakernaak, H. ; Sivan, R.: John Wi ley & Son, 1972 .
/4/
Schmidbauer, B.; Samuelsson, H.; Carlsson, A.: Satel lite control and stabil ization using on-board computers, Volume 2 Report under ESTEC Contract 1368/71 AA.
/5/
Kuo, B. C.: Analysis and synthesis of sampled-data control systems. Prentice-Hall 1963 .
/6/
Bruderle, E.; Roche, Ch .; Weingarten, H.: Attitude control of a geostationary telecommunication satell ite using stepwise motion of gimbal angles between satell ite and flywheel Symposium on incremental motion control systems and devices, 1974.
Linear optimal control systems
191
175
G. Hi rzinger Th. Lange 1.
INTRODUCTION
Large telecommunication satell ites to be built in the future wi 11 be complex systems with several antennas and large flexible solar panels in order to supply the considerable amount of electric energy needed for an increased communication capability. Required overall beam accuracies may be less than O. I degree. Clearly a sophisticated three-axis stabi 1 ization system wi 11 be used. In a geostationary orbit the disturbing torques are mainly due to solar pressure acting on the panels and thus are composed of secular and cycl ic (zero mean in inertial coordinates) terms. Using a reaction jet control system would require continuous mass expulsion to maintain precise attitude control and so for long time missions of ten years and more mass required would become excessive. A momentum exchange system. however. simply exchanges internal and external angular momentum to compensate the cycl ic disturbances. and only secular disturbances need be counteracted by other torques. for example gas jets which may be used for station keeping anyhow . This is one of the main reasons why double-gimballed momentum wheels seem to be the most promising kind of stabilization system for complex communication satell ites with high pointing accuracies. Though they may consume more power than mass expulsion systems because of their continuous rather than intermittent operation. this electri'c power will be only a fraction of that needed by the communication system and so does not represent a severe shortcoming . As is well known. a double-gimballed moment wheel allows to exert control torques in 3 axes by varying the wheel speed (pitch control). and by torquing the gimbals to rotate the direction of the momentum vector (roll-yaw control). It is assumed in this paper that an on-board computer is available for direct digital control. It is furthermore assumed that stepping motors are used as gimbal actuators as they a)
al low to perform small gimbal steps precisely and rei iably.
b)
ideally fi t into a di rect digi tal control scheme.
This kind of gimbal acutation. however. leads to some scheme of pulsefrequency modulation. Though pulse-frequency-modulated systems are very important especially in space appl ications. no general design method is given for them in the literature. So the main aspect of this paper is to show how modern control theory can help to develop general design concepts for computer-controlled pulse-frequency-modulated systems. Consequently the technique proposed is not restricted to the design of a DGMW system with stepwise gimbal motion. but is likewise applicable to other PFM-systems. Attitude estimation is assumed to be provided by a Kalman-Filter that makes use of any measurement available to estimate the state of the complete dynamical system involved. Surely infrared sensors will be employed for providing roll and pitch information. Due to the momentum bias strong coupling exists between roll and yaw and allows to estimate yaw from roll measurements. Nevertheless direct yaw information from star trackers or radio-frequency sensors would be advantageous. Clearly pointing accuracy is 1 imited by the estimation errors. but in this paper emphasis is layed upon show.ing how optimal use is made of the estimated state under the supposition of computer-controlled stepwise gimbal actuation. 176
G. Hi rzinger Th. Lange 2.
SYSTEM DESCRIPTION
Let us briefly derive the equations of motion for a rigid satellite with DGMW in a geostationary orbit (see Fig. 1). In body fixed coordinates the total angular momentum H writes (including the wheel bias but neglecting the gimbal and wheel moments of inertia).
H
-[
x + cos Yx siny z
w
"~
y - cos Yx cosYz HB
w
z - sin Yx HB
w
""
where Ix' Iy' I z w x' wy' Yx' Yz HB
r
x + yz
w
Iy wy - HB I
"J
(1)
z wz - Yx HB
ro 11, pitch, yaw moments of inertia respectively roll, pitch, yaw body rates respectively
W
z
-
roll, yaw gimbal angle respectively wheel momentum bias
n.J\T~
YAN GIUE.AL
~OLL
GU;SAi,--
VELOCITY
y 'I.
t
i> YAW AXIS
GEOCEtlTER
flg~
1
Reference Coordinate F'ramc
The second part of (1) holds for small gimbal angles and wi 11 be used in the following. From the kinematic relations the angular velocity w refers to the Euler angles as follows (assuming again small angles and angu 1a r rates) :
(2)
where 177
G. Hi rzinger Th . Lange
roll Euler angle
~
o
=
~
=
pitch Euler angle yaw Euler angle orbit rate (= 7,28 . 10- 5 rad/sec)
W o
The spacecraft's motion is determined by Euler's equation
where T is the external torque vector, d/dt( } denotes the time derivative with respect to the inertial space, while (') denotes the time derivative with respect to the spacecraft coordinate system. Substituting (1) and (2) into (3}, 1 inearizing the nonl inear set of equations and transferring the gravity gradient torques to the right hand side Ill, we arr i ve at Tx
Ix ii + a
T y
I
Tz
I z iJj + c
y
o+
~
+ b ~ + Yz Ha + W o Ha Yx
3 Wo2 (Ix - I z) o - Ha ~
+ d
~
(4)
- Ha Yx + W0 Ha Yz
where a
=4
b c d
=
2 W0 (I
Y
- I z) + Wo Ha
Ha - (I x - I y + I z) Wo W Ha + w02 (I y - Ix) 0 - Ha - Wo (I - Ix Y
-
I z)
Apparently the pitch motion is decoupled from roll-yaw and approximately represents a simple double integrator to be controlled via the wheel acceleration (contained i n Ha) . So we fully concentrate on the roll-yaw dynami c s which write in state space form:
o -a
~
o
o
o
o
-b
o
q>
T;
o 0 o -.!!,e
o
Ix
o
o
o
o
-d
r;
-c
T;
o
o
o
o
o
o
o
o
o
o o
o
o
o
o - I --
z
~
~
Yz
: [::l
o
:J
o
o
(5) 178
G. Hi rzinger Th. Lange Obviously the gimbal angle rates are the control variables . Note that this linear model would indicate complete controllabi lity, though from physical considerations the nonlinear system cannot be completely controllable. The gimbal angles may therefore be considered as uncontrollable state variables . Remember that we assume an on-board computer to steer the gimbal step motors as specified by a digital control law. 5tepwise gimbal motion implies that the gimbal rates and so the control variables occur in form of pulses. The gimbal step size and so the pulse areas are fixed, but the instants fo their occurrence are arbitrary apart from a certain dead time 6t between two pulses . So we have to handle a typical pulsefrequency modulated (PFM) system. In state space notation a 1 inear time-invariant single-input system
(6) is transferred from an initial state ing the transition equation
~(T) - efT ~(O)
+
T
f
~(O)
into a final state x(T) follow-
F(T-t} !;.S. u(t} dt
(71
o
Assume that u(t) is a pulse of width
+0
u(t}
T
T
for T1 - "2" t
occurring at t <
T 1 (0 < T 1 < T):
T
T1 +"2 (8)
o Then for
T
~(T) =
else
very small compared with the system time constants we get FT
F(T-Tl)
e- ~(O) + e-
where
S. T
h
f
(9)
T
"2 h
f
u(v) dv T
- "2 As mentioned in the introduction, PFM-systems are nonl inear systems and no general design techniques have been developed for them despite their practical importance. Well known techniques as integral-pulse-frequencymodulation (IPFM) rely on the assumption, that a pulse of area h occurring in the middle of an interval (Tl = T/2) causes the same final state as would do 11 constant input of same area u(t} "hiT. It is easy to show that apart from special cases (e.g. double integrating plant) this assumption is seriously hurt for greater T. So let us search for a more general design technique which in addition fits into a computer control scheme .
179
G. Hirzinger Th. Lange 3.
ON A GENERAL DESIGN CONCEPT FOR COMPUTER-CONTROLLED PFM-SYSTEMS
Computer control of a PFM-system implies that any T seconds -- where T is a fixed sampl ing interval -- measurements of the system outputs are processed (e.g. in a Kalman filter) to compute a sequence of control pulses for the next interval. If we presume that the computer should be employed with control tasks as little as possible, the questions arising are how large T can be made and how the pulses should be arranged. To ask these questions we go through the following considerations (for simpl icity in the single-input case): Apparen~ly we cannot assume that we might vary technique to be discussed controlled system show up pulse of arbitrary area h terval? A simple way to control; that is to say, a performance index
"~ ~. xT 1 k=O --...+
change the pulsed character of control. Yet the pulse area (height) arbitrarily by some later. Up to what sampling time T does the good performance, provided that one control may occur in the middle of each sampl ing inanswer this question is by means of optimal for different sampling times T we minimize
+ -2 uk r
g --... x'-+l
subject to the discrete system equation (see eqn (9) with Tl ~(O) = ~, ~(T) = ~+1)
(10) T/2,
(11) where A
b
--0
=
FT 12
f';
.ll.
T
Note that the discrete control variable uk has been chosen as hk/T to ensure that for doubled sampl ing interval a double pulse area is admitted. The prefactor T in eqn. (10) is due to the fact that for doubled T the number of contributions in the sum is halfed. Q and r are chosen preferably so that the maximal allowable deviations in the interesting state variables (e.g. roll and yaw angle) cause the same costs as do maximal control ampl itudes (e.g. gimbal angle rates). law uk =:;,.T ~ and in order to be independent of some special initial state -- computes as
J 1 is minimized by a linear control
J 1 = trace(~)
where the cost matrix
(12) ~
is easily computed by an iterative procedure 13/.
Clearly Jl increases for increasing T, and by starting with a smallest sampling time T = T, we practically compare with the ideal istic case that pulses may occur at any time (as the dead time 6t is impl ied for T > 6t only). However recal I that we assumed we might realize any pulse area hk (of 180
G. Hlrzing er Th. Lange Though this course only within a certain control amplitu de constra int). red by conside T each for ion assumpt the h approac we ble, impossi is ntal gimarrangin g thi element ary pulses of given area h* (e.g. increme area hk of pulse desired a Now . 2 fig. in shown as bal steps 6yx, 6y;) pulses which is approxim ated by an integer multipl e j of the element ary are set at times ti
+ i . 6t
= ;
where i =
- lJ..:..!l 2
t ' ... ,
i = j
=
.L:.!.
-2, -1, 0, 1, 2,
-2, -I, I, 2, ... , hk integer ( h* )
I I
t
I I
I I
I I
even
for
sampling period T I I I At At I At
... .....
At
for j odd
2
,
.., I I
At
lA!. I
2
I I
T
times of eventual unipolar pulse occurrence fig.
2
Propos ed arrange ment of pulses (j
4 here)
the compute r T should be an integer multipl e N of the spaeing 6t so that pulses) of number desired the to ng (accordi r re9iste a" set only "has to which is sensed via an externa l clock. The actual input vector
~(j) computes as
.L:.!. 2 1:
i=1
-1
b. + -I
1:
'-1
b.)/j
for
odd
(13)
-I
i=-(.Lf!-) 181
G. Hi rzinger Th. Lange
1
-1
2
L: b.
L: ..~)/j
even
(13)
where b. denotes the input vector for a pulse occurring at t to the-~elation F(I- ot ) I( r ot) - 2 FT/2
Due
i=1
-I
+
i=
2
~+e
e
for
_.L
~~2e-.~
holding for small o t, the actual input vectors b(j) may be expected to be close to~ . Of course the degree of misalignment would increase with increasing j. Therefore -- assuming that all j ~re equally probable -- we define a new representative input vector ~ replacing £0 :
b
~
~
b{j) . j)/ j j=1 j=1
=
(14)
So we have replaced the original, unattainable assumption of arbitrary pulse areas by the practically implementable scheme just described. But thereby we have introduced stochastic errors. Presuming again that all admissible control ampl itudes are equally probable, these errors are composed of a)
a quantization error ~lk (in general this is the main error) with zero mean and variance matrix
v
-1
b)
= _1_ (b h*) T (b h*) Jl2' - T - T
a input-misalignment error ~k with zero mean (due to the def i nition of eqn (14) and variance matrix 1
Y..2
=
N
N+T .Ll J=
A measure for the influence of these errors ~lk and ~~ (which occur on the right side of eqn. (11» on system performance I S given by the expected value J 2 (see /3/) : (15) Eqn . (15) presumes that the vectors ~Ik and ~k are uncorrelated. the same linear control law i s optima for Jl and J 2 .
Then
Though we must not add Jl and J2 due to their different defin i tions, a plot of J 1 and J 2 versus sampling t ime T allows to decide on a sampl ing time To for which good system performance can be expected without employing the on-board computer unnecessarily often . Fig. 3 shows a typical plot of this kind. The increase of J2 for small sampling times should not surpr i se as for small T the assumption of uncorrelated errors does no longe r hold and so a simple 1 inear controller would not be opt ima 1.
182
G. Hi rz i nge r Th. Lange
IJ)Ssib~
choicf of sampling period T
fig. 3
Typical plot of J 1 and J 2 versus sampling time
The results of this section ·are immediately appl icable to multi-input PFH-systems (adding the variances of the additional input erros in eqn. (IS)). So we summarize and generalize: It has been shown, how by proper arrangement of pulses a pulsefrequency modulated system can be transformed into a I inear discrete-time system
~+I
=
~~ +!~
(16)
and how an appropriate sampl ing time To can be found. The main difference to a normal discrete system may be seen in the quantization error 6~ due to the fact that only an integer number of pulses is possible. Ways to overcome this shortcoming are shown in the next section .
4.
CONTROL PRINCIPLES FOR STEPWISE ROLL-YAW GIHBAL ACTUATION
In the real appl ication the basic roll-yaw dynamics as developped in eqn. (S) have to be augmented by controllable subsystems (motions of flexible arrays) and uncontrollable subsystems (disturbance torque models, e.g. the mainly sinusoidal solar pressure). A complete discrete system description implying stepwise gimbal motion in a scheme as discussed in section 3 may be written in the form
183
G. Hirzinger Th. Lange
where ~
denotes the set of controllable state variables (~, ~, generalized coordinates of the flexible parts)
~, ~,
y
denotes the set of uncontrollable state variables (gimbal angles, modelled disturbance torques)
u denotes the control pulse areas per sampl ing time, that is here
" •
T~ [:::J
(18)
The computed ideal values of 6yx and 6l z are approximated by an integer number of incremental steps 6y= and 6yz arranged as shown in fig. 2. Those torques Tx , Tz which are known, but cannot be dynamically modelled, as are desaturation or station keeping thruster pulses, antenna steps etc., are introduced with their individual kind of actuation expressed by the input matriees Lx and Lvk' Let us discuss two control principles : a) Assume that
~
~ = .£.(~
is the desired set point.
-
A I inear control law
~s)
(19)
may be determined for the controllable part of system eqn. (17) by applying well-known design techniques for ensuring favourable closed-loop dynamics (e.g. decoupling of roll-yaw motion). A correction term 6~ should be added in order to counterbalance 1. the influence of the uncontrollable state variables and external torques which would cause a deviation (20)
2. the influence of the Quantizat ion error 6~_1 which was made in the preceding interval. After ~_I ha, been computed and the appropriate pulses set, we know the quantization error 6~_1 and so the state error
6~
= :[
6~_1
This state error propagates into a deviation
6~+1
where (21)
Though these adverse influences in general cannot be completely cancel led, they can be counterbalanced in the sense of minimizing an 184
G. Hi rzinger Th. Lange eucl idean norm of 6~+1 by a correcting control o~:
Of course, the matrix products occurring in eqn. (22) should be precomputed. Note that desaturation torques are seen as kind of disturbance torques which are known and so may be anticipated by proper gimbal motion. This implies that the thruster pulses, if possible, should occur in the middle of the sampling interval and theIr pulsewidth should correspond to an integer number of gimbal steps. b) The correction of a 1 inear control law as just discussed is not necessary, if one immediately refers to a reference state. Then for the k-th sampl ing interval the control ~ is computed thus that a norm L of the error between projected state ~+I and reference state 3k+1 is minimized: (23) Expressing
~ yields
~+I
~inL
by the right side of eqn. (17) and minimizing Lover ",T
- -I -T
I.! ~!)
! ~ (~~
+ ~ ~ +
-
.4 lK - ~+ I)
(24)
Eqn. (24) holds only if all saturation constraints -- as given by maximal gimbal rates -- are met; otherwise additional computations would be necessary. However for systems with fairly well-known disturbances it is no problem to keep the actual state close enough to the reference state. In the additude hold mode the reference state equals the set point state~. Yet assume that a set point change -- due to a new commanded roll angle -- occurs. All state variables should remain in rest apart from roll angle ~ and rate ~ which should follow a desired second order transient. In a) this transient is immediately determined by the feedback law . Here it is prescribed by the first two reference state v8riabl ~ s , which represent a second order model with chosen dynamics. This model is implemented in terms of a discrete algorithm, too, and it is driven by the roll angle commands. To ensure that full use . is made of the avai lable gimbal rates without exceeding the existing rate 1 imits, i t is proposed to choose the model eigenvalues thus that for a rather small "basic roll comrnand" 6 ~0 the gimbal rates approach but do not exceed their maximum values. For larger roll angle commands the maneuver may start wi th the "bas i c command" 6 ~0 . When the roll rate -- which nearly equals the required roll gimbal rate due to the coupling of roll rate into yaw acceleration (see eqn. (5)) -- attains its maximum value it may be "frozen in" by increasing the roll command input just with this maximum rate until it gets up to the actually commanded value. By the way a similar technique is possible for the 1 inear control law discussed in a). What concerns the matrix
~
it is proposed to set
185
G. Hi rzinger Th. Lange where ~ is the controllabil ity index. Then L = c represents that boundary of the state space that can be brought to the origin with a certain amount c of control energy (~ squared gimbal rates here) in a minimal number of sampling periods.
5. 5.1
DESIGN AND TEST OF A DGMW-CONTROL SYSTEM USING AN AIR-BEARING SIMULATOR Test set-up and disturbances
A DGMW-control system was designed and tested on an air-bearing simulator following the principles derived in this paper. The DGMW was fabricated under contract by Teldix Co., Heidelberg. It consists of a ballbearing momentum wheel mounted on two gimbals which are suspended by flex-pivots. The gimbals are driven by self-locking disc-type stepper motors. Three modes of operation were of particular interest: normal fine attitude control, maneuvers in roll and pitch, and momentum unloading by cold gas thruster firing. A process computer EAI PACER 100 was used for direct digital control . As shown in fig. 4, the air bearing platform contains the DGMW plus interface electronics, a power battery supply, four gas tanks connected through a common I ine, a FM/FM command receiver, an FM telemetry transmitter and attitude sensors . The latter ones comprise an infrared sensor for roll and pitch sensing and a sunsensor for yaw information. As an additional high precision optical reference, two 2-al!;is colI imators looking a> .. :r4atform fixed mi rrors are installed on the ground. Before closing the DGMW control loop, aA electromagnetic control system has to keep the platform in nul I position. The restoring torque, which is produced by electromagnetic coils equals the disturbance torques. So monitoring the current in the coils serves for an accurate bdlancing as wel I as for measuring the remaining disturbance torques. r-'I ~cEj F.:T, ;::: T ROi-;::;:Cf. I
fig. 4 186
Test setup
G. Hirzing er Th. Lange d in the Unfortu nately the remainin g torque level which can be achieve in space. ng laborato ry is orders of magnitu de greater than that prevaili error sounce This fact is mai·nly due to the presenc e of gravity . One it was caused of this kind is the imbalan ce drift. In our specifi c case actuato rs. gimbal the ing energiz after gimbals the of shift mass a by constan t with The effect was an exponen tial transie nt of one hour time i vely small t compara by 3 followed was It . Nm 10. 2 of an amplitu de in the labo~ changes with time (~ 10-4 Nm) due to tempera ture var i ations energiz e the ratory . So for precise measure ments it was necessa ry to gimbal actuato rs several hours before test. of mass of Another imbalan ce effect was caused by a shift of the center the flexibl e the DGMW when changin g the gimbal angle. This is due to radially fixed pivot gimbal bearing s. where the axis cannot be exactly shift along mass a was result The s. bearing ball with case as is the (see fig. 5) eristic charact ar nonline a up showing axis the nominal spin d. balance counter be not and accordi ngly could
'\ '\
7
if
v
-
.........
o
~
-
\
0
2°
-
I
./
4
4
1\ 4°
1---
o
2° --4. inner gimba l angle
ou t er gimbal an g le
fig. 5
V
~
J
l7
Gimbal pendu losity of the DGMW
Obvious ly the disturb i ng torque level torque due to ea r th rotation (~ 2.5 laborato ry the precess ion effect was dulos i ty of the DGMW and thus of the cession frequenc y wp computes as
is much higher than the precess ion . 10-5 Nm/O) . That is why in the increase d by a predeter mined penplatform . The correspo nding pre-
about 100 where p is to denote the pendulo sity torque. wp was chosen the nominal times earth rate precess ion frequenc y . As a consequ ence. 187
G. Hi rzinger Th. Lange pitch axis had to be aligned to the local vertical instead of the earth rotation axis, since here the precession torque is mainly d~e to the gravity field of the earth. The corresponding reference coordinate frame is dep,cted in fig. 6. As the gravity vector and the earth rotation vector are not coinci'dent, a constant precession torque Wo HB cos>. ( A m geographical latitude angle) due to earth rotation is present even in the null position; this torque had to be balanced.
NORTH
~:/~~r.l:"
I
.~ fig. 6
5.2
x,
ROLL AXIS
Y, PI'i'cn AXIS (::'UCAL VERTICAl.)
Reference coordinate frame
Controller specification and performance
By osci llation tests the main moments of inertia of the platform including DGMW were estimated as Ix - 33.6 Nms Iy
=
39.3 Nms 24.0 Nms
2 2 2
Nonzero products of inertia, especially Iyz, do exist but were neglected in order to test the parameter sensitivity of the chosen controller. The wheel nominal momentum amounts to 20 Nm. The step sizes ~Y ~ and ~y; of the gimbal stepper motors are 10 arcsec with a minimum dead time ~t = 50 msec. From a comparison of sampling periods as proposed in section 3, a sampl ing interval To = 0,5 sec turned out to be favourable for roll-yaw control and, of course, was used for pitch control, too: For a (rigid) s;'ltell ite with Ix = 300 Nms 2 , I z = 500 Nms 2 a sampl ing period To = 5 sec might suffice. Because of the high disturbance torques discussed above which cause large yaw estimation errors if yaw is not measured, the optical system mentioned in section 5.1 was used for measuring ~, 0 , 0/ in the first stage of tests which are reported here. Every To = 0,5 seconds the measurements are sampled, a scalar difference algorithm makes use of the pitch angle information to calculate the pitch control signal, while the roll and yaw information are processed in a Kalman filter algorithm. The estimate of the state of the rollyaw dynamics serves for determining the , optimal number ot gimbal steps.
188
G. Hirzinger Th. Lange The complete control algorithm takes about 10 ms computation time, follow\ng the concept as der(ved in section 3 and 4.
Pitch axis-control (vi'a decelerating and accelerating the wheel) is performed by a scalar difference equation (/5/) that real izes a discrete controller with transfer function C(z)
(25)
The fi rst factor (l)n the right side of eqn. (25) starids for a lead compensator, whose parameters were chosen thus that the (three) closed-loop poles are located at zl/2 = 0,5, z3 = O. By setting a2 = 0,9, b2 = 0,999 an additional lag compensator is real ized which reduces the static errors by a factor f = lOO . In case of a pitch angle cOOllland a timeoptimal controller is applied for avoiding that the linear controller might violate the saturation constraint and for making full use of , the available control torque. As the plant essentially represents a double integrator, the time-optimal switching curve is qiven analytically and so is implementable by a few multipli~ations. Computed control torques are transferred into actual torques by pulse width modulating the wheel motor drive. Fig. 7 shows the system's response to a commanded pitch angle of 10.
Roll-Yaw-Axis Control The dynamical system used in the control algorithm is of tenth order due to a disturbance model of fourth order (see section and 5. I). A correlation analysis was carried out after measuring the disturbance torques for a period of several hours. As a result of this analysis both in roll and yaw axis two independent first order systems with time constants TI ~ I hour and T7 ~ 0,5 min were modelled. They were assumed to be driven by white Gausslan noise processes, the variances of which were chosen thus that the stationary torque variances (without estimation) correspond to the autocorrelation characteristic values /3/. Another source of system noise to be accounted for in the Kalman filter is introduced by the gimbal angle stepsize, which may vary from 8 to 12 arcsec. Measurement noise is inferred by the optical sensors and amounts to 30 arcsec rms value. The gimbal angles are not measured but estimated. Both kinds of controllers as discussed in section 4 were implemented and yielded excel lent results. Fig. 8 shows the system's response to a roll an~~e command for controller b) of section 4 (minimizing a quadratic norm of the difference vector between actual and reference state). A second order reference model with (continuous-time) eigenvalues sl/2 = - 0,25 was chosen for roll maneuvers. Fig. 9 shows a section of the pulse pattern driving the roll gimbal stepper motor . Attitude errors during the transient as well as in the steady-state case amount to 30 arcsec and thus are just in the range of measurement noise.
189
G. Hirzinger
Th. Lange
1500
--
sec
....... 4 sec
J
I ~
/' / - --
0
L
L '-=t-
ROLL -- -- f c--
PITCH
-= c::- r:--f-
-+-- -
,- -
o - - r- , \
SeC
-t--- i- :-iL-- '-- ~
-~i---~ ~ _..- w
-- f- ~
T
--
- --
YAW
- - -
l -
-
t--- --
...
0
- -
___-L
--
,.........
150 sec
- - -+-
c-
-1500
-t--I--I-I i~ - - t- -~-- I....J
-:-r ~
PITCH 150
sec
I + --+--+-+-+---+---+-t----
~-r--,-.-r-r-.-, -
-I--+-+-I--!--!--I '-- --- 1- - -
f -f----1I-=-1~__1--f--f-+-f_-!--.
--
.
- t--+-+--I--+-+--t---t--I--f-- o
....
~~ 11!'2
--I- I----'f--i
.....
-
~.: I- .,
':"'f'f--
--=i-::; : ' --t-_+---1f-~-,.:.+--+-I--+--+- - - f - -1 50 se-H--+-+--i
1- - ~~~~~~c ~ ~_~ b~~-~~:~+-+-+-+-+-~-- ~ ~ ==
YAW
t:~~
fig. 7
Response to a pitch command of 1500 arcsec
fig.
II
•
8
-='F:= --
-re
c
- --
Response to a Roll Command of 1500 arcsec
=-
II ~==
fig.
190
9
A section of the pulse pattern driving the roll gimbal stepper motor
G. Hirzinger Th. Lange
6.
CONCLUSION
It is shown that an attitude control system using a double-gimballed momentum wheel with stepper motors as gimbal actuators turns out to be a pulse-frequency modulated system . Presuming that an on-board computer is available for direct digital control, a general design concept is developed for computer-controlled pulse-frequency modulated systems . The results are specialized to DGMW control and verified by a physical simulation indicating that the proposed concept provides very high pointing accuracies which in practice are I imited only by attitude measurement errors.
7.
ACKNOWLEDGEMENT
The authors gratefully acknowledge the valuable assistance of Mr. Humbert in the process computer implementation of the control schemes . Thanks also to the technical staff of the Institute for their assistance in various aspects of system simulation .
REFERENCES /1/
Lyons, M.G . ; Lebsock, K.L . ; Scott, E. D.: Double gimballed reaction wheel attitude control system for high altitude communications satel lites AIAA Guidance, Control and Fl ight Mechanics Conference 1971.
/2/
Mork, H.L .: Synthesis and design of a gimballed reaction wheel attitude stabilization package AIAA Guidance, Control and Fl ight Mechanics Conference 1971 .
/3/
Kwakernaak, H. ; Sivan, R.: John Wi ley & Son, 1972 .
/4/
Schmidbauer, B.; Samuelsson, H.; Carlsson, A.: Satel lite control and stabil ization using on-board computers, Volume 2 Report under ESTEC Contract 1368/71 AA.
/5/
Kuo, B. C.: Analysis and synthesis of sampled-data control systems. Prentice-Hall 1963 .
/6/
Bruderle, E.; Roche, Ch .; Weingarten, H.: Attitude control of a geostationary telecommunication satell ite using stepwise motion of gimbal angles between satell ite and flywheel Symposium on incremental motion control systems and devices, 1974.
Linear optimal control systems
191