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Attitude Stabilization of Geostationary Satellite with a Single Degree of Freedom Angular Momentum Wheel System
J. BROQUET ATTITUDE STABILIZATION OF GEOSTATIONARY SATELLITE WITH A SINGLE DEGREE OF FREEDOM ANGULAR MOMENTUM WHEEL SYSTEM J.BROQUET Engineer - Space Division, ENGINS MATRA BP 1 78140 - VELIZY - FRANCE SUMMARY Control of three-axis stabilized geostationary satellite without permanent yaw sensing requires an internal angular momentum control of yaw attitude. When the angular momentum is rigidly aligned with the pitch satellite axis, high pOinting accuracy requirements impose a North-South station keeping. When the angular momentum has at least one lateral degree of freedom in satellite axes, high pOinting accuracies can be achieved even without NorthSouth station keeping. The comparison of one-degree and two degrees of freedom systems shows that both have about the same performances while one-degree-of-freedom systems are much simpler from the hardware point of view. Various mechanizations of the latter are proposed with special emphasis on skewed momentum wheel configurations. Acknowledgement The author wishes to especially thank ESA which has sponsored part of the works - in particular during the contracts 1621, 2272 and 2349 - whose results are used for this presentation. INTRODUCTION For several years in Europe, operational geostationary satellites in optimized configuration appear as three-axis stabilized payload and antenna. From an attitude control point of view, the lack of suitable permanent yaw sensors for present and near future development leads to providing the satellite with a large angular momentum, enabling the control system to mak e an efficient use of the roll yaw coupling, without large yaw error. The angular momentum can be rigidly aligned with the pitch satellite axis and it has further to be controlled by external torquer actuations as soon as the satellite pOinting error exceeds a given threshold. This principle is used on "O.T.S". The major limitation of this system lies in the necessity of a Uorth/ South Orbit control leading to a large mass penalty. The angular momentum can be movable inside the satellite either around one or two orthogonal transverse axes in the roll yaw plane. The corresponding systems are respectively called "Single and double degree of freedom systems". They will be analysed more deeply here after. In all cases the control about the pitch axis is considered as practically uncoupled with the roll yaw control and is not discussed in this paper.
146
J. BROQUET NOTATIONS : DOF Degree of transverse freedom, FMW Fixed Momentum Wheel SGMW : Single Girnbal Momentum Wheel DGMW : Double Girnbal Momentum Wheel x, y, z satellite axes: X along the orbital velocity, y perpendicular to the orbit plane and z toward the earth center Ix, Iy, Iz : satellite moments of inertia H : wheel angular momentum Hy : angular momentum along y axis h
angular momentum in the x, z plane
J
wheel transverse inertia orbital rate spacecraft attitude angle about x, y, z gimbal angle angle of thruster torque with respect to roll axis.
PRINCIPLE OF CONTROL WITH A ONE-DEGREE-OF-FREEDOM SYSTEM The first main characteristic of the 1 DOF systems is the capability of operating an accurate attitude control on an inclined orbit. When the orbit inclination on the equatorial plane is i ~ 0, the 1 DOF systems allow to operate an ideal satellite pOinting with respect to the earth center whatever the direction of the degree of freedom in the x, z satellite plane. For example, if the degree of freedom is along the roll axis the angular momentum will be maintained close to the North/ South inertial direction. Similarly, if the degree of freedom is along the yaw axis, the angular momentum will be maintained close to the inertial direction normal to the orbit plane.
The the the and
second important characteristic of the 1 DOF system is capability of linearizing the roll control provided that DOF is not along the yaw axis, and of decoupling the roll yaw control if the DOF is along the roll axis.
Resulting from these two characteristics and from the fact that the yaw control stability and accuracy is not at all dependent on the orientation of the DOF, the optimized orientation appears generally as the roll axis as will be shown in this paper.
147
J. BROQUET Principle of yaw control As for the FMW system used on OTS, the yaw control with a 1 DOF system is based on the whecon law. In the optimized case where the DOF is along the roll axis, the yaw orientation is completely defined by the angular momentum orientation. Compared with the OTS case the H control about roll and yaw axis is obtained by ,the same thruster actuation logic using 'f + ~ Q! (0 instead of'-P as measurement of H misalignment about the roll axis.
Short term behaviour
Long term behaviour
Nevertheless the roll angle being not dependent on the H control the threshold on ~ can be selected relatively large in accordance with the acceptable yaw angle. Then the thruster actuation impulse bit can be widely increased. The roll transient due to nutation at time of thruster actuation can be minimized by an accurate open loop actuation of DOF. The short term behaviour as defined on the drawing corresponds to a perfect open loop compensation. Principle of roll control This principle is widely dependent on the mechanization of the DOF and will be discussed later. Compensation of yaw error due to known disturbing torques If large disturbing torques are identified, the effect of these torques on yaw angle can be cancelled by an appropriate modulation of the DOF angle measurement before the thl'uster logic. A constant disturbing torque MDC along roll axis can be cancelled by a constant bias on the DOF angle measurement (~C /W , li) . An inertial torque M,....I (M,....I = A sin Wt, M,....I = A cos Wt) can be cancelled by M sin~ ~ias on the DOF Mfi~le measurement: (00 = A (&in(Wt- 2o<.))jW.H. Then for large solar array satellite where large disturbing torques are applied, the 1 DOF systems allow to really cancell their effect on yaw angle without supplementary hydrazine consumption or number of thruster actuations.
148
J. BROQUET PRINCIPLE OF CONTROL WITH A 2-DEGREES-OF-FREEDOM SYSTEM This principle has been the subject of a lot of publications. See for example 14/. The main characteristics of this control are the linearization of both roll and yaw control, the minimization of hydrazine consumption if the DOF off-loading is operated at given pOint of the orbit and the capability of yaw error compensation for known disturbing torques. COMPARISON OF PERFORMANCE OF ONE AND TWO DEGREES OF FREEDOM SYSTEMS The main performances to be compared are the pointing accuracies, the hydrazine consumption and the number of thruster actuations, the mass, power and reliability. POinting accuracies Roll accuracy capability. Identical for the two systems Yaw accuracy capability. Only the effect of inertial torques is shown because for large solar array satellites this effect is the only significant one. . Expression of yaw error for a 2 DOF system : 't' = Mol Wo H . Expression of yaw error for a with K such that
DOF system :
er =
K. Mol WoH
YAW ERROR 4'
TYPICAL CURVE FOR H=50NMS AND REALISTIC ASSUlIPTIONS ON DISTURBING TORQUES
hres-\~ -hold f.bh an
~:(3~
~s = O. 1
2 SGMW ( ~s=O. 1°)
( ~s=0.05°
0
0 .1 0. 2 0. 4 0. 8 1.6 K2.3 1.
1.3 I • I 5 1.0
(-1= 0.0 5 1.7 1. 3 1.1
.07
SOLAR ARRAY POWER
0'---+-_ __+_ _ _ _-+-_--'''''.:0.5" -1 2 ~ "KW'
Number of thruster actuations and hydrazine consumption These curves are given for the same hypothesis as the yaw accuracy curves.
149
J. BROQUET 'lJ6 NUMBER OF ACTUATIONS
ON S YEARS
/
/
/
/
HYDRAZINE CONSUMPTION,S YEARS
/
/
SGMW (~=O.l°)?le)
//
/
~
5.40 DGMW 1 OFF ,/
lO~
day
",y"
~/
__ - -
:/" y
'G"GMW 2 OFF
_/~
/
/
/
/
/
~
~-::;r
loading/day
2
4 K'" POWER
JoKW
POWER Mass power and Reliability of hardware for attitude control excluding a hydrazine subsystem POWER
MASS
40 Kg
1 OOF
Ml
2 OOF
Ml + 10 Kg
~
PI
'>C
50 W
PI + 10 W
RELIABILITY/ 5 YEARS =«
0.887
Less than 0.75
The various curves and results of comparison show that the difference between the 1 and 2 OOF systems is not really significant as far as the performances are concerned except for the hydrazine consumption for large solar array satellites. But such a consumption can be certainly reduced to a large extent by using a direct torque compensation for most of the inertial torque. This compensation can be based on : - a compensation of torques about the normal to the solar array plane by a wind will torque resulting from a control of the angle between the panels, - a compensation about the transverse axis in the solar array plane by a magnetic torque obtained with a coil in the solar array plane. Thus the mass and reliability penalty for the 2 OOF systems and the remaining technological problems of OGMW systems lead to a general preference for the 1 OOF systems.
150
J. BROQUET
MECHANIZATION AND CONTROL OF THE ONE DEGREE OF FREEDOM SYSTEMS Skewed momentums wheels in the yz plane (DOF along roll axis) This mechanization uses two FMW such as those used for OTS. The angle 2 S between the wheels must be defined in the range 14 to 60 0 as a function of the acceptable speed variation range for each wheel.
~ I
Roll control principle The roll control is based on an internal angular momentum exchange and must be defined in order to insure stability of the satellite motion and particularly the nutation. Simplified equations of the system :
I
I-f> + H....... + Woh = MDX DZ
flY - H -f> + WoH't"'= M . •.
The roll short term motion appears as : f/r where r = H/I is the nutation frequency.
2
•
+'f= - h/ H
In order to stabilize the roll motion h must be put into the form : h/H = K1:P + K2 + K3'f' + possibly K4
+
!'-f'
So the control torques applied to the wheels theoretically use the 2nd order derivative of 'f. That is an important characteristic of the one DOF system with DOF about the roll axis. In fact the wheels can be controlled as internal servo onto the speed comman~~~ f (H, h) appears as a function of 'i', 'f' ,J'f instead of 'i', '1', 'i', but this does not change the problem of noise and electronics drift when the nutation periods are large (more than 200 sec for satellite power of 1 KW). To solve the problem of these 2nd order derivatives one can generally define a very simple control law based on a large decoupling between the nutation and the precession frequencies. This decoupling is verified for satellites with solar array power up to at least 1.5 Kw. For larger satellites the decoupling can be maintained through a direct torque compensation or by increasing the wheels angular momentum. whi~h
If the decoupling is not obtained, the -f term can be got with digital hardware through a Kalman filter,with only steady state gains for electronics minimization.
151
J. BROQUET Note on the selection of the OOF direction in x z plane. If the decoupling between H drift and nutation motion is not obtained one could avoid the requirement for -f by selection of the OOF orientation along a skewed direction of the x z plane. The stability margin when using a P.O. network is then the more effective the closer the OOF is to the z axis. But such a configuration with a skewed OOF has the disadvantage of increasing the yaw error : - as a function of ~ sinlS' where't is the angle between roll and OOF direction. Since a roll control A'f requires a var ia tion of ~: A ~ = 6.'f1 cos'lS the var ia tion of 'Vat the same time is A'Y = A'f tg CS'. This expression clearly shows that'1S'must be minimized. - as a function of fu tg'6 where fWB is the initial false zero of the OOF angle measurement which is corrected after in-flight identification in order to minimize hydrazine consumption. - aS,a function of (~C/WH) tg'lS x aX1S are compensate~.
when constant torques along
So e~cept for missions where the yaw accuracy requirement is not very severe, the preferred orientation of the OOF is the roll axis. Block diagram of the control. It defines the main functions of the system. Wheel velocity • close loop cont. frequency (I) Cd) 2
iJOF angle determination
(2)
Compensation ~T~h-r-u-s~t-e-r-a-c~t~.~l-o-g~i-c~~of thruster including compensat. I actuation hruster act.
Single gimbal momentum wheel with stepper moto~. As for 2 skewed FMW systems the roll control law must theoretically use a 2nd order derivative of 'f . But in addition the roll control actuation is necessarily quantized, leading to a decreasing of the roll accuracy as a function of the step size. A large simplification of the control law can be proposed when the nutation motion is widely decoupled from the H drift. The roll angle can then be controlled within a given threshold by a logic ordering a step change as soon as the threshold is exceeded. Each step change sequence can be operated without theoretical impact on the nutation by using a logic sequence as defined in reference III leading to a roll transient corresponding -to one step size. 152
J. BROQUET A slight modification of this step sequence in accordance with the logic control system allows the nutation to be really damped. This damping effect has a threshold whose magnitude is a function of the damping efficiency and of the uncertainty on the knowledge of the nutation period. Single gimbal momentum wheel with reversible torquer Such an equipment can be easily derived from the DGI-1:W with reversible torquers developed by TELDIX for ESA. Roll control principle. The roll control of a SGMW reversible torquer differs from the previous ones by the fact that the wheel is free to move with respect to the satellite about the gimbal axis (roll axis in the selected configuration) • The wheel nutation frequency then appears as W= H/'v'I'J. The stabilization of the satellite roll angle can be obtained by a phase lead network (~+~-f) for the gimbal torque determination, but this control does not stabilize the wheel nutation which must be necessarly damped by using the gimbal angle measurement. In fact, the problem becomes more complex when the gimbal friction torque is considered. Due to technological reasons for the equipment itself and to mass considerations, the SGMW free gimbal - seems to be at the present time less attractive than the SGMW stepper motor. Fixed momentum wheel close to -y wheel in the xyz plane This mechanization ant the associated control are not really different from the skewed wheel ones. The angle£is defined for two reasons :
,.:;..---+0::.
• it avoids the low velocities for the small wheel and then it simplifies the problem of speed measurement, • it avoids any transient in the roll control due to stiction torque which would appear, if the mean wheel velocity was varying about zero.
153
J. BROQUET COMPARISON BETWEEN THE VARIOUS 1 DOF SYSTEMS This comparison must be based on the configurations including redundancy provision. The most attractive configurations appear as shown on the next figures.
/;)~ SGMW Figure 1
~~
Skewed FMW Figure 2
~
h. Skewed + reaction
FMW wheel
Figure 3
The configuration 1 seems to be attractive from the mass and power point of view but requires a successful development of the basic equipment . The configuration 3 is attractive for mass and accuracy point of view but requires the development of a Reaction Wheel. The configuration 2 has a mass penalty but is attractive for accuracy . Furthermore it uses only fully developed and qualified equipment. Finally as a result of this comparison the three systems are attractive at various levels. The configuration 3 has nevertheless all the advantages of mass, accuracy and security of development. COMPARISON WITH OTHER CONTROL MOMENT GYRO SYSTEMS The mechanization and the control with CMG system widely depends on the application and especially on : - the inertia of the whole system - the disturbances to be compensated on the whole system - the pointing requirements. For example, the selected CMG system used on sky lab : 3 orthogonal control moment gyro 2 DOF or the one proposed in ref /3/ for CMG, are based on the requirement for three axis attitude. control of very large systems with various targets in a large range. In such cases, configurations with several fixed momentum wheels are not valid for the main reason that the instantaneous torque deliverable by a FMW is small compared to the one deliverable by a gimbal system. Thus, the case of geostationary satellite appears as a very particular case of application where disturbing torques are small ones, where inertia are relatively small and where the attitude must be maintained toward a same objective.
154
J. BROQUET CONCLUSION The comparison between 1 DOF and 2 DOF systems shows that for most of the applications there is not significant difference at performance level. As, in addition, the practical application of the OGMW remains difficult-even if the theoretical problems are perfectly solved-the 1 DOF should be generally preferred. The selection between FMW and 1 OOF system largely depends on the mission requirements for orbit inclination control or not and for simultaneous multi earth targets mission or not. Obviously, the DOF when required can always be placed at antenna level and this is a subject for a trade off according to each mission. Various mechanization of 1 DOF systems seems to be attractive. , At European level, the skewed wheels configuration (4 FMW with redundancy) has now the large advantage of being based on qualified hardware at this time. The 2 skewed wheels + Reaction wheel configuration requires the development of a reaction wheel able to work as a small FMW either in one way or in the other, depending on the large FMW failure. But this development seems to be straight-forward and this configuration could be the most attractive one. The SGMW with stepper motor configuration requires the development of the girnbal system. It could be the most attractive from mass and power points of view. BIBLIOGRAPHICAL REFERENCES /1/ Dr. U. Renner : Girnballed Momentum Wheels with self locking actuators - ESTEC working paper n° EWP/887 Noordwijk - January 1975 /2/ Dr. U. Renner : Attitude Control Requirements for future Communication Satellites - Presentation of a paper at IFAC Symposium "Automatic Control in Space" Noordwijk - October 1975 /3/ Tsuneo Yoshikawa : A steering low for a roof type configuration of girnbal control moment gyro system 6TH IFAC Triennal World Congress - August 24-30, 1975 /4/ Broquet J. : Selection and adaptation of a control law for a double girnballed momentum wheel system on a large solar array satellite. 6th IFAC Triennal World Congress August 24-30, 1975 /5/ Orbital Test Satellite performance capability study ESTEC Control 2272 /6/ Detailed assessment of the Orbital Test Satellite Platform for application to ECS.ESTEC Contract 2349 /7/ Study of attitude control and stabilization system using multiple fixed wheels. 155
146
J. BROQUET NOTATIONS : DOF Degree of transverse freedom, FMW Fixed Momentum Wheel SGMW : Single Girnbal Momentum Wheel DGMW : Double Girnbal Momentum Wheel x, y, z satellite axes: X along the orbital velocity, y perpendicular to the orbit plane and z toward the earth center Ix, Iy, Iz : satellite moments of inertia H : wheel angular momentum Hy : angular momentum along y axis h
angular momentum in the x, z plane
J
wheel transverse inertia orbital rate spacecraft attitude angle about x, y, z gimbal angle angle of thruster torque with respect to roll axis.
PRINCIPLE OF CONTROL WITH A ONE-DEGREE-OF-FREEDOM SYSTEM The first main characteristic of the 1 DOF systems is the capability of operating an accurate attitude control on an inclined orbit. When the orbit inclination on the equatorial plane is i ~ 0, the 1 DOF systems allow to operate an ideal satellite pOinting with respect to the earth center whatever the direction of the degree of freedom in the x, z satellite plane. For example, if the degree of freedom is along the roll axis the angular momentum will be maintained close to the North/ South inertial direction. Similarly, if the degree of freedom is along the yaw axis, the angular momentum will be maintained close to the inertial direction normal to the orbit plane.
The the the and
second important characteristic of the 1 DOF system is capability of linearizing the roll control provided that DOF is not along the yaw axis, and of decoupling the roll yaw control if the DOF is along the roll axis.
Resulting from these two characteristics and from the fact that the yaw control stability and accuracy is not at all dependent on the orientation of the DOF, the optimized orientation appears generally as the roll axis as will be shown in this paper.
147
J. BROQUET Principle of yaw control As for the FMW system used on OTS, the yaw control with a 1 DOF system is based on the whecon law. In the optimized case where the DOF is along the roll axis, the yaw orientation is completely defined by the angular momentum orientation. Compared with the OTS case the H control about roll and yaw axis is obtained by ,the same thruster actuation logic using 'f + ~ Q! (0 instead of'-P as measurement of H misalignment about the roll axis.
Short term behaviour
Long term behaviour
Nevertheless the roll angle being not dependent on the H control the threshold on ~ can be selected relatively large in accordance with the acceptable yaw angle. Then the thruster actuation impulse bit can be widely increased. The roll transient due to nutation at time of thruster actuation can be minimized by an accurate open loop actuation of DOF. The short term behaviour as defined on the drawing corresponds to a perfect open loop compensation. Principle of roll control This principle is widely dependent on the mechanization of the DOF and will be discussed later. Compensation of yaw error due to known disturbing torques If large disturbing torques are identified, the effect of these torques on yaw angle can be cancelled by an appropriate modulation of the DOF angle measurement before the thl'uster logic. A constant disturbing torque MDC along roll axis can be cancelled by a constant bias on the DOF angle measurement (~C /W , li) . An inertial torque M,....I (M,....I = A sin Wt, M,....I = A cos Wt) can be cancelled by M sin~ ~ias on the DOF Mfi~le measurement: (00 = A (&in(Wt- 2o<.))jW.H. Then for large solar array satellite where large disturbing torques are applied, the 1 DOF systems allow to really cancell their effect on yaw angle without supplementary hydrazine consumption or number of thruster actuations.
148
J. BROQUET PRINCIPLE OF CONTROL WITH A 2-DEGREES-OF-FREEDOM SYSTEM This principle has been the subject of a lot of publications. See for example 14/. The main characteristics of this control are the linearization of both roll and yaw control, the minimization of hydrazine consumption if the DOF off-loading is operated at given pOint of the orbit and the capability of yaw error compensation for known disturbing torques. COMPARISON OF PERFORMANCE OF ONE AND TWO DEGREES OF FREEDOM SYSTEMS The main performances to be compared are the pointing accuracies, the hydrazine consumption and the number of thruster actuations, the mass, power and reliability. POinting accuracies Roll accuracy capability. Identical for the two systems Yaw accuracy capability. Only the effect of inertial torques is shown because for large solar array satellites this effect is the only significant one. . Expression of yaw error for a 2 DOF system : 't' = Mol Wo H . Expression of yaw error for a with K such that
DOF system :
er =
K. Mol WoH
YAW ERROR 4'
TYPICAL CURVE FOR H=50NMS AND REALISTIC ASSUlIPTIONS ON DISTURBING TORQUES
hres-\~ -hold f.bh an
~:(3~
~s = O. 1
2 SGMW ( ~s=O. 1°)
( ~s=0.05°
0
0 .1 0. 2 0. 4 0. 8 1.6 K2.3 1.
1.3 I • I 5 1.0
(-1= 0.0 5 1.7 1. 3 1.1
.07
SOLAR ARRAY POWER
0'---+-_ __+_ _ _ _-+-_--'''''.:0.5" -1 2 ~ "KW'
Number of thruster actuations and hydrazine consumption These curves are given for the same hypothesis as the yaw accuracy curves.
149
J. BROQUET 'lJ6 NUMBER OF ACTUATIONS
ON S YEARS
/
/
/
/
HYDRAZINE CONSUMPTION,S YEARS
/
/
SGMW (~=O.l°)?le)
//
/
~
5.40 DGMW 1 OFF ,/
lO~
day
",y"
~/
__ - -
:/" y
'G"GMW 2 OFF
_/~
/
/
/
/
/
~
~-::;r
loading/day
2
4 K'" POWER
JoKW
POWER Mass power and Reliability of hardware for attitude control excluding a hydrazine subsystem POWER
MASS
40 Kg
1 OOF
Ml
2 OOF
Ml + 10 Kg
~
PI
'>C
50 W
PI + 10 W
RELIABILITY/ 5 YEARS =«
0.887
Less than 0.75
The various curves and results of comparison show that the difference between the 1 and 2 OOF systems is not really significant as far as the performances are concerned except for the hydrazine consumption for large solar array satellites. But such a consumption can be certainly reduced to a large extent by using a direct torque compensation for most of the inertial torque. This compensation can be based on : - a compensation of torques about the normal to the solar array plane by a wind will torque resulting from a control of the angle between the panels, - a compensation about the transverse axis in the solar array plane by a magnetic torque obtained with a coil in the solar array plane. Thus the mass and reliability penalty for the 2 OOF systems and the remaining technological problems of OGMW systems lead to a general preference for the 1 OOF systems.
150
J. BROQUET
MECHANIZATION AND CONTROL OF THE ONE DEGREE OF FREEDOM SYSTEMS Skewed momentums wheels in the yz plane (DOF along roll axis) This mechanization uses two FMW such as those used for OTS. The angle 2 S between the wheels must be defined in the range 14 to 60 0 as a function of the acceptable speed variation range for each wheel.
~ I
Roll control principle The roll control is based on an internal angular momentum exchange and must be defined in order to insure stability of the satellite motion and particularly the nutation. Simplified equations of the system :
I
I-f> + H....... + Woh = MDX DZ
flY - H -f> + WoH't"'= M . •.
The roll short term motion appears as : f/r where r = H/I is the nutation frequency.
2
•
+'f= - h/ H
In order to stabilize the roll motion h must be put into the form : h/H = K1:P + K2 + K3'f' + possibly K4
+
!'-f'
So the control torques applied to the wheels theoretically use the 2nd order derivative of 'f. That is an important characteristic of the one DOF system with DOF about the roll axis. In fact the wheels can be controlled as internal servo onto the speed comman~~~ f (H, h) appears as a function of 'i', 'f' ,J'f instead of 'i', '1', 'i', but this does not change the problem of noise and electronics drift when the nutation periods are large (more than 200 sec for satellite power of 1 KW). To solve the problem of these 2nd order derivatives one can generally define a very simple control law based on a large decoupling between the nutation and the precession frequencies. This decoupling is verified for satellites with solar array power up to at least 1.5 Kw. For larger satellites the decoupling can be maintained through a direct torque compensation or by increasing the wheels angular momentum. whi~h
If the decoupling is not obtained, the -f term can be got with digital hardware through a Kalman filter,with only steady state gains for electronics minimization.
151
J. BROQUET Note on the selection of the OOF direction in x z plane. If the decoupling between H drift and nutation motion is not obtained one could avoid the requirement for -f by selection of the OOF orientation along a skewed direction of the x z plane. The stability margin when using a P.O. network is then the more effective the closer the OOF is to the z axis. But such a configuration with a skewed OOF has the disadvantage of increasing the yaw error : - as a function of ~ sinlS' where't is the angle between roll and OOF direction. Since a roll control A'f requires a var ia tion of ~: A ~ = 6.'f1 cos'lS the var ia tion of 'Vat the same time is A'Y = A'f tg CS'. This expression clearly shows that'1S'must be minimized. - as a function of fu tg'6 where fWB is the initial false zero of the OOF angle measurement which is corrected after in-flight identification in order to minimize hydrazine consumption. - aS,a function of (~C/WH) tg'lS x aX1S are compensate~.
when constant torques along
So e~cept for missions where the yaw accuracy requirement is not very severe, the preferred orientation of the OOF is the roll axis. Block diagram of the control. It defines the main functions of the system. Wheel velocity • close loop cont. frequency (I) Cd) 2
iJOF angle determination
(2)
Compensation ~T~h-r-u-s~t-e-r-a-c~t~.~l-o-g~i-c~~of thruster including compensat. I actuation hruster act.
Single gimbal momentum wheel with stepper moto~. As for 2 skewed FMW systems the roll control law must theoretically use a 2nd order derivative of 'f . But in addition the roll control actuation is necessarily quantized, leading to a decreasing of the roll accuracy as a function of the step size. A large simplification of the control law can be proposed when the nutation motion is widely decoupled from the H drift. The roll angle can then be controlled within a given threshold by a logic ordering a step change as soon as the threshold is exceeded. Each step change sequence can be operated without theoretical impact on the nutation by using a logic sequence as defined in reference III leading to a roll transient corresponding -to one step size. 152
J. BROQUET A slight modification of this step sequence in accordance with the logic control system allows the nutation to be really damped. This damping effect has a threshold whose magnitude is a function of the damping efficiency and of the uncertainty on the knowledge of the nutation period. Single gimbal momentum wheel with reversible torquer Such an equipment can be easily derived from the DGI-1:W with reversible torquers developed by TELDIX for ESA. Roll control principle. The roll control of a SGMW reversible torquer differs from the previous ones by the fact that the wheel is free to move with respect to the satellite about the gimbal axis (roll axis in the selected configuration) • The wheel nutation frequency then appears as W= H/'v'I'J. The stabilization of the satellite roll angle can be obtained by a phase lead network (~+~-f) for the gimbal torque determination, but this control does not stabilize the wheel nutation which must be necessarly damped by using the gimbal angle measurement. In fact, the problem becomes more complex when the gimbal friction torque is considered. Due to technological reasons for the equipment itself and to mass considerations, the SGMW free gimbal - seems to be at the present time less attractive than the SGMW stepper motor. Fixed momentum wheel close to -y wheel in the xyz plane This mechanization ant the associated control are not really different from the skewed wheel ones. The angle£is defined for two reasons :
,.:;..---+0::.
• it avoids the low velocities for the small wheel and then it simplifies the problem of speed measurement, • it avoids any transient in the roll control due to stiction torque which would appear, if the mean wheel velocity was varying about zero.
153
J. BROQUET COMPARISON BETWEEN THE VARIOUS 1 DOF SYSTEMS This comparison must be based on the configurations including redundancy provision. The most attractive configurations appear as shown on the next figures.
/;)~ SGMW Figure 1
~~
Skewed FMW Figure 2
~
h. Skewed + reaction
FMW wheel
Figure 3
The configuration 1 seems to be attractive from the mass and power point of view but requires a successful development of the basic equipment . The configuration 3 is attractive for mass and accuracy point of view but requires the development of a Reaction Wheel. The configuration 2 has a mass penalty but is attractive for accuracy . Furthermore it uses only fully developed and qualified equipment. Finally as a result of this comparison the three systems are attractive at various levels. The configuration 3 has nevertheless all the advantages of mass, accuracy and security of development. COMPARISON WITH OTHER CONTROL MOMENT GYRO SYSTEMS The mechanization and the control with CMG system widely depends on the application and especially on : - the inertia of the whole system - the disturbances to be compensated on the whole system - the pointing requirements. For example, the selected CMG system used on sky lab : 3 orthogonal control moment gyro 2 DOF or the one proposed in ref /3/ for CMG, are based on the requirement for three axis attitude. control of very large systems with various targets in a large range. In such cases, configurations with several fixed momentum wheels are not valid for the main reason that the instantaneous torque deliverable by a FMW is small compared to the one deliverable by a gimbal system. Thus, the case of geostationary satellite appears as a very particular case of application where disturbing torques are small ones, where inertia are relatively small and where the attitude must be maintained toward a same objective.
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J. BROQUET CONCLUSION The comparison between 1 DOF and 2 DOF systems shows that for most of the applications there is not significant difference at performance level. As, in addition, the practical application of the OGMW remains difficult-even if the theoretical problems are perfectly solved-the 1 DOF should be generally preferred. The selection between FMW and 1 OOF system largely depends on the mission requirements for orbit inclination control or not and for simultaneous multi earth targets mission or not. Obviously, the DOF when required can always be placed at antenna level and this is a subject for a trade off according to each mission. Various mechanization of 1 DOF systems seems to be attractive. , At European level, the skewed wheels configuration (4 FMW with redundancy) has now the large advantage of being based on qualified hardware at this time. The 2 skewed wheels + Reaction wheel configuration requires the development of a reaction wheel able to work as a small FMW either in one way or in the other, depending on the large FMW failure. But this development seems to be straight-forward and this configuration could be the most attractive one. The SGMW with stepper motor configuration requires the development of the girnbal system. It could be the most attractive from mass and power points of view. BIBLIOGRAPHICAL REFERENCES /1/ Dr. U. Renner : Girnballed Momentum Wheels with self locking actuators - ESTEC working paper n° EWP/887 Noordwijk - January 1975 /2/ Dr. U. Renner : Attitude Control Requirements for future Communication Satellites - Presentation of a paper at IFAC Symposium "Automatic Control in Space" Noordwijk - October 1975 /3/ Tsuneo Yoshikawa : A steering low for a roof type configuration of girnbal control moment gyro system 6TH IFAC Triennal World Congress - August 24-30, 1975 /4/ Broquet J. : Selection and adaptation of a control law for a double girnballed momentum wheel system on a large solar array satellite. 6th IFAC Triennal World Congress August 24-30, 1975 /5/ Orbital Test Satellite performance capability study ESTEC Control 2272 /6/ Detailed assessment of the Orbital Test Satellite Platform for application to ECS.ESTEC Contract 2349 /7/ Study of attitude control and stabilization system using multiple fixed wheels. 155