1. Introduction
To realize accurate attitude control, the reaction wheel is commonly
used for many satellites. However, it has never used for very small
satellite such as micro-class or nano-class satellites. The main
reason is that existing reaction wheel is too heavy and take too much
space to carry, and also needs too much electricity to work for this
class of satellite.
In order for such satellites to have a high precision attitude control
capability, a novel attitude control device which is called three
dimensional reaction wheel (3DRW) has been proposed. 3DRW consists of
only one levitated spherical mass which can rotate around arbitrary axis,
while conventional reaction wheel consists of three or more one-axis
rotating mass to realize three axes control as shown in Fig.1. The
spherical mass consists of conductive metal, and is rotated by
rotational magnetic field. This leads to the reduction of the weight
and volume of the system. The spherical mass is kept in a given
position by electromagnetic force. This would reduce the failure
caused by the mechanical contact. For these reasons, 3DRW has many
advantages to conventional reaction wheel.
Conventional Reaction Wheel3DRW1-DOF Wheels3-DOF Rotation AxisRigid Ball
Fig.1: Conventional reaction wheel and 3DRW
In this paper, the results of the analysis and experiment on the
dynamics and control of 3DRW are shown. First, the concept of 3DRW
and analytical formulation is introduced. Then, the results of
experiment which demonstrates the dynamics of the rotation mass in a
magnetic field and the characteristic of open loop control of the
rotation. The control system to keep the ball in a given position is
also investigated. The results of the experiments are compared with
numerical simulations, by which the design method of 3DRW are
obtained. Finally, the characteristic of attitude control using 3DRW
is evaluated, and itfs advantage for microsatellites is revealed.
2. Rotation mechanism of 3DRW
Suppose that a magnetic field is rotating around the rigid ball, and
then an eddy current is generated on the surface of the ball. Due to
the interference between these magnetic field and eddy current, the
Lorentz force is produced on the ball and becomes the rotational
torque. The rotational magnetic field can be generated about arbitrary
axes; consequently, the rigid ball can be rotated about arbitrary axes.
In this section, the torque of 3DRW generated by magnetic field is
investigated.
2-1. Torque Generated by the Rotation Magnetic Field
Fig.2 indicates the analytical model. Suppose that the rigid ball has
superior electrical conductivity and rotates in uniform magnetic
field. is the angular velocity of the ball about z-axis, and B is
intensity of uniform magnetic field.
For the small element on the surface of the ball, let r the vector
from the center of the ball to the element, the angle between x-y
plane and the r vector, and the angle between x-axis and orthogonal
projection of r for x-axis plane.
The voltage induced by the magnetic field in the element is expressed by
()2coscosdVrBdփƃ=~?=?vBl (1)
where l is the characteristic length of the element. We assume that
electric current flows on the surface of the ball along the }z
direction. Resistance of element is expressed by ()sincosdrlddRSrd
d
ӃЃЃӃƃЃ=== (2)
where is the conductivity constant and S is the area of electrical
current path in the element, and is the effective volume of the
element. The electrical current along z-axis is derived as follows:
2coscosdVdirBddRЃփƃӃ==? (3)
փBrzxփBrzx
Fig. 2: A schematic diagram of analytical model
Lorentz force is induced on the element by the interaction between
the current and the magnetic field as follow:
)(Bi~?=dldF???????????=ƃӃӃӃƃӃƃփsinsincos0coscos223ddBr (4)
By integrating equation (4) for entire surface of the ball, the
following equation is derived:
???????????=~=?22/2/24208/300փЃӃƃBrdddTFr (5)
Assuming that the magnetic field is uniform through the whole sphere
for low angular velocity, the torque generated around z-axis is
integrated as follows: 252403BrTփЃ?= (6)
Equation (6) indicates that the torque is generated against the
direction of the rotation.
In the actual 3DRW, the ball rotates in inverse direction of the
rotating magnetic field. Therefore the rigid ball rotates in the
same direction of the rotation magnetic field.
Suppose that the magnetic field rotates relative to the ball, the
torque is expressed as 252)(403BrTփЃ?= (7)
where is the angular velocity of the rotation magnetic field.
2-2. Torque Generated by Magnet Field Orthogonal to the Rotational Axis
Electric magnets are also used to keep the ball in a given position.
If the magnetic field is parallel to the rotational axis of the ball,
the disturbance torque would be induced. The torque generated for this
relative position is derived in a similar way to the previous section.
(8) ???????????=~=?4/0022/2/2420փЃӃƃBrdddTFr
Assuming that the magnetic field is uniform through the whole sphere
as before, the torque generated around z-axis is integrated as
follows:
252201BrTփЃ?= (9)
Equation (9) implies that the torque is generated in a direction
opposite to the rotation. Thus, the magnetic field parallel to the
rotation axis reduces the spinning rate.
3. Rotation Control System
In this section, the formulation and the results of the experiments of
rotation system are shown.
3-1. Equation of Motion
Suppose that the rigid ball rotates at BR=[x, y, z]T in a
coordinate system, R, remain stationary relative to magnetic field.
From the equation (6) and (9), the torque is expressed in the coordinate system as ??????????????????????????????+??????????+???????????=zyxzyxtBtBtBrtTփփփЃ22242)(233)(323)(33281)( ??????????????????????????????+?=zyxzyxtBtBtBBrփփփЃ222242)()()(381E (10)
where E is unit matrix, and B is intensity of magnetic field and
satisfies following equation: const.)()()(222=++=tBtBtBBzyx (11)
Suppose that R rotates at relative to the inertial coordinate
system and set the coordinate transform matrix from inertial
coordinate
system to R, ARI, as
???????????????????????=ӃƃӃƃӃӃƃӃƃӃƃccssssccscsctctststcARI01 (12)
where t, and are euler angles of order about z, x then z.
Here we set B as
?????????????????????++??????????+=ӃƃӃƃӃӃƃƃccssstsctB)sin(0)cos(B (13)
The angular velocity vector of B, RI, is described as follows:
(14) ???????????=ӃƃӃƃsincoscossincosRI
Using
RIBIBRփփ?= (15)
then equation (10) is transformed as following equation: ()RIBIzyxtBtBtBBrTփփЃ?????????????????????+?=222242)()()(381E (RIBItBrփփЃ??)(81242A ) (16)
Assuming that A(t) is averaged in cycle of rotation, equation (16)
is transformed as following equation: (RIBIBrTփփЃ??=A24281 ). (17)
This equation denotes that we can control the torque by choosing
RI properly.
3-2. Experimental Setup
In the experiments, the rotation magnetic field is generated by
three or four electrical magnets. A schematic diagram of the rotation
system is indicated in Fig.3. Mag-CMag-EMag-DMag-BMag-AXYZSignal
Generator&Power AmplifierAC currentOutput Signal
ch1ch2Mag-AMag-AMag-BMag-CMag-DMag-CMag-EAC Currents flow in the
MagnetsFor rotation around Z-axis
For rotation around Z-axis Fig.3: A schematic diagram of the rotation
system
The five electrical magnets are positioned around the rigid ball.
To rotate the ball around z-axis, mag-A,B,C and D are used. To rotate
the ball around x-axis, mag-A,C and E are used. The magnets are
applied AC current which is generated by a signal generator and
amplified by a power amplifier. To simulate the rotation magnetic
field, the AC current has phase difference, /2, between adjacent
magnets. To investigate the dynamics of the sphere without mechanical
contact on the ground, an air levitation system is used. To measure
the angular velocity of the ball, laser tachometer is used. The rigid
ball is ticked so that the laser tachometer could easily detect the
pulse every rotation. Output signal by the tachometer is amplified by
operational amplifier circuit, and then logged.
3-3. Experimental Results
In experiments, the frequency of the rotation magnetic field and
intensity of it are changed as parameters. The amplitude of signal
is the maximum voltage of input signals for power amplifier generated
by the signal generator. Frequency of the rotation magnetic field is
the same of the signal generated by the signal generator. To observe
effects of the intensity and the frequency of the rotation magnetic
field, the amplitude of signal and the frequency of the rotation
magnetic field are changed.
Fig.4 shows terminal angular velocity in relation to frequency for
some input voltage. This denotes that terminal angular velocity of
the ball depends on both frequency of the magnetic field and voltage
of input signal.
Fig.4: Experimental result (Rotation around z-axis)
Through the experiments, it is found that the angular velocity of
the rigid ball can be changed by either the frequency of the rotation
magnetic field or intensity of it. Thus if we control the rigid ball
same as the conventional reaction wheel, there are 2 ways to control.
3-4. Verification by Numerical Simulation
To verify the formulation of rotation system derived in section 3-1,
the numerical simulation is carried out and compared with the results
of experiments.
Using equation (7), the rigid ballfs equation of rotation as follow:
փփЃaekskBrdtdJ???=))1((403252 (18)
where J is the inertia moment of the ball, ke is the efficiency of
input torque, s is the slip coefficient, and ka is the coefficient of
the air damper torque proportion to the angular velocity. The values
of constant numbers are estimated through some other experiments,
and determined as table 1.
Table 1: Determined values of constant number
Constant Number
Determined Value
ke
0.11
ka
9.7146~10-8 [Nms]
s
0.15
Using these values, the numerical simulations are carried out.
Figure 5 shows the results of the simulation about the transient
response of angular velocity for 30 and 90Hz, and compared with one of
experiments. The results of simulations agree with the one of
experiments closely. This denotes that the rotation system is properly
expressed by the analytical model of equation (18).
Fig.5: The results of Simulations and Experiments
4. Position control System
To increase the efficiency of the rotation for the lower power
consumption, the sphere mass should be set closely to the
electromagnets. Therefore, 3DRW needs high accuracy positioning
control. This section shows the formulation and experiment of position
control system. To keep the sphere mass in a given position,
electromagnetic power is used.
4-1. Dynamics and Control
Let x1, x2, x3 and u be position relative to the electromagnet,
velocity, current and control variable of voltage. Then the model
of the system can be expressed as
21xx=& (19) +?=mFxm2& (20) (33xcuk )
LRxii?+=& (21) ?????????=21123expffmKxKxF (22)
where m is weight of the rigid ball. denotes the acceleration of
external force, and sufficiently small compared to the gravity
acceleration on the ground. Fm is electromagnetic power, and Kf1, Kf2
are constant numbers which determine the efficiency of electromagnet.
Equation (21) denotes the equation of current on the AC circuit,
and R, L are resistance and inductance of the circuit. These equations
can be linearized around a given position (x10, x20, x30) as ukLRxxxLRaaxxxi????????????+?????????????????????????=??????????000000103213,21,2321&&& (23)
where
????????=2102212301,2expfffKxKKmxa (24) ?????????=210221303,2exp2fffKxKKmxa (25)
From equation (23), the transfer function G(s) from u to x1 can be
derived as follows: ))(()(1,223,2RLsasRkasGi+??= (26)
From equation (30), a2,1 is positive so the system is unstable.
Here we use PID controller, and set the compensator C(s) as ????????++=sTsTKsCDIP11)( (27)
where KP, TD and TI are constant gain. The closed loop transfer
function T(s) can be written as follow: IDIDTKsRaKsLaKTRsLsTKKssKTsT+?+?++++=)()()(1,221,2342(28)
where K=BRkiKP. By choosing KP, TD and TI appropriately, the position
control system would be stable and has required response.
4-2. Experimental Setup
The experimental position control system is shown in Fig.6.
Fig.6: Diagram of the position control system
An electrical magnet is placed over the rigid ball, and a displacement
sensor is placed on the magnet. A microprocessor unit receives
positional data from the sensor, and controls the current of the
electrical magnet through a power amplifier.
To negate gravity acceleration, position control system on the ground
needs large electromagnets and electric power. However, actual 3DRW used in space will need few power and small electromagnets for small acceleration.
To simulate the control system under small acceleration, air bearing
is used. This allows the ball to move with small power, though this is
limited within small range.
The force of the air bearing to levitate the ball decreases in
proportion to distance from the equilibrium position. For this case,
the external force terms in equation (20) is expressed as
)(102xxkmFxm?+?=& (29)
where k is constant coefficient and x0 is the the equilibrium
position. Correspond to this rewrite, equation (30) is transformed as
follows:
kKxKKmxafff?????????=2102212301,2exp (30)
This denotes that the transfer function G(s) on equation (26) doesnft
have the positive pole in case 201Fxx?>, and the system is stable,
while there remains steady state error. This error may vanish by PID
controller of equation (27).
To determine the values of constant number used in the previous
subsection, some calibration tests are performed. Consequently, the
values of constant number are determined as table 3.
Table 3: Constant number on position control system
Constant Number
Determined Value
M
0.1734 [kg]
Kf1
0.6374
Kf2
2.8031~10-3 [m]
R
5.0 []
L
0.1 [H]
k
1.3358~104 [s-2]
Using these values, the PID controllerfs gain are configured and
then determined after several trial and error tests. In the
experiment, gains are given by
KP = 0.02, TD = 125, TI = 0.01.
The simulation result of PID control using above gains are shown in
Fig.7. The initial position is 416.5m, and the desired position is
11m lower than it. In the transient response, the minimum position
is 405.487m, so the overshoot is about 0.1% of the displacement.
In the stationary response, amplitude of vibration is about 0.01% of it.
Fig.7: Simulation result of PID control
4-3. Experimental Results
Fig.8 shows the experimental result of PID control of the same
condition with simulation. In the transient response, the minimum
position is 405.1m, so the overshoot is about 4% of the
displacement. In the stationary response, amplitude
POWE
RAMP
MPU (PIC)
PID Controller
Electrical Magnet
Rigid Ball
Air Bearing
Displaceent
m Sensor
D/A
Control Signal
Positional Data
of vibration is about 5% of it. The electrical current is about 780mA
in the transient response and about 50mA in the stationary one.
This denotes that very small electrical power is needed to keep
the ball in a given position with high accuracy, compared to magnetic
levitation systems.
Fig.8: Experiment result of PID control
5. The characteristic of 3DRW
In this section, the characteristic of attitude control using 3DRW
is evaluated, and the advantage of 3DRW to conventional reaction wheel
for very small satellites is revealed.
5-1. Maximum torque of 3DRW
Here the maximum torque of 3DRW is estimated. The law of conservation
of energy is described as sPI?=221 (31)
where P is the electrical power and s is the time of supplying power.
From the derivation of this equation, the torque relative to the
electrical power can be written as
PKdtdITe== (32)
where Ke is the energy conversion efficiency. Take these parameters
as P = 5W, Ke = 1%, = 10Hz, then T = 7.96~10-4 Nm.This is
sufficient order to eliminate the environmental disturbance torque
for small satellites.
5-2. Comparing with Conventional Reaction Wheel
Here we estimate the rotorfs mass for 3DRW and conventional
cylindrical RW for which the same angular momentum, H, is generated
by the same angular velocity, . For 3DRW, the radius of rotor is
513815??????=HRDRW (33)
Meanwhile, the radius of rotor for RW is 5122??????=փHRRW (34)
where is the aspect ratio of the rotor. From these equations, the
weight ratio of rotors for conventional RW to 3DRW is obtained. Fig.9
shows the weight ratio relative to for both no redundancy case and
redundancy case. In the first case, the number of rotor is three for
RW and one for 3DRW, and 3DRW has the advantage in the weight
for >0.2. In the latter case, the number is four for RW and two
for 3DRW, and 3DRW has the advantage for >0.4. For commonly used
RW, is about 0.2 to 0.4, so 3DRW has the advantage for the non
redundant system. For the redundant system, 3DRW has the disadvantage
in weight. However, the redundancy is 2 for 3DRW and 4/3 for RW,
so 3DRW has the advantage in redundancy.
Fig.9: Weight ratio of traditional RW to 3DRW
6. Conclusion
In this paper, the novel attitude control device, 3DRW, is investigated.
The basic dynamic response of the rigid ball is obtained using air
bearing. The rigid ballfs behavior is dominated by the intensity and
the frequency of the rotating magnetic field.
The analytical model of torque and rotation magnetic field is
constructed. The results of simulations using analytical model is
agree with the results of experiments. Thus the analytical model can
be used for practical control.
The position control is also formulated and investigated through the
experiments. The ball is controlled with very high accuracy and low
electrical current.
Finally, the characteristic of attitude control using 3DRW is
evaluated,
and the advantage of 3DRW to conventional reaction wheel for very
small satellites such as micro-class or nano-class satellite.
7. References
[1] Atsushi Iwakura, Shinichi Tsuda, Yuichi Tsuda, gThe Investigation of 3 Dimensional Reaction Wheelh, The 57th International Astronautical Congress(IAC), IAC-06-B5.6.10, 2006.
[2] Atsushi Iwakura, gThe Basic Study of Rotation System using Lorentz forceh, The 25th International Symposium on Space Technology and Science, Kanazawa, June, 2006.