I. Introduction
This paper is concerned with the problem of
generating feedforward control inputs for flexible
spacecraft, robotic manipulator, and pointing system,
which are often required to maneuver as quickly as
possible without significant structural vibrations after a
maneuver.
Many researches have been examined about such a
maneuvering problem. Most researchers propose
algorithms based on feedforward bang-bang control
with constant-force discrete actuators1-6. However,
there are some drawbacks to these algorithms.
First, feedforward bang-bang control algorithms are
generated from formulation of input shaping as a zero
placement algorithm that can give rise to negative
shapers. Negative input shapers have notch-shaped or
bandpass-shaped frequency characteristic, so these
input shapers are sensitive to modeling errors, and
small modeling errors lead to significant amount of
residual vibrations. In other words, negative shapers do
not work well on most real systems because the system
frequencies need to be known very accurately1.
Second, negative input shapers consist of limited
number of positive and negative constant-amplitude
impulses, so these algorithms are effective for systems
with finite number of flexible-modes. This can be
restated in such a way that real flexible spacecraft
usually has a lot of flexible-modes which are varied by
each rotational axis, so negative input shapers can
cause considerable amount of residual vibrations which
originate from inescapable flexible-modes.
Third, constant-force discrete actuators such as onoff
reaction jets are usually used as bang-bang control
actuators. This implies low accuracy of control inputs
compared to linear continuous actuators such as CMGs
or RWs. This is mainly because on-off reaction jet is
naturally inferior to CMG or RW in output torque
accuracy.
This paper presents a preshaping profiler that
overcomes above-mentioned drawbacks. The
preshaping profiler has the function of computing
maneuvering profiles for feedforward control inputs.
This preshaping profiler is generated from extra
insensitive functions which are called sampling
function (also known as sinc function). Sampling
function is known to be insensitive to frequencies
above a designed one. This preshaping profiler
significantly reduces residual vibrations at the endpoint
of maneuvers. The reason is as follows. First,
residual vibrations are generated only as homogeneous
solutions of satellite dynamic-equations if there are no
control inputs after the maneuvers (control inputs exist
in only maneuvering durations). Second, the
homogeneous solutions can be disappeared if there are
no structural frequencies in the frequency range of
preshaping profiler. In consequence, all the residual
vibrations in the end-point of maneuvers can be
reduced by the preshaping profiler. This preshaping
profiler also has high robustness to modeling errors,
and this preshaping profiler is effective for systems
with infinite number of flexible-modes. The reason is
as follows. This preshaping profiler has lowpassshaped
frequency characterictic, hence, this causes no
residual vibrations as long as flexible-modes exist in
higher frequency than a certain frequency (sampling
function frequency). This preshaping profiler is
effective for high-accuracy control of satellites or
manipulators or pointing systems with large
complicated flexible structure.
This paper is organized as followings. Section II
shows the outline of the controller. Section III defines
preshaping profiler. Section IV shows effectiveness of
this preshaping profiler compared to past methods.
Section V describes the outline of VSOP2(ASTRO-G)
attitude control subsystem (i.e., ACS) which uses the
proposed preshaping profiler, and application example
to VSOP2(ASTRO-G) maneuvers is shown.
II. Outline of the Controller
The block diagram of ACS controller for
VSOP2(ASTRO-G) is shown in Fig. 1. It consists of
the feedforward control part of the nil-mode-exciting
(NME) profiler and feedback control part of C. The
input-output relation from the reference signal r and
disturbance w to the measurement output y is given by
y = Gyrr +Gyww (1)
where
2 1 ( ), ( ) yr yw yw G G J C s G I PC P = + = + ? (2)
P denotes the plant transfer function. J denotes the
spacecraft moment of inertia. We design C so that the
closed-loop system is robustly stable and has
disturbance attenuation capability to w. NME profiler
generates the feedforward profile of the attitude
maneuvers.
Fig. 2 shows the frequency assignments of each
controller. Feedforward profiler (NME profiler) has
higher frequency characteristic up to just below the 1st
flexible mode. Feedback controller C has lower
frequency and is used only for attenuating the attitude
drift which is generated in long-time constant.
Because of this frequency separation between
feedforward profiler (NME profiler) and feed back
controller C, the characteristic of the NME profiler is
not changed by the effect of feedback controller C. In
the next chapter, we explain how to design the NME
profiler.
III. Preshaping Profiler for Flexible Spacecraft
It is known that the shortest path of large angle
rotational maneuvers is performed by means of singleaxis
rotation. The single axis is called Euler-axis. The
Euler-axis is the only axis determined from initial
attitude and final attitude.
This paper presents a maneuvering method of
rotating around the Euler-axis (we are concerned here
with spacecraft maneuver and not concerned with
robotic manipulator and pointing system).
Sampling function (also known as sinc function) is
defined as
( )
t
t
y t
s
s
sin
( ) = . (3)
This function has no frequency response above s
.
Fig. 1 Block diagram of ACS controller
(VSOP2(ASTRO-G))
Fig. 2 Frequency assignments of each controller
(VSOP2(ASTRO-G))
It is possible to separate frequency characterictic of
ACS and flexible structures completely by means of
sampling function. Fig. 3 shows overplot of two
sampling functions (upper figure, there are two
waveforms which have 180deg offset and inverse
form) and combination (addition) of these two
waveforms (lower figure). Fig. 4 shows frequency
characteristic of sampling function. As shown in blue
solid line of Fig. 4, sampling function has in itself no
response above a certain frequency (here, 0.2Hz) and
flat response below the frequency, and as shown in
green solid line of Fig. 4, this basic character is not
changed by combination (addition) of sampling
functions. Therefore residual vibrations of flexible
structures at the maneuver end-point can be reduced to
almost zero by means of sampling function. As a result,
observation missions which are performed after each
maneuvers can be performed effectively in the shortest
time.
To separate frequency characteristic of ACS and
flexible structures, put the frequency of sampling
function s
as
lowest mode
s flex < (4)
where right-hand term is defined as the lowest flexiblemode
frequency of the spacecraft.
Angular acceleration profile for control inputs is
generated by addition of two sampling functions. As
shown in upper figure of Fig. 3, there are two sampling
functions in which peaks of these waves has offset of
one period in time axis, positive wave is used for
acceleration, negative wave is used for deceleration.
Angular acceleration profile for control inputs is given
by addition of these two waveforms. However, as
lower figure of Fig. 3 shows, there are small waves left
after deceleration, so it induces residual vibration of
flexible structures. For this reason, a window function
shown in middle figure of Fig. 5 is used to be added for
smooth damping at the end-point of maneuvers. Lower
figure of Fig. 5 shows angular acceleration profile for
control inputs drawn by above procedure.
In this way, angular acceleration profile for back
and forth rest-to-rest maneuvers is given as
( ( ))
( )
( ( ))
( )
( ( ))
( )
( ( ))
( )
?
?
?
?
?
+
?
?
?
?
?
=
22
22
21
21
2
12
12
11
11
2 1
2
sin sin
( )
sin sin
( )
( )
smpl offset
smpl offset
smpl offset
smpl offset
smpl offset
smpl offset
smpl offset
smpl offset
t t
t t
t t
t t
g t
t t
t t
t t
t t
A g t
dt
d t
(5)
where is the body-angle based on the Euler-axis, A
is the maximum acceleration, offset ij t is the offset time
of each peaks, and k g is the window function. Here,
the offset time offset ij t are expressed as
s
peakoffset s t t
2 = = (6)
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
time [s]
Amplitude [ ]
sampling function 11&12
smplfun11
smplfun12
0 10 20 30 40 50 60 70
-1.5
-1
-0.5
0
0.5
1
1.5
time [s]
Amplitude [ ]
sampling function 11+12
smplfun11+12
Fig. 3 Sampling function (upper) and
combined sampling function (lower)
0 10 20 30 40 50 60 70
-2
-1
0
1
2
time [s]
Amplitude [ ]
sampling function 11+12
smplfun11+12
0 10 20 30 40 50 60 70
0
0.5
1
time [s]
Amplitude [ ]
window function
hamming window1
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
time [s]
Amplitude [ ]
(sampling function 11+12)~(window function)
(smplfun11+12)~(window function)
Fig. 5 Combined wave (upper ? same as Fig. 3),
window function (middle), and
combined wave (lower)
10
-2
10
-1
10
0
10
1
10
2
10
3
0
0.2
0.4
0.6
0.8
1
1.2
Frequency [Hz]
Amplitude [ ]
sampling function 11 / 11+12
smplfun11
smplfun11+12
Fig. 4 Sampling function (frequency characteristic)
acceleration
deceleration
single sampling function
combined sampling function
mnv s t = 3t (7)
2 2 2 11
mnv obs peakoffset
offset
t t t
t = + ? (8)
offset offset peakoffset t = t + t 12 11 (9)
offset offset mnv obs t = t + t + t 21 11 (10)
offset offset mnv obs t = t + t + t 22 12 (11)
where mnv t is the maneuver time, obs t is the
observation time.
The window functions k g are hamming window
which expressed as
>
+
?
+
?
+
+ ?
=
2 2
0 for abs
2 2
for abs
2
2
0.54 0.46 cos
( )
1 2
1 2
1 2
offset i offset i win
offset i offset i win
offset i offset i
win
i
t t t
t
t t t
t
t t
t
t
g t
(12)
where win t is the time width of window function and
equals to maneuver duration mnv t . Fig. 6 shows a back
and forth maneuvering profile computed from Eq. (5).
We call this preshaping profiler the nil-mode-exciting
(NME) profiler.
So far we have outlined the way in which angular
acceleration profile is given. Now we would like to
reconsider the purpose of this research. The main
purpose is to perform rest-to-rest maneuver of flexible
spacecraft in the shortest time with linear actuators. For
that purpose, such maneuvers described below are
necessary.
[1] with a profile that does not cause residual vibrations
(Transient vibrations during maneuvers have no
influence on mission performance)
[2] make the most of torque output performance of the
actuator
Item [1] can be achieved by the characteristic of the
angular acceleration profile which is generated by the
NME profiler. As shown in Fig. 4, the angular
acceleration profile has no frequency response above a
certain designed frequency s
. Consequently, the
profile does not cause steady-state vibrations after the
attitude maneuvers.
As to item [2], as shown in upper figure of Fig. 3,
there are two sampling functions in which these waves
has a time offset, and positive wave is used for
acceleration, negative wave is used for deceleration.
Angular acceleration profile is defined by addition of
these two waveforms. Here, for the maximum uses of
the actuator ability, the peak of the angular acceleration
profile A should be set to the angular acceleration that
is computed from the maximum torque of the actuators.
This is how [2] is achieved.
In addition, a window function is combined with
sampling functions. This purpose is to reduce residual
vibration after maneuvers caused as particular solution
of satellite dynamic-equations with minimum loss of
the advantages of [1] and [2].
IV. Evaluation of NME Profiler
The effectiveness of the nil-mode-exciting (NME)
profiler compared to a past method is shown as follows.
Fig. 7 shows maneuver profile given by a past method7.
The profiler of this method proposes the maximum
acceleration and the maximum deceleration profiles
during maneuvers. Since frequency separation of ACS
and flexible structures is not considered in this past
method as shown in lower figure of Fig. 7, excitation of
0 10 20 30 40 50 60 70
-0.4
-0.2
0
0.2
0.4
Maneuver Profile @ Euler Axis
d2/dt2 [deg/s2]
0 10 20 30 40 50 60 70
-1
-0.5
0
0.5
1
d/dt [deg/s]
0 10 20 30 40 50 60 70
-1
0
1
2
3
[deg]
time [s]
Fig. 6 Acceleration (upper), body-rate (middle),
and body-angle (lower) profiles of
a back and forth maneuver
0 10 20 30 40 50 60 70
-6
-4
-2
0
2
4
6
x 10-3
time [s]
Amplitude [ ]
1st Maneuver
dTH2a
10
-2
10
-1
10
0
10
1
10
2
10
3
0
0.005
0.01
0.015
Frequency [Hz]
Amplitude [ ]
1st Maneuver
dTH2a
Fig. 7 Acceleration profile (upper) and
frequency characteristic (lower) of the past method7
flexible-modes cannot be avoided. Therefore, the
observation mission can not be started until residual
vibrations damp sufficiently. In other words, this
maneuver needs longer time to complete.
Open-loop dynamics simulations using these
methods are performed for 3 [deg] rest-to-rest
maneuvers. Each simulation is performed on the same
conditions which include the same flexible satellite and
same actuator performances. Table 1 shows major
parameters for analyses. Desired maneuver profiles
generated by each method are shown in Fig. 8 and Fig.
9. Open-loop dynamics simulation results are shown in
Fig. 10 and Fig. 11. The antenna gain-losses are
calculated as the sum square of flexible-modes
displacements. So magnitude of the residual vibrations
can be estimated by antenna gain-losses. As shown in
Fig. 10, residual vibrations after maneuver of the past
method do not damp for long time. On the other hand,
as shown in Fig. 11, residual vibrations at the end-point
of maneuver of the NME profiler are reduced to almost
zero. The effectiveness of NME profiler was verified
by open-loop dynamics simulations.
V. Application to VSOP2(ASTRO-G)-ACS
In this section, application to VSOP2(ASTRO-G)-
ACS is introduced. VSOP2(ASTRO-G) (VSOP: VLBI
Space Observatory Programme) is a satellite which
includes 2-CMGs and 4-RWs as control actuators.
Missions of VSOP2(ASTRO-G) are performed by way
of back and forth rest-to-rest high speed maneuvers
(switching maneuvers) in which the same couple of
target celestial sources are pointed repeatedly. The
satellite attitude is required to be highly-stable at the
end-point of maneuvers. Fig. 12 shows on-orbit image
of VSOP2(ASTRO-G). This satellite has two kinds of
flexible appendages, large deployable antenna and
flexible solar panel.
The feature of this system exists in the part of NME
profiler that computes the best profile of feedforward
control inputs in which the character frequencies of the
flexible appendages are not excited.
Condition for the switching maneuver analysis is
3[deg]/15[s]. Figure 13 shows the body angle errors
which are calculated as the differences between
evaluation results and target body angle. Each error is
within acceptable level.
VI. Conclusion
A new approach for computing feedforward control
inputs for flexible spacecraft rest-to-rest maneuvers
was proposed. The effectiveness, robustness, and
expandability of this preshaping profiler was confirmed.
Ideal frequency separation was realized between
flexible-modes and controllers.
Acknowledgments
This research was supported by all the members of
VSOP2(ASTRO-G) team in JAXA-ISAS and NEC
TOSHIBA Space Systems, Ltd.
The methods described in this paper are patent
pending. Commercial use of these methods requires
written permission from JAXA-ISAS and NEC
TOSHIBA Space Systems, Ltd.
References
1Singhose, W. E., Seering, W. P., and Singer, N. C.,
gTime-Optimal Negative Input Shapers,h ASME J. Dynam.
Syst. Meas. Control, Vol. 112, 1997, pp. 198-205.
2Singer, N. C., Seering, W. P., gPreshaping Command
Inputs to Reduce System Vibration,h ASME J. Dynam. Syst.
Meas. Control, Vol. 112, 1990, pp. 76-82.
3Singhose, W., Derenzinski, S., and Singer, N., gExtra-
Insensitive Input Shapers for Controlling Flexible
Spacecraft,h AIAA J. Guid. Control Dynam. Vol. 19, 1996, pp.
385-391.
4Liu, Q., Wie, B., gRobust Time-Optimal Control of
Uncertain Flexible Spacecraft,h AIAA J. Guid. Control
Dynam. Vol. 15, 1992, pp. 597-604.
0 50 100 150 200 250 300
-1
-0.5
0
0.5
1
1.5
2
x 10-3 Maneuver Error Angle (3-2-1 Euler Angle)
Roll [deg]
0 50 100 150 200 250 300
-1
-0.5
0
0.5
1
1.5
x 10
-3
Pitch [deg]
0 50 100 150 200 250 300
-1.5
-1
-0.5
0
0.5
1
1.5
x 10
-3
Yaw [deg]
time [s]
Fig. 13 Body angle errors
Fig. 12 VSOP2(ASTRO-G) on-orbit image
Table 1 Conditions for analyses
items conditions
maneuver angle 3 [deg]
maneuver time 15 [s]
CMG rotor momentum 60 [Nms]
CMG max. torque 42 [Nm]/2CMG
5Singh, G., Kabamba, P. T., and McClamroch, N. H.,
gPlanar, Time-Optimal, Rest-to-Rest Slewing Maneuvers of
Flexible Spacecraft,h AIAA J. Guid. Control Dynam. Vol. 12,
1989, pp. 71-81.
6Singh, G., Kabamba, P. T., and McClamroch, N. H.,
gBang-Bang Control of Flexible Spacecraft Slewing
Maneuvers: Guaranteed Terminal Pointing Accuracy,h AIAA
J. Guid. Control Dynam. Vol. 13, 1990, pp. 376-379.
7Saito, T., Maeda, K., Ninomiya, K., and Hashimoto, T.
H., gRate-Profiler Based Minimum-Time Control for
Spacecraft Attitude Maneuver,h IFAC Proceedings of 15th
IFAC Symposium on Automatic Control in Aerospace, 2001,
pp. 83-88.
0 5 10 15 20 25 30
-0.2
-0.1
0
0.1
0.2
0.3
Flex 1
mode i
1st mode
2nd mode
3rd mode
4th mode
5th mode
6th mode
7th mode
8th mode
0 5 10 15 20 25 30
-0.02
-0.01
0
0.01
0.02
Flex 2
mode
i
1st mode
2nd mode
3rd mode
4th mode
5th mode
6th mode
7th mode
8th mode
0 5 10 15 20 25 30
0
200
400
600
800
1000
1200
1400
1600
1800
Antenna Gain Loss
Gain Loss [dB]
time [s]
Fig. 10 Dynamics simulation of the past method7
0 5 10 15 20 25 30
-0.2
-0.1
0
0.1
0.2
0.3
Flex 1
mode
i
1st mode
2nd mode
3rd mode
4th mode
5th mode
6th mode
7th mode
8th mode
0 5 10 15 20 25 30
-0.02
-0.01
0
0.01
0.02
Flex 2
mode i
1st mode
2nd mode
3rd mode
4th mode
5th mode
6th mode
7th mode
8th mode
0 5 10 15 20 25 30
0
200
400
600
800
1000
1200
1400
1600
1800
Antenna Gain Loss
Gain Loss [dB]
time [s]
Fig. 11 Dynamics simulation of the NME profiler
0 1 2 3 4 5 6 7
-0.5
0
0.5
Acceleration[deg/s2]
0 1 2 3 4 5 6 7
0
0.5
1
Rate[deg/s]
0 1 2 3 4 5 6 7
0
1
2
3
Angle[deg]
0 1 2 3 4 5 6 7
0
50
100
150
H
cmg
total[Nms]
0 1 2 3 4 5 6 7
-50
0
50
T
cmg
total[Nm]
time[s]
Fig. 8 Maneuver profiles of the past method7
0 5 10 15
-0.5
0
0.5
Acceleration[deg/s2]
0 5 10 15
-0.5
0
0.5
1
Rate[deg/s]
0 5 10 15
-2
0
2
4
Angle[deg]
0 5 10 15
-100
0
100
200
H
cmg
total[Nms]
0 5 10 15
-50
0
50
T
cmg
total[Nm]
time[s]
Fig. 9 Maneuver profiles of the NME profiler