Aerospace and Electronic Systems Magazine July 2017 - 23
each command. This problem is called control allocation and arises when there is not a unique relation between control inputs and
commands. The aim of control allocation algorithms is to select
specific thrusters and calculate their firing durations through an
optimization to realize force and torque commands derived from
the control system of a spacecraft so that a performance index such
as power consumption or operational time is minimized. In general, this leads to constrained optimization that is a challenging
problem. There are several studies in the field of control allocation
methods. Some review articles that have done thorough studies of
algorithms, and their applications can be found in [19]-[21]. Practical studies on the control allocation issue can be tracked back to
1960s. Crawford [5], [6] showed how to select appropriate thrusters and adjust their time durations with the linear programming
(LP) method. Later, it was showed that the derived condition is not
applicable when the dimension of task is high and/or there are a
large number of thrusters in the attitude control system [22]. On the
whole, control allocation algorithms are divided into two categories: algorithms based on the optimization methods and algorithms
that are not based on the optimization methods. These algorithms
also can be classified as constrained and unconstrained algorithms.
There is some research in the area of control allocation methods,
but a topic that has not been noticed as much is the configuration
design of a redundant attitude control system. In addition to the control allocation algorithm that affects the total RCS efficiency, placing
thrusters in a certain position and angle can play an important role
in maximizing the efficiency. There are two studies concerning the
maximization of the attitude control system efficiency as a function
of thruster positions and angles [7], [23]. A number of studies indirectly address the problem of the attitude performance by different
strategies. In [24], a novel constrained L∞ method based on maximum angular momentum torque distribution and maximum output
torque was introduced to avoid the saturation and frequent unload.
Using this allocation strategy, the maximum allowable acceleration and velocity are calculated for a specific mission and a sample
actuator configuration in real time. The applied control allocation
strategy, along with sliding mode control, makes the designer sure
about the full controllability of the system using reaction wheels.
Another strategy to deal with three-axis thruster control performance
problem, in particular under failure conditions, is to use gimbaled
thrusters instead of fixed ones. In [25], a mathematical relationship
for a case of fixed thrusters, one and two gimbaled thrusters were
presented and the minimum number of required thrusters under the
JULY 2017
NOMENCLATURE
J
Number of thrusters
D
Task dimension or thrust space
LR
Level of redundancy
qs
Spacecraft quaternion vector contains qsi, i
= 1, 2, 3, 4
Qref
Reference quaternion matrix contains qrefi, i
= 1, 2, 3, 4
qe
Error quaternion vector contains qei, i = 1, 2,
3, 4
n
Eigenvector of rotation
ε
Angle of rotation
hBBI
Angular momentum vector of spacecraft
with regard to inertial frame
ω
Spacecraft angular velocity vector contains
ωμ, μ = x, y, z
Ω
Skew-symmetric matrix of spacecraft
angular velocity
I BB
Moments of inertial matrix
τB
External torques on spacecraft
τc
Control torque vector
Kpw
Gain of lag filter
τpw
Time constant of lag filter
Uon
Pulse-width pulse-frequency (PWPF)
modulator on threshold
Uoff
PWPF modulator off threshold
h
PWPF modulator hysteresis
A
Spacecraft thrusters' configuration matrix
R
Thruster's arm radius
L
Thruster's arm length
Ti
ith thruster
c
Efficiency factor of thrusters
href
Required angular momentum
t
Time
T
Control torque level
Kp
Proportional control gain
Kdi
Derivative control gains, i = 1, 2, 3
V
Lyapunov function
IEEE A&E SYSTEMS MAGAZINE
23
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