Aerospace and Electronic Systems Magazine July 2017 - 25
Pasand, Hassani, and Ghorbani
tions, the rotational kinematics of a typical spacecraft is usually
represented by parameters such as Euler angles, quaternions, and
direct cosine matrix. Because Euler angles encounter singularity at
π
± and the computational time of direct cosine matrix representa2
tion is expensive, the representation by quaternion is more applicable. The quaternion is able to deal with nonsingularity; meanwhile, it is free from the trigonometric components. All in all, this
representation is considered an appropriate candidate to present the
attitude behavior of a spacecraft. For this purpose, the quaternion
vector can first be presented as follows:
ε
sin .n1
2
n1
q1
ε
q = q2 = sin .n2 ; n = n2
2
q2
n3
q sin ε .n
4 3
2
ε
cos
2
(5)
(6)
T
CLOSED-LOOP CONTROL APPROACH
To design a control law, the control efforts must be provided so
that the stability of the closed-loop system is achieved. For this
purpose, the quaternion feedback control law is considered as follows [29]:
Figure 1.
The schematic diagram of the PWPF modulator.
JULY 2017
Parameters
(7)
Variations
Kpw
2.5:7.5
pw
0.1:1.0
Uon
0.1:1.0
h = Uon − Uoff
0.2:2Uon
where kp and kdi, i = 1, 2, 3, are positive control gains; T is the control torque level; and qe is the quaternion control error. qe can be
achieved by qe = Qrefqs, or in the expanded form
qref 3 − qref 2 − qref 1 qs1
qref 4
qref 1 − qref 2 qs 2
− qref 1 qref 4 − qref 3 qs 3
qref 2
qref 3 qref 4 qs 4
(8)
where Qref is the reference or commanded attitude quaternion matrix and qs is the current spacecraft quaternion. The following positive definite Lyapunov function can prove the finite time global
stability of the closed-loop system if kp, kdi > 0:
4
1 3
I ii2 Tk p qsi qrefi
2 i 1 i 1
V
where q = q1:3 q4 is an attitude quaternion that represents the
attitude of the spacecraft w.r.t. the local coordinate system.
τ x
k p qe1 + kd 1ω1
τ y = −T k p qe 2 + kd 2ω2
τ y
k p qe3 + kd 3ω3
The Recommended Parameters of the PWPF
Modulator [31]
qe1 qref 4
qe 2 = − qref 3
qe 3 qref 2
qe 4 qref 1
The quaternion vector including qi, i = 1, 2, 3, 4, has a real and
three imaginary elements, where ni, i = 1, 2, 3, and ε are taken as
the eigenvector of rotation and angle of rotation, respectively. The
time derivative of a quaternion is given by
1 −Ω ω
q = 2 −ω 0 q
Table 1.
2
(9)
REACTION THRUSTER MODULATION
Because a reaction thruster does not possess a linear relationship
between its input and its output, w.r.t. its on-off behavior, several
modulation methods have been used to link the level of the required
command with pulse width and frequency. To shape the nonlinear
output of an on-off thruster into a requested linear output, a number of thruster control methods are extensively exploited, of which
the most frequently used method is pulse-width pulse-frequency
(PWPF) modulator. The PWPF modulator introduces discontinuous and nonlinear control actions that may result in worse control
performance, but if it operates in a pseudolinear range, the thruster
will act as continuous effectors [23], [30], [31] . Other modulation
methods like the pseudorate modulator, integrated pulse-frequency
modulator, and pulse-width modulator are also used to shape the
output of constant thrusters. The PWPF modulator is here realized
because of its proved advantages over other types of modulators
[32], [33]. As can be seen in Figure 1, the PWPF modulator consists of a first-order lag filter, along with a Schmitt trigger inside a
negative feedback loop. PWPF parameters are shown in Figure 1.
These parameters are taken into account in the modulator design.
In [32], a thorough analysis was carried out to find the relationships between the characteristics of the PWPF modulator and the
selection of its parameters. The recommendations for the PWPF's
parameters are shown in Table 1. By static, dynamic, and system
IEEE A&E SYSTEMS MAGAZINE
25
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