Aerospace and Electronic Systems Magazine July 2017 - 24

Reaction Control Thruster Configurations for a Three-Axis Attitude Control System
nominal and single point of failure conditions were calculated. In
[26], the function object model (FOM) was presented as a performance criterion to evaluate the thruster system reconfigurability and
identify the component with the highest fault risks.
The subject of the present paper is to design a multijet control
system, emphasizing cases in which redundancy is employed to
improve reliability. Because thrusters are only capable of producing unidirectional torques, the number of required thrusters to fully
control the m degree of freedom is more than m; therefore, the
equivalent torque produced by a set of thrusters depends on not
only the level of thrust output but also the sum of pulse width when
thrusters are activated. It is straightforward to relate the geometry
of the thruster configuration to the degree of freedom and LR using
mathematic relations. The problem of designing a multijet system
for maximum overall effectiveness involves a compromise between the initial fuel supply and the number and arrangement of
jets. The geometrical arrangement has conflicting effects on fuel
efficiency and reliability.
As a rule, the first step to designing a new product is to investigate existing samples. Several samples of spacecraft can be used to
design a new product. It can be helpful to recognize pros and cons
corresponding to each thruster configuration by studying recent
trends and using their points of view in new designs. The results
will be useful by showing how well-suited configurations can be
selected for a given mission. Such information is not reported in
any publication, and obtaining it necessitates an extensive literature search. Similar research was done in the case of spacecraft
attitude control using reaction wheels, in which a number of attitude control performance indices were assessed [27]. The lack of
an article to compromise between these configurations and obtain
a summary about the most efficient one in the case of RTs is felt.
In this paper, several thruster configurations (whose thrusters are
located in the same plane) are investigated through the static and
dynamic behavior analysis in terms of some performance indices.
The rest of the manuscript is organized as follows. The preliminary
mathematical relationships are given first. Then the required kinematics and dynamic are presented, followed by the closed-loop control
structure, applied control allocation, and pulse modulation methods.
Finally, the considered samples are introduced, and the main static and
dynamic analyses, along with numerical results, are then presented.

J ≥ D + 2 LR + 1

(1)

The necessary and sufficient condition for а given configuration to cover a D-dimensional task with LR is as follows:
For every D − 1-dimensional hyperplane that divides the task
dimension in half, there must be at least LR + 1 thrust vectors in
each half.
The mentioned condition is equivalent to the following [5], [6]:
If at least one combination of thrusters can perform
the D-dimensional task, then each subset of D − 1
vectors that are linearly independent uniquely determines а D − 1-dimensional hyperplane containing all of them. For all such hyperplanes, the remaining J − (D − 1) vectors must include at least
2(LR +1) vectors not in the hyperplane , with at
least LR + 1 of them to each side.

SPACECRAFT ATTITUDE DYNAMICS AND KINEMATICS
The orbital transfer of a spacecraft includes the translational and
rotational motion. An acceptable assumption allows us to pursue
attitude dynamics independent of translational motion. According
to the Newton's second law, the summation of external moments
acting on a body is equal to the time rate of change of angular momentum of body B is equal to the time rate of change relative to the
inertial frame of the angular momentum of the body referred to its
center of mass (c.m.), or D I hBBI = mB . Transferring the rotational
time derivative of the preceding equation to the body frame B, we
have [28]

(

)

D I I BBω + ΩI BBω =  τ B

(2)

where τB is external torques on spacecraft, I BB is the spacecraft's
moment of inertia referred to the c.m. of B, ω is spacecraft's angular rate with respect to (w.r.t.) the inertial coordinate system, and Ω
is its skew symmetric matrix. Picking body coordinate ]B, we get
the following closed-form equation:
B

NECESSARY NUMBER OF THRUSTERS
The concept of redundancy is one of the most important design criteria in multi-input systems, because a high enough the LR guarantees system failure avoidance in practice. The LR is defined as one
less than the number of thruster failures, which leads the control
system to fail. For example, if the LR is R = 0, even one thruster's
failure leads to the control system's failure. As a rule, an LR of R =
n allows n + 1 thruster failures before system failure. From another
view, the higher LR means a greater expected lifetime and consequently a more reliable system. RTs can fail either by running out of
fuel or by suffering a fatal sequence of individual thruster failures.
To determine the minimum necessary number of thrusters for a
given LR and to determine the LR of a given configuration, we define D as the task dimension or thrust space, and J as the number of
24

thrusters or thrust vector. Achieving the D-dimensional task with
the minimum LR requires J thrusters, as follows [5], [6]:

B  dω 
B
B
B
B
 B
 B


 I B   dt  + Ω  I B  ω  =   τ B 



(3)

This can be simplified in the three directions, x, y, and z, as
follows:

ω x =
{ω y =

ω z =

τx
Ix

τy
Iy

τz
Iz

−
−
−

)ω ω

( Ix − Iz ) ω ω

− Iy
Ix

− Ix
Iz

)ω ω
x

(4)

The spacecraft dynamics can be represented by the attitude
kinematic and dynamic equations. Regarding the first set of equa-

IEEE A&E SYSTEMS MAGAZINE

JULY 2017

Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine July 2017