IEEE - Aerospace and Electronic Systems - April 2023 - 33

de Celis and Cadarso
along the trajectory. The goal is to determine the angular
parameters of the rocket as well as the line of sight
between the rocket and the target.
The IMU calculates position, velocity, and attitude
vectors by integrating accelerations and angular velocities
in body axes using Newton-Euler equations. As a result,
wind angles, such as bank ðmÞ, flight path ðgÞ, and heading
ðxÞ, may be calculated using the velocity vector and Euler
angles ðf; u; cÞ. The atmosphere model can also be used
to estimate the Mach number (M), together with the position
and velocity vector.
Line of sight ðLOSe
!
Þ is computed in Earth axes by
subtracting rocket position ðre
!Þ, which is determined by
IMU sensors, from target position tpe
!
, which is communicated
to the GNC system, for example, via an external data
link in the launcher.
GUIDANCE AND CONTROL
A modified proportional law is used for rocket guidance
and it is only turned ON once the rocket has burned through
all of the fuel it has on board [8]. The following equation
describes the mathematical expression for the recommended
guideline:
ndemh
ndemv
¼ N " # jj
_g
2
6
6
6
6
d
dt atan
4 d
" #
dt atan
!
!
LOSe
!
LOSe
!
je
!
ie
!
LOSe
!
LOSe
!
where, ndemh and ndemv
ke
!
ie
!
_x
3
7
7
7
7
5
(5)
are horizontal and vertical demanded
load factors, represented in body axes, respectively, and N is
the PN constant, which has been set to 3 (note that this value
has been obtained from experimental results).
Modern control theory is used to design the proposed
system. Its foundations are as follows. First, the system is
linearized at the working point. Second, a closed-loop
feedback system with adjustable gains is constructed.
Third, the inputs and outputs of the control system are
used to feed the training of a NN. Finally, the NN is used
to assist the system in determining the best gains for maintaining
high levels of stability and performance, even outside
the flight envelope conditions used during the
training process. This section ends with the introduction
of the employed actuator strategy.
LINEARIZATION
The system is linearized at 10 ofthe Mach number interval
shown in Figure 2. It is also linearized in five time intervals
during fuel burn. The aim is to consider variations in
rocket mass. The following assumptions apply when linearizing
equations (1), (2), (3), and (4).
APRIL 2023
2
6
6
6
4
2
6
6
6
4
jjLOSe
!
_
Rocket flight is bidimensional in order to simplify
the model equations.
Rocket's body is perfectly axis symmetrical.
Trigonometric functions can be linearized because
the flight angles are small enough.
The linearization of the equations of motion is only
shown for one of the planes of motion, namely the horizontal
plane, due to the rocket's axis-symmetric structure.
As the following equation shows, this model can be formulated
as a matrix equation:
_
Xp
!
¼ ApðM; tÞXp
!
ApðM; tÞ¼
BpðM; tÞ¼
2
4
2
4
þBpðM; tÞ Up
!
a11 a12 a13
001
a31 a32 a33
b1
b3
3
5
3
5
Xp
!
!
¼
2
4
x
c
c_
Up ¼ d½
where, the components of ApðM; tÞ and BpðM; tÞ are
functions of aerodynamic coefficients and mass, among
others. This implies that they are functions of Mach
number and time. Note that, r is a function of altitude,
but for this application it can be considered a constant,
as flight altitude is bounded in a small interval. Each of
the terms in ApðM; tÞ and BpðM; tÞ is showed in
expressions (7)-(9)
f1 ¼ 4mðtÞþrSdCNqðMÞ f2 ¼ rSdV2ðMÞ
2rS
2IyðtÞ
a11
a12
a13
b1
a31
a32
a33
b3
3
7
7
7
5
3
7
7
7
5
¼ f2
¼ f1
2
6
6
6
4
2
6
6
6
6
4
VðMÞCLa
VðMÞCLa
ðMÞ
ðMÞ
dCNqðMÞ
VðMÞCNdðMÞ
rSVðMÞ df1CMq ðMÞ
2
CMa ðMÞ df1CMq
ðMÞCNd
ðMÞ
3
7
7
7
5
CMa ðMÞþ df1CMq
CMa ðMÞ df1CMq
ðMÞCLa
ðMÞCLa
ðMÞ
ðMÞ
3
7
7
7
7
5
where, V is rocket speed (note that V ¼ aM, where a is
the speed of sound). The deflection of both control surfaces
is denoted by d, which is the contribution of both.
Note that, the rest of the variables have been introduced
before.
The command input to the control system in rocket
guidance is often a load factor, ndem, which may be represented
as a function of _x: ndem ¼ _xV
g . The accelerometers
in the IMU may be used to measure load factors directly.
Similarly, the IMU can measure c and its derivative.
can be calculated using n, c, and _c[15]. The measurement
in the following equation relates the state vector (Xp
!
)
with the measurements:
IEEE A&E SYSTEMS MAGAZINE
33
x_
(9)
(7Þ
(8Þ
3
5
(6)
IEEE - Aerospace and Electronic Systems - April 2023

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