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Study of a neural network-based system
for stability augmentation of an airplane Author: Roger Isanta Navarro
Annex 2
Airplane Dynamics
Supervisors: Oriol Lizandra Dalmases
Fatiha Nejjari Akhi-Elarab
Aeronautical Engineering
September 2013
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i
Contents
1 Introduction ....................................................................................................... 1
2 Coordinate reference frames ............................................................................... 2
2.1 Earth-fixed axes .................................................................................................. 2
2.2 Local horizon axes .............................................................................................. 2
2.3 Wind axes ........................................................................................................... 2
2.4 Body axes ........................................................................................................... 3
2.5 Stability axes ...................................................................................................... 4
3 Equations of motion ........................................................................................... 6
3.1 Euler angles and rotation matrices ..................................................................... 6
3.2 Kinematic equations ........................................................................................... 7
3.2.1 Linear motion ............................................................................................. 7
3.2.2 Angular motion ........................................................................................... 7
3.3 Dynamic equations ............................................................................................. 8
3.3.1 Linear motion ............................................................................................. 8
3.3.2 Angular motion ........................................................................................... 9
3.4 Summary of the equations of motion .............................................................. 10
4 Linearized equations of motion ......................................................................... 11
4.1 Previous considerations ................................................................................... 11
4.2 Reference steady state ..................................................................................... 11
4.3 Linearization procedure ................................................................................... 12
4.4 Linearized equations ........................................................................................ 12
4.4.1 Kinematic equations ................................................................................. 12
4.4.2 Dynamic equations ................................................................................... 13
4.5 Linearized air reactions .................................................................................... 13
4.5.1 Separation of motions and other hypothesis ........................................... 14
4.5.2 Linear forces and moments ...................................................................... 14
4.6 Linearized equations of motion ........................................................................ 15
5 Linearized system of equations for short period mode ....................................... 17
ii
List of figures
Figure 2.1 Earth-fixed axes (left) and local horizon axes (right). ...................................... 2
Figure 2.2 Wind axes. ....................................................................................................... 3
Figure 2.3 Body axes......................................................................................................... 4
Figure 2.4 Stability axes. ................................................................................................... 4
1
1 Introduction
This annex contains the procedures necessary to deduce the linearized equations of
motion of the aircraft, which will be used during system development.
First, the different reference systems often used in the mechanics of flight are
presented. Next, the equations of motion of the plane are derived to be, finally,
linearized. Obtaining the linearized equations of the longitudinal motion of an airplane is
the main objective of this Annex. After these are obtained, the Annex includes the
procedure to derive the system of equations corresponding to the short period mode, as
it will also be necessary for training and testing the stability augmentation system.
The contents and procedures within this Annex resulting from a summary of the
contents of [1] and [13].
2
2 Coordinate reference frames
The definition of the following frames of reference is required for the formulation of the
equations of motion of the airplane.
2.1 Earth-fixed axes
The Earth-fixed reference frame is the inertial reference frame to which the global
motion of the airplane is defined. Its origin is a fixed point of the surface. axis
points downwards to the center of the earth and axis points to a fixed direction
(usually North). axis forms a Cartesian right-handed coordinate system (pointing
usually to East).
The Earth is going to be considered flat, with no gravity gradients and the dynamic
effects of its rotation will not be taken into account.
Figure 2.1 Earth-fixed axes (left) and local horizon axes (right).
2.2 Local horizon axes
Local horizon axes ( ) are parallel to Earth axes with its origin being a fixed
point of the airplane. Usually, the center of gravity is chosen as origin.
2.3 Wind axes
Wind axes are fixed to the plane, with its origin in the center of gravity. points in
the direction of the oncoming relative wind. is perpendicular to , is contained in
𝑂𝐸
𝑧𝐸
𝑦𝐸
𝑥𝐸
𝑂𝐸
𝑧𝐸
𝑦𝐸
𝑥𝐸
𝑂𝐻
𝑧𝐻
𝑦𝐻
𝑥𝐻
3
the plane of symmetry of the airplane and points downwards. axis forms a Cartesian
right-handed coordinate system.
Three angles are defined with respect to the Earth-fixed or the local horizon axes:
velocity yaw angle , velocity pitch angle and velocity balance angle , which are
represented in Figure 2.2.
Figure 2.2 Wind axes.
Although these axes are suitable when dealing with point performance problems, they
present some advantages to take into consideration, for instance it is a useful frame to
present relevant aircraft performance data such as aerodynamic forces, and the relative
wind velocity is always null in and axes.
2.4 Body axes
Body axes are fixed to the airplane, with its origin in the center of gravity. points
forward and is contained in the aircraft plane of symmetry. is perpendicular to ,
also contained in the plane of symmetry and points downwards. axis forms a
Cartesian right-handed coordinate system.
Three angles are defined with respect to the Earth-fixed or the local horizon axes: yaw
angle , pitch angle and balance angle , which are represented in Figure 2.3. The
rotation matrix to perform transformations from Earth to body axes and vice versa is
defined in Section 3.1.
𝑥𝑊
𝜒
𝜒
𝛾
𝛾
𝜇
𝜇
𝑦𝐻
𝑥𝐻 𝑧𝐻
𝑧𝑊
𝑦𝑊
𝑽
4
Figure 2.3 Body axes.
2.5 Stability axes
Stability axes are chosen so that is aligned with the speed in a reference condition
of steady symmetric flight. With this choice the reference values and (reference
lateral and vertical speed) are zero, and important simplifications resulting in the
equations of motion and the expressions for the aerodynamic forces.
Figure 2.4 Stability axes.
𝑥𝐵
𝜓
𝜓
𝜃
𝜃
𝜙
𝜙
𝑦𝐻
𝑥𝐻 𝑧𝐻
𝑧𝐵
𝑦𝐵
𝑧𝑠
𝑥𝑠
𝑦𝑠
𝑉 𝛽
5
Since the orientation of the axes will vary depending on the initial flight conditions, the
values of the inertia tensor should be corrected as follows:
(2.1)
(2.2)
(
) (2.3)
where the subscript refers to the principal axes of inertia and is the angle between
(principal axis) and (stability axis), defined positive.
6
3 Equations of motion
3.1 Euler angles and rotation matrices
The following development uses the angles defined in body frame of reference, for they
will be the most widely used. However this development is general to any composition
of rotations.
Consider a vector expressed in Earth reference frame, three rotations will be
required to transform it to another reference frame, for instance, body axes. These
three rotations will be, in order of execution:
1. rotation around the axis, which defines a new reference frame: .
[
] [
] (3.1)
2. rotation around the axis, which defines a new reference frame: .
[
] [
] (3.2)
3. rotation around the axis, which reaches the body axes frame of reference.
[
] [
] (3.3)
These rotation matrices are orthogonal; therefore the inverse transformation can be
achieved using their transposed matrices. The combination of all three rotations results
in the rotation matrix .
[
] [
] [
] (3.4)
(3.5)
The resulting matrix to transform from Earth axes to body axes, which is also
orthogonal, is:
[
] (3.6)
and to transform from body axes to Earth axes:
7
[
] (3.7)
3.2 Kinematic equations
3.2.1 Linear motion
Consider the aircraft position with respect to the Earth reference frame defined within
the same frame:
{
} (3.8)
The time derivative of is:
{
}
(3.9)
where indicates this is the speed with respect to the ground (Earth) expressed in
Earth axes. This speed can also be computed transforming the speed expressed in body
axes to earth axes, using expression (3.7).
{
} (3.10)
{
} [
]{
} (3.11)
Ground speed can be written in terms of the aerodynamic speed and the wind
speed :
(3.12)
{
} {
} {
} (3.13)
3.2.2 Angular motion
To establish a relationship between the angular speed of the airplane and the time
derivatives of the angles one should consider that each turn is performed
over axes of different frames of reference, which are affected by previous rotations. The
angular speed is defined as follows.
8
{
}
(3.14)
and may be written in terms of the Euler angles time derivatives, considering their
corresponding frame of reference.
(3.15)
The following transformations allow expressing the components in the body reference:
{
} (3.16)
[
] { } {
} (3.17)
[
] { } {
} (3.18)
Therefore,
{
} {
} {
} (3.19)
Finally, the angular speed components may be written as
{
(3.20)
or, isolating the time derivatives:
{
( )
( )
(3.21)
3.3 Dynamic equations
3.3.1 Linear motion
The dynamic equations governing the linear motion of an airplane may be obtained
from Newto ’s seco l w:
∑
(3.22)
9
The forces acting on the body are aerodynamic and propulsive forces and weight (which
will require a transformation). The covariant derivative of the velocity must be taken
into account, since the reference frame is moving.
{ } (
| ) (3.23)
{ } {
} ({
} { } {
}) (3.24)
The following linear motion dynamic equations are obtained:
{
( )
( )
( )
(3.25)
3.3.2 Angular motion
The dynamic equations governing the angular motion of an airplane may be obtained
from the conservation of angular momentum:
∑
(3.26)
where is the angular momentum defined
(3.27)
with and being the inertia matrix and the contribution of the rotating parts of the
airplane respectively. The airplane presents symmetry around plane, therefore the
off-diagonal terms and are equal to .
[
] {
} (3.28)
The angular momentum yields:
{
} (3.29)
When applying the conservation of the angular motion the covariant derivative of the
angular momentum must be taken into account, since the reference frame is moving.
{
}
| (3.30)
10
{
} {
} {
} {
} (3.31)
The following angular motion dynamic equations are obtained:
{
( )
( ) ( )
( )
(3.32)
3.4 Summary of the equations of motion
The equations defining the aircraft motion obtained throughout the previous sections
are presented below:
{
( ) ( )
( ) ( )
( )
( )
( )
( )
( ) ( )
( )
(3.33)
11
4 Linearized equations of motion
The equations obtained above are frequently linearized when used in stability and
control analysis. This linearization is performed according to the Small Disturbance
Theory, in which the motion of the aircraft is assumed to consist of small deviations
from a reference condition of steady flight. This assumption has proven to provide good
results, especially dealing with stability problems of unaccelerated flight.
This theory presents, certain limitations; for instance when dealing with high pitch
values (
); a situation in which other approaches should be taken into account.
These situations, however, do not correspond to those discussed within this study;
therefore, the Small-Disturbance Theory model will be assumed valid.
4.1 Previous considerations
Within the development of this theory the reference values of the variables will be
denoted with a subscript, and the small perturbations will be preceded with the delta
symbol , i.e.:
(4.1)
Moreover, when the reference value is equal to zero, the delta symbol is usually
omitted.
All the disturbance quantities are assumed to be small, therefore, terms of order higher
than 1 when squared or multiplied by other disturbances:
( )( ) (4.2)
When dealing with trigonometric functions, the following relations are used considering
that the disturbances are much smaller than one ( ):
( ) (4.3)
( ) (4.4)
4.2 Reference steady state
The Small-Disturbance Theory is applied to a reference steady flight condition, which
may be used to eliminate all the reference forces and moments from the equations to
linearize. This steady flight condition is taken under the following assumptions:
Unaccelerated or steady.
Symmetric (no sideslip).
12
Straight (no angular velocities).
Leveled wings.
According to these assumptions, the reference steady flight condition equations yield:
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
4.3 Linearization procedure
The linearization procedure is exemplified with the dynamic equation along the axis.
( ) (4.13)
The products and are of higher order, therefore may be neglected. Expanding
each variable yields:
( ) ( ) (4.14)
( ) (4.15)
Subtracting the equation of the steady motion (4.8), the linearized equation is obtained:
(4.16)
4.4 Linearized equations
The resulting linearized kinematic and dynamic equations obtained by similar
procedures as the one shown in Section 4.3 follow.
4.4.1 Kinematic equations
(4.17)
(4.18)
13
(4.19)
(4.20)
(4.21)
(4.22)
4.4.2 Dynamic equations
(4.23)
(4.24)
(4.25)
(4.26)
(4.27)
(4.28)
4.5 Linearized air reactions
As performed with the variables in the previous sections, the aerodynamic loads may be
also considered as the addition of a value of reference condition of steady flight plus
small deviations from said value:
(4.29)
The perturbed quantity will depend on some of the state variables and their time
variables, measured at the reference condition:
(
) (
)
(
)
(4.30)
which for convenience will be shortened as
(4.31)
These quantities are the so-called stability derivatives. Stability derivatives
must be computed for each airplane and depend on the flight configuration and
14
characteristics. For the purpose of this study, these are considered data and will not be
computed. Their values are may be found in Annex 4 – Data for Boeing 747.
4.5.1 Separation of motions and other hypothesis
It is common practice to consider separately the longitudinal and lateral motion of the
airplane. This allows working with smaller expressions of easier handling and the results
do not differ significantly from the solution that would be obtained without considering
this separation of motion.
The separation of motion is based on the following two hypotheses:
- As a consequence of the symmetry of the airplane and the symmetry of the
reference motion, stability derivatives of asymmetric forces and moments with
respect to symmetric motion will equal zero.
- The opposite case: that the stability derivatives of symmetric forces and
moments with respect to asymmetric motion equal zero is not clear. However
for simplicity and for the purposes of this study will be considered true.
Further hypothesis are also considered:
- All derivatives with respect to rates of change of motion variables may be
neglected, except for and .
- The derivative is also negligibly small.
- The density of the atmosphere is assumed to be constant with altitude (for the
range of altitudes considered).
4.5.2 Linear forces and moments
After the previously stated hypothesis the forces and moments may be written:
(4.32)
(4.33)
(4.34)
(4.35)
(4.36)
(4.37)
where the terms with subscript refer to control forces and moments that result from
the control vector, which will be introduced lately.
15
4.6 Linearized equations of motion
Substituting equations (4.32) to (4.37) into equations (4.17) to (4.28) and considering
the separation of motions property just presented, the following systems of equations
for both longitudinal and lateral motion are obtained:
16
(4.3
8)
(4.3
9)
(4.4
0)
(4.4
1)
(4.4
2)
(4.4
3)
(4.4
4)
Lon
gitu
din
al m
oti
on
{
}
[
[
]
[
]
[
(
)
]
(
)
]
{
}
{
}
La
tera
l mo
tio
n
{
}
[
]
{
}
{
}
w
her
e
(
)
⁄
(
)
⁄
(
)
⁄
17
5 Linearized system of equations for short period mode
Since a short period stability augmentation system is to be designed, the simplified
equations corresponding to this motion are also derived.
The short period mode presents a longitudinal speed which is substantially constant,
especially when compared to the airplane pitching rate. Therefore, considering Equation
(4.38), the -force equation can be entirely omitted and can be set to .
{
}
[
[
]
[
( )
]
( )
]
{
}
{
}
(5.1)
In addition, the numerical examples and experiences indicate that is small compared
to , and that is also small when compared to the product . Including these
simplification yields to:
{
}
[
[
]
[ ]
]
{
}
{
}
(5.2)
Finally, simplifying the previous expression with , the linearized system of
equations for short period mode is obtained:
{ }
[
[
]
[ ]
]
{ }
{
}
(5.3)
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