Derivation and Definition of a Linear Aircraft Model Reference Publication 1207 1988 National...
108
NASA Reference Publication 1207 1988 National Aeronautics and Space Administration Scientific and Technical Information Division Derivation and Definition of a Linear Aircraft Model Eugene L. Duke, Robert F. Antoniewicz, and Keith D. Krambeer Ames Research Center Dryden Flight Research Facility Edwards, California https://ntrs.nasa.gov/search.jsp?R=19890005752 2018-06-01T17:57:29+00:00Z
Transcript of Derivation and Definition of a Linear Aircraft Model Reference Publication 1207 1988 National...
NASAReferencePublication1207
1988
National Aeronauticsand Space Administration
Scientific and TechnicalInformation Division
Derivation and Definition
of a Linear Aircraft Model
Eugene L. Duke,
Robert F. Antoniewicz,
and Keith D. Krambeer
Ames Research Center
Dryden Flight Research Facility
Edwards, California
https://ntrs.nasa.gov/search.jsp?R=19890005752 2018-06-01T17:57:29+00:00Z
CONTENTS
SUMMARY
INTRODUCTION
SYMBOLS
Vectors ..................................................
Matrices ..................................................
S ubscripts .................................................
Superscript ................................................
NONLINEAR SYSTEM EQUATIONS
1.1 Definition of Reference Systems ..................................
1.2 Nonlinear State Equations .....................................
1.2.1 Rotational acceleration ...................................
1.2.2 Translational acceleration ..................................
1.2.3 Attitude rates ........................................
1.2.4 Earth-relative velocity ....................................
1.3 Nonlinear Observation Equations .................................
1.3.1
1.3.2
1.3.3
1.3.4
1.3.5
1.3.6
1.3.7
1.3.8
Accelerations .........................................
Air data parameters .....................................
Flightpath-related parameters ...............................
Energy-related parameters .................................
Force parameters ......................................
Body axis rates and accelerations .............................
Instruments displaced from the vehicle center of gravity .................
Miscellaneous observation parameters ...........................
2
,1
d
5
5
5
6
ll
11
1-1
15
16
17
1S
20
20
21
21
21
21
22
LINEAR SYSTEM EQUATIONS 22
2.1 Linearization of the State Equation ................................ 23
2.2 Linearization of the Observation Equation ............................ 24
2.3 Definition of Matrices in Linearized System Equations ..................... 26
2.4 Elements of the Linearized System Matrices ........................... 27
3 CONCLUDING REMARKS 30
APPENDIXES 31
A--AERODYNAMIC FORCES AND MOMENTS 31
B--DERIVATION OF THE WIND AXIS TRANSLATIONAL
PARAMETERS V, &, AND _ 35
B.1 Preliminary Definitions ...................................... 35
B.2 Derivation of I}" Equation ..................................... 35
B.3 Derivation of 5 Equation ..................................... 36
B..I Derivation of/) Equation ..................................... 37
PRECEDING PAG_ RLANg N_ FILMED
iii
C--GENERALIZED DERIVATIVES 39
C.1 Generalized Derivatives of the Time Derivatives of State Variables .............. 39
C.2 Generalized Derivatives of the Observation Variables ...................... ,'13
D--EVALUATION OF DERIVATIVES 49
D.1 Preliminary Evaluation ...................................... 49
D.I.1 Rolling moment derivatives ................................. 49 °
D.1.2 Pitching moment derivatives ................................ 50
D.1,3 Yawing moment derivatives ................................. 50
D.1.4 Drag force derivatives .................................... 51D.1.5 Sideforce derivatives ..................................... 51
D.1.6 Lift force derivatives ..................................... 52 _
D.2 Evaluation of the Derivatives of the Time Derivativesof the State Variables ........................................ 53
D.2.1 Roll acceleration derivatives ................................ 53
D.2.2 Pitch acceleration derivatives ................................ 5,'t
D.2.3 Yaw acceleration derivatives ................................ 55
D.2.4 Decoupled roll acceleration derivatives ........................... 56
D.2.5 Decoupled pitch acceleration derivatives .......................... 57
D.2.6 Decoupled yaw acceleration derivatives ..........................D.2.7 Total vehicle acceleration derivatives ...........................
D.2.8 Angle-of-attack rate derivatives ..............................
D.2.9 Angle-of-sideslip rate derivatives ..............................D.2.10 Roll attitude rate derivatives ................................
D.2.11 Pitch attitude rate derivatives ...............................
D.2.12 IIeading rate derivatives ..................................D.2.13 Altitude rate derivatives ..................................
D.2.14 North acceleration derivatives ...............................
D.2.15 East acceleration derivatives ................................
D,3 Evaluation of the Derivatives of the Observation Variables ...................
D.3.1
D .3.2
D.3.3
D .3.4
D.3.5
D .3.6
D.3.7
D .3.8
D.3.9
D.3.10
D.3.11
D.3.12
D.3.13
D.3.14
57 158 1
59
60
62
62
63
64
65
65
66-
66
67 :
68
69
69
70
71
Longitudinal kinematic acceleration derivatives ......................Lateral kinematic acceleration derivatives .........................
Z-body axis kinematic acceleration derivatives ......................
x body axis accelerometer output derivatives .......................
y body axis accelerometer output derivatives .......................
z body axis accelerometer output derivatives .......................
Normal accelerometer output derivatives .........................
Derivatives of x body axis accelerometer output not at the vehicle center
of gravity ........................................... 72
Derivatives of y body axis accelerometer output not at vehicle center
of gravity ........................................... 72
Derivatives of z body axis accelerometer output not at vehicle center
of gravity ........................................... 73Derivatives of normal accelerometer output not at vehicle center
of gravity ........................................... 74Load factor derivatives ................................... 75
Speed of sound derivatives ................................. 76Mach number derivatives .................................. 77
iV
D.3.15
D.3.16
D.3.17
D.3.18
D.3.19
D .3.20
D.3.21
D.3.22
D.3.23
D.3.24
D.3.25
D.3.26
D.3.27
D.3.28
D.3.29
D.3.30
D.3.31
D.3.32
D.3.33
D.3.34
D.3.35
D.3.36
D.3.37
D.3.38
D.3.39D.3.40
D.3.41
Reynolds number derivatives ................................ 77
Reynolds number per unit length derivatives ....................... 78
Dynamic pressure derivatives ................................ 79
hnpact pressure derivatives ................................. 80Mach meter calibration ratio derivatives .......................... 81
Total temperature derivatives ............................... 82
Flightpath angle derivatives ................................ 82
Flightpath acceleration derivatives ............................. 83Vertical acceleration derivatives .............................. 84
Specific energy derivatives ................................. 85
Specific power derivatives .................................. 86Normal force derivatives .................................. 87
Axial force derivatives .................................... 87
x body axis rate derivatives ........................ . ......... 88
y body axis rate derivatives ................................. 89
z body axis rate derivatives ................................. 90
x body axis acceleration derivatives ............................ 90
y body axis acceleration derivatives ............................ 91
z body axis acceleration derivatives ............................ 92
Angle-of-attack sensor output derivatives ......................... 93
Angle-of-sideslip sensor output derivatives ........................ 93Altimeter output derivatives ................................ 9-t
Altitude rate sensor output derivatives .......................... 95
Total angular momentum derivatives ........................... 96
Stability axis roll rate derivatives ............................. 97
Stability axis pitch rate derivatives ............................ 97
Stability axis yaw rate derivatives ............................. 98
REFERENCES 101
V
SUMMARY
This report documents the derivation and definition of a linear aircraft model for a rigid aircraft of constant
mass flying over a fiat, nonrotating earth. The derivation makes no assumptions of reference trajectory or
vehicle symmetry. The linear system equations are derived and evaluated along a general trajectory and
include both aircraft dynamics and observation variables.
INTRODUCTION
The need for linear models of aircraft for the analysis of vehicle dynamics and control law design is well
known. These models are widely used, not only for computer applications but also for quick approximations
and desk calculations. Whereas the use of these models is well understood and well documented, their
derivation is not. The lack of documentation and, occasionally, understanding of the derivation of linear
models is a hindrance to communication, training, and application.
This report details the development of the linear model of a rigid aircraft of consta_t mass, flying over a
fiat, nonrotating earth. This model consists of a state equation and an observation (or measurement) equa-
tion. The system equations 5ave been broadly formulated to accommodate a wide variety of applications.
The linear state equation is derived from the nonlinear six-degree-of-freedom equations of motion. The
linear observation equa.tion is derived from a collection of nonlinear equations representing state variables,
time derivatives of state variables, control inputs, and fiightpath, air data, and other parameters. The linear
model is developed about a nominal trajectory that is general.-
Whereas it is common to assume symmetric aerodynamics and mass distribution, or a straight and level
trajectory, or both (Clancy, 1975; Dommasch and others, 1967; Etkin, 1972; McRuer and others, 1973;
Northrop Aircraft, 1952; Thelander, 1965), these assumptions limit the generality of the linear model. The
prhlcipal contribution of this report is a solution of the general problem o/" deriving a linear model of a rigid
aircraft without making these simplifying assumptions. By defining the initial conditions (of the nominal
trajectory) for straight and level flight and setting the asymmetric aerodynamic and inertia terms to zero,
one can easily obtain the more traditional linear models from the linear model derived in this report.
Another significant contribution of this report is the derivation and definition of a linear observation
(measurement) model. The observation model is often entirely neglected in standard texts. A thorough
treatment of common aircraft measurements is presented by Gainer and tIoffman (1972), and Gracey (1980)
provides a detailed discussion of speed and altitude measurements, llowever, neither of these references
present linear models of these measurements. This report reIies heavily on these two references and uses their
results as one of the bases for the nonlinear measurement equations from which the linear measurement
mode[ is derived. Also included in this report is a large number of other measurements or variables for
observation that have been found to be useful in vehicle analysis and control law design.
Duke and others (1987) describe a FORTRAN program called LINEAR tha.t derives a linear aircraft
model by numerical differencing (Dieudonne, 1978). The program LINEAR produces a linear aircraft model
(1ooth state and observation matrices) that is equivalent to the linear models defined in this report.
This report is divided into two main sections that define the reference systems and nonlinear state and
observation equations (section 1) and derive a linear model presented in the appendixes (section 2). The
appendixes contain a definition of tl_e linear aerody2mmic model used in this report (app. A), a derivationof the wind axis translational acceleration parameters (app. B), generalized Iinear derivatives of the non-
linear state and observation equations (app. C), and the individual derivatives of the state and observation
equations (app. D). The details of the principal results of this report are preseuted in appendix D.
i
7
SYMBOLS
A
a
Hn,i
(Ix
(lx,i
a_-,k
(19
ay,i
ay,k
az
(I z,i
(lz,k
b
C_
D
D:c
DyD_
E_F
fpa
g
goh
h,i
[
I1I2
I4I5[6L
e
M
total aerodynamic axial force, lb
speed of sound, ft/sec
normal accelerometer output, g
output of normal accelerometer not at vehicle center of gravity, g
output of accelerometer aligned with vehicle body z axis, g
output of accelerometer aligned with body z axis, not at vehicle center of gravity, g
kinematic acceleration in vehicle body x axis, g
output of accelerometer aligned with vehicle body y axis, g
output of accelerometer aligned with body y axis, not at vehicle center of gravity, g
kinematic acceleration in the vehicle body y axis, g
output of accelerometer Migned with vehicle body z axis, g
output of accelerometer aligned with body z axis, not at vehicle center of gravity, g
kinematic acceleration in vehicle body z axis, g
reference span, ft
generalized force or moment coefficient
derivative of generalized force or moment coefficient with respect to arbitrary variable x
reference aerodynamic chord, ft
total aerodynamic drag, lb
L-Lly - Ix
specific energy, ft
arbitrary force or moment
flightpath acceleration, g
acceleration due to gravity, ft/sec 2
acceleration due to gravity at sea level, ft/sec 2
altitude, ft
altitude measurement not at vehicle center of gravity, ftinertia tensor
moment of inertia about z body axis, slug-ft 2
product of inertia in x-y body axis plane, slug-ft 2
product of inertia in z-z body axis plane, slug-ft 2
moment of inertia about y body axis, slug-ft 2
product of inertia in y-z body axis plane, slug-ft 2
moment of inertia about z body axis, slug-ft 2
.rx:,-:L[fl .. + 1 :#xz:xI , -total moment about x body axis, fl-lb; or, total aerodynamic lift, Ib
unit length, ft
total moment about y body axis, ft-lb; or, Mach number
- 2
N
75
P
ps
pt
q
qc
qc/Pa
qsRe
Re r
F
rs
S
T
Ttt
u
V
v
ll7
X.
XT
x
Y
Y
ZTz
75i
0
tt
P0
vehicle mass, slugs
total moment about z body axis, ft-lb; or, total aerodynamic normal force, lbload factor
specific power, ft/sec
roll rate (about x body axis), rad/sec
static or free-stream pressure, lb/ft 2
stability axis roll rate, rad/sec
total pressure, lb/ft 2
pitch rate (about y body axis), rad/sec
dynamic pressure, lb/ff 2
impact pressure, lb/ff 2Mach meter calibration ratio
stability axis pitch rate, rad/sec
Reynolds number
Reynolds number per unit length, ft -1
yaw rate (about z body axis), rad/sec
stability axis yaw rate, rad/sec
surface area of wing, ft 2
total angular momentum; or, ambient or fi'ee-stream temperature, °R
total temperature, °R
time
velocity along x body axis, ft/sec
vehicle velocity, ft/sec
velocity along y body axis, ft/sec
velocity Mong z body axis, ft/sec
total aerodynamic force along x body axis, lb
total gravitational force along x body axis, lb
total thrust force along x body axis, lb
vehicle position along x earth axis, ft
total aerodynamic sideforce, lb
total aerodynamic force along y body axis, lb
total gravitational force along y body axis, lb
total thrust force along y body axis, lb
vehicle position along y earth axis, ft
total aerodynamic force along z body axis, Ib
total gravitational force along z body axis, lb
total thrust force along z body axis, lb
vehicle position along z earth axis, ft
angle of attack, rad
angle-of-attack measurement not at vehicle center of gravity, rad
angle of sideslip, rad
angle-of-sideslip measurement not at vehicle center of gravity, rad
flightpath angle, radith control surface deflection
pitch angle, rad
coefficient of viscosity, lb/ft-sec
density of air, lb/ft 3
arbitrary function
¢¢
Vectors
a
E
F
f
gH
h
M
R
U
V
X
Y_u
_x
fl
Matrices
A
A'
B
B'
C
F
F I
G
//
H /
I
j,
LBvR
T
0nX?_
lnxTn
bank angle, rad
heading angle, tad
body axis acceleration vector
attitude vector of Euler anglestotal force vector
state vector function
observation vector function
total angular momentum vector
sum of higher order terms in Taylor seriestotal moment vector
position vector in earth axis system
input or control vector
vehicle velocity vectorstate vector
observation vector
perturbation of control vector
perturbation of state vector
perturbation of time derivative of state vector
rotational velocity vector
state matrix of the generalized state equation, C'k = Ax + Bu
state matrix of the state equation, :_ = A'x + B'u
control matrix of the generalized state equation, C'_ = Ax + Bu
control matrix of the state equation, _ = A'x + B'u
system matrix of the generalized state equation, Ck = Ax + Bu
feedforward matrix of the generalized observation equation, y = Hx + G_ + Fu
feedforward matrix of the observation equation, y = H_x + 17_u
derivative observation matrix of the generalized observation equation, y = Hx + G_ + Fu
observation matrix of the generalized observation equation, y = Hx + G_ + Fu
observation matrix of the observation equation, y = H_x + F'uintertia tensor
scaling matrix for inertia tensor
transformation matrix from earth to body axes
transformation matrix from earth to body axes
angular velocity matrix in the generalized state equation, T::: = fix(t), :k(t), u(t)]n x m matrix of 0 values
an n x m matrix with values of 1 on the diagonal
4
Subscripts
bDgh
,i
,kL
n
n
P
S
T
t
V
'W
X
xyXZ
Y
Y
yzZ
0
aerodynamic; or static or, free stream
body axis system
drag
gravitation al
displacement of altitude instrument
displacement of altitude rate instrument
not at vehicle center of gravity
kinematic
lift
rolling moment
pitching moment
yawing moment
orthogonal
power plant induced
stability axis; or, specific
thrust
total
vehicle-carried vertical axis system
wind reference axis system
displacement in x body axis
x--y body axis plane
x-z body axis planesideforce
displacement in y body axis
y-z body axis plane
displacement in the z body axis
at sea level, standard day conditions; or, nominal conditions
Superscript
T transpose
1 NONLINEAR SYSTEM EQUATIONS
The motion of an aircraft as a rigid body can be described by a set of six nonlinear simultaneous second-
order differential equations. These equations, representing the translational and rotational motion of the
vehicle, call be formulated in tile notation of Kwakernaak and Sivan (1972) and Dieudonne (1978) as a
time-invariant system expressed as= r[x(t), u(t)] (1-i)
where x(t) is the 12-dimensional time-varying state vector (t being time), _(t) is the derivative of x(t) with
_respect to time, u(t) is the k-dimensional time-varying input or control vector, and f is a 12-dimensional
nonlinear function expressing the six-degree-of-freedom rigid body equations.
Measurements of the vehicle state can be represented by the observation equation
y(t) = g[x(t), u(t)] (1-2)
wherey(t) is an e-dimensionaltime-varyingobservationvectorand g is an C-dimensionalnonlinearflmc-tion expressingthe relationshipof the true vehiclestateand control vectors to the observed parameters.
Typically, the function g characterizes the dynamics and location of the sensors.
For the aircraft analysis and design problem, both the nonlinear and linear system equations are formu-
lated more broadly than just described (Edwards, 1976; Maine and Iliff, 1980, 1986). The nonlinear system
equations include _(t) terms in both the state and observation functions. In fact, in the most extended
form the state equation is expressed in terms of transformed variables (discussed in section 1.2.1). These
generalized equations form the basis of the analysis in this report. The generalized system equations are
T;c(t) = f[x(t), :k(t), u(t)]
y(t) = g[x(t), _(t), u(l)]
where T is a constant 12 x 12 angular velocity matrix.
(1-3)
1.1 Definition of Reference Systems
While numerous reference systems are used in aerospace applications, this report is lilnited to four reference
systems: the body, the wind, the vehicle-carried vertical, and the topodetic reference systems. The stability
axes are also defined even though this reference system is used only to define the stability axis rotational
rates (section 1.3.8).
Within this report the translational equations are referenced to the wind axes, and the rotational
equations are referenced to the body axes. Measurement equations are primarily referenced to the body -
axes when the use of a reference system is needed. The use of this mixed axis system definition in both
the nonlinear and linear models is related to the measurability and meaningfulness of quantities. Because
the aerodynamic forces act in the wind axes, this reference system is used for the translational equations. =
For instance, angle of attack, velocity, and angle of sideslip are either directly measurable or closely related
to directly measurable quantities, while the body axis velocities (u, v, and w in the x, y, and z directions,
respectively) are not. The body axis rotational rates are measured by sensors fixed in the body axes; wind
axis rates can be derived only from these quantities through axis transformations.
The first reference system to be described is the topodetic reference system, also called the earth-fixed
reference frame (Etkin, 1972), the earth axes (Thelander, 1965), and the Eulerian axes (Northrop Aircraft,
1952). The topodetic reference frame is considered fixed in space (and hence, inertial) with the orientation
of the axes as shown in figure 1; the x axis is directed north, the y axis cast, and the z axis down. The
vehicle position (x and y) and altitude (h) are measured from the origin of this reference systeln.
The vehicle-carried vertical axis system (fig. 2; Etkin, 1972) has its origin at the center of gravity of the
vehicle. The Xv axis is directed north, the yv axis east, and the z_ axis down. This axis system is obtained
by a translation of the topodetic axis system to the vehicle center of gravity. The attitude of the aircraft
(heading, pitch, and bank angles ¢, 0, and ¢, respectively) is described in terms of the orientation of the
aircraft body axes with respect to the vehicle-carried vertical axes.
The origin of the body axis system (fig. 3) is the vehicle center of gravity. The x axis is directed toward
the nose of the aircraft, the y axis toward the right wing, and the z axis toward the bottom of the aircraft.
The specific orientation of the actual body axes relative to the vehicle body is somewhat arbitrary. For =
symmetrical aircraft, the x and z axes are in the plane of symmetry; for asymmetrical aircraft, these axes
are located in a plane approximating what would be the plane of symmetry. The positive direction for the
body axis rates (roll, pitch, and yaw rates, p, q, and r, respectively), the body axis velocities (u, v, and w),
and the body axis moments (L, M, and N about the x, y, and z axes, respectively) are shown in figure 3.
/z
(Down)
Figure 1.
(North)
_._ y (Easl)
7256
Topodetie axis system.
(North) x 4
(North) xv
fyv
z v (Down)
(East)
.,_Y
Iz
(Down)
(East)
7257
Figure 2. Relationship between topode-
tic and vehicle.carried vertical axis sys-
tems.
_'Xb _____
L,p
N,r t
z
w 725_
Figure 3. Body axis system.
- 7
The relationshipbetweenthe vehicle-carriedvertical and body axesis shownin figure 4. The Eulerangles(_5,0, and _) define the orientation of the body axes with respect to the vehicle-carried vertical
axes. The rotations required to transform the vehicle-carried vertical axes to the body axes are shown in
figure .5. The heading angle g, is a rotation about the z vehicle-carried vertical axis into a new axis system
(designated (xl, 9_, Zl) in fig. 5); the pitch attitude 8 is a rotation about the y] axis into the (z2, 92, z2)axes system; the roll attitude _5is a rotation about the 92 axis into the body axes.
Figure 4.
x 2, x b
\
--XV_
//
z 2Zv' :'1
Yv
_Yl' Y2
- "--'_ Yb
7259
Relationship between vehicle-carried vertical and body axis systems.
These rotalions are described by
and the total rotation is described by
LBv = LvLeL¢_ =
COS_ --sint_, 0 7
L_ = sin _ cos 7¢, 0 J0 0 1
Lo = 0 1 0
-sinO 0 cosO
[,0 01L¢ = 0 cos¢ -sin¢
0 sin © cos ©
cos 0 cos g'
sin ©sin Ocos
- cos g5sin
cos _ sin 0 cos _5
+ sin _5sin _'
cos 0 sin C'
sin _5sin 0 sin ¢,
+ cos _5cos V;'
cos _5sin 0 sin g,
- sin _ cos _,
-sinO
sin d cos 0
sin 0 cos 0
(1-.5)
(1-6)
(l-r)
(1-8)
X v
x1,¢
//
//
//
//
//
/I
/
//
.//
//
(Zv) '..
"_" Yl
Yv
7260
(a) Rotation through %5 about zv axis.
x I
x2 ' x b
z2 #_., /
\ /\ /
",, /
\ /
\\ /\ /
\ /
\N\ .///
Z v' Z 1
7261
(b) Rotation through 8 about Yl axis.
Yl' Y2
Yb#
/
//
/
//
//
//
/
l/
/
/
(x2, Xb) "--.. +.]
b z2
7262
(c) Rotation through ¢ about Xb axis.
Figure 5. Rotation of axes through Euler angles.
Because LBV is a unitary matrix, the transformation from the body axes to the vehicle-carried vertical
axes is LTv .
The relationships between the body, wind, and stability axes are shown in figure 6. All three axis
systems have their orig}n at the center of gravity of the aircraft. The x axis in the wind reference system
(xw) is aligned with the velocity vector of the aircraft. The angle of sideslip fl and angle of attack o_define
the orientation of the wind axes with respect to the body axes. (The stability axes are shown in figure 6
also. This reference system is displaced from the wind axis system by a rotation fl and from the body axis
system by a rotation -c_.)
yw. _ •
I _ 5" _2 /\ Iz_, z. / -_...j" \ I . .
I _ w/- '_ _.'r'::_ \ v,'-v COS/)
s // zk \ i "_._<1
"-- . ,,11 _ xs
_ vi_ _-v sin/3
x W7253
=
Figure 6. Relationship of body, stability, and wind a_es.
Also shown in figure 6 are the components of the velocity vector V in the body axes (u, v, and w) -
and the definition of positive rotations for a and ft. It should be noted that fl is a positive rotation in a
left-handed coordinate system, whereas the positive sense of all other rotations used in aircraft analysis are
positive in a right-handed coordinate system.
The definitions of the body axis velocities (fig. 6) are
u = V cos c_cos _ (1-9)
v = Vsinfl (1-10)
w = Vsinacos/3 (1-11)
The total velocity V, angle of attack a, and angle of sideslip fl can be expressed in terms of these body axisvelocities as
v = [vl = (u2+ v + w2) 1/2 (1-12)
ct = tan -1 w (1-13)It
v (1-14)/_ = sin -1 _-
10
1.2 Nonlinear State Equations
For the aircraft problem, the state vector x is 12 x 1 vector composed of four 3 × 1 subvectors represenling thevehicle rotational velocity, the vehicle translational velocity', tile vehicle attitude, and the vehic]e location:
X = [X T X T X T xT] T (1-15)
\vhere
X 1 = [p q T] T
xa = [¢ e
x4 = [h x y]W
with xl, x2, xa, and x4 being the rotational velocity, translational velocity, attitude,
(1-16)
(1-17)
(1-18)
(1-19)
and position subvectors,
respectively. The vehicle rotational and translational velocity are defined within the aircragt-fixed axis
systems. In the formulation of the state used in this report, the vehicle rotations are body axis rates, whereas
the vehicle velocity terms are stability axis parameters. The vehicle attitude and location parameters are
earth relative.
The vector function f, relating the state vector its time derivative, and the control vector to the time
derivative of the state vector with respect to time, is a 12-dimensional vector function composed of four
3-dilnensional vector subfunctions:
f[x(t), _(t), u(t)] = [fT fT fat f TIT (1-20)
where t"1, f2, fa, and t".4are the vector functions that relate the x(t), _(t), and u(t) vectors to the rotational
acceleration, translational acceleration, attitude rate, and earth-relative velocity subvectors of _(t). In the
following sections, each of these subfunctions will be developed separately. The details of the derivationof these subfunctions can be found in any of the standard references on aircraft dynamics (Etkin, 1972;
McRuer and others, 1973; Thelander, 1965).
1.2.1 Rotational acceleration.--The subfunction fl of f from which the rotational acceleration
terms in the 5¢vector are derived is based on the moment equation
dH (1-21)M= dt
where M is the total moment on the vehicle and H is the total angular momentum of the vehicle. This
expression can be expanded to
M = _t ([Ft)+ a X (Ia) (1-22)
where 5/5t is the time derivative operator in a moving reference fl'ame (such as the vehicle body' axis system)and the substitution
H = If/ (1-23)1
has been used to replace the total angular momentum term with the product of the inertia tensor I and
the rotational velocity vector Ft. (The inertia tensor is assumed to be constant with time.) The definition
of the terms in equation (1-22) follow:
M= _M = M + kiT (1-24)
11
with L, M, and N being the aerodynamic total moments about the x, y, and z body axes, respectively, and
LT, A[T, and NT the sums of all power-plant-induced moments;
where I_., Iy, and Iz are the moments of inertia about the x, y, and z body axes, respectively, and Ixy, l_z, }
and Iyz are the products of inertia in the z-y, x-z, and y-z body axis planes, respectively; and
f_ = Xl = [p q r] T (1-26) :
where p, q, and r are the rotational rates about the x, y, and z body axes, respectively. Because it is
assumed that the inertia tensor is a constant with respect to time, equation (1-22) can be rewritten as
5--f_ =/-_(M - ft If_)x (1-27)bt
This is the vector subfunction for the rotational acceleration. Designating this subfunction as t"1, thefollowing definition applies:
f,[x(t),_(t), u(t)]- I-I[M- flx (/ft)] (1-28)
where
5
_-/a = fi[x(t),,(t), u(t)] (1-29)
5_a = [/5 O ÷]T (1-30)
where
Since the inverse of the inertia tensor 1-1 is given by
5 [_ /31i_1_ 1 I2 /4 /5
detI /3 /5 /-6
det I = ]xIyIz - IxI_z - IzI2_u-
h = L.ylz + Iyzzxz
5 = Ixyt_z + i_I_z
i5 = IxT_z + 1_&_
I6 = IxIy- I_y
(1-31)
(1-32)
(1-33)
(1-3_l)
(1-35)
(1-36)
(1-37)
(1-3s)
12
tile expression for the rotational accelerations can be expanded as a set of scalar equations:
1 [[_LI1 Jr _I[2 Jr _J¥13 - p2(fxzI2 - Ixyf3) -ac?Qq(fxz[1 - Iyz[2 - Dzf3)/')- det
- pr(IxJt Jr Dyh - Iuzh) Jr q2(IuzI1 - IzuI3) - qr(Dx[1 - IzyI2 Jr Ixzh)
- r2(I zI1 -
10 - det i [2Lh + _l"_Ih + _]-hr/s - p2(Ixzh - IzuIs) + pq(IxzI2 - Iy.L_ - Dds)
- pr(Ixvh + D_h - IyzIs) + q2(Iyzh - L:vIs) - qr(D_:h - I_I4 + ISa)
- - I...h)]
1 [_,L[3 Jr NMI5 4- _NI6 - p_iI_z[5 - [xy[6) 4- Pq([_d3 - [y:[5 - Dz[6)det I
- pr(I_J3 + DyI5 - IyzI6) + q2(Iyz[3 - Ixu[6) - qr(Dx[3 - [_y[5 + [zz[6)
where
(1-39)
(1-40)
(1-,11)
D_ = Iz- I_ (1-42)
Dy = I_- 1_ (1-43)
D_ = Iy-I_ (1-14)
Equation (1-3) defines the generalized nonlinear state equations a.s
T*(t) = f[x(t), :k(t), u(t)]
This equation, although more compficated than the nonlinear equations defined by equation (1-1), allows fora more tractable formulation of the state equation by using the matrix T to provide a means of addressing
the rotational accelerations in a deconpled axis system.
The derivation of the rotational acceleration terms is based on the moment equation (1-22):
M = 6_(If_)+ f_ x
Rearranging terms and assuming that the inertia tensor is constant with respect to time, the equation canbe written as
I_2 = M - £Z x If_ (1-45)
t"
The rows of this vector equation are now scaled using the following scaling matrix:
o oJ' = 0 1/[y 0 (1-46)
0 0 1/Zz
This matrix, when premultiplying equation (1-27), merely divides the first row by the roll inertia Ix, the
second row by the pitch inertia. Iv, and the third row by the yaw inertia I_. Using the definition
J = J'I (1-17)
13
the resultingequationis
andJ can be written as
J_f_ = J'M - J'(f_ x I_)
1.o - I_JGJ = - Ixy/I_ 1.0
GzlI_ - gzl& - I,:z/Ix ]zyz/ _1.0
Equation (1-48) can be expanded and expressed as
[] ][]p' 1.0 . - [xylI_ - &z/Ix Dq' - G,,/g 1.o - gzlg,:' - G:IIz - S_zlg 1.o ÷
=[_L/f_ - rp[_y/G + pqI_z/I_ + rqly/I_ + (q2 _ r2)lyz/ir _ qrI_/I_
_M/Iy - rpG/Iy + rqGy/Iy - pqly_/I_ + (r 2 - p2)G_/Iy + prI_/I_
_N/I_ + qpG/I_ - qrI_/I_ + prly_/I_ + (p2 _ q2)[_y/& _ pqIy/fz
where lY, _', and ÷' are the decoupled rotational accelerations of the vehicle.
Using the definition of J in equation (1-49), the matrix transformation T can be defined as
r
J 03×3
03x3 13x3
06X6
06x6
16X6
(1-48)
(1-49)
(1-5o)
(141)
which would be an identity matrix except for the presence of the inertia terms in the upper left-hand corner.
Thus, the vector subfunctions for the generalized state equation defining vehicle translational acceleration,
vehicle attitude rates, and earth-relative velocities are the same as those defined for the standard nonlinear
state equations in sections 1.2.2, 1.2.3, and 1.2.4, respectively.
1.2.2 Translational acceleratlon.--Derivation of the translational acceleration vector subfunction
f2 is based on the force equationJ
= -_t (mY) (1-52)F
where F is the total force acting on the vehicle and m is the vehicle mass. Tlfis expression can be expanded to
(_V )F=m _ +a x V (1-.53)
with the assumption of constant mass with respect to time and the following definitions of F and V:
F = [:SX XY SZ] T (1-5d)
where EX, EY, and EZ are the sums of the aerodynamic, thrust, and gravitational forces in the x, y, and
z body axes, respectively, and
V = [/',t v wl T (1-55)
ILl
Rearranging the terms of equation (1-52) gives an expression for the translational acceleration:
iv = IF- flx v (1-5G)_t m
This equation expresses body axis accelerations in terms of body axis forces, angular rates, and velocities.
IIowever, the desired form of this relation requires the translational accelerations in the wind axis system;
that is, in terms of the magnitude of the total vehicle velocity V, angle of attack a, and angle of sideslip fl,
which are expressed by equations (1-9) to (1-11)
u = Vcosacos/3
v = V sin/3
w = Vsin&cosfl
and equations (1-12) to (1-14)
v = IVl= (_=+ _=+ w2) 1/2
The wind axis translational acceleration terms (derived in app. B) are summarized as:
[f,-,s 3]T = r_[x(O,_(t), u(t)] (1-57)
where
_) = 1[ _ D cos/3 + Y sin/3 + XT cos a cos/3 + YT sin/3 + ZT sin a cos
-- rag(cos a cos/3 sin 0 - sin/3 sin ¢ cos 0 -- sin a cos/3 cos ¢ cos 0)1 (1-,5s)
_1. ¸ __
1
_-[-L + ZT cos c_ -- XT sin a + rag(cos & cos ¢ cos 0 + sin a sin 0)]Vra cos
+ q - tan/3 (p cos a + r sin a) (1-59)
1) = ,--_-[D sin _q + Y cos _3 - XT cos a sin 3 + Yr cos fl - ZT sin a
sin 3
+ rag(cos a sin/3 sin 0 + cos _ sin ¢ cos 0 - sin a, sin _qcos ¢ cos 0)] + p sin a - r cos a (1-60)
with D being total aerodynamic drag; Y total aerodynamic sideforce; and XT, ]_, and ZT total thrust
force along the x, y, and z body axes, respectively.
1.2.3
into body axis angular velocities is defined by
[1° 1R = 0 cos ¢ sin ¢ cos 0
0 -sine cos¢cos0
Attitude rates.--The matrix R that transforms angular velocities in the earth-fixed axis system
(1-6.1)
15
whereR is derived by Maine and Iliff (1986) from the total angular velocity of the aircraft expressed in
terms of the derivatives with respect to time of the Euler angles (¢, t), ¢):
I!] ° 0][!][100][= + cos¢ sine + 0 cos¢ sine
-sin¢cos¢ O-sin¢cos¢ oso0-:01[i]osin0 0 cos0 J
0 l[i]= cos¢ sin0cos0
0 - sin ¢ cos ¢ cos 0 _}
This transformation from earth-fixed to body axes can be expressed by the equation
(1-62)
_= _(dE) (1-63)
where E is an attitude vector whose components are the Euler angles:
= [(_ 0 _)]T (1-64)
Premultiplying both sides of equation (1-63) by R -1 and rearranging terms yields the equation for theattitude rates,
_eEdt = R-_fi (1-65)
which can be expanded into the scalar equations
= p+qsinCtan0+rcosCtan0 (1-66)
= qcos¢- rsin¢ (1-67)
= qsinCsec0 + rcosCsecO (1-68)
1.2.4 Earth-relatlve veloclty.--The matrix LBV that transforms earth axis system vectors into thebody axis system is defined by equation (1-8) as
Icos s n 01[cos00s,n01[ 0 0]= sine cos¢ 0 0 1 0 0 cos¢ -sin¢
0 0 1 -sinO 0 cosO 0 sine cos¢
cos 0 cos _ cos 0 sin _b - sin 0= sin ¢ sin 0 cos _ - cos ¢ sin ¢ sin ¢ sin 0 sin _/, + cos ¢ cos ¢ sin ¢ cos 0
cos ¢ sin 0 cos ¢ + sin ¢ sin ¢ cos ¢ sin 0 sin ¢ - sin ¢ cos ¢ cost cos 0
LBV
E
The specific relationship between earth-relative velocities and body axis velocities is expressed by
where R is the earth axis system vector defining the location of the vehicle:
(1-69)
1%= [x y z] T (1-70)
16
with z = -h.
The equation for the earth-relative velocity can be formulated as
dR= LB_V (1-71)dt
in which these velocities are expressed in terms of body axis velocities. Using equation (1-72) and the
definitions of the body a_s velocities in equations (1-12) to (1-14) allows the earth-relative velocities to be
expressed in terms of V, a, and/3:
J_ = V(cos a cos/3 sin 0 - sin/3 sin ¢ cos 0 - sin _ cos/3 cos ¢ cos 0) (1-72)
= V[ cos a cos/3 cos 0 cos _b+ sin fl(sin ¢ sin 0 cos ¢ - cos ¢ sin ¢)
+ sin a cos/3(cos ¢ sin 0 cos ¢ + sin ¢ sin 4)] (1-73)
= V[ cos a cos/3 cos 0 sin ¢ + sin/3(cos ¢ cos ¢ + sin ¢ sin 0 sin ¢)
+ sin a cos/3(cos ¢ sin 0 sin ¢ - sin ¢ cos _/,)] (1-7d)
1.3 Nonlinear Observation Equations
No standard set of observation variables exists for the aircraft analysis and control design problem. IIowever,
for any guidance and control problem, the main observation variables generally will be a subset of the statevariables. Other common observation variables are the vehicle body axis translational accelerations and
air data parameters. Thus, the dimension of g[x(t), zk(t), u(t)] is not fixed and varies from application to
application. The set of observation variables described in this section was selected to address a wide range
of problems. The basic composition of the observation vector y as used in this report is given by
Y = [x T _T u T y,T]T (1-75)
where x and :k are the state vector and time derivative of the state vector described previously, u is the
control vector, and y_ is defined by
y/__ [y_T y_T y_T y_T YV y_T y_T y_TIT (1-76)
where
y_ = [ax,k ay,k az,k ax a v az an ax,i ay,i az,i an,i n]T (1-77)
y_ = [a M Re Re' 77qc q¢/P_ P_, Pt T Tt] T (1-78)
Y; = [7 fpa _]T (1-79)
y_ = [E_ p_]T (1-80)
y_ = [L D N A] T (l-S1)
t = [l/ V W /t ?) _b] T (l-S2)Y6
y_ = [a,i /3,i h,,i h,i] T (1-83)
' = [T r_]T (1-84)Y8 P_ %
17
with the elementsof y_ beingtermsrelatedto thevehiclebody axisacceleration,the elementsof y_ beingair dataterms,the elementsof y_ beingflightpath-relatedterms,the elementsof y_ beingtermsrelatedtovehicleenergy,y_ beingavehicleforcevector,theelementsof y_ beingbody axistranslationalratesandthetime derivativesof thoseterms,y_ beinga vectorof variablesrepresentingmeasurementsfrominstrumentsnot locatedat thevehiclecenterof gravity,andtheelementsof y_ beinga collectionof miscellaneousterms.Obviously,thisgroupingof termsis somewhatarbitrary andis doneprimarily to easethedefinitionof thesetermsin the followingsectionsof this report. Thisgroupingof observationvariablesparallelsthat usedbyDukeandothers(1987).
Thevectorfunction g relating the statevector,the time derivativeof the statevector,and the controlvectorto the observationvectoris an_-dimensionaIfunctioncomposedof four subfunctions:
g[x(t),Sc(t),u(t)]= IxT _:TuT g,T] (1-85)
wherex, _, and u areidentity functionson the statevector,time derivativeof thestatevector,andcontrolvector,respectively,andg_is composedof vectorsubfnnctionsdefiningthey_vector.
The state vector, time derivativeof state vector, and control vector components of the observation
vector are not discussed in detail in this section of the report. The equations for the elements of the time
derivative of the state vector were developed in section 1.1. The observation equations for the state and
control variables are simply identities. The equations for the remaining observation variables are obtained
from a variety of sources. In addition to the previously cited sources, Clancy (1975), Dommasch and
others (1967), Gainer and IIoffman (1972), and Gracey (1980) provide the background and derivation oftile observation equations used in this report.
1.3.1 Accelerations.--The vehicle body axis accelerations and accelerometer outputs constitute the
set of observation variables that, after the state variables themselves, are most important in the aircraft
control analysis and design problem. These accelerations and accelerometer outputs are measured in units
of g and are derived directly from the body axis forces defined in section 1.2.2. The body axis acceleration
vector a can be expressed as
dr=a = dt V + _2 x V (1-86)
It is important to note here that the _, 73, and _b body axis velocity rates, derived in appendix 13 and
defined by equation (B-l), are not the body axis accelerations. The body axis accelerations contain not
only the body axis velocity rates but also the rotational velocity and translational velocity cross-productterms. Thus, expanding equation (1-86) yields
a = ay,k
az,kiz + qw- rv ]i_+ ru - pw
go + pv - qu(1-87)
where ax,k, ay,k , and az,k are the kinematic accelerations in the vehicle body x, y, and z axes, respectively.
Using
= + + + - (1-88)(v (1/m)(ZT + Za + Zg) + qu - pv
(stated as eq. (B-l) in app. B), equation (1-87) can be rewritten as
%,k = (1/m)(Yw + Y_ + Yg) (1-89)(l/m)(zs + & +
18
whereX_, Y_, and Z_ are total aerodynamic forces and Xg, ]_, and Zg are total gravitational forces along
the x, y, and z body axes, respectively. This can be expanded in terms of the gravitational and aerodynamic
forces to give (in units of g)
_ - YT+Y+gmsin¢cosO (1-90)a y,k goreaz,k aT -- D sin c_ - L cos a +gm cos ¢ cos 0
where go is the acceleration due to gravity at sea level.
The outputs of body axis accelerometers at the vehicle center of gravity are simply the body axis
accelerations due to the thrust and aerodynamic forces. The accelerometer output equations can be written
directly from equation (1-90) as
iax]a 1[xT coo+Lsin +Yaz gom ZT -- D sin a - L cos a
(1-91)
where ax, ay, and az are the outputs of accelerometers at the vehicle center of gravity and aligned with the
vehicle body x, y, and z axes, respectively. Because the normal acceleration an is defined by
an = -az (1-92)
an expression for this variable can be extracted from equation (1-91):
a,, = (--ZT + Dsina + Lcosa)/gom (1-9a)
The equations defining the output of accelerometers aligned with the vehicle body axes but displaced
from the vehicle center of gravity are derived by Gainer and IIoffman (1972) using the definition of inertial
acceleration given in equation (1-86)
a= _-V+fl x V
and the definition of inertial velocity5
V = _r+ fl x r (1-94)
The results from Gainer and IIoffman (1972) are reproduced here without rederivation:
au,i = ay + [(pq + i')Xy - (p2 + r2)yy _ (qr - p)zy]/9o (1-95)
az, az + [(pT- O)x + + )Yz - (q2+
where a_.i, ay,i, and az,i are outputs at accelerometers aligned with the x, y, and z body axes but not locatedat the vehicle center of gravity; the subscripts x, y, and z refer to the x, y, and z body axes, respectively;
and the symbols x, y, and z refer to the x, y, and z body axis locations of the sensors relative to the vehicle
center of gravity. Because the normal acceleration is the negative of the z body axis accelerometer, the
output of a normal accelerometer not at the vehicle center of gravity but aligned with the z body axis, an,i,
is given byan,i = an - [(pr - gt)xz + (qr + P)Yz - (q2 + p2)Zz]/g ° (1-96)
The final quantity included in the general category of accelerations is load factor n. This quantity is
defined without inclusion of the z body axis force component as
L_ (1-97)
mg
19
1.3.2 Air data parameters._The air dataparametershaving the greatestapplicationto aircraft idynanlicsandcontrolproblemsarethesensedparametersandthereferenceandscalingparameters.Chosen ]for inclusionasthesensedparametersareimpactpressureqc, static or free-stream pressure p_, total pressure !
pt, ambient or free-stream temperature T, and total temperature Tt. The selected reference and scaling !
parameters are Mach number M, dynamic pressure q, speed of sound a, Reynohls number Re, Reynolds "
number per unit length Re', and the Mach meter calibration ratio q¢/p_. The derivation of these quantities
is treated extensively by Gracey (1980).
The nonlinear equations defining these quantities are
a = [1.4 Po T] 1/2[ poTo J (1-98) =
VM = -- (1-99)
a
Re- pVg (1-100)#
Re'- pV (1-101)Iz
(t = lP V2
{ + O.2M ) - 1.0Mq_ = {1.2M215.70M2/(5.6M2 _
q__£__= f (1.0 + 0.2/}12) 3.5 - 1.0
P_ _ 1.2M215.76M2/( 5.6M2 - 0.8)] 2.5 -
Tt = T(1.0 + 0.2M _)
z
(1-102)
(M _< 1.0) (1-103)0.8)] 2"s- 1.0}p_ (M > 1.0)
(M _< 1.0) =1.0 (M _> 1.0) (1-104) -
(1-1o5)
where p is the density of the air, /z is the coefficient of viscosity, and the subscript 0 refers to sea level,
standard day conditions. Free-stream pressure, free-stream temperature, and the coefficient of viscosity are
properties of the atmosphere and are assumed to be functions of altitude alone.
1.3.3 Fllghtpath-related parameters.--Included in the observation variables are what might best
be termed flightpath-related parameters for lack of better nomenclature. These terms include flightpathangle 7, flightpath acceleration fpa, and vertical acceleration t_. The variables are defined by the following
equations:
7 = sin-1 1-7 (1-106)
fpa = -- (1-107)g
= a_,k sin 0 - ay,k sin ¢ cos 0 -- a_,k cos ¢ cos 0 (1-108)
2O=
1.3.4 Energy-related parameters.--Two energy-related parameters are included with the observa-
tion variables considered in this report: specific energy Es, and specific power Ps, defined as
y 2Es = h+ -- (1-109)
2g
Ps - des _ ]_ -t- (rV._.._ (1-110)dt g
1.3.5 Force parameters.--The set of observation variables being considered also includes four force
parameters. These quantities are total aerodynamic lift L, total aerodynamic drag D, total aerodynamic
normal force N, and total aerodynamic axial force A, defined as
L = CTSCL
D = qSCD
N = Lcosa+Dsina
A = -Lsina + Dcosa
where S is the surface area of the wing, CL coefficient of llft, and CD coefficient of drag.
(1-111)
(1-112)
(1-113)
(1-114)
1.3.6 Body axis rates and acceleratlons.--Because they are of interest in the control analysis and
design problem, six body axis rates and accelerations are included as observation variables. These include
the x body axis rate u, the y body axis rate v, and the z body axis rate w. Also included are the time
derivatives of these quantities, /t, _?, and _b, respectively.
The definitions of the body axis rates are given in equations (1-9) to (1-11) as
u = V cos a cos
v = Vsinfl
w = Vsinacosfl
The time derivatives of these terms can be defined using equation (B-l) and equations (B-8), (B-9), (B-10),
and (1-56) as
it = XT -- grasinO - Dcosa + Lsina + rV sin/3 - qV sin c_cos/3 (1-115)m
ij = YT + gmsin¢cosO + Y + pVsinacosfl- rVcosacosfl (1-116)D_
(0 --_ ZT "JVgm cos ¢ cos 0 -- D sin a - L cos a + qV cos a cos/3 - pV sin/3 (1-117)m
1.3.7 Instruments displaced from the vehicle center ofgravity.--The need to include measure-
ments from instruments displaced from the vehicle center of gravity arises from the fact that not all aircraft
21
instrumentationis locatedat thevehiclecenterof gravity. The mostimportantof thesequantitiesareun-doubtedlythe accelerometeroutputstreatedin section1.3.1.In this sectionfour additionalparametersarepresented:angleof attack (a,i), angleof sideslip(/3,/),altitude (h,i), and altitude rate (it,/) measurementsfrom instrumentsdisplacedfrom centerof gravityby somex, y, and z body axis distances. The subscripts
a,/3, h', and ]_ refer to the displacements of the angle-of-attack, angle-of-sideslip, altitude, and altitude rate -
instruments from the vehicle center of gravity. The equations used to compute these quantities are
a,i = a + qx_ - py_ (1-118)V
rx_ - pz_ (1-119)fl'; =/3+ v
h,i = h + Xh sin0 -- Yh sin¢cosO -- Zh cos ¢COS0 (1-120) -
J_,i = ¼+O(xhcosO+yhsinCsinO+zhcosCsinO)-(b(YhCOSCcosO-zhsinCcos8) (1-121)
1.3.8 Miscellaneous observation parameters.--The final set of observation parameters considered
ill this report is a miscellaneous collection of parameters of interest in analysis and design problems. These
parameters are total angular momentum T, stability axis roll rate Ps, stability axis pitch rate qs, and stability
axis yaw rate rs. The equations used to define these quantities are
= l(Ixp2 - 2Ixvpq - 2Ixzpr + [yq2 _ 2Iyzqr +/zr 2)T_5
Ps =pcosa+rsina
qs ---- q
r s = --ps]n_+ rcos
(1-122)
(1-123)
(1-124)
(1-125)
2 LINEAR SYSTEM EQUATIONS
The standard state equation for a linear differential system has the form
*(t) = A'x(t) + B'u(t) (2-1)
where, for a time-invariant system, A _ is a constant n × n matrix and B _ is a constant n × k matrix. The
standard output equation has the form
y(t) = tt'x(t) + F'u(t) (2-2)
where H' is a constant g x n matrix and F r is a constant g × k matrix. The generalized linear system
equations used with an extended formulation compatible with the generalized nonlinear equations (1-3) and(1-4) can be characterized by
eft(t) = ax(t) + Bu(t) (2-3)
y(t) = /Ix(l) + ak(t) + ru(t) (2-4)
where C and A are constant n x n matrices, B is a constant n x k matrix, H and G are constant g x n
matrices, and F is a constant gx k matrix. The nonlinear system equations developed in section 1 (eqs. (1-1)to (1-4)) can be linearized about a trajectory, and a linear model can be formulated that is similar to either
the standard or the generalized linear system equations.
22
2.1 Linearization of tlle State Equation
If u0(t) is giveninput to a systemdescribedby the statedifferentialequation(1-3),andif x0(t) is a knownsolutionof the state differentialequation,then appro_mationsto the neighboringsolutionscanbe foundfor smalldeviationsin the initial stateandin the input by usinga linearstate differentialequation.Thenonlinearstatedifferentialequation(1-3) canbe ]inearizedabout a generaltrajectory,asby KwakernaakandSivan(1972)and Dieudonne(1978),sothat x0(t) satisfies
T_o(t) = f[_0(t),xo(t), uo(t)]
Assuming that the system is operated at close to nominal conditions with u(t), x(t), and _(t) deviating
only slightly from uo(t), x0(t), and _o(t), the following expressions can be written:
u(t) = u0(t) + _u(t) (2-5)
x(t) = x0(t) + _x(t) (2-6)
,(t) = _0(t) + _,(t) (2-7)
where 5u(t), 5x(t), and 53¢(t) are small perturbations to the control, state, and time derivative of the state
vectors, respectively.
Substituting equations (2-5) to (2-7) into the nonlinear state differential equation (1-3), expanding in a
Taylor series about _o(t), xo(t), uo(t), and assuming T constant with respect to _(t) yields
0f 0f 0f
T[_o(t) + 5_(t)] = f[x0(t), ;k0(t), u(t)] + _xx 5x + _ 5_ + _uu 5u + h(t) (2-8)
where 0f/0x, 0f/01:, and 0f/0u are defined in equations (2-9) to (2-11) and h(t) represents the sum of
the higher order terms in the Taylor series, assumed to be small with respect to the perturbations. The
matrices used in the Taylor series expansion are defined by the following relationships:
0f 0f
0X -- 0X(Xo,_o,Uo)
(2-9)
0f 0f
01: - 0_:(Xo,_0,Uo)
(2-10)
the (i, j)th elements of which are defined as
0f 0f
0u - Ou(xo ,*o ,Uo)
(2-11)
( _x) ;,j - 5-Zxj°f' (2-]2)
(_)i,j-- O_jOfi (2-13)
(Oa-_fu)ij - OujOfi (2-14)
respectively, where fi is the ith simultaneous equation of the nonlinear state differential function in equa-
tion (1-3), xj the jth element of the state vector, _j the jth element of the time derivative of the state
23
vector,flj the jth element of the control vector, and all derivatives are evaluated at the nominal condition
(x0(t), ,0(t), .0(t)).
Subtracting equation (1-3) from (2-8), rearranging terms and neglecting the higher order terms yields a
liuearizcd state equation,
r[T__xx0fj, 5_¢(t)= _x0f5x(t)+ _uuOf5u(t) (2-15)
where the arguments of the matrix functions have been dropped to simplify the notation and where it is
understood that the matrices are to be evaluated along the nominal trajectory.
Letting
Of
C = T- a-_ (2-16)
Of
A = a---_ (2-17)
0f/3 = 0---u (2-18)
equation (2-15) can be written as
C 5_(t) = A 5x(t) + Z 5u(t) (2-19)
which is precisely the formulation of the generalized state equation desired.
Premultiplying both sides of equation (2-19) by C -1 results in the standard form of the linearized state
differential equation,
_,(t) = C-IA &(t) + C-_ _u(t) (2-20)
Letting
A'= C-_A (2-21)
B' = C-'B (2-22)
equation (2-20) can be written in the more usual notation
5_¢(t) = A' 5x(t) + B' 5u(t) (2-23)
2.2 Linearization of the Observation Equation
The technique used in section 2.1 to linearize the state equations can be applied to the nonlinear observation
equation (1-4),
y(t) = g[x(t), _¢(t), u(t)]
Performing a Taylor series expansion about the nominal trajectory (x0(t), _0(t), u0(t)) yields
0g 0g 0gyo(t) + 5y(t) = g[xo(t), *o(t), uo(t)] + _x 5x + 0-_ 5' + _uu 5u + h(t) (2-24)
where
Og _ Og
OX -- Ox(Xo ,Xo ,Uo )
(2-25)
2:1
(2-26)
0g 0g t0u _ 0u (xo,*o,Uo)
(2-27)
the (i, j)th elements of which are defined by
Og) Ogl (2-28)_xx i,j Ox j
= Og_ (2-29)
respectively, where gi is the ith simultaneous equation of the nonlinear observation equation (1-4). Again,
all derivatives are evaluated at the nominal condition (xo(t), :ko(t), u0(t)).
Subtracting equation (1-4) from equation (2-24), rearranging terms, and neglecting higher order terms
results in a linear observation equation,
0g 0g 0g5y(t) = _xx 5x + _xx 5zk + _uu 5u (2-31)
where the arguments of the matrix functions have been dropped to simplify notation. Letting
tt- Og (2-32)0x
Og (2-33)a = _--_
F = --0g (2-34)0u
equation (2-31) can be rewritten as
5y(t) = II 5x(t) + G 5_(t) + F 5u(t) (2-3._)
which is the generalized linear observation equation desired.
The standard form of the observation equation can be derived by substituting for 5_ from equation (2-23)
into equation (2-33). This substitution results in
5y(t) = lI 5x(t) + G[A' 5x(t) + B' 5u(t)] + F 5u(t) (2-36)
which can be written as
By letting
ey(t) = [tI + GA'I_×(t) + IF + cm] _.(t) (2-37)
H' = H + GA' (2-38)
F' = F + GB' (2-39)
equation (2-37) becomes
25
_y(t) = ti' _x(t) + r' _u(t) (2-40)
2.3 Definition of Matrices in Linearized System Equations
The results of sections 2.1 and 2.2 can be used to define the matrices in the linearized system equations
in terms of partial derivatives of the nonlinear state and observation functions taken with respect to the
state, time derivative of state, and control vectors. All derivatives are understood to be evaluated along thenominal trajectory.
Using the nonlinear state equation (1-3),
Tx(t) = f[x(_),_(t), u(0]
the terms in the generalized form of the linearized state equation (2-19),
C 5_(t) = A 5x(t) + B 5u(t)
can be defined as
Of
C -- T- 0--_ (2-41)
Of
A = 0--x- (2-42)
Of
B = 0-_ (2-43)
The terms in the standard form of the linearized state equation (2-20),
5_(t) = A' 5x(t) + B' 5u(t)
Of]-1 c0fA'= T-_ 0x
[ 0fl-' ofB'= T- 0_J 0-u
(2-44)
(2-45)
can be defined as
In a similar manner, the nonlinear observation equation (1-4),
y(0 = g[x(0, _(t),.(t)]
can be used to define the terms of the generalized linearized observation equation (2-35),
5y(t) = r,r 5x(t) + a 5:_(t) + F 5u(t)
26
{is
si = --°g (2-46)0x
Og (2-47)G=_x x
F = --°g (2-_s)0u
The terms in the standard form of the linearized observation equation (2-40),
5y(t) = H' 5x(t) + F' 5u(t)
can be defined as
Og Og [ Of] -1 Of (2-49)H'= 0---_+ _xx T-_x x 0---_
r' (9g 0g [ Of] -1 Of (2-50)= 0u + _ T- b2 0u
2.4 Elements of the Linearized System Matrices
The elements of the linearized system matrices derived in sections 2.1 and 2.2 are determined by applying the
linearization method employed with the vector equations in those sections to the individual scalar equations
constituting the vector equations that define the time derivatives of the state and observation variables.
Thus, for a matrix, such as the state matrix A defined by equation (2-42),
Of
0x
the element occupying the ith row and jth column of A, (A)i,j, can be represented as
Ofi(A)i5- Oxj (2-51)
where fl is the scalar function defining the time derivative of the ith state and xj is the jth state. The
individual terms used in the A, B, C, H, G, and F matrices are defined in appendix I) based on the
generalized derivatives derived in appendix C.
Using the state vector x defined in (1-7) as
x = [pq_-V,_Z ¢o Vohx y]T
the elements of the A matrix can be expressed as
I o(_,)lOp oO;')lOq... 0(s)')/0y ]
O(q'!/Op O(q')/Oq O(q')/Oy lA= . : "
I O(_ilOp O(i<)lOq O(i_ilOu|Lo(is)lOp o(h)lOq o(is)lO,uJ
(2-52)
27
Substituting for these partial derivatives using the terms in appendix D gives
A
[ (1/Ix)[(gtSb2/2Vo)C_p + OLr/Op
--Ixyr0 + Ixzqo]
(1/Iy)[(_tSbe/2Vo)Cmp + OMr/Op
-2I_,po - Ivzqo + ro(/_ - I_)]
(1/Ix)[(gtSb_/2Vo)Ceq + OqLT/Oq + [xzPO ..,
+2r_zqo+ _0(r_- &)](1/Iu)(_tS_.2/2VO)Cmq + OMT/cgq ...
+Ixyro -- Ivzpo]
The elements of the B, C, H, G, and F matrices can be determined in a similar fashion, although some
care must be taken in determining the elements of the matrices for the observation equation and the Cmatrix.
To determine the elements of the matrices for the observation equation, one must consider the definition
of the nonlinear vector function g defining the observation variables (eq. (1-85)),
g[x(t),,(t), u(t)] = IxT ,T uT g,T]
and the definitions of the matrices for the generalized linear observation equations (2-46) to (2-48),
0gt[.= --
Ox
a=O____g02
F=OgOu
These matrices may be expressed using a partitioning based on the vector subfunctions of g as
" OX
--D
o_
H= ] __
0ugK
a ,,,_
F
-0x'_g-ff
0_g-ff
Oub-ff
O___xOu
8u
0t!_Ou
DB
=
(2-55)
(2-56)
28
which become
H ._
a
r
" 112xl 2
012×12
Okxl2
012×12
112x12
0kxl2
O12xk
012Xk
lkxk
_ _2u-u.
upon evaluating the partial derivatives of the identity functions x, :_, and u.
The C matrix may be viewed as a partitioned matrix as
C
I
t C12Cll t.... I....
I' C2203X3 t
06x6
06X6
16X6
(2-57)
(2-5s)
(2-59)
(2-6o)
where, from equation (1-48),
[ i'0 -I_y/I_-I=_/l= 1c11 = J= -z=/6 I.o -1=/z_
-Ix=lI= -I_=lI= 1.0
and
(2-61)
C12 =[ - o0;')I o? -o(/,,) I oa -o(p,) Iob
-o( (_,)I of_ -O( q')I oa -O( q,)I o3-o( ;,) l Of/ -o( ,:')1o_ -o( ;') l OD
i -(OSbel2VoZ_)Ce_ -(OSb2121'SZx)Ce_ ]-( OSb_/2 VoIz )C_ a -( ilSb2 / 2Vo[z )C,w_ J
(2-o2)
29
1.o-o(_)/o_ -o(9)/o_ -o(9)/o_ lo;_ = -o(a)/of, _.o- o(_)/o_ -o(a.)/oql-0(_)/0I 7 -O(/:))/O& 1.0 - 0(,8)/0,8]
[i.O(_Sd2Vo'_)(cOsYoCD_-sin_oC,,)(_Sb/2Vo.O(cOsYoCD_)= 1.0 + (r]Sg/2V02rn cos t_O)CLa (_Sb/2Vo2m cos flo)CLb (2-63)
(_Se/2Vo2m)(sin t3o CD_ + cos/3o Cys) 1.0 - (iiSb/2Vo2rn)(sin 13o CDi3 + cos/30 C_3)
The inverse of the C matrix, C -a, can be expressed as a partitioned matrix in terms of the matrix subpar-
tltions of the C matrix as
I 06X6
, c_-_C -a = Oax3r __.......... (2-64)
06X6 16X6
The elements of the X, B _, H r, and F _ matrices can be determined using the C -1 matrix defined in
equation (2-64), the A, B, II, G, and F matrices, and the definitions for A r, B _, H _, and F _ given in
equations (2-21), (2-22), (2-38), and (2-39).
3 CONCLUDING REMARKS
This report derives and defines a set oflinearized system matrices for a rigid aircraft of constant mass, flying
in a stationary atmosphere over _ flat, nonrotating earth. Both generalized and standard linear system
equations are derived from nonlinear six-degree-of-freedom equations of motion and a large collection ofnonlinear observation (measurement) equations.
This derivation of a linear model is general and makes no assumptions on either the reference (nominal)
trajectory about which the model is linearized or the symmetry of the vehicle mass and aerodynamicproperties.
Ames Research Center
Dryden Flight Research Facility
National Aeronautics and Space Administration
Edwards, California, January 8, 1987
3O
APPENDIX A--AERODYNAMIC FORCES AND MOMENTS
The aerodynamic forces and moments acting on an aircraft are the result of multiple factors whose signif-
icance varies with flight condition as well as from vehicle to vehicle. In general, these forces and moments
are nonlinear functions primarily of Mach number, angle of attack, angle of sideslip, altitude, rotational
rates, and control-surface deflections. For the purposes of this report, the aerodynamic forces and moments
are assumed to be functions having the following form:
r = _(_,Z,V,h,p,q,r,e_,_,5l,...,5_) (A-l)
where F is an arbitrary force or moment, • is an arbitrary function, and the 5i are the n control surface
deflections. These forces and moments are related to the nondimensional force and moment coefficients by
the equations for the forces,
D = qSCD (A-2)
Y ----(¢SCy (A-3)
L = 4SCL (A-4)
and the moments,
L = qSbCe (:_-5)
M = CTSeC_ (a-6)
N -- qSbC_ (A-r)
where b is reference span and e is reference aerodynamic chord.
While the nondimensional aerodynamic force and moment coefficients are themselves nonlinear func-
tions of the vehicle states, time derivatives of the vehicle states, and the control surface deflections, these
coefficients are commonly expressed in linear form in terms of partial derivatives of these coefficients with
respect to the functional variables. These linear equations for the aerodynamic force and moment coeffi-
cients are derived in the same way as the linearized system equations (section 2); therefore, this derivation
will not be repeated here. These linear equations are
CL = CL0 Jr- CLa a "[- CLe_ -k CLhh -k CLv I/"
J- ECL615i J- CLpP -}" CLqq -Jr- CL_+ -[- CLa_ -[- CL_/_
i:1
CD ---- CDo '[- CD_ O: -{- CD_/3 + CDh h + CDv V
-[- ECD6i¢Si -[- CDp/_ + CDq 0 or- CDrr -{- CDa¢_ -k CD_J3
i=1
Cy : Cy o -}- Cy_a -_ Cy_/_ J- CYhh + CYvV
rL ,,
+ _ Cy_ _i + Cy,/_ + Cy_0 + Cy, _ + Cy_ h + C¥_i=l
Ce = Ceo + Ce_a + Cee_ + Cehh + CevV77, ^
+ __, ce_,_ + c_ + c_,o + ce_÷+ ce_a, + c_ji=1
(a-s)
(A-O)
(A-10)
(a-ll)
31
Cm = Cmo + Cm_,e_ + Cm_3 + Cmh h + C_v V
+ _ Cm_,Si + C,_,_ + CmqO + Cmr÷ + Cma& + CmS3 (A-12)/-----1
Cn = Cno + C,_ a + C_3 + Cnh h + Cnv V
+ _ 0,% 5i + C,_p_ + CnqO + Cn_" + Cn_ + C,,_3 (A-13)i=l
where C_o is the value of the coefficient along the nominal trajectory and the notation C_. is defined as
OC_ (a-1,l)C_.-- Ox
with C_ being an arbitrary force or moment coefficient and x being an arbitrary state, time derivative of state,or control-related parameter that for the usual derivatives is nondimensional. Itowever, the derivatives with
respect to altitude and velocity are not taken with respect to a nondimensional quantity. The definitionsof these nondimensional stability and control derivatives are given in terms of the coefficient C(. The
nondimensional stability derivatives are defined as
OC_ (A-15)C_- Oa
OC_ (A-16)Ce,_= 03
OC_ (A-17)C_v - O(bp/2Vo)
oc_ (A- _s)C_ - O(eq/2Vo)
oc_ (a-10)c_ = o(l,,,/2Vo)
oc_ (A-2o)c_c,- o(e,s/2_,,t,)
oc_ (Am)C_ = O(b3/2Vo)
The two other stability derivatives are not nondimensional and are defined as
oc_ (a-22)C_v =- OV
OC_ (A-23)C_h _ Oh
The control derivatives are defined as
OC( (A-24)
E==
E
|
=_
=
=_E
i
,ira
32
The rotational termsin equations(A-8) to (A-13)arenondimensionalversionsof the correspondingvari-ablewith
@ (t-25)- 2v0
eq A-26)q- 2Vo
br A-27)2Vo
•: C'& A-28)2Vo
- 2v0
Because the C_0 terms are included, the force and moment coefficients are total force and moment coefficients.
The state, time derivative of state, and control parameters on the right-hand side of equations (A-8) to
(A-13) are differentials.
33
APPENDIX B--DERIVATION OF THE WIND AXIS
TRANSLATIONAL PARAMETERS V, d, AND ¢)
The derivation of the wind axis translational acceleration parameters is based primarily on the definitions
in equations (1-9) to (1-14), the body axis translational acceleration equations (1-56), and the expression
of the force terms defined in equation (1-53). In the following sections, each of the wind axis transla-
tional acceleration terms is derived separately after stating some preliminary definitions applicable to allcalculations.
B.1 Preliminary Definitions
Equation (1-56),
5__V = 1F_ f/ x V5t rn
can be expanded, using equations (1-54), (1-55), and (1-26), to
= |(1/m)(YT+Ya +Yg)-l-pw-ru (B-l)
L(1/m)(Zw + za + zg) + qu- pv
The body axis aerodynamic forces can be rewritten in terms of the stability axis forces lift L, drag D, andsideforce Y:
X_ = -D cos oe + L sin ¢_ (B-2)
g_ = Y (u-a)
Z_ = -D sin a - L cosc_ (B-4)
The gravitational forces can be resolved into body axis components such that
Xg = -mgsin 0 (B-5)
}) =mg sin ¢ cos 0 (B-6)
Zg =mg cos ¢ cos 0 (B-7)
These equations will be used in the derivations of the V, &, and/) equations. Thus, the total forces in the
body axes can be defined and expanded as
EX = XT-Dcosa+Lsina-gmsinO
EY = YT + Y + gmsin¢cosO
EZ = ZT -- Dsina - Lcosa +gmcos¢cosO
(B-S)
(B-9)
(B-IO)
B.2 Derivation of I'z Equation
Beginning with the definition of V in terms of u, v, and w in equation (1-12),
V : (U 2 "_- V 2 -_ W2) 1/2
35PRECEDING PAGE BLANK NOT PILMED
the equation for 17`"becomes
= £g _- _7_ 732 w2)1/2dt ( u2 + +
which after expanding the derivative and cancelling terms, becomes
(B-11)
17 = l(u/t + v_ + w_2) (n-12)
By substituting the definitions for u, v, and w from equations (1-9) to (1-11) and cancelling terms, equa-tion (B-12) yields
l) = _/cos c_cos fl + 7?sinfl + zb sin acosfl (B-13)
The definitions for h, i,, and tb in equation (B-l) are now used with equation (B-13) to give
I}. __ COS O_ COS fl (Xa q- XT q- Xg) q- cos o_ cos/3(rv - qw)m
+ sin/3 (Ya + YT + gg) + sin fl(pw - ru)m
sin a cos fl (Z_-t- -1- Z T %- Zg) -at- sin c_cos/3(qu - pv)m
(B-14)
Expanding (B-l,1) in terms of equations (B-2) through (B-r) and cancelling yields
I? = 1[ _ D cos/3 + Y sin/3 + XT cos a, cos/3 + YT sin fl + ZT sin c_cos/3
-- rag(cos _ cos fl sin 0 -- sin/3 sin ¢ cos 0 - sin a cos/3 cos ¢ cos 0)]
+ rv cos a, cos/3 - qw cos c_cos/3 + pw sin fl - ru sin fl
+ qu sin a cos//- pv sin c_cos/3 (B-15)
Equation (B-15) can be simplified by recognizing that the terms involving the vehicle rotational rates are
identically zero, which becomes obvious after substituting for u, v, and w in these terms. Thus, the finalequation becomes
19
= m[ - D cosfl + Y sin/3 + X T cos ct co8/3 -4- YT sin/3 + ZT sin o, cosfl
- rag(cos a cos/3 sin 0 - sin/3 sin ¢ cos 0 - sin n, cos fl cos ¢ cos 0)]
=
B.3 Derivation of & Equation
The equation for & can be derived fl'om the definition of a in equation (1-13),
ct = tan -1 wu
Taking the derivative of ct with respect to time,
d d w
& = _7oe= dt tan-I-- /t
then expanding and cancelling terms, the equation becomes
1
5 - u2 + w 2 (u_b - izw)
(B-17)
(B-18)
36
Substitutingthe definitionsof u and w from equations (1-9) and (1-11) into equation (B-18) gives
& = _bcosa-/Lsina (B-19)V cos/3
Using equation (B-l) to substitute for _ and _b and equations (B-S) to (B-10) to define the forces,
equation (B-19) becomes, after rearranging terms,
1
& = Vmcos/3[-L + ZT cosa - XT sina -t- mg(cosa, cos¢cosO + sin a sin 0)]
1+ ---------_(qucoso_ - VV cos a - rvsin a + qW sin O¢) (B-20)
/_5"Vcos
which after substituting for u, v, and w from equations (1-9) to (1-11) and combining terms gives
1_ [-L + ZT cos_ - Xx sin _ + ._S(cos _ cos¢ cos0 + sin _ sin 0)]
Vm COS
+ q - tan _3(p cos a + r sin a) (B-2_)
B.4 Derivation of fl Equation
The equation for/3 is derived from the definition of/3 as given in equation (i-ld),
v
/J = sin -1 _-
Taking the derivative of/3 with respect to time yields
d d . -a v-Sln
(B-22)
which becomes, after expanding the derivative, substituting for 11, and cancelling,
1/3 = V[-/_cos asin/3 +/,cos/3 - _bsin a sin/3] (B-23)
Using equation (B-l) to substitute for _, /J, and zb and equations (B-8) to (B-10) to define the forces,
_= 1m--IT[ - cos oesin/3 (-D cos a + L sin a + Xx - my sin 0) + cos/3 (Y + YT + ngg sin ¢ cos 0)
-- sin cxsin/3 (-D sin ce - Lcosa + ZT + mg cos ¢ cos O)]
V[- cos a sin/3 (rv - qw) + cos _q(pw - ru) - sin a sin fl (qu - pv)] (B-24)+
Substituting into equation (B-24) for u, v, and w and rearranging terms yields the final equation
1/3 = _--_[D sin/3 + Y cos/3 - XT cos a sin/3 + YT cos/3 - ZT sin a sin/3
+ rag(cos a sin/3 sin 0 + cos/3 sin ¢ cos fl -- sin a sin/3 cos ¢ cos 0)]
+ p sin a - r cos a (B-25)
37
F
APPENDIX C--GENERALIZED DERIVATIVES
The equations defining the time derivatives of the state variables (derived in sections 1.2.1 to 1.2.4) and those
defining the observation variables (presented in sections 1.3.1 to 1.3.8) are used to determine the generalized
partial derivatives of the quantities with respect to a dummy variable _. The purpose of these generalized
derivatives is primarily to facilitate the derivation of the terms in the linearized equations presented ill
section 2.4; however, these equations have also proved to be useful for computer programs and were used
to verify the results obtained using LINEAR (see Duke and others, 1987).
C.1 Generalized Derivatives of the Time Derivatives of State Variables
Equations (1-39) to (1-41) define the rotational accelerations of the vehicle. These equations are used to
determine the generalized derivatives of these quantities.
OL OM ON OLT OJ_IT ONTO(p) 1 I1 + I 2 + I3 + I1 + I2 + 13 --
Op
- [2p(IxzI2 - I_vI3) - q(I_zh - I_zI: - DzI3) + r(Ixyh + D_[2 - Iw_)] 0--_
Oq
+ [P(Ixzh - IvzI2 - DzI3) + 2q(IvzI1 - IxyI3) - r(Dxla - [xv5 + Ixu/3)] 0_
- [p(L:yh + Dv[2 - Iyzh) + q(Dh - I_I2 + I_zI3) + 2r(I_zI1 -/_zI2)] _-_ (C-1)
0((i) 1 ( OL OM ON OLT OJ_IT ONT0_ detI _:_+x4-b-Y+_-_-+_:--_-+q--_+t_ 0--Y-
- [2p(I_J4 - I_I_) - q(I=I2 - r_zi4 - DJ_) + ,.(I_I_ + DJ4 - I_I_)] O__vO_
+ b,(I_I2 - I_I_ - D_Is) + 2q(I_I2 - I_yls) - r(D_I: - IxJ4 + Ix_Is)] 0qO(
Or)- [p(I_J2 + DJ4 - I_zI5) + q(D_I2 - L:/4 + I_J5) + 2r[I_zr2 - I=I_] _ (C-2)
O(i') 1 { OL 02ll ON OLT OJ_IT ONT0_-aet_ ±_3-(+_--_-+I_-_-+r_--gC+±s-bY -+I_ 0-2--Op
- [2p(/_J_ - Ij6) - q(Ij3 - I_Js - D_I_) + ,'(_I_ + D_ - Z_ [_)]
Oq+ [p(ix__r_- i,,_I_ - DJ_) + 2q(I_I_ - I_:,,I_)- r(D_I_ - I_[_ + I=I_)] O_
Or t- [p(,%[_ + D_t_ - _I_) + q(Dz[_ - [_ + _z_) + 2,,([_J_ - I_z_)] -_ (C-3)
The quantities I1, 12, I3, I4, I5, I6, D_:, Dr, D_, and det [ are defined in equations (1-32) to (1-38) and(1-42) to (1-44).
Equation (1-50) defines the decoupled rotational accelerations of the vehicle (1_', (i', and r'), which are
used to determine the generalized derivatives of the decoupled quantities:
O(p') 1 [OL OLT (rtxy - qI_:_) Op Oq
Or] (C-4)- (pixy - q[u + 2r[_:z + qI_) -_
PREC_ING PAGE BLANK NOT FILMED
39
a(;;,) 1[aM cop Oq
o_]- (pi_ - qi_. - 2,%_ - pr=) _- (c-,5)
o(,:') l[O.,v o:v op aqO¢ - Iz [ a_ + --o-f- + (qL: + rIyz + 2pI= u - qlv) -_ + (pI= - rI== - 2q[= v - ply) -_
- (qI z- ptyz) (c-6)
Equations (1-58) to (1-60) define the translational accelerations of the vehicle. These equations are usedto determine the generalized derivatives of these quantities:
O(I::) 1
O( m
o(_)o(
OD OXT OY O0__%T __- cos f1-5( + cos _ cos9 --_ + sin fl 5-( + sin _ cos9 -- + sin/3 0_#0¢
+ [ - XT sin a cos/3 + ZT cos a cos t3 + m9(sin 0 sin a cos/3
Oa
+ cos0 cos¢ cosa cos#)1 O--(
+ [D sin/3 + Y cos/3 - XT sin/3 cos a + YT cos/3 - ZT sin a sin/3
0/3+ rag(sin 0 cos a sin/3 + cos 0 sin ¢ cos/3 - cos 0 cos ¢ sin a sin/3)] 0(
0¢- rag(- cos 0 cos ¢ sin/3 + cos 0 sin ¢sin a cos/3) _--_
- mg(cosOcosacos/3 + sin 0sinCsin/3 + sin 0 cos Csin c_cos/3) 0_}
1 ( OL OZT OXT'_ Op Oq OrmVcos/3 -_-+coscr O----c-sin_ O( j-tan/3c°scr_-_+_-_-ta'n/3sinao_
{ }or1 [-L + ZT cos a - XT sin a + rag(cos 0 cos ¢ cos a, + sin 0 sin o_)] 0(mV 2 cos/3
+ V cos/3 [- ZT sin _ -- XT cos a -- rag(cos 0 cos ¢ sin _ -- sin 0 cos c_)]
+ tan fl (p sin c_ -- r cos _) } 0_0(
[ tan_q+ [m-_7-f-os/3[-L + ZT cosc_ -- XT sina
+ rag(cos 0 cos ¢ cos c_ + sia 0 sin a)]
V c_s/3
o/31 .(pcosa+rsin oz)cos 2
cos0sin¢cosa _--_- (sin 0 cos ¢ cos o_- cos 0 sin a) O-_-
(c-7)
(c-s)
2
2
4O
o(3) 1 f_ |sin9
O_ mY L -_- + cos/7 _ - cos _ sin fl -- + cos fl _59I_ _ sin c_sin fl --_c]0Zr]
Op 0 r
+ sin_ _ - cosc_0_
1
mV_[ D sin fl + Y cos/3 - XT cos _ sin/3 + YT cos fl -- ZT sin _ sin/?
OV
+ m9(sin 0 cos a sin/3 + cos0sin 4)eosfi - cos0cos 4)sin asinfl)] 0_
+ n---_[- T sin c_sin fl -- ZT cos a sin fl + rag(- sin 0 sin a sin/3 -- cos 0 cos 4) cos a sin fl)]
+pcosa + rsin_ _-
+ m--_[D cos/3 - Y sin/3 - XT cos c_cos/7 - }_ sin/_ - ZT sin a cos
05+ rag(sin 0 cos c_cos fl - cos 0 sin 4)sin t_q- cos 0 cos 4)sin ct cos 13)]
+ (cos 0 cos 4)cos/3 + cos 0 sin 4)sin (1sin/7) _-
__ 00+ (cos0cos asin]3 - sin0sin 4)cos_ + sin 0cos4)sin_ sin_) _-_ (c9)
Equations (1-66) to (1-68) define the vehicle attitude rates. These equations are used to determine the
generalized derivatives of these quantities:
Op Oq Or0_ + sin 4)tan 0 _ + cos 4)tan 0 _-_ + (q cos 4)tan 0 - r sin 4)tan 0) 0_bo_
+(q sin 4) sec 2 0 + r cos 4)sec 2 0) 590
o(0) oq o_ o¢o---(= cos_ N - si_, N - (qsin, + _cos4))
Oq Or 04)
- sin 4)sec0 _ + cos4)sec0 _ + (q co_4)sec0 - _sin _ sec0)
00+ (qsinCsecOtanO + rcosCsecOtanO) -_.
(C-10)
(C-11)
(c-_2)
Equations (1-72) to (1-74) define the earth-relative velocities of the velficle.
to determine the generalized derivatives of these quantities:
- [cos/3 cos c_sin 0 - sin/3 sin 4 cos 0 - cos fl sin c_cos 4 cos 0] c)V0f
f
- V(cos fl sin a sin 0 + cos p cos o_cos 4 cos 0) 0a
0]3- V(sin/3 cos a sin 0 + cos fl sin 4 cos 0 - sin fl sin c_cos 4 cos 0) _-
These equations are used
- V(sin fl cos 4 cos 0 - cos ]3sir, a sin 4 cos 0) 04
00
+ V(cos/3 cos a cos 0 + sin/3 sin 4 sin 0 + cos t3sin ozcos 4 sin 0)
- [cos _qcos c_cos 0 cos ¢ + sin/3 (sin 4 sin 0 cos ¢ - cos 4 sin ¢)
OV
+ cos 13sin o_ (cos 4 sin 0 cos _b+ sin 4 sin _b)] 0f
0_
- V[eos_sin _ cos0cos _, - cos/3 cos o_ (cos C sin 0 cos _ + sin ¢sin _)] 0f
- V[ sin ]3cos c_cos 0 cos _b - cos ]3sin 4 sin 0 cos _b - cos 4 sin _b
0Z+ sin fl sin a (cos 4 sin 0 cos ¢ + sin 4 sin _)] 0f
04+ V[sin fl (cos 4 sin 0 cos _b+ sin 4 sin _/,) - cos ]3sin c_ (sin 4 sin 0 cos _ - cos 4 sin _,)]
- V [cos fl cos a sin 0 cos ¢ - sin fl sin 4 cos 0 cos 0 - cos fl sin a cos 4 cos 0 cos ¢] 000f
- V[cos fl cos c_cos 0 sin _b+ sin _ (sin 4 sin 0 sin _ + cos 4 cos _)
+ cos _ sin a (cos 4 sin 0 sin _ - sin 4 cos g,)] 0_,
0(/)____))= [cos _ cos _ cos 0 sin _ + sin/3 (cos 4 cos _b+ sin 4 sin 0 sin g,)0f
0V
+ cos fl sin a (cos 4 sin 0 sin _ - sin 4 cos _)] 0f
0o_
- V [cos/3 sin o_cos 0 sin _ - cos fl cos o_ (cos 4 sin 0 sin _ - sin 4 cos _)] -_-
- V[ sin fl cos _ cos 0 sin _ - cos fl (cos 4 cos _ + sin 4 sin 0 sin g,)
0Z+ sin ]3sin c_ (cos Csin 0 sin _b - sin 4 cos _)] 0f
(C-13)
(c-14)
!
2_
7_
42
0¢- V[sin _ (sin ¢ cos¢ - cos ¢ sin e sin ¢) + cos_ sin _' (sin ¢ sin e sin _, + cos¢ cos¢)] 9-_
0#
- V(cos fl cos a sin 8 sin _b- sin/_ sin ¢ cos 0 sin ¢ - cos fl sin a, cos 4' cos 8 sin ¢) _-_
+ V[cos fl coso_co_0 cos_, - sin _ (cos ¢ sin ¢ - sin ¢ sin 0 cos ¢)
+ cos_ sin _ (cos ¢ sin 0 cos¢ + sin ¢ sin ¢)] (C-15)
C.2 Generalized Derivatives of the Observation Variables
The vector equation (1-90) defining the body axis kinematic accelerations is used to determine the gener-
alized derivatives of the individual body axis accelerations:
O(a_,k) 1
O( gom
O(%,k) 10_ gom
O(az,k) 1
O_ gore
OXTo(
OD OL _ (D cos a - L sin a) _-
OD OL 00]-cosa _- +sina _ + (Dsina + Lcosa) _0a _ gmcosO _ (C-16)
0Y o¢ 00] (o_7)+ -_- + gmcos0cos¢ _ - gmsinOsin¢--_
0¢ 00]-gm cos Osin ¢ -_ - gm sin Ocos ¢(c-is)
Vector equation (1-91) defines the output of body axis accelerometers at the vehicle center of gravity
and is used to determine the generalized derivatives of the individual body axis accelerometers:
0(%) 1 [OXT(9_ gem [ O_
0(%) 1 (OYrO_ - go._ \--5_-
O(a.) 1 [OZTO( - gem L-_
OD OL __]--cosa_+sina_+(Dsina+Lcosa) 0a (c-19)
+ o_)
OD OL _ (D cos a - L sin a) cO(x]- - sin a _ - cos a -_- -_(C-21)
Using equation (1-93), the generalized derivative of the output of a normal accelerometer at the vehicle
center of gravity can be expressed as
(9a. 1 + sin a + cos a + (D cos a - L sin a) Oa
The vector equation (1-95) defining the output of orthogonal accelerometers aligned with the body axes
; but displaced fi'om the vehicle center of gravity is used to determine the generalized derivatives of these
" 43
quantities:
110-_ = O---_-+ g--_ (q yx + rz_) -_ + (py_: - 2qxx) -_ + (pz= - 2rx_) -_ + z_: -_ - y_: _-_ (C-23) !
[ ] -O(ay,i) Oau 1 (2py u _ qxu) Op Oq Or Ol) Oi" (C-24)o_ - o_ 9-0 _ - (pz_+ r_) _ - (qz_- 2_v_)_ + _ _ - z_
0(_,,) 0_o( o_ (o25)
Equation (1-96) defines the output of a normal accelerometer aligned with the z body axis but not located
at vehicle center of gravity, an,i. This equation is used to determine the generalized derivative of an,i:
O(a,_,i) Oa_ 1 [ Op Oq Or 0)5 0c)] (C-26)O_ - -g-( + 97 (2;z_ - rxz) _ + (2qzz- rUz)-_ - (px_+ qYz) -_ - yz -_ + x_ O(J
In equations (C-20) to (C-23), the partial derivatives of the vehicle rotational rates with respect to the
dummy variable _ are defined by equations (C-1) to (C-3). The partial derivatives of the outputs of the -
body axis accelerometers at the vehicle center of gravity are defined by equations (C-16) to (C-19). In these -
equations, as before, the subscripts x, y, and z refer to the x, y, and z body axes, respectively, and thesymbols x, y, and z refer to x, y, and z body axis locations of the sensors relative to the vehicle center
of gravity.
Using equation (1-97), the generalized derivative of the load factor can be defined as
0(n) 1 orO_ mg O_ (C-27)
Equations (1-98) to (1-105) define the air data parameters of interest for this report. These equations
are used to determine the generalized derivatives of the air data parameters:
O(a) o.Tpo OT
O_ poTo [1.4(p0/p0Tu)] 1/2 0_
O(M) 10V V Oa
O_ a O_ a2 O_
0(Re) p( OV V( Op pV( O#- +
O_ tL O_ IL O_ #2 0_
O(Re') p OV V Op pV O_- +
O_ # O_ p O_ #2 O_
O(q) OV v 2 Opo_ - pv -g-(+ 2 o_
(c-2s)
(c-29)
(c-3o)
(o30
(c-32)
kl,1
[(1.0+ 0.2M2)3'5- 1.0]
+ 1.4M(1.O+ 0.2M2)2"Sp_,O_
l,,)_,r2( 5.76M 2 ,_2.s ]L...... k .6M - o.sJ - 1.oj
+p_ {2.4M ( 5"76M2 _2.5
\ 5.6M 2 - 0.8;
[(5.6M_ - 0.8)_]
( 5.76M2 ,_1.5+ 3"0M2 \5.6M 2 - 0.8J
(M _<1.o)
(M" >_ 1.0)
(c-33)
(M > 1.0)
1.4M(1.O + 0.2M2) 2.s _/
O(qc/p_) ")a_ {" 5 .76M2 _2.5. {' 5"76M_ "_ (C-3d)o_ - {..... _\5.6M 2-0.8] "aoM_\5.6M _-o.s7
[ 9.216M ]'_ 0M[(5.6M 2 - 0.8)2J f -_
O(Tt) _ (1.0 + 0.2M 2) OT OM (c-a5)O_ --_ + 0.4T1ll 0--#-
In the preceding equations, the generalized derivative of Mach number appears several times. This term
can be expanded using equation (C-29).
The definitions of the flightpath-related parameters are presented in equations (1-106) to (1-108). These
definitions are used to derive the generalized partial derivatives of the flightpath-related parameters:
0(7)_ 1 [ ¼ OV O]_] (C-36)0---_- (V 2 --)/2)1/2 _" V 0 e "+ O_J
O(fpa) 10l)" (C-37)o_ g o_
o(_)O_ ---- [--ay,k COS ¢ COS O + az,k sin ¢ cos O] 0¢0_
+ [ax,k cos 0 + ay,k sin ¢ sin 0 + az,k cOS ¢ sin O] O0o_
Oax,k cOay,.._kk_ i)az,k (C-38)+sinO O# -sin¢cosO O_ cos¢cosO O_
The partial derivatives of altitude rate ]z and velocity rate 1) that appear oil the right-hand side of these
equations are defined in equations (C-13) and (C-7), respectively. The partial derivatives of the body axis
accelerations appearing in equation (C-38) are defined in equations (C-16) to (C-18).
45
Usingequations(1-109)and (1-110),the generalizedderivativesof the energy-relatedparametersaredefined.Thepartial derivativesof altituderate andvelocityrate appearingin equation(C-40)aredefinedin equations(C-13)and(C-10),respectively:
o(E.) v oy oh (o39)-5_ - g 0,_+0_
o(Ps) __ov v o9 oi_0_ - g 0( + -_ --o_+ --o¢ (o4o)
The derivatives of the force parameters, lift (eq. (1-111)) and drag (eq. (1-112)), are defined in sec-
tion D.1. The generalized derivatives of the normal force (eq. (1-113)) and the axial force (eq. (1-114)) are
presented in terms of the generalized derivatives of the lift and drag forces:
O(N) OL OD _ (L sin _ - D cos a) as0_ - cos_ _- + sin_ _- _ (C-41)
0(A) _ OL OD _ (L cos a + D sin a) Oa0{ sin a -_ + cos a -_- 0_
(o42)
The body axis rates are defined in equations (1-9) to (1-11). The time derivatives of these terms are
defined in equations (1-115) to (1-117). These equations are used to derive the generalized derivatives of
the body axis rates and accelerations:
0#O(u) _ OV 0a _ V o_sin fl _-0--(-- cos_cosZ5-T- Vs:n_cos# N _os (C-43)
o(,) ov ozo_ - sin_ N + y Cos# 0_ (c-44)
o(_) ov o_ o_ (c-45)o-T = _i._ _osa ?-( + v cos_ cosa y( - y sin_ sin_ O-7
O(_) 1 {OX_ OD OZ) Oq O_O¢ - ._ k N cos_ -g-(+ si. _ -_ - V sin_ cos_ -_ + Vsin_ 0-_
+(rsin#_qsinc_cos/3) OV [1 ]-_ + (D sin a + L cos a) - qV cos a, cos ]3 0a0_
o/3 00+ (rVcos/_ + qVcosasinfl) --_ - gcos0 0--_
Or0(_) 1 /'0]:r OY Op v cos# 0¢(9_ - m \ 0_ ÷-_- +Vsino_cosp_- cos_ --
OV Oa+ (p sin # r cos _ cos/3) _ + (pV cos _ cos/3 + rV sin _ cos fl)o_ cos
uq
o/3 o¢ 00- (pV sin c_sin/_ - rV cos _ sin/_) _-_ + g cos 0 cos ¢ _- - 9 sin 0 sin ¢ 0--_
(c-46)
(o4:)
46
1 (00__%Tsin 0D OL)_vsin/30p cos/38qm -- _- co_,_-_- _ +v cos_ o_
+ (qcoso_cos_-;sin_)°V [-_ I °°_-_- - (D cos a - L sin a) + qV sin a cos/3 -_-
- (qV cos a sin t3 + pV cos 13) 0/3 0¢ 00- gcos 0sin ¢ _- gsin 0cos¢
(c-4s)
The outputs of various instruments displaced from the vehicle center of gravity are defined in equa-
tions (1-118) to (1-121). These equations define angle of attack, angle of sideslip, altitude, and altitude rateinstrument outputs. The generalized derivatives of the quantities are based on these equations:
0(_,i)0f
0(_,d0f
O(h,dOf
y, Op xa Oq (qx_2py_) OV Oa--= -V o-_+ v of -- 5-(+ o--(z_ Op x_ Or (rx_zpz_" _ OV Off
--= y of + V o_ \ v2 ]-_ + o_
o¢- (--Yh cos ¢ COS0 + Zh sin ¢ cos 0)
O0 Oh
+(Xh cos0 + yhsinCsinO + Zh cosCsin 0) _ + 0---_
(c-49)
(c-50)
(C-51)
o¢[()(Yh sin Coos0 + z.h cos Coos 0) + t}(y/_ cos Csin 0 - z h sin Csin 0)]
[ ]o0+ - 0(_ sin 0 - u_sin ¢ cos0 - _ cos¢ cos0) + _(U_cos¢ sin 0 - _ sin ¢ sin0)
0¢ 00- (yhcos¢ cos0 - zAsin¢ cos0) N + (x_ cos0 + yhsin¢ siu 0 + _ cos¢ sin 0)
+_- (c-52)
The generalized derivatives of bank angle rate, pitch attitude rate, and altitude rate with respect to the
dummy variable _ are defined in equations (C-10), (C-11), and (C-13), respectively.
The final set of observation variables is defined in equations (1-122) to (1-125). These equations, defining
total angular momentum and the stability axis rotational rates, are used to determine the generalized
derivatives of these quantities:
O(T) op
o(p_) opof _ cos_ -_
O(q_) Oqof of
o(_) opO( - sin,_gg
Oq Or
+ (Iuq - r_:yp - Iuzr ) -_ + (/_r - I_p - Iy_q) -_
Or Oa
+ sin a _-- (psin a- r cos a)_-
Or
-t- cos o_ _ + (-p cos a - r sin a) 0_0_
(C-53)
(c-54)
(c-55)
(c-56)
47
r
L
APPENDIX D--EVALUATION OF DERIVATIVES
The generalized partiM derivatives presented in equations (C-l) to (C-56) contain partial derivatives of the
state variables, thrust forces, and total aerodynamic forces and moments with respect to the dummy variable
_. In this appendix, these partial derivatives are defined with respect to specific state, time derivatives of
state, and control variables. The derivatives of atmospheric parameters are also discussed.
D.1 Preliminary Evaluation
First, the partial derivatives of the state variables with respect to the state, time derivatives of state, and
control variables are considered. All partial derivatives of the state variables with respect to the state
variables are either equal to zero or unity. Thus,
Op Oq Or OV Oa Off 0¢ O0 0¢ Oh Ox Oy
O--p= Oq = 0---_= O----V: O--_= O---fl= 0-¢ = 0--0= 0--_ : Oh--_= b x - Oy- 1 (D-1)
and all other derivatives of state variables with respect to state variables are equal to zero. The partial
derivatives of the state variables with respect to the time derivatives of the state variables (5' and _, in
particular) are equal to zero. This is also true of the partial derivatives of the state variables with respectto the control variables.
Second, the partial derivatives of the aerodynamic forces and moments with respect to the state, time
derivatives of state, and control variables are evaluated. Using the definitions of the force and moment
coefficients presented in appendix A, the partial derivatives can be explicitly evaluated in terms of the
stability and control derivatives.
D.I.1 Rolling mornent derivatives.--
Op 2V
OL qSb_ C
O__L_L_qSb2caOr 2VOL
O----V= SbpVC_ + _SbCev
3L
Oa - qSbCe_
OL
03 - qSbCe
OL 2 OpO----h= SbV_C_ -_ + qSbCeh
OL qSb_ C
OL qSb2 C-
OL
O_i - _SbCe_,
(D-2)
(D-3)
(D-4)
(D-5)
(D-6)
(D-7)
(D-S)
(n-9)
(D-IO)
(D-11)
PRECEDING PAGE BLANK NO_ FILMED
49
D.1.2 Pitching moment derivatives.--
OM FTSb_d7Op - _ _'_
OM__ _S_2Cm qOq 2V
OM (tSb_
OM- S_pVC_ + _IS_C_vOV
OM FtSe c7cga - 2V _°
OM
0/3 - CTS_Cr_,
OM _ OpOh - SW2Cm -_ + _lS_Cmh
03,I F1Se2O& - 2V C_,
031 (iSb_
OM
O_i - qS_C._,
(D-12)
(D-13)
(D-14)
(D-15)
(D-16)
(D-17)
(D-18)
(D-19)
(D-20)
(D-21)
D.1.3 Yawing moment derivatives.--
ON _ gtSb_c,_pOp 2V
ON qSb_
Oq - -_ ¢;nq
ON (tSb 2Or - 2V C'_T
ON- SbpVCn + qSbCnvOV
ON
Oa - (ISbC.¢,
ON
03 - qSbC,_
ON _ OpOh - SbV2C'_ _ + qSbCnh
ON qSb_c-_- 2V '_'_
ON qSb_ Co3-ON
06i - #SbC,_,
(D22)
(D-23)
(D-24)
(D-25)
(D-26)
(D-27)
(D-28)
(D-29)
(D-30)
(D-31)
5O
D.1.4 Drag force derivatives.--
OD _Sb C_--i-VD.OD _Se C
OD (tSb C0r-5-i7 o_019
O-"-Y :- S pVCD -_- gtSCDv
OD- OSCD_
O_OD
- OSCD_o_
OD _ OpO-h = SV2CD _ + clSCDh
OD c_S_C
OD qSb Co_ - _Y o_OD
08i CtSCD_
(D-32)
(D-33)
(D34)
(D-35)
(D-36)
(D-37)
(D-38)
(D-39)
(D-40)
(D-41)
D.1.5 Sideforce derivatives.--
OY OSb Clop -- "-_ YP
OY _ qS_ Cy qOq 2V
OY _ glSb Cy _Or 2VOYO---V= SpVCv + qSCYv
OY
lOa qSCy=
OY
o---_= _scv,
OY 1 OpOh - SV2Cy -_ -t- qSCy h
OY _Se COd- 2-V v_
OY clSb C
OY
O_i _tSCy_,
(D-42)
(D-43)
(D-44)
(D-45)
(D-46)
(D-47)
(D-48)
(D-49)
(D-50)
(D-51)
51
D.1.6
OL @SbC- KV (DS2)
cOL _S_ _ (D-53)N- 2--VcOL qSb COrr - _ Lr (D-54)
cOL-- Sf)VCL -1- (_,-.,¢CLv (D-55)
cOVOL
OOl q°CCL_ (D-56)
OL
COG- qSCLz (D-57)
COL 1 copOh - SV2CL _ + (tSCLh (D-58)
cOL 4S_ CcO&- _ La (D-59)
cOL 4Sb CcOj- _-_ L_ (D-60)
cOL
cOSi -- 4_CL6i (D-61)
Lift force derivatives.--
Next, the partial derivatives of the powerplant-induced forces and moments with respect to the state,
time derivative of state, and control variables are considered. The partial derivatives of the powerplant-
induced forces and moments are assumed to be zero except for moments taken with respect to the body
axis rates (p, q, 7"), moments and forces taken with respect to the velocity and velocity orientation terms
(V, a, fl), and forces taken with respect to the control variables. These terms, assumed to be nonzero, are
taken as primitives and not evaluated further. Thus, using Fp to represent a powerplant-induced force (XT,"fiT, and ZT) and 2_[p to represent a powerplant-induced moment (LT, _[T, and ART),
cOp - O----q-- COt. - CO¢ cO---O-- cO¢ cOt, - cOx - cOy -0 (D-62)
cO_ - cOO - cO_ - Oh - COx 0----_- 05---_- = 0 (D-63)
an d
op; coF;oF; cot; OM; COM;OM; OM;---, _, , , , , , , ,and _-fiCOV Oc_ coil 0 5_ COp COq Or COV Oc_
are taken as primitives and not evaluated further.
The final set of partial derivatives to be discussed are the derivatives of atmospheric parameters with
respect to the state, time derivative of state, and control variables. In this report, all atmospheric parametersare assumed to be functions of altitude only. Thus, except for
OT Op O# Op_Oh' 0--£' O---h_'and Oh'
all derivatives of ambient temperature, density, viscosity, and ambient pressure are assumed to be equal
to zero. The nonzero quantities listed previously are dependent on an atmospheric model. Clancy (1975),Dommasch and others (1967), Etkin (1972), and Gracey (1980) present discussions of atmospheric models.
In this report, the quantities will be taken as primitives and not evaluated further.
E
=
=
=
52
D.2 Evaluation of tile Derivatives of tile Time Derivatives
of the State Variables
The generalized derivatives of the time derivatives of the state variables are defined in appendix C, equa-
tions (C-1) to (C-15). In this section, these generalized derivatives are evaluated in terms of the stability
and control derivatives, primative terms, and the state, time derivative of state, and control variables. In
this section, the notation c9(_i)/Oxi is used to represent the more correct notation Ofi/cgxj that is employed
in the discussion at the beginning of section 3. This notation is used because there is no convenient no-
tation available to express these quantities clearly--particularly not the usual notation employed ill flight
mechanics texts such as Etkin (1972) and McRuer and others (1973). The notation tha.t defines quantities
such as Lp = c9(fi)/Op and }Vlq = c9(gl)/0 q is misleading in this context because the definitions of those terms
(such as L;, Mq) are based on assumptions of symmetric mass distributions, symmetric aerodynamics, and
straight and level flight, and additionally do not include derivatives with respect to atmospheric quantities.
D.2.1 Roll acceleration derivatives.--
0(i)) 1
Op det I
O(/)) 1
Oq det I
0(/)) 1m
Or det I
0@) 1m
0V det I
0(p) 1
0a det I
0(fi) 1w
0fl det I
0(p) _ 00¢
0(p) _ o00
off,) _ oo¢,o(_) s
w
Oh det I
VtSb 'I bC OLT 03JT (-.95'rT
"_0 (. 1 fP "]l- Z2c'Cmp "J- [35Cnp) "J- _ J- _ J- 6_l--_
- 2p0(I_J2 - I_yh) + qo(I_J1 - IwI2 - D_h) - %(L_5 + D_h - z_5)]
qSerr bf , OLT O_IT 0NT2-i_0t_,_,, + beck, + I_bc_)-N-q + W + 0--_
+ Po([xz[i -- IyzI2 - DzI3) + 2qo([yz[1 - IxuIa) - ro(DJ_ - IxyI2 + [xzla)]
[_ O_T OM_r O_VT,:_o(IlbCa +/2eCm_ + IabC,_,.)+ ---_-r + ---g-_r + 0-"7
- po(l_y[1 - Dyh - IyzI3) - qo(DJ_ - Ixvh + I_,.I3) - 2ro(IvzI_ - IxJ2)]
IiSb(pVoCe + qCev ) + [2S_(pVoCm + _tCmv )
oL_ oM_ ox_]+ I3Sb(pVoC,, + &_C,,v) + 1_ _ + I2 -_- + I3 _j
OLT OMT ONT]FIS(IlbC&, + [2eCm_ -b I3bCn_) + I1 -_a + 12 T_ + h oa ]
OL T OMT ONT ]OS(hbC_, + hec_ + hbC,_) + h -_ + h -_- + h Off J
[Ilb(_17o2C_ cop (_Vo2Cm OD•
(D-64)
(D-65)
(D-66)
(D-67)
(D-6S)
(D-69)
(D-V0)
(D-71)
(D-72)
(D-73)
53
aU,)Ox
0(_)Oy
0(i,)O&
a(_)
06i
-0
=0
/(ZlbC_ + hec,._, + z3bc,,_)2Vo det
qSb
2Vo det i (hbCgi3 + I2_Cmi3 + I3bCni_)
qsi(_r,bcg_, + heG,,_, + &bC._,)dct
D.2.2
0(,)) 1
Op det I
0(4)Oq
o(4) 1Or det I
0(0) 1w
OV dot I
0(0)0_
0(0)Off det I
0(0___2)= o0¢
0(4__2)= o0o
0(,-;,)_ oae0(0) sOh det I
Pitch acceleration derivatives.--
qSb I bC OLT OMT ONT_0 ( 2 _, + I4eC,np + IsbCnp) + I2 -o-y ÷ I4 --_-_-p+ Is -ap
- 2po([_:z[4 - I_vIs) + qo(I_,I2 - IvJ4 - DJs) - ro(IxvI2 + DyI,, - Iv_Is)]
1 [(1S_.ci ,.,._, OLT O3IT O]VTdet I [ggo t _u,_eq + hec.,q + lsbC,_q) + 12 _ + LI -_q + Is O----q-
+ po(L=h - l_,Z4 - rids) + 2qo(I_,I2- I_ds) - ,'o(D_I2- I_h + l_,&)]
[_ OLT OMT ONTo (hbC& + hgCm. + I5bCnr) + h -_r + h _ + 15 0---_-
- po(Gyh - Dyh - I_,Is) - qo(Dxh - Ixyh + L:Js) - 2ro(/v_h - Ixz/4)]
hSb(pI4Ce + qCtv) + QSa(pVoCm + qCmv)
oLs OMT 0NT l+ IsSb(pVoCn + qC. v ) + I2 _ + h _ + 15 _j
1 [ OLT O.[I,[T ONT]det I qS(I2bCG + I4eCmo + Isban_) + I2 _ + I4 _ + I5 --_ ]
OL_ O_¢s ONTlqS(hbC_,+ Z4eCm,,+ &be.,,) + h _ + h -5-Y + & -_J
[I2b(_V2C, Op'-_-{- (tCgh) +14c.(1V'2Cm Op-_-'_-'[- qCmh)
1 2+ &b(_v6 c,_ ap
0(4)OX
(D-74)
(D75)
(D76)
(D-77)
(D-78)
(D-79)
(D-80)
(D-81)
(D-82)
(DS3)
(D-84)
(D-85)
(D-86)
(D-87)
(D-88)
(D-89)
5.|
a(_--!= oOy
0(o__2)_0_
o(_)o#
°(_--2_ ,#-/hbc,, + _r4_c,,,,,+ IsbC,,,,)O6i cte_ _ '
q°_c f(hbC_ a + [4cCma + IsbC_a)2Vo det
qSb "I bC"21_o'_et [1" 2 !k + I4cCrn b + IsbCnb)
(D-90)
(D-91)
(D-92)
(D-93)
D.2.3
o(÷)0p
Yaw acceleration derivatives.--
1 [_lSb,f bC OLT OMT ONTdeti[_oo ( 3 _,+Is_C._p+I6bC'_p)+h-_p+IS--_p+Ie'op
o(+) 1Oq det I
o(÷)Or
o(÷)OV
o(÷)Oa
o(+)o_
°(÷--!= oo¢
0(÷--2= oO0
o(÷) = oo_o(÷) sOh det I
0(÷---2)= oOz
0(÷--2= oOy
1 FEtSb- b OLT OMTd_tI [EVo/& ce_ + ,rs_Cm_+ [_Cn_) + 5 _ + I_ _ + I_ --
- po(I_Ja + D_,I5- I_,_I6)- qo(D_&- IxJ5 + I=I&
- 2_o(I_zI3- z=Isi]
- 2po(IxzI5 - I,:vI6) + qo(I_zI3 - IyzIs - D_I6) - ro(I=J3 + Dv/s - I_zla)] (D-94)
OLT OMT ONT
- _[_(I3bCG+I5_Cmq+I6bC_q)+I3--_q +I5 _-q +I6"OqLzvO
+ po(l=13- ly_15- z)z16)+ 2qo(&_&- I_16)- To(mJ3- I_15+ l=r6)1 (m95)
ONT
Or
1 [IsSb(pVoC_ + CIC_v) + [sSS(pVoCm + qCmv)det I
__ o_IT aNT]+ &Sb(pVoCn+ qc,_v) + h + z5-O-V-+ z6-oV-j
i [ OLT OMT 0NT]act I OS(I3bC_o + IseCm_, + I6bCn_,) + I3 _ + I5 _ + I6 ---_-_-]
l [ %__LflT 03_/T aNT]det I flS(hbC_ + IseCm o + I6bC_o) + h -- + Is --_ + I6 -_--]
-_ + -_ +<"0
(D-96)
(D-97)
(D-98)
(D-99)
(D-100)
(D-101)
(D-102)
(D-103)
(D-104)
(D-105)
55
o(i-)0,_
o(,'.)
0,5i
qSe (I3bCe_, + IseCr_ + IBbCn_)2V0 det I
qSb (i3bCei_ + Is_Cm_ + IBbC,b)217o det I
giS (IabCe6, + IseCma, + [6bC,% )det I
(D-106)
(D-107)
(D-108)
D.2.4 Decoupled roll acceleration derivatives.--
(ISb 2 ,._ OLT
Op 1-,: Op
O(f)') 1 ros<_ OLT
N = KtVigo_ +O(fo') 1 [_lSb2,_, OLT
-- o--Si7yw e,. + -- _Or I_L2Io Or
Ixyro + Ix_qo]
+ G_po+ _[_qo + ,.o(r_- &)]
[zyPo + qo([y - Iz) - 2Iu_ro ]
oo;')OV
o0;')0Oz
o0;')o3
o0;')o¢
o(,;,)
Oz
o(p)Oy
o(p)oa
o(p)o_
o(P)O&
OLT]i Sb(pVoCe + _tCev) + -_-j&
i ((tSbCg_ + OLT_
1
-0
-0
qSb_
- 21,%L: Cea
qSb 2
- _ og_
-- q-_xbCQi
(D-109)
(D-110)
(D-111)
(D-112)
(D-11a)
(D-114)
(D-115)
(D-116)
(D-117)
(D 11S)
(D-119)
(D-120)
(D-121)
(D-122)
(D-123)
!
7
56
D.2.5 Decoupled pitch acceleration derivatives.--
o(O')Op- _ L-Y_ol[(tSb_Cmv + OMTo___p_2[_po - [u.qo++ ro(lz - I_)]
O(O') I ((ISg2c OMT )Oq - _!ry -_o m+ + _ + Izyro - ZyzPo
0(i1' ) l [glSb6. OMT ]
[ °M+l0(0') 1 S_(pVoCm + qCmv) + OV J--_= _
0(0') 1 ( 02I_IT_--Oa = -_u gtS gC m_ + _ ]
o(o')_ 1
o(0') -0o¢
o(0') -0O0
o(o')o¢
0(09
OM_)gtSeCm_ + --_--2
---0
Oh - % _o(o') -0
Oz
o(o') -0Oy
0(09 c+se_Cma
oa 2_/o%
0(09 qSbe C
o(0') os_ Co_---_-.-% m+,
(D-124)
(D-125)
(D-126)
(D-127)
(D-128)
(D-129)
(D-130)
(D-131)
(D-132)
(D-133)
(D-134)
(D-135)
(D-136)
(D-137)
(D-138)
D.2.6 Decoupled yaw acceleration derivatives.--
+ _ + 2_%po+ q0(I+- %) + %z,'o
ONT+ -_q + po(Z_- %) - 2f_qo - _%+,'o]
o(e) 1 [c_Sb2ov - _ 5-froc_"O( i,') 1 r qxbe
- g
0(÷')0r-L-zl ( qSb2-_oCn_+ ONT--Ixzq°+[Y_P°)Or
_ [ OX+lO( ÷'____J= 1 Sb(pVoCn + _C,w ) + --o-i-/-JOV Iz
(D-139)
(D-140)
(D-141)
(D-l,12)
57
a(÷')8a
o(÷')a_
a(÷') _o¢
a(÷')O0
a(÷')a¢
a(÷')Oh
a(÷')Ox
a(÷')Oy
a(÷')8&
a(÷')ok-
o(÷') _5 cO& - I. ';_i
_ 1 {'(ISbCn_, + ONT_I: t, --G-a-_)
] (_SbCn_\ q- ONT)770
-- -- 0
-- -- 0
& Wc. y£+ocn_
-- -- 0
0
4Sb_
2 VoI_ C_
qSb 2
(D-143)
(D-144)
(D-145)
(D-146)
(D-147)
(D-HS)
(D-149)
(D-150)
(D-15i)
(D-152)
(D-15a)
D.2.7 Total vehicle
ado _Sbas, - 2,-Vo_(-
o(?) _ qse (_Oq 2Vorn "
ado _Sb- iVoo_(-01"
O('(_") 1
8V m
0o_ m
acceleration derivatives.---
cos/30 Cnp + sin/30 Cyp)
COS/30 CDq nt- sin/3o Cyq)
cos/30 Co, + sin/3o CYr )
- Scos/3o (pVoCD + qCDv) -_ S sin/3o (pVoCy + 0CY_)
oxs+ cos ao cos/3o _ + sin ao cos/3o -_ + sin/30 0V ]
OXT-- qS COS/30 CD¢, OC qS sin/30 CYa + cos ao cos/30 Oa
OaT OYT
+ sin ao cos/30 _ + sin/30 _ - XT sin ao cos/30
+ Zm cos ao cos/3o + rag(sin 0o sin ao cos/30 + cos 0o cos ¢o cos ao cos/3o)[J
(D-154)
(D-155)
(D-156)
(D-157)
(D-158)
58
If
- _/_s(-co_oco,+_ngoCo+_,ngoCY_+_o_oc_)0XT 8ZT 8YT
+ cos ao cos _o _ + sin ao cos/30-fff + sin/_o 8fl
- XT sin/30 cos ao -- ZT sin ao sin/30 + YT cos flo1
+ rag(sin 0o cos ao sin/30 + cos 0o sin ¢o cos/30 - cos 0o cos ¢o sin cto sin flo)/,J
8¢o(v) _00
o(?)o¢
0(?) _oh
a(t)cox
a(?)Oy
a(#)8&
o(% _o_
a(?)8,5_
-- - g(cos 0o cos ¢o sin _o - cos 0o sin ¢o sin C_ocos/30)
g(- cos Oocos ao cos/30 - sin Oosin ¢o sin/50 - sin 80 cos ¢o sin ao cos/30)
0
Op
_- _+
qS_' (__ COS/_0 CDa + sin flo CYa)2Vom"
#tSb (Q_700m, -- cos/_0 CD_ + sin _o Cy_)
_-(- cos/3o CD6i + sin/3o Cy_i )
1 ( OXT 69YT+ -- cosao + sin/3o +m cos_o_
ozs__i__o cos/_o-377]
(D-159)
(D-160)
(D-161)
(D-162)
(D-163)
(D-164)
(D-165)
(D-166)
(D-167)
(D-16S)
D.2.8 Angle-of-attack rate derivatives._
o(a)8p
O(&)
Oq
0(_)
_F
o(_)OV
qSb
- 2V02m cos/50 CLp - tan 3o cos ao
qSe
_ 2V_mcosfloCLq,, + 1.0
qSb
- 2Fk2mcos/_oCL,o - tan/3o sin ao
OZT OXT__ --_ _1 [s(V°pCL + qCLv) -- cOSaO _ + sinao 017
mVo cos flo
V--7[-glSCL + ZT cos a0 -- XT sin+ o: 0
- tj
+ rag(cos 0o cos ¢o cos ao + sin 0o sin a'o)]_)
(D-169)
(D-It0)
(D-171)
(D-172)
59
a(&)0o:
o(_)o/3
o(_)o¢
o(_,)O0
o(_)
Oh
o(_)O=
o(_)
05i
-0
1 [ OZ T OX TmVo cos/30 _gtSCL_' - cos C_o _ + sin C_o _ + ZT sin o'o + XT cos ao
+ rag(cos 0o cos ¢o sin ao -- sin 0o cos ao)
+ tan/3o (Po sin ao -- ro cos ao)[
1 ( OZT OXTrnTv})cos 13o_ (tSCL_ - cos C_o--_ + sin C_o 0---fi-
- tan/3o [-OSCL + ZT cos a'o - XT _in ao
+ rag(cos 0o cos ¢o cos C_o+ sin 0o sin a'o)]}
1
cos _/3o (Po cos C_o+ ro sin ao)
g cos 0o sin ¢o cos aoVocos_o
g (sin 0o cos ¢o cos C_o- cos 0o sin _o)Iiocos3o
mVocos3o V°2CL -0--£+ qCi_
qS_
Vo2mcos/3oCL_clSb
2Vo2m cos ,30 CL_
0ZT1 -_SCLe, + cos O'o 06,mVocos3ooxz]
-- - sin C_o"-_6i J
(D- 173)
(D-17d)
(D-175)
(D-176)
(D-177)
(D-]78)
(D-179)
(D-180)
(D-181)
(D-182)
(D-lS3)
D.2.9 Angle-of-sldesllp rate derivatives.--
0(3)Op
0(3)
Oq
0(3)or_
_tSb . .
- 2Vo-_m(sin/3o CDp + cos/3o Cvp) + sin O_o
OSe
- 2Vo2m(Sin/3o CDq + cos/3o Cyq)
clSb ,. ,_ ,._-- = _(sm _o _D, + cos¢_oCy,) - cos_o
_Vd,n
(D-184)
(D-185)
(D-186)
6O
-.¢-
a(a)1 [Ssinflo (pVo + 4CDv) + Scos/3o (pVoCy + (]CYv)OV m Vo
O XT (YYT 0 ZT ]
- cos ao sin/3o --_ + cos flo _ - sin ao sin flo --_]
1 [qS(sin/3o CD + cos flo Cy) - XT cos ao sin flom Vff
+ Vr cosflo - ZT sin ao sin flo] (D-187)
1 [ OXT OYrmVo qS(sinflo CD_ + cosflo Cy_) - cos ao sinflo _ + cosflo 0a
OZT- sin % sin flo _ + XT sin ao sin flo -- ZT cos % sin flo
-- my(sin 0o sin ao sin flo + cos 0o cos ¢o cos ao sin flo)]
+ Po cos C_o+ ro sin ao (D-188)
Off mVo qS[sinflo (CDfi -- Cy) -t- cosflo (CD + Cyfi)]
OJ_:r OZT0YT sin % sin flo
-- cos ao sin flo -- + cos flo O----f-- Off
-- XT cos ao cos flo -- YT sin/30 -- ZT sin ao cos flo
+ my(sin 0o cos _o cos flo -- cos 0o sin ¢o sin flo -- cos 0o cos ¢o sin % cos rio)} (D-189)
0(/)_____)= _-_(cos 0o cos ¢o cos flo + cos 0o sin ¢o sin ao sin flo)o¢
o(3_2)o0= _( _o_0ocos_o_n #o- s_n0os_n¢ocos#o+ sin 0o cos ¢o sin ao sin rio)
o(j)
o(_)
o(_) _ oo¢
0(j__)) - S [sinfloOh mVo
O(j) _ 0Oz
O(j) _ 0Og
0(_)
O_
o(_)
O) -
o(j)06_
lvo2C D Op-ff_+ qCD_
(IS_ .
2v-_m(sin_ocD_+ cos9oCy_)
._ (sin flo CD_ + cosflo Cy_)zvdm
1[mVo qS(sin flo CD6, + cos flo CY_ i
oz_ 1- sin ao sin flo --_-/j
+cos_o (1VdCv Op• _-#+ (TCy.)]
OXT OYT
-- cos _o sin flo -_/ + cos flo 06i
(D-190)
(D-191)
(D-192)
(D-193)
(D-194)
(D-195)
(D-196)
(D-197)
(D-]9S)
61
D.2.10 Roll attitude rate derivatives.--
8p
8q
8r
ov
8_
o3
8¢
ae
8_
oh
8x
0(4,)8y
o_
o_
-1
- sin ¢o tan Oo
-0
-0
- qo cos ¢o tan Oo - ro sin ¢o tan 0o
- qo sin ¢o sec 2 _o + ro cos ¢o sec 2 0o
-0
-0
-0
-0
=0
(D-199)
(D-200)
(D-201)
(D-202)
(D-203)
(D-204)
(D-205)
(D-206)
(D-207)
(D-208)
(D-209)
(D-210)
(D-211)
(D-212)
(D-213)
=
z
D.2.11 Pitch attitude rate derivatives.--
a(o) _ 0Op
8(0)-- COS _)0
Oq
- sin ¢oOr
o(o) _ o8V
(D-214)
(D-215)
(D-216)
(D-217)
¢;2
o(o__))= o
0(0---2)= 009
o(#)-- -qo sin ¢o - ro cos ¢o
o¢
o(0--2)= ooo
°(0--!)= o0¢
o(#) _ ooh
o(_) _ oOx
o(0--2)= oOy
o(_) _ ooa
o(o) _ o
°(°--2)= o0,5_
(D-218)
(D-219)
(D-220)
(D-221)
(D-222)
(D-223)
(D-224)
(D-225)
(D-226)
(D-227)
(D-228)
D.2.12 Heading rate derivatives.--
o(¢)Op
0(¢)
Oq
0(¢)
Or
0(¢)
OV
o(¢)
0(¢)
0Z
o(_;)o¢
o(_)oo
o(¢)Oh
-0
- sin &o sec Oo
- cos 6o see Oo
-0
-0
-0
- qo cos 0o see Oo - ro sin ¢o sec Oo
- qo sin ¢o see Oo tan Oo+ ro cos ¢o see Oo tan Oo
-0
(D-229)
(D-230)
(D-231)
(D-232)
(D-233)
(D-234)
(D-235)
(D-236)
(D-237)
63
D.2.13
o(_) _ o(gz
o(¢) _ oegg
o(_) _ oo_
o(_) _ ooa
o(_) _ o05i
Altitude rate derivatives.--
0(t0 _ oOp
O(t_)_ ooz
a(h) _ oOr
a(h)OV
o(t_)Oa
o(t0o3
o(h)o¢
adoae
o(iO _ o
a(/,) _ oOh
ado _ oOx
o(i_) _ o8u
ado _ oOd
o(i_) _ oo3ado _ o05_
- sin 0o cos flo cos ao - sin ¢o cos 00 sin/3o - cos ¢o cos 0o cos flo sin ao
- -Vo(cOS flo sin So sin 0o + cos flo cos So cos ¢o cos 00)
- -Vo(sin flo cos So sin 00 + cos flo sin ¢o cos 00 - sin flo sin o'o cos ¢o cos 00)
- -Yo(sin/30 cos ¢o cos 0o - cos/_o sin So sin ¢o cos 0o)
- Vo(cos flo cos So cos 0o + sin flo sin ¢o sin 0o + cos flo sin O'o cos _o sin 00)
(D-238) :
(D-239)
(D-240)[
(D-241)
(D-242) :
=
(D-243) _
(D-244) -
(D-245) -
(D-246)
(D-247)
(D-248) Z
(D-249)
(D-250)
(D-251)
(D-252)
(D-253)
(D-2.54)
(D-255)
(D-256)
(D-257)
64
D.2.14 North acceleration derivatives.--
Op
Oq
Dr
a(:_ )
OV
a(:_)
8(:_-)o/3
-0
-0
-0
- cos/30 cos ao cos 0o cos ¢o + sin/3o (sin ¢o sin 0o cos ¢o - cos ¢o sin ¢o)
+ cos/30 sin ao (cos ¢o sin 0o cos Co + sin ¢o sin ¢o)
- Vo[ cos/30 cos ao (cos &o sin t_ocos _bo + sin ¢o sin ¢o) - cos/3o sin so cos t_ocos _bo]
- Vo[ cos/30 (sin ¢o sin 0o cos ¢o - cos ¢o sin _'o) - sin/30 cos _o cos 0o cos ¢o
- sin/30 sin ao (cos ¢o sin eo cos ¢o + sin ¢o sin ¢o)]
- Vo[ sin/3o (cos ¢o sin 0o cos ¢o + sin ¢o sin g'o)
+ cos/30 sin Veo(cos ¢o sin _o - sin _o sin 00 cos ¢o)]
- Vo( sin/3o sin ¢o cos 0o cos _bo- cos/3o cos C_osin 0o cos _bo
+ cos/3o sin c_ocos ¢o cos Oocos ¢o)
0-7-_--= Vo[ - cos/30 cos C_ocos 0o sin ¢o - sin/30 (sin ¢o sin Oosin ¢o + cos ¢o cos ¢o)
- cos/3o sin ao (cos ¢o sin 0o sin ¢o - sin ¢o cos ¢o)]
o(_) _ o
o(_) _ o8x
O(_) _ 0Oy
o(_) _ orga
o(_) _ o0/3
o(:_) _ o
(D-258)
(D-259)
(D-260)
(D-261)
(D-262)
(D 263)
(D-264)
(D-265)
(D-266)
(D-267)
(D-268)
(D-269)
(D-270)
(D-271)
(D-272)
D.2.!5 East acceleration derivatives.-
0(/'--2)= oOp
o(i/) _ oOq
o(_) _ oOr
(D-273)
(D-274)
(D-275)
65
c.3(_)) _ cos 0o sin _/'ocos/3o cos a'o + sin/3o (cos _bocos _bo + sin gSosin 0o sin _o)8V
+ cos/30 sin a'o (cos _bosin 0o sin _bo- sin _bocos _bo)
0(____)= Vo[ cos/3o cos C_o(cos _bosin 0o sin g'o - sin _bocos _bo)0c_
o(_))00
(D-276)
- cos/_o sin O_ocos 0o sin _bo]
- Vo[ cosflo (cos qSocos _o + sin _bosin 0o sin go) - sin/3o cos a'o cos 0o sin _bo
- sin/3o sin C_o(cos _bosin 0o sin g_o - sin q5o cos _o)]
- Vo[ sin flo (cos 4)0sin 0o sin _o - sin _bocos Ibo)
- cos/3o sin c_o (sin qSosin 0o sin _bo+ cos qSocos _bo)]
- 1Io( sin flo sin _bocos 0o sin _o - cos/3o cos ao sin 0o sin _bo
+ cos _o sin C_ocos 0o cos 0o sin _bo)
- Vo[cos/_o cos ao cos 0o cos _bo- sin/30 (cos _o sin g'o - sin q5o sin 0o cos _'o)
+ cos,_o sin _o (cos 4)osin 0o cos g'o + sin $o sin _bo)] (D-281)
0(#) _ 0 (D-282)Oh
o(_) _ o (D-2sa)Ox
°(_----2)= o (D-2S4)0v
o(,)) _ o (D-28s)8&
o(_) _ o (D-286)o3
o0)) _ o (D-2S7)0(5;
(D-277)
(D-278)
(D-279)
(D-9_80)
D.3 Evaluation of the Derivatives of the Observation Variables
........ Tim generalized derivatives of the observation variables are defined in appendix C, in equations (C-16) to(C-56). In this section, these generalized derivatives are evaluated in terms of the stability and control
derivatives, primative terms, and the s_ate, time derivative of state, and control variables.
D.3.1 Longitudinal kinematic acceleration derivatives.--
0(a_,k)Oq
O(_x,_)Or
OSb (2V-_ora'- cos ao CDv -b sin C_o CLp)
qSe (_2Vogom" cos ao CDq + sin ao CLq)
qSb .2V--_om (- cos ao CD_ + sin eto CL_ )
(D-28S)
(D-289)
(D-290)
66
==
0(ax,k) 1
OV gore
O(ax,k) 1Oa gore
c0(ax,k) 1
Off gore
O(a_,k) _ 00¢
0(az,k) _ _goose00 go
O(az,k) _ 0o¢
0(ax,k) 1
OXT ]--S cos ct0 (pVoCD -_-(1CDv) --1-_ sin ao (pVoCL + c]CLv) + "-O--_J
{ ox ;0S[-cosao (CD,_ - CL) + sinao (CL_ + CD)] + --_-j
[ °X lqS(- cosao CD_ + sin ao CLz) + 0fl J
Oh gom
O(ax'k) -- 08Z
O(ax,k) _ 0Oy
0(_,k) _Se (0& 2V-_om -- COS C_0 CDa -_-sin CtoCLa)
O(a_,k) ¢%0¢) 2Vogom (- cos C_oCoo + sin _o CL/_)
[ °x l0(a_,k) 1 qS(- cos c_oCD_, + sin _o CL_, ) + --_/j08i gore
(D-291)
(D-292)
(D-293)
(D-294)
(D-295)
(D-296)
(D-297)
(D-29S)
(D-299)
(D-300)
(D-301)
(D-302)
D.3.2 Lateral kinematic acceleration derivatives.--
O(ay,k) glSbOp - 2Vogom CYp
0(ay,k) qScOq -- 2Vogom CYq
O(ay,k) _Sb--Cy r
Or 2Vogom
(O(ay,k) 1 SpVoCy + gtSCy v + -_]OV gore
O(ay,k) 1 qSCyo + --O-£-a]Oe_ gore
(0(ay,k) 1 OSCy ° + --_-]Off gore
O(ay,k) __ g cosOocoS¢O0¢ go
O( ay,k ) _ _ g__.sin 0o sin ¢oO0 go
0(ay,k) _ 0o_
(D-303)
(D-304)
(D-305)
(D-306)
(D-307)
(D-30S)
(D-309)
(D-310)
(D-311)
67
68
D.3.3
Oh gore
6_(ay'k) = 0Ox
0(ay,k)_=0
8y
O(au,k ) FISCCy_
O& 2Vogom
0(amk ) _Sb Cy_0j 2Yogom
O(ay,k.__.__._) 1 ( OYm "_06i - gore \_SC'_ + --_-i J
+ QSCyh)
Z-body axis kinematic acceleration derivatives.--
O(a:k)Op
O(az,k)
Oq
O(az,k)
OF
(_9(az,k)
OV
O(_,k) _Oa
O(a_,k)_O_
O(az,k) --
o¢0(._,k)
c90
O(az,k) _ 0
0"_,
_[ (1 )0(a:,k) __ 1 sin so SVo2CD _ + gtSCDh + cos C_oOh gore
O(a=,k)= oOx
0(_z,k! = oou
O(a_,k_:__=)_ - qS_ (sinsoCDa + cosao CLa)O& 2Vogom
0(az,k) qSb , .
o--f- = 2V-_omw_°c_, + _o_oC_)
[ oz ]O(az,k) _ 1 0S(sin ao CD,, + cos so Cb6, ) - --_-/j05i gore
_tSb , .2V_oqom (sin so CDp + cos a0 CLp)
qSe ..2V--a-_n-qnm(sin so CDq + COS O_0 CLq)
qSb , ."V--'_ {,sin so CD_ + COS ¢t 0 CLr )z _ ogom
[ °Z l1 S sin ao (pVoCD + CtCDv ) + S cos C_o(DVoCL + ClCLv ) -- -0V'-Jgore
c7S [sinc_o(CD_--CL)+COSao(CL_ +CD)]+ 1 9ZTgore gore Oa
OS- 1 OZT_- (sin so CDz + cos ao CL_) +
gore gore Off
fig- cos 0o sin ¢ogo
g---sin 0o cos ¢ogo
(D-312)
(D-313)
(D-314)
(D-315)
(D-316)
(D-317)
(D-318)
(D-319)
(D-320)
(D-321)
(D-322)
(D-323)
(D-324)
(D-325)
(D-326)
(D-327)
(D-328)
(D-329)
(D-330)
(D-331)
(D-332)
D.3.4 x body axis accelerometer output derivatives.--
ctSb (2Y-_om "- cos(toCDp + sin (to CLp)
qXa(-- COS aO CDq + sin ao Caq)
2 Vogo m
CTSb (- cos ao CD_ + sin (t0 CL_)2Vogom
1
gore1
gore1
go m
,[g0 m
Op(1SVo2CD Op qSCDh) +sin(to(1SVo2CL-_--h +(tSCLh)]cos (to _-_ +
(D-333)
(D 334)
(D-335)
(D-336)
(D-337)
(D-338)
(D-339)
(D-340)
(D-341)
(D-342)
(D-a4a)
(D-344)
(D-345)
(D-346)
(D-347)
D.3.5 y body axis accelerometer output derivatives.--
c)(ay) _ qSb C-
0(%) qSe COq 2V---_om Yq
8(%) qSbCy_
Or 2Vogom
(O(av) I SpVoCy q-- qSCy v -1- -_7-JOV gom
(D-348)
(D-349)
(D-350)
(D-351)
69
o9
8(a_) _ o84)
O(a_) _ 080
8(a_) _ o8_
a(ay)Oh
8(a_)
Oy
8(%)O_
8(a_)
O(a_)
05i
gore qSCy¢, + --_ I
1 (gO m qSCy# -i- ---_-]
go Wt(½SVo C¥°p-ffi + 4SCy_ )
-0
-0
4Se
2Vogo rn Cva
qSb2 Vogo 77-t Cy_
1 (()k_)gora qSCy_, + --ff-_[]
(D-352)
(D-353)
(D-354)
(D-355)
(D-356)
(D-357)
(D-358)
(D-359)
(D-360)
(D-361)
(D-362)
7
D.3.6
O(az)Op
o(a_)Oq
O(a_)Or
O(az)
OV
O(az)Oa
o(._)83
O(a_
0¢
O(a_O0
o(_)o_,
O(.z)Oh
z body axis accelerometer output derivatives.--
qSb ,.- 2V--_om ksm ao CDp + COS (_0 FLy )
4Se ..
-- 2V-_-om (sin ct 0 CDq -t- COS a' 0 Cbq )
qSb ,.
- 2V_.qom(SmaoCD_ + cos c_oCL_)
[ oz ]- goml S sin ao (/)VoCD + (tCDv) -1- ,_c cos a, 0 (DVoCL + qCLv ) - _]
i{qS[sin ao (CD_ - CL) + cos C_o(CL_ + CD)] - --0ZT}gore Oa
[ ozT]- gore (_S(sin a'o CD, + cos a_o CL,) - --_-j
--0
--0
--0
go,nl[sina°(_ SIffCDOp_ + (]SCDh) + cos ao (lkZSI,_2C L Op + (]SCL,,) ]
(D-363)
(D-364)
(D-365)
(D-366)
(D-367)
(D-368)
(D-369)
(D-370)
(D-371)
(D-372)
70
O(a_)= 0Oz
O(a_)_ oOy
O(az)_ _s_ (sin ao CDa -t- cos c_0 CLa )0& 2Vogom
O(az) qSb (sin _0 CDb -t- cos 0_0 CL#)
Off 2Vogom
[ °zTtO(az) 1 0S(sin ao CD6, + cos ao CL6, ) -- --_-/]06i gore
(D-373)
(D-374)
(D-375)
(D-376)
(D-377)
D.3.7 Normal accelerometer output derivatives.--
O(an)
Op
a(_,,)oq
0(a,,)OP
O(a.)OV
O(a.)Oa
a(a,,)0t3 gore
O(an) _ 0o¢
O(an) _ 080
O(a,,) _ 0o_
O(a,) 1Oh
a(_)Oz
o(_,,)8y
a(_,,)0&
0(an)
05i
_Sb(sin so CDp + cos ao CLp)
2 Vogo m
C_S_(sin ao CDq + cosao CLq)
2Vogom
_Sb , .2V-_anm[Sm ao Cmr + cos So CLr)
1
gore1
gore1
go m
-0
-0
OZT]sin So ( S pVoCD + qSCDv ) + cos ao ( S pV'oCL -t- (TSCLv) -- --_-]
{ oz ;OS[sin ao (CDo - CL) + cos ao (CL_ + CD)] _ ]
OZT ](]S(sin + cos ao CL¢) -o_0 CD_
Op
[sinao(_SVo_CD Op_+ {SCD0 + cos c_o(2SVo2CL-_+e]SCLh)]
gTS_ . .-_ _smao CDa + cosao CL_)
2Vogo m "
OSb (sin Ct0 CDb -_ cos o_0 CLB)2 Vogo m
[ °Z l1 qS(sin ao CD6, + cos ao CL_i ) - --_-/]gore
(D-378)
(D 379)
(D 380)
(D-381)
(D-382)
(D-383)
(D-aS4)
(D-385)
(D-386)
(D-387)
(D-388)
(D-389)
(D 39(1
(D-391
(D-392)
71
D.3.8 Derivatives of x body axis accelerometer output not at the vehicle center
of gravity.--
O(a_,i)Op
O(a_,dOq
O(ax,i)OT
O(ax,i)
qSb , 1
2V-_0m [- cos ao CDv + sin ao CLp) + --(qoY_ + rozx)go
qS_ (_ 12V0gom" cos ao CDq + sin _0 C G) + (PoYx - 2qoz_.)
qSb , }o2V-_om [- cosa0 CDr + sinao CL_) + --(poz_ -- 2roxx)
1
gore
1
gore1
OV
O(a_,i)
Oa
O(ax,i)Off gore
a(a_,i) = 00¢
o(a_,_) _ oO0
O(a.,i)
O(ax,i)
-0
Oh gore
O(a_,i) _ 0Ox
O(a_,d _ oOy
O(a_,O _ oOi,
O(a_,i) z_00 go
0(_,_) y_m
0 i_ go
cg(az,i) qS_
0& 2Vogom
O(ax,i) (tSb
03 - 2Vogo-_
O(ax,i) _ 1 [_qS(_06i gore
[--SCOS G'0 (pVoCD + ([CDv) + Ssin ao (DVoCL + qCLv ) + (_l, j
{ ox,r CS'[- coseo(CD,, - CL) + sinao (CL,, + CD)] + --9--_--_J
[ °XTl_S(- cosao CD_ + sin_o OLd) + -0-f-J
"_ "4- q S C D h /i "_- C'_O _l "JrCOS _0
(- cos o0 CDa + sin ao CLa)
(-- cos ao CDb + sin ao CL_)
COS a 0 CD6 i + sin ao CL h) + (_6i ]
(D-393) i
(D 394) :
(D-395) _
(D-396) _
(D-397
(D-398
(D-399)
(D-_00) L
(D-401) _-
(D-402) !
(D-403)
(D-404) -
(D-405)
(D--I06)
(D-407)
(D-40S)
(D-409)
(D-410)
D.3.9 Derivatives of y body axis accelerometer output not at vehicle center
of gravity.--
O(a_,i)Op
qSb C - l(2poyy- 2_ YP go
i
q0 ,Ty ) (D-411)
72
L
O(a_i)Oq
O(ay,i)
Or
O(a_,i)OV
O(ay,i)
ozo(_,i)
o¢o(_,i)
O0
O(_,_)0¢
O(au,i)
2VogomC7S----_- l (pox_ + roZy)- O'vq +
_.-(ISb C _oo(qoZy 2roy,)- 2Vn-7_m %+
1 SpVoCy + qSCy. + OV ]gore
( or )1 qSCYo + --5-g-_/go m
_ ( aye)1 qSCy_ + --5-f]gore
-0
-0
-0
Oh gore
O(a_,_)_ 0Ox
O(ay,i) _ 0Oy
O(au,i ) Zy
(_SV2C Y Op-_ + ctSCy,,)
Oi_ go
O(ay,i) _ 00_
O(ay,i) xy
(9i" go
O(ay,i) gtSc- cy_
O& 2Vogom
O(au,i ) qSb
OtJ - 2Vogom C'%
O(a_,J_ 1 (C_scv_,05i gore + 05i )
(D-4_2)
(D-413)
(D-41,_)
(D-4_5)
(D-416)
(D-417)
(D-418)
(D-419)
(D-420)
(D-42J)
(D-422)
(D-423)
(D-424)
(D-425)
(D-426)
(D-427)
(D-428)
D.3.10 Derivatives of z body axis accelerometer output not at vehicle center
of gravity.--
O(_z,i)Op
O(a_,i)
Oq
O(az,i)
Or
gSb 1
(sin O' 0 CDp + COS (2 0 CLp) -- o--_(2poz_ - roXz)2Vogo m
2v-_-_-o._(_in_,ocD. + _o_ooeL.) - (2qo_- ,'ou_.)rlSb
CD. + cosao CL,) + l(pox_ + qoYz)(_ino,o2 Vogo m 9o
(D-429)
(D-430)
(D-_31)
73
O(a_,d 1OV gore
O(_z,d _ iO_ gore
O(a_,i) 1
Off gore
0(_,_) _ o0¢
O(az,i____2= oO0
O(a_,d _ o0¢
O(az,i) _ 1
Oh gore
O(az,d _ oOz
O(a_,,) _ oOy
O(az,i) _ Y__2
O# go
O(az,i) xz
oo0(a_,i)
Oi.
O(az,i)
O&
O(az,d
O(az,i)
go
--0
[ oz ]S sin so (pVoCD + qCDv ) + S cos Oeo (pVoCL + _CLv ) - --o-V-]
,iS[sin So (CDo -- CL) + cos C_o(CL_ + CD)] -- 0----_--
oz ][qS(sinso CD_ + cos so CL¢) - --_-j
1 2
_Se , .
2V-_qamtSm So CD_ + cos Uo CL_)
qSb (sinso CD_ + cosaoC%)2 Vogo m
[ oz ]1 qS(sin c_o CDe, + cos so CL6i ) - --_-/jgore
(D-432)
(D-433)
(D-434) "
(D-435)
|
(D-436) !!
(D-437) _
(D-43S) _
(D-439)=
=
(D-440) "
x
(D-441) ..
(D-442) _
(D-443) -
(D-444) -
(D-445)
(D-446)
D.3.11 Derivatives of normal accelerometer output not at vehicle center
of gravity.--
0(a.,i)
Op
O(an,i)
Oq
O(a_,d_r
O(an,i)
OV
Oa
gtSb , .2V-_gom (sin c_o CDp + cosso CLp)+ gol(2p°Zz- r0zz)
= 2V--_0m (sin so CDq + cos So CLq) + (2qoZz - royz)
ctSb . _o- 2V-_om(SlnaoCD_ + cosso eLf) -- (poXz +qoYz)
[ oz ]1 S sin so (pVoCD + (lCDv) -1- S cos So (pVOCL _- qCLv ) - -_-jgore
{1 4S[sin c_o (ODe, - CL) + cos So (CL¢, + CD)] -- --gore Oa
(D-447)
(D-448)
(D-449)
(D-450)
(D-451
O(a_,i)0_
O(an,_)= 00¢
O(an,_)= 080
8(an,_) _ 00¢
O(a_,i) 1
Oh gore
O(a_,_)_ 0Ox
O(a_,i) _ 0Oy
O(_,_,i)ab
O(a_,_)0_
a(a_,i)
O&
O(an,i)
o_O(a_,i)
05i
1 [_S(sinc_o CD_gore
oz_]+ cos_o CL_) -- --5-SJ
[sin _o (_SVo2CD Op-_ + qSCDh) -It- COS a'0
YZ
go
Xz
go
__-0
{S_ (sin O_0 CDa + COSO_0 CL_)2Vogom
(tSb (sin O_0 CD# + COS O_0 CLa)2 Vogo m
[ °Z l1 0S(sin so CD_, + cos So CL_I ) - --_iJg0m
S_,02C,L0p
(D-452)
(D-453)
(D-454)
(D-455)
(D-456)
(D-457)
(D-458)
(D-45O)
(D-460)
(D-461)
(D-462)
(D-463)
(D-464)
D.3.12 Load factor derivatives.--
o(_)Op
o(_)Oq
o(_)Or
o(_)OV
cg(n)
Oa
O(n)
0_
o(_) =0¢
O(n) -_00
(tSb C2V--_m Lp
qS_
2Vog m CLq
rlSbCL,
2Vogm
- !(SpVoCL + _scL.)gm
_lS CL_gm
(IS CL zgm
(D-465)
(D-466)
(D-467)
(D-468)
(D-469)
(D-470)
(D-471)
(D-472)
75
o(_) _ o0¢
a(n) aOh gm
a(_) _ oOx
0(_) _ oOy
o(_) qse cO& - 2V---_ L_
o(n) _SbO# - 2VogmCb_
O(n) qS-- CL_,
06i gm
1 2
(_sv6 CL Op-_ + qSCLh )
(D-,t73)
(D-474)
(D-475)
(D-476)
(D-477)
(D-478)
(D-479)
76
D.3.13 Speed of sound derivatives.--
o(_) _ oOp
O(a) _ oOq
O(a) _ oOr
o(a) = oOV
O(a) = o0o_
O(a) _ 0Off
O(_) _ o"0¢
O(a) _ oO0
o(_) = o
o(,,)Oh
o(_)Ox
O(a)OU
a(,)06
O(a)oA
0.7po 0T
poTo(1.4po/poTo)_/z Oh
-0
-0
-0
-0
(D-480)
(D-481)
(D-482)
(D-483)
(D-484)
(D-485)
(D-486)
(D-487)
(D-488)
(D-489)
(D-490)
(D-491)
(D-492)
(D-493)
(D-494)
D.3.14 Mach number derivatives.--
O(M) _ 0op
O(M) _ oOq
O(M) _ oOr
O(M) iOV a
O(M) _ 0Oa
O(M) _ o09
O(M) _ o0¢
O(M) _ ooe
O(M) _ o0_
O(M)_ Vo[ _O.Tpo ] 0TOh a 2 LpoTo(1.4po/poTo)l/2J 8h
O(M) _ oOx
O(M) _ oOy
o(M) _ oO&
O(M) = oo3
O(M) _ o06_ .
(D-495)
(D-496)
(D-497)
(D-498)
(D-499)
(D-500)
(D-501)
(D-502)
(D-503)
(D-504)
(D-505)
(D-506)
(D-507)
(D-508)
(D-509)
D.3.15 Reynolds number derivatives.--
O(Re) _ oOp
a(R_) _ oOq
8(Ro) _ oOr
(D-510)
(D-511)
(D-512)
77
O(Re) peOV It
O(r_e)_ oOa
O(R_) -0ap
0(Re) _ o0¢
0(Re) _ 000
a(_) _ o0¢
a(R_) Voe Op pVoe 0#
Oh p
O(Re) _ 0Ox
O(Re) _ oOy
0(Re) _ 00a
0(Re) _ 0o3
0(Re) _ o06_
Oh #2 Oh
(D-513)
(D-514)
(D-515)
(D-516)
(D-517)
(D-518)
(D-519)
(D-520)
(D-521)
(D-522)
(D-523)
(D-524)
D.3.16 Reynolds number per unit length derivatives.--
a(Re') _ oOp
0(Re') _ 0Oq
0(Re') _ oOr
(9(Re') p
OV t t
a(Re') _ o0_
0(Re') _ 0aZ
O(Re') _ 0a¢
O(R_') _ o00
O(R_') _ oo¢
(D-525)
(D-526)
(D-527)
(D-528)
(D-529)
(D-530)
(D-531)
(D-532)
(D-533)
78
O(Re)Oh
a(Re')Ox
a(Re')Oy
0(Re')
O&
o(Re)o3
0(Re')
05_
Vo Op pVo Op
# Oh #2 Oh
-0
-0
-0
-0
-0
(D-534)
(D-535)
(D-536)
(D-537)
(D-538)
(D-539)
D.3.17 Dynamic pressure derivatives.--
a(o) _ o0
a(q) _ oOq
0( _) _ oOr
°(_---2)= proov0(_--2)= oOa
a(_-A)= oo3
o(_) _ o0¢
a(_) _ o00
°(_-A)= oo¢
Oh 2 Oh
o(o) = oOx
°(_-A)= oOy
°(°--2)= oo_a(o) = oo_
o(_) _ o
(D-540)
(D-541)
(D-542)
(D-543)
(D-544)
(D-545)
(D-546)
(D-547)
(D-548)
(D-549)
(D-550)
(D-551)
(D-552)
(D-553)
(D-SS4)
79
D.3.18 hnpact pressure derivatives.--
O(qc)_ oOp
,9(qc)_ oOq
O(qc) _ oOl"
o(q_)OV
[ 1_'_-[_/2_'_f_l(1.0 + 0.2M_) 2"5
( 5.76M2 _.5P--_a2._Laf\5.6.-_---5.8]
+ 3.0M2 ( S.TBM__\,_.6M_- 0.8J
0¢
O(q¢)_ o80
O(qc) _ o0¢,
O(q¢)_ oOa
O(q_)_ o0_
O(q_)_ o
0(%)Oh
O(qc) _ oOx
O(qc)-0
Oy
O(qc) _ 00_
O(%) _ 0o_
O(qc)-0
(_f < 1.0)
,.5 9.216M ] (M > 1.0)(5.6_-_.s):j -
[(1.0 + 0.2M2) 3"s - 1.0] _0_ I_M(1. 0 + 0.2;1/2)2.s7,,_ Oa
s.6M 2-o.8j -
a 2 5.6--M-_= _).8 /
+ 3.0M 2 ( 5.76M2 ,_l.S 9.216M ] Octk5.6M:- 0.8/ (5.6_--_.8)2J 85,
(:a_r_<1.o)
(_f >_1.o)
(D-555)
(D-556)
(D-557)
R
(D-558)
(D-559) "
(D-560) _
(D-561) :
(D-562) _
(D-563) -
(D-564)
(D-56S)
(D-566)
(D-567)
(D-568)
(D-569)
80 =
D.3.19 Mach meter calibration ratio derivatives.--
O(qJp_) _ oOp
o(qc/p_) _ o0q
O(qJp_) _ oOr
OV
o(q_/p_) _ oOa
o(%/p_) _ o03
O(q_/p_)_ oo¢
O(qc/p_) _ 0O0
O(qc/p_,) _ 0o¢
L_-M(1.O + 0.2M2) 2'5
1 [2.4M (" 5"76M2 ,_2.5
[ 5.76__I2 -_1.5+ 30M_ \5.6-_-5.8)
Oa- x'--_-_l(1.O + 0.2M2) 2"5 N7
9.216M ](5.6,,_-- O.S)2 J
(_t _< 1.0)
(M _>2.0)
(M < 1.o)
O(qc/p_)
Oh_ a____)[2.4 M ( 5.76MZ 2.5\ 5.6-_- b.8)
O(q_/p_) _ oOx
O(q_/p_) _ oOy
O(qjp_) _ oOd
O(qc/pa) _ 0
O(qc/pa) _ 005i
(, 5.76312 ,_1.5+ 3.oM: \5.UM-_-i_.s/
9.216M ] Oa(3I >__1.0)
(D-570)
(D-571)
(D-572)
(D-573)
(D-574)
(n-575)
(D-576)
(D-577)
(D-578)
(D-579)
(D-SS0)
(D-581)
(D-582)
(D-583)
(D-584)
81
D.3.20 Total temperature derivatives.--
O(Tt) _ oOp
O(T_) _ oOq
0(Tt) _ 00r
0(Tt) 0.4 T3f
OV a
O(T,) _ 0
O(T,) _ o0_
0(Tt)0h - {1.0
O(Tt) _ 0Oz
0(Tt) _ 0Oy
0(Tt) _ 00&
0(Tt) _ o
0(Tt) -_ 0
0&
+ 0.2M 2 0.4 TMVo [ 0.7p ] 10Taz LpoTo(1.4po/poTo)a/2J J Oh
(D-SSS)
(D-586) "
(D 5S7) -
(D-588) :
(D-589) :
(D-590) "
(D-591)
(D-592)L
(D-593)
(D-594)
(D-595)
(D-596)
D.3.21 Flightpath angle derivatives.--
a(_) _ oOp
o(_) _ oOq
a(_) _ oOr
a(_) hoOV Vo(_% 2 -- 62)1/2
o(_) _ oOa
a(_--A): o
o(-y) _ oo¢
o(-y) _ o00
(D-597)
(D-598)
(D-599)
(D 600)
(D-601)
(D-602)
(D-603)
(D-604)
82
o(_) _ oo_
°(v--A): ox_
a(_---2: oOx
o(.y) _ oOy
a(_) _ oo_
o(_) _ oa_
o--7-= (Vo_ - _)ln
a(_,) = o05_
(D-605)
(m-6o6)
(D-607)
(D-60S)
(D-609)
(O-6_0)
(D-B11)
(D-6J2)
D.3.22 Fllghtpath acceleration derivatives.--
_9(fpa) _ 08p
a(fp_) _ oOq
O(fpa) = 0cqr
a(fpa) _ oOV
O(fpa) _ 0Pa
O(fp_) _ 0o_
_(fpa) _ 0o¢
(9(fpa) _ 0O0
(9(fpa) _ 08#,
c_9(fpa) _ 0Oh
O(fpa) _ 0Ox
0(fpzL) _ 0Oy
o(rp_) 107 #
(D-613)
(D-614)
(D-615)
(D-616)
(D-617)
(D-61S)
(D-619)
(D-620)
(D-621)
(D-622)
(D-623)
(D-624)
(D-625)
83
a(fp_) _ o (I)-626)0&
0(fpa) _ 0 (D-627)03
O(fp_) - 0 (D-628)
D.3.23 Vertical acceleration derivatives.--
Op 2I'_go m
Oq 21/_go m
o(il) ¢Sbw
Or 2Vogom
o(il) 10 V go m
O(h,) 1
Oa gore
O(h) 1
0 /3 gora
--[sin Oo(- cos a'o CDp + sin ao CLp)
-- sin ¢o cos 0o C¥ v + cos ¢o cos 0o (sin c_oCDp + cos (_o Cbp)] (D-629)
--[ sin Oo (- cos ao CDq + sin ao CLq) - sin ¢o cos OoCyq
+ cos ¢o cos 0o (sin ao CDq + cos oz'0 CLq)] D-630)
[ sin Oo (- cos ao CDr + sin ao CL_) -- sin ¢o cos OoCy_
+ cos¢o cos0o (sin ao CD_ + cos ao CL,.)] D-63J)
- -- sin 0o -Scos a'o (pVoCD + _tCDu) + Ssin ao (pVoCL + (tCbv) + -'_-< j
( aye)- sin 4o cos 0o SpVoCv + (tSCy v + --_j
+ cos ¢o cos 0o IS sin ao (pVoCD+ qODv )
OZT]} D-632)+ S cos aO (pVoCb + FtCLv ) -- -0-_7-7j
{ [- -- sin0o -_Scosao (CD_, - CL) + 4Ssinao (Cb_ + Co) + --_ j
- sin ¢o cos 0o (_ISCy_ + Oa ]
[ OZT]} (D-633)+ cos ¢o cos 0o 0S sin C_o(CD_, - CL) + _S cos ao (CLo + Cm) - -_a,J
+ cos ¢o cos 0o 4S sin ao CDz + g/S cos ao CLz +
o(?i)0¢ : -av° cos ¢o cos 0o + a_] sin ¢o cos 0o (D-635)
o(il)0--'_ = a*° cos 0o + avo sin 0o sin ¢o + a_o cos ¢o sin 0o (D-636)
O(i;____)= 0 (D-637)o_
=
=
84
O(h) 1
Oh gom
__°()i)= o0x
a(Y____)= 0Oy
o(f_)0&
o(h)
a(_)05i
{ sin 0o [-- cos O_o(_SVo2C D _cgP+ qSCDh) q- sin Cto (1SVo2CLOp. _ -t- qSCLh)I
o, )- sin ¢o cos 8o SVo2Cy -_ -I-qSCyh
cos _-£ + qSCD,,)
qs_ [2Vogom sin 6o (- cos ao CDa + sin ao CL_) -- sin ¢o cos 0o Cy_
+ cos ¢o cos 0o (sin C_oCDa + cos ao Cha)]
qSb2Vogo----_[sin 0o (- cos ao CDb + sin ao CLt_) -- sin ¢o cos 0o Cy_
+ cos ¢o cos 0o (sin ao CD/_ + cos O'o CLb)]
(D-638)
(D-639)
(D-640)
(D-641)
(D-6,12)
[ <,1_ (go--_ sin 0o qS(- cos ao CDa, q- sin ao Cba, ) + --_-/] sin _0 COS 0 0 (7oCCyG -]- -_i//
CgZT l+ cos¢o cos0o [OS(sinaoCDel + cosao Cn,i) - --_/j } (D-643)
D.3.24 Specific energy derivatives.---
O(E_) _ 0 (D-644)Op
0(Es) _ 0 (D-645)Oq
O(E_) _ 0 (D-646)Or
O(Es) Vo- __ (D-647)OV g
O(E_) _ 0 (D-648)Oct
O(Es) _ 0 (D-649)
O(E_) _ 0 (D-650)o¢
O(Es) _ 0 (D-651)00
O(E_) _ o (D-652)0_
0(Zs) = 1 (D-653)Oh
85
O(E_) _ 0Oz
O(Es) _ oou
O(Es) _ o
O(E_)_ oo3
O(Es) _ o
(D-654)
(D-655) i
z
(D 656) i
(D-657)
(D 65S)_
D.3.25 Specific power derivatives.--
o(P_) _ oOp
O(Ps) _ 0Oq
O(Ps) _ 0Or
o(Ps) 9OV g
O(P_) _ 0Oc_
o( Ps) _ o8_
o(P_) _ o0¢
o(P,) _ o00
o(P_) _ oo_
O(Ps) _ 0Oh
o(Ps) _ oOz
cO(P_) _ 0Oy
o(Ps) vOv - g
O(p_) _ oo4
o(ps) _ ooh
O(P_) 1Oi_ -
(D-659) :
(D 660)
(D-661) "
(D-662)
(D-663)
(D-664)
(D-OGS)
(D-666)
(D-667) -
(D-668)
(D-669)
(D-670)
(D-671)
(D-672)
(D-673)
(D-67_1)
86
(D-675)
D.3.26 Normal force derivatives.--
0(N)Op
O(N)
cgq
O(N) _Or
O(N)
OV
O(N)
c_ga
O(W)
oZO(N)
o¢o(N)
O0
O(N)
o¢o(g)
Oh
O(N)Ox
O(N)
Oy
O(N)w
O&
O(N)
O(N)05i
qSb,_-V-_otcos ao CLp + sin ao CDp)
c]S_",_o (cos ao CLq + sin ao CDq)
_vob(cos CL_ + sin CD,.)O_o o_0
S[cos ao (pYoCL + (tCLv) + sin ao (pVoCD + ,tCDv)]
_S(cos ao CLo + sin ao CD,_ - sin ao CL + cos ao CD)
qS(CO80_ 0 CLz "[- sin ao CD_)
0
0
0
0
0
_S_.
_oo (cos a o CL_ + sin C_oCDa
_tSb,_,cos ao CL_ -I- sin ao CDp)
(IS(COS O:0 CL6, "_- sin ao CD6, )
(D-676)
(D-677)
(D-678)
(D-679)
(D-680)
(D-6Sl)
(D-682)
(D-683)
(D-684)
(D-685)
(D-686)
(D-687)
(D-688)
(D-689)
(D-690)
D.3.27 Axial force derivatives.--
O(A)
Op
O(A)
Oq
O(A)
_r
_Sb(- sin ao Cbp + cos ao CDp)
2Vo
_S_._o (- sin_o c_ + _os,_oCD_)
qSb(- sin a0 CLr + cos ao CD_)
2Vo
(D-691)
(D-692)
(D-693)
87
O(A)OV
O(A)0a " = _S(- sin a o CLe, "{-¢O80_0 CDc, --
O(A)OZ
0(A)a¢
o(A)ao
a(A)O_O
O(d)
Oh
O(A)
Ox
O(A)8y
a(A)oa
a(A)
a_O(A)
-- ,5'[--sinG' 0 (RVoCL + (1eLy) + cos a'o (pVoCD + (1CDv)]
-- eTS(- sin _o CL_ + cos o_ 0 CD_ )
-0
-0
-0
-S[-
-0
-0
sin ao (_Vo2CL Op
cos ao CL -- sin ao CD)
) op )]+ (tC'Ln -{- COS aO Vo2CD _ + (ICl)h
- 27o" sin ao CLa + cos C_oCDa)
CtSb
- 2Vo (- sin C_oCL e + cos C_oCDo)
- tiE(- sin ao CLa, + cos ao CDai )
(D-694)
(D-695) "
(D-696)
(D-697)
(D-698)
(D-699)
(D-700)
(D-r01)
(D-702)
(D-703)
(D-704)
(D-705)
D.3.28 x body axis rate derivatives.--
o(_)Op
o(_)8q
o(_)Or
o(_)OV
o(_)Oa
o(_)0/3
o(_)o¢
o(u)O0
o(_)o¢
-0
=0
-0
-- COS Og0 COS t_0
- Yosin ao cosflo
- -Vo cos ao sin/30
-0
-0
-0
(D-706)
(D 707)
(D 70S)
(D-709)
(D-710)
(D-711)
(D-712)
(D-713)
(D-714)
88
o(_) _ o8h
O(u) _ oOx
o(_) _ oOy
O(u) _ o05
a(_) _ o09
o(u) _ oO&
(D-715)
(D-716)
(D-717)
(D-718)
(D-719)
(D-720)
D.3.29 y body axis rate derivatives.--
0¢)8p
a(v)Oq
a(v_2)=Or
a(.___2)=OV
a(v_2)=8a
a(vA)=aZ
O@) o0¢
0(_) _ o00
a(_--2)= o8¢
a(_--2)= oOh
a(v--2)= o8x
a(_) _ oOy
a(v) _ oO&
a(,_)_ oa_
O(v) _ o05_
-0
-0
0
sin/3o
0
Vocos/_o
(D-721)
(D-722)
(D-723)
(D-724)
(D-725)
(D-726)
(D-727)
(D-728)
(D-729)
(D-730)
(D-731)
(D-732)
(D-733)
(D-734)
(D-735)
89
D.3.30 z body axis rate derivatives.--
o(_) _ o3p
o(w) _ oOq
a(w) _ oOr
8(w) _ sin O_ocos floOV
O(w) _ Vocos_o cos_o0a
O( w) _ - Vo sin ao sin/3o09
o(_) _ oo6
o(_) _ o8O
o(w) _ oo¢
O(w) _ oOh
o(_) _ oOx
O(w) _ 0Oy
o(_) _ oOd
o(_) _ o
O(w) _ o05i
(D-736)
(D-737)
(D-738) :
(D-739)
(D-740) -:_
(D-741)_
(D-742)
(D-743) -
(D-74,1) --
(D-745)
(D-746)
(D-747)
(D-748)
(D-749)
(D-750)
D.3.31 x body axis acceleration derivatives.--
off,) 4SbOp 2I,bm
o(_) qScOq 2Vom
o(_) qSb0r 2Vom
Off,)
OV
(-- COS C_0 CDp + sin a o CLp)
(- cos Cao CDq + sin ao CLq) - Iib sin C_ocos flo
(- cos ao CDr + sin So CL_) -t- Sb sin flu
{ os ;1 S[- cosao (pVoCD -t- (]CDv) + sinao (pI,SCL +/jCLv)] + -_j
+ro sin 3o -- qo sin so cosflo
(D-751)
(D-752)
(D-753)
(D-754)=
9O=
OXT]__ =lm q°c(- cos ao CD_ + sin C_oCL_ + sin ao CD + cos C_oCL) + --_aJ
- qoVo cos So cos/30
[ ox ]1 c]S(- cos ao Cm a + sin so CLz) + --_jm
+ roVo cos/30 + qoVo cos a'o sin/3o
o(_) _ oo¢
o(_)c90 - -g cos Oo
0(_) _ 0o¢
-- -- COS Oz0 I/':CD _ + qODn + sin a'o Vo2CL _ + qCLhOh m
O(_) _ 0Ox
o0,) _ oOu
o(_) Os_Od
0(_)
2Vom (- cos ao CDa + sin ao CLs )
4Sb
- 2Vom (- cos ao CD b + sin so CLb)
[ oxg1 qS(-- COS Ct 0 CD6 i -b sin so CL_, ) + --_]732
o(_)05_
(D-755)
(D-756)
(D-757)
(D-r_S)
(D-759)
(D-760)
(D-761)
(D-762)
(D-763)
(D-764)
(D-765)
D.3.32 y body axis acceleration derivatives.--
o(_)Op
o(_)cgq
o(_)Or
o(_)ovo(_)cga
o(_)o/3
o(_)o¢o(_)oo
o(_)o¢
qSb21-VoomwYP + Vo sin Ceocos/3o
qSe _
2-Vo m Cyq
ClSb C- _ v_ - Voco__o _o_/so
[ °Y*I_ lm S(pVoCy -b #tOy v ) + --_1 + po sin ao cos _qo- ro cos a'o cos/30
( os5-- m FISCy_ + ace ] + poVo cos C_ocos/3o + roVo sin ao cos flo
1 ( cgYT_-- m #ISCye + 0/3 ,I - pol'{3 sin ao sin flo - roVo cos ao sin flo
- g cos Oocos ¢o
- gsin0osin¢o
-0
(D-766)
(D-767)
(D-768)
(D-769)
(D-770)
(D-7n)
(D-772)
(D-773)
(D-774)
" 91
o(_)Oh
0(_)
Ox
o(_)09
o(_)O&
o@)
a(_)86,
- mS _,;2Cy _ + qty.
-0
-0
qS_ C2T7-om Y+
qSb C2I_om Y_
OYT)1 ,iSCy_, + _-/]m
(D-775)
(D-776)
(D-777)
(D-778)
(D-779)
(D-780)
D.3.33 z body axis acceleration derivatives.--
o(_) qSb (_0p -- 21/_ore, sin a' 0 CDp -- cos o_0 CLp) - IZo sin/30
- (- sin a'o CDq -- cos o: 0 CLq) Jr Vo cos o' 0 cos floOq 21,'_m
O(,b) _Sb- (- sin ao CD_ - cos ao CLr )
Or 2Vom
[ °z"lO(&)Ol.... - m,1 -Ssinao (pI'_)CD + qCDv) -- ScOSCto (pI'SCL + OCLv) + 017 j
+ qo cos aO cos/3o -- Po sin _qo
[ °z'"lO(_b)Oct-- ml (IS(- sinao CD_ - cOSao CL,, - cosa'o CD + sino_o CL) -[- Oct j
-- qoVo sin ao cos/3o
_ _[ _ oz ]O(tb)__Ofl ml 0S(-sino'oCDz COsa0CLz)+ OflJ-q°I';c°s°'°sin/J°-P°lT°c°sfl°
o(,c_,)- g cos0osin¢o
0¢o(_)
00 - -gsinO°c°s¢°
0('_,) _ 00_,
_ [ (1 )O( £)Oh mS _ sin ao Vo2CD "_ + qCDh --
0( _,) _ oO:L"
o(,_,) _ oo,j
o(.,_) 0=s_(_0& - 21ZVom sin ct 0 CDa - cos a 0 Cba)
o(.(Q OSb ,0_ - 2t-Vom( sin ao CD/_ -- cosa'0 CL/_)
co_,_o( _ _,_2cL°p, _+ OcL.)]
(D-781)
(D-782)
(D-783)
(D-784)
(D-rSS)
(D-786)
(D-787)
(D-rSS)
(D-789)
(D-790)
(D-790
(D-792)
(D-rga)
(D-79:i)
92
0(_) _ 1 [_S(- sin CD,,,(2 0O,Si m [
1- cos% CL,,) + -0-_/1 (D-795)
D .3.34 Angle-of-attack sensor output derivatives.
OY - HI
O(a,i) _ 1Oa
0(_,0 _ ooz
O(_,i) _ oo¢
0(%) _ oO0
O(c_,O_ oO_
0(_,0 _ oOh
O(_,_) _ oOz
O(a,d _ oO_
0(<0 _ oO&
O(_,d _ oo)
O(_,d _ o06_
(D-796)
(D-797)
(D-798)
(D-799)
(D-S00)
(D-801)
(D-802)
(D-803)
(D-804)
(D S0_)
(D-806)
(D-SOt)
(D-SOS)
(D-809)
(D-Sl0)
D.3.35 Angle-of-sidesllp sensor output derivatives.--
o(Z,d zzOp Vo
o(a,j _ oOq
0(_,0 _pOr Vo
(D-Sll)
(D-S12)
(D-Sl3)
93
0(&) _OV
o(&) _ oOa
o(_,_)_ i09
o(&) _ o
o(&) _ o
o(&) _ oo,#
o(&) _ ocgh
O(fl,J _ oOx
o(&) _ oOy
o(&) _ o0,5
o(&) = ooh
o(&) _ o
roxz - pozp (D-814)
(D-815)
(D-816)
D-817)
D-818)
D-819)
D-S20)
D-821)
D-822)
D-823)
D-824)
D-825)
D.3.36 Altimeter output derivatives.--
o(h_) _ o01-,
8(h,;) _ o8q
O(h i) = ocgr
O(h,_)_ oOV
O(h,d _ oOa,
O(h,d = oo_
0(h,i)0¢
o(#,,_)00
O(#,.,_)
- --Yh cos ¢o cos Oo + zh sin 540cos Oo
-- Xh COSO0+ Yh sin ¢o sin Oo + zh cos ¢o Sill 0 0
-0
(D-S26)
(D-S27)
(D-828)
(D-S20)
(D-830)
(D-831)
(D-832)
(D-833)
(D-83,t)
9l
o(h_) _ tDh
O(h,d _ oOx
a(h,_) _ oOy
a(h,_) _ oO&
O(h,d = oo)
o(h,d _ 0
(D-835)
(D-836)
(D-837)
(D-S3S)
(D-839)
(D-840)
D.3.37 Altitude rate sensor output derivatives.--
o(i,;) _ oop
o(id _ oOq
O(]_,d _ oOr
o(i_) _ oOV
o(i,,,d _ o
o(h,d03
o(i,,do¢
o(l,.d00
O(i_,i)0¢
o(i_,d _ oOh
o(i_,_)_ oOx
o(i_,d _ ooy
o(i_,d _ o8&
o(h,d _ o03
-0
- $(Yh sin q5ocos 0o + z h cos 45ocos 0o) + O(YL cos ¢o sin 0o - z h sin q5osin 0o)
- O(x h sin 0o - Y'hsin ¢o cos 0o -- z h cos ¢o cos 0o)
+ _P(Yhcos ¢o sin 0o - zh sin ¢o sin 0o)
-0
(D-841)
(D-842)
(D-843)
(D-844)
(D-S45)
(D-846)
(D-847)
(D-848)
(D-849)
(D-SS0)
(O-SS_)
(D-S52)
(D-853)
(O S.S,l)
95
C_
a(i,,O0¢ - -Yj, cos 6o cos Oo+ zh sin ¢o cos Oo
o(i_,_)00 = x_ cos 0o + Yh sin ¢o sin 0o + z h cos ¢o sin 0o
a(i,,j IaJ_ -
a(/,,J = o
(D-855)
(D-856)I
(D-857) !
(D-SSS):,
D.3.38 Total angular momentum derivatives.--
O(T)- I_.po - Ixyqo - L:zro
Op
O(T)Oq - [_qo -- [_po - [y_ro
O(T)Or -- Ij'o - l_po - Iv=qo
O(T) = oov
O(T) _ 0
O(T) = 0a3
a(T) _ o06
O(T) _ 000
O(T) _ oa¢
0(T) _ 0Oh
a(T)-0
Ox
a(T) _ 0Oy
O(T) _ ooa
a(T) = oo_
O(T) _ o
(D-859)::
(DSG0):
(D-SG1):
(D-862)_
(D-863)_
(D-864)"
(D-SGs)!
(D-866)_
(D-S67)I
(D-868)
(D-869)
(D-870)
(D-871)
(D-872)
(D-873) _
96
D .3.39 Stability axis roll rate derivatives.---
O(ps)Op
o(ps)aq
o(ps)Or
O(p_)ov
o(;_)8a
o(ps)8_
o(p_)0¢
a(p_)oo
a(p_)0¢,
o(p_)oh
o(p_)8x
a(;_)Oy
O(ps)oa
a(p_)
o(w)
- sin ao
-0
-0
-0
-0
-0
Po sin ao + ro cos ao
(D-874)
(D-875)
(D-876)
(D-877)
(D-STS)
(D-879)
(D-880)
(D-881)
(D-882)
(D-SS3)
(D-884)
(D 885)
(D-886)
(D-887)
(D-888)
D.3.40 Stability axis pitch rate derivatives.--
a(qs) _ 08p
Oq
O(qs) _ 0Or
a(q_) _ oav
O(q_)_ o
a(qs) _ o&3
(D-889)
(D-890)
(D-891)
(D-892)
(D-S93)
(D-89,1)
97
O(q_)_ oo¢
O(q_) _ 000
a(qs) _ oo_
O(qs) = ooh
o(q_) _ oOx
o(q_) _ oOy
O(q_)_ o8&
o(q_) _ o
O(qs) _ 0
(D-895)
(D-896)
(D-897) :
(D-898) ..
(D-899)
(D-900)
(D-901)
(D-902) :
(D 903)
D .3.41
98
Stability axis yaw rate derivatives.--
O(r_) _ sin aoOp
0(_) _ oOq
Or -- COS O_0
0(r0 _ oOV
o(_)Oc_ - -Po cos O'o - ro sin O'o
0(7"0 _ oo_
o(,'0 _ oo¢
0(_) _ oO0
0(_) _ oo_
0(,,_) _ oOh
0(_) _ o8x
O(r_) _ o8y
(D-904)
(D-905) _
(D-906)
(D-907)
(D-908)
(D-909)
(D-910)
(D-911)
(D-912)
(D-913)
(D-914)
(D-915)
o(_,) _ oO&
Off,) _ ook
°(_--2)= ooq6_
(D-916)
(D-917)
(D-9_8)
99
REFERENCES
Clancy, L.J.: Aerodynamics. John Wiley & Sons, New York, 1975.
Dicudonne, James E.: Description of a Computer Program and Numerical Technique for Developing LinearPerturbation Models From Nonlinear Systems Simulations. NASA TM-78710, 1978.
Dommasch, Daniel O.; Sherby, Sydney S.; and Connolly, Thomas F.: Airplane Aerodynamics. Pitman
Publishing Company, Marshfield, Massachusetts, 1967.
Duke, Eugene L.; Patterson, Brian P.; and Antoniewicz, Robert F.: User's Manual for LINEAR, A
FORTRAN Program to Derive Linear Aircraft Models. NASA TP-2768, 1987.
Edwards, John E.: A FORTRAN Program for the Analysis of Linear Continuous and Sampled-Data
Systems. NASA TM X-56038, 1976.
Etkin, Bernard: Dynamics of Atmospheric Flight. John Wiley & Sons, New York, 1972.
Gainer, Thomas G.; and Hoffman, Sherwood: Summary of Transforrnation Equations and Equations of
Jlfotio_ Used in Free-Flight and Wind-Tunnel Data Reduction and Analysis. NASA SP-3070, 1972.
Gracey, William: Measurement of Aircraft Speed and Altitude. NASA ITP-1046, 1980.
Kwakernaak, Huibert; and Sivan, Raphael: Linear Optimal Control Systems. John Wiley & Sons, NewYork, 1972.
Maine, Richard E.; and Iliff, Kenneth W.: User's Manual for MMLE3, a General FORTRAN Program for
Maximum Likelihood Parameter Estimation. NASA TP-1563, 1980.
Maine, Richard E.; and Iliff, Kenneth W.: Application of Parameter Estimation to Aircraft Stability andControl--The Output-Error Approach. NASA RP-1168, 1986.
McRuer, Duane; Ashkenas, Irving; and Graham, Dunstan: Aircraft Dynamics and Automatic Control.
Princeton University Press, Princeton, New Jersey, 1973.
Northrop Aircraft, Inc.: Dynamics of the Airframe. Bureau of Aeronautics, Navy Department, Report AE-61-4 II, Sept. 1952.
Thelander, J.A.: Aircraft Motion Analysis. Air Force Flight Dynamics Laboratory, FDL-TDR-64-70, 1965.
PRECEDING PAGE BLANK NO_I' FILMED
101
SOace_O'nr'R_raonReport Documentation Page
1. Report No.
NASA RP- 1207
2. Government Accmion No.
4. Title and Subtitle
Derivation and Definition of a Linear Aircraft Model
7. Author(s)
Eugene L. Duke, Robert F. Antoniewicz, andKeith D. Krambeer
9. Performing Organization Name and Address
NASA Ames Research Center
Dryden Flight Research Facility
P.O. Box 273, Edwards, CA 93523-5000
12. SponsodogAgencyName and Addresa
National Aeronautics and Space Administration
Washington, DC 20546
3, Raciplent's Catalog No.
5. Report Date
August 1988
6. Performing Organization Code
8. Performing Organization Report No.
H-1391
10. Work Unit No.
RTOP 505-66-11
11. Contract or Grant No.
13. Type of Report and Period Covered
Reference Publication
14. Sponsoring Agency Code
15. Supplementary Notes
16. Abstract
This report documents the derivation and definition of a linear aircraft model for a rigid aircraft of constant
mass flying over a flat, nonrotating earth. The derivation makes no assumptions of reference trajectory or
vehicle symmetry. The linear system equations are derived and evaluated along a general trajectory andinclude both aircraft dynamics and observation variables.
17. Key Words (Suggested by Author(s))
Aircraft models
Flight controlsFlight dynamicsLinear models
18, Distribution Statement
Unclassified -- Unlimited
Subject category 08
19. Security Classif. (of this report)
Unclassified
NASA FORM 1626 ;CT B6
20. Security Classif. (of this page)
Unclassified
21. No. of pages 22. Price
108 A06
*For sale by fhe National Technical Infovmatlon Service, Springfield, VA 22161-2171.
NASA-Langley, 1988
