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NASA Reference Publication 1207
August 1988
Derivation and Definition of a Linear Aircraft Model
Eugene L. Duke, Robert F. Antoniewicz, and Keith D. Krambeer
NASA
NASA Reference Publication 1207
1988
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
NASA
National Aeronautics and Space Administration Scientific and Technical Information Division
CONTENTS
SUMMARY
INTRODUCTION
SYMBOLS
Vectors 2
Matrices . 0. 1
ee
Subscripts 2. 2...
Superscript 2...
1 NONLINEAR SYSTEM EQUATIONS
1.1 Definition of Reference Systems...
1... 1.
1.2 Nonlinear State Equations... 2...
1.2.1 Rotational acceleration. . 2...
1.2.2 Translational acceleration...
2... 2. ee ee
1.2.3 Attitude rates, 2...
1.2.4 Earth-relative velocity...
0. 2 0 ee
1.3 Nonlinear Observation Equations .. 2... 1
1.3.1 Accelerations... 2.2...
1.3.2 Air data parameters...
. 0... . 0-2 ee
1.3.3 Flightpath-related parameters. .... 2... .....0-2. 000020 ee0 e 4ees
1.3.4 Energy-related parameters. 2... 1 0. ee
ee
1.3.5 Force parameters. 2... 0 6.
1.3.6 Body axis rates and accelerations. ©... 2... ee ee
1.3.7 Instruments displaced from the vehicle center of gravity... ......-.-.-..-0-.
1.3.8 Miscellaneous observation parameters. ...... 0-0. 02 . eee.ee-ee. e ee
2 LINEAR SYSTEM EQUATIONS
2.1 Linearization of the State Equation... 6... ee
2.2 Linearization of the Observation Equation...
0... ee ee ee
2.3 Definition of Matrices in Linearized System Equations ...........0002 .e0e .eae
2.4 Elements of the Linearized System Matrices
3 CONCLUDING REMARKS
APPENDIXES
A—AERODYNAMIC FORCES AND MOMENTS
B—DERIVATION OF THE WIND AXIS TRANSLATIONAL
PARAMETERS V, a, AND 6
B.1 Preliminary Definitions a
Sr
Sr
SO
B.2 Derivation of V Equation
B.3 Derivation of d& Equation i
B.4 Derivation of B Equation
iil
C—GENERALIZED DERIVATIVES
39
C.1 Generalized Derivatives of the Time Derivatives of State Variables ..............
39
C.2 Generalized Derivatives of the Observation Variables...
2... 0... eee
43
D—EVALUATION OF DERIVATIVES
49
D.1 Preliminary Evaluation .. 0... 1.
ee
49
D.1.1 Rolling moment derivatives... 2...
49
D.1.2 Pitching moment derivatives. .. 2... 1
ee ee es
50
D.1.3 Yawing moment derivatives... 2... ee es 50
D.1.4 Drag force derivatives... 0...
51
D.1.5 Sideforce derivatives...
2 0.
51
D.1.6 Lift force derivatives... 1... ee
52
D.2 Evaluation of the Derivatives of the Time Derivatives
of the State Variables 2... 1.
ee
53
D.2.1 Roll acceleration derivatives, 2.2... ee ees
53
D.2.2 Pitch acceleration derivatives...
2... 0. ee
54
D.2.3 Yaw acceleration derivatives. 2...ee
55
D.2.4 Decoupled roll acceleration derivatives...
2...
ee
ee
56
D.2.5 Decoupled pitch acceleration derivatives...
2... 0.. ee .
57
D.2.6 Decoupled yaw acceleration derivatives. ... 2... 0.0.00 eee ee eee eee ee
57
D.2.7 Total vehicle acceleration derivatives, 2... 20.0.0.e. e ee. e es 58
D.2.8 Angle-of-attack rate derivatives. ©... 2..e. e ee ee eee
59
D.2.9 Angle-of-sideslip rate derivatives... 2... ee es
60
D.2.10 Roll attitude rate derivatives... 2. ee
ee ee 62
D.2.11 Pitch attitude rate derivatives. 2... 2... ee ee ee ee 62
D.2.12 Heading rate derivatives, 2.. ee . ee es
63
D.2.13 Altitude rate derivatives, 2.2...
es
64
D.2.14 North acceleration derivatives. 2... 2..e. e ee ne 65
D.2.15 East acceleration derivatives. ©... 0...
ee
ee eee ee 65
D.3 Evaluation of the Derivatives of the Observation Variables... ............000.
66
D.3.1 Longitudinal kinematic acceleration derivatives...
2... 20.0.0... .0.00004
66
D.3.2 Lateral kinematic acceleration derivatives...
2... 0.0.0.0 000.. .e.ee
67
D.3.3 Z-body axis kinematic acceleration derivatives. .. 2.0.0.00000.c.ee.eae 68
D.3.4 x body axis accelerometer output derivatives...
2.2... ..0..00..0..2.-.2.2004 69
D.3.5 y body axis accelerometer output derivatives...
2... 00... 0... ee ee ee
69
D.3.6 z body axis accelerometer output derivatives. ©... .......0 000000 bea
70
D.3.7 Normal accelerometer output derivatives. .. 0... 0.
ee
ee es
71
D.3.8 Derivatives of x body axis accelerometer output not at the vehicle center
of gravity, ee
ee
72
D.3.9 Derivatives of y body axis accelerometer output not at vehicle center
Of gravity, ee
72
D.3.10 Derivatives of z body axis accelerometer output not at vehicle center
of gravity. ©.
ee
ee 73
D.3.11 Derivatives of normal accelerometer output not at vehicle center
of gravity, ©.
ee
ee 74
D.3.12 Load factor derivatives. 2...
ee
ee
75
D.3.13 Speed of sound derivatives. ©... 0. eeee ee ee eee
76
D.3.14 Mach number derivatives. . 2... 2...ee ee 77
iv
D.3.15 Reynolds number derivatives. ©...ee
77
D.3.16 Reynolds number per unit length derivatives. 2...
ee
ee
es
78
D.3.17 Dynamic pressure derivatives...
0. 6
ee
es
79
D.3.18 Impact pressure derivatives...
0. oe
ee
ee
80
D.3.19 Mach meter calibration ratio derivatives... 2... ee ee ee 81
D.3.20 Total temperature derivatives. 2... 0
ee
es 82
D.3.21 Flightpath angle derivatives, 2...es
82
D.3.22 Flightpath acceleration derivatives...
0 0. ees
83
D.3.23 Vertical acceleration derivatives. . 2... 2
es
84
D.3.24 Specific energy derivatives, ©... es
85
D.3.25 Specific power derivatives... 0.0
86
D.3.26 Normal force derivatives. 2... 2... ee ee 87
D.3.27 Axial force derivatives...
0... ee
ee
ee 87
D.3.28 x body axis rate derivatives... 1...ee ee
88
D.3.29 y body axis rate derivatives...
0.
ee
ee es
89
D.3.30 z body axis rate derivatives... 6.
ee
ee
90
D.3.31 x body axis acceleration derivatives. ©... 0. ee
ee
ee
90
D.3.32 y body axis acceleration derivatives. .. 2... ee es
91
D.3.33 z body axis acceleration derivatives. 2... 0. ee ee 92
D.3.34 Angle-of-attack sensor output derivatives. 2..e . e eee ee 93
D.3.35 Angle-of-sideslip sensor output derivatives. ©... ee es 93
D.3.36 Altimeter output derivatives, 2... 0. ee ee ee ns 94
D.3.37 Altitude rate sensor output derivatives. 2. 2... 2
ee ee
95
D.3.38 Total angular momentum derivatives. 2... 1. ee
96
D.3.39 Stability axis roll rate derivatives. ...2. -0-0.ee. e e. ee 0 eee. ee ene 97
D.3.40 Stability axis pitch rate derivatives. ............------.--+-2+0-048.
97
D.3.41 Stability axis yaw rate derivatives. .. 2...
ee
98
REFERENCES
101
SUMMARY
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 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 constant mass, flying over a
flat, nonrotating earth. This model consists of a state equation and an observation (or measurement) equation. The system equations have 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 equation is derived from a collection of nonlinear equations representing state variables, time derivatives of state variables, control inputs, and flightpath, 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 principal contribution of this report is a solution of the general problem of 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 Hoffman (1972), and Gracey (1980)
provides a detailed discussion of speed and altitude measurements. However, neither of these references present linear models of these measurements. This report relies heavily on these two references and uses their results as one of the bases for the nonlinear measurement equations from which the linear measurement
model 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 that derives a linear aircraft model by numerical differencing (Dieudonne, 1978). The program LINEAR produces a linear aircraft model
(both 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 the linear aerodynamic model used in this report (app. A), a derivation of the wind axis translational acceleration parameters (app. B), generalized linear derivatives of the nonlinear 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 presented in appendix D.
SYMBOLS
8
bobs
N
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 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 z axis, g reference span, ft generalized force or moment coefficient derivative of generalized force or moment coefficient with respect to arbitrary variable z reference aerodynamic chord, ft
IIIt,,ot,a—-l-IyIIzaz erodynamic drag, lb
specific energy, ft
arbitrary force or moment
flightpath acceleration, g acceleration due to gravity, ft/sec? acceleration due to gravity at sea level, ft/sec? altitude, ft altitude measurement not at vehicle center of gravity, ft inertia tensor moment of inertia about 2 body axis, slug-ft? product of inertia in s—y body axis plane, slug-ft? product of inertia in x—z body axis plane, slug-ft? moment of inertia about y body axis, slug-ft? product of inertia in y-z body axis plane, slug-ft? moment of inertia about z body axis, slug-ft? II, — 12,
Leyle + Lyzlez
Ioylyz + Tylez I,1— ,I2, Ty Tyz + Leylez I,Iy — I, total moment about unit length, ft total moment about
z body y body
axis, ft-lb; axis, ft-lb;
or, total aerodynamic or, Mach number
lift, 1b
a S
xR ¢
Siar star ery
N X
poe
iar
vehicle mass, slugs total moment about z body axis, ft-lb; or, total aerodynamic normal force, lb load factor specific power, ft/sec roll rate (about « body axis), rad/sec static or free-stream pressure, lb/ft? stability axis roll rate, rad/sec total pressure, lb/ft? pitch rate (about y body axis), rad/sec dynamic pressure, lb/ft? impact pressure, lb/ft? Mach meter calibration ratio stability axis pitch rate, rad/sec Reynolds number Reynolds number per unit length, ft7} yaw rate (about z body axis), rad/sec stability axis yaw rate, rad/sec surface area of wing, ft? total angular momentum; or, ambient or free-stream temperature, oR total temperature, °R time velocity along x body axis, ft/sec vehicle velocity, ft/sec velocity along y body axis, ft/sec velocity along 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 z 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, lb 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, rad ith control surface deflection pitch angle, rad coefficient of viscosity, lb/ft-sec density of air, lb/ft? arbitrary function
~ .
CHR WWRN ARN
> .
GRE
ew ae e oe ey by
O HRH eSHed
bank angle, rad heading angle, rad
Vectors
body axis acceleration vector attitude vector of Euler angles total force vector state vector function observation vector function total angular momentum vector sum of higher order terms in Taylor series total moment vector position vector in earth axis system input or control vector vehicle velocity vector state vector observation vector perturbation of control vector perturbation of state vector perturbation of time derivative of state vector rotational velocity vector
Matrices
state matrix of the generalized state equation, Cx = Ax+ Bu state matrix of the state equation, x = A’x+ B'u control matrix of the generalized state equation, Ck = Ax + Bu control matrix of the state equation, x = A’x + B’u system matrix of the generalized state equation, Cx = Ax+ Bu feedforward matrix of the generalized observation equation, y= Hx+Gx+Fu feedforward matrix of the observation equation, y = H’x + F’u derivative observation matrix of the generalized observation equation, y= Hx+Gx+ Fu observation matrix of the generalized observation equation, y= Hx+Gx+ Fu observation matrix of the observation equation, y = H'x + F’u intertia 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, Tx = f[x(t), x(t), u(t)] n X m matrix of 0 values an n X m matrix with values of 1 on the diagonal
a es
Subscripts
a
aerodynamic; or static or, free stream
b
body axis system
D
drag
g
gravitational
h
displacement of altitude instrument
h
displacement of altitude rate instrument
yt
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
z-z body axis plane
sideforce
displacement in y body axis
yz
y-z body axis plane
z
displacement in the z body axis
0
at sea level, standard day conditions; or, nominal conditions
5 a
w n
e t
a 8
i
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, can be formulated in the notation of Kwakernaak and Sivan (1972) and Dieudonne (1978) as a
time-invariant system expressed as
x(t) = f[x(Z), u(2)]
(1-1)
where x(t) is the 12-dimensional time-varying state vector (t being time), x(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)
where y(t) is an ¢-dimensional time-varying observation vector and g is an £-dimensional nonlinear function expressing the relationship of the true vehicle state and 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 formulated more broadly than just described (Edwards, 1976; Maine and Iliff, 1980, 1986). The nonlinear system equations include x(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
TH(t) = f[x(t), X(t), u(2)]
(1-3)
y(t) = g[x(t), (2), u(Z)]
(1-4)
where T is a constant 12 x 12 angular velocity matrix.
1.1 Definition of Reference Systems
While numerous reference systems are used in aerospace applications, this report is limited to four reference
systems: the body, the wind, the vehicle-carried vertical, and the topodetic reference systems. The stability
axes are also defined rates (section 1.3.8).
even
though
this
reference
system
is
used
only
to
define
the
stability
axis
rotational
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, respectively) are not. The body axis rotational rates are measured by
and w sensors
in the x,y, and z fixed in the body
directions, axes; wind
axis rates can be derived only from these quantities through axis transformations.
The first reference system reference frame (Etkin, 1972),
to be described the earth axes
is the topodetic reference (Thelander, 1965), and the
system, also called the earth-fixed Eulerian axes (Northrop Aircraft,
1952). The topodetic reference frame is considered fixed in space (and hence, inertial) with the orientation
of the vehicle
axes as shown in figure position (x and y) and
1; the x axis is altitude (h) are
directed measured
north, from
the y axis east, the origin of this
and the z axis down. reference system.
The
The vehicle-carried vertical axis system (fig. 2; Etkin, 1972) has its origin at the center of gravity of the
vehicle. The zy axis is directed north, by a translation of the topodetic axis (heading, pitch, and bank angles 7, 6, aircraft body axes with respect to the
the y, axis east, and the z, axis down. This system to the vehicle center of gravity. The
and 4, respectively) is described in terms vehicle-carried vertical axes.
axis system is obtained attitude of the aircraft of the orientation of the
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6
x (North) A
(North) x,
» y (East)
(North) x <—
yy (East)
zy (Down) y (East)
2
(Down)
7256
Figure 1. Topodetic axis system.
z
(Down)
7257
Figure 2. Relationship between topodetic and vehicle-carried vertical azis sys-
tems.
Yo L,p
7256
Figure 3. Body azis system.
é<+>o N
The relationship between the vehicle-carried vertical and body axes is shown in figure 4. The Euler angles (7%, 8, 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 ~ is a rotation about the z vehicle-carried vertical axis into a new axis system (designated (21, y1, 21) in fig. 5); the pitch attitude @ is a rotation about the y; axis into the (22, Y2, 22) axes system; the roll attitude ¢ is a rotation about the y2 axis into the body axes.
Xo: Xb
7259
Figure 4. Relationship between vehicle-carried vertical and body azis systems.
These rotations are described by
cosy —sinw 0
Ly =
snp cosy 0
| 0
0
1
Ig = Lg =
[ cosé 0 sind
0
1 O
| —sin@ 0 cosé
rT 1 60
0
0 cos@ —sing
| 0 sing cos¢
and the total rotation is described by
[py = Ly Lely =
cos @cos sin@sin@cosw
— cos ¢dsin
cosésin@cosw +sin dsin w
cos@sin wv singsinOsiny
+ cos @cos wv
cosdsin@sinw —sin cos wv
—sin 6 sin dcosé sin dcos 6
(1-6) (1-7)
a )
A
Pv
Ka 7 7 4 7
X4 a
7
7 7
7
/ 7 7
7 4
¢// 7
4
(?v)
N ~ ~ ~~ aN ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~
4 sy ~
> W 7260
(a) Rotation through » about zy azis.
2
xry
Xa Xb
2,
@
‘XN
/
_
y
aN
/
\
Los
Ny
4
N
7
2, ~<
fs
_
VA
(¥4)
7261
(b) Rotation through 6 about y, avis.
Yy Yo
(Xa: Xp) &
v
29
(c) Figure 5.
Rotation Rotation
through of azes
¢ about through
7262
zr, azis. Euler angles.
Because Lpy is a unitary matrix, the transformation from the body axes to the vehicle-carried vertical axes1 is L#BV:
The relationships between the body, wind, and stability axes are shown in figure 6. All three axis systems have their origin at the center of gravity of the aircraft. The z axis in the wind reference system (tw) is aligned with the velocity vector of the aircraft. The angle of sideslip @ and angle of attack a 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 § and from the body axis system by a rotation —a.)
7263
Figure 6. Relationship of body, stability, and wind azes.
Also shown in figure 6 are the components of the velocity and the definition of positive rotations for a and f. It should left-handed coordinate system, whereas the positive sense of all
vector V in the body axes (u, v, and w) be noted that @ is a positive rotation in a 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 cosacosB v= Vsinf
w = VsinacosZ
(1-9) (1-10)
(1-11)
The total velocity V, angle of attack a, and angle of sideslip @ can be expressed in terms of these body axis velocities as
V = [V] = (u? + 0? + w?)/?
(1-12)
a = tan? —Ww
(1-13)
B = si. n -1 1 VU
(1-14)
10
1.2 Nonlinear State Equations
For the aircraft problem, the state vector x is 12x 1 vector composed of four 3 x 1 subvectors representing the vehicle rotational velocity, the vehicle translational velocity, the vehicle attitude, and the vehicle location:
x = (xp x? x? xi
(1-15)
where
x1 =([pqr]t
(1-16)
x2 = [V a py"
(1-17)
x3 = [¢ 6 pt
(1-18)
x4 = ([h x yl?
(1-19)
with x1, X2, X3, and x4 being the rotational velocity, translational velocity, attitude, and position subvectors, respectively. The vehicle rotational and translational velocity are defined within the aircraft-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-dimensional vector subfunctions:
f[x(t), x(t), u(¢)] = [fy fy fe fy)"
(1-20)
where f,, fz, f3, and f, are the vector functions that relate the x(t), x(¢), and u(t) vectors to the rotational acceleration, translational acceleration, attitude rate, and earth-relative velocity subvectors of x(t). In the following sections, each of these subfunctions will be developed separately. The details of the derivation of 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 accelerationTh—e subfunction f, of f from which the rotational acceleration
terms in the x vector are derived is based on the moment equation
M = —d H
(1-21)
where M is the total moment on the vehicle and H is the total angular expression can be expanded to
M = © (12) +x (IQ)
momentum
of the vehicle. This
(1-22)
where 6/6¢ is the time derivative operator in a moving reference frame (such as the vehicle body axis system)
and the substitution
H=I0
(1-23)
has been used to replace the total angular momentum term with the product of the inertia tensor J and the rotational velocity vector 9. (The inertia tensor is assumed to be constant with time.) The definition of the terms in equation (1-22) follow:
SL
L+1¢
M=/0M/=|M+ Mr
(1-24)
UN
N+ Nr
11
with L, M, and N being the aerodynamic total moments about the Ly, My, and Ny the sums of all power-plant-induced moments;
x,y,
and
z body
axes,
respectively,
and
I= | -IIn,y IzIyy —LIyzez
—Iyz
—Tyz
I,
(1-25)
where J,, Iy, and I, are the moments and J,, are the products of inertia in
of inertia about the z, y, and z body axes, respectively, and the z-y, x—-z, and y-z body axis planes, respectively; and
Iny,
Iz,
Q=x,=[p qr]?
(1-26)
where p, g, and r are the rotational rates about assumed that the inertia tensor is a constant with
the z, respect
y, and z body axes, respectively. Because it to time, equation (1-22) can be rewritten as
is
“0 =I7“'(M-2 x IQ)
(1-27)
fy[x(2),X(#), u(d)] = I7[M — @ x (19)] (1-28) This is the vector subfunction for the rotational acceleration.
following definition applies:
Designating this subfunction as f,, the
where
a60 Q == f,f[ix(ltx)(,t),Xx(1), w(t)
(1-29)
0 = [pil
(1-80)
Since the inverse of the inertia tensor [~! is given by
1 RT|(4R4F 4
where
det I = Iplyl, — Il}, — 1,13, — IyI2, — = TI, - 1,
Tz = Ipylz + Tyzloz T3 = Igylyz + Tyler: I, = II, — 12,
Ls = Iglyz + Teylez Ig = Ip, — 12,
yz TezTry
oe
(1-32) (1-33) (1-34) (1-35)
(1-36)
(1-37) (1-38)
12
the expression for the rotational accelerations can be expanded as a set of scalar equations:
1 b= det qeLh + EMI, + UNIs - p(Ir2d2 ~ InyI3) + pq(Uezh — Ty2To — Dols)
~ pr(Lnyly + DyI2 _- Iyz13) + Q (Lyely _ IzyI3) _ qr(D. _ IgyTo + I,2I3)
— P(Iyzl — Ivela)]
@ = qq1 lELh t+ EMIy + UNI ~ p?(Ieels — Inyls) + pq(Iezl2 ~ Iyelas — Des)
_ pr(Icyle + DyI4 _ IyzTs) + q’(LyzTe _ IzyIs) _ gr( Dele _ Igy + Izz1s)
— 1? (Tyelo — IpeTs)]
1
r= det pelts + UMIs + UNIe — p*(InzIs ~ Inyle) + pq Tez Is - Iyzts — DzIe)
— 1 (Iy2T3 — Inels)] - pr(InyIs + D,Is _ IyzIe) + Q (Tyo _ IpyIo) _ qr( Dats _ IgyTs + Iz,Ig)
where
D, = 1,-I, Dy = I, —I, D,=I1,-I,
(1-39)
(1-40)
(1-41)
(1-42) (1-43) (1-44)
Equation (1-3) defines the generalized nonlinear state equations as
TX(t) = f[x(t), x(t), u(t)]
This equation, although more complicated than the nonlinear equations defined by equation (1-1), allows for
a more tractable formulation of the state equation by using the matrix T to provide a means of addressing the rotational accelerations in a decoupled axis system.
The derivation of the rotational acceleration terms is based on the moment equation (1-22):
M= (UM) +0 x IQ
Rearranging terms and assuming that the inertia tensor is constant with respect to time, the equation can
be written as
[6Q=M-2 x ID
(1-45)
The rows of this vector equation are now scaled using the following scaling matrix:
1/Ir 0
0
J'=
0 l/f, 0
0
0 8 1/I,
(1-46)
This matrix, when premultiplying equation (1-27), merely divides the first row by the roll inertia J,, the second row by the pitch inertia I,, and the third row by the yaw inertia J/,. Using the definition
J=J'I
(1-47)
13
the resulting equation is and J can be written as
20 = J'M-J'(Qx IQ)
1.0
_ ey/ Te
—™ Tez [Te
J=
~ Ipy/TIy
1.0
— Iyz/Iy
~ vr/ Is
-I,,/T,
1.0
(1-48) (1-49)
Equation (1-48) can be expanded and expressed as
pr¢q|
=
—LTeIy|1e.eT0/yTz ——IIyn1zy0[/IIz> ——I-nT1e.y0e//IIny || |[ p4i¢
DL/Ty = tpley/IL: + plz| Te + rqly/In + (¢? — 9? )Iyze/Te — ar le/Te
=
UM/Iy — rple/Ty + rey]Ty — palyz/Ty + (r? — p* Tez /Ty + pri,/TIy
i UN/Iz + (ple/Iz — QrInz/Iz + ptlyz/Ie + (p? — Q Tey/ Lz — pql,/I,
(1-50)
where p’, q’, and 7 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
:
.
==J --L1 03y3222 | Oexe
T= | oxo ites
t
Oexe
lexe
t
|
(1-51)
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 acceleration—Derivation of the translational acceleration vector subfunction f2 is based on the force equation
F= S (mv)
(1-52)
where F is the total force acting on the vehicle and m is the vehicle mass. This expression can be expanded to
F=m (<v +2 xX v)
(1-53)
with the assumption of constant mass with respect to time and the following definitions of F and V:
F=[SX SY yz]
(1-54)
where 2X, NY, and LZ are the sums of the aerodynamic, thrust, and gravitational forces in the z, y, and 2 body axes, respectively, and
Valu v wit
(1-55)
14
Rearranging the terms of equation (1-52) gives an expression for the translational acceleration:
oét ye m p_oaxv
(1-56)
This equation expresses body axis accelerations in terms of body axis forces, angular rates, and velocities. However, 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 6, which are expressed by equations (1-9) to (1-11)
u = Vcosacosf v= Vsinf w= VsinacosB
and equations (1-12) to (1-14)
V= [Vv| —_ (u? + vy? + w?)i/?
a = tan7} (=)
3 = sin7} (+)
The wind axis translational acceleration terms (derived in app. B) are summarized as:
[V & A)" = f[x(t), X(t), ud]
(1-57)
where
V. = 71b Deosh + ¥ sin B + Xp cosacos B+ Yrsin B + Z sin a cos p
— mg(cos a cos G sin @ — sin G sin ¢cos @ — sin acos § cos ¢ cos A)]
(1-58)
a= Vmco1 sbl -L+Zycosa — Xz sina + mg(cosac¢ocoss @ + sin asin 6)] +q- tan B(pcosa + rsina)
(1-59)
B: = y1l sin 8 + ¥ cos — Xp cosasin B + Yr cos 6 - Zr sin asin B + mg(cos asin Bsin 8 + cos B sin dcos # — sina sin G cos ¢cos @)| + psina —rcosa (1-60)
with D being total aerodynamic drag; Y total aerodynamic sideforce; and Xz, Yr, and Zr total thrust force along the z, y, and z body axes, respectively.
1.2.3 Attitude rates.—The matrix R that transforms angular velocities in the earth-fixed axis system into body axis angular velocities is defined by
1
kR={0
0
0
cos¢_
-sing
—sin 6
singcosé
cos¢cosé
(1-61)
where F 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 (4, 6, ):
D
lb
1 0
0
0
1 0
0
cos# 0 —sin6
0
q| = {01+ {0 cos¢d sing
6{+]0 cos¢@ sing
0 1 #0
0
r
| 0
0 —sind cos¢ | | 0
0 —sing cos¢
sin@ 0 cosé
o
‘1 (OO
— sin 6
db
= |0 cos¢ singcosé | | 6
0 —sin ¢ cos ¢cosé
o
(1-62)
This transformation from earth-fixed to body axes can be expressed by the equation
a=R(5B)
(1-63)
where E is an attitude vector whose components are the Euler angles:
E = (60 y]"
(1-64)
Premultiplying both sides of equation (1-63) by R71 and rearranging terms yields the equation for the attitude rates,
{B= R10
(1-65)
which can be expanded into the scalar equations
bd = p+qsindtand + rcosdtand | 6 = qcos¢—rsing p= qsin sec 6 + rcos dsecé
(1-66) (1-67) (1-68)
1.2.4 body axis
Earth-relative system is defined
velocity—The matrix by equation (1-8) as
Lpy
that transforms
earth
axis
system
vectors into the
Lpy =
| cos snw
| 0
—sin w 0
cosy 0
0
1
cos? 0
—sinO
Q sind 1 O 0 cosé@
1 0
0
0 cosd@ —sing
0 sing cos¢
r
cos # cos
cos @ sin a
—sin@
=
singsin# cosy —cosdsiny singdsinOsiny+cosdcosy sin ¢cos@
| cosdsinPcos+singsiny cos dsin 6 siny—singcosy cosdcosé
The specific relationship between earth-relative velocities and body axis velocities i8 expressed by
=V=L Inv (4( SR)
(11--6699)
where R is the earth axis system vector defining the location of the vehicle:
R=[z y 2]?
(1-70)
16
with z = —A. The equation for the earth-relative velocity can be formulated as
R = L5yV
(1-71)
in which these velocities are expressed in terms of body axis velocities. Using equation (1-72) and the definitions of the body axis velocities in equations (1-12) to (1-14) allows the earth-relative velocities to be expressed in terms of V, a, and ~:
h = V(cosacos (sin @ — sin J sin ¢ cos 6 — sina cos f cos $ cos 9)
(1-72)
z = V[cosacos cos @ cos ~ + sin G(sin ¢ sin 6 cos ~ — cos dsin ) + sin a cos (cos dsin 8 cos # + sin dsin p)]
(1-73)
y = V[cosacos 6 cos @ sin p + sin B(cos pcos p + sin dsin 6 sin 7)
+ sin a cos G(cos ¢ sin 6 sin ~ — sin d cos p)]
(1-74)
1.3. Nonlinear Observation Equations
No standard set of observation variables exists for the aircraft analysis and control design problem. However,
for any guidance and control problem, the main observation variables generally will be a subset of the state
variables. Other common observation variables are the vehicle body axis translational accelerations and
air data parameters. Thus, the dimension of g[x(t), x(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= [x7 xt ut y7}t
(1-75)
where x and x are the state vector and time derivative of the state vector described previously, u is the control vector, and y’ is defined by
ywe=lyt ys ys ys ys ye yr ys)
(1-76)
where
yi = [deck Gyk Gzk Gz Gy Gz Oy Apy dy; Gz Ong rm"
(1-77)
y) = [a M Re Re’ @ 4 4c/Pa Pa & T Tr)"
(1-78)
y3 = [7 fpa h]?
(1-79)
y, = [E, P,J*
(1-80)
y,=[L DNA ©
(1-81)
ye=[uvwid wt
(1-82)
y7 = [az Bi hy kyl"
(1-83)
ys =[T Ds % Ts)"
(1-84)
17
with the elements of y{ being terms related to the vehicle body axis acceleration, the elements of y} being air data terms, the elements of y3 being flightpath-related terms, the elements of y, being terms related to vehicle energy, y{ being a vehicle force vector, the elements of y4 being body axis translational rates and the time derivatives of those terms, y7 being a vector of variables representing measurements from instruments not located at the vehicle center of gravity, and the elements of yg being a collection of miscellaneous terms.
Obviously, this grouping of terms is somewhat arbitrary and is done primarily to ease the definition of these terms in the following sections of this report. This grouping of observation variables parallels that used by
Duke and others (1987).
The vector function g relating the state vector, the time derivative of the state vector, and the control vector to the observation vector is an £-dimensional function composed of four subfunctions:
glx(t), x(t), u(t)] = [x* x7 ut g!7)]
(1-85)
where x, x, and u are identity functions on the state vector, time derivative of the state vector, and control vector, respectively, and g’ is composed of vector subfunctions defining the y’ vector.
The state vector, time derivative of 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.T”he 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 Hoffman (1972), and Gracey (1980) provide the background and derivation of the 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
a= ad va )AV taxV
(1-86)
It is important to note here that the u, 4, and w body axis velocity rates, derived in appendix B and defined by equation (B-1), 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-product terms. Thus, expanding equation (1-86) yields
ark
au+qu—rv
a=} aye | =| o+ru—pw
zk
w+ pv — qu
(1-87)
where Using
dz,
@yk,
and
a,
are the kinematic
accelerations
in the vehicle
body
z, y, and
z axes,
respectively.
a
(1/m + X) a+X( g) X + r7 v -quw
ob) =| (/m)(¥r+¥.+ ¥,) + pw-ru
(1-88)
w
(1/m + Za)+ ( Zg) Z+ qy u— pv
(stated as eq. (B-1) in app. B), equation (1-87) can be rewritten as
ek
(1/m)+ (X,X+rXx)
ay k
=
az k
(1/m)(¥r + Ya + Y;)
(1/m)(Zp + Za + Zg)
(1-89)
18
where X,, Ya, and Z, are total aerodynamic forces and X,, Y,, and Z, 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)
az k
1
dyk | =
Gzk
Jo
Xz — Deosa+ Lsina — gmsin@ Yr+Y+gmsin ¢cos
Zy — Dsina — Lc+ogms cosa ¢cos
(1-90)
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
ar
1
ay | = ——
dz
gom
X77, - Dcosa+ Lsina Yr+Y
Zr — Dsi—nLa cosa
(1-91)
where az, ay, and a, are the outputs of accelerometers at the vehicle center of gravity and aligned with the vehicle body z, y, and z axes, respectively. Because the normal acceleration a, is defined by
Gy, = —a,
(1-92)
an expression for this variable can be extracted from equation (1-91):
a, = (-Zr + Dsina+ Lcosa)/gom
(1-93)
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 Hoffman (1972) using the definition of inertial acceleration given in equation (1-86)
and the definition of inertial velocity
a-~ét Viaxv Ve g6 rtQxr
(1-94)
The results from Gainer and Hoffman (1972) are reproduced here without rederivation:
ddazy,,iii | = | adazyz ++—[([[q((p?prg++— ¢1#?))ta)zyte +—— ((qp(?rpq++7? B))*yy)2y¥e ~—— (¢((?aprr+ ~+P”B9)))22z2yx]])///999000
(1-95)
where a,j, 4y;, and a,,; are outputs at accelerometers aligned with the z, y, and z body axes but not located
at the vehicle center of gravity; the subscripts z, y, and z refer to the z, y, and z body axes, respectively;
and the symbols z, 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, ani,
is given by
Oni = On ~ [(pr — g)az + (ar + By — (a° + vp) 221/90
(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
n= mLg
(1-97)
19
1.3.2 Air data parameters.—The air data parameters having the greatest application to aircraft dynamics and control problems are the sensed parameters and the reference and scaling parameters. Chosen for inclusion as the sensed parameters are impact pressure q., static or free-stream pressure pa, total pressure
pt, ambient or free-stream temperature T, and total temperature T;. The selected reference and scaling
parameters are Mach number M, dynamic pressure g, speed of sound a, Reynolds number Re, Reynolds number per unit length Re’, and the Mach meter calibration ratio q./pa. The derivation of these quantities
is treated extensively by Gracey (1980).
The nonlinear equations defining these quantities are
¢ =
| |1.4
Po
Polo
r|
1/2
(11--9988)
M = —V
(1-99)
Re = PVLeE
Re’ = 7ivV
g. = tlaid,
(1-100)
(1-101)
(1-102)
fe _=) J {(11..20M+7[05..27M6?M)?5/(5—.61M.0?]p,— 0.8)]?5 —1.0}p, ((MM <> 11..00))
(1-- 103)
qe _ J (1.04 0.2M*)3* — 1.0
(M < 1.0)
Pa ‘| 1.2M*[5.76M?/(5.6M? — 0.8)]?5—1.0 (M > 1.0)
1-104 (1-104)
T, = T(1.0+ 0.2M?)
(1-105)
where p is the density of the air, yu 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 Flightpath-related parameters.—Included in the observation variables are what might best be termed flightpath-related parameters for lack of better nomenclature. These terms include flightpath angle ¥, flightpath acceleration fpa, and vertical acceleration h. The variables are defined by the following
equations:
y== sin7! AV
fpa = —V g h = ag, sind — dy, sin @ cos @ — az, cos cos 8
( 1-. 106 )
(1-107)
(1-108)
20
1.3.4 Energy-related parameters.—Two energy-related parameters are included with the observation variables considered in this report: specific energy E,, and specific power P,, defined as
E,== h+ _y2?_g2
( 1-109 )
Po== dE, Sa h:y + —V3V
( 1-110 )
1.3.5 Force parameters.—The set of observation variables being considered also includes four force parameters. These quantities are total aerodynamic lift LZ, total aerodynamic drag D, total aerodynamic normal force N, and total aerodynamic axial force A, defined as
L = qSCy
(1-111)
D = 4SCp
(1-112)
N = Leosa+Dsina
A= —-Lsina+Dcosa
(1-113)
(1-114)
where S' is the surface area of the wing, C{, coefficient of lift, and Cp coefficient of drag,
1.3.6 Body axis rates and accelerations.—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 z 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, a, 0, and w, respectively.
The definitions of the body axis rates are given in equations (1-9) to (1-11) as
u = Vcosacosf
v=Vsinf w= Vsinacos PZ
The
and
time derivatives
(1-56) as
of these
terms
can
be
defined
using
equation
(B-1)
and
equations
(B-8),
(B-9),
(B-10),
i Xv -—gmsind —_ Dcosa+ Lsina +rV sin B — qgV sinacos@
(1-115)
a=
Yr
+ gmsing@cosé+Y m
+ pV
sin acos
3 — rV cosacos 8
w= Zr + gm cos $ cos @— siDsina na —~ L cosa +qV cosacos 3 — pV sin B m
(1-116) (1-117)
1.3.7 ments from
Instruments displaced from instruments displaced from the
the vehicle center of gravity—The need vehicle center of gravity arises from the fact
to include measurethat not all aircraft
21
instrumentation is located at the vehicle center of gravity. The most important of these quantities are un-
doubtedly the accelerometer outputs treated in section 1.3.1. In this section four additional parameters are presented: angle of attack (a,;), angle of sideslip (,;), altitude (h;), and altitude rate (h.;) measurements
from instruments displaced from center of gravity by some z, y, and z body axis distances. The subscripts
a, B, h, and h 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; = a+ Pa Pua
Vv
(1-118)
Bi = B+ 2’V
(1-119)
hi = h+2,sin—@y, sin dco6 s— z, cos cos 8
(1-120)
hy =hA+t O(a; cos @ + y; sin dsin 6 + z; cos sin @) — HY}, cos $cos 6 — z; sin pcos @)
(1-121)
1.3.8 Miscellaneous observation parameters—tThe final set of observation parameters considered
in 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 p,, stability axis pitch rate q., and stability
axis yaw rate rs. The equations used to define these quantities are
,
T= 5 ep? — QIeypq — 2Leept + Iyg? ~ 2Tyzqr + Ier*)
(1-122)
Ps = pcosa+rsina Gs = 4
(1-123) (1-124)
re = —psina+rcosa
(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 x n matrix and B’ is a constant n x k matrix. The standard output equation has the form
y(t) = H’x(t) + F’u(t)
(2-2)
where H’ is a constant ¢ x n matrix and F’ is a constant ¢ x 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
Cx(t) = Ax(t) + Bu(t)
(2-3)
y(t) = Hx(t) + Gx(t) + Fu(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 fx n
matrices, and to (1-4)) can the standard
F be or
is a constant £x linearized about the generalized
k matrix. The nonlinear system a trajectory, and a linear model linear system equations.
equations developed in section 1 (eqs. (1-1) can be formulated that is similar to either
22
2.1 Linearization of the State Equation
If uo(t) is given input to a system described by the state differential equation (1-3), and if xo(t) is a known solution of the state differential equation, then approximations to the neighboring solutions can be found for small deviations in the initial state and in the input by using a linear state differential equation. The nonlinear state differential equation (1-3) can be linearized about a general trajectory, as by Kwakernaak and Sivan (1972) and Dieudonne (1978), so that x(t) satisfies
TXo(t) = f[Xo(t), xo(t), uo(4)] Assuming that the system is operated at close to nominal conditions with u(t), x(t), and x(t) deviating only slightly from uo(t), xo(t), and Xo(t), the following expressions can be written:
u(t) = uo(t) + du(t)
(2-5)
x(t) = xo(t) + &x(t)
(2-6)
x(t) = Xo(t) + 6x(t)
(2-7)
where éu(t), 6x(t), and 6x(t) are small perturbations to the control, state, and time derivative of the state
vectors, respectively.
Substituting equations (2-5) Taylor series about Xo(t), Xo(t),
to (2-7) into the nonlinear state uo(t), and assuming T constant
differential equation with respect to x(t)
(1-3), yields
expanding
in
a
T[Xo(° t) t + 6 6°x(t)] _= ff[xo(t), Xe o(t), u(t)] + a Ioxf 6x + mDoxsf 6xx: + “ aouf 6u + h(t)
(2-_8)
where 0f/Ox, Of/Ox, and Of/Ou 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:
ax
Ox}(X,o ,Xo,Wo)
(2-9)
de = El. of
of
(Xo0,Xo0,Uo)
(270)
BOuf = aOuf (X0,Xo ,Uo)
the (7, 7)th elements of which are defined as
(2-11)
Of
Of;
(a). a (5), 7 ax; Ox/);,5; Ox;
21(212)
(2-13)
(Sofaag = BOaf;s
en)
respectively, where f; is the ith simultaneous equation tion (1-3), x; the jth element of the state vector, x;
of the nonlinear the jth element
state differential function of the time derivative of
in the
equastate
23
vector, i; the jth element of the control vector, and all derivatives are evaluated at the nominal condition
(xo(t), Xo(t), uo(t)).
Subtracting equation (1-3) from (2-8), rearranging terms and neglecting the higher order terms yields a
linearized state equation,
Ir — =5o|f E(6x(t)t) == —o=f x(t)+ —of Sul)
(22--115)
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
C=T- xof
(2-16)
A= =of
(2-17)
B- aoaf
(2-18)
equation (2-15) can be written as
C 6x(t) = A 6x(t) + B du(t)
(2-19)
which is precisely the formulation of the generalized state equation desired.
Premultiplying both sides of equation (2-19) by C7! results in the standard form of the linearized state
differential equation,
6x(t) = C71A x(t) + C71B 6u(t)
(2-20)
Letting
A'=C71A
(2-21)
B’=C™!B
(2-22)
equation (2-20) can be written in the more usual notation
6x(t) = A’ 6x(t) + B’ Su(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), x(t), u(d)]
Performing a Taylor series expansion about the nominal trajectory (xo(t), Xo(t), uo(t)) yields
yo(t) + 6y(t) = glxo(t), k. o(t), uo(t)] + 56]= 6x + SOge 6x. + 5O% ou + h(?)
(2-24)
where
O6)g _ Og
x
.
(Xo0,X0,Uo)
(2-25)
24
og _ 9B
(2-26)
Ox
Ox (X0,X0,Uo)
odug _ 9Ou8 (20,3%0, 100)
(2-27)
the (i, 7)th elements of which are defined by
(<d5=g)), = =OOgxi j
(22--2288)
CaaaSg n =>OOxgi;
(22--2299)
po=g} = oOgti
Cae
Ou;
2-30
(
)
respectively, where g; is the ith simultaneous equation of the nonlinear observation equation (1-4). Again, all derivatives are evaluated at the nominal condition (xo(t), Xo(t), uo(t)).
Subtracting equation (1-4) from equation (2-24), rearranging terms, and neglecting higher order terms
results in a linear observation equation,
dy(t) _= ODgx 6x+ Oage 6x,. +, Oagu éu
(2-31)
where the arguments of the matrix functions have been dropped to simplify notation. Letting
H= xdg
(2-32)
G= aoxg
(2-33)
Fe= adug
(2-34)
equation (2-31) can be rewritten as
éy(t) = H éx(t) + G 6x(t) + F du(t)
(2-35)
which is the generalized linear observation equation desired. The standard form of the observation equation can be derived by substituting for 6x from equation (2-23)
into equation (2-33). This substitution results in
éy(t) = H 6x(t) + G[A’ 6x(t) + B’ éu(t)] + F éu(t)
(2-36)
which can be written as
éy(t) = [H + GA’) 6x(t) + [F + GB’] éu(t)
(2-37)
By letting
H'’=H+GA'
(2-38)
F' = F4+GB'
(2-39)
equation (2-37) becomes
25
dy(t) = H’ bx(t) + F’ bu(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 the nominal trajectory.
Using the nonlinear state equation (1-3),
Tx(t) = f[x(t), x(2), u(t)] the terms in the generalized form of the linearized state equation (2-19),
C 6x(t) = A 6x(t) + B u(t)
can be defined as
C=T- Doxf
.
A= aoxf
B= aouf
(2-41) (2-42) (2-43)
The terms in the standard form of the linearized state equation (2-20),
6x(t) = A’ 6x(t) + B’ bu(t)
can be defined as
A='
- a ax Of]~' df
—_—_
_
B=' =
5 du of)]—! of
—_
_—
In a similar manner, the nonlinear observation equation (1-4),
y(t) = g[x(t), X(t), u(t)]
can be used to define the terms of the generalized linearized observation equation (2-35),
Sy(t) = H 6x(t) + G 6&(t) + F du(t)
(2-= 44) (2-45)
H=2
(2-46)
g-2
(2-47)
F= =
The terms in the standard form of the linearized observation equation (2-40), by(t) = H’ éx(t) + F’ du(t)
can be defined as
H' U == O— 3g" +5— o2g lr ——— =o—f\~! aanf——— Fefao “sdg2.tar“Oog lT- _ 2 aa)t) _5Of
(2-48)
(2-- 49) (2-_ 50)
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),
A=_ —Ooxf
the element occupying the ith row and jth column of A, (A);,;, can be represented as
(Adis = 5af;
(2-51)
where fj is the scalar function defining the time derivative of the ith state and x; is the jth state. The individual terms used in the A, B, C, H, G, and F matrices are defined in appendix D based on the generalized derivatives derived in appendix C.
Using the state vector x defined in (1-7) as
x=[pqrVaBgophzy)
the elements of the A matrix can be expressed as
[ 8(p/)/ Op A(p')/Oq «++ A(p’)/By] O(q')/Op O(4')/Oq «++ A(q')/Oy
(2-52)
O(x)/Op O(x)/Oq --+ O(x)/dy | A(y)/Op A(y)/Aq --- A(y)/Oy|
27
Substituting for these partial derivatives using the terms in appendix D gives
(1/Te)[(G5b?/2Vo)Ce, + OLr/Op
(1/Le)[(GSbe/2V0)Ce, + OL7/dq + InzPo
—LryTo + T2290]
+2Iy290 + ro(ly ~ I,)|
A=
(1/Iy)[(G@Sbe/2V0)Cm, + OMr7/dp
(1/Iy)(GSé/2Vo)Cm, + OMr/0q
wae
—2Iz2Po — Iyz40 + ro(l, _ I;)]
+InyTo - IyzPo]
(2-53)
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 C
matrix.
of
To
the
determine
nonlinear
the elements of
vector function
the matrices
g defining the
for the observation equation, one must
observation variables (eq. (1-85)),
consider
the
definition
g[x(t), x(t), u(t)] = [xT x7 ut g7]
and the definitions of the matrices for the generalized linear observation equations (2-46) to (2-48) ,
H= 3
o = 98
m y
H T
i
These matrices may be expressed using a partitioning based on the vector subfunctions of g as
ooxx 7
ax
ox
! |
(2-54)
t
a a
L
S a
G=|--
(2-55)
Fa} __
(2-56)
28
which become
lizxie |
012x12
H=|___-
Oxx12
| xd oe!
012x12 |
G=|__li_2-x12 Oxx12 3g L @x 4d O12xk
F=|-_O-1e-xk
liexk
|= |
upon evaluating the partial derivatives of the identity functions x, x, and u. The C matrix may be viewed as a partitioned matrix as
L e olt y | Oexe
Ca | Sex| Ce I
O6x6
L
i}
lexe
1
where, from equation (1-48),
Cu =J=|-Iee/1.I0 y 1.0 —Inz/ly _ cy/Tr —Ip2/Tr —Ig./I, —Iyz/T, 1.0
and
-A(p!)/AV —A(p')/A& —A(p')/OB Ci2 = | -—AO((rq'’))//AOVV ——AO((qr/'))//0A&& ——8A((qr''))//0A8B
0 —(GS'be/2Volz)Ce, —(GSb?/2Volz)Ce, 00 ——((GG55eb?e//22VVoollzy))CCnm,a ——((GG55bbE?//22VVoollzy))CCnm,,
(2-57)
(2-58)
(2-59)
(2-60)
(2-61)
(2-62)
29
[ 1.0 - a(V)/aVv ~O(V)/a ~a(V)/aB
—8(&)/AV -0(8)/aV
1.0 -—8A((aB))//de6
1.
0
—0(a)/08 — A(8)/dB
[1.0 (GSé/2Vom)(cos Bo Cp; — sin Bo Cy,) (G5b/2Vom)(cos Bo Cp;)
= [0 1.0+(gS2/2VemBcoo)CsL;
(qSb/2V2em cos Bo)CL;
(2-63)
10 (GSe/2VFm)(sin Bo Cr, + cos Bo Cy,) 1.0 — (G5b/2V2m)(sinBo Cp, + cos Bo Cy,)
The inverse of the C matrix, C~!, can be expressed as a partitioned matrix in terms of the matrix subpartitions of the C matrix as
(2-64)
The elements of the equation (2-64), the A,
A’, B,
B’, H,
H’, and F’ matrices G, and F matrices,
can and
be determined the definitions
using the C-! matrix defined for A’, B’, H’, and F’ given
in in
equations (2-21), (2-22), (2-38), and (2-39).
3 CONCLUDING REMARKS
ieTnqhiusaa tisrotenpasotritoanraedreyridveearsitvmeoadsndphferdroefmeinenosovnearlisnaeetarfolfat,sliinxne-oadnrerigzoretedaet-isonyfgs-tferemeaerdtmoha.mtriecBqeoustahtfioorgnesna errioafgliidmzoeadtiirocrnaanfdtanosdftacnaodnalsratrdagnetlicnmoealaslrse,ctsiyofslnytienmogf nonlinear observation (measurement) equations.
This derivation of a trajectory about which properties.
linear model the model
is is
general and makes linearized or the
no assumptions on either symmetry of the vehicle
the reference (nominal) mass and aerodynamic
Ames Research Center Dryden Flight Research Facility
National Aeronautics and Space Administration Edwards, California, January 8, 1987
30
APPENDIX A—AERODYNAMIC FORCES AND MOMENTS
The aerodynamic
icance varies with
forces
flight
and moments acting on an aircraft
condition as well as from vehicle to
are the
vehicle.
result of multiple
In general, these
factors
forces
whose signif-
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:
F = 8(a,G,V,h, p,q, 7, &, B, 51, ..-,6n)
(A-1)
where F is an arbitrary force or moment, ® is an arbitrary function, and the 6; 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=qSCp Y = qSCy
L = GSC,
(A-2) (A-3)
(A-4)
and the moments,
L = qSbC, M = GS, N = qSbC,
(A-5) (A-6) (A-7)
where 6 is reference span and é 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 coefficients 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
Ch = Cry + Cia@ + Chgf + CLR + CLyV
+ 32 C146: + Chph + Ctgd + Cha? + Chak + CL,B
(A-8)
i=1
Cp = Co +Cp,at+ CoP + Cp,h + Copy V
+ i=>1 Cp,,6: + Co, B + Co,4 + Co. ? + Co,&+ Co, 8
(A-9)
Cy = Cy,+Cy,a+ Cy,8 + Cy,h+ CyyV
n
a
+> Cy, 6: + Cy,p + Cy,¢ + Cy,? + Cy,4 + Cy,6
i=1
(A-10)
Ce = Cay + Ce + Cag 8 + Co,h t+ Ce,V
+ 32n C05, 5: + Cob + CoG + CoP + Cea + Ce,8 $
(A-11)
i=1
31
Cin = Cin + Cra + Cg + Cyl + CmyV
+ > Crs, 6; + CmpP t=1
+ Cm4
+ Cn,?
+ Cm
Cn = Crp + Canad + Cng8 + Cnyh + CnyV
+ C'm;8
(A-12)
n + > Ong, 65+ Cn,D + Cr
i=1
+ Cn?
+ Cra
? + CrP
(A-13)
where C¢, is the value of the coefficient along the nominal trajectory and the notation C¢, is defined as
Ce 7
_
Ie
Oz
(A-14)
with Cg being an arbitrary force or moment coefficient and z being an arbitrary state, time derivative of state,
or control-related parameter that for the usual derivatives is nondimensional. However, the derivatives with
respect of these
to altitude and velocity are not taken nondimensional stability and control
with respect derivatives
to are
a nondimensional quantity. The given in terms of the coefficient
definitions Ce. The
nondimensional stability derivatives are defined as
Ce = ace
(A-15)
Ce, = =
(A-16)
Ce, = Hay
(A-17)
Ce, = Fein
(A-18)
Cz, = xb
(A-19)
Ca = aD
(A.20)
Ce, = Tao
(A-21)
The two other stability derivatives are not nondimensional and are defined as
The control derivatives are defined as
Ce, = oct Ce, = a
(A-22) (A-23)
Ces, = a
(A-24)
32
The rotational terms in equations (A-8) to (A-13) are nondimensional versions of the corresponding vari-
able with
p= 32
(A-25)
i=
(A-26)
f= se
(A-27)
a= a
(A-28)
p- 8
(420)
Because the Cg, 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, a, AND 6
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 translational acceleration terms is derived separately after stating some preliminary definitions applicable to all calculations.
B.1 Preliminary Definitions
Equation (1-56),
6
ét
v-sFr-axv m
can be expanded, using equations (1-54), (1-55), and (1-26), to
u
(1/m + ) X,+(X,X ) 7 + rvp— qu
o| = |(1/mY) at(YY ,) ¥+ r pw+ —ru
(B-1)
w
(1/m + Za)+ ( Zz)Z+ r qu— pv
The body axis aerodynamic forces can be rewritten in terms of the stability axis forces lift L, drag D, and sideforce Y:
X, = —Dcosa+Lsina
(B-2)
Y, = Y
(B-3)
Z, = —Dsina — Lcosa
(B-4)
The gravitational forces can be resolved into body axis components such that
X,g = —mgsin@
(B-5)
Y, = mgsin ¢cos 6
(B-6)
Z, = mg cos ¢cos 8
(B-7)
These equations will be used in the derivations of the V, &, and B equations. Thus, the total forces in the body axes can be defined and expanded as
DX = Xy—-— Dcosat Lsina — gmsin§
(B-8)
LY = Yr +Y + gmsin dcos6
(B-9)
“LZ = Zr — Dsina — Lcosa+ gmcos ¢cos 8
(B-10)
B.2 Derivation of V Equation
Beginning with the definition of V in terms of u, v, and w in equation (1-12),
35
the equation for V becomes
V,_ =—r7d~adV=—FdAaC +v* 24 + 9 2)w*)1/2
|
which after expanding the derivative and cancelling terms, becomes
V. = p1 (uu + vd + ww)
(B--1111) (B-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
;
V = tcosacos# + bsin 8 + wsinacos ZB
(B-13)
The definitions for u, 9, and w in equation (B-1) are now used with equation (B-13) to give
Ve= cos a.cos By + X7 + Xz) + cosacos B(rv — qw)
+ ney, + Yr + Y,) + sin O(pw — ru)
sin a cos 3 (Za + Zp + Zz) + sin acos B(qu — pv)
(B-14)
Expanding (B-14) in terms of equations (B-2) through (B-7) and cancelling yields
V:e= T1l Pos B+ ¥ sin 8 + Xr cos acos@ + Yrsin§ + Zr sin acos
— mg(cosa cos # sin 6 — sin 6 sin ¢cos @ — sin a cos co¢scos 8)]
+ rucosacos § — qwcosacos 8 + pwsin 8 — rusinB
+ qusinacos 8 — pusin acos B
(B-15)
Equation (B-15) can be simplified by recognizing that the identically zero, which becomes obvious after substituting equation becomes
terms for u,
involving v, and w
the vehicle rotational in these terms. Thus,
rates are the final
V. = 71,17 Deosh + ¥ sin B + Xp cosacosf + Yrsin 8 + Zr sin acos B — mg(cos a cos f sin @ — sin 8 sin ¢ cos 8 — sin a cos 3 cos cos 6)]
(B-16)
B.3 Derivation of & Equation
The equation for @ can be derived from the definition of a in equation (1-13),
a = tan} *U Taking the derivative of a with respect to time,
~ dt. dtu
then expanding and cancelling terms, the equation becomes
1
.
36
(B-17)
Substituting the definitions of u and w from equations (1-9) and (1-11) into equation (B-18) gives
°+=
w cosa — usina
V cos B
(BB--119)
Using equation (B-1) to substitute for % and w and equations (B-8) to (B-10) to define the forces, equation (B-19) becomes, after rearranging terms,
a= e Vm cs os8 + V ot
+ Zp cosa — Xp sina + mg(cosacos ¢ cos # + sin asin 6)] cosa — pucosa@ — rvsina + qwsina)
(B-20)
which after substituting for u, v, and w from equations (1-9) to (1-11) and combining terms gives
Gr ~
1
Vmcos B
[-L
+ Zy cosa
— Xz sin a + mg(cosacos cos
# + sin asin )|
+q-—tan 6 (pcosa+rsina)
(B-21)
B.4 Derivation of § Equation
The equation for B is derived from the definition of 6 as given in equation (1-14),
GB==si - sin 1 _7U—V Taking the derivative of 3 with respect to time yields
B» = ade = 3d 5a- 1 VU which becomes, after expanding the derivative, substituting for V, and cancelling,
(B-22)
B= 5 [-itcos asin B + bcos — wsinasinf]
(B-23)
Using equation (B-1) to substitute for u, 0, and w and equations (B-8) to (B-10) to define the forces,
BA = 7d1 i cos asin 8 (—-DcoLss inaa+ + X7 — mgs 6) +icon sB (Y + Yr + mgsin ¢cos 8)
— sinasin # (—Dsina — Lcosa + Zp + mg cos ¢cos6)] + y1l- cosa sin6 (rv — qw) + cos 8 (pw — ru) — sinasinB (qu — pv)|
(B-24)
Substituting into equation (B-24) for u, v, and w and rearranging terms yields the final equation
B: =
Ty1
lh sin 6 + ¥ cos 6 — Xqcosasin#+
Yreos 8 — Zrpsinasin 8
+ mg(cos asin 6 sin 0 + cos B sin ¢ cos 6 — sin asin 8 cos ¢cos 4)]
+ psina—rcosa
(B-25)
37
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 in 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.
aOEp) > det4 l I bOaLE + I. =O0ME + 1 O_=@EN +h| OBlEy +12| OMBEy + _I3 OBNE
~ [2p(Inel2 _ IryI3) a qezhy ” Tyzdo _ D,Is) + r(Inyly + DyIn - Tyz13)| =
+ [peel _- Iyzte _ D,I3) + 2q(Lycl _- IgyI3) _- (Dh _- IpyIo + Teyla)] D5e
_ (poy + Dyn _ Iyz13) + q(Di _ Inylo + Iy2I3) + 2r(Lyzh ~ Ip21o
a»,}
ag) _= Ta1l { l=BOLE +14 pOeM ts gOeN 12 OLarEy +I, OaMe + Is OaNeT
(C-1)
— [2p(Iezl4 — Inyls) — q(Ieel2 — Iysts — Dz Is) + (Teyl2 + Dyla — IyeTs)] 50
6)
+ (p(Inzo - Tyzl4 - D,IJs5) + 2q(Lyzl2 _ IpyIs) 7 r(Dzle ~ Igy, + TzyI5)) 3
- [p(LryI2 + Dy Ig _ Iy2Is) + (Dele
_ Ipyls
+ IzzJs5) + 2r[Lyzlo
- I,z 14]
o
ai
Ba(ei) = det{l 2I. DE 4 Is oOeM te_ oONe. +h O5lgy ts| OMBey + |Ig OBNEq
_ [2p(IneIs _ IzyIe) - qr
6)
I3 _ TyzIs _ D_Je) + r(IsyI3 + D,Is _ IyzIe)] 5e
(C-2)
6)
+ [pUz2ls 7 LyzTs _- D_Ig) + 2q(Lyel3 _ Izy Ie) _ r( Del _ Igyls + I,2I6)| 3
0
7 [(p(Leyls + DyIs — TyzI6) + q( D213 — Lryds + Tr2I6) + 2r(Lyzt3 _ I,215)} x}
(C-3)
The quantities I), I2, I3, I4, Is, (1-42) to (1-44).
Ie,
Dr,
Dy,
Dz,
and
detJ are
defined in equations
(1-32)
to
(1-38)
and
Equation (1-50) defines the decoupled rotational accelerations of the vehicle (p’, q’, and r), which are
used to determine the generalized derivatives of the decoupled quantities:
Ap’) _ 1 7OL , abr 0
clase + O3E 0 —(rtly ales) $2et (pez + rly + 2qInz — rl,) 7e
_ (pIny _ ql, + 2rlee + qI,) %y
(C-4)
39
“0 _ r Es 4 oe — (rly + Gl yz + 2plzz — Iz) 3 + (rIey — plyz) i
~ (ply — ley — 27Izz — plz) Al
(C5)
1) z Ee + aE + (In + tl yz + 2pley — aly) 3 + (ple — rhe ~ 2qIay ~ Ply) 5e
~ (qlaz — plyz) 4
(C8)
Equations (1-58) to (1-60) define the translational accelerations of the vehicle. These equations are used
to determine the generalized derivatives of these quantities:
ae = ={-« cos
5 + cosacos 9 x Pasi ing OOEeX t sin cos
FE 4 si np St
+[- Xrsinacos 8 + Zp cosa cos 3 + mg(sin O sin a cos B
+ cos # cos ¢ cos a cos 3)] 5—
+ [Dsin 6+ Y cos 6 — Xqsin B cosa + Yp cos GZ — Zysinasin B
+ mg(si6 ncos asin GB + cos @ sin ¢cos B — cos 6 co$ssin asin B)] Be
— mg(—cos8 cos din 8 + cos 8 sin d sin a cos ) D9E%
~ mg(cos @ cos a cos Z + sin @ sin ¢sin 8 + sin 6 cos d sin a cos f) x00e
(C-7)
O(aae) =_ TaV 1cosB ( ~ OBLE T0084 OaZEy — si. na aaE) - tan Bcosa OSpe + O5q - tan Bsina 5,
_ { mV? 1 cosB nag T COS o — XTpssin in+ amgg((ccooss@6 ccooss p c¢coossaa ++ ssiinn 8@ sina i} ae
+ {ra1 g 2 si. na — XT cosa — mg(ccoos s¢s@in. a — s. i8 ncos a)
+ tan@ (psian — rcosa) \0g
+ {atran eG al + Zpcosa — Xysi\na
+ mg(cos @ cos ¢cos a + sin sin a)] —
q(P cosa +rsin ay} o
_ Cassg . cos 9 si. n ¢ cos a) ODge ~ in g te . ¢ cos @ — cos @ si. n a)| B06e8
(C-8)
40
ao(ep) _7 1 V [: sin B dDDE + cos 8 ODYE
+ sina P5p rCoOsS aO oFne
cosa si_ n 8, ODXEt + cos 8 ODEr sin asin B 2—1e)
+ mg(sin @ cos asin 3 + cos @sin ¢ cos 8 — cos 6 cos dsin asin f)] aOV
+ {ltr sinasin G — Zp cosasin # + mg(— sin @ sinsian B — cos @ cos ¢ cos asin @)}
+ pcosa + rsina} oa
+ —,[D cos 6 ~ ¥'sin 8 ~ X.c0s cos 8 — Yr sin 6 ~ Zr sin acos f -+- mg(sin @ cos a cos 8 — cos @ sin ¢sin 8 — cos @ cos sin a cos f)| o
+ Z(cos 6 cos ¢cos 3 + cos @sin dsin asin 3) o0—¢g
+ (cos 6 cos asin 8 — sin @sin cos Z + sin Ocos Psin asin 3) —0
Equations (1-66) to (1-68) define the vehicle attitude rates. These equations are used to determine the generalized derivatives of these quantities:
4) _¢3 + sin dtan8 st + cos¢tan0 at (qcos ¢tan @ — rein ptan@) i
+(qsin ¢sec” 6 + rcos psec” 9) n2e
(C-10)
=e os¢ 52e4 sing 528 — (qsin d + rcos ¢) ae
(C-11)
el = sing@sec?B—E14 cosdseco a2e%~ + (qcos psec O — rsinosec 6) i2ef
+ (qs @seci#ntan 6 + rcos @sec # tan @) 0D—E6
(C-12)
to
Equations (1-72) to (1-74) define the earth-relative velocities determine the generalized derivatives of these quantities:
of
the
vehicle.
These equations are used
a(€h) = [cos # cos asin 6 — sin Bsin d cos 6 — cos § sin a cos ¢ cos 6] o0vg
— V(cos Z si. n asin. 6 + cos 8 cos a cos ¢ cos 6) B0Ea
— V(si. n6 cos asin. 6 + cos B si. n ¢cos 6 — s.in6 si.n a cos ¢ cos 6) BOEB
~ V(si. n 6 cos ¢cos @ — cos Gsin. asin. ¢ cos 6) 3OEg
+ V(cosB cosa cos 6 + si.n si.n dsin. 6 + cos Bsin. acos @sin. 0) D0E0
(C-13)
“ne = [cosB cos a cos 8 cos w + sin 8 (sin ¢sin 6 cos ~ — cos dsin w) + cos f si.n a (cos dsin. 6 cos p + si.n dsin. p)] DOEV
— V[cos 8 sin a cos 8 cos op — cos f cos x (cos $ sin 9 cos y + sin gsin p)] ae ~ V[sin 8 cos a cos @ cos p — cos #sin dsi8 cnos — cos dsin
+ sin # sin @ (cos Psin @ cos ~ + sin dsin wy]
+ V[sin @ (cos ¢sin 6 cos + sin d sin ~) — cos sin a (sin dsin 6 cos — cos psin ~)] -
— V[cos @ cos a si.n 6 cos » — s.in§ si.n ¢ cos 6 cos ~ — cos @ si.n a cos ¢ cos 6 cos x] B0E0 — V[cos 6 cos a cos Osin w + sin J (sin @sin 0 sin + cos @cos )
+ cos @ sin a (cos Psin 8 sin ~ — sin cos w)] ie
(C-14)
a = [coB scos a cos sin p + sinB (cos dcos ~~ + sin dsi@nsin p)
+ cos § sin a (cos ¢sin @ sin w — sin ¢cos b)] a
— V[cos sin a cos 6 sin — cos 8 cos a (cos ¢ sin @ sin — sin dcos w)| b0ae — V[sin 6 cos a cos 6 sin py — cos 7 (cos décos p + sin dsi6nsin w)
+ sin 6 sin a (cos sin 8 sin y — sin ¢cos p)] ae
42
— V[sin 6 (sin ¢ cos p — cos ¢sin O sin p) + cos # sin a (sin dsin A sin p + cos ¢ cos )] i — V(cos 2 cos asin Osin y — sin B sin ¢ cos 6 sin p — cos @ sin a co¢scos 6 sin p) D0E0
+ V[cos @ cosa cos6 cos # — sin § (cos dsin p — sin dsin A cos )
+ co# s sin a (cos dsin 0 cos p + sin dsin y)| ne
(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:
A(aDrE x)
go1m |[xOse
_
cos
@
aD
OE
+ sina
L
O=E~ +(Dsina
+L cosa)
e2% — gmcos6
a6
B—E
.
(C-16)
O(ayx)
dE
1 Ee
~ gom| ae
ay
+ DE
+
gmcos6
cos
ad
¢ B—E
—
gmsin
8 sin
an
oe
;
(C-17)
O(daE2)
_ gom1| Eeoe
— si: n a bODe
— cos & OBLE
~ (Deosa
— Lsi. na)
da
oe
—gm co6 ssin len gmsi6 cnos na
(C-18)
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:
7 _ a Ke ~ cosa oe +sina ne + (Dsina+ L cosa) ae
(C-19)
we ~ om (Fe + )
O(daEz) _ go1m
Oa2E7
— si. na OOED
— cosa BaEL
— (Dcosa
— Lsina)
da
Fe
(C-20)
(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
Gdn | —oOgZ tsina aOEeD + cosa OB=Le> + (Dcosa— Lsina) se
o€ ~ gom
(C-22)
The vector equation (1-95) defining the output of orthogonal accelerometers aligned with the body axes but displaced from the vehicle center of gravity is used to determine the generalized derivatives of these
43
quantities:
Oa0z€i)
O(a, oa ;
_ OGaEz, 1
op
_ og
Or,€
= f3e) ~ b1o [(2puy — qty) Bre}E — (pty + rzy) Brey) — (q2y — 2ryy) f3e} +z
OG | |
OD
BE
~
fy
Or
|
;
(C-24)
OdBzEi)
OOaE, d1o (ep, _ rZz) dBpE + (2qzz _— ryz) gDqe | (p22 + qyz) Oar6 ¥ oBpe +2, =oqa (C-.25)
Equation
at vehicle
(1-96)
center
defines the
of gravity,
output of
Gn,i- This
a normal
equation
accelerometer aligned with the z body axis but not located
is used to determine the generalized derivative of ani:
O(n)
d€
_
Odn
“OE
+o1
[(ep-. — 7Zz)
jOep + (2qz2 — rye) oBEq
(prz + qyz) aOre v2 oBpg +
a5g
(C-26)
oesbIdfnqyouumdmgabyertmoqailyuvoaisanxttisyis,.voza,nrasiacsyac,bel(bleCaeen-frd2oo€0rm)ee,aztreetrrtoehsfdeee(rfCais-tnut2oeb3tds)hc,ezr, ibyvptyeth,hseiecqalunpzeaad, rtticyioze,annltsbaeonrddd(eyCro-fiz1va)agxrtrieiasfvvteoeirsltoy(tcCoao-aft3ri)teo.htnehsdeeT2f,hivofneeehy,ditchpleaaebrnytdisraeelonqztsuoaardtbtseioirodoinynvarsaletlaiaxtv(reieCasvst-,ee1s6o)rtfoewsittpotthehhece(tCior-vvue1eets9lhpp)yieu.,cctltesIanntdcoofetnhtettteshhhreeee
Xn) 1 aL Using equation (1-97), the generalized derivative of the load factor can be defined as OE = hg bE
(C-27)
are
Equations (1-98) to (1-105) define
used to determine the generalized
the air data parameters of interest for
derivatives of the air data parameters:
this
report.
These
equations
O(a) _
0.700
ar
2§ —— poTo[1.4(po/poLo)]'/? 9€
(C-28)
Avi
_ Ya
(C-29)
+— so - Fe
|
(C-30)
a6 OE pe a? BE
3)
Ht = pV +e
(C-32)
( [(1.0 + 0.2M!?)35 — 1.0] rOia
+ 1.4M(1.0 + 0.2M?)?5p, oM
(M < 1.0)
8(7 qe) = 4 [12M 2((5_56.M7?6M—?_)_\*” ~ 1.0) JOdea
(C-33)
+Pa
{2
4M
Ga5z.76M
va)
2.5
+
3.0M 2 (=
8(_85.76—M?0
1.5
3)
{
fea9r.21-6M|_] aOeM}
( 1.4M(1.0 + 0.2M?)25 ou
(M 2 1.0)
(M < 1.0)
A(qea/Pea)) _ | fo onaM ( 565M.?76M-?0.8 ) + 3.00 ( 5.65M.276M—?0.8 )"
(C-34)
L Fea9.21—6Mfl} “aOME
(M > 1.0)
OT5e) _= (10+ 02M?)2) OZTe + 0ATM OM
(C-35)
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:
Ay) ____1 _[_h av | dh
OE
(V2 — f2y1/2 | V O€ + 7 |
(C-36)
A(fpa)
og
=o g109V
(6h) _= [—a,,, cos dcos 6 + a, si. n ¢ cos 8] ODdE
(C3-737)
+ fa, cosO + ay, si.n dsin. 6 + az, cos psin. 9»] 30E8
+ si. n 6 OaBEr _ sin cos 8 OdByE, _ cos $ cos 8 04B2E%
(C-38)
The partial derivatives of altitude rate A and velocity rate V that appear on 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
Using equations (1-109) and (1-110), the generalized derivatives of the
defined. The in equations
partial derivatives (C-13) and (C-10),
of altitude rate respectively:
and
velocity
rate
appearing
energy-related parameters are in equation (C-40) are defined
(Bs) _V OV, Oh
0g
g9 0§ OE
(C-39)
O(0Fg,) = Vg9 o08v Vvg aOFv, © oAhE
(C-40)
The tion D.1. presented
derivatives of the The generalized in terms of the
force parameters, lift (eq. (1-111)) derivatives of the normal force (eq. generalized derivatives of the lift and
and drag (eq. (1-113)) and the
drag forces:
(1-112)), are axial force
defined in (eq. (1-114))
secare
oO(eN) _= cosa OaeL.sina aGDe — (Lsianae D cosa) daae
(C-41)
Ov6A)= . sin aeAL + cosa OaDe _ (L cosa + Dsi;na) dBaE
(C-42)
The defined
body axis rates are in equations (1-115)
defined in equations (1-9) to to (1-117). These equations
the body axis rates and accelerations:
(1-11). are used
The time to derive
derivatives of these terms are the generalized derivatives of
“pe = cos acon FE — Vin acosp $2 — V eosasing 8
(C-43)
= = sin B Fe + Veosp oe
(C-44)
= = sin cos Se + V con cos p 22 — V sinasina 22
(C-45)
a - we (Ger 2080 FP + sina 32) —Vsinavcos 8 SE + Vsin 8 oP
+ (rsin 8 gsin a.cos 8) 5 4 [Fr(Dsina + Leos) ~ qV eos aos | ne
+ (rV cos 8 + qV cosasinB) — 9 cos@ z
(C-46)
aD = 2 (Get Ge) + V sine cos $2 — V eosacos
+ (Psin cos — rcoscxcos 8) Fe + (pV cosacos + rV sina cos) 2
~ (PV sin asin ~ r¥ cosasin 6) Fe + geoss cos 5° — sin Bsin go Me
(C-47)
46
Aw) 4 (2s _.. OD
an
7
og
ag 0g sin a =~ — cosa
V si np 2 +V cosacoss $f
+ (qcaocoss § — ps.inB) OovE [n1 oe — Lsi;na) + qV si.n acos | 0BaE
— (qV4 cosasi6 n+ pVV cos a) =2i~e — g gcos@sin ¢ o=D¢~E —g sin 8 cos 0b6e
(C-48)
The outputs of various instruments displaced from the vehicle center of gravity are defined in equations (1-118) to (1-121). These equations define angle of attack, angle of sideslip, altitude, and altitude rate instrument outputs. The generalized derivatives of the quantities are based on these equations:
“ae 0(a@,:) _ vYoa eOptvLeoeOg e O(Bi) _ V _% Op T,Vxg ObrE
\( Vov?pe)) oS0v7 dBaE \orWBE) OV. Tt O8B
aOhei)
( y,.cos cos 6 + 2, si. n gcosd) d©5¢g
(C4-949) (6-;50)
+(a;, cos 0 + yp, sin dsin 6 + z_ cos psin 9) 55 + oF
(C-51)
one = en sin pcos 8 + 2; cos dcos 8) + acy, cos fsin 8 — z, sin dsin 6)| ae
+ |- 6.(x; sin 8 — y; sin dcos6 — z; cos pcos6) + $.(yj, cos dsin 9 — asin gsin8)| i0
— (yj, cos pcos# — z; sin dcosA) ae + (xj, cos
+3ad6hS
+ y, sin dsin6 + z; cos ¢sinA) -6
(CC--5522)
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:
= (Iep ~ Tey ~ Tear)32EEB + (Iyd— Tey? — Iyer) 5 + (er ~ Inzp — Iyz4) a (C-53)
ee = cosa se +sina 7 (psian— r cosa) ae
(C-54)
ae _ a
(C-55)
ae = -—sinag st + cosa er + (-pcosa— rsin a) 522
(C-56)
47
APPENDIX D—EVALUATION OF DERIVATIVES
The generalized partial derivatives presented in equations (C-1) 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 O¢q Or OV @da OB O¢6 40 Ab Oh Ax Oy Op Oq Or OV Oa OB Ad 86 AY Oh Ox dy 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 (a and £, in particular) are equal to zero. This is also true of the partial derivatives of the state variables with respect
to 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.1.1 Rolling moment derivatives,—
on - re
(D-2)
be = re
(D-3)
or = orc
(D-4)
ae = SbpVCr + G5bCp,
(D-5)
oe = 4SbC:,
(D-6)
io = 7SbC1,
(D-7)
ad = 550V7C% 7p + 95bCo,
(D-8)
53 - 1 oe
(D-10)
x = gSbC%,.
(D-11)
49
D.1.2
Pitching moment derivatives.—
GSbe
av"? ge 2V Cig
qSbe 2V Cme
—GQSVcEC me GSCm,
i2 St_ V?Cm A0P + G_S—@Cm,
qc? 2V
Cima
GSbe
2V Cm,
GSCng
D.1.3
Yawing moment derivatives.—
qSb? 2V Cn qs be 2V Cng qSb? 2v°""
SbpVCy, + GSbCry
~SbV°C, => + GSbCn,
50
(D-12) (D-13) (D-14) (D-15) (D-16) (D-17) (D-18) (D-19) (D-20) (D-21)
(D-22)
(D-23)
(D-24) (D-25) (D-26) (D-27)
(D-28)
(D-29)
(D-30)
(D-31)
D.1.4 Drag force derivatives.—
SpVCp + Cp,
gSCp, gSCb, 5lo0i, Cp 5OP7, + -WSC,
D.1.5 Sideforce derivatives.—
SpVCy + @SCyy
qSCy,
gSCy,
lon
OP, _
35V Cy an t 1 O¥n
(D-32) (D-33)
(D-34)
(D-35) (D-36) (D-37) (D-38) (D-39) (D-40) (D-41)
(D-42) (D-43) (D-44) (D-45) (D-46) (D-47) (D-48) (D-49) (D-50) (D-51)
ol
D.1.6 Lift force derivatives.—
——aadLLp- =
Gqsb #q22eV6 CLp
Oq
2V CL,
OOLn
OOVD
_ =
x
sqoSpybVCCLle +
aSCLy
a = GSCL,
a
aOLg = ICs
DOLh = g1.o.. Cu 5Opp. + I_ SCun 3OLa = qaSyé Cle aOLp _7 q2Sb Ck
(DD--5522)
(DD--5533)
o(Dd-5)5)
(D-56)
(D-57)
(D-58) (D-59) (D-60)
OOsL: = GSCL,. ;
(D-61)
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 powerplantinduced 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, 8), 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 F, to represent a powerplant-induced force (X7, Yr, and Zr) and M, to represent a powerplant-induced moment (Lr, Mr, and Ny),
OF, OF, OF) OF, OF, OF, OFp _
_
_
(D-62)
dp = 0g)s—i«< rs si Hsi—si sO
siCiRC CY
:
OM,
ad
t t
OM,
06.
OM,
db
9M,
Oh
OM,
Oz
OM,
dy
OM
06,
P
9_ °
(D--63)
“ d
OF, OF) OF, OF, OM, OM, OM, OM, OM, OM,
OV’ da’ OB’ 06;’ Op’ dq’ Or’ OV’ da’ a
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 parameters
OT Op OH. 4 APs are assumed to be functions of altitude only. Thus, except for
8h? Oh’ dh
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.
52
D.2 Evaluation of the Derivatives of the 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 0(x;)/0z; is used to represent the more correct notation Of;/Oz; 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 in flight
mechanics texts such as Etkin (1972) and McRuer and others (1973). The notation that defines quantities such as Lp = O(p)/Op and M, = O(q)/0q is misleading in this context because the definitions of those terms
(such as Ly, M,) 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.—
ADpp) = det1 T «[aavseo tC + Ihe_ Cm, + I3bCn,) + O5Ly + OMDpy + ODNp
7 2po(Tezte _ IzyI3) + go(Iezl _ Lyzto _ D_Is3) _ To(Layli + Dylo — Iyz)|
(D-64)
OP)
EpgySé eC + IneC'm, + I3bCn) BF OMOg, + OONgT
+ Polezh _ IyzIn _ D_Is) + sual h _ IeyIs) _ ro De _ IeyT2 + IpzI3)
(D-65)
a4 O(p) _ 1 raSb
OL
Or
det J | ayy of1bCe + 1neC m ,+ b Can.N ) + or
OMy T, ON OAT
Or
Or
0) —
— poUleyly — Dyl2 - IyzIzs) — qo(Deh - IpyTo + I,13) - 2ro(Iyzh _” Inz12)
1 S0(pVeCe + Gey) + FSepVoCm + Imy)
(D-66)
+ Is8B(pVoCn + TCny) + Ih “a +h - +4 1
(D-67)
ae) = a0{[as(nice, + In6Cm. + IsbCn.) +h eas +t Fe tds Fe
(D-68)
a
O(6)
a¢
=
_
Fopp
|B
UibCey
+ 122mg
+
IabCng)
+
I
a
+h
a8
+i
7
|
(D-69) (D-70)
” =0
(D-71)
20 = Sali (uate cn) +4 (en M40 we =0 + Isb (G1 v Ch odhp + Cnn)
(D-72) (D-73)
53
O(p) _
Odpx) _ 0
Oy O0(Gp) ~ BV gSdee=
O(D
GSb
(hbCt, + IntCm, + I3bC;,., )
Ae AP) = deca
++ InC ing, + I36C ng. )
(D-74) (D-75) (D-76) (D-77)
(D-78)
D.2.2 Pitch acceleration derivatives— OO(p4) =_ Geolil |=ay, 20Ce + I40C'm, + I56Cn,) + Io ODLp
+ Ig OMOpr
+ Is OONpT
— 2poTezls — TeyIs) + qo(ezle = Tyz14 — DzIs) — roTeyl2 + Dyls — Iyels)| (D-79)
0(d4q) _> de1tl| a 2V, 20% + [46Cm, + IsbCn,+)Ip ODLgy +14 OMBqy + Is OaNq?
+ PoUIrzl2 _ LyzT4 _ D,Is) + 2qo(Tyz12 _ IzyIs) - ro(Dzl2 - IeyI4 + Ieels)
(O4r) _~ de1tT{ 2D2V, d20 Ce, + [4eCm, + I56Cn,.) + Io OaLr +14 OMDry + Is OaNrT
(D-80)
~~ Po Loyle _ Dyls -~ IyzIs) _ go( Dele _ Igy Ig + Iz2Is5) _ 2ro(Lyzl2 7 Irs)
a4 ey = sal [S6(pVeCe + Ce) + LSe(pVoCm + FComy)
+ I5ISsb88(pVeCn n + I_Cny) n +o OSlPy + Ig OMMEy 4 1, aONT
BAqa) __ a1r7 [es(nbcr, + [4éCm. + IsbC,.) + Ip aDLay + 1, aMJray + Is aFNay
SF0(q) _= Taqil [PSsCty + Leng + TebCng) + h OSLp ts GeOMry ts OSNTT
A) _ 4
Og
3Ag)°_
Og) _
OaOgyh) ” _
_$ ap 0
detT [ap (5=VeCeDh + Cx.)
+ 14(0ly(an5V8Cm
opDh + my)
+ Isb(S+ VEyC,2 n OopP 4 im)
oaqz) _9
(D-81)
(D-82) (D-83) (D-84) (D-85) (D-86) (D-87)
(D-88) (D-89)
54
Oy
0
aa4) _ WeGeS- ppl habCes + I4Cm, + IsbCng)
ai4e = mW,GSsbey (labCe, + L4eCmy + IsbCn,)
364;) = Fh InbCe, ts, + eC. + t4COm,, + 4T556C ns, )
(D-90) (D-91) (D-92) (D-93)
D.2.3 Yaw acceleration derivatives.—
at?) Op
= =
ja1 l pVe9
30Cee + IséCm, + IebCn,) + Is OOLpy + Is ODMp
+ Ie ODNp
— 2po(lzels ~ Ixyle) + do(Ieela ~ Iyels ~ Dele) ~ rollayla + Dyls ~ Iyele)| (D-94)
= = det I E| aGyS,C lsbCe + I5éCm, + IgbCn,) + Ig =OdLqT + Is OMaqr 18I, OaNqT
(D-95) + Po(IrzI3 7 Iyzts _ D_I¢) + 2¢0(Lyz13 _ IzzIg) _ ro(Del3 - IgyTs + Irzla)|
41 EeSd (IsbCe, + Isr, + lobCn,) + Ig 9OL22 4 1 OOMMyE 4 7, SONNye
— po(Inyls + DyIs _ IyzI6) _ go( Del _ Igy Ts + IzzI6)
_ 2ro(Lyz13 _ Ieels)|
- 1 [2258(pVeCe + GCu,) + Ise pVoCm + ICmy) det I + IeS8(p+ V_Fn0yC) n+ Ig OSLF + fy VEOMr 4 _ONT
- Fag |B5(bC, + IstCn, + IgbCn,.) + Is ee + Is a + Is ad
33
saaJy [B5U28Ce, + Is€Cm, + IgbC,,) + Is a + Is eo
Ig ad
=0
(D-96)
(D-97) (D-98) (D-99) (D-100)
= 0
(D-101)
= 0
(D-102)
= —_ [20(5 VoCe oe aC, + Is0(5 VoCm <o + ICs
+ 1eb( 53 Cn se + 1Cm,.)]
= 0
(D-103) (D-104)
= 0
(D-105)
55
Or) Oa ~~
«gS 2Vodet
I (136C'e, + Is€Cm,
+ IgbCn, )
Or)
dB
_
—-gSb
2Vodet I
(IsbCe,
+
IstCm,
+
IebCr;)
A(*) _ 48 06;
detl (136Ce,. + IséC mg, + IgbC ng, )
D.2.4
Decoupled roll acceleration derivatives.—
mle ete ele a oe 3
ast
OL
“OV Ce” t 7 — dyeyTo + Test)
f[ ste
3 .
OL
at a + InzPo + 2Ly240 + To(Ly _ 1)|
sh
Oly
| s2rVo Ce += Or — 4zryP0 + go(Ly ~ I,) ~ 2Iyer9
s b(pVoCe + GCn,) + aOaLVly
(1800, + SOta) ~ (a8b8C, + SOLFty)
(D-106) (D-107) (D-108)
(D-109) (D-110) (D-111) (D-112) (D-113) (D-114) (D-115) (D-116) (D-117) (D-118) (D-119) (D-120) (D-121) (D-122) (D-123)
D.2.5
Decoupled pitch acceleration derivatives.—
aq) _ z [Cg 4+
— 2Iz2P0 — Iyz40 + To(Tz — I.)
z (Geen + oe + IzyTo ~ Lt)
r [Fon + aa + po(lz — Ir) + Ieyqo + 21.79
z [$e oVoCm + my) + | i (452mg + oat)
i (asm, + a)
D.2.6
Decoupled yaw acceleration derivatives.—
_ 1 Gre , oNe
20
=”
Op + 2IeyPo + go( Ir _ Iy) + her
z [Br Cn, + ae + pole — Ty) — 2Teyqo — Testo]
z (Seo + oN — Izz40 + 100)
Oo 7“EL [S000VCn + WCny) + |
(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-139) (D-140) (D-141) (D-142)
57
ar) _
(D-145)
a) =0
(D-146)
Oh ~ 9
(D-147)
“Se = Fe (Zvien SE + Cm)
(D-148)
ae =o
(D-149)
‘ =o
(D-150)
- Cn,
(D-151)
= _ oe ne
(D-153)
D.2.7
Total vehicle
av = ogmSb n
acceleration derivatives.—
cos Bo Crp, + si. n Bo Cy,)
“ av = VqoSmé — cosJo Cr, + si;n Bo Cy,)
aav) = aVSbom- cos J Cp, + si;n Bo Cy,)
BavV) = m1| 7 5608 G0 (eVoCn + F_ Cd,) + Ssin; Bo (pVoCy + G- Cy,)
+ cos ag cos Bg Oax + si. n ag cos Bo rOiZa + si.n Bo | OY:
aayv) =~1 | — gS‘ cos Bo Cp, + G5 sin Bo Cy, + cos ao cos Bo aaxa
+ sin a cos Bo oer + sinSo aYt" — X7 sin ao cosfo
+ Zp cos ag cos By + mg(sin O sin &p cos Bo + C08 Op cos dy COs ag COS 6o)}
(D-154) (D-155) (D-156)
(D-157)
(D-158)
OavB)
i1 las(- cos Bo Cg + sin Bo Cp + sin Bo on + cos BooY)
mM
+ c08 00.008 BoOax: + si a0 608 Fo5eT+ sin 6 oXPt
— Xp sin Bp cos ag — Zz sin ag sin Bo + Yz cos Bo
+ mg(sin 9 cos ag sin Bo + cos 9 sin do cos Bo — cos Op cos go Sin ao sin 6o)} (D-159)
g(cos 99 cos dg sin Bo — cos 9 sin $o sin ag cos fo)
(D-160)
g(— cos Mo cos ao Cos Go — sin Go sin ¢o sin Go — sin Ao cos ¢o sin Ao Cos fo)
(D-161)
0
(D-162)
S| oso (3v8CD 2 + aCo,) + sino (AveCy 2 + acv,)|
(D-163)
0
(D-164)
0
(D-165)
Hom — cosfo Cp,, + sinBo Cy, )
4i50 (_ cos By Cp, + sin Bo Cy,)
(D-166) (D-167)
2 cos 39 CDs, + sin Bo Cy,, )
+ ~1 (cos Qo cos Bg O5X—6,T—
in Ao SE= + sin Q cos Bo FO5ZE)
(D-168)
D.2.8
Angle-of-attack rate derivatives.—
(a) Op
=
qSb
~2Vem cos Bo CLy
—
tan fp cos ao
O(a) _
qse
dq O(a
2VemcosBy aSb
“4 +10
~ Wm cos 3g Ch, — tan Bo sin ao
4d) V =
mVocos Bo { s(VoeCt + GAT CL) — cos ag oOZyy t sin ag OaXy?
1
+ G0 OL
+ Zp cosag — XT sin ag
+ mg(cos 69 cos dg cos ag + sin Og sin ao)
(D-169) (D-170)
(D-171)
(D-172)
59
OF(aa) __ mVo c1 os Bo [_asc _ COS Ao OOae27T + si. n ao O3Xay + Zp si. n ag + xX Xp cosag
+ mg(cos 4 cos ¢o sin ao — sin 0 cos ao)
+ tan Bo (po sin ao — To cos 20)|
.
O(aaa) _ ~ inp 1cos Ao {aGSCL, — Cos ag OOZBT + sin ao OOXB
— tan Bo [-gSCL + Zr cosa — X7 sin ao
+ mg(cos 69 cos $9 cos ag + sin Mp sin a0)]}
- cos? Bo (Po COs Ag + To sii n ag)
(Oad) __ Vo cogs Ao cos & si; n do cos a9
O(3a0) = Voco9sg Ba (si.n 99 cos ¢g cos ag -— cos O7%] si: n ao)
5 =0
O(a) _
S
lian OP, . )
Oh
~ mVo cos Bo (Svec dh t (Cts
ae)z = 0
|
8) = 0
y
O(a) _
qGSé
Oa
Vem cos Jo Cha
O(a) _
gGSb
ap = ~ 2V2m cos By UA
O(a) _ 1
-
OZp
or)
06; — mVocos Bo [-250., + £05 40 05, ~ Bin eo “5,
(D-173)
( D-174 ) ( D-175 )
( D-176 )
(D-177)
;
(D-178)
(D-179)
(D-180) (D-181)
(D-182)
(D-183)
D.2.9 Angle-of-sideslip rate derivatives—
(6
Gsb
oe ~ amin Bo CD, + cos fio Cy,) + sin a
O(6e
= spGiSé
0
(sin Bo Cp, + cos Bo Cy,)
(66) = aaGSpb oin Bo Cp, + cos Bo Cy,) — cos ag
(D-184)
(D-185)
(D-186)
60
(a8v) _ mV1o [s si. n Bo (pVo + G_ Cp,,) + S cos Bo (pVoCy + G_ Cy,) — ¢€08 ao si. n Bo axX: + cos Go ra e) 7 si. n ag si. n Jo a OZ:
_ mvz1 a5 (si,n Bo Cp + cos Bo Cy) — Xq cos ag si. n Go
+ Yr cos Bo — Zr sin ag sin Bo]
(8)
Oa
mVo [aas'sin fo Cp, + cos Bo Cy, ) — cos ag si. n Bo Oax + cos fp aOY:
— si. n ao si. n By —OdZ—aT + Xpsin. ag si; n Bo — Zr cos ag si. n Bo
— mg(sin sin ag sin Bo + cos A cos do cos ao sin 6o)|
+ po cos ag + Tg Sin ao
(a8p)
aE {qsisin Bo (Cp, — Cy) + cos Bo . + Cy,)|
— COS Qo sin fg —aX—pT
cos fp OaXF — sin ao sin G9 O—dZ—Br
— X7 cos ag Cos Bo — Yer sin Bp — Zr sin ag cos fo
(D-187) (D-188)
+ mg(sin 89 cos ag cos Bp — cos 9 sin Po sin Bp — cos 9p cos go sin ao cos #0) } (D-189)
(cos 99 cos dp cos Bo + cos Op sin do Sin ao sin Bo)
0
.
JVo ( cos 9 cos Qo Sin Bo — sin Op sin $g cos Bo + sin 8 cos ¢o sin ao sin Bo)
0
mVSof,[sin Bo (S1 ven 3DOph + iC, ) + cosfio (S1 vécy o$Dhpe + Cv, )|
(D-190)
(D-191) (D-192) (D-193)
0
(D-194)
(D-195)
AaPE (sin Bo Co, + ¢08 Bo Cy,,) WgeSmb sin Bo Co, + cos Bo Cy,)
~ mVo [as(sin Bo CD,, + cos Bo Cy,,) — cos ag sin By OaAT
— sinQgoosin2 OG|
08 Bo OYa:e=
(D-196) (D-197)
(D-198)
61
D.2.10
Roll attitude rate derivatives—
ed(p4)
xe) = sin go tan 9
ora(¢) = CoS dg tan A aOgv) _~° GOga) _~°
aOdp) =_ °
“o = qo cos pp tan 9 — 7 sin dp tan A
ae) = qosin do sec? O + 79 cos do sec” Oo
aadp) _ aadh) _= 9
Og) _
dx =0
O(9)y” ba(a¢) °
aa9)5.
(db) _
06;
62 —
(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)
(D-214) (D-215) (D-216) (D-217)
qo =0 ) =0
au = —go sin do — ro cos do
46) =0 1 =0
a6) 9 oO) -9
uu) _ a) = 0 ae ~
a) _
av) 0
2) = cos 49sec
3) 0 Ae) ~ 9 a) =0 a) = qo cos $y sec Oy — rosin do sec Ap Ae) = qo sin Po sec Mp tan By + 79 cos go sec Optan I Ae) 0
(D-218) (D-219)
(D-220)
(D-221) (D-222)
(D-223) (D-224)
(D-225) (D-226) (D-227)
(D-228)
(D-229)
(D-231)
(D-232) (D-233) (D-234) (D-235) (D-236) (D-237)
63
a ~° y=
Fe 79 8 0 18) 2 9
D.2.13
cal
Altitude rate derivatives.—
= sin 99 cos Go cos ao — sin % cos 9 sin Bp — cos $0 cos 9 cos Bg sin ag 4— = —Vo(cos fo sin ag sin 09 + cos Bo cos ag cos go cos 89)
aq = —Vo(sin Bo cos ag sin Op + cos Go sin ¢o cos Og — sin Jo Sin aig cos $o cos Op)
>= ~ = —Vo(sin Bo cos ¢o cos Ig — cos fig sin ag sin oo Cos Op)
= Vo(cos fo cos ag cos 69 + sin Jo sin do sin 9 + cos Ao sin ag cos $o sin Op)
64
(D-238) (D-239) (D-240) (D-241)
(D-242)
(D-243) (D-244) (D-245) (D-246) (D-247) (D-248) (D-249) (D-250) (D-251) (D-252) (D-253) (D-254) (D-255) (D-256)
(D-257)
D.2.14
North acceleration derivatives.—
cos Jo Cos A cos Op cos Ho + sin Bo (sin go sin A cos Wo — cos Go sin Yo) + cos Jo sin ag (cos do sin Op cos Hq + sin Po sin Yo)
Vo[ cos Bo cos a (cos go sin 89 cos Ho + sin do sin Wo) — cos Bo sin ap Cos Op cos wo] Vo[ cos Bo (sin do sin 9 cos wo — cos do sin yo) — sin Bo cos ag cos Bp cos ho
— sin Bo sin ao (cos go sin A cos Ho + sin do sin Ho)]
Vo[ sin Bo (cos go sin 49 cos Ho + sin $g sin Wo) + cos Bo sin ao (cos go sin Ho — sin do sin Op cos Ho)]
Vo( sin Go sin go cos 9 cos to — cos Bg COS ag Sin Op Cos Yo + cos Jig sin ag cos do Cos Op cos Yo)
Vo[ — cos Bo cos ag cos 8 sin Ho — sin Bo (sin do sin 9p sin Ho + cos go cos 0) — cos Bo sin ag (cos o sin Op sin Yo — sin do cos Wo)]
(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-273) (D-274)
(D-275)
65
OV = cos 9 sin Yo cos Bo cos Ao + sin Bo (cos do cos Yo + sin go sin Ho sin Yo) + cos Bo sin ag (cos dg sin 4 sin Hp — sin ¢o cos po)
oa) = Vo[ cos Go cos ag (cos go Sin Oo sin Ho — sin go cos Yo) — cos Jo sin a Cos Mp sin wo]
ee = Vo[ cos fo (cos go cos Wo + sin do sin 9 sin to) — sin Bp cos ao Cos Mp sin Wo
— sin Bo sin ao (cos go sin % sin Yo — sin go cos o)]
au = Vo[ sin Go (cos go sin 8 sin Wo — sin do cos Yo)
— cos Jo sin ao (sin do sin A sin %o + cos go cos Ho)]
aD) = Vo(sin- Gp sin $9 cos 99 sin wp — cos Bg cos ag sin Mp sin wo
+ cos J sin ag cos do cos Gp sin Wo)
7 = Vo|[ cos fo cos ap cos 9 cos Ho — sin Jo (cos go sin Yo — sin go sin Op cos yo)
aayn) _ 9
AY) _
Ox 0
Oy) _ Oy 0 Oy) _ aa 7°
+ cos Bo sin ap (cos do sin 69 cos Ho + sin Po sin Yo)]
aoyp) = 0
aay6), _~ °
(D-276)
(D-277)
(D-278)
(D-279)
(D-280)
(D-281) (D-282) (D-283) (D-284) (D-285) (D-286) (D-287)
D.3 Evaluation of the Derivatives of the Observation Variables The 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 state, time derivative of state, and control variables.
D.3.1
Longitudinal kinematic acceleration derivatives.—
O(aaye = WaGScb om cos ao Cp, + si.n a0 CL,)
A(— az
= WaGSgé on cos a9 Cp, + si: n ao C,)
r,) (ex) = WagScb om COs Ag CEH, + si: n ag CL)
(D-288) (D-289) (D-290)
66
Mee)
Mee2 )
=
. = 0
a 1
a | |- S cos a
(eVoCD
+
GCp,)
+
S sin ao (PVoCL
+ GCL)
+
al OV
{asI- cos ag (Cp, — Cy) + sin ap (CL, + Cp)} + —OOX—aT:
O(4azp,k) = Jo1m [25'(— cos ao Co+ g sin ao CLz+ )OOx>By>
O(az.k)
0¢
O(a7zk0). _
0
Gqgo cos 9
MeODehbs) go1m]
O(@z,k) _
Or
O(az) _
Oy
0
(Asvace2 — COS Qo 5° ¥o Cpa,
+4+ GSCp, )
+ sin 26 (5~SVECL=OO—hp + 5Ct,)|
ee) = i (— cos ap Cp, + sin ao CL, )
“eee
Tren (- cos do Crp, + sin ao CL,)
O(caze, n
= Jo1m las(- COS Qg Cos, + sin ao Che, y+ i OX:
|
(D-291) (D-292) (D-293) (D-294) (D-295) (D-296) (D-297) (D-298) (D-299) (D-300) (D-301)
(D-302)
D.3.2
Lateral kinematic acceleration derivatives—
O( ay) _ q5b
Op ———-2Vogom Yp
O(ayx) _ qSC
Oq
— 2Vogqm Yq
A(dyk) _ Sb
Or
7 2Vogom Yr
wv) 6) Se . =
(Sevice
+
a5cy, +
OYy OV
O(dy,k) _ il (a
=)
das ggm g5Cy, + Oa
O(ay x) _ 1
_
=)
dB ~ gom (ascv, + 9B
wee = “00s 9 cos do
“Cu
i O( e ay)
= “= sin 89 sin op
0 _= 0
(D-303) (D-304)
(D-305)
(D-306) (D-307) (D-308) (D-309) (D-310)
(D-311)
67
+ SC) Ody)
1 € an OP
Oh ~ gom 2° Vo CY 5oh
O( dy)
Ox 0
O(ay,k)
Oy
0
O(ayx)
qc
0a
O(dy,k) Ts)
2Vogom
@
qSb
2Vogom
(D-312) (D-313) (D-314) (D-315) (D-316) (D-317)
D.3.3 Z-body axis kinematic acceleration derivatives—
asl - Gin a Ch, + 608 00 CL,)
Mex) ee
-_ sin ao Cp, + cos ao CL, )
a2, ~~ gom S sin ao (pVoCp + GCp,) + S cos ao (pVoCL + GCL) - OV
Mee “ean
~ _
ela o0 -- (sin a
(Cb, Cp,
~ Ci) + cos. a0 + cos a9 Chg) +
(CL, =
+ Co)] oh
+
wm
Fe
Oe = -* cos 8 sin do
Aes) - = sin 0 cos dg
ae =9
“esu) - — = OE = 0 ae = 0
sin a (SsveCp se + SC,
+ COs a (Gsvecu ai + iSCu)]
“Ce “eo
_ Tr (sin a9 Cp, + cos a CL,) __ Tein ao Cp, + cos a9 CL)
Ae: =-— 95(sin a9 Cr,, + c08 ao CL, ) - ]
(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)
68
D.3.4
x body axis accelerometer output derivatives.—
Oar)
Op
_
~
qSb
2Vogom
(—
cos
ao
Cp,
+
sin
ao
CL, )
(— cos ag Cp, + sin ao CL, )
BV; 0 jam cos Qo Cp, + sin ao CL, ) i —|- S cos ag (pVoCpd + Coy) + S sin ao (PVoCL + GCLy)
Jom
—P —1 { a5[- cos 9 (Cp, ~ CL) + sin ao (CL, + Cb)] + =—
om
—j1—m) |as(- cos a9 Cp, + sin ao Cus) + | OaXp>
+ =
Jo1m | — COS Ag G1aia: oe + aSCp,) + sin Qo (S1 svieu sf + 1SC., |
aSé
Wagon cos Ao Cp, + sin ao CL, )
DWoggSbom
cos a9 Cp, + si. n ag CL;)
= j1om G5(— cosa Cp,, + sin ao CLs, ) + O04X6;T—
(D-333)
(D-334) (D-335) (D-336) (D-337) (D-338) (D-339) (D-340) (D-341) (D-342) (D-343) (D-344) (D-345) (D-346)
(D-347)
D.3.5
y body axis accelerometer output derivatives.—
Oay)
Sb
Op — 2Vogom Ye
Olay) _—-g@St__,
Oq
7 QVogom ~*?
Oay)
Sb
Or — 2Vogom Y
(ay) =_ ——_1 (Spvocy +50, + FOYFy )
OV
gom
(D-348) (D-349) (D-350) (D-351)
69
Adday) ~ g1om (ascy oT n a9 e
ee = — (ascv, + 5A)
(D-352)
(D-353)
"He = 0
(D-354)
“ta) = 0
(D-355)
Oe = 0
"HR = gm (35¥8Cr Gp + 950%)
((D>--35560)
oa = 0
(D-358)
ee = 0
(D-359)
Ke = Tans
(D-360)
a = Tees
(D-361)
ee = — (ascv,, + <=)
(D-362)
D.3.6 z body axis accelerometer output derivatives—
“ye _ -5
(sin a9 Cp, + cos ag CL,)
Mas) _ - (sin a9 Cr, + cos ao Ch.)
Ae) =- Tors (sin ao Cp, + cos a CL,)
Aes) _ oe E sin a9 (pVoCp + GCp,,) + 5 cos ao (pVoCL + GCL) - at
Mee) _ -— {asisin ao (Cp, — CL) + cos ao (CL, + Cp)] - oe
Mac) _ -s. [7S (sin a9 Cog + cos ao C,,) - =
= 0
i =0
Aias) _= 0a
[sin ax (55vep 7b + 1SCp,) + cos.ag (S5vec, 7 + SC, )|
(D-363)
(D-364)
(D-365)
(D-366)
(D-367)
(D-368)
(D-369)
(D-370) (D-371)
(D-372)
=Oz = 0
ve =0
Aas} = 5
(sin ap Cp, + cos ao CL;)
ae =— Teen (sin ag Cr, + cos a CL,)
ae = -— las (sin a9 Crp,, + cos a0 Chg, ) — |
|
(D-373)
(D-374)
(D-375)
(D-376)
(D-377)
D.3.7.
Normal accelerometer output derivatives.— ae _ a (sin a9 Cp, + cos ao CL,)
— _ aE (sin ap Cp, + cos a9 CL,)
Hen) _ sie (sin ag Cp, + cos a9 CL, )
*Gn) _ = [sin ao (SpVoCp + FSCp,) + cos ao (SpVoCi + FSCLy)— ra
oan) = a { a8{sin ao (Cp, — Ci) + cos a0 (CL, + Co)] - “oth
te: = _ |as'sin ao Cp, + cos ao CLg) — a
om =0
Gn) = 0 ee = 0
Mea) = — [sin on (55v8C oe + 1SCp,) + cos a9 (55vcu ve + 75C1)|
Aen) =0 ao) =0
en) = (sin ao Cp, + cos ao CL,)
a = 5 (sin ao Cp, + cos ao Ci,) a _ — @5(sin ao Cog, + cos a0 CLs, )- oe
(D-378) (D-379)
(D-380)
(D-381) (D-382) (D-383) °
(D-384)
(D-385) (D-386)
(D-387)
(D-388) (D-389)
(D-390
(D-391:
(D-392)
71
D.3.8
Derivatives of x body of gravity.—
axis
accelerometer
output
not
at
the
vehicle
center
Odeye; =
O(Saez; ) -
WaGagSb on
Waibga an
cos ag Cp, + si. n ao CL,) + P1 AC + Tozx)
cos a9 Cp, + si. n ag CL,) + 991 (Poe — 2q02r)
(D-393)
(D-394)
Gs) = Fee (- cos ag Cp, + sin ag CL,) + ~(Pozs — 2ror,)
*O((aGze,si) = —oi—mt |- S'cos ag (pVoCp + FCdy) + S sin ao (pVoCL + GCL, ) + | OX
eO(saszj) — 1 {ast- cosa (Op, ~ Ch) + sin ao (Ct, + Cp) + SO%xc}
dO(aBz,i) = Jo1m ie
0 oeLt = 9
cos a9 Cp, + sin ag Cz) + a OaXpy
0 Csix,)t =
e0 e)zt =0
OO(aez,;)
0 CeaZt)
= Jo1m
=0
[- COS Qg
(5 SVACDDOhp= + 75Cp,
) + si. n ag
(S1 svgcr
DO3hp6 + SC, )|
O(aGz,4 =0
“0 Gaerie = 0
aOarqi). q2o
a(aee) = -Ye
0 ie,Lt ) = WaGSgE on cos a Cp, + si: n ag CL,)
re] aAy j ) = WaGScb om cos ao Cp, + si.n ag Cis)
(D-395)
(D-396)
(D-397)
(D-398)
(D-399) (D-400)
(D-401)
(D-402)
(D-403)
(D-404)
(D-405)
(D-406) (D-407)
(D-408)
(D-409)
O8O((45azz,,i) = —jao=m(- G5'(— cos ag CDs, + sin ag CLs,
O6X,T
(D-410)
D.3.9
Derivatives of of gravity.—
y body
axis accelerometer
output
not at vehicle
center
=
Cy, — 910 (Poy — Gory)
(D-411)
O(adyq,i)
DVGoSgEom Cy, + a10 (poty + Tozy)
O(aayr,i)
DVgoGSgbome + i70 (O79 _ 2royy)
Meus) 1 (ascy, 4 2) O(oaoy; ) = Jo1m (spvocy + q_ SCy, + SOYa: h
Hod9ua8s) = g1goomm ((gca\,sPc%vs. ++4 2“O9a)p
O(aay6,i)
a. O(ay,i) _ ap 9 O(ay,i) _
Dee O(a, ; Ga1m (581VR
CY 50p + 4_ 8Cys)
O(ay i)
Oz
0
O(ay i)
Oy
0
O(ay,i) yy
Op
Jo
O(aaygi) _
O(ay,i) _ ty
Or
Jo
O(ay,i)
GSé
0a
2Vogom °°
O(ddyB,i) 2Vqobgom~_*, 6
O(Saye,i 2 =
(a_ scv,
+ OiY:)
(D-412) (D-413) (D-414) (D-415) (D-416) (D-417) (D-418) (D-419) (D-420) (D-421) (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(aOzpi)
O(az)
Oq
Olazi) Or
_——-2VoggGSobm (si.n a9 Cp, + cos ag CL,) — P 1 a — Totz)
_
GSE (si.n ag Cp, + cos ag CL,) - 91 00 — ToYz)
~ ~ 2Vogom
_
gSb
—- BVogom
(si. n a9
Cp,
+
cos ag
CL,.)
+
1 go0
PO"?
+
goyz)
(D-429) (D-430) (D-431)
73
azn wv Ges)
=
-—om
[s sin a
(pVoCp
+ GCp,,) +
S cos ao (pVoCL
+ GCL, ) -
OZr
OV
ee) O(a, o ; ) =
-— tom1
{as[sin ao (Cp, — CL) + cos ag (CL, + Cp)] [7fSo'.sin ao Cp, + cos ag Cig) — a OZ
Oa
O(az;)
dg
O(@az6,i) _ °
O(az,i) Oy
O(Sae,; )
~Fom1
O(az,i)
Ox 0(az,:)
dy
[si. n Qo ($1 5veCp
+ SC] DOhp + iSCo,) + COs Ao (5=SV¢ CL a=Ohp
O(az,i)
Yz
(aosps) _ tJoe
og
Jo
O(aaze,i) =_ 9
Ge)
it (sin a9 Cp, + cos ag CL)
0 i Zt )
WaqcSobm (si:n ao Cp, + COS Qo Ci,)
O(Da6z;,)
~ Jo1m [2[5_'.(.sin Qo CDs, + cos ao CLs.) — FOZEy
(D-432) (D-433) (D-434) (D-435) (D-436) (D-437) (D-438) (D-439) (D-440) (D-441) (D-442) (D-443) (D-444) (D-445)
(D-446)
D.3.11
Derivatives of normal of gravity.—
accelerometer
output
not
at vehicle
center
O(aan; ) = WagaSeb in. ag Cp, + cos ag Ch.) + 901 (PO — Totz)
Oloaen; ) _ WeGoSoEm (sin ao Cp, + cos ag CL,) +e~ (24022 — ToYz)
O(oann3 ) = WeaeSbn (sin ao Cp, + cos ag CL,.) — 1 ue + qoyz)
O((aonrj )
O(<aan; )
=
=
Jo1m
Jo1m=
Is sin ao (pVoCp + GCp,,) + S cos ag
{aS{sin ao (Cp, — CL) + cos a (CL,
(PVoCL
+ Cp)]
+ GCL,,)
Fa}—
OLZy
Oa
-
wOLy
OV
74 |
(D-447) (D-448) (D-449) ~» (D-450) (D-451)
= ) )
n,t
Jo1m [7_s (si.n ao Cr, + C08 Ap CL) — a OZ:
“ O(a an,ei) _= 0
a9. O(ani) _
n O(at n, i) = 9
Oy
O(adnhi) _ go1m
O(an,i) _
dz
O(an,i) _
Oy
O(an,i) _
Op
O(an,i) _
0
0
_Yz 9o
Lz
Og ~ Jo
ar O(an,i) _
[si. n ao
(5
5°
2
Vo Cp
op
Dh
+
_
)
@SCp, } +
cos
ao
(3
;
55Vo CL
op Oh
+
5Ct,)|
Aen) = —
7) a nt ) = WegqSob
(sin ao Cb, + COS Xp Che)
*i.n ao CD, + Cos ao Ci)
O(onn ) = jm [7459s(}sin ao CDs, + cos ao CL, ) _ aOZi
(D-452) (D-453) (D-454) (D-455) (D-456) (D-457) (D-458) (D-459) (D-460) (D-461) (D-462) (D-463) (D-464)
D.3.12
Load factor derivatives.—
O(n) _ _G5b_,
ap 2Vogm'”
An) _ _gSet_,
dg
2Vogm“4
O(n)
gb
Or
2Vogm
Bin) 95g, a0 ) = gm1 cSeVoCr + GSCLy)
a0(an) _ 48gm
ag
gmc
O(n) _
0g
O(ano) _ °
(D-465) (D-466) (D-467) (D-468) (D-469) (D-470) (D-471)
(D-472)
75
op ~°
OO(hn) —> 9m1 (fslvac
O(n) _
Ox
O(n) _
dy
0
O(n) gSe
Oa ~~ 2Vogm Le
An) _ _gSb_,
0B Wogm ve
DO(bn) ggSme's
oOhp + SC.)
D.3.13 Speed of sound derivatives—
O(a) _
dp
0
O(a) _
Oq
y
oO(ra) _79
aO(va) = °
GO(aa) _ 9
oO(pa) _~°
Oa(ea) _
3O(6a0) _7°
AOa((apa)) _ 9
0.770
aOTT
Dh ~— poTo(1.4 po/poTo)!/? Oh O(a) _
dx
O(a)
Oy
y
Od(a)a?
(4) _ 9
Op
(D-473) (D-474) (D-475)
(D-476)
(D-477) (D-478) (D-479)
(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)
0O(a5) ,_= °
D.3.14 Mach number derivatives.
Oa 0(M)
OB
O(M)
O¢
0(M)
00
Od(bM)
a(M) _ _Vo
0.7po
OT
Oh
a? poTo(1.4 po/poTo)1/? Oh
0(M)
Ox
0(M) Oy
O(M)
Oa
O(M)
op a(M)
D.3.15
Reynolds number derivatives—
“O(aRep) 7_ °
O(Re)
oq
0
“O(oRer) ~°
(D-494)
(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-510) (D-511) (D-512)
77
(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-525) (D-526) (D-527) (D-528) (D-529) (D-530) (D-531) (D-532) (D-533)
O(Rdeh’) ~_ Vpow 9Ohp _ ppVeo OOhp
Re) =0
aoe =0
oe) =0
Aa(R0eB!)
_ 9
-0
D.3.17 Dynamic pressure derivatives.— oo = 0 oe =0
ty =0
ao = pVo aD 0 a 0
0 ao =0 oo =0
oa) = Mi se
ey = 0 a =0 a =0
se =0
we 0
(D-534)
(D-535)
(D-536)
(D-537)
(D-538)
(D-539)
(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-554)
79
D.3.18 Impact pressure derivatives.—
p= 0 14Pa M(1.0 + 0.2M2)2
(Ge) _
Ov
Pa
5.76M2
2.5
a [2-4 (Gar - =a)
+ 3.0M? ( 565M.?76M—208 i
(5.69M.?21—6M0.8)?
(M < 1.0)
(M > 1.0)
Bae) 9
oR 0
(D-555) (D-556) (D-557)
(D-558)
(D-559)
(D-560)
(D-561) (D-562)
(D-563)
Ae) _ |
Cee)o!-
Me =o
op
oR =
80 —
5.6M7?6M—?0.8 i
| + 3.0M?
( 5.65M.?76M—? 0.8 \"
9.216M
(5.6M? — 0.8)?
Oa Oh
(D-564)
(M > 1.0)
(D-565) (D-566) (D-567) (D-568)
(D-569)
D.3.19
Mach meter
Ad/Pa) _ 4
Op
calibration
ratio
derivatives.
O(dc/Pa)
= 0
0q
O(dc/Pa) _ 4
Or
.
14 M(1.0 + 0.2M?)25
(M < 1.0)
oOA(dgeaeiv/fPeaa)s
gia1 [24M
(285.76M—
2
0.8
)
2.5
-
| + 3.0M 2f(_5i.76M-?_o\a")" (56M__?9.21—6oM8|
Ade/Pa) _ 4
da
0(4c/Pa)
ag
= )
O(qc/Da)
a¢
= 0
0(4-3/0Pa) -0
A(de/Pa) _ 4
an
— LffoM(1.0 + 0.2M?)?5 92
(M210) (M < 1.0)
a8(dcR/Pa.)
-3/2 {24M \( GM5.?76M—208 y
O(qc/Pa) = 0
Ox
O(dc/Pa) _ 4
Oy
OOd(cdd//PPaaa))”
_
_
4
9
ag
O(4c/Da) -0
06;
+ 30M 2((_55.876M—2_ss\)** (6M_?9.216M—_o_]sy| o@na “4 21.0)
(D-570) (D-571)
(D-572)
(D-573)
(D-574) (D-575) (D-576) (D-577)
(D-578)
(D-579)
(D-580) (D-581) (D-582) (D-583)
(D-584)
81
D.3.20
Total
temperature
am) 6
5
derivatives.—
an) _
a) _
1) _ 0-4 7M
a =0 an) _
ae ~ {10 +0.2M~ ° a Feces ual or oh) = 0 an) _
a) =0 = 9
— =0
D.3.21 Flightpath angle derivatives— a -0
oo = 0 29) =0
ra ~ a at) =0 "ye -9
CAME
ne =9
7) _ 9
(D-585) (D-586) (D-587) (D-588) (D-589) (D-590)
(D-591) (D-592) (D-593)
(D-594) (D-595) (D-596)
(D-597) (D-598) (D-599)
(D-600) (D-601) (D-602) (D-603) (D-604)
7) 2
ay) =0 oo) = 0
a =0
oo) = 9 oP =0
— =
ae =0
|
AT a
D.3.22 Flightpath acceleration derivatives.—
(D-605) (D-606) (D-607)
(D-608)
(D-609) (D-610)
(D-611)
(D-612)
(D-613) (D-614) (D-615) (D-616) (D-617) (D-618) (D-619) (D-620) (D-621) (D-622) (D-623) (D-624)
(D-625)
83
O(fpa) _ 4
Ofpa) _ O(ipa) = 0
(D-626)
(D-627)
(D-628)
atah
Weggsbo [sin 69(— cos ag Cp, + si; n ao CL.)
— sin go cos 6 Cy, + cos do cos Op (sin ag Cp, + COs ao CL,)]
ce
Fe0e9e0 lsin 99 (— cos ag Cp, + sin ao C,) — sin do cos 9 Cy,
+ cos Gg Cos Op (sin ag Cp, + cos ao C1,)]
“
se 090 [sin 0 (— cosap Cp, + sin ag C,.) ~ sin do cos 0p Cy,
a(h)
+
{sin
cos do
[-s
cos
Ao (sin ap
(pVoCp
Cp, +
+ aC
cos
ag CL,
Ss
|
VoCL + GCu,+ )OX?
(D-629) (D-630) (D-631)
— sin do cos A (Spvecy + G@SCy, + OaV)
+ cos dg cos I E sin Qo (pVoC/p + GCp,)
+ S cos ao (pVoCL + GCL, ) — a OZ \
ach)
pe1al si. n 09 [-2_s cos a (Cp, — CL) + G_5a'esin ag (CL, + Cp) + OiX a
(D-632)
— sin dg cos 89 (ascv, + oOa) + cos go cos Ip las sin Qo (Cp, — CL) + GS cos ag (CL, + Cp) — 5 i
(D-633)
om1 [si,n 60 (-7_s cos ao Cp, + GaS:sin ao Cus + aOXHT) — si. n do cos Ap (a_ scv, + aOYRr)
+ cos go cos 8 (as sin a9 Cp, + GS cos ag Cig + |Z
(D-634)
— Ay, COS Go COS Ay + a,4 | SiN Pp Cos Oo Az COS Ay + dy, Sin Oo sin do + Az, COS gg sin Io
= 0
(D-635) (D-636) (D-637)
84
mOhn)
—1s= ,{ sin 6 |- COS Ag (5l5ova8nCp aOrP ,+ _4Cps) + siinnaaoo (5S5V%C.Ch D2h ++ aqS8CCi.,.))
— sin ¢g cos 6p (s1 svacy oaOhfp + aSCy, )
+ cos $9 cos Ap [sin Qo (5=SV¢Cp ce + 15Cp,)
si = 0 z
an) _0
7]
+ COs Ag (52 sv3 CL oO=ph + SC, )| \
(D-638)
(D-639) (D-640)
ae = lst
09o
9 (— cos ag Cp, + sin ao CL.) — sin do c08 9 Cy,
+ cos ¢o cos 9 (sin a9 Cp, + cosa CL, )]
(D-641)
= esa
oJo
90 (— cos ao Cp, + sin ao CL,) — sin do cos Ap Cy,
+ cos go cos 9 (sin ao Co, + cos ag Ci, yj
(D-642)
—a(h = —a1 \ sin 9 las(- cos Qo CDs, + sin ao CL,,) + FaXey — sin do cos Oy (a_ scv, + 5dYEy)
+ cos do cos 09 [75(sin Qo Cos, + COs Qo CLs, ) - =| \
(D-643)
D.3.24
Specific energy derivatives—
us = 0
(D-644)
oe =0
oe) =0
sea _ =
ee =0 ae =0
(D-645)
(D-646)
(D-647)
(D-648) (D-649)
a =0
fs) =0
AS = 0
(D-650)
|
(D-651)
(D-652)
Ae) =1
(D-653)
85
D.3.25 Specific power derivatives.— 86
(D-654) (D-655) (D-656) (D-657) (D-658)
(D-659) (D-660) (D-661) (D-662)
(D-663)
(D-664) (D-665) (D-666) (D-667) (D-668) (D-669) (D-670) (D-671) (D-672) (D-673) (D-674)
D.3.26
Normal force derivatives.—
oO(SN
“OaNq)
= iGSb (cos ag CL, + sin ag Cp, )
= SGV1iS7,E (608 ao CL, + sin ao Cp,)
a O(N = TqAb (cos ag CL, + sin ao Cp,)
aO(”N) = S$[cos ao (pVoCL + GCL) + sin ao (pPVoCDd + GCp,)]
a) = G5 (cos ap CL, + sin ao Cp, — sin ao Cl, + cos ag Crp)
— O(N = G_ S(cosag CL, + si. n ag Cog)
AN) _ 9
ag
“aONN3))0
_
_
"
OY
ON)
Oh
_
lis» Op
=S$ [c0s ao (5¥5 CL d=h + Cis)
+ si. n a9 (5 Ve Cp dO=ph> + iC, )]
O(N) _
Oz 0
“OaN)y _~°
aO()N = BGSyE (0820 CL, + sin ao Cp,)
rO(rN = ~oq—vb; (COs a9 Ci, + sinag Cp. a)
N = G5(cosa CL,, + sin a Cp, )
(D-675)
(D-676) (D-677) (D-678) (D-679) (D-680)
(D-681)
(D-682) (D-683) (D-684) (D-685) (D-686) (D-687) (D-688) (D-689)
(D-690)
oO(eA _ AqSb (— sin ao CL, + cos ao Cp,)
~ O(A = TGAS (— sin ag CL, + cos a Cp,)
(A) =_ ~gb—(-sinao CL, + cos ap Cp, )
Or
2Vo
(D-691) (D-692)
(D-693)
87
By = Sl-sinao (pVoCL + FCiy) + c08 a0 (eVeCD + FCdy)]
“4 = 95(—sinap CL, + cos ao Cp, — cos ag Cy — sin ag Cp)
“S = GS(—sin ao C.., + 08.00 Cog)
a4) 9
a) -o
ae =9
AA) = 5 [-sinac (Svecr $4 Cin) + cosa (5VECD $F + aCd,))
24) =0
oe = 0
a4) = e- sin ap CL, + cos ag Crp,)
a“ = e- sin a0 CL, + cos a0 Cp,)
"ae = GS(—sinao CLs, + COS Qo Cp,,)
(D-694)
(D-695)
(D-696)
(D-697) (D-698)
(D-699)
(D-700)
(D-701)
(D-702)
(D-703)
(D-704)
(D-705)
“A = ie 9
a) ~9
Me) — cos a c08 fo
a) = —Vosin a cos Bo
we = —Vocos ag sin So
a =0
4) -9
Au) _ 4
(D-706) (D-707)
(D-708)
(D-709)
(D-710)
(D-711)
(D-712) (D-713)
(D-714)
Dh
0
Oapu)_
Ae = 0
D.3.29 y body axis rate derivatives—
22) = 9
eo
a0,oU) = sin fo
AD) = Vacos fo
2) -_
ae)
Se =
2) _ 9 20) <9
0tae)p =
=
(D-715) (D-716) (D-717) (D-718) (D-719) (D-720)
(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.—
oh = 0
(D-736)
“eo =0
(D-737)
ca = 0
(D-738)
a) = sin ag cosBo
(D-739)
ow) = Voc0s ag cos fo
ue = —Vosinag sinBo
(D-740) (D-741)
“) =0
(D-742)
ae) = 0
(D-743)
—aaa=0
(D-744) (D-745)
(D-746)
aoaaw) 6
Au) 4
5OB =
(D-747) (D-748) (D-749) (D-750)
qSb (— cos ao Cp, + sin ao CL,)
~ 2Vom
qSc (— cos ag Cp, + sin ag Ci,) ~ Vosin ag cos Bo
~ 2Vom
GSb (— cos ao Cp, + sin ao CL,) + Vosin Bo
2Vom
=>1 {s{- cos a (pVoCp + GCpy,) + si.n ao (pVoCL + GCLy)] + aOOXrV}
+ro sin Bo — go sin ag cos Bo
(D-751) (D-752) (D-753)
(D-754)
90
.
Ox
Od(%a) = ~ lzs(- cos ag Cp, + sinag CL, + sin ao Cp + cos ap CL) + <2
— goVo COS Ag cos Bg
3O(Ba) = =1 |as(- cos ag Cr+ , sin ao Cus) + aaOXBeR
+ roVpo cos Bo + gGoVo cos ag sin Bo
a =0
) = ~9c0s 9
a =0 sO(ou =>S - COS Qo (5 Vo Cp FO=hp + Cb, ) + si. n Qo (v1 ec
a)x =0 om =0
¥y
aOet) = sqoScmo cos Qo Cp, + si. n ao CL, ) 1OG) = ogSm b CO8 Ag Co, + si. n ag Ci;) On(te i— g5(— cos ao Cp,, + sin a0 CL,,) + | Ox
5AO°hp + Cin)
(D-755)
(D-756)
(D-757)
(D-758)
(D-759)
(D-760)
(D-761) (D-762)
(D-763)
(D-764)
(D-765)
D.3.32
y body axis acceleration derivatives.—
sa =
Cy, + Vosin ao cos Bo
‘O(o = MGoSmt Cy,
x) = a0 Cy, — Vo cos ag cos Bo ea(st) = = | S(eVacy +qCy,)+ O=aYy|y
+ po Sin ap COS Bo — To COS Go COS Bo
a0) = - (ascv, + We)
Aaop) = 1~ (950%, + =55B)
+ poVo cos ag Cos Bp + ToVo Sin ag cos Bo
— poVosin. a si. n 8p — ToVo cos ag si. n Bo
a = gcos 8 cos do
“ = —gsin 8% sin do
~ =0
(D-766)
(D-767)
(D-768)
(D-769) (D-770) (D-771)
(D-772) (D-773)
(D-774)
91
OOOO(ovh6):) _7° mS$ (5fl%o. CY oORp +ICy, )
Ov) |
Oy = 0
ao) _ gse
aas)a __g3SVbam~ Ye
agp ~ 2Vom
Ys
Oo)
1
OYr
a6; =m (a8cy, 3)
(D-775) (D-776) (D-777) (D-778) (D-779)
(D-780)
D.3.33 z body axis acceleration derivatives—
a
7Gb0
= DgVeSm
~ si. n ag Cp, — cos ag Cy) — Vo si. n Bo sin Qo Cp, — cos ag CL,) + Vo cos ag cos Bo
= aqSobm (— sin a9 Cp, — cos a CL,.) =a 1 | —S sin a9 (pVoCp + GCp,,) — S cos ag (pVoCL + GCL) + | OALVZy
+ 40 COS Gq COS Bo — po sin Bo = —1 ie sin ag Cp, — cos ao CL, — cosag Cp + sin ag CL) + lO7LaZlT
- Vo sin A Cos Bo 33 =m1 f_g5(— si. n ag Cp, — cos ag Chg) + 5 A
— qoVo cos ag sin Bo — poVo cos Bo
= —g cos Go sin do
— = —gsin % cos gg
= 0 Ae) 2 - sin ag € VeCp e + Cs) — cos ag (Svc. oe + i )|
=0 =0
= 7
= a
(—sin ap Cp, — cos ag CL, )
(— sinag Cp, — cosag Cus)
(D-781) (D-782)
(D-783)
(D-784)
(D-785) (D-786) (D-787) (D-788) (D-789) (D-790) (D-791) (D-792) (D-793) (D-794)
92
1 m
g5(—
sin ao CDs,
— COS Qo
CL.)
+
OLT 06;
D.3.34
Angle-of-attack sensor output derivatives.—
O(a) Yor
dp
Wo
Ooi) _ La
dg Vo
O(a,i) _
or
O(a,i) _ Go®a
ave
Oa) _
da 1
O(aap.i)
O(aae) _9
Aaaj0 ) _ O(aap,:) _9 O(aah )
0(a,i) _
Ox 0
O(a) _
Oy
0
daO(a,i) _
O(a) 0
0B
36, 9 O(a.) _
— PoYa
A(8:) _ _ 20
Op
Vo
(8,1)
Oq
0
ABs) _ 2g
Or
7 Vo
(D-795)
(D-796) (D-797) (D-798) (D-799) (D-800) (D-801) (D-802) (D-803) (D-804) (D-805) (D-806) (D-807) (D-808) (D-809)
(D-810)
(D-811)
(D-812)
(D-813)
93
O(Bi) _
ave
_ ToXe V—o Pozs
“ee 9
*od 4
ne =0
ea) =0
ee 0
6) 0
AB.) =0 ae -9
aes) =0
oe =0
ee) =9
Ze
a 9
ahi) =0
Ans) 4
ha) 9
ae 9
ne = ~yn cos do cos O + zp sin $9 cos %
ii) = 2, c0s Oy + yp sin bo sin 89 + 2n cos do sin Bo Ahi) _ 9
(D-814)
(D-815)
(D-816) (D-817) (D-818)
(D-819) (D-820)
(D-821) (D-822)
(D-823)
(D-824)
(D-825)
(D-826) (D-827)
(D-828)
(D-829)
(D-830)
(D-831)
(D-832)
(D-833) (D-834)
oh
O(h,) _
Oz = 0
Ohi) _
Oy =0
OAhhdiia)) _=_ O
Op
(h06i,) _9
(D-835) (D-836) (D-837) (D-838) (D-839)
(D-840)
D.3.37
Altitude rate sensor output derivatives.—
Oha) _
Op
0
Ahi) _
oq 0
Ohi) _
Or
0
Oa(vha) _~°
ACh) _
da 0
O(aap) _ °
oe = $(y; sin ¢o cos
+ 2 cos ¢o cos %) + A(y; cos do sin Oo — z sin do sin M)
mea) = —O(2;, sin 00 — yj, sin $y cosIp — 2), cos do COs My)
+ (yj, cos do sin Oo — 2, sin do sin Oo)
O(ahps) _= 9
OhO)h”_
Ohi) _
Ox = 0
Ohi) _
Oy
0
0(ha)a_
Ohi) _ 0
ap
(D-841) (D-842) (D-843) (D-844) (D-845) (D-846)
(D-847)
(D-848)
(D-849) (D-850) (D-851) (D-852) (D-853)
(D-854)
95
D.3.39
1) cosa Stability axis roll rate derivatives—
2Ps00— )s.na
O(ps)
0a
MP)
op
D.3.40
Stability axis pitch rate derivatives—
O(qs)
Op
=
0
JCA)
0q
=1
O(Gs)
Or
=
0
O(4s)
OV
=0
(Gs)
da
=0
O(9s)
op
=0
(D-874) (D-875) (D-876) (D-877) (D-878) (D-879) (D-880) (D-881) (D-882) (D-883) (D-884) (D-885) (D-886) (D-887)
(D-888)
(D-889) (D-890) (D-891) (D-892) (D-893)
(D-894)
97
Od = 0
82a _ myAae)= =9o0 =o
T0EB =o
D.3.41 Stability axis yaw rate derivatives.— —— = —sinag
98
(D-895) (D-896) (D-897) (D-898) (D-899) (D-900) (D-901) (D-902)
(D-903)
(D-904) (D-905) (D-906) (D-907) (D-908) (D-909) (D-910) (D-911) (D-912) (D-913) (D-914) (D-915)
(D-916)
(D-917)
(D-918)
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