Aircraft Equations of Motion:
Flight Path Computation!
Robert Stengel, Aircraft Flight Dynamics, MAE 331,
2016
•!
•!
•!
•!
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•!
Learning Objectives
How is a rotating reference frame described in an
inertial reference frame?
Is the transformation singular?
Euler Angles vs. quaternions
What adjustments must be made to expressions
for forces and moments in a non-inertial frame?
How are the 6-DOF equations implemented in a
computer?
Aerodynamic damping effects
Reading:!
Flight Dynamics!
161-180!
Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.
http://www.princeton.edu/~stengel/MAE331.html
http://www.princeton.edu/~stengel/FlightDynamics.html
1
Assignment #5"
due: November 11, 2016!
•! Takeoff from Princeton Airport, fly over
Princeton and Lake Carnegie, and land at
Princeton Airport!
•! “HotSeat” cockpit simulation of the Cessna 172!
•! 3- and 4-member teams; each member
successfully flies the circuit!
•! Individual flight testing reports!
2
Review Questions!
!! What are the differences between NTV, NTI, LTV, and LTI
ordinary differential equations?!
!! What good is a rotation matrix?!
!! What is a “3-2-1” rotation sequence?!
!! What are the pros and cons of Euler angles for representing
rotational attitude(i.e., position)?!
!! What is a “cross-product-equivalent” matrix?!
!! How is the “inertia matrix” defined?!
!! What is the difficulty in transforming from an inertial to a
body frame of reference?!
!! How did increasing engine power affect the stability and
Control of World War II airplanes?!
3
Euler Angle Rates!
4
Euler-Angle Rates
and Body-Axis Rates
Body-axis
angular rate
vector
(orthogonal)
" !x
$
!B = $ !y
$
$# ! z
Euler angles form
a non-orthogonal
vector
Euler-angle
rate vector
is not
orthogonal
%
" p %
'
'
$
' =$ q '
'
$ r '
&
'& B #
% " (
'
*
!=' # *
' $ *
&
)
% "! ( % ,
* ' x
'
! = ' #! * + ' , y
!
' $! * '
) '& , z
&
(
*
*
*
*) I
5
Relationship Between Euler-Angle
Rates and Body-Axis Rates
•! "˙ is measured in the Inertial Frame
•! "˙ is measured in Intermediate Frame #1
•! "˙ is measured in Intermediate Frame #2
•! ... which is
! p $ ! 1
0
& #
#
# q & = # 0 cos )
# r & # 0 'sin )
% "
"
! '! $
! p $
!
&
#
&
#
#
B
# q & = I3 # 0 & + H2 #
#" 0 &%
# r &
#"
%
"
'sin ( $!# )! $&
&
!
sin ) cos( &# (! & = LBI +
cos ) cos( &%#" *! &%
0
(!
0
$
! 0 $
&
&
B 2#
& + H 2 H1 # 0 &
&%
#" )! &%
Can the inversion
become singular?
What does this mean?
Inverse transformation [(.)-1 " (.)T]
$
&
&
&
%
!!
"!
#!
' $
) & 1 sin ! tan "
)=& 0
cos !
) &
( % 0 sin ! sec"
cos ! tan " '$ p
)&
*sin ! )& q
cos ! sec" )(&% r
'
) I
) = LB+ B
)
(
6
Euler-Angle Rates and Body-Axis Rates
7
Avoiding the Euler Angle Singularity
at ! = ±90°
!! Alternatives to Euler angles
-! Direction cosine (rotation) matrix
-! Quaternions
Propagation of direction cosine matrix (9 parameters)
! IB h B = !
" I H IB h B
H
# 0
!r ( t ) q ( t )
%
! BI ( t ) = ! "" B ( t ) H BI ( t ) = ! % r ( t )
H
0
! p (t )
%
%$ !q ( t ) p ( t ) 0 ( t )
Consequently
H BI ( 0 ) = H BI (!0 ," 0 ,# 0 )
&
(
( H BI ( t )
(
(' B
8
Avoiding the Euler Angle Singularity
at ! = ±90°
Propagation of quaternion vector: single rotation
from inertial to body frame (4 parameters)
!! Rotation from one axis
system, I, to another, B,
represented by
!! Orientation of axis vector
about which the rotation
occurs (3 parameters of a unit
vector, a1, a2, and a3)
!! Magnitude of the rotation
angle, ", rad
9
Checklist!
"!Are the components of the Euler Angle rate
vector orthogonal to each other?!
"!Is the inverse of the transformation from
Euler Angle rates to body-axis rates the
transpose of the matrix?!
"!What complication does the inverse
transformation introduce?!
10
Euler Rotation of a Vector
Rotation of a vector to an arbitrary new orientation
can be expressed as a single rotation about an axis
at the vector’s base
! a1 $
&
#
a ! # a2 & :
# a &
" 3 %
Orientation of rotation axis
in reference frame
' : Rotation angle
11
Euler’s Rotation Theorem
Vector transformation
involves 3 components
rB = H BI rI
= ( aT rI ) a + "# rI ! ( aT rI ) a $% cos & + sin & ( rI ' a )
= cos & rI + (1! cos & ) ( aT rI ) a ! sin & ( a ' rI )
! a1 $
&
#
a ! # a2 & :
# a &
" 3 %
Orientation of rotation axis
in reference frame
' : Rotation angle
12
Rotation Matrix
Derived from
Euler’s Formula
rB = H BI rI = cos ! rI + (1" cos ! ) ( aT rI ) a " sin ! ( a! rI )
Identity
( a r ) a = ( aa ) r
T
I
T
I
Rotation matrix
H BI = cos ! I 3 + (1" cos ! ) aaT " sin ! a!
13
Quaternion Derived from Euler
Rotation Angle and Orientation
Quaternion vector
4 parameters based on Euler’s formula
!
#
#
q=#
#
#"
q1 $
&
q2 & ! q 3 $ ! sin ( ' 2 ) a
&=#
!#
q3 & # q4 & # cos ( ' 2 )
% "
& "
q4 &
%
!
( a1
#
$ # sin ( ' 2 ) * a
* 2
&=#
* a3
& #
)
% #
cos ( ' 2 )
#"
+
,
$
&
&
&
&
&
&%
( 4 ! 1)
4-parameter representation of 3 parameters;
hence, a constraint must be satisfied
q T q = q12 + q2 2 + q32 + q4 2
= sin 2 ( ! 2 ) + cos 2 ( ! 2 ) = 1
14
Rotation Matrix Expressed with
Quaternion
From Euler’s formula
H BI = "# q4 2 ! ( q 3 T q 3 ) $% I 3 + 2q 3q 3T ! 2q4 q! 3
Rotation matrix from quaternion
H BI =
" q2 ! q2 ! q2 + q2
2
3
4
$ 1
$ 2 ( q1q2 ! q3q4 )
$
$ 2 ( q1q3 + q2 q4 )
#
2 ( q1q2 + q3q4 )
!q12 + q22 ! q32 + q42
2 ( q2 q3 ! q1q4 )
2 ( q1q3 ! q2 q4 )
%
'
'
'
'
&
2 ( q2 q3 + q1q4 )
!q12 ! q22 + q32 + q42
15
Checklist!
"!What is an “Euler Rotation”?!
"!Why would we use a quaternion vector to
express angular attitude instead of an Euler
Angle vector? !
"!How many components does a quaternion
vector have?!
16
Quaternion Expressed from
Elements of Rotation Matrix
Initialize q(0) from Direction Cosine Matrix or Euler Angles
" h11 ( = cos !11 ) h12
$
H BI ( 0 ) = $
h21
h22
$
h31
h32
#
q4 ( 0 ) =
h13 %
'
h23 ' = H BI ((0 ,) 0 ,* 0 )
h33 '
&
1
1+ h11 ( 0 ) + h22 ( 0 ) + h33 ( 0 )
2
Assuming that q4 " 0
!
! q (0) $
1
#
&
#
1 #
q 3 ( 0 ) ! # q2 ( 0 ) & =
& 4q4 ( 0 ) #
#
q
0
(
)
#
&%
#" 3
"
!" h23 ( 0 ) ' h32 ( 0 ) $% $
&
!" h31 ( 0 ) ' h13 ( 0 ) $% &
&
!" h12 ( 0 ) ' h21 ( 0 ) $% &
%
17
Quaternion Vector Kinematics
d ! q 3 $ 1 ! q4 ' B ( '" B q 3
&= #
q! = #
T
dt # q4 & 2 #
('
'
q3
B
"
%
"
$
&
&%
( 4 ! 1)
Differential equation is linear in either q or #B
!
#
#
#
#
#"
! 0
q!1 $
#
&
q!2 & 1 # (' z
= #
q! 3 & 2 # ' y
&
# ('
q! 4 &
x
%
"
'z
0
(' x
(' y
(' y ' x $ !
& #
'x 'y & #
&
0
'z & #
#
&
#
(' z 0
%B "
q1 $
&
q2 &
q3 &
&
q4 &
%
18
Propagate Quaternion Vector
Using Body-Axis Angular Rates
! p (t ) q (t ) r (t ) # ! ! % % % #
y
z
"
$ " x
$
!
#
dq ( t ) #
=#
dt
#
#
#"
$ !
& #
& #
& #
& #
& #
&% B #"
! 0
r ( t ) 'q ( t ) p ( t )
q!1 ( t ) $
&
#
0
p (t ) q (t )
q!2 ( t ) & 1 # 'r ( t )
&= #
0
r (t )
q! 3 ( t ) & 2 # q ( t ) ' p ( t )
# ' p ( t ) 'q ( t ) 'r ( t )
0
q! 4 ( t ) &&
#"
%
q1 ( t ) $
&
q2 ( t ) &
&
q3 ( t ) &
q4 ( t ) &&
%
Digital integration to compute q(tk)
q int ( t k ) = q ( t k!1 ) +
dq (" )
#t dt d"
k!1
tk
19
Euler Angles Derived from Quaternion
{
} ')
{
}
$
2
2
$
'
$ ! ' & atan2 2 ( q1q4 + q2 q3 ) , %1* 2 ( q1 + q2 ) (
&
) &
sin *1 $% 2 ( q2 q4 * q1q3 ) '(
& " )=&
& # ) &
%
( & atan2 2 ( q3q4 + q1q2 ) , $1* 2 ( q2 2 + q32 ) '
%
(
%
)
)
)
)
(
•! atan2: generalized arctangent algorithm, 2 arguments
–! returns angle in proper quadrant
–! avoids dividing by zero
–! has various definitions, e.g., (MATLAB)
*
" y%
tan !1 $ '
,
# x&
,
,,
!1 " y %
!1 " y %
atan2 ( y, x ) = + ( + tan $ ' , ! ( + tan $ '
# x&
# x&
,
,
( 2 , !( 2
,
0
,-
if x > 0
if x < 0 and y ) 0, < 0
if x = 0 and y > 0, < 0
if x = 0 and y = 0
20
Checklist!
"!Can propagation of quaternion become
singular?!
"!Can quaternion replace Euler Angles to
express angular orientation?!
"!Can we define one from the other?!
21
Rigid-Body
Equations of Motion!
22
Point-Mass
Dynamics
•! Inertial rate of change of translational position
r!I = v I = H IB v B
! u $
#
&
vB = # v &
#" w &%
•! Body-axis rate of change of translational velocity
–! Identical to angular-momentum transformation
v! I =
1
FI
m
v! B = H BI v! I ! "" B v B =
! X $ ! C X qS
#
& #
FB = # Y & = # CY qS
#" Z &%B # CZ qS
"
$
&
&
&
%
=
1 B
H I FI ! "" B v B
m
1
FB ! "" B v B
m
23
Rigid-Body Equations of Motion
(Euler Angles)
•! Translational
Position
! x $
&
#
rI = # y &
#" z &%
I
•! Angular
Position
% " (
'
*
!I = ' # *
' $ *
&
)I
•! Translational
Velocity
! u $
&
#
vB = # v &
#" w &% B
•! Angular
Velocity
" p %
'
$
!B = $ q '
$ r '
&B
#
•!
Rate of change of
Translational Position
r!I (t ) = H IB (t ) v B (t )
•!
Rate of change of
Angular Position
! I (t ) = LIB (t ) " B (t )
!
•!
Rate of change of
Translational Velocity
•!
Rate of change of
Angular Velocity
v! B (t ) =
1
" B (t ) v B (t )
FB (t ) + H BI (t ) g I ! "
m (t )
!! B ( t ) = I B "1 ( t ) #$ M B ( t ) " !" B ( t ) I B ( t ) ! B ( t ) %&
24
Aircraft Characteristics
Expressed in Body Frame
of Reference
! Xaero + Xthrust
#
FB = # Yaero + Ythrust
# Z +Z
thrust
" aero
Aerodynamic
and thrust
force
Aerodynamic and
thrust moment
Inertia
matrix
! Laero + Lthrust
#
M B = # M aero + M thrust
# N +N
aero
thrust
"
" I xx
$
I B = $ ! I xy
$
$# ! I xz
! I xy
I yy
! I yz
! CX + CX
$
aero
thrust
#
&
C
+
C
& = # Yaero
Ythrust
#
&
% B #" CZaero + CZthrust
(
(
(
$
! CX
& 1
#
2
& 'V S = # CY
& 2
# C
&%
" Z
B
$
&
& qS
&
%B
)
)
)
$
! C +C
laero
lthrust b
$
! Cl b $
&
#
&
&
#
& 1 2
#
& = # Cmaero + Cmthrust c & 'V S = # Cm c & q S
2
&
# Cb &
&
#
% B # Cnaero + Cnthrust b &
" n %B
%B
"
! I xz %
'
! I yz '
'
I zz '
&B
Reference Lengths
b = wing span
c = mean aerodynamic chord
25
Rigid-Body Equations of Motion:
Position
Rate of change of Translational Position
x! I = ( cos! cos" ) u + ( # cos $ sin " + sin $ sin ! cos" ) v + ( sin $ sin " + cos $ sin ! cos" ) w
y! I = ( cos! sin " ) u + ( cos $ cos" + sin $ sin ! sin " ) v + ( # sin $ cos" + cos $ sin ! sin " ) w
z!I = ( # sin ! ) u + ( sin $ cos! ) v + ( cos $ cos! ) w
Rate of change of Angular Position
!! = p + ( q sin ! + r cos ! ) tan "
"! = q cos ! # r sin !
$! = ( q sin ! + r cos ! ) sec "
26
Rigid-Body Equations of Motion:
Rate
Rate of change of Translational Velocity
u! = X / m ! gsin " + rv ! qw
v! = Y / m + gsin # cos" ! ru + pw
w! = Z / m + g cos # cos" + qu ! pv
Rate of change of Angular Velocity
(
{ (
)
(
{ (
)
(
)
} ) (I
(
)
})
p! = I zz L + I xz N ! I xz I yy ! I xx ! I zz p + "# I xz2 + I zz I zz ! I yy $% r q
q! = "# M ! ( I xx ! I zz ) pr ! I xz ( p 2 ! r 2 ) $% I yy
r! = I xz L + I xx N ! I xz I yy ! I xx ! I zz r + "# I xz2 + I xx I xx ! I yy $% p q
(I
Mirror symmetry, Ixz # 0
xx
I zz ! I xz2 )
xx
I zz ! I xz2 )
27
Checklist!
"!Why is it inconvenient to solve momentum rate
equations in an inertial reference frame?!
"!Are angular rate and momentum vectors
aligned?!
"!How are angular rate equations transformed
from an inertial to a body frame?!
"!After all the fuss about quaternions, why have
we gone back to Euler Angles?!
28
FLIGHT -!
Computer Program to
Solve the 6-DOF
Equations of Motion!
29
FLIGHT - MATLAB Program
http://www.princeton.edu/~stengel/FlightDynamics.html
30
FLIGHT - MATLAB Program
http://www.princeton.edu/~stengel/FlightDynamics.html
31
FLIGHT, Version 2 (FLIGHTver2.m)
•! Provides option for calculating rotations
with quaternions rather than Euler angles
•! Input and output via Euler angles in both
cases
•! Command window output clarified
•! On-line at
http://www.princeton.edu/~stengel/
FDcodeB.html
32
Examples from FLIGHT!
33
Longitudinal Transient
Response to Initial Pitch Rate
Bizjet, M = 0.3, Altitude = 3,052 m
•!
For a symmetric aircraft, longitudinal
perturbations do not induce lateraldirectional motions
34
Transient Response
to Initial Roll Rate
Lateral-Directional Response
Bizjet, M = 0.3, Altitude = 3,052 m
Longitudinal Response
For a symmetric aircraft, lateraldirectional perturbations do induce
longitudinal motions
35
Transient Response to
Initial Yaw Rate
Lateral-Directional Response
Longitudinal Response
Bizjet, M = 0.3, Altitude = 3,052 m
36
Crossplot of Transient
Response to Initial Yaw Rate
Longitudinal-Lateral-Directional Coupling
Bizjet, M = 0.3, Altitude = 3,052 m
37
Checklist!
"!Does longitudinal response couple into lateraldirectional response? !
"!Does lateral-directional response couple into
longitudinal response? !
38
Daedalus and Icarus, father and son,
Attempt to escape from Crete (<630 BC)
Other human-powered
t work
airplanes that didn
Bob W
C Davis
Edward Frost (1902)
AeroVironment Gossamer Condor
Winner of the 1st Kremer Prize: Figure 8 around pylons half-mile apart
(1977)
AeroVironment Gossamer Albatross
and MIT Monarch
2nd Kremer Prize: crossing the English Channel (1979)
3rd Kremer Prize: 1500 m on closed course in < 3 min (1984)
Gene Larrabee, 'Mr. Propeller' of human-powered flight, dies at 82 (1/11/2003)
“The Albatross' pilot could stay aloft only 10 minutes at first. With (Larrabee’s)
propeller, he stayed up for over an hour on his first flight …. There was no way the
Albatross could cross the Channel, which took almost three hours, without
(Larrabee’s) propeller.” (D. Wilson)
MIT Daedalus
Flew 74 miles across the Aegean Sea, completing Daedalus
s
intended flight (1988)
Aerodynamic Damping!
44
Pitching Moment due to Pitch Rate
(
)
M B = Cm q Sc ! Cmo + Cmq q + Cm" " q Sc
$
'
#C
! & Cmo + m q + Cm" " ) q Sc
%
(
#q
45
Angle of Attack Distribution
Due to Pitch Rate
Aircraft pitching at a constant rate, q rad/s, produces a
normal velocity distribution along x
!w = "q!x
Corresponding angle of attack distribution
!" =
!w #q!x
=
V
V
Angle of attack perturbation at tail center of pressure
!" ht =
qlht
V
lht = horizontal tail distance from c.m.
46
Horizontal Tail Lift
Due to Pitch Rate
Incremental tail lift due to pitch rate, referenced to tail area, Sht
(
!Lht = !C Lht
)
ht
1
"V 2 Sht
2
Incremental tail lift coefficient due to pitch rate, referenced
to wing area, S
( !C )
Lht aircraft
(
= !C Lht
)
- " ( CL %
" Sht % *" ( C Lht %
" qlht %
ht
=$
!
)
=
,
/
$
'
$
'
$
'
'
ht # S &
,+# () & aircraft
/. # () & aircraft # V &
Lift coefficient sensitivity to pitch rate referenced to wing area
C Lqht !
(
" #C Lht
)
"q
aircraft
% " C Lht (
% lht (
='
' *
*
& "$ ) aircraft & V )
47
Moment Coefficient
Sensitivity to Pitch Rate of
the Horizontal Tail
Differential pitch moment due to pitch rate
! "M ht
1
%l ( 1
= Cmqht #V 2 Sc = $C Lqht ' ht * #V 2 Sc
&V)2
2
!q
,% ! CL (
% lht ( / % lht ( 1
2
ht
= $ .'
'& *) 1 '& *) #V Sc
*
2
!+
V
c
&
)
.10
aircraft
Coefficient derivative with respect to pitch rate
Cmqht = !
" C Lht
"#
" C Lht $ lht ' $ c '
$ lht ' $ lht '
&% )( &% )( = !
& ) & )
V
c
"# % c ( % V (
2
Coefficient derivative with respect to normalized pitch rate
is insensitive to velocity
Cmq̂ht
! Cmht
! Cmht
! C Lht
=
=
= "2
! q̂
!#
! qc 2V
(
)
$ lht '
&% )(
c
2
48
Pitch-Rate Derivative Definitions
•! Pitch-rate derivatives are often expressed
in terms of a normalized pitch rate
•! Then
Cmq̂ =
q̂ !
qc
2V
! Cm
! Cm
" 2V %
=
=$
C
# c '& mq
! q̂ ! qc
2V
(
)
Pitching moment sensitivity to pitch rate
! Cm " c %
=$
'& Cmq̂
#
2V
!q
C mq =
But dynamic equations require $Cm/$q
2
# "VSc 2 &
!M
# c & # "V &
2
Sc = Cmq̂ %
= Cmq ( "V 2 ) Sc = Cmq̂ %
$ 2V (' %$ 2 ('
!q
$ 4 ('
49
Roll Damping Due to Roll Rate
2
" !V 2 %
" b % " !V %
Sb
=
C
Sb
Cl p $
l p̂ $
# 2V '& $# 2 '&
# 2 '&
" !V % 2
= Cl p̂ $
Sb
# 4 '&
< 0 for stability
•!
Vertical tail, horizontal tail, and wing
are principal contributors
( )
Cl p̂ ! Cl p̂
•!
Vertical Tail
( )
+ Cl p̂
Horizontal Tail
pb
2V
( )
+ Cl p̂
Wing
Wing with taper
(C )
l p̂
•!
p̂ =
Wing
=
! ( "Cl )Wing
! p̂
Thin triangular wing
NACA-TR-1098, 1952
NACA-TR-1052, 1951
=#
C L$ & 1 + 3% )
(
+
12 ' 1 + % *
(C )
l p̂
Wing
=!
" AR
32
50
Roll Damping Due
to Roll Rate
Tapered vertical tail
(C )
l p̂
vt
=
CY
! ( "Cl )vt
= # $vt
! p̂
12
% Svt (% 1+ 3+ (
' *'
*
& S )& 1+ + )
p̂ =
pb
2V
Tapered horizontal tail
( )
Cl p̂
ht
=
CL
! ( "Cl )ht
= # $ht
! p̂
12
% Sht (% 1+ 3+ (
' *'
*
& S )& 1+ + )
pb/2V describes helix angle
for a steady roll
51
Yaw Damping Due to Yaw Rate
2
" !V 2 %
" b % " !V %
Sb = Cnr̂ $
Sb
Cnr $
# 2V '& $# 2 '&
# 2 '&
< 0 for stability
" !V % 2
= Cnr̂ $
Sb
# 4 '&
r̂ =
rb
2V
52
Yaw Damping Due
to Yaw Rate
( )
Cnr̂ ! Cnr̂
Vertical Tail
( )
+ Cnr̂
r̂ =
Wing
rb
2V
Vertical tail contribution
( )
! ( Cn )Vertical Tail = " Cn#
(C )
nr̂ vt
=
Vertical Tail
! " ( Cn )Vertical Tail
(
)
! rb 2V
( )
rlvt
=
( )
V = " Cn#
Vertical Tail
! " ( Cn )Vertical Tail
! r̂
$ lvt ' $ b '
&% )( &% )( r
b V
% lvt (
' *
Vertical Tail & b )
( )
= #2 Cn$
Wing contribution
(C )
nr̂ Wing
= k0C L2 + k1C DParasite, Wing
k0 and k1 are functions of
aspect ratio and sweep angle
NACA-TR-1098, 1952
NACA-TR-1052, 1951
53
Checklist!
"!What
"!What
"!What
"!What
is
is
is
is
the
the
the
the
primary source of pitch damping?!
primary source of yaw damping?!
primary source of roll damping?!
difference between !
Cmq and Cmq̂ ?
54
Next Time:!
Aircraft Control Devices
and Systems!
Reading:!
Flight Dynamics!
214-234!
Learning Objectives!
Control surfaces!
Control mechanisms!
Powered control!
Flight control systems!
Fly-by-wire control!
Nonlinear dynamics and aero/mechanical instability!
55
Supplemental
Material
56
Airplane Angular Attitude (Position)
(! ," ,# )
•! Euler angles
•! 3 angles that relate one
Cartesian coordinate frame to
another
•! defined by sequence of 3
rotations about individual axes
•! intuitive description of angular
attitude
•! Euler angle rates have a
nonlinear relationship to bodyaxis angular rate vector
•! Transformation of rates is
singular at 2 orientations, ±90°
57
Airplane Angular Attitude (Position)
B
$ h11 h12
&
H BI (! ," ,# ) = & h21 h22
& h
h
% 31 32
h13 '
)
h23 )
h33 )
(I
H BI (! ," ,# ) = H 2B (! )H12 (" )H1I (# )
T
$% H BI (! ," ,# ) '( = $% H BI (! ," ,# ) '( = H IB (# ," , ! )
*1
•! Rotation matrix
•! orthonormal transformation
•! inverse = transpose
•! linear propagation from one attitude to
another, based on body-axis rate vector
•! 9 parameters, 9 equations to solve
•! solution for Euler angles from parameters
is intricate
58
Airplane Angular Attitude (Position)
Rotation Matrix
H BI (! ," ,# ) = H 2B (! )H12 (" )H1I (# )
H BI (! ," ,# ) =
% 1
0
0
'
' 0 cos ! sin !
' 0 $ sin ! cos !
&
0 $ sin " ( % cos#
*'
1
0
* ' $ sin#
0 cos" *) '
0
&
( % cos"
*'
*' 0
* '& sin "
)
sin#
cos#
0
0 (
*
0 *
1 *)
H BI (! ," ,# ) =
%
cos" cos#
'
' $ cos ! sin# + sin ! sin " cos#
' sin ! sin# + cos ! sin " cos#
&
cos" sin#
cos ! cos# + sin ! sin " sin#
$ sin ! cos# + cos ! sin " sin#
$ sin "
(
*
sin ! cos" *
cos ! cos" *
)
H BI H IB = I for all (! ," ,# ), i.e., No Singularities
59
Airplane Angular Attitude (Position)
Rotation Matrix = Direction Cosine Matrix
Angles between
each I axis and each
B axis
" cos !11
$
H BI = $ cos !12
$ cos !
13
#
cos ! 21
cos ! 22
cos ! 23
cos ! 31 %
'
cos ! 32 '
cos ! 33 '
&
60
Euler Angle Dynamics
%
! = ''
!
'
&
"!
#!
$!
( % 1 sin " tan #
* '
cos "
*=' 0
* ' 0 sin " sec#
) &
cos " tan # ( % p (
*'
*
+ sin " * ' q * = LIB , B
cos " sec# * '& r *)
)
LIB is not orthonormal
LIB is singular when ! = ± 90°
61
Rigid-Body Equations of Motion
(Euler Angles)
r!I ( t ) = H IB ( t ) v B ( t )
! ( t ) = LI ( t ) " ( t )
!
I
B
B
v! B ( t ) =
H IB , H BI are functions of !
1
FB ( t ) + H BI ( t ) g I ! "" B ( t ) v B ( t )
m (t )
!! B ( t ) = I B "1 ( t ) #$ M B ( t ) " !" B ( t ) I B ( t ) ! B ( t ) %&
62
Rotation Matrix Dynamics
! IB h B = !" I h I = !" I H IB h B
H
# 0
%
!! = % ! z
%
%$ "! y
"! z
0
!x
!y &
(
"! x (
(
0 (
'
! IB = !" I H IB
H
! BI = ! "" B H BI
H
63
Rigid-Body Equations of Motion
(Attitude from 9-Element Rotation Matrix)
r!I = H IB v B
! BI = !"
" B H BI
H
v! B =
1
FB + H BI g I ! "" B v B
m
!! B = I B"1 ( M B " !" B I B! B )
No need for Euler angles to solve the dynamic equations
64
Successive Rotations Expressed
by Products of Quaternions and
Rotation Matrices
Rotation from Frame
A to Frame C through
Intermediate Frame B
Matrix Multiplication Rule
q BA : Rotation from A to B
qCB : Rotation from B to C
qCA : Rotation from A to C
HCA ( qCA ) = HCB ( qCB ) H BA ( q BA )
Quaternion Multiplication Rule
C
! q C q B + q B q C ' q" C q B
! q3 $
# ( 4 )B 3A ( 4 )A 3B ( 3 )B 3A
& = qCB q BA ! #
qCA = #
C
B
C T
B
#" q4 &% A
q
q
'
q
(
)
(
)
(
4 B
4 A
3B ) q 3A
#"
$
&
&
&%
65
Rigid-Body Equations of Motion
(Attitude from 4-Element Quaternion Vector)
r!I = H v B
I
B
q! = Qq
H IB , H BI are functions of q
1
v! B = FB + H BI g I ! "" B v B
m
!! B = I B"1 ( M B " !" B I B! B )
66
Alternative Reference
Frames!
67
Velocity Orientation in an Inertial
Frame of Reference
Polar Coordinates
Projected on a Sphere
68
Body Orientation with Respect
to an Inertial Frame
69
Relationship of Inertial Axes
to Body Axes
•! Transformation is
independent of
velocity vector
•! Represented by
–! Euler angles
–! Rotation matrix
! vx $
! u $
&
#
&
#
B
=
H
v
v
I #
y &
&
#
&
#
#" w &%
v
z
&%
#"
! vx $
! u $
&
#
&
I #
# vy & = H B # v &
&
#
#" w &%
#" vz &%
70
Velocity-Vector Components
of an Aircraft
V, ! , "
V, ! , "
71
Velocity Orientation with Respect
to the Body Frame
Polar Coordinates
Projected on a Sphere
72
Relationship of Inertial Axes
to Velocity Axes
•! No reference to the body
frame
•! Bank angle, %, is roll angle
about the velocity vector
! vx $ ! V cos ' cos (
& #
#
# vy & = # V cos ' sin (
& # )V sin '
#
#" vz &% I "
$
&
&
&
%
#
vx2 + vy2 + vz2
# V & %
%
( %
)1 #
2
2 1/2 &
% ! ( = % sin $ vy / vx + vy '
% " ( %
$
' %
sin )1 ( )vz / V )
$
(
)
73
Relationship of Body Axes
to Wind Axes
•! No reference to
the inertial frame
! u $ ! V cos ' cos (
& #
#
V sin (
# v &=#
#" w &% # V sin ' cos (
"
$
&
&
&
%
#
# V & %
%
( %
% ! (=%
%$ " (' %
$
&
u 2 + v2 + w2 (
sin )1 ( v / V ) (
(
tan )1 ( w / u ) (
'
74
&
(
(
(
(
(
'
Angles Projected
on the Unit Sphere
Origin is airplane
s
center of mass
" : angle of attack
# : sideslip angle
$ : vertical flight path angle
% : horizontal flight path angle
& : yaw angle
' : pitch angle
( : roll angle (about body x ) axis)
µ : bank angle (about velocity vector)
75
Alternative Frames of Reference
•! Orthonormal transformations connect all reference frames
76