Aircraft Equations of Motion: Flight Path Computation!

Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2016

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

•! 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

Learning Objectives

Reading:!Flight Dynamics!

161-180!

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)

!! B ==

! x

! y

! z

"

#

$$$$

%

&

''''B

=

pqr

"

#

$$$

%

&

'''

Euler-angle rate vector is not orthogonal

Euler angles form a non-orthogonal vector !! =

"

#$

%

&

'''

(

)

***

!!! =

!"!#!$

%

&

'''

(

)

***+

, 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

Inverse transformation [(.)-1 " (.)T]

˙ "

˙ "

˙ "

pqr

!

"

###

$

%

&&&= I3

!'00

!

"

###

$

%

&&&+H2

B

0!(0

!

"

###

$

%

&&&+H2

BH1200!)

!

"

###

$

%

&&&

pqr

!

"

###

$

%

&&&=

1 0 'sin(0 cos) sin) cos(0 'sin) cos) cos(

!

"

###

$

%

&&&

!)!(!*

!

"

###

$

%

&&&=LI

B !++

!!!"!#

$

%

&&&

'

(

)))=

1 sin! tan" cos! tan"0 cos! *sin!0 sin! sec" cos! sec"

$

%

&&&

'

(

)))

pqr

$

%

&&&

'

(

)))=LB

I ++B

Can the inversion become singular?What does this mean?

•! ... which is

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)

!HBI hB = "!! IHB

I hB

!H IB t( ) = ! """ B t( )H I

B t( ) = !

0 !r t( ) q t( )r t( ) 0 ! p t( )!q t( ) p t( ) 0 t( )

#

$

%%%%

&

'

((((B

H IB t( )

Consequently

8H I

B 0( ) = H IB !0,"0,# 0( )

Avoiding the Euler Angle Singularity at ! = ±90°

Propagation of quaternion vector: single rotation from inertial to body frame (4 parameters)

9

!! 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

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

Rotation of a vector to an arbitrary new orientation can be expressed as a single rotation about an axis

at the vector’s base

11

Euler Rotation of a Vector

a !a1

a2

a3

!

"

###

$

%

&&&

: Orientation of rotation axisin reference frame

': Rotation angle

12

Euler’s Rotation TheoremrB = H I

BrI= aTrI( )a + rI ! a

TrI( )a"# $%cos& + sin& rI ' a( )= cos& rI + 1! cos&( ) aTrI( )a ! sin& a ' rI( )

Vector transformation involves 3 components

a !a1

a2

a3

!

"

###

$

%

&&&

: Orientation of rotation axisin reference frame

': Rotation angle

13

Rotation Matrix Derived from

Euler’s Formula

rB = H I

BrI = cos! rI + 1" cos!( ) aTrI( )a " sin! !arI( )

aTrI( )a = aaT( )rI

H IB = cos! I3 + 1" cos!( )aaT " sin! !a

Identity

Rotation matrix

Quaternion Derived from Euler Rotation Angle and Orientation

q =

q1q2q3q4

!

"

#####

$

%

&&&&&

!q3q4

!

"##

$

%&&=

sin ' 2( )acos ' 2( )

!

"

##

$

%

&&=

sin ' 2( )a1a2a3

(

)

***

+

,

---

cos ' 2( )

!

"

######

$

%

&&&&&&

4-parameter representation of 3 parameters; hence, a constraint must be satisfied

14

qT q = q12 + q2

2 + q32 + q4

2

= sin2 ! 2( ) + cos2 ! 2( ) = 1

Quaternion vector4 parameters based on Euler’s formula

4 !1( )

Rotation Matrix Expressed with Quaternion

15

H I

B = q42 ! q3

T q3( )"# $%I3 + 2q3q3T ! 2q4 !q3

H IB =

q12 ! q2

2 ! q32 + q4

2 2 q1q2 + q3q4( ) 2 q1q3 ! q2q4( )2 q1q2 ! q3q4( ) !q1

2 + q22 ! q3

2 + q42 2 q2q3 + q1q4( )

2 q1q3 + q2q4( ) 2 q2q3 ! q1q4( ) !q12 ! q2

2 + q32 + q4

2

"

#

$$$$

%

&

''''

From Euler’s formula

Rotation matrix from quaternion

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

17

Assuming that q4 " 0

H IB 0( ) =

h11 =cos!11( ) h12 h13h21 h22 h23h31 h32 h33

"

#

$$$

%

&

'''= H I

B (0,)0,* 0( )

Initialize q(0) from Direction Cosine Matrix or Euler Angles

q4 0( ) = 121+ h11 0( ) + h22 0( ) + h33 0( )

q3 0( ) !q1 0( )q2 0( )q3 0( )

!

"

####

$

%

&&&&

= 14q4 0( )

h23 0( )' h32 0( )!" $%h31 0( )' h13 0( )!" $%h12 0( )' h21 0( )!" $%

!

"

####

$

%

&&&&

Quaternion Vector Kinematics

18

Differential equation is linear in either q or #B

!q = ddt

q3q4

!

"##

$

%&&= 12

q4'' B ( "'' Bq3('' B

Tq3

!

"##

$

%&&

!q1!q2!q3!q4

!

"

#####

$

%

&&&&&

= 12

0 ' z (' y ' x

(' z 0 ' x ' y

' y (' x 0 ' z

(' x (' y (' z 0

!

"

#####

$

%

&&&&&B

q1q2q3q4

!

"

#####

$

%

&&&&&

4 !1( )

Propagate Quaternion Vector Using Body-Axis Angular Rates

19

dq t( )dt

=

!q1 t( )!q2 t( )!q3 t( )!q4 t( )

!

"

######

$

%

&&&&&&

= 12

0 r t( ) 'q t( ) p t( )'r t( ) 0 p t( ) q t( )q t( ) ' p t( ) 0 r t( )' p t( ) 'q t( ) 'r t( ) 0

!

"

######

$

%

&&&&&&B

q1 t( )q2 t( )q3 t( )q4 t( )

!

"

######

$

%

&&&&&&

qint tk( ) = q tk!1( ) + dq "( )dt

d"tk!1

tk

#

Digital integration to compute q(tk)

p t( ) q t( ) r t( )!

"#$ ! % x % y % z

!"

#$

Euler Angles Derived from Quaternion

20

!"#

$

%

&&&

'

(

)))=

atan2 2 q1q4 + q2q3( ), 1* 2 q12 + q2

2( )$% '({ }sin*1 2 q2q4 * q1q3( )$% '(

atan2 2 q3q4 + q1q2( ), 1* 2 q22 + q3

2( )$% '({ }

$

%

&&&&&

'

(

)))))

•! atan2: generalized arctangent algorithm, 2 arguments–! returns angle in proper quadrant–! avoids dividing by zero–! has various definitions, e.g., (MATLAB)

atan2 y, x( ) =

tan!1 yx

"#$

%&' if x > 0

( + tan!1 yx

"#$

%&' , !( + tan!1 y

x"#$

%&' if x < 0 and y ) 0, < 0

( 2 , !( 2 if x = 0 and y > 0, < 00 if x = 0 and y = 0

*

+

,,,,

-

,,,,

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

•! Body-axis rate of change of translational velocity–! Identical to angular-momentum transformation

!rI = v I = HBI vB

!v I =

1mFI

vB =uvw

!

"

###

$

%

&&&

23

!vB = H IB !v I ! """ BvB =

1mH I

BFI ! """ BvB

= 1mFB ! """ BvBFB =

XYZ

!

"

###

$

%

&&&B

=

CXqSCYqSCZqS

!

"

###

$

%

&&&

!rI t( ) =HBI t( )vB t( )

!vB t( ) = 1

m t( )FB t( )+H I

B t( )g I ! """B t( )vB t( )

!!! I t( ) =LB

I t( )""B t( )

!!! B t( ) = IB"1 t( ) MB t( )" "!! B t( )IB t( )!! B t( )#$ %&

•! Rate of change of Translational Position

•! Rate of change of Angular Position

•! Rate of change of Translational Velocity

•! Rate of change of Angular Velocity

rI =xyz

!

"

###

$

%

&&&I

!! I =

"

#$

%

&

'''

(

)

***I

vB =uvw

!

"

###

$

%

&&&B

!! B =

pqr

"

#

$$$

%

&

'''B

•! Translational Position

•! Angular Position

•! Translational Velocity

•! Angular Velocity

Rigid-Body Equations of Motion(Euler Angles)

24

Aircraft Characteristics Expressed in Body Frame

of Reference

IB =I xx !I xy !I xz

!I xy I yy !I yz

!I xz !I yz I zz

"

#

$$$$

%

&

''''B

FB =Xaero + Xthrust

Yaero +YthrustZaero + Zthrust

!

"

###

$

%

&&&B

=

CXaero+ CXthrust

CYaero+ CYthrust

CZaero+ CZthrust

!

"

####

$

%

&&&&B

12'V 2S =

CX

CY

CZ

!

"

###

$

%

&&&B

q SAerodynamic and thrust force

Aerodynamic and thrust moment

Inertia matrix

Reference Lengthsb = wing spanc = mean aerodynamic chord

MB =

Laero + LthrustMaero + Mthrust

Naero + Nthrust

!

"

###

$

%

&&&B

=

Claero+ Clthrust( )b

Cmaero+ Cmthrust( )c

Cnaero+ Cnthrust( )b

!

"

#####

$

%

&&&&&B

12'V 2S =

ClbCmcCnb

!

"

###

$

%

&&&B

q S

25

Rigid-Body Equations of Motion: Position

!xI = cos! cos"( )u + # cos$ sin" + sin$ sin! cos"( )v + sin$ sin" + cos$ sin! cos"( )w!yI = cos! sin"( )u + cos$ cos" + sin$ sin! sin"( )v + # sin$ cos" + cos$ sin! sin"( )w!zI = # sin!( )u + sin$ cos!( )v + cos$ cos!( )w

!! = p + qsin! + r cos!( ) tan"!" = qcos! # r sin!!$ = qsin! + r cos!( )sec"

Rate of change of Translational Position

Rate of change of Angular Position

26

Rigid-Body Equations of Motion: Rate

!u = X /m! gsin" + rv!qw!v =Y /m+ gsin# cos" ! ru+ pw!w = Z /m+ gcos# cos" +qu ! pv

!p = I zzL + I xzN ! I xz I yy ! I xx ! I zz( ) p + I xz2 + I zz I zz ! I yy( )"# $%r{ }q( ) I xxI zz ! I xz

2( )!q = M ! I xx ! I zz( ) pr ! I xz p2 ! r2( )"# $% I yy

!r = I xzL + I xxN ! Ixz I yy ! I xx ! I zz( )r + I xz2 + I xx I xx ! I yy( )"# $% p{ }q( ) I xxI zz ! I xz

2( )

Rate of change of Translational Velocity

Rate of change of Angular Velocity

Mirror symmetry, Ixz # 0 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

http://www.princeton.edu/~stengel/FlightDynamics.html

FLIGHT - MATLAB Program

30

http://www.princeton.edu/~stengel/FlightDynamics.html

FLIGHT - MATLAB Program

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 lateral-directional motions 34

Transient Response to Initial Roll Rate

Lateral-Directional Response Longitudinal Response

Bizjet, M = 0.3, Altitude = 3,052 m For a symmetric aircraft, lateral-directional 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

Bizjet, M = 0.3, Altitude = 3,052 m

Longitudinal-Lateral-Directional Coupling

37

Checklist!"!Does longitudinal response couple into lateral-

directional 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 airplanes that didn t work

Edward Frost (1902)

C Davis

Bob W

AeroVironment Gossamer CondorWinner 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 DaedalusFlew 74 miles across the Aegean Sea, completing Daedalus s

intended flight (1988)

Aerodynamic Damping!

44

Pitching Moment due to Pitch Rate

MB = Cmq Sc ! Cmo+Cmq

q +Cm""( )q Sc

! Cmo+ #Cm

#qq +Cm"

"$%&

'()q Sc

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

!" = !wV

= #q!xV

Corresponding angle of attack distribution

!" ht =qlhtV

Angle of attack perturbation at tail center of pressure

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

Lift coefficient sensitivity to pitch rate referenced to wing area

!Lht = !CLht( )ht

12"V 2Sht

CLqht!" #CLht( )

aircraft

"q=

"CLht

"$%&'

()* aircraft

lhtV

%&'

()*

!CLht( )aircraft

= !CLht( )ht

ShtS

"#$

%&' =

(CLht

()"#$

%&' aircraft

!)*

+,,

-

.//=

(CLht

()"#$

%&' aircraft

qlhtV

"#$

%&'

Incremental tail lift coefficient due to pitch rate, referenced to wing area, S

47

Moment Coefficient Sensitivity to Pitch Rate of

the Horizontal TailDifferential pitch moment due to pitch rate

!"Mht

!q= Cmqht

12#V 2Sc = $CLqht

lhtV

%&'

()*12#V 2Sc

= $!CLht

!+%&'

()* aircraft

lhtV

%&'

()*

,

-..

/

011

lhtc

%&'

()*12#V 2Sc

Cmqht= !

"CLht

"#lhtV

$%&

'()lhtc

$%&

'() = !

"CLht

"#lhtc

$%&

'()2 cV

$%&

'()

Coefficient derivative with respect to pitch rate

Coefficient derivative with respect to normalized pitch rate is insensitive to velocity

Cmq̂ht=!Cmht

! q̂=

!Cmht

! qc2V( ) = "2

!CLht

!#lhtc

$%&

'()2

48

Pitch-Rate Derivative Definitions•! Pitch-rate derivatives are often expressed

in terms of a normalized pitch rate

Cmq= !Cm

!q= c2V

"#$

%&'Cmq̂

Cmq̂= !Cm

! q̂= !Cm

! qc2V( ) =

2Vc

"#$

%&'Cmq

q̂ ! qc

2V

•! Then

But dynamic equations require $Cm/$q

49

!M!q

= Cmq"V 2 2( )Sc = Cmq̂

c2V

#$%

&'(

"V 2

2#$%

&'(Sc = Cmq̂

"VSc 2

4#$%

&'(

Pitching moment sensitivity to pitch rate

Roll Damping Due to Roll RateClp

!V 2

2"#$

%&'Sb = Clp̂

b2V

"#$

%&'

!V 2

2"#$

%&'Sb

= Clp̂

!V4

"#$

%&' Sb

2

Clp̂( )Wing

=! "Cl( )Wing

! p̂= #

CL$

121+ 3%1+ %

&'(

)*+

•! Wing with taper

•! Thin triangular wingClp̂( )

Wing= !

"AR32

Clp̂! Clp̂( )

Vertical Tail+ Clp̂( )

Horizontal Tail+ Clp̂( )

Wing

•! Vertical tail, horizontal tail, and wing are principal contributors

< 0 for stability

NACA-TR-1098, 1952NACA-TR-1052, 1951 50

p̂ = pb2V

Roll Damping Due to Roll Rate

Tapered vertical tail

Tapered horizontal tail

p̂ = pb2V

Clp̂( )ht =! "Cl( )ht! p̂

= #CL$ht

12ShtS

%

&'

(

)*1+ 3+1++

%

&'

(

)*

pb/2V describes helix angle for a steady roll

Clp̂( )vt =! "Cl( )vt! p̂

= #CY$vt

12SvtS

%

&'

(

)*1+ 3+1++

%

&'

(

)*

51

Yaw Damping Due to Yaw Rate

Cnr

!V 2

2"#$

%&'Sb = Cnr̂

b2V

"#$

%&'

!V 2

2"#$

%&'Sb

= Cnr̂

!V4

"#$

%&' Sb

2

< 0 for stability

52

r̂ = rb2V

Yaw Damping Due to Yaw RateCnr̂

! Cnr̂( )Vertical Tail

+ Cnr̂( )Wing

Cnr̂( )Wing

= k0CL2 + k1CDParasite,Wing

k0 and k1 are functions of aspect ratio and sweep angle

Wing contribution

Vertical tail contribution! Cn( )Vertical Tail = " Cn#( )

Vertical Tail

rlvtV( ) = " Cn#( )

Vertical Tail

lvtb

$%&

'()

bV

$%&

'() r

r̂ = rb2V

Cnr̂( )vt=!" Cn( )Vertical Tail

! rb2V( ) =

!" Cn( )Vertical Tail! r̂

= #2 Cn$( )Vertical Tail

lvtb

%&'

()*

NACA-TR-1098, 1952NACA-TR-1052, 1951 53

Checklist!"!What is the primary source of pitch damping?!"!What is the primary source of yaw damping?!"!What is the primary source of roll damping?!"!What is the difference between !

54

Cmq and Cmq̂

?

Next Time:!Aircraft Control Devices

and Systems!Reading:!

Flight Dynamics!214-234!

55

Control surfaces!Control mechanisms!Powered control!

Flight control systems!Fly-by-wire control!

Nonlinear dynamics and aero/mechanical instability!

Learning Objectives!

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 body-axis angular rate vector

•! Transformation of rates is singular at 2 orientations, ±90°

! ," ,#( )

57

Airplane Angular Attitude (Position)

H IB(!," ,# ) =

h11 h12 h13h21 h22 h23h31 h32 h33

$

%

&&&

'

(

)))I

B

H IB(!," ,# ) = H2

B(!)H12 (" )H I

1 (# )

H IB(!," ,# )$% '(

*1= H I

B(!," ,# )$% '(T= HB

I (# ," ,!)

•! 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)

H IB(!," ,# ) = H2

B(!)H12 (" )H I

1 (# )

H IB(!," ,# ) =

1 0 00 cos! sin!0 $sin! cos!

%

&

'''

(

)

***

cos" 0 $sin"0 1 0sin" 0 cos"

%

&

'''

(

)

***

cos# sin# 0$sin# cos# 00 0 1

%

&

'''

(

)

***

H IB(!," ,# ) =

cos" cos# cos" sin# $sin"$cos! sin# + sin! sin" cos# cos! cos# + sin! sin" sin# sin! cos"sin! sin# + cos! sin" cos# $sin! cos# + cos! sin" sin# cos! cos"

%

&

'''

(

)

***

Rotation Matrix

H IBHB

I = I for all (!," ,# ), i.e., No Singularities 59

Angles between each I axis and each

B axis

Airplane Angular Attitude (Position)Rotation Matrix = Direction Cosine Matrix

H IB =

cos!11 cos!21 cos!31cos!12 cos!22 cos!32cos!13 cos!23 cos!33

"

#

$$$

%

&

'''

60

Euler Angle Dynamics

!!! =

!"!#!$

%

&

'''

(

)

***=

1 sin" tan# cos" tan#0 cos" +sin"0 sin" sec# cos" sec#

%

&

'''

(

)

***

pqr

%

&

'''

(

)

***= LB

I ,, B

LBI is not orthonormal

LBI is singular when != ± 90°

61

!rI t( ) = HBI t( )vB t( )

!vB t( ) = 1

m t( )FB t( ) +H IB t( )g I ! """ B t( )vB t( )

!!! I t( ) = LB

I t( )"" B t( )

!!! B t( ) = IB"1 t( ) MB t( )" "!! B t( ) IB t( )!! B t( )#$ %&

Rigid-Body Equations of Motion(Euler Angles)

HBI ,H I

B are functions of !!

62

Rotation Matrix Dynamics

!!! =

0 "! z ! y

! z 0 "! x

"! y ! x 0

#

$

%%%%

&

'

((((

!HBI hB = "!! IhI = "!! IHB

I hB

!HBI = "!! IHB

I

!H IB = ! """ BH I

B

63

!rI = HBI vB

!vB =

1mFB +H I

Bg I ! """ BvB

!!! B = IB"1 MB " "!! BIB!! B( )

Rigid-Body Equations of Motion(Attitude from 9-Element Rotation Matrix)

!H IB = ! """BH I

B

No need for Euler angles to solve the dynamic equations64

Successive Rotations Expressed by Products of Quaternions and

Rotation Matrices

65

qAB : Rotation from A to BqBC : Rotation from B to CqAC : Rotation from A to C

Rotation from Frame A to Frame C through Intermediate Frame B

HAC qA

C( ) = HBC qB

C( )HAB qA

B( )

qAC =

q3

q4

!

"##

$

%&&A

C

= qBCqA

B !q4( )B

C q3AB + q4( )A

B q3BC ' "q3( )B

C q3AB

q4( )BC q4( )A

B ' q3BC( )T q3A

B

!

"

###

$

%

&&&

Quaternion Multiplication Rule

Matrix Multiplication Rule

!rI = HBI vB

!vB =

1mFB +H I

Bg I ! """ BvB

!!! B = IB"1 MB " "!! BIB!! B( )

Rigid-Body Equations of Motion(Attitude from 4-Element Quaternion Vector)

!q =Qq HBI ,H I

B are functions of q

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

vxvyvz

!

"

####

$

%

&&&&

= HBI

uvw

!

"

###

$

%

&&&

uvw

!

"

###

$

%

&&&= H I

B

vxvyvz

!

"

####

$

%

&&&&

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

•! No reference to the body frame

•! Bank angle, %, is roll angle about the velocity vector

V!

"

#

$

%%%

&

'

(((=

vx2 + vy

2 + vz2

sin)1 vy / vx2 + vy

2( )1/2#$

&'

sin)1 )vz /V( )

#

$

%%%%%

&

'

(((((

vxvyvz

!

"

####

$

%

&&&& I

=

V cos' cos(V cos' sin()V sin'

!

"

###

$

%

&&&

Relationship of Inertial Axes to Velocity Axes

73

Relationship of Body Axes to Wind Axes

•! No reference to the inertial frame

uvw

!

"

###

$

%

&&&=

V cos' cos(V sin(

V sin' cos(

!

"

###

$

%

&&&

V!

"

#

$

%%%

&

'

(((=

u2 + v2 + w2

sin)1 v /V( )tan)1 w / u( )

#

$

%%%%

&

'

((((

74

Angles Projected on the Unit Sphere

" : 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)

Origin is airplane s center of mass

75

Alternative Frames of Reference•! Orthonormal transformations connect all reference frames

76