Euler Angle Rates - Princeton University

[Pages:36]Aircraft Equations of Motion:

Flight Path Computation

Robert Stengel, Aircraft Flight Dynamics, MAE 331, 2018

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 2018 by Robert Stengel. All rights reserved. For educational use only.



1

Euler Angle Rates

2

1

Euler-Angle Rates and Body-Axis Rates

Body-axis

angular rate vector (orthogonal )

" $

x

% '

" p% $'

B = $ y ' = $ q '

$ #$

z

' &'B

#$ r &'

Euler angles form a non-orthogonal vector

% ( '* =' * &' )*

Euler-angle rate vector is not

%

=

' '

( *

% '

x

* ' y

( * *

orthogonal

' &

* )

' &'

z

* )*I

3

Relationship Between EulerAngle 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 q

$! &# &= I3#

0

$!

& &

+

H

B 2

# #

0

$

!

& &

+

H2B

H12

# #

0 0

$ & &

"# r %& "# 0 %& "# 0 %&

"# %&

! # #

p q

$! &# &=#

1 0

0 cos

-sin sin cos

$! &# &#

$

& &

=

LBI

Can the inversion become singular?

"# r %&

"# 0

- sin

cos cos

%&"#

& %

What does this mean?

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

$ & &

'$ )& )=&

1 0

& %

) (

& %

0

sin tan cos

sin sec

cos tan

'$ )&

p

' )

-sin )& q ) = LIBB

cos sec ()%& r ()

4

2

Euler-Angle Rates and Body-Axis Rates

5

Avoiding the Euler Angle Singularity at = 90

? Alternatives to Euler angles

- Direction cosine (rotation) matrix - Quaternions

Propagation of direction cosine matrix (9 parameters)

H

I B

h

B

=

I HBI hB

Consequently

H!

B I

(t

)

=

-

"

B

(

t

)

H

B I

(

t

)

=

-

0

r(t) -q(t)

-r(t)

0

p(t)

q(t) - p(t) 0(t)

H

B I

(t

)

B

( ) H

B I

(

0

)

=

H

B I

0,0, 0

6

3

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

7

Checklist

q Are the components of the Euler Angle rate vector orthogonal to each other?

q Is the inverse of the transformation from Euler Angle rates to body-axis rates the transpose of the matrix?

q What complication does the inverse transformation introduce?

8

4

Rigid-Body Equations of Motion

9

Point-Mass Dynamics

? Inertial rate of change of translational position

rI

=

vI

=

H

I B

v

B

!

v

B

=

# #

u v

$ & &

"# w %&

? Body-axis rate of change of translational velocity

? Identical to angular-momentum transformation

v! I

=

1 m

FI

v! B

=

HIBv! I

-

" Bv B

=

1 m

H

B I

FI

-

" Bv B

!

FB

=

# #

"#

X Y Z

$ & & %&B

=

! # # # "

C X qS CY qS CZ qS

$ & & & %

=

1 m

FB

-

" Bv B

10

5

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

? Rate of change of Angular Position

rI

(t

)

=

H

I B

(t

)

v

B

(t

)

I (t) = LIB (t)B (t)

?

Rate of change of Translational Velocity

v B

(t)

=

1

m (t )

FB

(t)

+

H

B I

(t)

gI

-

B

(t

)

vB

(t)

? Rate of change of Angular Velocity

! B (t ) = IB-1 (t )MB (t ) - " B (t )IB (t ) B (t )

11

Aircraft Characteristics Expressed in Body Frame

of Reference

Aerodynamic and thrust force

! # FB = # # "

Xaero + Xthrust Yaero + Ythrust Zaero + Zthrust

$ &

!# C + C Xaero

Xthrust

& =# &# %B "#

C + C Yaero

Ythrust

C + C Zaero

Zthrust

$

!

& & &

1 2

V

2S

=

# # #

%&B

"

CX CY CZ

$ & & qS & %B

Aerodynamic and thrust moment

! # MB = # # "

Laero + Lthrust M aero + M thrust Naero + Nthrust

$

( ) !

#

C + C b laero

lthrust

( ) &

&

=

# #

C + C c maero

mthrust

& %B

( ) #

"#

C + C b naero

nthrust

$

&

!

& & &

1 2

V

2S

=

# # #

%&B

"

Clb Cmc Cnb

$ & & qS & %B

Inertia matrix

I xx

-I xy

-I xz

Reference Lengths

IB

=

-I xy -I xz

I yy -I yz

-I yz I zz

B

b = wing span c = mean aerodynamic chord

12

6

Rigid-Body Equations of Motion: Position

Rate of change of Translational 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

Rate of change of Angular Position

= p + (q sin + r cos) tan

= q cos - r sin

= (q sin + r cos)sec

13

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

( { ( ) ( ) } ) ( ) p! =

I zz L + I xz N -

I xz

I yy - I xx - I zz

p

+

I

2 xz

+

I

zz

I zz - I yy r

q

I

xx I

zz

-

I

2 xz

( ) ( ) q! = M - I xx - I zz pr - I xz p2 - r2 I yy

( { ( ) ( ) } ) ( ) r! =

I xz L + I xx N -

I xz

I yy - I xx - I zz

r

+

I

2 xz

+

I

xx

I xx - I yy p

q

I

xx I

zz

-

I

2 xz

Mirror symmetry, Ixz 0

14

7

Checklist

q Why is it inconvenient to solve momentum rate equations in an inertial reference frame?

q Are angular rate and momentum vectors aligned?

q How are angular rate equations transformed from an inertial to a body frame?

15

FLIGHT Computer Program to

Solve the 6-DOF Equations of Motion

16

8

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