Lateral-Directional
Approximation Models
State-Space Form (Straight & Level)
 U0 


 Y
Ixz
 p

r

I xx   L


  Ixz p  r  N 
 I zz
  0





Yp
Yr  U 0
Lp
Lr
Np
Nr
1
tan  0
g cos 0      Y a
0   p   L a
 
0   r  N  a
  
0     0
Y r 
L r   a 

N  r   r 

0 
Rearrange the left hand side
00
U
0 U0 0 0
1/
     
Ixz
Ixz
     0 0 1 1 
IIxxxx
 p p   
 r r   
IxzIxz
11
     0 0 
I
zzI zz
     
 0
 0 0 0 00
1
00
  Y
cos0 0  
    YY

Y  YYp p YY
UU0 0 ggcos
r
 a YY

r

a
 r r

00  
  

  LL  LLp p
LLr r
00   pp   LL a a LL r r a a 
   
  
  N
 r 
N
N
0
r
N
N




N
N
0
r
N





 Npp
r
 a N  r 
r 

r

a

00  
     
 r



tan0 0
  00 11 tan
00        00
00  
11

State-Space Form (Straight & Level)
1

Y

U0
  
   1  L  Ixz N 
 
 p   D   I
xx



r 

 

1  Ixz
    D  L I  N  
zz



0

Where 𝐷 =
1
Yp
U0
1
Yr U 0 
U0

1
Ixz
Np 
 Lp 
D
I xx


1
Ixz
Nr 
 Lr 
D
I xx


1  Ixz
Np 
 Lp
D
I zz


1  Ixz
 Nr 
 Lr
D  I zz

1
tan  0
1
𝐼𝑥𝑧2
1−𝐼𝑥𝑥 𝐼𝑧𝑧
1
1


g cos 0 
Y a

U0
U0
   
   1 

Ixz
0
N a 
  p    L a 
I xx

 r   D 
   1 

Ixz
      L a
0
 N a 
I zz

D 



0
0








1
Ixz
N  r    a 
 L r 
D
I xx
  
 r 

1
Ixz
 N r 
 L r
D
I zz


0

1
Y r
U0
Lateral-Directional Eigenvalues
Lateral-Directional Eigenvectors
(phasor diagram format)
Roll Mode Approximation
The approximation is based on the following observations:
• Dominant in roll rate
• Side slip and yaw rate have no components and can be assumed to be constant
zeros
Thus, roll mode approximation is obtained by ignoring side slip and yaw rate dynamics
(deleting β 𝑎𝑛𝑑 𝑟)
1 
1 
 

Ixz
Ixz
 p    Lp 
N p  0   p    L a 
N a 
I xx
I xx
   D 
      D 


  
  

0
1
0




1
Ixz
L

N
 a
 a    a 
D
I xx
   
  r 
0



1
Ixz
1
Ixz
1
Ixz
p   Lp 
N p  p   L a 
N  a   a   L r 
N  r  r
D
I xx
D
I
D
I
xx
xx





A roll damper thus provides augmentation from the aileron to increase the roll damping
Dutch Roll Mode Approximation
The approximation is based on the following observations:
• Dominant in side slip, yaw rate, and roll rate
• The vehicle bank angle has no components and can be assumed to be zero
Thus, Dutch roll mode approximation is obtained by ignoring vehicle roll angle
(deleting 𝜑)

1
Y

U
0
   

Ixz
  1 
p

L

N




  D
I xx


r  
 
 1  L Ixz  N 
 
 D   I zz


1
Yp
U0

1
Ixz
Np 
 Lp 
D
I xx


1  Ixz
Np 
 Lp
D
I zz



1
1
Y a
Yr U 0  

U0
U
0


  



1
Ixz
1
Ixz
N r    p     L a 
N a 
 Lr 
 
D
I xx
I xx
  r   D 

  

1  Ixz
 1  L Ixz  N 
 N r 
 Lr
a 
 D   a I zz
D  I zz
 





   a 
1
Ixz
N r   
 L r 
D
I xx
   r 


1
Ixz
 N r 
 L r
D
I zz
 
1
Y r
U0
Dutch Roll Mode Approximation
Now we also know that the roll rate is dominated by the roll mode
Thus, in the Dutch roll mode approximation the roll rate derivative should be set
to zero (by singular perturbation theory)





1
Y

U
0
  


1
Ixz
 
 0    D  L  I N  
xx

r   
  
L'

 1  Ixz

 N 
  L
I zz

D 
'

N
1
Yp
U0

1
Ixz
Np 
 Lp 
D
I xx

L'p

1  Ixz
Np 
 Lp
D
I zz

N 'p





1
1

Y a
Yr U 0  

U0
U
0
 
  

   1 

1
Ixz
Ixz

L

N
N a 
 r
 L a 
r  p 
D
I xx
I xx
  r   D 





L'r
L' a



Ixz

1  Ixz
1 
L

N
 N r 



a

a
 Lr
D
I zz
D  I zz




'
'


Na
Nr



1

Y r

U0

   a 
1
Ixz
L

N
 r
r  

D
I xx
   r 

L' r


1
Ixz
 N r 
 L r
D
I zz

'

Nr
Solving the algebraic equation, we get the following relation between roll rate, sideslip
and yaw rate
1
p  '  L'   L'r r  L' a a  L' r  r 
Lp
Dutch Roll Mode Approximation
Substitute for the roll rate to obtain the Dutch-roll approximation
1
L'
1
 Y  Y p '
U0
Lp
   U 0
 
L'
r  
'
'
N  N p '

Lp
1
1
L'r 
Yr U 0   Yp ' 
U0
U0
Lp   
 r 
'
'
' Lr
 
Nr  N p '

Lp
1
L' a
1
U Y a  U Y p L'
0
0
p

'

'
' L a
 N a  N p '
Lp

L' r 
1
1
Y r  Y p ' 
U0
U0
L p   a 
'
  r 
'
' L r
N a  N p '

L p 
Navion Aircraft
Dutch roll eigenvalue using the 4X4 lateral-directional state: -0.4867 ± 2.3349i; 𝜁 =
0.204, 𝜔 = 2.39𝑟𝑎𝑑/𝑠𝑒𝑐
Text obtains Dutch roll approximation by neglecting Np’ and Lr’
(Np’=0.3498 and LrP’ =2.193)
Eigenvalues: -0.5074±2.105𝑗; 𝜁 = 0.234, 𝜔 = 2.16𝑟𝑎𝑑/𝑠𝑒𝑐
Using the derived approximation: -0.5531 ± 2.2519i; 𝜁 = 0.239, 𝜔 = 2.32𝑟𝑎𝑑/𝑠𝑒𝑐
Navion Aircraft (Text approximation)