Leon P. Gefert and Theodore W. Wright

Glenn Research Center, Cleveland, Ohio

Polyhedral Interpolation for Optimal

Reaction Control System Jet Selection

NASA/TM—2014-218317

June 2014
NASA STI Program . . . in Profi le

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA Scientifi c and Technical Information (STI) 

program plays a key part in helping NASA maintain 

this important role.

The NASA STI Program operates under the auspices 

of the Agency Chief Information Offi cer. It collects, 

organizes, provides for archiving, and disseminates 

NASA’s STI. The NASA STI program provides access 

to the NASA Aeronautics and Space Database and 

its public interface, the NASA Technical Reports 

Server, thus providing one of the largest collections 

of aeronautical and space science STI in the world. 

Results are published in both non-NASA channels 

and by NASA in the NASA STI Report Series, which 

includes the following report types:

• TECHNICAL PUBLICATION. Reports of 

completed research or a major signifi cant phase 

of research that present the results of NASA 

programs and include extensive data or theoretical 

analysis. Includes compilations of signifi cant 

scientifi c and technical data and information 

deemed to be of continuing reference value. 

NASA counterpart of peer-reviewed formal 

professional papers but has less stringent 

limitations on manuscript length and extent of 

graphic presentations.

• TECHNICAL MEMORANDUM. Scientifi c 

and technical fi ndings that are preliminary or 

of specialized interest, e.g., quick release 

reports, working papers, and bibliographies that 

contain minimal annotation. Does not contain 

extensive analysis.

• CONTRACTOR REPORT. Scientifi c and 

technical fi ndings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. Collected 

papers from scientifi c and technical 

conferences, symposia, seminars, or other 

meetings sponsored or cosponsored by NASA.

• SPECIAL PUBLICATION. Scientifi c, 

technical, or historical information from 

NASA programs, projects, and missions, often 

concerned with subjects having substantial 

public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientifi c and 

technical material pertinent to NASA’s mission.

Specialized services also include creating custom 

thesauri, building customized databases, organizing 

and publishing research results.

For more information about the NASA STI 

program, see the following:

• Access the NASA STI program home page at 

http://www.sti.nasa.gov

• E-mail your question to help@sti.nasa.gov

• Fax your question to the NASA STI 

Information Desk at 443–757–5803

• Phone the NASA STI Information Desk at

 443–757–5802

• Write to:

 STI Information Desk

NASA Center for AeroSpace Information

 7115 Standard Drive

 Hanover, MD 21076–1320
Leon P. Gefert and Theodore W. Wright

Glenn Research Center, Cleveland, Ohio

Polyhedral Interpolation for Optimal

Reaction Control System Jet Selection

NASA/TM—2014-218317

June 2014

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Available from

NASA Center for Aerospace Information

7115 Standard Drive

Hanover, MD 21076–1320

National Technical Information Service

5301 Shawnee Road

Alexandria, VA 22312

Available electronically at http://www.sti.nasa.gov

Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

Level of Review: This material has been technically reviewed by technical management. 


 	

 

 

 

 



 

 


	
 

 	
	
	
   

 		

	
	



                 

 	
	
	


                     

 	

	

	
!

	
	
                 "

	

	
#	$
!

	
	
              "

 

$

	
                                 %

 &	




'	                     %

 (

	
    %

 
#	$
'	
	

	                  )

&
*
	
	

	                    

 	
!



" 

# 	
$" 

% 

&
 

' !
$(
&
"


() 

* &
"
	
 #

% 	
	
	
                                 

% 

$

	
                                 "

% 
'$
$


	

''
	                  "

%#
	
	
'$
$

+
		

              %

+ &
	
!



" +

 &
	
$" +

 
" 

NASA/TM—2014-218317 iii

!



,
!
	


'#'


	
	

	


		
	

 

-$
.
!
'$
$
 &
	



$	

$



	




'

#

	

	
	'
'		
	
*

	




	
	#


	


	


	


/


$
'

		 &



	
	
'
0 	


	$0 	'






Polyhedral Interpolation for Optimal 

Reaction Control System Jet Selection 

Leon P. Gefert and Theodore W. Wright 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

NASA/TM—2014-218317 v

 	



	
	'
	


	
	

	



	
1,-,2 	

		

3 4(5 	
'
#$
!
	
 

-$
3 -5 . &
.
	
	
!'
0



'$
$


	'
#+
)
	'
))




 &
.
	
!'

	6


	'
	


		
#'$0 *
#	

.
!
+
	
	

'$
$



	
	#
	
	
	
	

	
	

	
	
	
'

&
	


 - .
	
$	

$

	
'
+
		
	0 	7
!'



#	

.

!

	
	
	
6 &
.
!



	
	

''


		2
	0 

	0 	0 	'


$
	#
'



''	
 8	


	0 	

	


''

'

!
'$
$
 &
	



'$
$


	
	
	
	

	
	
	




	
	


	$
3	
	5
+





	'

	
'


9+0 


		




 - .





		2
#'''



 '	

$0 
#'''




'
.

	
	#


7



	
6	

	'


	

 - .
'$
$


	
''
	

	
	
	 &

''


	





		2

	

	'
	#
'



#

	

	#


	


		
$
	
	





&
	
'#
	
+
		


7

	

 - .
'$
$

	
'

*


'		
	#


		$0 	'
	

	
	
		


$
3	'
#$

#




	
	5
	

		#

+


	#


7
	'

	

:
7
	
$	

	#


70 


''
#$

+
		
	
	
	
	



 2
 

 
 
 	
	 

 
 
	
	
 	
 	

 '0

#
	


	



!$ 
	''0 
	
	
	

	

'	
	

*

#
	
	''

	
	'

	
	




'$
$


	


'
	

 



&
	''



	

	
+

 - .

	



'

$

''
	

	 &

	'



'

'

	
6

	
	#

	 &



	

	



'$
$
0 

	



	#
'
 - . &
'
	

	

''


		2
	0 

	

	'
	0 +
	

$






3
	
0 
	
	'
	

	

'
'



	6
	

#

	


+


#
$
'

	

	
	'
	
	
4	
6$


+	$
#	7


5

;
	

	
0 

!'

#
#	

 - '$
$

	
	!

NASA/TM—2014-218317 1
0.1

0

0.1

0.4

0.2

0

0.2

0.4

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

Max Wdot

0.2

0.25

0.3

0.35

0.4

; < ='
		

	
	
	'
	
	


	
	
>deg/s2?


	
0 
'

$<

 ,	$
6
&
'




		

	
	

#

$
	'


	''
	0 +

	
	
	
	

 

,$
6
&
	'




	
	

	' 
+
'$

$


#
	
	
	#


	0 

+

	
	
	


'

 ;

4!$
6
&
	



'
3+

'
$
	






	


'$
$
5

' 
+
'$
$


#
	
	
	#



	
+
*	

	
	0 

+

	




'$
$


'


	

'	0 

'	

	
	

		

$
	

$
+
'


'< 
	
	'
	
	

 3	'


 !	5 (	$

	
	'

	

	


#

	

3	 4π 	'	
5
+


'
$
#


	



 &
,&@,8 @1A=B 
	

+	
'

	
	






=
 4( )'
	


;  +







	'


	
	
'


'
	

 - '$
$

	
	




	 &
	
'!

	


		




	$
3
	
		

	
	
	

#

	#
$
	5

NASA/TM—2014-218317 2
;  +




	



'$
$

	

	
	




	0 +

	


'


	


		

	
	
	'

;  +




.



 - .
'$
$

	
	




	0


	'


	$

&
	

'$
$

	

3	
	



''
	
	5
	

#
	


/


$
'

*7
$
'

	





 





, 

		


$


	

 	


+
	

	
#


'
'#
	
	
	

'
	 		< 	
$
	
+

'0 	


+


	
7
	'
	 &
	


+



'



4	


7
	'
	
	
#



+

		
	
	
	

	

+

	$

	
	

	

	
	

&
	
	'
CC


#
''	'
	
#	'


	


	'
	#
'


3


	D 
$



	
	

	#
'50 #


#	
$$
	'

$
	

	

&
'$
$

	


	
	
	

	

	
$
	

	


#
''6

	' ;
	

0 $	

$

	
+


#


$
)E
3

!5
	

	



	0 
	
+


#
))E
3
	
+	$
5
	
	



	0 	'

	
+


	

	

	
		
#+
)E
	'
))E

&
$
	
	'

	
		
	
	


7$

!
$

	
'$

$

	


	$
	
	

	



$
	




&
'$
$

	

	


	
	



)E

))E0 	7

#


	

	




		
$	

$ 8$
	

$


'

0 	
F




	

	' 

$


	
#$

!'

''
	
	
	6


	
	
'



	 =



	
3+
	

	


	



5
	'

	

3'

'
5
	
7+0 	

	


'
!'
#$
	$
	
	
#

	'

 


	


&
	



	
'$
$

	


'		


	

	
#
''	6

'
	
#



+
	

'
	

	

$<

 	
	
6
&
	
	
'


+

	
!	
	



	


	#
'


30 #
	
	.
	

/5 	
	

NASA/TM—2014-218317 3
0.10 0.1

0.4

0.2

0

0.2

0.4

0.4

0.2

0

0.2

0.4

Max Wdot

0.2

0.25

0.3

0.35

0.4

0.10 0.1

0.4

0.2

0

0.2

0.4

0.4

0.2

0

0.2

0.4

Duty Cycle for S4FB

0

0.2

0.4

0.6

0.8

1

0.10 0.1

0.4

0.2

0

0.2

0.4

0.4

0.2

0

0.2

0.4

Duty Cycle for S4RB

0

0.2

0.4

0.6

0.8

1

0.10 0.1

0.4

0.2

0

0.2

0.4

0.4

0.2

0

0.2

0.4

Duty Cycle for S2FB

0

0.2

0.4

0.6

0.8

1

0.10 0.1

0.4

0.2

0

0.2

0.4

0.4

0.2

0

0.2

0.4

Duty Cycle for S3RB

0

0.2

0.4

0.6

0.8

1

0.10 0.1

0.4

0.2

0

0.2

0.4

0.4

0.2

0

0.2

0.4

Duty Cycle for S2FB

0

0.2

0.4

0.6

0.8

1

0.10 0.1

0.4

0.2

0

0.2

0.4

0.4

0.2

0

0.2

0.4

Duty Cycle for S2RB

0

0.2

0.4

0.6

0.8

1

0.10 0.1

0.4

0.2

0

0.2

0.4

0.4

0.2

0

0.2

0.4

Duty Cycle for S1FB

0

0.2

0.4

0.6

0.8

1

0.10 0.1

0.4

0.2

0

0.2

0.4

0.4

0.2

0

0.2

0.4

Duty Cycle for S1RB

0

0.2

0.4

0.6

0.8

1

; < A
	
'
'$
$

	

	
	




	
>?

NASA/TM—2014-218317 4
0.1

0

0.1

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

Duty Cycle for S4FB

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

; < =

	
'
 - .
'$
$

	
	




	
>?

NASA/TM—2014-218317 5
'
	

	
 
 0 	'

 
	
!
	'
#	$

	

!

	
	
	
	
#'

 

$

6
&


$

'
	7
	
	
	

	
0 !'

	





$'
	
	


	
'0 	'

	

'$
$6




	
	 &	



	'
#	$
'	
	
	



	



 =
$

!	


	
	

	

*

	


, 	-(


&
	
	



#
''	'

	
	




	





	

	
	
'

'

#
	'
+

$

'
	

	

,, 









&

	
 L ̄i 	'

 F ̄i 

"

$
	#
'

	

	'' &

	6


	'



	

	
	0 #



	#
'

	$
	

&
	 m0 
 
	 L ̄cm0 	'
	  ̄I 

 

	

	'' &
	
 
	
$

	
'
	
	
	
	

/

&
	
		

	
	 a ̄i '

	
 i 

+
!
	
))E
'$
$


'


*	 

a ̄i = F ̄i/m 35

&
	

		

	
	
	 a ̄ 
 3 × 8 	
	
	


 a ̄i 

&
* τi '

	


'

*	 

τi = 

L ̄i − L ̄cm

× F ̄i 35

&
		

	
	 ω ̄ ̇ i '

	


'

*	  1
	

*



	
	

ω ̄ ̇ i =  ̄I−1 · τi 35

&
	

		

	
	
	 ω ̄ ̇ 
 3 × 8 	
	
	


 ω ̄ ̇ i 

NASA/TM—2014-218317 6
,, .


/
$

,
	






#
''	'0 	
	
	

	
+	$
	
	
)E


))E
#


'

 &

	
#'0 	'
	
2
6

#
	
	


'
3#$
	

5

#

	
		$ DCsOfV erticiesR0

+

*	 

DCsOfV erticiesR =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

11001100

11101101

10100101

10000100

10010110

11011110

01011010

01001000

01101001

00110011

00100001

10110111

00010010

01111011

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

35

&

	

	
 V ̄i 
'
+
*	 G

V ̄i = DCsOfV erticiesR · ω ̄ ̇

3G5

&



#
''	'
	
'

''

	


	


&
	
 '#'
 #$

 
 
'0 +
	
 '
 3#$
 	

5

 #


	
		$ trianglesR0 +

*	  3

 24 × 3 	0 


+




	
	'5

&



	
	
	
	
 i 	
'
#$
	
 i
+
 trianglesR





'

	
	
0 	'



'

	

'





	 V ̄ 

NASA/TM—2014-218317 7
trianglesR =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

234

241

456

461

678

681

892

821

12 3 11

12 11 10

13 5 12

13 12 10

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

14 7 13

14 13 10

11 9 14

11 14 10

12 5 4

12 4 3

13 7 6

13 6 5

14 9 8

14 8 7

11 3 2

11 2 9

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

35

,, 

0



(

&
 '
 
	

	0 

 



!'
 
	
!
 
 	
	




	
	
	
 ;
	
	
0 


'!'
#$

 V ̄1, V ̄2, V ̄3

'
	

	
'#'
#$

	
! a, b, c, d +

ax + by + cz + d = 0 35

&

	
!
	
	

	'




*	 " 	' %

[abc]=(V ̄2 − V ̄1) × (V ̄3 − V ̄1) 3"5

d = V ̄1 · [abc] 3%5

,, 

"



0



(

8	$
!

	
	

	
	

#
6	

	'

'

	

	6

 ;
	
	
0 


 V ̄1, V ̄2, V ̄3 	
'



%
#	$

! c1x, c1y, c1z, c2x, c2y, c2z, c3x, c3y, c3z 
*	 )

⎡

⎣

c1x c1y c1z

c2x c2y c2z

c3x c3y c3z

⎤

⎦ =

⎡

⎣

V 1x V 1y V 1z

V 2x V 2y V 2z

V 3x V 3y V 3z

⎤

⎦

−1

3)5

NASA/TM—2014-218317 8
, 

"
(


&


$




#
''	'

	
	

	

$


	

	
	
	
 '
 
 +
 	

 

 '
	
 ,


 

	

	'
	
3



	

/0 +
+
	


	#
'








	
	
5

,, &
(
(


, #
''	'
	
#



	

#$

#	


'

	
	

	



	'
!
3	

$
0 #
	5

	 V ̄10 V ̄20 	
+


! 

B
	
'
 P ̄ 3 [Px Py Pz]5
	
	


	'
	
	


	
'!







	 [V 1x V 1y V 1z] 	' [V 2x V 2y V 2z]0 


	
'	
+


'	

+
'



	

'

'


4*	  +
	
'	






	 &

+


#


	





	
'


'



	0 	'
	


	



	
'



















V 1x V 2x Px

V 1y V 2y Py

V 1z V 2z Pz



















≤ 0 35



'	


'
+
	




	0 



	



8
	

	

	
 	

 	
 
 $
 	
 '
	
 	

 	
 	
	
	
 ;

	
0 	



>&0 ;	
0 ;	
0 &0 ;	
0 &?

	


'


	

	



, 	

8
	
 



	

#
	
#
	'
#$
	

	

8
	





'

	
	
 &
	

	0 	'
	
#

'
	
	
	#



7
3#$
	

8
	
		$
	
	
#	$
#0 	'

	
#
	
	
	#

'




'
	

#5

&

	
'



	
	

	
'

	

	0 


$
	



	
' 
	


'	
*
"


	
	'
G
	''0

	'

'	
	
*'0 




	


	
	
'
*

	
	


G



	'
)
	''

,, 1





=


	


'0 



#+

'

	'


	
	
	

' &


	

	
 Q ̄0  [Qx Qy Qz] 	
+


NASA/TM—2014-218317 9
; < -

	





	
#
''	'

	


! G

&
 
	
! a, b, c, d 
	
	

+
 '
 '

	
	
	


&

	
!
	
'


	
	

	 t 	
' Q ̄  P ̄0 	
+


*	  	'
*	 

t = −d

a Px + b Py + c Pz

35

Q ̄ = t P ̄ 35



*	  *



	0 
	''
	'
	
'
3& d !



'
	$+


	'
	
#
'


	'
5 4*	  *



6

	 ;'

	

	

*
	
	





	0 
	''
	'


'
	

,, 
"








8	$
'	
	


	
#

	

	




	'
	

	


	

	


NASA/TM—2014-218317 10
origin

V ̄1

V ̄2

V ̄3

P ̄

Q ̄

; G< &
	

	






'

	'

	


&
#	$
! c1x, c1y, c1z, c2x, c2y, c2z, c3x, c3y, c3z 
	
	

+
 '

'

	
	
	
 &
#	$
!
	
'



#	$

'	 b1, b2, b3 	
+

*	  

b1=[Qx Qy Qz] · [c1x c2x c3x] 35

b2=[Qx Qy Qz] · [c1y c2y c3y] 3G5

b3=[Qx Qy Qz] · [c1z c2z c3z] 35

4	


'
'
*



	
	'

	''0 
!'

#	$

'	
*
	
	


%


	
	'

	''

,, &

2
(



&

	'
'$
$

	






'
'


'$
$

	

	


	


	'

#	$
'	0 	
+

*	 

interpolatedDC = [V 1DC V 2DC V 3DC] · [b1 b2 b3] 35

&
	

	

'

	



 - .
3
#	$
'	
	


	0 #

'$
$

	

	

	


	$5 4	
'
'
*


6


	
	'

	''0 
!'


	'
'$
$


	



 - .
*
	

	





	
	'

	''

NASA/TM—2014-218317 11
 	
!



"

H





#
''	'

	
	
	'


'	








	


	
'
0 
+
	
'



	 10−12E &


	

$
	

	

;

	0 #
''	'

	

	
6	
	

$
'
6

'
'$
$

+
*
	
 &


	

	
	
6	
0

	


	$0 	'





# 	
$"

&
	




	



	
#
''	'

	

	

	
	'

	#
 

=	  

$ ,'' I'

;'
		

	
	


.  %G ) %

	
		

	
	

   %)

;'

	
!

	
  ) )

;'
#	$
!

	
  %

&	
  ) G

&	#
 < 	



	
	

&
	







$

	



 - .
	

'$

$



	
	
	'

	#
 

=	  

$ ,'' I'

'$
	

3
'5  G) )

A


	    

8	$
'	  %  )


	 "   )

&	
 % GG 

&	#
 < 	






$


8$
6
	
	$
	

	
#



	
	
'0 



$


	


	

	
#
''

	

	
$
	


#



NASA/TM—2014-218317 12
200

150

100

50

0

50

100

150 150

100

50

0

50

100

150

150

100

50

0

50

100

150

y

Pure Translation Interpolated DC Acceleration Magnitude

x

z

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

; < ='

	
	
	
	
	'
	
	


	
	
>f t/s2?

% 

&


&
	''



	

	
	
+

 - .

	



'6


$

''
	
	
' @7

	
	0 
$


	
	'

	



	
!	 &
	
	


	
	
*
	




	
	


$


	''
'0 +

		

 &


	'




	
	

	'0 	'
!	

$

	



'


'

;  +

'

	
	


	$
3	

	
	
	
	
	



	''
'5

NASA/TM—2014-218317 13
150

100

50

0

50

100

150

100

50

0

50

100

150

100

50

0

50

100

150

Impure Translation Acceleration

0.12

0.14

0.16

0.18

0.2

0.22

; < ='
CC
3+
''
	5
	
	


	$
>f t/s2?

:	
$0 
	



	
	


	$


	$


'




'
#$
'	
0 



#




	
	

	



	
	

' !
$(
&
"


()





C
		

C
	
	
	



	'0 
	


6

	



	

+
	



	



	
	

3	




#

	7

	


+	'
	5 ;  +

'
CC
	
	



	$
3+
''
		

	
	
	
	

+'5

; " +




	




	
	
'$
$

	

	
	





	0 +

	

'


	


		

	
	
	' ; %

+




.




	
	
 - .
'$
$

	
	





	0 
	'


	$

&

	
	
 '$
$

 	

	
'
 
$'
	
#

 #

''	'
+
	
#	' 
	
%
'
	'
G)
0 #

+

	


NASA/TM—2014-218317 14
1000 100

100

0

100

100

0

100

Max acceleration

1000 100

100

0

100

100

0

100

Duty Cycle for S4FB

1000 100

100

0

100

100

0

100

Duty Cycle for S4RB

1000 100

100

0

100

100

0

100

Duty Cycle for S2FB

1000 100

100

0

100

100

0

100

Duty Cycle for S3RB

1000 100

100

0

100

100

0

100

Duty Cycle for S2FB

1000 100

100

0

100

100

0

100

Duty Cycle for S2RB

1000 100

100

0

100

100

0

100

Duty Cycle for S1FB

1000 100

100

0

100

100

0

100

Duty Cycle for S1RB

120

140

160

180

200

220

240

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

; "< CC
	
	
'
'$
$

	

	
	




	
>?

NASA/TM—2014-218317 15
150

100

50

0

50

100

150

100

50

0

50

100

150

100

50

0

50

100

150

Duty Cycle for S3RB

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

; %< =
CC
	
	
'
'$
$

	
	




	
>?

		
	
	7

!
	
	

#
<

J &
	

$
	'
	
#
''
+
'	

J I$
$

	

	


	

)E

))E

J I$
$

	

	$

	
$
	

/	
	



$'

,



'$
$

	

	
'0 $
	
#
'

*	$

''

		

	
	 I$
$

	


	

		

	
	
	

#
6

	'


	

	
	
0 	'


	
	
	'

	

'$
$

	
#
#'

	




'$
$

	
		

	
	

3+




	
$5

* &
"
	



	

'$
$

	



	



	
	

$'

6


+

$

	

	

	

	0 

$

'
+


#
'#'

NASA/TM—2014-218317 16
*, 	-(


,
	


G)



	
	

$'0 	
	
	

	
+	$
	
	
)E


))E
#


'

 &

	
#'0 	'
	
2
6

#
	
	


'
3#$
	

5

#

	
		$ DCsOfV erticiesT0

+

*	 " 3

 50 × 8 	0 






	
	'5

DCsOfV erticiesT =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

10101010

00101110

00111100

10111000

11110000

11100010

11000011

10001011

00001111

01010101

00011101

01110100

11010001

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

01000111

01010100

00010100

00010101

00000101

01000101

01000001

01010001

00101010

00101000

10101000

10100000

10100010

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

10000010

10001010

00001010

00001100

00110000

11000000

00000011

01010000

10110000

01110000

11010000

11100000

00001110

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

00001011

00000111

00001101

00111000

00101100

00011100

00110100

10000011

11000010

11000001

01000011

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

3"5

&

	

	
 V ̄i 
'
+
*	 %

V ̄i = DCsOfV erticiesT · a ̄3%5

&



	
	

$'
	
'

''

	

%
	


&
	
 '#'
 #$

 
 
'0 +
	
 '
 3#$
 	

5

 #


	
		$ trianglesT0 +

*	 ) 3

 96×3 	0 







	
	'5

NASA/TM—2014-218317 17
trianglesT =

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

1 22 23

1 23 24

1 24 25

1 25 26

1 26 27

1 27 28

1 28 29

1 29 22

10 15 16

10 16 17

10 17 18

10 18 19

10 19 20

10 20 21

10 21 34

10 34 15

9 39 29

9 29 40

9 40 33

9 33 41

9 41 18

9 18 42

9 42 30

9 30 39

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

5 36 34

5 34 37

5 37 32

5 32 38

5 38 25

5 25 35

5 35 31

5 31 36

3 43 23

3 23 44

3 44 30

3 30 45

3 45 16

3 16 46

3 46 31

3 31 43

7 47 27

7 27 48

7 48 32

7 32 49

7 49 20

7 20 50

7 50 33

7 33 47

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

11 17 16

11 16 45

11 45 30

11 30 42

11 42 18

11 18 17

12 46 16

12 16 15

12 15 34

12 34 36

12 36 31

12 31 46

4 43 31

4 31 35

4 35 25

4 25 24

4 24 23

4 23 43

2 44 23

2 23 22

2 22 29

2 29 39

2 39 30

2 30 44

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

,

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

8 40 29

8 29 28

8 28 27

8 27 47

8 47 33

8 33 40

6 26 25

6 25 38

6 38 32

6 32 48

6 48 27

6 27 26

13 49 32

13 32 37

13 37 34

13 34 21

13 21 20

13 20 49

14 50 20

14 20 19

14 19 18

14 18 41

14 41 33

14 33 50

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

3)5

*, 

"
(


&
	
	
 
$'
	
 #
 


	

 	
 #$

#	
 
 
 '


	
	
	



	'

3	

$
0 #
	5

	 V ̄10 V ̄20

	
+

! ) =
!	

'	






	

	


!



#+

+
	
	

3
'
#+

'
	'
	

	


! )5 
	
0 )
'	
	
*'




	
	
	



	



$'

	

	
	
'

*, "
"





=

'$
$



	
	 DCit 	
'0 
''
	

	 ω ̄ ̇ u

	
#
'
+
*	 

NASA/TM—2014-218317 18
; )< 1

	





	
	
	

$'

	

	

ω ̄ ̇ u = DCit · ω ̄ ̇

35

1	

''
	 ω ̄ ̇ u $
'

	
	
' P ̄c ''

	6



	0 +

'
	




#
''	'

	
	


!'




'$
$



	


	
'< DCc 8	

	




'$
$


	
	$0 
'$
$

+


	

	
		


	
	 ω ̄ ̇ c 	#

	
	0 	
+

*	 

ω ̄ ̇ c = DCc · ω ̄ ̇

35

&
 	
 '$
 $

 '

 #
 	
'
 '+


	'
 *'

 


+	'
	0 

 DCr 	
+

*	 

DCr = DCc · −1

3

(

ω ̇ ux

ω ̇ cx

+

ω ̇ uy

ω ̇ cz

+

ω ̇ uy

ω ̇ cz

) 35

*, 
"
"
3






&
!'

		

	
	
'$
$
 DCt0 
'$
$



	
	

DCit 	
#'
+

'$
$



+	'
	 DCr #$

$
	''


NASA/TM—2014-218317 19

	'
	





	

	


"
'$
$


))E0 	
+


*	 	' G

DCsum = DCit + DCr 35

DCt = DCsum/ 	(DCsum) 3G5

&
C
	'
	
C
*
	
	

#
'

#
	#	$
	
	
	'
	6


	'




'$
$
 

*	






#+

+

	'

+ &
	
!



"

,
+

	

	
	
0 
	


	
	
'$
$




	


	
	

$'



	

	
+
'	









	
	

9+0 #


	
	
'$
$

+
	

	



		

	
	
'$
$

'




	

	
+


	
	







H
	


#

	
	
'
	
''0 
	
'

	
	
	6

'
'


	'
		

'
'$
$



	
	

)E0

+
	
+
	
'

)E &
+
	
'




	 10−5 '

H


	
	



	
	


*


0 

	
$









	
	
6	

	'


	$

A
	
	

	


	










!$ H
	



#

	
	
'
	
''0 
	
'



!$
3	'
#$





'$
$
5

E
+



	'
'$
$
0 +
	
+
	
	






G)E

&
'
#+

	'
	'
	



!$


	
	
'$
$

	

	


	
	
'

+

!  &


+
	





'
'


	


	0 	'
	
	
	
	#
G
'

	

 &
	
$"

&
	







$

	



 - .
	

'$

$

 
 
	
	
	
 	'

	#
  &


'

 #



	
	
	'


	
'$
$
0 	
+


	
#


NASA/TM—2014-218317 20
1.5

1

0.5

0

0.5

1 1

0.5

0

0.5

1

1

0.5

0

0.5

1

y

Pure Translation Fuel Consumption Error (max: 1.32273))

x

z

0

0.2

0.4

0.6

0.8

1

1.2

; < A
	
	



'
#+
	

	'

	'

=	  

$ ,'' I'

'$
	

3)
'5  %) G) )

A


	    

8	$
'	  %  )


	 "   )

& 	
	
	
  )

	
  % GG 

 	
	
	
   )

-	

	'
#  " " %

&	
 " " 

&	#
 < &	
	






$


NASA/TM—2014-218317 21
 
"

=	

 - .



'	#


		
#	


	
	
'

0 	


	$0 	'



 8
6#
	

	

	
	7


0 	'
		
	#


7

	


	7



	 &
	
'#
	
+
!
		


	
	


	
	7

	'		



$


	




;

	0 
+
	

$
'
	

	
+
	


 
	

	
6

	
6	
0 	


	$0 


0 	'
	
	
6

	0 *
	
	



$
%
	
	


	



A
	
	
	
 #

+

 +
	
0 #
 *
#
	
	
	
+

''
	
	'
	
	
	 

	0 


	

	

	

+

 	
0 #
	
 $




 
	
	
6	

	'


	6

$ 9+0 

	
	
3E
	0 G)E
	5




	'





&
		


$


+
	


 	
$

+0 #


6
'



''


$$

	



	
	
#
$	





		 &


'

	
#	'


$


=
 4( 		0 	'


	
	#
$


	



		

''


$

&	'	

.


	

	
$

0 #
	

	
 &$

	


G)
'

	
	
'
0 	'
$
	7

	






H
	'

	'	

.

0 
	'''

$


+
		

#

.!'

	
+
		$
	'
!$
	

	 ;
	
0 '
6

$
	
+

		0 
	#
$

	
	



'
	
	'

!	 &
	#
$

'
		

	
	'
	
	
+
'	#

#	6




+

'
#





		





		
#$
		

,'
!	

$0 

'	
 	
7
	
	

	

'
 




NASA/TM—2014-218317 22