MCDOIiIIELL DOUGLAS TECHNICAL SERVICES CO. HOUSTON ASTRONAUTICS DIVISION

SPACE SHUTTLE ENGINEERING AND OPERATIONS SUPPORT

DESIGN NQTE NO. 1.4-8-020

EULER ANGLES, QUATERN IONS, A140 TRANSFORMATION MATRICES FOR SPACE SHUTTLE ANALYSIS

MISSION PLANNING, MISSION ANALYSIS, AND SOFL6I~IARE FORMULATIOIJ . .

This Design Note i s Submitted t o NASA Under Task Order D0710, Task Assignment D, Contract NAS9-14960.

. Prepared by:

Approved by:

Task Manager 488-5560, Ext. 248

' d ? Approved by: - , , F , , > ~ . ' & e / J . .

P. M:'Efugge ' Work Package Manager

488-5660, Ext. 222 vl M A & G 3 = ' C ) T;=mO) - 4 W H

Approved by: U ! (f db!* Approved by: W. E. Wedlake

%?'a/[- W. €. yltyes

In teg ra t ion Manager 488-5660, Ext. 293

WBS Manager 488-5660, Ex t . 266

DN N O . : 1 ,4-8-020 Page: 1

Due t o the extensjve use o f the quaternion i n the onboard Space

Shuttle Coniputer System, consf derable analysis i s being performed using

relationships betwclen the quaicrnion, the transformation matrix,

and t h e Euler angles. This Design Note offers a brief mathenlatical

development of the relationships between the Euler angles and the

transformation matrix, the quaternion and the transforma'tion

matr ix, and the Euler angTes and the quaternion. The analysis

and equations presented here apply d i rec t ly t o current Space

Shuttle problems. Appendix A presents t h e twe l ve three-axis

Euler transformation matrices as functions o f the Eul er angles, . .

the equations f o r the quaternion as a function o f t he Euler angles,

and the Euler angles as a function of the transformation matr ix

The equations o f Appendix A are a v a l u a b l ~ reference il: Shut t le

analysis work and this Design Note i s t h e o n l y known document where

each o f the twelve Euler ang le t o quaternion relationships a re

given, Appendix B presents a group o f u t i l i t y subroutines t o

accomplish the Euler-matrix, quaternion-matrix, and Eul er -

quaternion relationships o f Appendix A,

DN NO. : 1.4-8-020 Page: 2

2.0 DISCUSSIOt4

2.1 Euler Angle Transforlnation Matrices

The f o l l o w i n g analysis and util i t y subrout ines are offered

t o simp1 i f y computer programs when working w i t t l coordinate

transformation matrices and their relationships w i t h t h e

Euler Angles and the Quaternions . The coord ina te t r a n s f o r -

mation m a t r i c e s discussed .here are def ined using the f o l iowing

figure,

FIGURE 1

The t ransfornlat ion m a t r i x M , i s def ined t o transf0r.m v e c t o r s - - -

i n the ?- system (x, y , z ) i n t o t l i c . o r i g i n a 1 x-system (x , y,

2 ) and i s givcn by the equa t ion ,

x = MST

where (1 )

- - - x = (x , y, z) and 51 = (x, y, 2 ) .

Using the right-hand rule f o r p o s i t i v e r o t a t i o n s , t h e M mat r ix

i n (1 ) above i s c o n s t r u c t e d by t h e fo l l owing analysjs. The

f i r s t r o t a t i o n i n Figu re 1 above i s about the x-ax is by the

amount The s i n g l e r o t a t i o n about t h e x-ax is r e s u l t s in , I

t h e fo l lowing t r ans fo rma t ion ,

or x = X? i n ma t r ix -Form. Rota t ion about the y ' - a x i s by t h e

amount O2 y i e l d s t h e i n t e rmed ia t e transformation matr ix :

or x' = Yyb i n m a t r i x form. F i n a l l y rotation about the ?'-axis

by the amount O3 yields the in t e rmed ia t e t r ans fo rma t ion 'ma t r ix ,

DN NO. : 1.4-8-020 Page: 4

and i n n ~ a t r i x forn i = zZl. NOW using the th ree equations,

by substitution

x = ( X Y z) x. Then frorn equation 1,

M = ( X Y Z)

Computation f o r the M m a t r i x f r o m the indicated matrix multi-'

p l i c a t i o n i n equa t ion (7) yields,

(- C O S O ~ sine3) ( ~ i n 8 ~ )

sinsl sinee cose3) (case, cose3 - sine, sine2 sineg)(- pine, 3 3 ~ 9 ~ ) (S l - cosel sine2 c 0 s 8 ~ ) ( s i n 0 ~ coseg + cosel sine2 ~ i n e ~ ) ( c o s 8 ~ cvse2)

The m a t r i x M in equation (8) i s a function of;

(1 ) The three Euler angles el O2 and e3 and

(2) The sequence o f rotations used t o generate the 'matr ix .

By examination of equation ( 7 ) i t i s possible t o show t h a t t he re

are twe lve possible Euler rotat ional . sequences. If the . ( X Y I )

notation i n equation (7) represents a ro ta t ion abou t the X axis,

then the Y a x i s and f i n a l l y t h e Z a x i s , then t he following per-

Df4 NO. : 1.4-0--020- Page: 5 . ,'

niutations o' t h e r o t a t i o n a l o r d e r represents t he twel ve possi t l o

Eu7 er angle s e t s using three r o t a t i o n s ,

X Y Z Y X Z Z X Y

X Z Y Y Z X Z Y X

X Y X Y X Y Z X Z

x z x Y Z Y

Any repeated a x i s r o t a t i o n such as XXY does not represent a

t h ree a x i s rota t ion b u t reduces t o t h e two axis rotation XY. .

Hence the rotations described i n (9) above represent a17

twe] ve poss ib l e sets o f Euler angle defini.ng sequences. Con-

versely then , f o r a given transformation matr ix there are

twe l ve Euter angle sets which can be extracted from the matrix.

The 'Eul er matrices corresponding t o a1 1 possible rota t ional

sequences o f ( 9 ) above are presented i n Appendix A.

The util i t y sub rou t ines "EULHAT" generates t h e trans'formation

matr ix from a given Eufer sequence and the Evler angles . The

util i ty subroutine "EIIATEUL" extracts the Euler angles from a give^

Euler rotational sequence. The convention i s es tab l ished t h a t

the Euler angles occur i n the same sequences as the a x i s

rota t ions . U s i ~ ~ g this coccept and functional n o t a t i o n , equat ion (7)

could be expressed as

M = X Y Z = M(BX, Qy, ElZ)

and from (9 )

Dl4 Na. : 7.4-8-020 Page: 6

M = X Z X = M ( o , , o,, 8;) e t c . (11 1

This convention i s assurncd i n both subroutines and a l s o

used t h r o u g h o u t this design note when Euler angles are

used. A brief expfainatjon of the use of these two

u t i l i t y subroutines i s gjven i n Appendix B.

I t i s interest ing t o note tha t a negative rotation in

t h e s ingle a x i s rotation matrices of equations (21 , ( 3 )

and ( 4 ) will r e su l t in the formation of the transpose

o f the matrix. However the transpose o f M in equation

(7) i s f o r m ~ d from reversing the order o f multiplication

' ,and transposing the ind iv id~ ,1 a x i s rota t ion equations,

i .e.

Hence the transposes of the matrices o f (9) are eas i ly

formed by reversing the order o f mu1 t i p7 ication and

transposing each s ingle axis rotation equation.' Using

the notation i n equations(l0) and (11 ) above equation

(12) could be nr i tten, -

I t i s recornended t o avoid confusion t h a t the 'forward

transformation be conputed and simply transposed t o yield

the reverse transforrnation m a t r i x . All matrices of Appendix

A are i n t h e forward form, i , e . X blx and formed from (9).

DN NO. : I . 4-8-020 Page: 7

2.2 TRAPISFORI-IATIOtI I4ATRICES USING THE tIAI.11 I-TON QUAl'EREIIOfI

The transformation mat r i x o f equat ion (1) can be wri t ten as

a function o f the Hamil ton Quaternion ;

q2 = cos a s i n w/2

93 = cos Psin u/2

q4 = cos y s i n w / 2 , where o is the rota t ion an i l e about t h e rota t ion axis w i t h

a, 8 . and y d i r e ~ t i & ' a < ~ l e s with the x, and z axes re- 2 2 2 . 2 - spebtively:. Notice a l so t h a t ql -I- qp + q3 + q4 = 1, s ince

2 2 2 cos u + cos p+ cos y 1 . The rotatiori angle, w, i s assumed

p o s i t i v e according to the r ight-hand ru l e o f axis ro ta t ion . . .

The mat r i x M becomes

For a more detailed discussion. of the derivation of equation

(16), see Reference 1 , Using functional notation, equation

(15) can be written,

M = M h l . q2. q3. q4). (76)

Ulll i ke the Eul er angle rota t ional sequences to describe the

transformation matrix o f equat ion (1 ), only two quaternions

can be . . found from equat ion (15). The two 'quaternions are:

DN NO. ; 1 ,4-8-020 Page: 8

and 93 -93

These two quaternions represent a po.si t i v e r o t a t i o n abou t

the r o t a t i o n a x i s po in t ing i n one d i r e c t i o n and a p o s i t i v e

r o t a t i o n about the same line o f r o t a t i o n po in t ing i n the

opposiPe direct lon. Both quatern ions o f (17) s a t i s f y equa t ion

(15).

The, u t i l i:ty s u b r o u t i n e "QMAT1' generates t h e t r ans fo rma t ion

matrix from a given quatern ion. The ''91t4Tfl a lgo r i t hm gene ra t e s

the matrix as given i n equation (15) w i t h o u t dup l i ca t i ng any

a r i t h m e t i c operat ions. The subroutine "MATQ" e x t r a c t s the p o s i t i v e

quaternion, i .e., q, > 0, from the t ranzformat ion m a t r i x and

normal izes the r e s u l t s t o guaran tee an orthogonal matr ix . In

order t o avo id any d i scon t i nu i t y i n ex t rac t i ng t h e quaternion

from t h e t r ans fo rma t i on matr ix, the procedure as described i n

Reference 2 i s used.

E a r l y works by Hamil ton (Reference 3) presented t h e qua te rn ion

as having a sca la r and a vector p a r t , i.e.,

and equation (76) c o u l d be expressed as, -b

. M = Nql, q 2 9 q3, q4 ) = M(5, V ) .

DU NO. : 1.4-8-020 Page : 9

For a given quntcr t i ion the fo l l owing r e l a t i o n s h i p i s

true (from (17) above), -t -C

M(S, V ) = l. l(-S, 4) . (20)

The transpose o f t he t rans format ion m a t r i x is given by,

2.3 EULER ANGLE AND QUATERNION RELATIONSHIPS.

By examination o f equat ions !lo) 2nd (16) the equality,

can. be erritten. Based on an equality for ,each element o f the

m a t r i x t h e following nine equations murt be t rue ;

sinel C O S ~ ~ +- cosel sine2 s in8 3 = Z(q3q4 + q1q2)

I t i s p o s s i b l e to solve for the values o f the quatern ion using

the t r i gonomet r i c h a l f angle i d e n t i t i e s . For t h i s Euler sequence,

i .e. X(el ) Y (e2) Z (ej), the f o l l o w i n g quaternion resul ts ;

DN NO. : 1 ,4-8-020 Page: 10

Appendix A gives the quatern ion as a function o f t h e Euler

angles for each of t h c tvrelve Euler r o t a t i o n a l sequence pre-

sented in Section 2.1. The special equat ions for t h e qua te rn ion

s func t ions zr" the Euler angles, 1 i ke equat ions (24) above,

are sometimes cumbersome to use, especially when multiple

~ u l e r sequences are utilized. Less coding, ' bu t perhaps more . . computer apera t ions , a r e r equ i r ed by using the more general

method of f i r s t gene ra t ing the mat r ix M fronl thz given Euler

angle set and then simply extracting the quatern ion from the

matrix. This method i s e f f e c t e d by f j r s t a cal l t o ,"EULt44T"

and then a c a l l t o "MASQ"..

ON NO.: 1.4-8-020 Page: 11

3.0 REFEI1T;lICES -we-.--

1. Working Paper: MDTSCO, TI1 No, 1.4-MPB-304, E914-8A/B-003,

"Quaternions and Quaternion Transformations ,I1 Dav id M.

tlcnderson, 23 June 1976,

2. Transmi t t a l i4cmo: MDTSCO, "I .4-MPB-229, "Irnprovi ng Con~pu ter Accuracy

i n Ex t rac t ing Quaternions," Dav id M. Henderson, 9 March 7976.

3. Sir W i l l i a n ~ Rowan Hamilton, LLD, LL,D. MRIA, D.C.L. CANTAB.,

"Eiements of Q u a t ~ r n i o n s " 2 Vo1 m e s , Chelsea Put1 i s h i n g Company,

New York, N. Y., 3rd Edi t i on 1 969, L ibrary o f Congress 68-5471 1

ON NO. : 1,4-8-020 Page: 12

APPENDIX A

The twelve Euler matrices for each ro ta t ion sequence are gSven based.

on t he s i n g l e a ~ i s rotation equations ( 2 ) , ( 3 ) and (4). The Euler

m a t r i c e s transform vectors from the system t h a t has been ro ta ted i n to

vectors i n the s ta t ionary system. Also presented here are the equaticns

f o r t h e quatern ion as a function o r t h e Euler angles and the*Euler

angles as a funct ion o f the matrix elements for each rotation sequence.

DN NO. : 1 -4-8-020 Page: 13

A x i s Rota t ion Scquecre: 1, 2, 3

-1 "13 O2 = t a n ( 2)

'-ml 3

Dt-i NO. : 1.4-8-020 Page: 14

Ax is Rota t lon Sequence: I , 3 , 2

DN No. : 1 -4-8-020 Pagc: 15

Ax i s R o t a t i on Sequence: 1, 2, 1

e2 = tan -1 (F)

DN NO, : 1 .4-0-020 Page : 16

(4 ) El = M ( X ( e l ) , z(e2) . X ( 9 ) = XZX

Axis Rota t i on Sequence : 1, 3 , 1

B2 = t an - 1 ( V ) "1 1

Axis Rotat ion Sequence: 2, 1 , 3

Axis Rotat ion Sequence: 2, 3, 1

- C O S O ~ C O S ~ ~ - C O S B ~ ~ ~ ~ ~ ~ C O S D ~

+ ~ i n 0 ~ s i n O ~

sine2 C O S O ~ C O S ~ ~

-sinOl cose2 sine, s i ne2cosf3, -sine, sin02sin03

+cosol sin€$ -

@1 = tan-' ( z)

DN NO. : 1.4-8-020 Page: 20

Axis Rotation Sequence: 2, 3, 2

Axis Rotation Sequence: 3, 1 , 2

O3 = t a n -1

DN 110. : 1 -4-0-020 Page: 22

Axis Rotation Sequence: 3 , 2 , 1

A x i s Rota t ion Sequence: 3, 1 , 3

DN NO. : 1.4-8-020 Page: 24

A x i s Rotation Sequence: 3, 2, 3

€I2 = tan -1 (J- )

Thc f o l l o w i n g subrou t ines w i t h a b r i e f description oc t h e i r use are

presented i n t h i s appendix.

(1) "EULMAT" - Generates the transformation m a t r i x from a given s e t of

Euler angles and an a x i s rotation sequence.

( 2 ) "MATEUL" - Extracts the Euler angles from the given transformation

matrix and an a x i s rotation sequence.

( 3 ) "QMAT1' - Generates the t ransformat ion ,natnix fron~ a g iven

quaterni on.

(4) l ' ~ ~ f b l t - Extracts t h e quaternion from a given transformation

matrix.

(5) "YPRQ" - Generates t he quatern ion di rec t ly from t h e yaw-pitch-

roll Euler angles.

( 6 ) '"POSNOR" - Computes t h e posi t ive-normal i zed quaternion fram the

given quaternion.

INPUT:

DN NO, : 1.4-8-020 Page: 26

EULMAT

Generates a 3 x 3 transfor~nation matrix from a

given sequence and Euler angle s e t ,

ISEQ - Ro ta t i on Sequence (Integer Array ( 3 ) ; i . e . ,

1 , 2, 3 )

EUL - Euler Angles in radians, i n "ISEQ"

Order ; ARRAY ( 3 )

OUTPC T : A - he 3 x 3 transformation matrix

ALGORlTHM REFEREWE: Appendi .~ A; Eul er Sequences (1 ) thru (12),

DN NO.: 1.4-8-020 Page: 27

EULER AlIGLES TO TI1E TRt\llSFOR'lATION MATRIX .

--11.1.---.---..*---- - -I.C-I.l".f.-.--l-l - . * . I . I C - ---..--. --I .-.----I.- -.. -.----__...- ..-. ----.. --.--- -.. s u j r t o ~ l ~ l r ! t LUL'IAT C N T R Y POI K T p~ir237

. .. . STGxALE , lSSIGF.~MENT t 2 t O C k , , T Y P E , V E t h T 1 V C L C C ~ T I D E S , lr ir l4C) . . " .

- - - 7 - - t 'Vl7. , : - . . (r- A *.a- -.- .--- -'

L ~ G J ? ~ . - ; - 78; IT IL.ET; .J 1 ~!cL[:'=x(K,J,;I-------'.' . 3.;; - 3F C L . i ( , . i 1 H O l . i . = , i I l : , J 1 - . y j ~ 7 9 ?+ If [ i ~ ~ 1 ! - I i r L i l ) . I - T m l . . . : L - l ,.I 60 T O 253 -a. 'L.Y. IF.(-t?F.5) 7 l;.,.I\ v - : ' ) . ] - : 1.T * 1 T 0 25-2

i 1s ~ € I ~ T I ' = T L . ~ P + X ( I r K t I - : l f : t I l i L ~ t 2+ 2 5 i C C Y T I :rUE 1 zg: l , F ( t . T - u , i ! i l ( ? , J f = T k - r Y P -

-.u ---_c! ;I +p T F ( L-m*~.&..* 1 , J l=J r PIP $ =.$

*- XPIZ 3 2 2 . CD:;TI XU:

EtJD DF C O K P J L A T I O l 4 : -.-

NO. .ET C\Gh!USTlCS. *.-

NAME: - PURPOSE:

INPUT :

MATEUL

Ex t rac ts t h e Euler angles f rom t h e g iven t rans -

formation m a t r i x and t h e r equ i r ed Eulcr

r o t a t i o n a l sequence.

I S E Q - R o t a t i o n sequence, (Integer Array ( 3 ) ,

A - The 3 x 3 transformztion'

OUTPUT: EUL - The Euler angles, in " ISEQ" ordcr;ARRAY(3).

ALGORITHM REFEREPIC€: Appendix A; Euler angles as a function o f the

m a t r i x elements, sequences (1 ) thru (12 ) .

TRANSFORf4ATIOlI -- IIATRIX TO THE LULER ANGLES -

STni;ftGf E55TENE!EP!T IFLOCK, T Y P E , R E L A T I V C L O C A T I O P . : , i (PFtE 1 . ..

- - r s i ?;.?L':! L 2 1 "3l27 r ; j O R 33Rl;l f 'jt!7.-ttrc3 L3Ll 1 '3L335

DN No. : 1.4-0-020 Page: 31

.- . -- - T TRANSFORlllATION WATIZI X TO THE EULER ANGLES - ( C O N T I ~ ~ I I F . ~ ) ,----- -

1__-__1_

E N D OF C O R P I t b T I O V : NO D I L 5 N O S T I C S .

DN NO, : 1.4-8-020 Page : 32

NAME : . " - QWT PURPOSE : . - Generates t h e transformation rnatri:, f r r r m the g iven

quatcrni on,

INPUT : - Q - The quater.r ran; A R R A Y ( 4 ) .

OUTPUT : A - The 3 x 3 transformation matrix

ALGORZTllMP\EFERENCE: - - - . . - - . Equation ( 1 5 ) f r o m S e c t i o n 2 . 2 .

Dl4 NO. : 1.4-8-020 Page: 33

QUATERt310N YO TtiE TRl\tlSFORi!ATIOI4 I IATRIX

NAME: - PURPOSE:

IIiPUT:

OUTPUT :

ON NU. : 1 .4-8-020 Page: 34

MATQ

Ex t rac ts t h e p o s i t i v e quaterni on f rom t h e given

transforn~ation m a t r i x .

A - The 3 x 3 t rans fo rn~a t ion m a t r i x

Q - The p o s i t i v e quaterni on;ARRAY (4 ) .

'ALGORITHM REFERENCE: See Reference 2.

DN 110. : 1.4-8-020 Page: 35

- S T O 2 h C . F ,555 Ibi.NEIkT ( I r L g C K , T Y P E , Gf L A T I V ~ L O ' C ~ T 1014, t ; A ~ : f 1 -

513iSI l e SUBF.GLlTlhC ! 4 A T C ! ( R , C 1 .. - ?* Jlt.:Ei\S'IRr4 A 13 , 3 1 , C ( q 1 , T ( i t 1 3C113.3 --- .--

jsictt 7 -+ -- -r PJ t ? O l C i 9JG=Sc S -. 48 30105 5 + SO 4L J = i r 4

6$ C I ( J ) = - ? [.-hL+--* , b * L ! bo7-m.*?-

--- a I 1-2 71= 8+ 1 F t J . - Z P . 3 ) LO I D 2 3 - ?+ - I F { J s L ' C , 4 I 50 TO 33 . . -.

3 ~ 1 2 2 120 G I J l = : .I# rgq - - -- ' --

c-9-1-2-1 3 l* = A t : , : ) 4 A ( ; , ; - l + & t 2 , ~ ~ + 3 * : 2 D 1 Z L 1 ?G T f J ) = J , ? -. E D 1 2 3 13* EO TCj

13 TEHPZ4tl 1 ) - A [ Z , z ) - s [ ; , , f l + l T.[-J"j-=..x-~.;-2 I-- &-.[ ;-,:.I-- -- - GO Tc Sf 15=

179 2; fE%?=-h[l ' ) + t . ( 7 r 7 ) - ' h ( j , 3 i + l . S - 13s T - t d b = A i 1 I 4 ; - A (5, il

3~ 12 10:: G C T O 3 5 ufil32 2Z* 3fj TEblP=-A( I ! l)-ht2;?lCk ( ; , 3 I * ? . ; - SO133 2 1 * T [ J ) = % [ ? , i l - f t f i , - l

35 T F I T E ? ' I - ' . L T . i ; I S ] b O T O q'2 ,; ~ - ~ - = - 1 ~ , p '-- - -- -.

iliiL3l *. c'Q .;r 1'3 35145 ~ 5 3 b O COhT I:.bE

1-F-t ?-.-L-Q.i-l+.GU- T O -5" . -

LLlG 4 27+ 4 1 1]= , . !~05 ,2?~- ( -? -15] - 331~1 5 L3+ I F ( I . ; . ~ ' . ~ I ! ) = A ~ - ' s . I ~ ' . ~ ~ ~ T ~ I ) / o ( I ) I - - CG147 2Q+ TE.?P=;2.25 / V t 1 1 30-1.5:'I ::$ L!P_-~L. J=2 tu 30153 33 u U( ~ 1 z ~ E . w ="ri~-l - 3G15q 325: Sfi C O N 1 ~ ~ J U F

Z Q 1 5 5 3 3~ 6 3 RETURN -- ;a157 34= EN13

&.

E P I D OF C O H P I L A T ~ O S : NO OT&CNOST 1 C S -

- '

-- ---

DN No. : 1 .4-8-020 Page: 36

NAME : - YPRQ

PUPPOSE: Generates the quaterni on d i rect ly frcm the yaw-

pitch-roll Euler angles, i .e . , a 3, 2, 1 Euler

sequence.

IPIPUT: YPR - The yaw-pitch-roll Euler angles J ARRAY ( 3 ) .

OUTPUT: Q0 - The p o s i t i v e qua te rn ion , ARRAY (4 ) .

ALGORITHM REFERENCE: Appendix A, t h e quaternion equations for Euler

sequence (lo), a 3, 2, 7 sequence.

NOTE: Th i s subrout ine ca l ls "POSNOR" t o t ake the normal o f t h e p o s i t i v e

quaterni an.

---- YAW-PITCIi-ROLI. EULER ArIGLES TO THE QUATERrJION -

5953 PUS i.:'J K ,73611 C O S r!n:3 SX.'*! - ---" -

I- ,7gCb N C R 2 3 ' i

DN NO* : 1.4-8-020 Page: 38

NAME: - POSHOR

PURPOSE: To o u t p u t t h e positive and nonnal i zed quaterni on

f rom t h e given quaternion.

INPUT: - Q - The quaternion; ARRAY (4).

OUTPUT: QO - The posi1;ive-normal ized quakern ion ;

ARRAY (a)., ALGORlTHM REFERENCE:

I . If the sign of Q(1) is negat ive:

Se t QO(T) = -Q(I) for I = 1, 2, 3, 4.

2. Se t QOfl) = QD(I)/TEMP

where TEMP = ~ Q O ; c (10; + QO; + Q O ~

SELECTS TCiE ,"3SITIVE Q!lATERf.IIOIl - AtlD ~IOP,MALIZES -.

S T O P , 1 s f 'OStcCrZ, POt , tJQr? FOIi 5 Z Z 3-1 ,2 / r G /7 7 - ; j G 2 : t t ! I

E N D 9F C O I I P I L A T I O U : NO 131 A G f : O S T i C S .