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28 October 1968 

RELA IONSHIP BETWEEN MIRROR GIMBAL ANGLES AND STEREO AND 

OBLIQUITY ANGLES 

W. C. ENGLEHART 

In this report, relationships between the mirror gimbal angles and the photographic 

or line-of-sight angles are derived for three coordinate systems : S 

v 

 , S

r 

 and S

r

'. 

Rotations as shown in Figure 1 : 

S

y 

—S1 rt

v 

about x

v 

Si 2 Evabout yi 

The transformation equations are: 

[xyl 

2 

z2 

y2 

z2_. 

= 

= 

cE

v 

0 

i 

LSE v 

cf. 

0 

sE

v xvi

x2- 

0 

1 

-sE

v 

0 

0 cE

v 

:-.,D

v 

en

v 

-cE sn 

v v 

where 

S

v 

 Coordinate System 

The photographic angles are defined in the vehicle coordinate system, Sv, as a roll 

about the x

v 

axis (D

v

), then a pitch about the displaced y

v 

axis (Ey). The definition 

of these angles in terms of main tracking mirror gimbal angles and the ATS tracking 

mirror girdua.1 angles follows. 

Stereo and Obliquity Angles : 

-sE 

cE 

sn

v 

1 

o cn

v 

o -so 

v

C2 

v 

c 

v 

cn 

 v 

0 

s o

y 

coy. 

Yv 

x 

yv 

z 

V 

(1) 

(2) 

s = sin 

C = cos 

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Main Tracking Mirror Gimbal Angles: 

The angle (40is a roll about the main tracking mirror roll gimbal and 

8 is a pitch about the main tracking mirror pitch gimbal. The angle al 

is the -2 degree pitch rotation from the S

v 

to the Sb coordinate system.- 

t 

Rotations : 

S

v 

S

b -asm 

S 

m m 

81 about the yv 

about xb 

2 8 about ym 

The rotations and the rotations for stereo and obliquity are shown in 

Figure 2. 

The transformation equations are: 

or 

X 

m 

Ym 

z 

m 

c2 8 

0 

s28 

- 

= [2 

0 

1 

0 

] [ Ym 

-s2 0 

0 

c2 el 

J 

xb 

1 

0 

0 

- 

•• 

S 

Yv 

0 

cc) 

-scp 

0 

scp 

ccp 

.... 

•■•••■ c81 

0 

se1 /.... 

0 

1 

0 

^ 

-s81 

0 

ca l 

a- - 

x

v 

v

' v 

z

v 

A* Om. 

(3) 

(4) 

where 

which yields 

1 = -2 deg. 

x

m 

c20 c81 - s20 ccp s81 s28scp -c28581 - s2eccpc9 i x

v 

scpse ccp scpce i (5) 

z

m 

s28 c81 + c28 ccp sal -c29 scp -s28 sal-1- P c2G ccpce 1 z

v 

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1' 

ze 

TM OA, 

MAIN TZACI.Air.; /q /??J a/iv; 5/94 fiNGLES 

F7' G 

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Since z coincides with -x 2 ml 

sE

v

= s2Occp sa l- c20c0i 

cEV = s20scp 

V 

cE

v 

 ca

v 

 = + c2e se, + s2eccpc0 / 

From Equation (7) 

= s20 scp 

v cENT 

Note from Equation (6) that for Ey= 0 with Qv= 

tan z e = cot e i 

Since 0 1 = -2 deg, e = 46°. 

Introducing AO, which is the angular deviation from the Ev 

= 0 mirror 

into Equations (6) and (9) yields: 

0= 46° + AO 

1 

sE

v 

= s209(ccps2

e1 - 

+ -2

0

1 ) + c2Ae se1 ce 1 (ccp - 1)! 

sn = .cE) (s2AOse1 + c2/10 ce 1

) 

v 

v 

(6) 

(7) 

(8) 

(9) 

(10) 

position, 

(11) 

(12) 

(13) 

The following expressions to obtain gimbal angles given 

E

v 

and Qv can be 

derived using Equations (6) through (8). 

c29= cZ cf:2 s0 - sE' ce v v 1 v 1 

where 0 s e 90 deg 

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cE sC2 

V V 

s20 

1,6,0 = 0 - 46 degl 

DORIAN 

(15) 

(16) 
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ATS Tracking Mirror Gimbal Angles: 

The angle cp

a 

 is a roll about the ATS tracking mirror roll gimbal and 

eais é. pitch about the ATS tracking mirror pitch gimbal. The angle *2 

is thej 9 degree yaw rotation from the S

v 

to the S

a 

coordinate system. 

otations: 

S

y 

S

a 

S

ma 

S

a 

*

2 about zv 

S

ma 

waabout xa 

S O y 

a 

about 

ma ma 

The above rotations and the rotations for stereo and obliquity are shown 

in Figure 3. 

or 

{xma l 

yma

t 

z

ma 

where *2 

[xma ll 

y

ma

' 

z

ma 

The transformation 

sma t = [8 a 

c ea 0 -s e

a

- 

0 1 0 

-

s

a 

0 c 

a- 

= 9 deg, which 

ceac*2 - sea sTa 

-cepasit2 

se

a 

+ CO scp 

a 

3 yma 

yields 

equations are: 

[(pi 4 s 

a x

a 

a zv v 

1 0 0 c* 2 

0 cep a scpa -s*2 02 

L0 -scpa ccpa o 0 

s*2 COa 

s + se

a 

stp

a 

0

2. 

c(Pa c4/ 2 

s B aor z - ce a stpa icitz 2 

01 

0 

1 

-se

a 

yv 

z

v 

cep

a 

scPa 

ce a cepa 

(17) 

(18) 

- 

xv 

Yv 

t_z\r_ 

(19) 

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ATS TREIC tepi$ fruRF:GR almSAL ANG1...eS 

FIGURE 3 

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S

v 

-• SI Q

v 

about x

v 

S

1

-+ S2 0 about yi 

E + 90° 

v 

x

v 

Yv 

zy 

(20) 

where 

is0 = se

a 

 cir 2 + c8 ascp

a 

I 

sCi

v 

se

a 

sit -c8aspac*

z 

cl3 cQ

v 

= ce

a 

ccp

a 

r

E

v 

 r 2;3- 90 deg 

- 

SECRET DORIAN 

BIF'-107-50239-68 

Page g 

To relate E

v 

and Q

v 

to the mirror gimbalangles, the mirror normal 

will be placed in the plane containing and with an equal angle between the 

optic 1 1 axis and the line-of-sight (see Figure 3). The rotations shown in 

Figure 3 are: 

2 

The transformation equations are: 

03 • sf3sn

v 

-s 0 cO

v

1 

= 0 6-2

v 

sn 

v 

si3 -cf3 sn

v 

03 cQ 

conicides with z ' 

ma 

Using Equations (21) _d (22) yields: 

sO 2 - sOa 

sC2 = sit 2 

(25) 

where 

X 

y2 

z2 

Since z 
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Gimbal a ngles are found int erms of Evand nvfrom Equations (21) through 

(24) as: 

E + 90° 

v 

2 

se

a 

 = sf3c*2 - ci3sn

v. 

s4t2 

sO s4tz+ cO'DvclItz 

sYa 

ce 

a 

(26) 

(27) 

(28) 

Note from these equations that when 

E

y 

= 0 and Q = 0, 

O

a 

= 44,299 deg and 

w a= 8.891 deg. 

SECRET / DORIAN 
Stereo and Obliquity Angles: 

Rotations: 

S

r 

-*S1 O

r 

about x

r 

S

1 'S2 Erabout yi 

From previous 

x

2 

Y2 

z2 

= 

relations, 

cE

r 

sE sc2 

r r 

0 c(I

r 

sE

r 

-cE s(-2 

r r 

-sE cc-2 

r r 

sn

r 

cE cCI 

r r 

x

r 

Yr 

z

r 

(29) 

1 

where the unit line-of-sight vectcr, , is z 2 . 

The attitude matrix, A, relates S

r 

to S

y

, i.e. , 

A: S -'S 

T V' 

is 

A= [eliy []x l vi z 

(30) 

where the order of rotation is not important since small angles will be 

assumed. cp , 9 , and tir are respectively, the roll, pitch and yaw angles 

v v v 

of the body. 

A= 0 

[1 

8 

v 

0 

1 

0 

,-8

v 

0 

1 

1 

0 

0 

0 

1 

-9v 

0 

cd 

1 

1 

[4/v 

0 

1 

() 

0 

1 

(31) 

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S

r 

Coordinate System Page 11 

If the stereo and obliquity angles are to be expressed in the LViRVV 

coordinate system, S

r

, the following relations exist between the tracking 

mirror gimbal angles and the stereo and obliquity angles. 
rr 

[ 

Yr 

z

r

_ 

yv 

z

v 

(32) 

(33) 

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Assuming multiples of small angles equals zero 

(34) 

(

cd - stZ c4) (cf2vcpv+ sf2v) 

v vv/ / 

From previous 

n 

Y2 

z2Iresults 

v 

Yv 

1 41 

-4iv 1 

z -cPy 

• s E vs v sf-2 

= 0 cfl

v 

[s2:

v 

-cE sf2 

v v 

-8 

v 

zPv 

1 

-sE 

v 

sn

v 

cE 

v 

v 

cf.2 

v 

where z , is 

Combining Equations 

the unit LOS vector ''. 

(32) and (33) yields: 

I 

x

2 

Y2 

i 

t 

0 

cE 

v 

sl; sc2 

ci2

v 

v v 

-sE cf2 1 1 1 

sC2

v 

1-111v 

v v 4i v 

1 

Tv] ri 

Yr 

/ 

z

2 7_,-' - v 

-cE sC2 ✓ v cE 6-2 v v Le

v -(Pv 1 z

r 

(

zE -sE cfl 8 )(cE + sE sf2 +sZcS2 cp (-c2.; + sE sC2 - s cc 

v v v vv v v v v v v v v/ v v v \iv v v• 

- cClv*v+ s e v) 

+ + cE cc2 Bv) (sE -cE sf2 -cE 62 cp) (-sE-0 -cE sf2 tI)+CF-:6) t I z 

v vv. if v v v v-v v. v. v vi j L 

Since z2

' coincides with z2 

sE =sE + cL: sCl* + cE c C2 0 

✓ v v VV v VV 

-cE sc2 = - cE sC*2 cE cf2 

r r v v v v v v v 

cE cS2 = -sE 8 - cE sf2 cr, cat 

✓ r ITV V V V V 

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Therefore, from Equation (36), 

I sZr= sE

v 

+ cE (s0 6'1 )1 

v v v, v v 

(39) 

BIF-107-50239 -68 

Page 13 

+ cLir(cpv+84)] (45) 

0 0 

c2.]

v 

 sQv 

 [1 + 

r - 

- cE [-E.A!r(1 + 

Equation (45) can be reduced to 

cZr sE 

sn

v cE (sO

r 

cO

r

:

v

) + cE 41

v 

and from Equation (37), 

cE sZ 

sr = CLv (sp + 

v 

 cp ) 

v v cE 

r r 

Multiplying Equation (36) by Tvand (37) by ev, then summing yields: 

sEr:pr- cErsC2 r9 = sEv(cPv+ + cEvsav(lfycPv- 9v) V 

Multiplying Equation (38) by pvand summing with Equation (37) yields: 

-cE sp + cE cf2 cP = sE Cif - 9 cp ) - cZsC2(1 + cP 2

) r r r ry vv vv 

v v v 

Multiplying Equation (42) by (tifv(Pv- 0v) and (41) by (1 + cPv

2

), then 

summing yields: 

SE [1 + C) + 11A + = SE + %,1 - cE [sar(Vve\Px z)+ cflr(9v \\pv] (43) 

v 

N0 0 N:10 

vc40 r 

0 0 

Equatioh (43) can be reduced to 

= sE - cE (sS2 c CI

r 

 0 

V r rry v

) 

Multiplying Equation (41) by (iry- 9v:pv) and (42) by -(cpv+ *ved, then 

summing yields : 

(40) 

(41) 

(42) 

(44) 

(46) 

—S-E-G-ii-E-T4 DOR IAN 
BIF-107-50239-68 

Page 14 

rx 

[Yr 

z ' 

(48) 

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S

r 

Coordinate System 

If the stereo and obliquity angles are to be expressed in the LV/OP coordinate 

system, S the following relations exist between the tracking mirror gimbal angles 

and the stereo and obliquity angles. 

Stereo and Obliquity Angles 

Rotations : 

S

r 

S

1 O

r 

about x

r 

S

1 -4S2 Erabout yi 

From 

Yz 

z2 

previous relations 

[cE sE sd 

r r r 

0 

r

' 

sE -cE

r 

sr:

r 

sE cS2 r r 

sir 

cE

r 

 /cU

r [xir 

Yr 

r_ 

(47) 

where the unit LOS Vector, k, is z2. 

Assume the crab angle, yaw angle between the orbit plane and the 

relative velocity vector, is defined as 71. Then 

S

r 11 about z

r 

x

r 

sr) 

Yr = -sr; 0 

z

r 

0 0 1 

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From. Equation (35)' assuming rlis a small angle 

(cE sn V -sE 61 8 )(c> +sE sS2 +sE cS2 cp ) (-cE 9 +sE sS-2 cp -sE cif ) 

v V vv V vv vv v v V vv vv v vv v v 

( -cC2 + sC2 9 ) 

v v v v 

(sE +cE sC1 +cE ci2 9) (sE sC2 -cE cS1 cp ) (-sE 9 -cE sf2 cp +cE cD ) 

v V vv V yv vv v v V V V vv V v V. v 

x., 

Y2 

z

2 

(49) 

1 Ti 0 

-n 1 0 

0 0 1 

(cO sS2 cp ) 

v v v 

(cS2 

v 

 cp

v 

 + 

where z is the unit LOS vector k. 

Since z21coincides with z 

= sEv[l tvn] + cEvrsil(*v+ n) + cC2v(ev+ c-pvT))1 ( 5 0) 

Q = s Zy. + T1] - cE r-i- -so - + coy(cpv-Or1)] (51) 

cE cc-2

r 

-sE 9 c2 

V 

- cE [s cp - cS2 I (Si) 

r ' V V V VV 

Therefore, from Equation (A-42), 

sE = sE + cE [sQ (111 + TI) cO 9 

r v v V v v v 

and from Equation (A-43), 

= cEr

v

i snv + Qvcpv) v + n1)

c s E 

In order to simplify the following expressions, let 

A = sE'

r 

G n NI NT 

B = sc./ H = cp e 

v v 

D = CE c c2 = r r ev 

E = 1 - K = otp 

F = + 11 

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(53) 

(54) 
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Substituting these expressions into Equations (50) through (52) yields : 

A = sE

r 

E + cE sC2 F + cE ca G 

v v ' v v (55) 

B = sE F - cE sit E cE ci2 H (56) 

v V v v 

D = -sE J - c2; sQ K + cE cC2 (57) 

v v v v 

Multiplying Equation (57) by H and summing with Equation (56) yields : 

B + DH = sE

v 

(F - JH) -cE

v 

 sc/

v 

 (E + KH) (58) 

Multiplying Equation (55) by H and (56) by G, then summing yields : 

AH + BG = sE

v 

(EH + FG) +v 

v 

sC.2 (FH-EG) (59) 

Multiplying Equatior. (58) by (FH-EG) and (59) 

summing yields : 

by (E + KH) , then 

sl:

v 

[F2- FHJ + EGJ + E2+ EHK + FGK] = A(E+ HK) + B(F+ GK) (60) 

+ D( -EG+ FH) 

Substituting for the dummy variables yields: 

+ B Div+ r+ eV + cf) 2 

71j + ID [-e - a\r1 + cPv\VII+ 's:PvT1 J 

sE [1 + + 

v 

2( 1+ 1k2 2 _A‘.21,. A r1_ 

0 .110 0 0 - 0 

vsltv 

0 0 0 0 

v4NO 

' 

0 2 

,, . 

- cPv V13 

21 0 (61. ) 

Equation (61) can be reduced to: 

v r r v r 

= SE cE [sn + -n1+ d (62) 

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Multiplying Equation (58) by (EH + FG) and (59) by -(F - J1-1), then 

summing yields: 

-cE

v 

 sC2 [E2 + F2 + E(HK + GJ) + F(GK - 1-1J)] = A CH3 - F] + B GJ + El • (63) 

+ FG] 

.Substituting for the dummy variables yields: 

A[4v - 11+ 

Equatioin 

-cs [1+ + 9+ + (1 ++ gs +02 

"cs 0 '‘O 10 2 , B[1 

0 

- Isv\t(1 + evcp 

(64) 

0 0 

e v 41 \ f(1 

0 

reduces to 

+ 9 )] = 

o 0 

r+ + D[9 + 

0 0 

40 

c; sE I 

=

c 

 (s r '- cOricf)v

) + cE (icy+ r1) Ev 

(64) 

(65) 
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SUMMARY 

131F-107-50239-68 

Page 18 

The expressions derived in this report to go between all cordinate 

systems are summarized below. 

S

v 

Coordinate System 

Main Tracking Mirror 

sE

v 

s2Ae(csos2

9

1 + c

2

9

1

) + c2A9 se tc81(ccp- 1) (66) 

sSl = sW (s20 se1 + cne ce1

) (67) 

v cE

v 

c2e = cEv 1 cn se sE v 1 ce where 0 8 + 90 deg (68) 

v 

cE sQ 

v v 

" scp =s2e 

(69) 

. 

IA e = e - 46 deg (70) 

e

l = -2 deg 

ATS Tracking Mirror 

se = s8ac.t 2+ ce ascpa sl1r 2 (71) 

E

y 

= - 90 deg (1Z) 

where 

sil

o 

= 1s* 

(sI3 c*z- se a) (73) 

2 

1 

= (E v + 90 deg) (74) 

se

a 

= se cir2- cesic sir (75) 

scpa = cs(3 stz+ so c*Z) (76) 

a 

= 9 deg. 

SECRET DORIAN 

where 
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Page 19 

(77) 

S

r 

Coordinate System 

sE

r 

= sE

v 

+ cE

v 

(s0 i + cal 0 ) 

v v v v 

sO

r 

= 

cEv (sc2 + co cp ) - sE

v 

v v v v cE cE

r 

sE

v 

= sE

r 

- cE

r 

(sC1 + cQ ) 

r v r v 

cF

r 

sE 

T sc2 

v cE

v 

 (s Sir - cC2r cPv) cE Tv 

S

r

' Coordinate System 

(78) 

(79) 

(80) 

sE

r

' = sE

v 

 + cEv 

v 

[s CI 

v+ 

+ cC2 0

v 

(81) 

v 

cE sE 

sC2 -s) ' = 

- cE

r 

—Y-, (sn + cC2

v 

 up

v ) cE

r

' "v+'n) 

(82) 

sE

v 

= sE'-cE' [sC2 I (* + TO + cC) ' 8 ] (83) 

r T r v r v 

cE

r

' . sE 

r

. ' 

sC-2 cE = (sDr

' cCI ' cp 

v 

 ) + cE (4 v 

+ 11 ) ( 84) 

v 

V 

r 

v 

ADRIAN