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TECHNICAL PUBLICATION

NPIC/TP-22/63

August 1963 /.

1111191 IIEUM1111 I

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DETERMINATION OF PITCH AND ROLL

FROM HORIZON EXPOSURES

ON KH-4 SATELLITE PHOTOGRAPHY

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NPIC/TP-22/63

DETERMINATION OF PITCH AND ROLL

FROM HORIZON EXPOSURES ON KH-4 SATELLITE PHOTOGRAPHY

•a. INTRODUCTION

linrizon photographs arc used to determine.

the pitch and roll components of the K11-4

vehicle's attitude or. more properk of an orbiting camera's attitude. llorizon photographs

do not give any yaw information in cases where

the primary or pan camera is aimed at the nadir

.1' !gore I i. The determination of pitch and roll

is simple and straightforward, and good approxi

mat ions of both pitch and roll are easily obtained

from simple measurements and computatson.•

When the primary tamers is not oriented toward

the nadir and the horizon cameras are attached

to toe primary camera anthat the horizon photos

have an apparent swing. as is the case with K11-4.

the measurements are restricted to r smaller

section of the horizon image and the data reduction becomek more complicated (Figure 21.

Many reduction systems have been proposted and used. They range from the extremely simple ones where simple scales are

used to measure the image and the computation

is done graphically, to the complex ones wnere

precise machines are used to measure the image

and the computation is accomplished with a

computer. lhe following is presented as a

solution falling between the extremes, but definitely toward the more complex approach.

lhis determination of pitch and roll from

horizon photographs is tailored to be practical

for the reduction of a large number of photographs. it is assumed that measurements arc

made with a comparator, such as a Mann

instrument, and that the computation is performed by a computer.

••••: on•• n •• • •••

FIGURE I. NORUON IMAGE FOR VERTICALLY ORIENTED

PRIMARY CAMERA.

FIGURE 2. HORIZON IMAGE FOR PITCHED PRIMARY

CAMERA.

DEFINITIONS

Yaw, pitch, and roll are taken as orthogonal

rotations about the three mutually perpendicular

primary photo axes having their origin at the lens

(Figure 31. Yaw is considered primary . pitch

secondary . and roll tertiary. Since a mechanical

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gimbal ayntem is not involved. the ordering of the

orientation rotations is arbitrary though necessary to adequately define them. Swing, S. is a

rotation of the x-y-axes about the z-axis in

the plane of the photograph. Swing is positive

when the x-y-axes are rotated lathe left as seen

from a point on the positive z-axis.

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NFIC/TP-22/63

FIGURE 3. ATTITUDE COMPONENTS.

rre • 

• •

•

ev.". -%* •• • • 1' • ••••

s2ti°

••

• y "-Ylio•

and

In

..?"•••• •••• •••• T..

plaucial. • . .

2%. •

rill

issiNSC • •• moliS

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s

f1,4- • 0"--4

er.31.5311„hea,-

FIGURE 5. ROLL COMPONENT MI AND SWING ESL

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far apart as possible, and points 6 through 9

should be approximately evenly spaced between

them.

COMPUTATION

Assume the following photo measurements:

MEASUREMENTS

The measurement of image points is done

with respect to an axis system on thephntograph

having its origin at the principal point. Ten

points are measured on each horizon frame, as

shown in Figures 4 and 5. The4 fiducial marks

arc read first and then 6 points on the horizon

image. Points 5 and 10 should be chosen as

FoGuRE 4 MEASUREMENT POINTS.

., - end fiducial mark

f•, side fiducial mark

) 6 . - opposite end fiducial mark

P G •• n • ) 4 ,- opposite side fiducial mark

- approximately evenly spaced

points on the horizon image.

The principal point is determined from the intersection of two straight lines. Instrument tolerances in placing the fiducial marks prohibit the

method of taking the average coordinates of opposite fiducial marks. From the equation of a

straight line:

and

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These equations represent the fiducial linesdefined by the opposite fiducial marks. Their

simultaneous solution gives the coordinates of

the principal point. It is convenient at this step,

but not necessary, to reduce the coordinates of

the points on the horizon curve to the principal

point origin, i.e.: let

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•

•

•

Assuming the horizon image to be circular,

the center is found from the circle of best fit in

the sense of least squares. it is worthwhile to

note that the horizon image is not circular but,

because such a small section of the actual hyperbolic curve is given, there is no practical difference. It is convenient to find the center of

the circle. From the general equation of a

circle: if ,s „ A. t•

There are six ohnervat kin equations of the form

A 1..21.... . v,. 1. Lie. ilf• (If • )fl 1 . 7 • ):I

which gi.e the following normal equations:

A ii C

I IR 1 I • . • i

1

f .. 1 r ) 1 1 ) I 1 )

n•n.n.., • I • w • n • • ••n•• • , n• n • • • = • • • im • = • n••• Mmi al it .1 . „: ii

i„.1 . ,s .a i.1 . ,1.11J • [.• ..1

' J

and

1,0 ..101 * .1.111

I

•

• . [ 1

).))* 0 . 1,2

I .0.

The simultaneous solution of these normal equations gives the coefficients of the equation tithe

circle. 1 hen the center coordinates of the circle

are: .,.4 and

The roll component, • , on the photo is merely the

difference between the radius of the circle and

the distance from the principal point to the center

of the circle (Figure 5):

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The roll component g• . for the horizon camera is

determined by:

P..-,r_. 4

T •

where t is the calibrated focal length of the

horizon camera.

Swing is given by:

. met Lii•

A •

Because the y-axes of the horizon camera

and the primary camera are not exactly parallel,

the calibration data are used to rotate the horizon camera axes to parallel those of the primary

camera. This rotation is significant in swing.

The correction to the swing is

16 um"

I)

where u, N, and x are calibration numbers given

in the 101-4 calibration a or t port

camera of the forward set. Then

$. !I •

The computation thus far is computed for

both the starboard and port horizon cameras

and then the average roll component,

P:

2

is used to compute the dip angle, n • ,for the port

horizon camera measured in the principal plane.

This operation begins an iterative loop which

usually yields good results in three cycles. The

protection of the preset or "built-in" dip, a, to

the principal plane is

D' me" D ran 6) . (1)

where 6 is replaced by S for the first approximation. Then the roll in the principal plane is

I' . ti. (2)

and the roll in the pitched plane is

I elerS (11L— eaa (3)

where 4 is replaced be S for the first approximation. The azimuth of the principal plane is

computed from the expression

...tari a min lax (4)

where d is replaced by S for the first approximation. Delta is really half the tilt of the

horizon photo, therefore

a. •. (5)

which gives all of the components required to

compute the pitch:

We l Elam taa e••• •, tan a]. • . • • (6)

New values for r , a , . , and d are computed

using the last computed values in equations

zoo

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NPIC/IP-22/63

;It through (61. Three iterations are usually

enough. The values of (equation 31 and (equation 61 refer to the left horizon photo.

The pitch and roll for the pri mars camera is

determined hr adding the calibration data. The

KII-a calibration data provides the following•

I. - the v -coordinate of the pan camera

calibration mark. "I hen the pitch for the pan

camera it given hr

V,

C., - co • Ise t. A

where a Is the calibrated focal length of the

in camera. In the case of roll. the KII-4

calibration data is:

2... the %-coordinatetif the hormon camera calibration mark, which Is co-planar with

3. %, . the -coordinate of the pin camera

to tn • corresponding mark.

1 hen the roll for the pan camera is

P t. 0 . •I • um' I •I • ?•;:r1t-•

P • 1:. • um" • 41.1% (8)

If only one horizon photo is used, the altitude is used in the computation. In this case

0 is computed as above. and thci determined from 

1 • s•sr.

It • tt

where It is the mean radius of the earth and II

iss the altitude above the surface. The insensitivity of this solution indicates that the radius

approximation is valid. Then

A r

and

e • I 0 •

Now the three-cycle iteration with equations

It through 011 will yield the pitch and rollfor

the horizon camera.

. .D.1111.• IONS.

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FIGURE 6. GEOMETRY OF TILTED HORIZON PHOTO.

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NPIC/TP-22/63

APPENDIX

Derivation of Equations 3, 4, and 6 (Fig- 3. Equation (6):

ure 6). Cat ICQ-GQ

I. Equation (3): us

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used, nee tiss tam

awe I 

tam 6

2. Equation (4): Rearranging:

V. P. I •fis am di ma 6 •• 6 tan it • Ian tan •

I raw I,

um" Han P sus eh LIN .) cali • Ian a

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