TWO APPROACHES TO GRAVITY REFERENCED ORIENTATION

• Document Type: CREST
• Collection: NGA Records (Formerly NIMA)
• Document Number (FOIA) /ESDN (CREST): CIA-RDP80T00703A000800040001-0
• Release Decision: RIFPUB
• Original Classification: K
• Document Page Count: 39
• Document Creation Date: December 28, 2016
• Document Release Date: May 9, 2001
• Sequence Number: 1
• Content Type: REPORT

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TWO APPROACHES TO GRAVITY REFERENCED ORIENTATION
Declassification Review by NIMA / DoD
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Page
1.
First Approach ..............................................
1
2.
Second Approach .............................................
12
3.
Precision of the Gravity Direction Angles ...................
18
4.
Further Adjustment ...........................................
24
5.
Application of the Gravity Direction Cosines to a Gravity
Referenced Datum ...........................................
27
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LIST OF FIGURES
FIGURE Page
1 Pitch, Roll and Yaw ................................... 2
2 Change in 'i and ~ with p = 0 for Two Exposures ........ 4
3 Geometry of Solution for Direction of Zg in Xa, Ya, Za
Coordinate System ..................................... 5
4 Orthogonal Rotations .................................. 10
5 Sinusoidal Cones Generated by x, y and z Axes about ZG 14
6 Simplest Concept of Gravity Reference Plane........... 28
7 Projection of Arbitrary Axis Xa to XG in Plane ZGXa... 30
8 Projection of Any Line 0 to the XG YG Plane........... 34
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TWO APPROACHES TO GRAVITY REFERENCED ORIENTATION
1. First Approach
Given camera parameters and overlapping exposures of an object the
shape and configuration of an object may be determined by the use of
conjugate images in suitable reciprocal collinearity equations. If only
the shape of the object is required all exposures may be referred to any
orthogonal coordinate system inherent to the object. However, if the
object's orientation is a function of the direction of gravity then an arbitrary
coordinate system with an unknown relation to the direction of gravity
is unacceptable.
In the absence of any gravity referenced control data the search
for the gravity direction in the arbitrary coordinate system must be
found in those motions under the influence of gravity. The motions of
the camera associated with pitch, roll, and yaw are a coupling consequence
of the drag resistance of the camera through a space under continuous
restoring force of gravity. Translations from camera station to camera
station are mechanically imposed while the changes in orientation occur--
ring during translation have the direction of gravity as the axis of
equilibrium. These axes of equilibrium are illustrated in Figure 1. If
xv, yv, and zv are the axes of a camera vehicle Z9 is the direction of
gravity and Yg is the horizontal direction of motion. At rest xv, yv'
zv coincides with Xg, Yg, and Zg. Pitch (7r) is the vertical angle y
axis defines with the horizontal X Y v
g, g plane. Roll (p) is the rotation
of xv, zv plane about the yv axis from the Xg, Y9 plane. Thus both pitch
and roll refer to the Xg Yg plane but i is a vertical angle that always
contains Z g and p is not a vertical angle except when 7 = 0. p lies in
the x v 0 Zv plane when it 0. Yaw (~) is the rotation about Zg of the yv
axis or the horizontal angle the yv axis makes with Y axis.
g
Assuming the camera to be rigidly attached to its transport vehicle
the camera axes may be treated as vehicle axes with constant angles relating
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Figure 1 PITCH, ROLL AND YAW
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the vehicle axis to the camera axes. Let Tr O, po, and ~o be the constant
pitch, roll, and yaw of the camera with the vehicle at equilibrium whence
changes in vehicle orientation do not alter the camera relation to the
vehicle.
It is believed that a given vehicle had undergone normal pitch and
yaw rotations with minimal or nominal zero roll rotations. This means
the xv axis lies in the Xg, Y9 plane and defines a level line which with
horizontal change associated with yaw defines a horizontal plane whose
perpendicular is the direction of gravity. This geometry is shown in
Figure 2 with two different orientations. It is evident if yaw is a
constant xv does not change its horizontal direction without which the
direction of Zg cannot be determined. Figure 3 illustrates the proposed
solution based on the assumption of significant pitch and yaw and small or
neglectable roll.
Assume p = 0 and X'g or the axis of pitch is a level line. X'g
in the presence of yaw about Zg defines a plane whose perpendicular is
Zg . Xa, Ya, Za is the arbitrary static object space coordinate system to
which the camera undergoing yaw and pitch rotations refer. The camera
orientations with respect to the arbitrary datum is established. It is
desired to refer the camera orientation to the direction of gravity and
X' g 31 Y' g system normal to Z g . X' g and Y' g lie in a horizontal plane but
because of yaw are different for each exposure. However, the camera axes
defines constant angles x 0 X', Yo X', and z o X' with the yawing X' g (xv)
axis. It is necessary to determine both these constant angles and the
g g
instantaneous direction of X' in order to determine the direction Z
Given the orientation matrix of a camera axes referred to the arbitrary
datum.
X
a
Z
a
cos ax
cos (3x
cos (3x
~y
cos a.z
cos Sz
cos (3z
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Figure 2 CHANGE IN iT AND f WITH P =0 FOR TWO EXPOSURES.
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Figure 3 GEOMETRY OF SOLUTION FOR DIRECTION OF
Z.g IN Xa,Ya,Za COORDINATE SYSTEM.
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As usual the columns are the direction cosines of the camera x, y, z axes
with respect to the Xa, Ya, Z a axes and the rows are direction cosines
of the Xa, Ya, Za axes with respect to the camera x, y, z axes. If one
knew the direction cosines of the instantaneous X' axis. referred to the
g
arbitrary datum one could write for the constant camera-vehicle angles.
the following equations:
cos aX, COSax + cos 3X, cos Qx + cos '+fX, cos Yx = cos xoX'
cos aX, cos ay + cos SX, cos Rv + cos J_X, cos yy = cos yoX'
cos cX, cos az + cos (3X, cos ~z + cos yX, cos Yz
Since the right hand terms are constant
cos xoX' cos ax + cos yoX' cos ay + cos 2oX' cos az
cos xoX' cos 0x + cos yoX' cos (3y + cos zoX' cos
cos xoX' cosy x + Cos yoX' cosy y + cos zoX' cos
Now since Zg is perpendicular to X'g
Yz
cos aX,
1X'
YX,
cos ax, cos aZ + cos (3X, cos RZ + cos YX, cos YZ = 0.
g g g
While u j, 1X,, and y,,, vary for each exposure they are expressible
in terms of the known arbitrary orientation matrix and the constant angles
of the vehicle level X' (xv) axis with respect to the camera x, Y. z axes.
Substituting
(cos aZ cos xoX') cos ax + (cos aZ cos yoX')
g g -
+(Cos aZ cos xoX') cos ~x + (cos RZ cos yoX')
g g
cos ay + (cos aZ cos zoX') cos az
g
Cos (3y + (cos aZ Cos zoX') cos az
g
+(cos YZ cos xoX') cos yx + (cos YZ cos yoX') Cos yy + (cos yZ cos zoX') cos Yz
g g g
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Division by cos yz cos zoX' gives
g
(tan nzg tan nx,) cos ax + (tan nz tan EX,) cos ay + (tan ri z ) cos a
z
+(tan Ezg tan rlx,) cos Rx + (tan Ez tan Ex,) cos ay + (tan Ezy) cos
g n z
+(tan nx, ) cos yx + (tan , ) cos Y y - cos
X y yz
which contains 8 unknowns in parentheses. Assume an 8 x 8 is solved
to obtain preliminary values of tan nz , tan Ez , tan r)X,, and tan ,
g g X
cos az
g
tan nz
+ tan ? n +
g
tan2E-,) 1/2
g
tan Cz
Nz -
g
z
g
(1
+ tan
r1z+
g
tan z ) 1/2
g
cos yz =
1
g
(1
+ tan nz +
tan,-'EZ ) 1/2
g
tan nx,
(1 + tan nY, + tan C X,)
1/2
cos yoX' =
tan X,
l+tannx, +tan~ r)
1/2
cos zoX' =
1
(1 + tan rIx, + tan x, )
1/2
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Inasmuch as p = 0 any 8 x 8 formed from sequential exposures in
flight will yield poor results because of the small changes in Eo. To
get good results 40 to 50 exposures must be normalized to an 8 x 8 where
by Least Square values may be obtained and residuals generated. Since
the right hand term of the original equation is zero or cos 90' the
residuals obtained are the cosines of angles less than or greater than
90? or cos (90? + p) _ + sin p. Thus the residuals are a consequence
of the assumption p = 0. A large number of condition equations will
include the necessary spread in yaw - especially if the values are
selected from different flight directions.
Assuming those values of sin p having 3 times the mean deviation
from the average sin p are rejected from the solution an iterative form
of the equations is normalized and solved. The iterative form is a
4 x 4
A ATIZ + B ACZ + C AnX, + D AEX, = sin p
g g
A = (tan n'X, cos ax + tan C'X, cos ay + cos az) sec2rl'Z
g
B = (tan T7'X, cos ~x + tan C'X, cos Sy + cos Rz) sec2v Z
g
C = (tan fl'Z cos ax + tan 'Z cos x + cos yx) sec2n'X,
g g
~
D = (tan T1' Z cos ay + tan Z cos y + cos yy) sec 2v X,
g g
The primed Ti' and ' denote the best values obtained from the
normalized 8 x 8 which are still considered preliminary. Solving
normalized equations for AT)Z, AEZ, Any? and A~X,
= n'z + EAnZ
g g g
8
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z z + E1Ez
g g g
= n'X, + ZAfX,
EX, = 'X' + Et1CX,
Naturally values tan n'z , tan ~'Z , tan n'X,, and tan TI'X, are revised
g g
with each iteration until the E sin2 p a minimum.
An important point not mentioned is the fact that the cameras cannot be
interchanged nor can the camera x, y axes be different from exposure to
exposure. Each camera defines different constant angles with the vehicle
and any one camera x, y axes defines constant angles if the camera x, y
axes are not rotated from frame to frame.
Suppose for some reason the camera x, y axes were rotated 90?, 180?,
or 270? as is indicated in Figure 4.
In each case for convenience in relative orientation the camera
axes has been rotated from the original orientation xa, yo to x, y for
which the matrix is given. It is only necessary to restore the matrix
to the original axes when the rotation is 90?, 180?, or 270? to make the
sign and/or column change indicated below:
90? Rotation
a a a
x0 yo z0
b b b
x YO z
0 0 0
c c c
x y z
0 0 0
-a a a
y x z
-b b b
y x z
9
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x
Yo
90?Rotation
x -xo
= -yo
1800Rotation
yo
x Yo
y = -xo
270?Rotation
Figure 4 ORTHOGONAL ROTATIONS
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a
a
a
-a
-a .
a
z
X0
y0
Z 0
x
b
b
b
-bx
-by
bz
x
o
yo
z
0
c
x
c
z
-c
x
-c
c
z
0
0
ax a a
o Yo z
0
b b b
x
0 0 z 0
a
y
-a a
x z
cx cy cz
0 0 0
by -bx bz
A test solution of an 8 x 8 for each of two cameras yielded both
encouraging and discouraging results. The standard coordinates tan nZ-
and tan ~Z were roughly the same in magnitude and sign for both cameras.
This would not be possible unless the solution was based on correct
theorems. The values tan r1'X and tan EX, showed large variations as might
be expected from 8 closely grouped exposures and/or containing an unknown
camera x, y rotation,. The signs of tan rlX, and tan CX, were opposite
for the two cameras as was expected.
The consistent values of = and CZ and the consistent signs of the
pX, and EX, values suggest the solution is valid and the assumptions are
valid. The large spread in the values of rlX, and CX, obtained by two
camera solutions indicates the x, y axes must be rotated back to the
original camera orientation and a large over determination is necessary
to insure a spread in yaw and to average out the variation of p from zero.
In view of the sensitivities of the 8 x 8 and the linearized 4 x 4
to both unknown x, y rotations and departures from the assumption of p = 0
an unduly large effort is necessary to pre-process each orientation matrix
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employed. To surmount this difficulty a solution is proposed not dependent
on p = 0 and independent of x, y rotations.
2. Second Approach
It may be noted that the z columns of the orientation matrix are
unaffected by an x, y rotation. The following approach is suggested to
provide a solution that is not dependent on p = 0 and is independent
of an x, y rotation. The 8 x 8 and linearized 4 x 4 indicate p is not
quite as small as thought and r is not quite as large as thought. If
this is true, each axis (x, y, z) of each camera with n exposures
generates a right cone with a sinusoidal surface and whose axis is the
direction of gravity. An equation is written for each exposure of each
camera:
+/cos (3
cos a G
1 tcos x
cos +
ax cos x
cos aG cos a +cos 13 cos ( + cos yG
cos x_g x2 cos x g x2 cos x g
cos a cos a cos Y
G cos ax + 'Cos cos Rx + G
cos x g n cos x g n cos x g
y axis
cos aG cos a + cos RG cos a + cos yG
cos yg yl cos yg yl cos yg
cos aG cos a, +'cos SG
cos yg Y2 cos y
Yx = 1
1
Yx2 = 1
Y
x
n
Yy
1
cos (3 +cos Y, N cos Y = 1
y2 cos yg y2
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cos a cos 13 cos Y
G
cos a +
G
cos 3 +
G
cos Y = 1
cos yg
Yn
cos yg
Yn
cos yg
Yn
cos aG
cos a -i COS aG
cos a
+ cos YG cos Y = 1
cos z
zl cos z
zl
cos z zl
g g g
cos aG
cos az
+ cos aG
cos R
+ cos YG cos Y
= 1 __
cos z
n
cos z
z
2
)
z
cos z 2
g g g
cos a cos cos Y
( Cos
(Cos
G
cos
+
cos R
+
cos Y
1
t
cos z J n
1
cos z
z
l
z =
cos z n
The geometry of the above three solutions is illustrated in Figure
5. xg,' yg and z9 are the cone angles each axis generates with the
direction and gravity. aG, (3G, and YG are the direction angles of the
direction of gravity in the arbitrary coordinate system.
Only the third array is independent of an x, y rotation. However,
if there were no x, y rotation the first and second array would yield
the same direction of gravity except for the cone angle generated. The
approximate orientation of the system at rest is
xg = 114? (65?)
= 00
y g
z = 24?
g
This means that the x solution in the absence of an x, y rotation would
yield results comparable to the z solution whereas the y solution being very
approximately a level line (direction of motion) would yield a poor
solution. The y cone degenerates to a sinusoidal plane. Since the
departures from cosine 90? are equal to or greater than several degrees,
the cosine of 90? + 2? has twice the numerical magnitude -as the cosine
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Figure 5 SINUSOIDAL CONES GENERATED BY x,y and z AXES ABOUT ZG
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of 900 + 10 for example. cos yg being a denominator can affect the
numerator cos aG, cos RG, cos yG by several 100% with an error of several
degrees. Assume the values in parentheses have been solved from a
normalized 3 x 3
cos x
g rfcos
Cos
cos z =
g
Yg ' cos
cos
cos aG = cos aG cos xg
C-7O s X
cos R = cos aG
G cos x
g
( cos SG Cos z 9)
Cos
Cos
z
cos x * cos (3G
Cos x = (Cos yG
g Cos y
g
1/2
Yg
Yg
cos
COS
(cos
CCS
aG lcos z
z g 1 -
yG \ cos z
zg ) -
The particular values of xg, yg, and z9 for each exposure are obtained
by substitution of the equated values of cos aG, cos 13G, cos YG back
into the original condition equations:
Cos aG Cos ax + Cos aG Cos (3x + Cos yG Cos yx = Cos xn
n n n g
=I -cos YG
cos y G I`
COs X
x
a
G
aG
Yg
1
+ cos RG +
yG l~ 1/2
x
g
1
+ cos 1G\ + Cos YG 1/2
cos yg) cos yg
Cos aG cos y = cos aG Cos z
g cos z g
g
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cos aG cos ay + cos aG cos Ry + cos yG cos yy = cos yn
n n n g
cos aG cos aZ + cos aG cos az + cos yG cos yz = cos zn
n n n g
(pitch) Air n = yng - y
g
0
(tilt) At = z - z
n n9 g
0
(roll) Apn pn po
/sin x
tan po = go
cos z
_0
sin x
n
tan p n =(cos g
These quantities are only of academic interest.
To test the above equations, the x, y, and z columns of eight
camera orientation matrices were normalized and solved by the method
of Least Squares. The results are given below:
x
y
z
cos aG
.0250543
-.1388537
.1172539
cos ~G
.2918100
.9568414
.2801572
cos yG
.9561481
-.2552922
.9527663
As anticipated, the direction cosines of x and z are consistent
to the first order where as those of y are absurd. By treating cos y
g
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as zero the solution for cos aG, cos I3 G, and cos yG is as follow:
cos aG = .0962929
cos aG = .4908134
cos YG = .8503334
which means treating y as sweeping out a sinusoidal plane in place of
a conic removes the absurd aspects of the values even though the results
are not as logical as those obtained with z.
Were it not for the problem of x, y rotation a solution embracing
a conic could be employed with the rows as well as the columns.
cos x cos y cos z
cos x + g cos + $ cos a = 1
cos aG cos aG y cos aG z
cos x
cos + cos yg cos + cos zg cos = 1
cos a cos aG y cos a z
G G
cos x
g cos x + Cos y9 cos + cos zg cos = 1
cos yG cos yG y cos yG z
Solution of normalized 3 x 3 in each of the above forms yielded
reasonable results for the third row but for the reason of weak geometry,
absurd results in the first and second row.
The evidence points toward the most reliable solution being obtained
with the third column. To this end 67 starboard and 71 port third colummn.
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matrices were employed in a Least Square determination of cos aG, cos
cos YG, and cos zg. The results are given below:
Starboard (67)
Cosine
Angle
zg.
.957277
24?
52'
aG
.1948335
78?
46'
aG
.2858874
73?
23'
YG
.9382475
20?
14'
Port (71)
Cosine
Angle
.9085228
24?
43'
.1669294
80?
23'
.3628651
68?
43'
.9167680
23?
33'
G'
Inasmuch as the interlocking angle between the two cameras is
some value greater than 48? and inasmuch as the bisector of this angle
approximates the vertical, solved for values of z g starboard 24? 52' and
z9 port of 24? 43' establishes confidence in the above direction angles
of the vertical. The difference in the starboard and port gravity angles
may be attributed entirely to the noise of the sinusoidal assumptions.
3. Precision of the Gravity Direction An les
In order to demonstrate the precision of the determination a
numerical example of 8 starboard 3rd column values normalized to a 3 x 3
is given:
cos aG
'cos caG
cos YG
cos z
g
cos +
z
cos zg
cos +
z
cos zg
= 1
z
.2978860 .5787331 .7591650 =
1
-.2073995 .4288371 .8792521 =
1
-.2193539 .4242990 .8785523 =
1
.1588703 .6334470 .7573012 =
1
-.0026143 -.0984186 .9951417 =
1
.1853262 .5969892 .7805498 =
1
-.2711262 .3807350 .8840427 =
1
.4051708 .0442197 .9131710 =
1
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Normal Equations
cos aG
[
aa] +
( cos aG
1 [ab
] +( cos YG
[ac] = [a]
- ~ -OS-Z -7,
)
Cos z
cos aG
[
ab] +
cos RG
[bb
] +
cos YG )
[
bc] = [b]
cos zg
cos zg
cos zg
cos aG
[
ac] +
cos aG
[bc
] +
cos YG [
cc] = [c]
cos z
cos z
cos z
g
g
g
.4771323 .1.166048 .2437452 = 0.3467600
.1166048 1.6131149 2.4138944 = 2.9888415
.2437452 2.4138944 5.9097528 = 6.8471758
Solving by Crammer's Rule
cos YG = 1.0297208
co zg )
cos z = 1
g cos aG + cos aG + cos YG 1/2
(cos zg, cos z g cos z g)
1
= 0.9252666
+ 22? 17'.5
I cos aG = .1267239
cos aG .3027854
cos aG = .1172539 aG = 83? 17'
cos 3G = .2801572 aG = 73? 44'
cos YG = .9527663 YG = 17? 41'
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To determine the residuals the values of cos aG , cos (3G, and cos -yG
are substituted in the original equations:
cos z9 - (cos az cos aG + cos az cos RG + cos yz cos y G )
n n n
g gn
where z. is the Least Square value and zh is the value obtained from any
condition equation n
now cos z = cos z cos A + sin z sin A
gn g gn g z
gn
To the first order cos A = 1 therefore cos z - cos z = - sin z . A
z gn g gn g z
gn
Cos z - Cos z
gin
sin z - sin 1'
g
A z'gn (in minutes of arc)
With this formulation the following residuals were obtained:
Az =
44'.0
'9l
Az =
-75.0
g2
Az =
-44.8
'9'3
Az =
69.3
g4
Az =
45.4
g5
z =
-67.0
g6
z =
73.5
f7
77
Az
=
-42.3
f
t7 8
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generated by the starboard optical axis and are equal in magnitude to
what might be expected. Since the EAz = 3'.5 the value of z9 obtained
g
is the most probable value exhibited by the 8 starboard exposures.
EA = 3.5
z
g
(Y = EA
zg = 57'.7 (ignoring signs)
n
and the mean error of e
z
g
ED 2
e = = 75'.0
z
g n-u
where n = the number of equations and u = the number of direction angles.
In order to determine the mean error of aG, aG, and yG it is
necessary to compute the relative weights of the gravity direction angles
from a modified form of the normalized 3 x 3.
cos aG [aa] + cos aG [ab] + cos yG [ac] = 1, 0, 0
cos aG [ab] + cos ~G [bb] + cos yG [bc] = 0, 1, 0
cos aG [ac] + cos ~G [bc] + cos yG [cc] = 0, 0, 1
Using the first right hand column for PaG, the second for PRg and the third
for P'yg the following equations and numerical values are obtained
PaG = A = 1.7293830 .4666159
[bb] [cc] - [bc] 3.7062241
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P a = A =
1.7293830 =
.6265149
G [cc].[aa]F - [ac] 2
2.7603222
= A =
P Y
1.7293830 =
2.2873243
G
[aa].[bb] - [ab]2
.7560725
A = [aa]{{[bb] ? [cc] - [bc]2}- [ab] {[ab] . [cc] - [bc] ? [ac])+ [ac] {[ab] ? [bc] - [bb] ? [ac])
An alternate method of computing weights was employed as a check. To
determine the weight of variable a g G and y G are expressed in terms of ag in
the ~G and yG normal equations. These values are substituted in the normal
equation aG. The resulting coefficient of aG is the weight of aG. To determine
the weight of variable aG, aG and yG are expressed in terms of aG in the aG
and YG normal equations. These values substituted in the normal equation in
aG form the weight of R , in the resulting coefficient. Similarly the coefficient
of yG gives the weight of yG when ag and ag expressed in terms of yC are
substituted in the normal equation in yG.
[bb ]
cos G + [b c]
P a
cos
g
G
= - [ ab ] cos aG
[bc] cos G + [cc] cos G
= -[ac] cos a'G
cos Q =
[ac]. [bc] - [ab]. [ac]
cos a
G
[bb]. [cc] - [bc]' f
G
:
cos 1
[bc]? [ab] - [bb]. [ac]
cos a
G
[bb ]. [cc] - [b c]
G
Substituting in the aG normal equation
P a = [ab] + [ab] [ac]. [bc] - [ab]? [cc] + [ac] [bc]. [ab] - [bb].-[ac,
G {[bb]. [cc] - [bc [bb]- [cc] - [bc]Z
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P aG
[aa] cos aG + [ac] cos G = -[ab] cos aG
[ac] cos aG + [cc] cos C = - [b c] cos aC
cos aG [b c] . [ ac] - [ab ] ? [ cc] cos aG
[aa]?[cc] - [ac]
cos [ac] ? [ab] - [bc] ? [aa] l cos aG
YC [aa] ? [cc] - [ac]
P R = [ab] [bc]' [ac] - [ab] ?' [cc]1+ [bb] + [bc] r [ac] - [ab] [bc] ? [aa]
G {[aa]?[cc] - [ac] J 1 [aa]?[cc] - [ac]
P YG
cos aG
5G
[bc] ? [ab ] - [ac]. [bb] cos YG
t [aa].[bb] - [ab]
[ab].[ac] - [aa].[bc] cos y
.{ [aa].[bb] - [ab] ?
2
P -y [ac] f [bcj . [ab] - [ac] ? [bb] + [bc]~ [ab] ? [ac] - [aa] . [bc]
C ] - [ab
r _ _I
b] [aa].[bb
[bb] [a
Exactly the same numerical values were obtained P aG = .4666151)
P aG = .6265150
P YG = 2.2873241
[aa] cos aG + [ab] cos 3G = -[ac] cos yC
[ab] cos aG + [bb] cos RG = - [bc] cos YC
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The mean error of each direction angle is computed with
e
e aG =
75.0 =
109'.8 =
10
49.8
-
vP aG
.683
e
e aG =
75.0 =
94'.8 =
10
34.8
fP R
.792
e
e y0 = Zg =
75.0 =
49'.6 -
fP Y
1.512
-
G
While these errors by ordinary standards are large those based 67 on star-
board exposure and 71 port exposures are much smaller. Even these error values
are adequate for the purpose of referring the object space coordinate
system to the direction of gravity.
4. Further Adjustment
Inasmuch as cos aG, cos aG, and cos yG are dependent functions
it would seem that improved values would be obtained if the condition
equation were linearized in terms of two independent Eulerian angles.
cos aG = cos 0G sin YG
cos SG = sin 0G sin YG
cos YG = cos YG
(cos 0G sin y G ) cos az + (sin 0G sin YG) cos Rz + cos YG cos Yz = cos z
g
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cos z9 - (cos aG cos (x z + cos 1G cos 13z + cos yG cos yz) = -sin zg Az
g
A . AAG + B . AyG = A z
g
A = (-cos S.G cos az + cos aG cos (3 ) / sin zg
B = [(cos aG cos az + cos (3G cos ~ z) cot yG - (cos yG cos yz) tan yG)] / sin z9
n condition equations of the above form may be normalized to a 2 x 2 in AUG
and Ay G' A test solution demonstrated that the best values had already been
obtained inasmuch as on the first iteration AUG and AyG were significantly
zero. Had the condition equation been linearized, it would have the following
form:
sin aG cos az AaG + sin 1G cos (3z AI3G + sin yG cos yz AyG = sin Zg Az
g
Assume there is a particular value of aG, RG, and yG that satisfies the equation:
cos aG cos az + cos aG cos 0z + cos yG cos Y - cos Z = 0
n n n n n n g
Since cos zg - cos z g= -sin z Az
n g g
n
cos aG = cos aG + A cos aG
n n
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cos aG = cos aG + A cos (3G
n n
cos yG = cos yG -~- Acos yG
n n
A cos aG = cos aG - cos aG = -sin aG AaG
n n
A cos aG = cos aG - cos aG = -sin aG A3G
n n
A cos yG = cos yG - cos yG = -sin yG AYG
n n
Substituting we obtain to the first order the form obtained by linearization:
A cos aG cos az + A cos ~ G cos az + A cos yG cos yz
n n n
= cos z9 - (cos aG cos a z + cos ~G cos az + cos yG cos yz) _ -sin z g Az
sin a sin a sin y
or + G cos az (La,G) + G cos s z (Ar G) + G cos yz (AYG) _ + A z
sin z sin z sin z g
g g g
New normal equations in AaG, fl~G , and AYG are easily formed since all
coefficient changes are constant. It is necessary to accept the values of
aG, (3G, and yG as the best possible if the correction determined from the
revised normal equations are significantly zero.
sin2 aG sin a sin R sin a G sin y G [ sin a
sin z [aa] L~aG + G G [ab] SSG + [ac] ~yG= [a.~z ] G
g sin zg sin zg g sin zg
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 Approved For Release 2001/08/02 : CIA-RDP80T00703A000800040001-0
sin 4 sin 'G
G G
2
+ sin SG [bb]
A
sin R sin sink G
=[b.
]
+ G ~G [b c] ~~
A1
[ab]
sin zg
aG
sin1z
G
s1n2 zg
G
z
g sin z9
sin aG sin YG [ac] Aa sin RG "'in . 'YG
[b c] AR + sin? G
o /q
G
[cc] ?~'G
[ C. Az ] sinY G
G
i
z
g sin z
sin z g
sin zg
s
n
g
g
3.2707100
.7726223
.5110397 =
.0414955
.7726223
10.3315155
4.8919810 =
.1753744
.5110397
4.8919810
3.7896887 =
.0940528
Solution of the above normal equations gives in minutes of aTe
AaG = 0'.009
AEG = 0'.013
These quantities show no improvement is justified.
5. Application of the Gravity Direction Cosines to a Gravity Referenced
Datum
There are a number of possibilities. The simplest notion is to allow
G Za plane with the XG YG plane to be the YG axis
the intersection of the Z'
and the intersection of the Xa Ya plane with the XG YG plane be the XG axis.
This concept is illustrated in Figure 6.
cos aG' cos I3Go cos YG
tan 0G = cos aG
cos aG
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Figure 6 SIMPLEST CONCEPT OF GRAVITY REFERENCE PLANE
28
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cos aG = cos 0G
X
cos aG =-sin 0G
X
cos yG = 0
X
cos aG = sin 0G cos yG
Y
cos ~ = cos 0G cos yG
Y
cos yG = sin yG
Y
Such a datum orientation is not likely to be entirely satisfactory
inasmuch as the arbitrary datum selected is probably inherent to the object
such as the X axis defining the axis of the object. In such a case, it may
be desirable to let Xa or Ya be the preferred axis projected to the XG YG
plane from ZG. This selection is illustrated in Figure 7. Since /Z_G XG = 90?
cos a4 = sin aG
X
0 = cos aG cos RG + sin aG sin aG cos sXY
0 = cos RG cos yG + sin aG sin yG cos sYZ
0 = cos yG cos + sin,-yG sin aG cos SZX
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Figure 7 PROJECTION OF ARBITRARY AXIS Xa TO XG IN PLANE ZGXa
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cos sXY -
cos aG cos aG
sin aG sin ~
cos s = YZ _.cos.RG.cos YG
sin G sin G
cos sZX
. cos YG cos aG
sin G sin G
cos 0 cos 0 cos a cos
cos R = -sin R G G=- G G
G x G sin aG sin SG sin aG
cos = -sin YG cos YG cos aG = - cos aG cos yG
~GX sin YG sin aG sin aG
The direction of YG is deduced from
cos aG cos Did + cos 0 cos aG'+ cos yG cos YG = 0
Y
cos a cos a + cos Q cos S + cos y cos Y = 0
GY GX GY GX GY GX
cos aG = cos G (-cos YG cos aG ) - (-cos SG cos aG ) cos yG = 0
Y sin aG sin aG
cos G = cos yG (sin aG) - cos a (-cos YG cos aG )
Y sin aG
cos y G (sin2aG + Cos2aG)
sin aG
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cos y
sin aG
cos YG = cos aG (-cos RG cos aG) - cos R sin a
Y sin aG G G
cos aG
sin aG
Summarizing for the projection of either X or Y:,
a a
Xa Projected
cos aG = sin aG
X
cos aG = -cos RG cos aG
X sin aG
cos = - cos YG cos aG
yGX sin aG
cos R-G = cos yG
Y sin aG
cos YG = cos aG
Y sin aG
Ya Projected
cos aG =
cos YG
X sin G
cos aG = 0
X
cos YG =
X
cos aG
sin sG
cos aG = -cos aG cos SG
Y sin G
cos IG = + sin aG
Y
cos yG = - cos YG cos aG
Y sin RG
In the most general sanse, any preferred line o may be projected to
the XG YG plane. Assume it is desired to project any selected line to the
XG YG plane and let the projected line be the XG or YG axis. The direction
cosine of ZG and o are known whence
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cos a = cos (3G cos yo - cos ~ o cos yG
GX sin G
0
cos g M cos YG Cos ao.- cosy. COS aG
G sin G
0
cos.aG cos ao.- cos.ao.cos.RG.
`'`X sin G
0
cos G 0 = cos aG cos a0 + cos RG cos a0 + cos' YG cos y0
cos aG = cos I3 G cos yG - cos RG cos yG
Y X X
cos aG = cos YG cos aG Y cos aG
Y X X
cos yG = cos aG cos ~G - cos aG cos aG
Y X X
The projection of any line is illustrated in Figure 8. If it is
desired to let YG be the perpendicular to the Z0 plane, only the subscripts
are changed.
In any case, whatever the decision the matrices and coordinates are
rotated to a gravity reference:
cos aG cos aG cos aG
X Y
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Figure 8 PROJECTION OF ANY LINE 0 TO THE XG YG PLANE
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cos ~G cos aG cosc RG
X Y
cos yG cos YG cos YG
X Y
cos a cos ay
x
cos ax cos 13
cos ;fi cos Y
cos a
z
cos (3
z
cos Y
The gravity reference matrix is reduced as follows:
cos ax = cos ax cos aG + cos Rx cos aG + cos YX cos YG
g X X X
ay cos aG + cos S cos SG + cos Yy cos YG
X y X
cos a = cos az cos aG + cos Sz cos [3G + cos Yz cos YG
X X X
cos 0x = cos ax cos aG