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AIRBORNE KINEMATIC GPS POSITIONING FOR PHOTOGRAMMETRY THE DETERMINATION OF THE CAMERA EXPOSURE ... - page 5 / 11

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time3 is the central time and

t1 . . . t5 are the five consecutive time tags of GPS antenna phase center positional data

Y=a

y

Z=a

z

Similar equations could be written for the other two models:

(4) (5)

2 2

+ b y t + c y t + b z t + c z t

The unknown parameters for each model can be related to distance, velocity and acceleration by differentiating the

above equations as follows:

2

-X=0

2

-Y=0

2

-Z=0

.

distance from origin = a, velocity = dX/dt = b + 2ct, and acceleration = dX2/d2t = 2c at t3

The observation equations are:

a

x

a

y

a

z

v

x

v

y

v

z

=

x x + bt + ct

=

y y + bt + ct

=

z z + bt + ct

(6) (7) (8)

(9) (10) (11)

The coefficient matrix elements for all three

the partial derivatives of unknowns. The coefficient models, as follows:

the model matrix is

with respect the same for

models are to the all three

1 3 -(t -t )

2

2 3 -(t -t )

2

3 3 -(t -t )

2

4 3 -(t -t )

2

5 3 -(t -t )

2

-1

1 3 -(t -t )

-1

2 3 -(t -t )

-1

3 3 -(t -t )

-1

4 3 -(t -t )

-1

5 3 -(t -t )

A=

(12)

From this point forward in the discussion, time differences will be denoted as simply "t" to simplify the expressions.

The observation vectors composed of the observed coordinate values for each GPS epoch are:

for X;

-x

1

-x

2

-x

3

-x

4

-x

5

for Y;

-y

1

-y

2

-y

3

-y

4

-y

5

for Z;

-z

1

-z

2

-z

3

-z

4

-z

5

(13)

A least squares solution minimizing the function

PHI = V' P

V

(14)

is used to solve for the unknown parameters (Uotila 1986). The P matrix is the scaled inverse of the variance-covariance

matrix (Sigma Lb) is represented by

for the observed quantities. Sigma02, the variance of unit

The scaling weight, which

in this case has the value of 1.

parameters

begins

with

the

normal

The solution for

equations

noted

the unknown as follows:

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