introduction theory of operation...theory of operation of the software. following that is the...

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Introduction This section discusses the overall theory of operation of the Real Time Navigator (RTN)

and presents test results to verify its operation. The first part presents an overview of the

theory of operation of the software. Following that is the mathematical details of the

navigation and filtering functions that comprise the RTN software package. Finally a

synopsis of the tests and test results are given.

Theory of Operation

System Block Diagram

Figure 1 gives a block diagram of the basic RTN function. Some of the processes are not

pertinent to a discussion of the Navigation function and its performance but are included

for completeness. The processes are as follows

• GPS Server: This process receives the time tagged position, velocity and timing

data from the GPS NovAtel ProPak-LB receiver via a serial interface. It also

generates, via three Kalman Filters utilizing GPS velocity, estimates of track,

pitch and roll and inertial acceleration for the IN-AIR restart mode.

• IMU Server: This process receives the time of validity interrupt from the Kearfott

KI-4901 and the incremental velocities and angles via a serial interface..

• SocketTranslator: This process provides the Ethernet communication link

between the RTN and the Monitor/Control program.

• ModeManager: This process controls the automatic mode changes of the system.

• Logger: This process provides an output capability to dynamically save the raw

IMU and GPS data, the navigation data and other pertinent data.

• Align: This process implements the three Alignment Kalman filters used in the

RTN.

• Nav: This function is the heart of the RTN and implements a strapdown

mechanization of the inertial navigation algorithm.

• Camera Server: This process interfaces with the ROI camera. It receives and time

tags the camera exposure interrupt. The process also receives via a serial channel

the Image Data Message and the Relative Roll Angle Data Message from the

camera SIU and it sends to the camera the SIU Orientation Data Message.

• Time Server: This process receives the GPS receiver 1PPS interrupt and via a one

state Kalman filter calibrates the processor clock. The resultant clock bias

estimate is used in a Timer Software Class to time tag the various events.

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Figure 1: RTN Block Diagram

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Mathematical Notation

Throughout this text the following conventions are used, except were noted.

1. Direction cosine matrices are denoted f

iC where the subscript indicates the initial

coordinate frame and the superscript indicates the final coordinate frame.

2. All coordinate frames are right-handed orthogonal and all Euler angles are right-

handed.

3. Three element Vectors are shown with an over arrow, e.g. Vr

.

4. A skew symmetric matrix is denoted ar

which indicates a matrix with the form

−

−

0

0

0

xy

xz

yz

aa

aa

aa

Inertial Navigation

Inertial navigation uses inertial information to compute position and speed over the

surface of the earth and attitude with respect to north and local level. The inertial

information is in the form of linear acceleration and rotational rate as measured by

instruments (accelerometers and gyroscopes) which utilize the affects predicted by

Newton’s laws of motion. The purpose of the inertial navigation algorithms is to cancel

out from the inertial measurements those effects not due to relative motion over the

earth’s surface. These effects include the rotation of the earth with respect to space,

gravity and coriolis forces. The resultant accelerations and rates are used to compute the

relative speed and position with respect to the earth and attitude with respect to level and

north.

There are many ways to mechanize the concept of inertial navigation [1, 2, 3] using

different types of instruments and configurations. For the RTN, a “strapdown”

mechanization is used. In this approach the inertial instruments are fixed to the vehicle

body and the measurements are sampled and processed by algorithms within a digital

processor. Note that in this mechanization there is no physical level platform, i.e. the

level coordinate frame (platform) only exists as a set of numbers in the computer. The

gyroscopes sense the rotation of the body with respect to inertial space and after an

adjustment for earth rotation and motion over the surface of the earth are used to update

the direction cosine matrix from the instrument coordinates to a locally level frame. This

direction cosine matrix is used to transform the accelerometer measurements from the

instrument axes to the level frame. The resultant components are then adjusted for gravity

and coriolis effects and numerically integrated to from velocity relative to the earth’s

surface. The level frame is called the “navigation” frame.

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Mathematically the strapdown navigation mechanization is described by the following

differential equations [1]. For level acceleration

Lesf

L

iL VgaCVrrrrr&r

×Ω+−+= )2(ρ

and the derivative of the L

iC matrix is given by

.⟩⟨+⟩⟨−= L

i

L

IL

i

Ii

L

i

L

i CCC ωωrr&

Where

• LVr

: Level velocity with respect to earth, meters per second.

• L

iC : Direction cosine matrix from instrument coordinates to level navigation

frame.

• sfar

: Specific force measured by accelerometers in i coordinates.

• gr

: Plumb gravity at current latitude and altitude.

• ρr

: Motion over the surface of the earth (transport rate in radians per second).

• eΩr

: Current earth rate components in navigation frame.

• i

Iiωr

: Gyroscope output; space rate of i coordinates with respect to inertial ( I )

frame coordinatized in instrument frame ( i ).

• L

ILωr

: Rate of change of the level coordinates (navigation frame) with respect to

inertial space. This is the summation of transport rate and earth rate.

By correctly initializing and numerically integrating the above equations, level velocity

can be determined. From velocity, position on the earth surface can be calculated by

performing another numerical integration.

The RTN exclusively uses the WGS84 physical constants and gravity model given in [4].

The following describes the navigation equations used in the RTN. Specifically the RTN

uses a Wander Azimuth mechanization which allows navigation over the poles without

loss of information or mathematical singularities in the computation [1].

Figure 2 presents the coordinate frames used in the RTN.

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Position

Position in the RTN is represented by a quaternion ( qP ) and ellipsoidal altitude. The

quaternion is used because of its compact form, numerical accuracy and efficiency. The

position quaternion represents the direction cosine matrix from an earth centered frame to

the navigation frame which is a local level wander azimuth frame. Altitude is computed

by open-loop integration of vertical velocity.

The earth centered frame is right handed orthogonal, is fixed to the earth with it’s x axis

along the earth spin axis and positive through the north pole. The z axis is orthogonal to

the x axis and its positive axis is coincident with the Greenwich meridian. The y axis is

orthogonal to x and z, (see Figure 2).

Greenwich Meridianeω

N

E

D,

Long

Lat

EZ

EX

EY

Locally Level Frame coincidentwith ISA position.

αX

αY

α

αZ

Wander Angle

Figure 2: Coordinate Frames

The wander azimuth frame is a right handed orthogonal frame whose x axis is pointed

north when wander angle is zero. The z axis is pointed down. The direction cosine matrix

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from the earth centered frame to wander azimuth is given by the following ordered set of

single axis Euler rotations

)()()( LongRLatRRC xyze −= αα

where

• α is wander angle.

• Lat is latitude

• Long is latitude

• e as a subscript or superscript denotes the earth centered frame

• α as a subscript or superscript denotes the wander azimuth frame

Position Initialization

The position quaternion represents the direction cosine matrix αeC . The RTN is

initialized with the GPS latitude ( 0l ) and longitude ( 0long ) and wander angle ( 0α ) is

determined by the Coarse Alignment Filter, i.e. gyrocompassing. The position quaternion

is only initialized with the latitude and wander angle. Since initial longitude and the

change in longitude during navigation are both represented by single rotations about the

same axis (earth centered x) we need only initialize the position quaternion to an initial

longitude of zero. After that we simply add the initial longitude to the change in longitude

as computed from the quaternion during navigation. The initial value of qP is given by the

following operation

−

−

=

=

)2

cos()2

cos(

)2

cos()2

sin(

)2

sin()2

cos(

)2

sin()2

sin(

00

00

00

00

0

0

0

0

l

l

l

l

h

g

f

e

Pq

α

α

α

α

.

Position Update

The position quaternion is time updated using transport rate which is computed from the

current velocity, altitude and earth radii. Transport rate indicates the relative motion of

the navigator over the surface of the earth. Because the RTN is mechanized as a wander

azimuth navigator we need only update the quaternion for level components of transport

rate, i.e. xρ and yρ . This means that the update can be simplified by the elimination of

the z component of integrated transport rate. The update algorithm is

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Γ

Γ

−−

−

−

=

=

+

+

4

3

3

1

)1(

F

F

F

hfe

gef

fhg

egh

h

g

f

e

P y

x

nn

nq

where

)2

cos(

)2

sin(

4

3

Γ=

Γ

Γ

=

F

F

and

• yx,Γ : Integrated value of transport rates over position update interval minus the

position correction from the alignment process.

• Γ : Magnitude of integrated level transport rate.

Now the integrated transport rate is calculated via

−

=

Γ

Γ

z

x

y

x

y

x

δθ

δθ

λ

λ

where

x

x

δθ

δθis the position correction in radians.

and

−

∆=

∆=

α

αα

ρ

ρ

λ

λ

0

1

1

0

1

1

Y

x

NED

N

E

NED

y

x

y

xV

V

CR

R

Ctt

where

• t∆ is the integration time

• ER is the prime radius of curvature

• NR is the meridian radius of curvature

Position Direction Cosine

The position correction cosine matrix is calculated from the position quaternion via

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++−−−+

++−+−−

−++−−

=2222

2222

2222

)(2)(2

)(2)(2

)(2)(2

hgfeehfgfheg

ehfghgfeghef

fhegghefhgfe

Ce

α

Position Angle Extraction

Latitude, longitude and wander angles are extracted from the position direction cosine

matrix via

)(tan

)(tan

))(1

(tan

)1,1(

)2,1(1

0

)3,3(

)3,2(1

2

)3,1(

)3,1(1

e

e

e

e

e

e

C

C

longC

Clong

C

Clat

α

α

α

α

α

α

α −

−

−

−=

+−=

−−=

.

Altitude

Altitude is computed via trapezoidal integration of the vertical component of velocity

which includes corrections for altitude error and vertical velocity error which are

calculated by the Alignment process.

Velocity

Velocity is computed by transforming the accelerometer measurements into the level

coordinate frame and subtracting the computed coriolis acceleration and gravity effects.

The result is then, after appropriate initialization, integrated to form level velocity.

Velocity Initialization

The three velocity integrals are initialized to zero when the navigator is started using the

Ground Align mode. When an In-Air alignment is performed the GPS velocities are used.

Velocity Update

Velocity is updated using the compensated accelerometer measurements. The raw

accelerometer measurements are compensated by bias estimates supplied by the

Alignment process. The accelerometer hardware performs an integration of the observed

inertial acceleration (specific force) and provides integrated samples to the navigation

processor. These samples represent the incremental velocity change over the sample time.

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The velocity algorithm simply adds the incremental changes with the appropriate

corrections to the previous velocity. Thus

−

−

−

∆−

∆−

+

∆

∆

∆

+

=

+ yznxyn

xznzxn

zynyzn

z

y

x

z

y

x

i

nz

y

x

nz

y

x

VV

VV

VV

t

altlatG

t

V

V

V

V

V

V

C

V

V

V

V

V

V

ωω

ωω

ωω

δ

δ

δα

),(

0

0

1

where

Ω

Ω+

Ω+

=

e

z

e

yy

e

xx

z

y

x

2

2

2

ρ

ρ

ω

ω

ω

and

• zyxV ,,∆ are the compensated accelerometer measurements.

• ),( altlatG is Plumb gravity as a function of latitude and altitude.

• yx,ρ is the transport rate.

• eΩ is the earth rate vector in wander azimuth coordinates.

• t∆ is the time interval.

• zyxV ,,δ are velocity corrections supplied by the Alignment process.

Attitude

The attitude of the system is described to the outside world by heading (ψ ), pitch (θ )

and roll (φ ) which are seen pictorially in Figure 3. In this figure the axes labeled

ZYXI ,, are the instrument axes.

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North

East

Down

ψ

θ

φ

XI

YI

ZI

Figure 3: Attitude Angles

These angles are only for the convenience of the user and are not the ones the navigation

computer uses to describe attitude. Attitude information in the RTN is contained in the

quaternion qA . We use the quaternion here for the same reasons as in the position case.

The direction cosine matrix relating the instrument axes ( i ) to the navigation frame (α )

is αiC . This matrix is used to transform the accelerometer measurements to the navigation

frame.

Attitude Initialization

The attitude quaternion must be initialized in either the Ground or In-Air alignment

mode. For our purposes here we will assume that either of these processes results in a

direction cosine matrix from the level frame to the instrument frame and is made

available to the navigation process. Initialization comprises forming from this matrix an

equivalent quaternion ( qA ). Reference [3] gives the general procedure which is more

complicated that the position quaternion initialization.

Attitude Update

The attitude quaternion must be updated to account for the motion of the inertial

instruments with respect to the level frame. This is done by using the compensated

gyroscope measurements. The raw gyroscope measurements are compensated for bias

estimates supplied by the Alignment process, earth rate and transport rate. In terms of

direction cosine matrices the attitude update is represented as

)(

)1()()1(

ni

ninini CCC ++ = αα .

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The quaternion representation of this expression is

Φ

Φ

Φ

−−−

−

−

−

=

=

+ 4

3

3

3

1F

F

F

F

dcba

cdab

badc

abcd

d

c

b

a

Az

y

x

nn

q

where

• [ ]Tdcba ,,, are the quaternion elements

•

)2

cos(

)2

sin(

4

3

Φ=

Φ

Φ

=

F

F

• zyx ,,Φ are the compensated gyroscope measurements

• Φ is the magnitude of Φr

.

The level frame rotates in inertial space due to earth rate and transport rate. Thus in order

to keep the navigation frame locally level these rates must be subtracted from the

measured rates. The compensation for gyroscope measurements is

α

αδrrr

i

cor C−Θ∆=Φ

where

Ω

Ω+

Ω+

∆+Ψ−=e

z

e

yy

e

xx

t ρ

ρ

δ αrr

and

• corΘ∆r

is the gyroscope measurements corrected by the Alignment process bias

estimates.

• Ψr

is the attitude correction vector from the Alignment process.

Attitude Direction Cosine Matrix

The accelerometer measurements must be transformed from their coordinate frame to the

locally level navigation frame. The direction cosine matrix to do this is calculated from

the attitude quaternion elements via

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+−−+−

−−+−+

+−−−+

=2222

2222

2222

)(2)(2

)(2)(2

)(2)(2

cbadadbcbdac

adbccbadcdab

bdaccdabcbad

Ci

α .

This matrix is also used to transform Alignment process corrections, transport rate and

earth rate to instrument coordinates.

Attitude Angle Extraction

The RTN has as an initial attitude output matrix c

iC which can be used to calculate the

standard attitude angles from any frame fixed with respect to the instrument frame to the

local level North frame. Heading, pitch and roll are calculated via

i

ciC CCCαα =

and

=

−=

=

−

−

−

α

α

α

α

α

α

φ

θ

ψ

)3,3(

)2,3(1

2

)1,3(

)1,3(1

)1,1(

)1,2(1

tan

)(1tan

tan

c

c

c

c

c

c

C

C

C

C

C

C

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Alignment Kalman Filters

The purpose of the Alignment filters is to correct the navigation function for position,

velocity and attitude and calibrate the inertial instruments. The navigation corrections, as

generated by the various alignment filters, are incremental corrections that are integrated

by the navigation function. The instrument corrections are only generated by the FAF and

are summed externally to the filter and applied to the instruments as total corrections.

There are three Alignment filters in the RTN. The first is the Coarse Align Filter (CAF)

which is used to gyrocompass the system, i.e. find true north. The CAF is only used when

performing a ground alignment and provides only level velocity, tilt and earth rate

corrections to the navigation function. The Fine Align Filter (FAF) has two modes. The

first mode is used during ground alignment and after the CAF has completed and also

utilizes the Navigator velocity as a measurement. Velocity, attitude and instrument

corrections are generated by this filter and are applied to the navigation function. The

second mode of the FAF uses GPS position measurements. The FAF using GPS

measurements is the normal mode of operation when the system is moving and provides

position, velocity, attitude and instrument corrections to the system.

Kalman Filter General Equations

The CAF and FAF procedures utilize the Kalman filter algorithm to process the data. The

Kalman filter is a formal procedure for calculation of state estimates of a linear system

given a measurement that has Gaussian additive noise. The procedure guarantees

optimality in a minimal least squares sense. See reference [6] for a complete discussion.

The equations used in the filter are given here for completeness. In the following sN is

the number of states and mN is the number of measurements.

• nP is the state covariance matrix for iteration n of dimensions sN square.

• nΦ is the state transition matrix for iteration n of dimensions sN square. This

transforms the state vector nx from time at 1−n to time at n .

• nF is the matrix of differential equations of state that are integrated to form nΦ of

dimensions sN square.

• nQ is the Model Noise covariance matrix, of dimensions sN square, that is used

by the filter to represent un-modeled errors in the system.

• nK is the Kalman gain matrix for iteration n of dimension sN by mN .

• nH is the measurement transformation matrix of dimensions mN by sN . This

relates the system state vector to the measurement observation.

• nx is the system state vector of dimension sN .

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• nz is the measurement vector of dimensions mN .

• nR is the measurement covariance matrix of dimensions mN square.

In the following equation set a hat over a term, e.g. P , implies time propagation, and a

bar, e.g. P , implies a measurement update.

The time propagation of the state covariance is given by

n

T

nnnn QPP +ΦΦ=+1ˆ

where the state transition matrix is

∆=Φ F

n e

and the Model noise matrix is

τττ dqQT

nnn ),(),(0

∆Φ∆Φ= ∫∆

where q is the White Noise spectral density.

The Kalman gain is calculated via,

[ ] 1ˆˆ

−

+= n

T

nnn

T

nnn RHPHHPK

which is used to update the state covariance via

[ ] [ ] T

nnn

T

nnnnnn KRKHKIPHKIP +−−= ˆ .

The state is time propagated using

nnn xx Φ=+1

and updated using the Kalman gain and

[ ]nnnnnn xHzKxx −+=ˆ .

Coarse Align Filter (CAF)

The CAF filter is used to initialize the navigation system heading and provide leveling of

the strapdown platform. The CAF is invoked immediately following the initial attitude

determination of the system using the accelerometers. The differential equations defining

the CAF process model the velocity errors introduced into the navigator due to an

incorrect heading. An incorrect heading causes the components of Earth Rate to be

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incorrectly subtracted from the measured space rates. This causes the computational level

platform to tilt which causes a velocity error. The differential equations that define this

process are

YY

XX

XY

YX

Y

X

gV

gV

Ω−=

Ω−=

−=

=

=Ω

=Ω

δψ

δψ

ψδ

ψδ

δ

δ

&

&

&

&

&

&

0

0

Where

• YX ,Ωδ are incremental Earth Rate in Wander coordinates.

• YXV ,δ are velocity errors in Wander coordinates.

• YX ,ψ are the navigator tilt errors about the level Wander coordinates.

• g is the acceleration due to gravity.

The CAF process initially sets the system wander angle (α ) to zero and position to the

GPS values. Then velocity observations are formed every second from the navigator and

used as measurements to the CAF. The CAF supplies incremental velocity and attitude

corrections to the navigator and adjusts the wander angle via

)tan(X

YaΩ

Ω−=α .

The earth rate estimate is arrived at by summing the incremental estimates ( YX ,Ωδ ).Note

that all the states in the CAF are reset to zero at the beginning of each filter cycle.

In Air Alignment

In-Air alignment is used to re-start the system when in flight. In-Air alignment can be

performed under the following conditions:

1. GPS must be in at least single-point mode and provide position and ground speed.

2. Speed must be greater than five meters per second.

3. We assume speed is constant and that the aircraft side slip and angle of attack are

small. Note that the aircraft may execute a coordinated turn during In-Air

Alignment.

Under these conditions the process is as follows:

1. Estimate acceleration from GPS velocity. Three two state Kalman filters are used

to accomplish this task. The filters are run using one second velocity observations

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from the GPS. The filters are run continuously. The acceleration estimates

are [ ]T

DEN vvv &&& ,, . Specific force is formed from the acceleration estimates by

subtracting gravity from the down component of the acceleration estimates.

Specific force vector is then [ ]T

DEN aaa ,, .

2. Heading is initialized to ground track using GPS velocities (

=

N

E

V

Va tanψ ).

3. Pitch is initialized using GPS velocities (

+

−=

22tan

EN

D

VV

Vaθ ).

4. The direction cosine matrix from the local level NED frame to a frame fixed to

the aircraft that is not rolled (body prime) is calculated via, ( ) ( )ψθ zy

B

N RRC =′

.

5. Transform the specific force estimates to the body prime and calculate the roll

angle via

−=

′

′

B

Z

B

Y

a

aa tanφ .

6. Form the NED to body direction cosine matrix ( ) ( ) ( )ψθφ ZYX

B

N RRRC = .

7. Using the aircraft body to Pod DCM ( P

BC ), the Pod to camera DCM ( ( )P

C

PC φ )

and the camera to IMU DCM ( I

CC ) calculate the initial navigation DCM via

( TB

N

P

B

C

P

I

C

N

I CCCCC )(= . Initialize the navigation DCM to this value.

8. Initialize the navigation position to the GPS position and the navigation velocity

to GPS velocity.

9. Command the PADS to navigation closed loop mode.

Fine Align Filter (FAF)

The inertial navigation function in the RTN utilizes a local level platform to compute the

system attitude, velocity and position. The purpose of the Kalman filter in the RTN is to

correct the navigation function for errors that arise because of erroneous initialization or

instrument (gyroscope and accelerometer) errors. The FAF is a Kalman filter having the

following states

• Nine INS error states, position error, velocity error, tilt error and heading error.

• Three gyroscope bias errors coordinatized in the instrument frame.

• Three accelerometer bias errors coordinatized in the instrument frame.

• Three lever arm errors coordinatized in the body frame.

• Three GPS position errors coordinatized in the Wander frame.

The FAF can use as a measurement vector either velocity error or position error. It is

intended that when using the Navigator velocity as the error measurement the system is

not moving. The FAF using velocity as a measurement is essentially an extension of the

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ground alignment procedure. When the FAF uses position error as a measurement the

system can be in motion. In the RTN this is called the “Closed Loop” mode.

The following sections describe the generalized INS error equations. The specific

equations used in the RTN FAF are then given. This development includes the state

differential equations, instrument and lever arm error equations, Model noise functions

and the measurement equations.

Generalized INS Error Equations

This section presents the generalized INS error equations as developed in reference [5].

The error equations given here are derived via first order perturbation of the inertial

navigation equations. Essentially these equations describe mathematically how specific

errors propagate through the inertial navigation system to generate position, velocity and

attitude error. These equations provide the basic building blocks for the Kalman filter

design because they describe how errors propagate through the navigation function.

Velocity error propagation is given by

AVAVrrrrrrrrr

& δρδρδφδ +Ω+−Ω++= 22

where

• Vr

δ is the velocity error.

• φr

is the small angle vector of the level platform with respect to the true reference.

• Ar

is the true specific force applied to the system.

• ρδr

is the error in transport rate.

• Ωr

δ is the error in earth rate.

• Vr

is the true velocity of the system.

• ρr

is the true transport rate.

• Ωr

is the true earth rate.

• Ar

δ is the accelerometer error.

The attitude error propagation is given by

εωψψrrrr

& +×=

where

Ω+=rrr

ρω

and

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• ψs

is the small angle vector of the platform with respect to where the navigation

system computed reference is.

• εr

is the gyroscope error.

The small angle error between the true reference and the computed reference is θδr

. This

is a position error and its differential equation is

θδρρδθδrrr

r& ×−= .

Note that

ψθδφrrr

+= .

The error in earth rate is given by

θδδrrr

×Ω=Ω .

These generalized error equations are used to derive the specific error equations in the

RTN by substituting the appropriate constraints which include the Wander azimuth

mechanization and choice of coordinate system.

RTN Filter State Equations

The following subsections provide the details of the equations used in the FAF in the

RTN.

INS Errors

Table 1 presents the INS error equations that are used in the FAF. In the table zyxSf .. is the

specific force in Wander coordinates as measured by the accelerometers. plumbg is plumb

gravity as given in reference [4].

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Table 1: INS Error Equations

States

XR∆ YR∆ ZR∆ XVδ YVδ ZVδ Xψ Yψ Zψ

XR&∆

yρ− 1

YR&∆

xρ 1

ZR&∆

yρ xρ− 1

XV&δ

R

g plumb

~

ZΩ2 )(2 YY ρ+Ω− ZSf YSf−

YV&δ

R

g plumb

~

ZΩ− 2 )(2 XX ρ+Ω ZSf− XSf

ZV&δ

R

g plumb

~2

)(2 YY ρ+Ω )(2 XX ρ+Ω−

YSf XSf−

Xψ&

ZΩ )( YY ρ+Ω−

Yψ&

ZΩ− )( XX ρ+Ω

Zψ&

=

)( YY ρ+Ω )( XX ρ+Ω−

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Instrument Errors

The FAF in the RTN has states for gyroscope and accelerometer biases. These are

coordinatized in instrument coordinates.

Gyroscope

Bias

There are three bias states which are transformed in the state transition matrix via

Bi gCrr αε = .

Accelerometer

Bias

There are three bias states which are transformed in the state transition matrix via

Bi aCArr

αδ = .

Lever Arm Errors

The lever arm is the physical distance between the GPS phase center of the antenna and

the point of navigation of the RTN. The lever arm, which is defined in the instrument

coordinates, must be transformed to the navigation frame for use in calculating the

position error between the GPS and the RTN. When transforming this vector two error

sources can corrupt the resultant vector, these are INS tilt and heading error and the error

in measuring the lever arm in the instrument frame. An analysis of this transformation

results in the following expression for the transformed lever arm. In this expression BL is

the true lever arm in the instrument coordinates, Lδ is the measurement error of the lever

arm in instrument coordinates and φ is the tilt and heading error of the INS and L is the

erroneous lever arm in the navigation frame.

BBBBB LCLCLCLrrrrr

ααα δφ ++=ˆ

This expression is used to model the affects of these errors on the position observation as

described in a following section.

Model Noise

There are several instrument error sources that are not specifically modeled in the FAF.

Typically these errors cause position errors in the INS that have short time affects. For

example gyroscope scale factor error only comes into play during a turn. The design goal

of the FAF is to account for these errors in such a manner that the FAF will reduce its

Kalman gain matrix when these un-modeled errors are affective. In this manner the FAF

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states will not be corrupted by the variations in position error caused by the un-modeled

error sources.

The Kalman filter Model Noise covariance matrix ( nQ ) is the mechanism for adjusting

the system state covariance matrix to account for the un-modeled error sources.

Essentially a covariance function is designed to represent each of the un-modeled errors

and included in the Model Noise covariance elements that are applicable. This means that

gyroscope errors are added to the tilt and heading (ψ ) covariance elements and

accelerometer errors are added to the velocity error ( Vδ ) covariance elements.

Gyroscope

The gyroscope is subjected to drifts due to a plethora sources. The following are the most

important for the ring laser gyroscope as is used in the Kearfott KI-4901 IMU which the

RTN uses.

The following sections provide the mathematical models used to represent the gyroscope

un-modeled errors as covariance matrices. These matrices all represent the individual

errors in the navigation frame and are used in the FAF state covariance time propagation

nQ matrix ψ state elements. The summed gyroscope un-modeled errors are

grwgcBgn QQQQ ++= βψ )( .

Axis Misalignment

This error is caused by the non-orthogonal nature of the instrument cluster assembly.

Essentially some small fraction of the rotation about one principal axis is coupled to

another. In this model 2

βσ g is the uncertainty in the misalignment and βg∆ is an

effectiveness term adjusted by simulation and field testing so that the FAF ignores the un-

modeled error.

The expression that models this error is

ββαα

β σ

ωω

ωω

ωω

gg

it

xy

zx

zy

ig dtCCQ ∆

+

+

+

= ∫∆

2

022

22

22

Note that this model is driven by the current space rates the system is being subjected to

and that the error source is coordinatized in the instrument frame. This results in a

covariance matrix with off-diagonal terms.

Correlated Noise

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This error source represents time correlated error, i.e. errors that follow a first order

Markov model. In this model 2

gcBσ is the uncertainty in the drift and gcB∆ is an


modeled error. The time correlation of the error is included in the effectiveness term.


gcBgcBgcBQ ∆= 2σ .

Note that this error can be modeled directly in the navigation frame since the similarity

transformation is not required. This covariance matrix is diagonal.

Wide Band Noise

Wide band noise refers to the classic Angle Random Walk specification given for

practically any gyroscope. In this model 2

grwσ is the random walk and t∆ is the FAF

iteration time increment.

tQ grwgrw ∆= 2σ



Accelerometer

The following sections provide the mathematical models used to represent the

accelerometer un-modeled errors as covariance matrices. These matrices all represent the

individual errors in the navigation frame and are used in the FAF state covariance time

propagation nQ matrix Vδ state elements. The summed accelerometer un-modeled errors

are

arwacBaSFan QQQQVQ +++= βδ )( .

Misalignment

This error is caused by the non-orthogonal nature of the instrument cluster assembly.

Essentially some small fraction of the specific force along one principal axis is coupled to

another.

In this model 2

βσ a is the uncertainty in the sensitivity and βa∆ is an effectiveness term

adjusted by simulation and field testing so that the FAF ignores the un-modeled error.


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ββαα

β σ aa

it

xy

xia dtC

aa

aCQ ∆

+

= ∫∆

2

022

2

0

Note that this model is driven by the specific force the system is being subjected to and

that the error source is coordinatized in the instrument frame. This results in a covariance

matrix with off-diagonal terms.

Scale Factor

Scale factor error affects the axis of measurement and is a function of the specific force

being applied to the system. In this model 2

aSFσ is the uncertainty in the sensitivity and

aSF∆ is an effectiveness term adjusted by simulation and field testing so that the FAF

ignores the un-modeled error.


aSFaSF

it

z

y

x

iaSF dtC

a

a

a

CQ ∆

= ∫∆

2

02

2

2

σαα

Note that this model is driven by the specific force the system is being subjected to and

that the error source is coordinatized in the instrument frame. This results in a covariance

matrix with off-diagonal terms.

Correlated Noise

This error source represents time correlated error, i.e. errors that follow a first order

Markov model. In this model 2

acBσ is the uncertainty in the drift and acB∆ is an


modeled error. The time correlation of the error is included in the effectiveness term.


acBacBacBQ ∆= 2σ



Wide Band Noise

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Wide band noise refers to the classic Velocity Random Walk specification given for

practically any accelerometer. In this model 2

arwσ is the random walk and t∆ is the FAF

iteration time increment.

tQ arwawB ∆= 2σ



Zero Velocity Measurements

When the FAF and CAF use velocity error as an observation the system is constrained to

be stationary. Thus any velocity generated by the navigation function can be attributed to

an error. The measurement for the FAF and CAF filters is therefore

α

nn Vzrr

= .

The transformation of the FAF and CAF states to the measurement is then

=

z

y

x

xn

V

V

V

XH

δ

δ

δ

1

1

1

.

GPS Position Measurements

In the Pinpoint system there are two components of the lever arm from the GPS antenna

to the Inertial Sensor Assembly navigation point. The first is defined in aircraft body

coordinates and the second is defined in the inner camera axis system. Figure 1 presents

the coordinate frames and lever arm components. The GPS position information is

measured at the antenna phase center which is located at the origin of the aircraft body

fixed coordinate frame. Essentially the GPS position information has to be translated to

the origin of the Navigation frame which is the origin of the ISA ( zyxI ,, ) frame. Note that

the ISA frame is fixed to the inner camera axis frame ( zyxC ,, ) and rotates with respect to

the camera pod frame ( zyxPOD ,, ) through the pod roll angle Pφ . The pod roll angle is

measured by a resolver and sampled at a five Hertz rate. At the GPS observation time the

total lever arm is transformed to ISA axes via

I

C

I

B ρρρrrr

+=

where I

Cρr

is the fixed distance from the origin of the camera inner axis system to the

Navigation axes origin and is coordinatized in the ISA frame.

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B

P

CAOP

C

P

I

C

I

B CTCC ρφρrr

/))((=

where )( OP Tφ is the roll angle at the observation time as interpreted from the five Hertz

data stream.

The total lever arm is then transformed to geographic axes via

ρρ αα

rrI

GGCC= .

The Kalman filter position observation is

G

OO TGpsTNavf ρε −−= ))()((

where ()f transforms the angular difference (latitude and longitude in radians) to linear,

i.e. meters. Note that there are several DCM’s and the navigator position involved in this

calculation that are time dependant. The GPS observation time OT is the UTC time that

the GPS receiver time tags the GPS position. This time is typically two hundred

milliseconds behind real time. This necessitates that the navigation data, DCM’s and

position, be placed in a circular buffer so that the data can be retrieved at the observation

time OT . A second order curve fit of the data is used to calculate the position at the

observation time. The position error is finally transformed into the Wander azimuth

frame for inclusion in the Kalman filter as an observation.

To complete the observation equation we need to model the source of the errors in the

observation. The filter models the following

• Position errors due to the integration of gyroscope and accelerometer errors by the

inertial navigation function. These are the zyxR ,.∆ error states

• Position errors in the GPS reference, these are the zyxGPS ,,δ error states.

• Position errors due to GPS to instrument lever arm errors, these are the

zyxL ,,δ states.

• Position errors due to incorrect transformation of the lever arm into navigation

coordinates.

The measurement matrix relates the FAF error states to the error measurement and is

+

+

−

−

−

+

∆

∆

∆

=

z

y

x

z

y

x

i

i

z

i

y

i

x

i

xy

xz

yz

z

y

x

nn

GPS

GPS

GPS

L

L

L

C

L

L

L

C

R

R

R

XH

δ

δ

δ

δ

δ

δ

ψψ

ψψ

ψψαα

0

0

0

1

1

1

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XC

YC

ZC

Cρr

XIZI

YI

YPOD

ZPOD

Pφ

XCA /

YCA /

ZCA /

Bρr

Flight Direction

GPS Antenna Phase Center

ρr

Figure 4: Coordinate Frames and Lever Arm

RTN Operation

The RTN at startup is initialized in Standby mode. The operator must command the RTN,

via the monitor program, to initiate ground alignment or in-air alignment. After either

alignment is complete the RTN can be commanded to closed-loop operation.

Monitor Program Operation The monitor program can be run on any Windows machine that is connected to a LAN

that is connected to the PADS. This is a standalone GUI process. Once a connection is

made to the RTN the monitor can be used to change the mode of the RTN processes or

monitor the key system parameters such as attitude and position. Note that in the

following the acronym “PADS” refers to the machine the RTN is running on.

Details of Operation

When the Monitor program is first initiated it starts in the Setup mode. The Setup page is

shown in Figure 5. The Monitor will attempt to connect to the last location it used.

Changes to the connection address are made via the “Location” item in the “PADS

Communications” box. The “Font Size” box can be used to change the overall font size

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of the GUI. Clicking on the “Note” button causes an edit dialog box to be presented that

the user can type to. When this box is closed the note is time tagged and written to the

“.note” file in the PADS. Once the Monitor has established communication to the PADS

the “Initial Conditions” dialog box will be presented. In normal operation the, if the GPS

is operating, the current GPS position will be presented on the left side of the box. The

operator can change the current initial position by using the arrow buttons or enter the

data in the appropriate box. There are three other items that the operator must enter in this

dialog.

1. The root filename that will be used to name all the data files.

2. The amount of time, in seconds, used to level the system.

3. The amount of time, in seconds, to run the Coarse Alignment Filter. This process

performs Gyro-compassing to establish true heading.

Once these parameters have been entered the “OK” button is clicked on and the system

enters in the Leveling mode. Clicking on “CANCLE” keeps the system in Standby. The

operator should then switch to the “Main” page by clicking on the “Main” tab. The

“Main” page is shown in Figure 7. This page allows the operator to monitor the following

information

1. The time the system is in Level and Coarse align.

2. The system position and attitude and speed.

3. The GPS position and the GPS derived track, pitch and roll.

4. The GPS solution type, GDOP and HDOP and the number of satellites used in the

solution.

5. The IMU time and a count of the check sum errors.

In addition a time counter is displayed to the right of the Mode pull-down that gives the

running time. When in leveling and heading alignment modes this timer counts down

from the initial values entered in the “Initial Conditions” dialog box.

Figure 9 presents the mode “pull-down” items. Using this menu the RTN can be

commanded into various modes. Table 2 presents the valid mode transitions for the RTN.

Table 2: Mode State Transition Diagram†

Transition TO Mode Current

Mode Standby Level/Coarse

Align

Fine

Align

InAir

Align

Closed

Loop

Suspend Open

Loop

Stop

Standby N X X X

Level/Coarse

Align

X N T X

Fine

Align

X N X X X X

InAir

Align

X N X X X X

Closed X N X X

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Loop

Suspend X X N X

Open

Loop

X N X

Stop N

† Valid transitions indicated with X, N indicates no transition, T is an automatic

transition.

The only automatic mode transition is from Level/Coarse to Fine Align. The sequence of

mode changes, after the Level/Coarse Align command is given, is

1. Perform the initial level alignment by averaging the accelerometer measurements

over a time interval specified by the user.

2. Initiate the Coarse Alignment Filter (CAF) for a time interval specified by the

user. This initializes true heading via gyro-compassing and refines the level

alignment.

Typically the operator initiates Level/Coarse Align via the Initial Conditions dialog box.

The system will automatically transition through Level and Coarse Align to Fine Align.

During the Level/Coarse and Fine Align modes the Aircraft/Camera should not be

moved. The system will stay in Fine Align until the operator commands the system to a

valid mode. The most common mode to transition to is “Closed Loop” in which the

ALIGN Kalman filter uses GPS to continually correct the navigation function. If GPS is

not available the system can be commanded to “Open Loop” or Suspend. Once in “Open

Loop” the system can not be reverted to any mode other than “Standby” or “Stop”. The

“Suspend” mode invokes the ALIGN Kalman filter but the navigation function is

operated without using corrections. The operator can transition, however, into the

“Closed Loop” mode from “Suspend”. Note that there is no minimum time limit that the

system has to be in “Fine Align” and the most common commanded mode transition is

directly to “Closed Loop” as soon as Level/Coarse Align is complete.

Figure 10 gives the Align page selected by clicking on the “Align” tab. This page

presents data pertinent to the Fine Align and ALIGN filter performance. The box labeled

“Corrections” gives the total instrument corrections being applied to the navigator. The

“Standard Deviations” box shows the Kalman filter state standard deviations. The units

for both displays are given next to the variable name. The “Residuals” box contains the

Kalman filter measurement residuals. Note that in “Fine Align” the units are meters per

second and in ALIGN the units are meters. The Circular Buffer index is the index in the

buffer holding the navigation data where the current GPS measurement is aligned in time.

The Time variable is the GPS time of the observation.

Figure 11 presents the GPS data page. The current position, speed and track angles are

shown at the top of the page. The “Dilutions of Precision” box gives the various DOPs

and the time of validity. The “Satellites” box gives the solution type, satellite count,

OminStar signal strength and the satellite signal strength (C/N) by PRN. Also shown are

the azimuth and elevation angles of the satellites in view.

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Figure 12 shows the Time Filter page. This page displays information about the time

synchronization process in the RTN. The “Time Filter Parameters” box shows key

elements of the single state Kalman filter that calibrates the processor clock to the GPS

clock via the 1PPS interrupt. The “Time Stamps” box presents the current time stamp for

the various processes.

Figure 13 gives the “IMU” page as selected by clicking on the “IMU” tab. The “Data for

1 Second” box gives the time, message count and the average instrument outputs over the

last second. The “Status” box shows the status byte, checksum error count and dropout

count.

Once the RTN has been initialized and running the operator need only monitor the key

parameters as the need arises.

If the system must be restarted while moving (in air) the operator must first command the

RTN to “Standby”. The GPS must be operational and speed greater than five meters per

second. The “Track”, “Pitch” and “Roll” parameters must be displayed on the GPS data

line of the “Main” page. The operator commands the RTN into in-air-alignment mode by

selecting the “INAIRALIGN” item in the Mode pull-down. While in this mode the

“Heading”, “Pitch” and “Roll” items in the System data line should follow the same

items in the GPS data line. The system can be commanded to “INAIRALIGN” in any

orientation of the aircraft or camera. The navigation system will be reset to the GPS

derived parameters every second and perform free inertial navigation between the GPS

observations. The operator can command the RTN to “OPEN LOOP”, “CLOSED

LOOP” or “SUSPEND” at any time. Typically the system is commanded to “CLOSED

LOOP” soon after selecting “INAIRALIGN”.

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Figure 5: Monitor Setup Page

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Figure 6: Monitor Initial Conditions Dialog Box

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Figure 7: Monitor Main Page: During Leveling

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Figure 8: Monitor Main Page: During Heading Alignment (CAF)

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Figure 9: Monitor Mode Control Pull-down

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Figure 10: Monitor Align Page

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Figure 11: Monitor GPS Page

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Figure 12: Monitor Time Filter Page

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Figure 13: Monitor IMU Page

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Figure 14: Monitor Main Page: Commanding RTN to "Closed Loop"

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Testing of the RTN The inertial strapdown navigator and alignment Kalman filters are embodied in a

software package called the Real Time Navigator (RTN). Testing and validation of this

function is based on well known differential error equations describing the strapdown

navigator behavior when subjected to various instrument and alignment errors, see

references 1 through 3 and 5. The development path of the RTN involved first

programming it in C and stimulating it with known controlled inputs, i.e. using a

simulation of the instruments which generated perfect instrument outputs. This allowed

the RTN to tested over a range of profiles and instrument errors. Results of these tests are

not reported here. After the RTN passed these initial tests the program was embedded in a

real time wrapper which resulted in the RTN process which is executed on the PADS

hardware under the QNX operating system. The wrapper allowed the process to acquire

data from the actual instruments and output the navigation data. A discussion of this

wrapper is not included here.

Testing of the RTN in its real time form is reported in the following sections. Before any

conclusions could be drawn from the results of the various tests the instruments that were

to be used had to be characterized. This was accomplished with Cluster Analysis [7] of

the accelerometers and gyroscopes. The results of this testing is reported first. These

results were used to determine what performance we should expect and also used to

“Tune” the Alignment Kalman filters to the real instrument noise.

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After the instruments had been characterized several types of stationary tests were

conducted. The first was the stationary free inertial test, then a closed loop test, followed

by rotation tests and lastly heading alignment tests.

Following the stationary tests the system was installed in a truck for road testing. This

involved general navigation, ground and simulation in-air alignment.

A short flight test was conducted by AAI to evaluate the RTN in the flight environment.

This is reported on after the truck testing.

Finally the last section gives specifics about the NRL flight tests.

Stationary Testing

Cluster Analysis Procedure

Cluster Analysis is another name for the methodology of generating an Alan Variance.

Reference 7 gives a discussion of Cluster Analysis as it applies to RLG Noise analysis.

Essentially Cluster Analysis provides an easy way to determine the noise variance at a

given correlation time. This is accomplished by breaking the time sequential data into

adjoining clusters of the same time span. Within each cluster the data is averaged. This

results in a series of compressed data packets where each packet or cluster contains the

averaged data over the same length of time. Each cluster is then differenced with its

neighbor and the variance of the series of differences calculated. Thus for one cluster

length we get one variance which represents the variation in the data at that correlation

time. This process is repeated for different cluster lengths. Obviously, for a given fixed

length data set, the longer the cluster length is the fewer clusters are available which

means the estimation accuracy of the variance decreases with increasing cluster length.

Reference 7 describes a technique that allows one to graphically determine from a Cluster

Analysis plot parameters such as random walk, exponentially correlated noise, bias

instability etc. This technique was used to analyze the Cluster Analysis plots.

The data set used to generate the Cluster Analysis data was approximately sixteen and a

half hours long and was collected in the AAI laboratory.

The data was post-processed which resulted in Cluster Analysis plots.

Cluster Analysis Results

For the sixteen and half hour IMU data set, Figure 15 presents the estimated percentage

error in the Cluster standard deviation as a function of the correlation time. This reflects

the smaller number of samples available as the correlation time gets larger. Figure 16 and

Figure 17 presents the Accelerometer and Gyroscope Cluster Analysis plots derived from

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the KI-4901 data. Figure 18 and Figure 19 present the Cluster Analysis plots derived

from the Phalanx IMU.

Table 3 summarizes the significant parameters from the Cluster Analysis plots. Note that

this table does not include all the parameters that could be estimated from the Cluster

Plots such as rate random walk, quantization and rate ramp errors.

Comparison of the Kearfott and Phalanx parameters and Cluster Plots shows that all the

Kearfott accelerometers are comparable to the “z” axis Phalanx accelerometer and that

the “y” and “x” axis Phalanx accelerometers have significantly less random walk then the

Kearfott accelerometers. The Phalanx gyroscope is not even in the same class as the

Kearfott device. All of the gyroscope parameters are significantly better then the Phalanx.

Table 3: Parameter Estimates from Cluster Analysis

Instrument Random Walk Bias Instability Correlated Error

Measured Specification Measured Specification Time

Constant

Uncertainty

KI-4901

Gyroscope

(1.6±0.4)e-3

°/√Hr

0.003 °/√Hr (2.87±0.6)e-3

°/Hr

0.003 °/Hr n.a. n.a.

KI-4901

Accelerometer

0.14±0.04

ft/s/√Hr

0.164 ft/s/√Hr (5 ± 0.5)µg 50 µg 10 sec 40 µg

Phalanx

Gyroscope

(3±0.8)e-3

°/√Hr

n.a. (0.02±0.0014)

°/Hr

n.a. n.a. n.a.

Phalanx x & y

Accelerometer

< 0.01

ft/s/√Hr

n.a. X (1 ± 0.05)µg

Y (1.5 ±

0.08)µg

n.a. n.a. n.a.

Phalanx z

Accelerometer

0.13±0.03

ft/s/√Hr

n.a. (2 ± 0.1)µg n.a. n.a. n.a.

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100

101

102

103

104

105

10-1

100

101

102

Correlation Time T (sec)

Es

tim

ate

d P

erc

en

t E

rro

r

Figure 15: Cluster Analysis Estimated Percentage Error

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100

101

102

103

104

105

10-2

10-1

100

101


σ(T

) (f

t/s

ec

/hr)

Cluster Analys is of KI4901 SN0008 Accelerometers26-Dec-2003

Ax

Ay

Az

Figure 16: KI-4901 Accelerometer Cluster Analysis

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100

101

102

103

104

105

10-4

10-3

10-2

10-1

100


σ(T

) (d

eg

/hr)

Cluster Analysis of K I4901 SN0008 Gyroscopes.26-Dec-2003

Gx

Gy

Gz

Figure 17: KI-4901 Gyroscope Cluster Analysis

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100

101

102

103

104

105

10-2

10-1

100

101


σ(T

) (f

t/s

ec

/hr)

Cluster Analys is of Phalanx Accelerometers06-Jan-2004

Ax

Ay

Az

Figure 18: Phalanx Accelerometer Cluster Analysis

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100

101

102

103

104

105

10-4

10-3

10-2

10-1

100


σ(T

) (d

eg

/hr)

Cluster Analys is of Phalanx Gyroscopes.06-Jan-2004

Gx

Gy

Gz

Figure 19: Phalanx Gyroscope Cluster Analysis

Stationary Bench Test Free Inertial Results

The purpose of this test was to demonstrate the free inertial performance by allowing the

inertial navigator to run for an extended period of time in the free mode, i.e. no

corrections from the Alignment Kalman filter. In this case the run lasted sixteen and half

hours. The free performance was preceded by three minutes of ground alignment. The

IMU x axis was initially pointed south and the unit was not rotated. Vertical position was

not computed. Figure 20 presents the position error, latitude and longitude, as a function

of time. Note the expected eighty four minute period modulation and the peak error at

twelve hours. These are related to the Schuler tuned inertial navigation algorithm. See

reference 1 for a discussion. Typically an inertial navigator performance is quantified by

its radial position error drift rate in nautical miles per hour. Figure 21 presents the radial

position error for this test. The line labeled “linear” represents a least squares curve fit of

the radial error. The slope of this line is representative of the drift rate of the system and

is seen to be 7.8 meters per minute which is 0.25 nautical miles per hour. This is

consistent with the inertial instrument performance derived from the Alan variance

analysis. Figure 22 presents the velocity error as a function of time. Again, the magnitude

of the velocity errors and the modulation are all well within what is expected from an

inertial navigator using instruments of this quality.

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0 100 200 300 400 500 600 700 800 900 1000-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

Errors wrt reference

La

t, L

on

, A

lt E

rro

r (m

)

Time (min)

Lat

Lon

Alt

Figure 20: Free Inertial Performance Position Error

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0 100 200 300 400 500 600 700 800 900 10000

2000

4000

6000

8000

10000

12000

Radial Error

Err

or

(m)

Time (min)

y = 7.8*x + 2.4e+003

data 1

linear

Figure 21: Free Inertial Performance, Radial Error

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0 100 200 300 400 500 600 700 800 900 1000-1.5

-1

-0.5

0

0.5

1

1.5

2

Veloc ity Errors wrt to Reference

Time (min)

Ve

loc

ity

Err

ors

(m

/s)

ε Vx

ε Vy

ε Vz

Figure 22: Free Inertial Performance Velocity Error

Stationary Bench Test Closed Loop Results

In this test the sixteen and half hour data set was processed by the Nav/Align program

using GPS data in the closed loop mode. GPS data was simulated by using the fixed

known position of the test stump at AAI as the reference measurement. The purpose of

the test was twofold. First, to demonstrate that the Nav/Align process works with real

IMU data. Second, to determine the closed loop position performance for the stationary

case. Figure 23 shows the system latitude and longitude position error in meters. Note

that there is no noise on the position reference so the variations seen in the figure are only

due to the IMU noise and processing. For this test the GPS observation noise level was

set to 0.3 meters for latitude and longitude and 0.6 meters for altitude (one sigma).

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-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Latitude vs Longitude (m)

Longitude Error (m)

La

titu

de

Err

or

(m)

Figure 23: Closed Loop Position Error

Bench Rotation Test Results

The purpose of the rotation test was to verify that the RTN software correctly uncouples

the accelerometer bias from the initial tilt error. In addition the stability of the attitude

measurement, after instrument error estimation, was demonstrated.

The RTN system performed a sixty second ground alignment followed by forty seconds

of Fine Alignment and then transition into Nav Closed loop using GPS observations. The

test sequence was as follows

1. Initial heading of unit, East.

2. Rotate to West at 300 seconds.

3. Rotate to East at 720 seconds.

4. Rotate to West at 1080 seconds.

5. Rotate to East at 1620 seconds.

6. Run terminated at 5280 seconds.

Note that the ISA axes are as follows:

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1. The z axis is along the center of the cylindrical case. The positive direction is the

end of the case where the high voltage connector is. The z axis lies in the plane

described by the isolator mounting holes.

2. The x axis is in the plane described by the isolator mounting holes and pointed to

the right as one looks down on the cylindrical case with the positive z axis up.

3. The y axis is positive down and is right-handed orthogonal to x and z.

Note that heading is defined as the angle from north to the projection of the ISA x axis

onto the level plane.

Figure 24 presents the attitude variation about the ISA x axis (φε ). Note that as the ISA

is rotated the angle variation becomes more stable. This is because, first the

accelerometer bias is estimated and then the gyro bias. The result is the stable angle

shown from the last rotation time to the end of the test. The variation is well within ± 25

µ radian. Figure 25 shows the attitude variation about the ISA z axis (θε ) which has

similar characteristics. Figure 26 gives the Align Kalman filter attitude and heading

uncertainties. These are commensurate with the observed variations. Figure 27 presents

the heading variation. In this case the heading drift rate has not been calibrated to the

extent the level gyroscopes were so there is a dynamic drift. The heading error is being

minimized by the system gyro-compassing as can be seen by the decrease in heading

uncertainty in Figure 26. Finally, Figure 28 presents the accelerometer bias estimates and

uncertainties. Of note in the accelerometer bias plot is the fact that the vertical channel

bias estimate is large, in fact it’s not shown on the plot. This was eventually traced to an

incorrect gravity model. This error was not uncovered in simulation testing because the

same gravity model was used in the simulation as in the RTN. The other accelerometer

biases were commensurate with the KI-4901 specification.

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500 1000 1500 2000 2500 3000 3500 4000 4500 5000

9.9

9.95

10

10.05

10.1

10.15

10.2

10.25

10.3


He

ad

ing

, R

oll a

nd

Pit

ch

Err

or

(mr)

ψ ε

φ ε

θε

Variation about X ax is

Figure 24: Rotation Test Attitude variation about the ISA case x axis

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500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-9.4

-9.3

-9.2

-9.1

-9

-8.9

-8.8

-8.7


He

ad

ing

, R

oll a

nd

Pit

ch

Err

or

(mr)

ψ ε

φ ε

θε

Variation about z ax is

Figure 25: Rotation Test Attitude variation about the ISA case z axis

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1000 2000 3000 4000 5000 6000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Attitude Error S tate Std Dev

σψ

x,

σψ

y,

σψ

z (

mr)

σ ψx

σ ψy

σ ψz

Figure 26: Rotation Test Align Attitude and Heading Uncertainties

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1500 2000 2500 3000 3500 4000 4500 500016

16.2

16.4

16.6

16.8

17

17.2

17.4

17.6

17.8

18


He

ad

ing

, R

oll a

nd

Pit

ch

Err

or

(mr)

ψ ε

φ ε

θε

Variation in Heading

Figure 27: Rotation Test Heading variation

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1000 2000 3000 4000 5000

-250

-200

-150

-100

-50

0E

st

(Mic

roG

s)

Accel B ias Estimates

Time (sec)

Abx

Aby

Abz

0 2000 4000 60000

50

100

150

200

250

300

350

400

Es

t (M

icro

Gs

)

Accel B ias Std Dev

Time (sec)

σ Abx

σ Aby

σ Abz

Figure 28: Rotation Test Accelerometer bias estimates

Coarse Alignment (CAF) Testing

The CAF provides an automatic alignment of the system with true north via wide-angle

gyro-compassing when stationary.

One problem that was uncovered in translating the CAF from the simulation environment

to the RTN was the systems inability to correctly accomplish multiple alignments without

completely restarting the system. In other words every time we terminated the alignment

by commanding Standby and then restarting the alignment the resultant alignment got

progressively worse. This was traced to a counter not being reinitialized in the RTN on

subsequent Standby commands.

Multiple tests of the CAF mode were accomplished, however only three examples will be

presented here. Figure 29 and Figure 30 present the heading and CAF velocity residuals

for a -90 degree heading. Figure 31 and Figure 32 give the +90 degree case and Figure 33

and Figure 34 present the 180 degree case.

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0 20 40 60 80 100 120-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Time (sec)

Wander

Angle

(deg)

Figure 29: CAF Test

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0 20 40 60 80 100 120-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1x 10

-3

Resid

ual(m

/s)

Filter Residuals

Time (sec)

εX

εY

Figure 30: CAF Test

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0 20 40 60 80 100 1200

10

20

30

40

50

60

70

80

90

Time (sec)

Wander

Angle

(deg)

Figure 31: CAF Test

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0 20 40 60 80 100 120-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025R

esid

ual(m

/s)

Filter Residuals

Time (sec)

εX

εY

Figure 32: CAF Test

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0 20 40 60 80 100 120-20

0

20

40

60

80

100

120

140

160

180

Time (sec)

Wander

Angle

(deg)

Figure 33: CAF Test

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0 20 40 60 80 100 120-6

-5

-4

-3

-2

-1

0

1x 10

-3

Resid

ual(m

/s)

Filter Residuals

Time (sec)

εX

εY

Figure 34: CAF Test

Ground Truck Testing

The purpose of truck testing is to subject the RTN to dynamic inputs such as linear

acceleration, constant speed and rotations. Two tests are reported on here, first a short test

that demonstrated the ability of the system to track the GPS input and a gross check on

the report heading and attitude and overall response to dynamic variables such as GPS

position uncertainty. The second test subjected the system to a long road test. Key

parameters that were observed here were heading and overall stability. Secondarily the

RTN software was being tested for memory leaks or any other term error anomaly.

Short Test

The first truck road test of the system was conducted on February 25, 2004. The test

included the Ampro development system, the Kearfott KI-4901 ISA it’s electronics and

the NovAtel ProPak-LB receiver. A laptop PC was connected to the system via the

Ethernet. The PADS/C control and monitor program was run on the laptop and used to

control the PADS software residing on the Ampro development system. The PADS

software included executive functions to collect data from the GPS receiver and the KI-

4901 and log all the required data to the hard drive and manage the navigation and

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Kalman filter processes. The system was started by transitioning from standby to coarse

align. Coarse align levels the navigation frame and initializes heading and position.

Coarse align was followed by Fine align which continued to refine heading and attitude.

Finally the system was commanded to Nav-Closed-Loop. This invokes the ALIGN

Kalman filter that used GPS as the position reference. At the time of this test the CAF

was not available so an initial heading had to be manually input into the system.

Figure 35 shows the test vehicle and with the equipment pallet. Figure 36 presents the test

pallet. The ISA was orientated on the pallet with the X axis pointed to the front of the

truck. The GPS antenna was mounted on a tripod in the truck bed. The pallet requires 115

VAC 60Hz power to run the AMPRO development system and the KI-4901 power

supply. The NovAtel ProPak-LB connects directly to the truck 12 VDC power plug. AC

power comes from a Tripp Lite DC to AC converter, model PV1000FC. This unit

supplies a low distortion sine wave power source that can sustain 1000 watts. For our

purposes 180 watts was sufficient.

Figure 37 gives the test profile. Initial heading was north. Figure 38 presents the vehicle

heading. Figure 39 through Figure 42 present Kalman filter performance data during the

test. Note in Figure 40 and Figure 41 the transition from Fine Align to Nav Closed Loop

at around 40 seconds. At this point the measurement to the filter changes from navigator

velocity to GPS position. The jump in innovation uncertainties (Figure 40) at around 530

seconds was due to the vehicle passing under a railroad bridge. The loss of GPS signal is

apparent in the residuals (Figure 41). Note, however, that whereas the innovation

uncertainty jumped to a meter or more the position uncertainty (Figure 42) only increased

to a quarter meter. This is attributable to the inertial navigator’s ability to provide very

good short term position information.

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Figure 35: Test Vehicle

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Figure 36: Test Pallet

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-85.585 -85.58 -85.575 -85.57 -85.565 -85.56 -85.555 -85.5542.886

42.888

42.89

42.892

42.894

42.896

42.898

42.9

42.902

Longitude (deg)

La

titu

de

(d

eg

)

Road Test 1

Figure 37:Short Road Test Profile

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0 100 200 300 400 500 600 700

-150

-100

-50

0

50

100

150H

ea

din

g (

de

g)

Time (sec)

Figure 38:Short Road Test Profile Heading

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0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Es

t (m

illr

ad

)

Tilt and Heading Error Std Dev

Time (sec)

Figure 39:Short Road Test Kalman Filter Tilt and Heading Error Uncertainties

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100 200 300 400 500 6000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Filte

r In

no

vati

on

s (

m)

Kalman Filter Innovations

Time (sec)

Figure 40:Short Road Test Kalman Filter Innovation Covariance Diagonal Elements

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-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Filte

r R

es

idu

als

(m

)

Kalman Filter Residulas

Time (sec)

Figure 41:Short Road Test Kalman Filter Residuals

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0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Es

t (m

)

Pos ition Error Std Dev

Time (sec)

Figure 42:Short Road Test Kalman Filter Position Error Uncertainties

Long Road Test

On April 19, 2004, the PADS system was loaded into the AAI test vehicle and driven to

Illinois for camera vibration testing. We took this opportunity to perform a long road test

of the system. The main purpose of this test was to see if there was any degradation of the

system with time as could be caused by a memory leak. Figure 43 and Figure 44 presents

position and heading for the complete test. Figure 45 shows the GPS position uncertainty.

This was not up to expectations but is believed to be due to interference from traffic and

other obstructions.

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-88.5 -88 -87.5 -87 -86.5 -86 -85.541.5

42

42.5

43

Longitude (deg)

La

titu

de

(d

eg

)

Road Test

GR to ROI

Figure 43: Long Road Test PADS position

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50

100

150

200

250

300

350

400

He

ad

ing

(d

eg

)

Time (sec)

Road Test

AAI to ROI

Figure 44: Long Road Test PADS Heading

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0 2000 4000 6000 8000 10000 120000

2

4

6

8

10

12

14

16

18

20

Po

sit

ion

Std

. D

ev.

(m

)

PADS GPS Data

σ Lat

σ Lon

σ A lt

Figure 45: Long Road Test GPS Position Uncertainty

In Air Alignment Ground Testing

The In Air Alignment mode is implemented in the RTN as a commanded mode. Note that

the mode can only be commanded when there is GPS data and the speed of the system is

greater than five meters per second. The operation of the mode is as follows:

1. At the GPS data rate (currently one Hertz) the velocity data is filtered via three

two state Kalman filters. These filters provide estimates of acceleration along the

North, East and Down axes. The GPS velocity and the derived acceleration

estimates are used to continuously provide heading, pitch and roll estimates for

initialization of the navigation function during In Air Alignment.

2. The user invokes the In Air Alignment mode via the monitor program. The

monitor will not allow transition to this mode if the speed is below five meters per

second and there is not valid GPS data. There are no other restrictions on

transitioning into this mode. The system will stay in In Air Alignment mode until

the user commands the system to another mode. Typically the user would

transition to Closed Loop mode, however, for testing purposes all other modes are

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available. While in In Air Alignment the navigation function is initialized every

second (synchronous with the GPS data) to the current attitude estimate, GPS

position and velocity. Between initializations the navigation function is run in an

open loop manner.

3. The user invokes a transition to Closed Loop mode. As stated above this is the

most useful transition and the one that will be used in flight as it forces the system

to the GPS reference. The mode was ground tested using the truck. Figure 46

through Figure 48 present the pertinent data. Figure 46 shows the profile of the

test. The test started at the lower right corner and progressed westerly. Figure 47

and Figure 48 present the north and east velocity components. Note that the mode

was commanded when the vehicle speed was about seventeen meters per second.

Figure 50, Figure 51 and Figure 52 give the Align filter state standard deviations

for position velocity and attitude. These plots start at the time of the transition to

Closed Loop. Figure 53 shows the summed tilt and heading corrections applied to

the navigation function by the Align filter. Note that the initial heading error

uncertainty is over 100 mill radians. This is significantly higher than the initial

value used when transitioning to the Fine Align Filter from the CAF and reflects

the larger heading uncertainty due to aircraft crab angle. Essentially, in the In Air

Alignment mode, the heading is initialized to the ground track angle as derived

from the GPS velocity components which will introduce a heading error

commensurate with whatever the wind components are. Figure 54 presents the

position residuals from the Align filter.

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-85.562 -85.56 -85.558 -85.556 -85.554 -85.552 -85.55 -85.54842.883

42.884

42.885

42.886

42.887

42.888

42.889

42.89

42.891

42.892

42.893

Longitude (deg)

Latitu

de (

deg)

InAir Align Ground TestPADS Nav Data

Figure 46: Ground In-Air Test

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0 20 40 60 80 100 120 140 160 180-10

-5

0

5

10

15

20

Velo

city N

ort

h (

m/s

)

Time (sec)



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0 20 40 60 80 100 120 140 160 180-20

-15

-10

-5

0

5

10

15

20

Velo

city E

ast

(m/s

)

Time (sec)



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0 20 40 60 80 100 120 140 160 180-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Velo

city D

ow

n (

m/s

)

Time (sec)



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0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Est

(m)

Position Error Std Dev

Time (sec)

σδ X

σδ Y

σδ Z


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0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Est

(m/s

)

Velocity Error Std Dev

Time (sec)

σδ VX

σδ VY

σδ VZ


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0 20 40 60 80 100 120 140 1600

20

40

60

80

100

120

Est

(mill

rad)

Tilt and Heading Error Std Dev

Time (sec)

σψ X

σψ Y

σψ Z


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0 20 40 60 80 100 120 140 160-30

-20

-10

0

10

20

30

Corr

ections (

mill

rad)

Tilt and Heading Corrections

Time (sec)

Σψ X

Σψ Y

Σψ Z


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0 50 100 150-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Filt

er

Resid

uals

(m

)

Kalman Filter Residulas

Time (sec)

Res X

Res Y

Res Z


AAI Flight Test

A flight test of the PADS system was accomplished on Friday December 3. Figure 55

through Error! Reference source not found. present data from the flight. Of note is the

GPS position performance which is given in Figure 56. Overall the GPS did not perform

to the expected OmniStar HP level. This was most likely due to the position of the

antenna which was tapped to the inside of the windscreen on the right side. Obviously

this was not an optimal position for the antenna and performance most certainly suffered.

The number of satellites used, Figure 57, shows that during the high position uncertainty

the number of satellites used was minimal.

Performance of the In-Air GPS processing is given in Error! Reference source not

found. through Error! Reference source not found.. Track and pitch angles are

calculated from the GPS velocity vector and the roll angle and acceleration are derived

from the estimated derivatives of GPS velocity. Note that the roll angle estimate is roll of

the aircraft with respect to the level North-East plane whereas the Nav roll angle is of the

IMU axes which are not aligned with the aircraft axes.

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Coarse Align Filter (CAF) performance is demonstrated in Figure 58 through Error!

Reference source not found.. The CAF process was run with the aircraft engines off.

Align performance is given in Figure 60 through Figure: AAI Flight Test 66. Attitude

uncertainties are given in Figure 62: AAI Flight Test Align Attitude Uncertainties.

Heading uncertainty reached a minimum of 350 µrad. Performance did not reach the

expected 50 µrad level because of the elevated position uncertainties.

Navigation performance is given in Error! Reference source not found. through Error!

Reference source not found.. Error! Reference source not found. show roll angle. We

see that during the three major coordinated turns a bank angle of about 25 degrees was

maintained. The speed during two of these turns was about 140 knots which resulted in a

3.5 degree per second turn rate and acceleration of about 0.4 g’s. Error! Reference

source not found.Error! Reference source not found. show estimated wind parameters

and finally Error! Reference source not found. gives the measured specific force.

Note that in the gyro bias estimates, Figure: AAI Flight Test 65, the gyro x axis bias

estimate appears to run out to -2.5e-3 degrees per hour. This drift, though not excessive,

is greater than what is expected. Note also the excessive accelerometer bias along the z

axis in Figure: AAI Flight Test 66 . This of coarse was later determined to be due to a

gravity model error.

−85.7−85.65

−85.6−85.55

−85.5−85.45

43.1

43.2

43.3

43.40

200

400

600

800

1000

Longitude (deg)

Dec 3,04GPS start time 504141

Latitude (deg)

Altitude (

m)

Figure 55: AAI Flight Test Profile

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0 200 400 600 800 1000 1200 1400 16000

1

2

3

4

5

6

7

8

9

10

Time (sec)

GP

S U

ncert

ain

ties (

m)


σN

σE

σD

Figure 56: AAI Flight Test GPS Position Uncertainties

0 200 400 600 800 1000 1200 1400 1600 18003

4

5

6

7

8

9

10

Time (sec)

Num

ber

SV

’s u

sed in s

olu

tion


Figure 57: AAI Flight Test GPS SV’s used in Solution

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0 20 40 60 80 100 120 140 160 180−15

−10

−5

0

5

10

15

Time (sec)

CA

F L

evel E

art

h R

ate

Estim

ate

s (

deg/h

r)

Dec 3,04CAF start time 504171.8218

σ Ωx

σ Ωy

Figure 58: AAI Flight Test CAF Level Earth Rate Estimates

0 20 40 60 80 100 120 140 160−0.01

−0.008

−0.006

−0.004

−0.002

0

0.002

0.004

0.006

0.008

0.01

Time (sec)

CA

F R

esid

uals

(m

/s)

Dec 3,04CAF start time 504171.8218

Resx

Resy

Figure 59: AAI Flight Test CAF residuals

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0 200 400 600 800 1000 1200 14000

2

4

6

8

10

12

14

16

18

20

Time (sec)

Alig

n P

ositio

n U

ncert

ain

ties (

m)

Dec 3,04FAF start time 504350.8276

σ δ Px

σ δ Py

σ δ Pz

Figure 60: AAI Flight Test Align Position Uncertainties

0 200 400 600 800 1000 1200 14000

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Time (sec)

Alig

n V

elo

city U

ncert

ain

ties (

m/s

)


σ δ Vx

σ δ Vy

σ δ Vz

Figure 61: AAI Flight Test Align Velocity Uncertainties

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0 200 400 600 800 1000 1200 14000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (sec)

Alig

n A

ttitude U

ncert

ain

ties (

milr

ad)


σ ψx

σ ψy

σ ψz

Figure 62: AAI Flight Test Align Attitude Uncertainties

0 200 400 600 800 1000 1200 1400−10

−8

−6

−4

−2

0

2

4

6

8

10

Time (sec)

Alig

n R

esid

ula

s(m

)


Resx

Resy

Resz

Figure 63: AAI Flight Test Align residuals

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0 200 400 600 800 1000 1200 14000

1

2

3

4

5

6

7

8

9

10

Time (sec)

Alig

n Innovations(m

)


Resx

Resy

Resz

Figure 64: AAI Flight Test Align Innovation Uncertainties

0 500 1000 1500−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

Gyro

Bia

s E

stim

ate

s (

Deg/H

r)

Dec 3,04Gyro Bias EstimatesFAF start time 504350.8276

Time (sec)

Gbx

Gby

Gbz

0 500 1000 15004.955

4.96

4.965

4.97

4.975

4.98

4.985

4.99

4.995

5

Gyro

Bia

s S

tandard

Devia

tion (

Deg/H

r)

Dec 3,04Gyro Bias Std DevFAF start time 504350.8276

Time (sec)

σ Gbx

σ Gby

σ Gbz

Figure: AAI Flight Test 65 Align Gyro Bias Estimates

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0 500 1000 1500−2000

−1500

−1000

−500

0

500

1000

1500

Accel B

ias E

stim

ate

s (

µ G

)

Dec 3,04Accel Bias EstimatesFAF start time 504350.8276

Time (sec)

Abx

Aby

Abz

0 500 1000 15000

50

100

150

200

250

300

350

400

Accel B

ias S

tandar

Dev (

µ G

)

Dec 3,04Accel Bias Std DevFAF start time 504350.8276

Time (sec)

σ Abx

σ Aby

σ Abz

Figure: AAI Flight Test 66 Align Accelerometer Uncertainties

NRL Flight Tests

The NRL flight test of the Pinpoint system was started on April 24, 2006. This section

provides navigation data for selected flight segments.

Davison to Patuxent River

There were two flights on April 25th

. The first was from Davison Field to the Naval Air

Station at Patuxent River, MA. This flight was for the purpose of a crew briefing at the

Naval Air Station concerning flights around Webster Field. For completeness a plot of

this flight is given in

Figure 67. It was during this flight that the lack of OmniStar HP service was first noted.

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−77.5

−77

−76.5

−76

38.2

38.4

38.6

38.8

39−200

0

200

400

600

800

Longitude (deg)

Davison to PaxGPS start time 217627

Latitude (deg)

Altitude (

m)

Figure 67: April 25 Davison to Pax Profile

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Figure 68: April 25 Davison to Pax GPS Position Error States

Figure 69: April 25 Davison to Pax Lever Arm Error States

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Figure 70: April 25 Davison to Pax Accelerometer Bias States

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Figure 71: April 25 Davison to Pax Gyro Bias States

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Figure 72: April 25 Davison to Pax Residuals and Innovations

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Figure 73: April 25 Davison to Pax Attitude Uncertainties

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Figure 74: April 25 Davison to Pax Velocity Uncertainties

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Figure 75: April 25 Davison to Pax Position Uncertainties

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Figure 76: April 25 Davison to Pax CAF Earth Rate Estimates

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Figure 77: April 25 Davison to Pax CAF Residuals

April 25…Patuxent to Davison Over Webster Field

The second flight on April 25 was from the Naval Air Station to the vicinity of Webster

Field for orbiting and conducting the Pinpoint tests. Figure 78 presents the profile for the

first part of this flight. At a certain point the system had to be shutdown and restarted

because the TPE had crashed. This happened at about ninety minutes from takeoff. When

an In-Air restart was attempted the system could not be aligned. The remainder of the

flight the RTN was run but only for the purpose of generating camera event records and

saving the raw IMU and GPS data for post-processing. Figure 97 presents the Kalman

filter position residual and Innovations for the first part of the flight. Note the large

uncertainties after takeoff. Figure 85gives the attitude and heading uncertainty from the

Kalman filter. Again we are not quite reaching 100 µrad in heading uncertainty, mainly

due to the higher than expected GPS position uncertainty.

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−76.5−76.45

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April25 AGPS start time 229611

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Figure 78: April 25 Pax Take-Off Profile

Figure 79: April 25 Pax Take-Off Speed

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Figure 80: April 25 Pax Take-Off GPS Position Error States

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Figure 81: April 25 Pax Take-Off Lever Arm Error States

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Figure 82: April 25 Pax Take-Off Accelerometer Bias States

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Figure 83: April 25 Pax Take-Off Gyro Bias States

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Figure 84: April 25 Pax Take-Off Residuals and Innovations

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Figure 85: April 25 Pax Take-Off Attitude Uncertainties

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Figure 86: April 25 Pax Take-Off Velocity Uncertainties

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Figure 87 Pos: April 25 Pax Take-Off Position Error Uncertainties

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Figure 88: April 25 Pax Take-Off CAF Earth Rate Estimates

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Figure 89: April 25 Pax Take-Off CAF Residuals

April 27…Davison to Webster Field

By April 27 the OmniStar HP service had been restored and the receiver software

updated. The flight was from Davison Field to the vicinity of Webster Field. At around

110 minutes after takeoff the system was restarted due to a need to recycle power on the

camera (this shuts off the power to the Kearfott ISA which essentially kills the

navigator). Again the system failed to accomplish an In-Air restart so data was simply

gathered for post-flight calculation. It turned out that during this last portion of the flight

there were no event data records logged therefore the images that were taken during this

part of the flight can not be used in the analysis.

Figure 90presents the first part of the flight profile. Figure 97Error! Reference source

not found. gives the Kalman filter residual and Innovations. Note the uncertainty is less

than the 0.2 m level. This is commensurate with OmniStar HP performance. Figure 98

shows the attitude and heading uncertainty. Note that the heading uncertainty is

approaching 100 µ rad.

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Figure 90: April 27 Take-Off Profile

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Figure 91: April 28 Take-Off GPS Uncertainty

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Figure 92: April 27 Take-Off GPS SV’s Used

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Figure 93: April 27 Take-Off GPS Position Error States

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Figure 94: April 27 Take-Off Lever Arm Error States

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Figure 95: April 27 Take-Off Accelerometer Bias States

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Figure 96: April 27 Take-Off Gyro Bias States

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Figure 97: April 27 Take-Off Residuals and Innovations

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Figure 98: April 27 Take-Off Attitude Uncertainties

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Figure 99: : April 27 Take-Off Velocity Uncertainties

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Figure 100: April 27 Take-Off Position Uncertainties

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Figure 101: April 27 Take-Off CAF Residuals

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Figure 102: April 27 Take-Off CAF Earth rate estimates

April 28…Davison to Webster Field…Third In Air Start

The April 28th

flight started with camera problems immediately. Among other problems

the PADS was not receiving the event pulses from the camera. In addition there was a

power problem with the Honeywell INS. After several attempts at re-starting various

control programs for the camera it was decided to do a complete restart of the camera.

This of coarse this meant that the Kearfott ISA would be turned off which then entailed

an attempt to do an In-Air restart of the RTN. It was known that the In-Air process had a

bug and that we would not therefore get a good navigation solution, but it was felt that we

could at least collect camera images and the event data and the raw navigation data and

we could always recover the required navigation data via post-processing. A couple

different attempts were made at In-Air restart. Finally, by reviewing the source code for

part of the Navigator process that performs the In-Air alignment a coding error was

uncovered. This was corrected and the Navigator process rebuilt and an In-Air restart of

the RTN performed. Figure 103 shows the flight profile after the In-Air restart. Error!

Reference source not found. and Error! Reference source not found. give the Kalman

filter Innovations and position residuals during the In-Air restart. Error! Reference

source not found. gives the attitude and heading uncertainties. Note that the heading

uncertainty does not reach the low level achieved on the April 27 flight. This is due to the

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degraded nature of an In-Air alignment. Heading uncertainty could have been reduced if

higher G turns could have been executed, however this was not feasible.

−77.2−77

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April 28 In Air 3GPS start time 495338

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Figure 103: April 28 In Air profile

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Figure 104: April 28 In Air GPS Uncertainties

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Figure 105: April 28 In Air GPS SV’s Used

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Figure 106: April 28 In Air Speed

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Figure 107: April 28 In Air GPS Position Error States

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Figure 108: April 28 In Air Lever Arm Error States

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Figure 109: April 28 In Air Accelerometer Bias States

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Figure 110: April 28 In Air Gyro Bias States

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Figure 111: April 28 In Air Residuals and Innovations

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Figure 112: April 28 In Air Attitude Uncertainties

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Figure 113: April 28 In Air Velocity Uncertainties

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Figure 114: April 28 In Air Position Uncertainties

References [1] K. R. Britting, “Inertial Navigation Systems Analysis”, John Wiley and Sons, 1971.

[2] G. R. Pitman, editor. “Inertial Guidance”, John Wiley and Sons, 1962.

[3] P. G. Savage. “Strapdown Inertial Navigation Lecture Notes”, Strapdown Associates,

Inc, 10201 Wayzata Boulevard, Minnetonka, MN 55343, 1983

[4] WGS1984, NIMA TR8350.2, Third Edition, Amendment 1, 3 January 2000.

[5] J. R. Huddle, “Inertial Navigation System Error Model Considerations in Kalman

Filtering Applications”, chapter 13, AGARD No.256, “Advances in the Techniques and

Technology of the Application of Nonlinear Filters and Kalman Filters”, 1982.

[6] editor Arthur Gelb, “Applied Optimal Estimation”, M.I.T. Press 1974.

[7]M.M.Tehrani, “Ring laser gyro data analysis with cluster sampling technique”, Proc.

of Fibre Optic and Laser Sensors SPIE, Vol 412, pp. 207-220, 1983.

introduction theory of operation...theory of operation of the software. following that is the...

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