National Geodetic Survey...use raw accelerometer and gyro data to obtain free-inertial navigation...

5.1 Theoretical Fundamentals of Inertial Gravimetry, C. Jekeli, OSU Airborne Gravity for Geodesy Summer School , 23-27 May 2016

V. Theoretical Fundamentals of Inertial Gravimetry • Basic gravimetry equation • Essential IMU data processing for gravimetry • Kalman filter approaches • Rudimentary error analysis – spectral window • Instrumentation

National Geodetic Survey


Inertial Gravimetry

• Use precision accelerometer triad, instead of gravimeter

− OTF (off-the-shelf) units designed for inertial navigation rather than gravimetry

• Usually consider strapdown mechanization, instead of stabilized platform

• Vector gravimetry, instead of scalar gravimetry

− determine three components of gravity in the n-frame

• Need precision gyroscopes to minimize effect of orientation error on horizontal components

• Two documented approaches of data processing

− either integrate accelerometer data or differentiate GPS data


• Recall strapdown mechanization

ba – inertial accelerations measured by accelerometers in body frame

ix – kinematic accelerations obtained from GPS-derived positions, x, in i-frame.

GPS transformation from inertial frame to n-frame

transformation from body frame to inertial frame gyros

accelerometers

Gravitational Vector in n-frame ( )n n i i b

i b= −C Cg x a

◦ where is the data interval, e.g., δt = 1/50 s 1k kt t tδ −= −

− accelerometer data typically are ( )1

k

k

t

bk

t

t dtδ−

= ∫v a b kk t

δδ

⇒ ≈va (e.g.)

better approximations can be formulated

Direct Method


Determination of (1) ibC

• Let eζ be the unit vector about which a rotation by the angle, ζ, rotates the b-frame to the i-frame ζ

ζe

3b

2b

1b ( ) ( )

( ) ( )( ) ( )

2 2 2 21 2 3 4 2 3 1 4 2 4 1 3

2 2 2 22 3 1 4 1 2 3 4 3 4 1 2

2 2 2 22 4 1 3 3 4 1 2 1 2 3 4

2 22 22 2

ib

q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q q q q

+ − − + − = − − + − + + − − − +

C

− this is a linear D.E. with no singularities

• Quaternions satisfy the differential equation: 12

ddt

= Aq q

1 2 3

1 3 2

2 3 1

3 2 1

00

00

ω ω ωω ω ωω ω ωω ω ω

− − = − − − −

A− where and 1

2

3

gyro databib

ωωω

= =

ω

bcd

ζ

=

e

( )( )( )( )

1

2

3

4

cos 2sin 2sin 2sin 2

qb qc qd q

ζζζζ

= =

q→ quaternion vector:


• Solution to D.E., if A is assumed constant,

− using gyro data, typically given as 1

k

k

tb

k ibt

dtδ−

= ∫θ ω

− where 1

k

k

t

kt

dt−

= ∫Θ A

2 31

1 1 1ˆ ˆ , 1,2,2 8 48k k k k kI k−

= + + + + =

Θ Θ Θq q

Determination of (2) ibC

− note: A is assumed constant only in the solution to the D.E., not in using the gyro data (model error, not data error in Θ)

− solution is a second-order algorithm, neglecting terms of order δt3

− higher-order algorithms are easily developed (Jekeli 2000)

− is given by an initialization procedure 0q


Determination of δgn (1)

• Kalman filter approach to minimizing estimation errors

− estimate IMU systematic errors and gravity disturbance vector

− formulate in i-frame and assume negligible error in niC

− system state updates (observations) are differences in accelerations

( )( ) ( ) ( )

i i i

i i i i i i

i b i i b i ib b

δ δ δ

δ δ δ

= + −

= + + − − +

= − × − − C C

y a g x

a a g g x x

a a x gψ

iδ a

i i i i iie ie= + Ω Ωg xγ

− over-script, ~, denotes indicated (measured) quantity

− δai includes accelerometer errors and orientation errors


• System state vector, εD

− IMU biases, scale-factor errors, assumed as random constants


− orientation errors, ψ i; i i bb ib

ddt

δ= −Cψ ω

22

22

2

0 0 0 02 0 0 0 0

0 0 0 0

N Nn n n

E E g

D D

d ddt dt δ

β βδ β δ β δ

β β

= − − +

g g g w

◦ wδg is a white noise vector process with appropriately selected variances

◦ βN , βE , βD are parameters appropriately selected to model the correlation time of the processes , ,2.146 N E Dβ=

− gravity disturbance components, modeled, e.g., as second-order Gauss-Markov processes in n-frame, with i i n

nδ δ= Cg g

D D D D Dddt

= +F G wε ε where wD is a vector of white noise processes

• System dynamics equation



• Alternatively, omit gravity disturbance model

( )( ) ( )

i i i

i i i i i

i b i i b ib b

δ δ

δ δ

= + −

= + + − +

= − × − C C

y a g x

a a g x x

a a xψ

iδ a

i i i i iie ie= + Ω Ωg xγ

− observations assume normal gravitation is correct

− optimal estimates of IMU systematic errors by Kalman filter yield y

− gravity disturbance estimates: ˆiδ ≈ −g y y

◦ assumes residual IMU systematic errors are small and white noise can be filtered

− successfully applied technique (Kwon and Jekeli, 2001)


Indirect Method • Recall inertial navigation equations in n-frame

( )n

n b n n n nb ie in

ddt

= − + +C Ω Ωv a v ge

e nn

ddt

= Cx v

− integrate (i.e., get IMU navigation solution) and solve for using a model and GNSS tracking data

ng

− analogous to traditional satellite tracking methods to determine global gravitational field, except gravity model is linear stochastic process, not spherical harmonic model

− navigation solution from OTF INS should not be integrated with GNSS!

◦ use raw accelerometer and gyro data to obtain free-inertial navigation solution

◦ IMU and GNSS must be treated as separate sensors, just like in scalar gravimetry

- one could pre-process IMU/GNSS data to solve for IMU systematic errors, neglecting gravity


• Solve for gravity disturbance treated as an error state (among many others) in the linear perturbation of navigation equations


− typical error states collected in state vector, εΙ , include: ◦ position errors, velocity errors, orientation errors ◦ IMU systematic errors (biases, etc.) ◦ gravity disturbance components

I I I I Iddt

= +F G wε ε where wI is a vector of white noise processes

− integration is done numerically (e.g., using linear finite differences)

( ) ( ) ( ) ( ) ( )1 1,I k I k k I k I k I kt t t t t t− −= +Φ G wε ε where Φ = state transition matrix

− observations are differences, IMU-indicated minus GNSS positions, treated as updates to the corresponding system states

( ) ( ) ( ) ( )k k I k kt t t t= +Hy ε v where v is a vector of discrete white noise processes



• Gravity disturbance model − stochastic process; e.g., second-order Gauss-Markov process,

22

22

2

0 0 0 02 0 0 0 0

0 0 0 0

N Nn n n

E E g

D D

d ddt dt δ

β βδ β δ β δ

β β

= − − +

g g g w

◦ wδg is a white noise vector process with appropriately selected variances

◦ βN , βE , βD are parameters appropriately selected to model the correlation time of the processes , ,2.146 N E Dβ=

− theoretically the gravity model is an approximation since the gravity field is not a linear, finite-dimensional, set of independent along-track signals as required/modeled in the system state formalism

• Kalman filter/smoother estimate, is optimal in the sense of minimum mean square error

− successful estimation depends on stochastic separability of gravity disturbance from accelerometer errors and coupled gyro errors


Rudimentary Error analysis

• Assume that n- and i-frames coincide (approx. valid for < 1 hour)

n n n≈ − ⇒g x a

gravitation errors

errors in kinematic

acceleration

errors in inertial

acceleration

gravitational gradients

errors in sensor

orientation

position errors

n n n n b n bb b

n nδ δδ δ= + − −Ψ C C Γg x a a pnegligible

• δg PSD is obtained from models of IMU and position error PSDs and PSD of vehicle acceleration

1 1 3 2 2 3 1g x a a aδ δ ψ ψ δΦ Φ Φ Φ Φ= + + +

3 3 2 1 1 2 3g x a a aδ δ ψ ψ δΦ Φ Φ Φ Φ= + + +


PSD Models

EGM96

1'x1' ∆g

model

frequency [cy/m]

Φδg

(f) [

mga

l2 /(cy

/m)2 ]

MathCad: degvarfit4.mcd Gravitational Field

frequency [Hz]

1aΦ

2aΦ

3aΦ

Accelerations* dashed-line: psd model

* data from Twin Otter aircraft

MathCad: accpsdfit.mcd

[(m

/s2 )

2 /Hz]

− Aircraft speed = 250 km/hr; altitude = 1000 m

– Accelerometers: white noise (±25 mGal)

– Gyros: rate bias (± 0.003°/hr) plus white noise (±0.06 °/hr/√Hz)

– Orientation: initial bias (0.005°)

approximated by simple PSD models

− Position: white noise (±0.1 m)


PSDs of Gravity Errors, δg1, δg3, Versus Along-Track PSDs of Signals, g1, g3

• Signal-to-noise ratio > 1 over larger bandwidth for g3

signal-to-noise ratio > 1

+ g3 :

g1 :

• Significant error source is orientation bias, especially for g1, g2

• Signal PSD’s move to right and down with increased velocity

Spectral Window:

1 10 4 1 10 3 0.01 0.11 10 9

1 10 8

1 10 7

1 10 6

1 10 5

1 10 4

1 10 3

0.01

Φδg3

Φδg1

Sg3

Sg1

[m2 /s

4 ]/H

z

Frequency [Hz]

1,3xδΦ


• Schematic: Pendulous, Force-Rebalance Accelerometer (e.g.)

• Torque needed to keep proof mass in equilibrium is a measure of acceleration

input axis

Case Hinge Permanent Magnet Forcer Coil

Light Emitting Diode

side view

plan view

Photo Detector

Shadow Arm Proof Mass

Forcer Direction

pulse train Pulse Rebalance Electronics Amplifier

1c

3c

2c

3c


• Honeywell (Allied Signal, Sundstrandt) QA3000

Pendulous, Force-Rebalance Accelerometer (e.g.)

• Litton (Northrop Grumman) A-4 Miniature Accelerometer Triad


INS Used for Airborne Gravimetry

Honeywell Laseref III

University of Calgary

Strapdown Platforms

Ohio State University

Litton LN100 Honeywell H-770

Intermap Technologies

ITC-2 Inertial Survey System

Bauman Moscow State Technical University

Stabilized Platform

iMAR RQH 1003

National Geodetic Survey...use raw accelerometer and gyro data to obtain free-inertial navigation...

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