Attitude Determination - Using GPS. 20/12-2000 (MJ)Danish GPS Center2 Table of Contents Definition of Attitude Attitude and GPS Attitude Representations

Attitude Determination

- Using GPS

20/12-2000 (MJ) Danish GPS Center 2

Table of ContentsTable of Contents

• Definition of Attitude• Attitude and GPS• Attitude Representations• Least Squares Filter• Kalman Filter• Other Filters• The AAU Testbed• Results• Conclusion


What is Attitude?What is Attitude?

Orientation of a coordinate system (u,v,w) with respect to some reference system (x,y,z)

V

U

W

Z

Y

X


When is Attitude information needed?When is Attitude information needed?

• Controlling an Aircraft, Boat or Automobile• Onboard Satellites• Pointing of Instruments• Pointing of Weapons• Entertainment industri (VR)• Etc...


Attitude sensorsAttitude sensors

Currently used sensors include:

• Gyroscopes• Rate gyros (+integration)• Star trackers• Sun sensors• Magnetometers• GPS


Advantages of GPSAdvantages of GPS

• Adding new functionality to existing equipment• No cost increase• No weight increase• No moving parts (solid-state)• Measures the absolute attitude

DisadvantagesDisadvantages• Mediocre accuracy (0.1 - 1º RMS error)• Low bandwidth (5-10 Hz maximum)• Requires direct view of satellites


Interferometric PrincipleInterferometric Principle

MasterAntenna

IntegerComponent

k

SlaveAntenna

Carrier Wave (from GPS satellite)

e(3x1)

FractionalComponent,

Baseline, b(3x1)

Unit Vector to GPS Satellite


Interferometric PrincipleInterferometric Principle

Measurement equation:

The full phase difference is the projection of thebaseline vector onto the LOS vector:


Attitude MatrixAttitude Matrix

V

U

W

Z

Y

X

9 parameters needed:

When (x,y,z) is a referencesystem:


Properties of “A”Properties of “A”

“A” rotates a vector from the reference systemto the body system

The transpose of “A” rotates in the oppositedirection (back again)


Properties of “A”Properties of “A”

Rotation does not change the size of the vectors:

Every rotation has a rotation-axis (and a rotation-angle)

The rotation-angle is the eigenvalue of “A”


Euler sequencesEuler sequences

A sequence of rotations by the angles (,,) aboutthe coordinate axes of the reference system

Single axis:

Multiple axes:


QuaternionsQuaternions

A quaternion consists of four composants

Where i,j and k are hyperimaginary numbers



A quaternion can be thought of as a 4 dimensionalvector with unit length:



Quaternions represent attitude as a rotation-axisand a rotation-angle



Quaternions can be multiplied using the specialoperator defined as:



The attitude matrix can be formed from thequaternion as:

Where


Least Squares SolutionLeast Squares Solution

Including attitude information into themeasurement equation

Linearization of the attitude matrix



Forming the phase residual





Estimate update


Extended Kalman FilterExtended Kalman Filter

A Kalman filter consists of a model equation

and a measurent equation



And their linearized counterparts….

And



EstimateUpdate

TimePropagation

z(t k) x(t k-), P (t k

-)

x(t k+), P (t k

+)

x(t k+1-), P (t k+1

-)

x(t k), P (t k)k = k + 1

Estimate of

Algorithm



Tuning of the filter

Noise variance determined experimentally



Tuning of the filter

Noise variance determined by ‘trial-and-error’



Determining the system model

Dynamicmodel

Kinematicmodel

N q


Other FiltersOther Filters

Line bias calibration

Vector Determination

Attitude Determination

Integer resolution

Raw phase measurements

Extended phase measurements

Corrected phase measurements

Vector estimate

Attitude estimate

Ka

lma

n f

ilte

r

AL

LE

GR

O

Co

he

n

Le

ast

Sq

ua

res

Ba

r-It

zha

ck

Ne

w A

lgo

rith

m

Eu

ler-

q

Ge

om

etr

ic D

esc

en

t

Ba

r-It

zha

ck 2

Non-iterative Iterative


TestbedTestbed

Labtop PC

GPS receiver

Controller box

Surveyor platform

Antenna array

LON network

RS232

Motorcommands

Encoderreadings


SoftwareSoftware

Start

Load simulatedattitude

Start User Interface+ Initialize Testbed

angles.txt

phase.outlos.out

GPSreceiver

angles.out LON nodes

Store angle-valuesfrom encoders in file

Proces 2

Store phases andLOS vectors from

receiver in file

Proces 1

Send anglecommands to LON

nodes

Proces 3

Stop


Motor ControlMotor Control


Motor AnglesMotor Angles

0 50 100 150 200 250 300 350 400 450 500-200

-150

-100

-50

0

50

100

150

200

Time [seconds]

An

gle

[d

eg

ree

s]Angle of Antenna Array


Local Horizontal SystemLocal Horizontal System


ResultsResults

Based on actual and simulated data, the followingperformance parameters were evaluated

• Accuracy• Computational efficiency• Ability to converge


AccuracyAccuracy


SpeedSpeed

2 3 4 5 60

1

2

3

4

5x 10

3

Number of satellites

ALLEGRO New Algorithm + Euler-qNew Algorithm + SVD Single-point Cohen Bar Itzhack + SVD

2 3 4 5 60

1

2

3

4

5

6x 10

4 Speed Analysis with three baselinesN

um

be

r o

f F

loa

ting

Po

int

Op

era

tion

sKalman filter w. bias est.Kalman filter Geometric Descent Bar Itzhack 2


ConvergenceConvergence

1 2 3 4 5 6 7 8 9 1011 1213 14150

1000

2000

3000

4000

5000

6000Kalman Filter

Convergence Time (Samples)

Nu

mb

er

of

Sa

mp

les

Number of baselines = 3

Convergence = 100%

Average = 5.75

1 2 3 4 5 6 7 8 9 1011 1213 14150

1000

2000

3000

4000

5000

6000Single-point Algorithm


Nu

mb

er

of

Sa

mp

les


Convergence = 100%

Average = 4.25

1 2 3 4 5 6 7 8 9 1011 1213 14150

1000

2000

3000

4000

5000

6000ALLEGRO Algorithm


Nu

mb

er

of

Sa

mp

les


Convergence = 100%

Average = 4.56

1 2 3 4 5 6 7 8 9 1011 1213 14150

1000

2000

3000

4000

5000

6000Geometric Descent Algorithm


Nu

mb

er

of

Sa

mp

les


Convergence = 100%

Average = 5.33


ConvergenceConvergence

1 2 3 4 5 6 7 8 9 1011 1213 14150

1000

2000

3000

4000

5000

6000Kalman Filter


Nu

mb

er

of

Sa

mp

les


Convergence = 96.61%

Average = 7.87

1 2 3 4 5 6 7 8 9 1011 1213 14150

1000

2000

3000

4000

5000

6000Single-point Algorithm


Nu

mb

er

of

Sa

mp

les



Average = 4.64

1 2 3 4 5 6 7 8 9 1011 1213 14150

1000

2000

3000

4000

5000

6000ALLEGRO Algorithm


Nu

mb

er

of

Sa

mp

les



Average = 4.76


ConclusionConclusion

• Kalman filter is by far most accurate, but also computationally very heavy

• Single-point (LSQ) offers good accuracy + high speed

• Vector matching algorithms has the lowest accuracy but does not suffer from convergence problems

• Performance depend on satellite constellation

• Results were affected by mechanical problems with levelling of the testbed

Documents

Attitude Determination - Using GPS. 20/12-2000 (MJ)Danish GPS Center2 Table of Contents Definition of Attitude Attitude and GPS Attitude Representations