Attitude Determination - Using GPSold.gps.aau.dk/downloads/attitude.pdf · 20/12-2000 (MJ) Danish...

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, ∆ϕ

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 (φ,θ,ψ) 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(tk) x(tk-), P(tk-)

x(tk+), P(tk+)

x(tk+1-), P(tk+1

-)

x(tk), P(tk)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 ω 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

Kal

man

filte

r

ALL

EG

RO

Coh

en

Leas

t Squ

ares

Bar

-Itzh

ack

New

Alg

orith

m

Eul

er-q

Geo

met

ric D

esce

nt

Bar

-Itzh

ack

2

Non-iterative Iterative


TestbedTestbed

Labtop PC

GPS receiver

Controller box

Surveyor platform

Antenna array

LON network

RS232

Motorcommands

Encoderreadings


SoftwareSoftwareStart

Load simulatedattitude

Start User Interface+ Initialize Testbed

angles.txt

phase.outlos.out

GPSreceiver angles.out LON nodes

Store angle-valuesfrom encoders in file

Proces 2Store phases andLOS vectors from

receiver in file

Proces 1Send angle

commands to LONnodes

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]

Angl

e [d

egre

es]

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 103

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 104 Speed Analysis with three baselinesN

umbe

r of F

loat

ing

Poi

nt O

pera

tions

Kalman 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)

Num

ber o

f Sam

ples

Number of baselines = 3Convergence = 100%Average = 5.75

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

1000

2000

3000

4000

5000

6000Single-point Algorithm


Num

ber o

f Sam

ples


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

1000

2000

3000

4000

5000

6000ALLEGRO Algorithm


Num

ber o

f Sam

ples


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

1000

2000

3000

4000

5000

6000Geometric Descent Algorithm


Num

ber o

f Sam

ples



ConvergenceConvergence

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

1000

2000

3000

4000

5000

6000Kalman Filter


Num

ber o

f Sam

ples

Number of baselines = 2Convergence = 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


Num

ber o

f Sam

ples


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

1000

2000

3000

4000

5000

6000ALLEGRO Algorithm


Num

ber o

f Sam

ples



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

Attitude Determination - Using GPSold.gps.aau.dk/downloads/attitude.pdf · 20/12-2000 (MJ) Danish...

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Transcript of Attitude Determination - Using GPSold.gps.aau.dk/downloads/attitude.pdf · 20/12-2000 (MJ) Danish...