# attitude by qingyunliuliu

VIEWS: 4 PAGES: 39

• pg 1
```									Attitude Determination

- Using GPS

• Definition of Attitude
• Attitude and GPS
• Attitude Representations
• Least Squares Filter
• Kalman Filter
• Other Filters
• The AAU Testbed
• Results
• Conclusion
20/12-2000 (MJ)            Danish GPS Center   2
What is Attitude?

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

Y

X

20/12-2000 (MJ)            Danish GPS Center                          3
When is Attitude information needed?

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

20/12-2000 (MJ)           Danish GPS Center                4
Attitude sensors

Currently used sensors include:

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

20/12-2000 (MJ)            Danish GPS Center     5

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

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

20/12-2000 (MJ)         Danish GPS Center                6
Interferometric Principle
Carrier Wave (from GPS satellite)

Integer
Component

Master
Antenna        k                      Fractional
Component, 

Baseline, b(3x1)
Slave
Antenna

e(3x1)
Unit Vector to GPS Satellite

20/12-2000 (MJ)           Danish GPS Center                               7
Interferometric Principle

Measurement equation:

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

20/12-2000 (MJ)          Danish GPS Center                 8
Attitude Matrix
Z
V                  9 parameters needed:
U

W

Y

When (x,y,z) is a reference
X                                    system:

20/12-2000 (MJ)                Danish GPS Center                   9
Properties of “A”

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

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

20/12-2000 (MJ)         Danish GPS Center              10
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”

20/12-2000 (MJ)         Danish GPS Center                  11
Euler sequences

A sequence of rotations by the angles (,,) about
the coordinate axes of the reference system

Single axis:

Multiple axes:

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

A quaternion consists of four composants

Where i,j and k are hyperimaginary numbers

20/12-2000 (MJ)       Danish GPS Center          13
Quaternions

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

20/12-2000 (MJ)        Danish GPS Center                14
Quaternions

Quaternions represent attitude as a rotation-axis
and a rotation-angle

20/12-2000 (MJ)        Danish GPS Center                15
Quaternions

Quaternions can be multiplied using the special
operator            defined as:

20/12-2000 (MJ)        Danish GPS Center              16
Quaternions

The attitude matrix can be formed from the
quaternion as:

Where

20/12-2000 (MJ)        Danish GPS Center         17
Least Squares Solution

Including attitude information into the
measurement equation

Linearization of the attitude matrix

20/12-2000 (MJ)          Danish GPS Center     18
Least Squares Solution

Forming the phase residual

20/12-2000 (MJ)        Danish GPS Center   19
Least Squares Solution

20/12-2000 (MJ)    Danish GPS Center   20
Least Squares Solution

Estimate update

20/12-2000 (MJ)        Danish GPS Center   21
Extended Kalman Filter

A Kalman filter consists of a model equation

and a measurent equation

20/12-2000 (MJ)         Danish GPS Center           22
Extended Kalman Filter

And their linearized counterparts….

And

20/12-2000 (MJ)         Danish GPS Center   23
Extended Kalman Filter

Algorithm

z(tk)                    x(tk-), P(tk-)

Estimate
Update
Estimate of

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

Time
Propagation

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

20/12-2000 (MJ)                         Danish GPS Center   24
Extended Kalman Filter
Tuning of the filter

Noise variance determined experimentally

20/12-2000 (MJ)            Danish GPS Center    25
Extended Kalman Filter
Tuning of the filter

Noise variance determined by „trial-and-error‟

20/12-2000 (MJ)            Danish GPS Center          26
Extended Kalman Filter
Determining the system model

N    Dynamic                      Kinematic   q
model                          model

20/12-2000 (MJ)             Danish GPS Center                   27
Other Filters
Raw phase measurements                                    Non-iterative                                                          Iterative

Integer resolution

Extended phase measurements

Line bias calibration

Corrected phase measurements

New Algorithm
Bar-Itzhack

Geometric Descent
Vector Determination

Least Squares

Bar-Itzhack 2
Kalman filter

ALLEGRO

Cohen
Vector estimate

Euler-q
Attitude Determination

Attitude estimate

20/12-2000 (MJ)                            Danish GPS Center                                                                                                                 28
Testbed

Controller box
LON network

Labtop PC
Motor                    Encoder
RS232

Antenna array

Surveyor platform

20/12-2000 (MJ)                          Danish GPS Center                                 29
Software
Start

attitude                     angles.txt

Start User Interface
+ Initialize Testbed

Proces 1                  Proces 2                      Proces 3
Store phases and                                            Send angle
Store angle-values
LOS vectors from                                         commands to LON
from encoders in file

Stop
GPS            phase.out   angles.out           LON nodes

20/12-2000 (MJ)                             Danish GPS Center                                       30
Motor Control

20/12-2000 (MJ)       Danish GPS Center   31
Motor Angles
Angle of Antenna Array
200

150

100

50

Angle [degrees]
0

-50

-100

-150

-200
0   50   100   150   200       250        300   350   400   450   500
Time [seconds]

20/12-2000 (MJ)                                                                                                    Danish GPS Center   32
Local Horizontal System

20/12-2000 (MJ)   Danish GPS Center   33
Results

Based on actual and simulated data, the following
performance parameters were evaluated

• Accuracy
• Computational efficiency
• Ability to converge

20/12-2000 (MJ)          Danish GPS Center               34
Accuracy

20/12-2000 (MJ)     Danish GPS Center   35
4
x 10                            Speed Analysis with three baselines
6
Kalman filter w. bias est.
5          Kalman filter

Speed
Geometric Descent
4          Bar Itzhack 2

3

Number of Floating Point Operations
2

1

0
2                          3                      4                      5   6

3
x 10
5
ALLEGRO
New Algorithm + Euler-q
4
New Algorithm + SVD
Single-point
3          Cohen
Bar Itzhack + SVD

2

1

0
2                          3                     4                       5   6
Number of satellites

20/12-2000 (MJ)                                                                                                                   Danish GPS Center   36
Convergence
Kalman Filter                                                Single-point Algorithm
6000                                                            6000
Number of baselines = 3                                          Number of baselines = 3
5000              Convergence = 100%                            5000               Convergence = 100%

Number of Samples

Number of Samples
Average = 5.75                                                   Average = 4.25
4000                                                            4000

3000                                                            3000

2000                                                            2000

1000                                                            1000

0                                                               0
1 2 3 4 5 6 7 8 9 1011 1213 1415                                1 2 3 4 5 6 7 8 9 1011 1213 1415
Convergence Time (Samples)                                      Convergence Time (Samples)

ALLEGRO Algorithm                                            Geometric Descent Algorithm
6000                                                            6000
Number of baselines = 3                                          Number of baselines = 3
5000              Convergence = 100%                            5000               Convergence = 100%

Number of Samples

Number of Samples
Average = 4.56                                                   Average = 5.33
4000                                                            4000

3000                                                            3000

2000                                                            2000

1000                                                            1000

0                                                               0
1 2 3 4 5 6 7 8 9 1011 1213 1415                                1 2 3 4 5 6 7 8 9 1011 1213 1415
Convergence Time (Samples)                                      Convergence Time (Samples)

20/12-2000 (MJ)                                                                                                                               Danish GPS Center   37
Kalman Filter                                                Single-point Algorithm

Convergence
6000                                                            6000
Number of baselines = 2                                          Number of baselines = 2
5000              Convergence = 96.61%                          5000               Convergence = 69.34%

Number of Samples

Number of Samples
Average = 7.87                                                   Average = 4.64
4000                                                            4000

3000                                                            3000

2000                                                            2000

1000                                                            1000

0                                                               0
1 2 3 4 5 6 7 8 9 1011 1213 1415                                1 2 3 4 5 6 7 8 9 1011 1213 1415
Convergence Time (Samples)                                      Convergence Time (Samples)

ALLEGRO Algorithm
6000
Number of baselines = 2
5000              Convergence = 69.75%

Number of Samples
Average = 4.76
4000

3000

2000

1000

0
1 2 3 4 5 6 7 8 9 1011 1213 1415
Convergence Time (Samples)

20/12-2000 (MJ)                                                                                                                               Danish GPS Center   38
Conclusion

• 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
20/12-2000 (MJ)          Danish GPS Center               39

```
To top