Measuring the Dynamic Soaring of Albatrosses by Time

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Measuring the Dynamic Soaring of Albatrosses by Time
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Colloquium Satellite Navigation 2009
Measuring the Dynamic Soaring of Albatrosses
by Time-Differential Processing of Phase Measurements
from Miniaturized L1 GPS Receivers
Dipl.-Ing. Johannes Traugott
Institute of Flight System Dynamics
Technische Universität München
[email protected]
+49 (0)89 289-16066
June 15, 2009
Johannes Traugott
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The Project
Dynamic Soaring of the Wandering Albatross
The Albatross – a perfect glider
• Flight over thousands of kilometers over open seas (97% gliding without
flapping)
• Approx. 80% of lifetime airborne (sleeping while flying?)
• High wingspan (max. 3.5m), wing loading (~13 kg/m2), aspect ratio (~18)
and gliding ratio (~20)
• Very weak flight muscles (< 6% of body mass only);
tendon mechanism to lock wings while gliding
Wandering Albatross (diomedea exulans) in the skies
[Photo: Traugott]
June 15, 2009
Johannes Traugott
Track of a Wandering Albatross on
migration [Tickell 2000]
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Dynamic Soaring
• No thermal lift over open seas • Extraction of wind energy in boundary layer above sea surface
tcyc = 7.3s
DS trajectory and shear wind profile
Energy neutral dynamic soaring cycle
[Sachs 2004]
[Sachs 2004]
Controversy in community:
• Wave-slope soaring?
• Shear wind effect not sufficient for
continuous flight?
• Upper curve?
June 15, 2009
Johannes Traugott
Real measurements required…
• …for finding answers to these
questions
• …for validating simulation &
optimization results
3 / 62
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GPS Measurement Campaign (Winter 2008/09)
Sate of the art:
• Measurement of global tracks for studies of migration and foraging behavior
(e.g. BirdLife International 2004)
• Low sampling rates / modest precision (ARGOS)
Our objectives:
• Measurement of individual flight
cycles with high precision sufficient
for flight mechanical analysis on
maneuver scale
• Global trajectories of secondary interest
only.
Cooperation with CNRS / Centre
d’Ecologie Fonctionnelle et Evolutive
(Montpellier)
Expedition to Kerguelen Archipelago
The distribution of Wandering Albatrosses. Prince Edward Is. (1), Is. Crozet (2), I. Amsterdam (3), Is.
Kerguelen (4), Heard I. (5), Macquarie I. (6), Auckland Is. (10), Campbell I. (11), Antiposes Is. (12),
South Georgia (18), Tristand da Cunha (19) and Gough Island (20) [Tickell 2000]
June 15, 2009
Johannes Traugott
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Navigation Requirements and Constraints
Precision in the decimeter range for
individual maneuvers limited from 15
seconds up to a view minutes
High sampling rate (10 Hz) due to
high dynamics / short cycle times
No baseline restriction due to far
distances travelled
• shadowing during bird equipment
• shadowing during prey catching
[Photo: Traugott]
• loss-off-lock due to exceeding
bank angles, …
June 15, 2009
Johannes Traugott
[Photo: National Oceanic and
Atmospheric Administration/
Administration/DoC]
No (static) initialization procedures
due to practical constraints in the
field:
[Photo: Archive Traugott]
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Sensor Requirements and Constraints
Form-factor < 100 x 50 x 15 mm
Mass < 100 g
Lifetime: Several days or triggering mechanism for high rate sampling
• Memory no off-the-shelf solution
• Battery limiting factor concerning weight
No wiring on the bird! self-contained, sealed & rugged unit
Approach
Processing time differences
of L1 phase observations
from low cost, miniaturized
GPS receiver modules
Find & test suitable hardware
Develop & implement algorithm
Find & equip bird (and get receiver back)
June 15, 2009
Johannes Traugott
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Technische Universität München
Hardware
Receiver Module
U-blox LEA-4T
• Access to 10 Hz raw-data via
binary UBX protocol (UBX-RAW)
• 4 Hz online solution
• Promising results in ZBL-tests
• L1 only
Zero-baseline test result of the TIM module
(TIM-LL -TIM-LP, PRN 26-12, 07/08/07, DLR Oberpfaffenhofen,Germany)
June 15, 2009
Johannes Traugott
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Integrated Logger
B
C
2 GB (Micro) SD Card as external memory (8 MB integrated
flash not sufficient))
approx. 10 MB / h RXM-RAW @ 10 Hz, NAV-SOL @ 1Hz
8 MB flash only (30 min recording)
3 LS 1450 (C) lithium-thionyl chloride primary cells
2 LS 1450 cells sufficient for all applications
Operated by reed contact
Wireless data downloaded to base station
Recording time: several days
3 axis MEMS accelerometer
June 15, 2009
Triggering logic for high-rate raw-data recording
Johannes Traugott
C
A&B
A
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Casing (for B)
• Custom-made casing milled by
AKA-MODELL of TUM
• Transparent for reading status
LED when sealed
• Material: Makrolon
• Mass: approx 20 g
• Sealing by SikaFlex 221
1st prototype
June 15, 2009
Final design
Johannes Traugott
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Power supply
• Ilog = 65 mA; Ulog = 3.0 V; Plog = 195 mW
• Ambient temperature 0°C – 13°C
• Required lifetime
corresponding to memory
up to 6 days
U [v]
?
LS
17500
LS
14500
L 91
14.5 g 16.7 g
21.9 g
18 g
Test results
Li/FeS2
LiSoCl2
primary
primary
?
Datasheet Saft LS 14500 [manufacturer]
LiPoLy rechargeable
June 15, 2009
• Info from datasheets
not sufficient to make
proper decision
Johannes Traugott
• Tests under expected
conditions for various
candidates
• Final choice:
Lithium-thionyl chloride
primary cells
10 / 62
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3D MEMS accelerometer (C only)
Calibration line
Calibration test results
8
4000
low sensitivity
normal sensitity
3500
6
3000
digital output [-]
acceleration [g]
4
2
0
• Determination of scale
factor, bias, temperature
sensitivity,… currently in
progress for each axis of
the tags applied in the field
+1g
2500
2000
• One sample per GPS raw
data set (10 Hz)
1500
-1g
-2
1000
-4
-6
0
low sensitivity
normal sensitity
• No anti-aliasing filter in
front of ADC
500
2000
digital output [-]
June 15, 2009
4000
0
0
• σ = 0.1 m/s (experimental
results @ room
temperature)
2000
sample [-]
4000
Johannes Traugott
• Adjustment of sensitivity
settings in the field by flight
data analysis
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Software / Algorithm
Core Algorithm
THE OBSERVABLE
Individual carrier phase measurement
[Montenbruck 2006]
[Montenbruck 2006]
(
)
Φ(t ) = ρ (t ) + cδ R (t ) + λ1 ϕ S (t0 ) − ϕ R (t0 ) + N = f (ξ i , X i , ti )
144424443
June 15, 2009
N ' ≠ f (t )
Johannes Traugott
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Technische Universität München
Forming time-differences
bi
∇x := x(ti ) − x(tb )
bi
∇Φ = bi ∇ρ + c bi ∇δ + λ1 ∇N '
bi
provided continuous phase-lock
in receiver PLL,
unknown ambiguity cancels!
June 15, 2009
Johannes Traugott
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BASIC NAVIGATION TASK
Geometry of the problem
X bS
X iS
ρi
ρb
eb
• Only relative solution of interest
• Change of unit vectors due to
elapsed time not neglected
ei
xb tb
∆xi
bi
b
xi ti
Receiver position and time (PT)
at time ti
x i = (xi , yi , zi )
T
ξ i = (x i , cδti )
T
June 15, 2009
Baseline between “virtual” base
epoch at time tb and ti
b bi = x i − x b
β bi = ξ i − ξ b
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Equations of the problem
• Models enhanced by atmospheric corrections (troposphere, ionosphere)
)
)
R
Φ i = ρ i + cδ i + λ1 N '+Tˆi − I i
bi
(
) (
ˆ = (ρ − ρ ) + c(δ − δ ) + Tˆ − Tˆ + Iˆ − Iˆ
∇Φ
i
b
i
b
i
b
i
b
) bi
bi
bi
bi
= ∇ρ + c ∇δ + ∇Ti − ∇Iˆi
(
= f ξ i , X i , ti , ξ b , X b , t b
)
)
unknown
• “Observed = Computed” for all satellites in view (m>=4)
bi
~
bi
ˆ
∇Φ = ∇Φ
(Over-determined) set of nonlinear equations in receiver PT
June 15, 2009
Johannes Traugott
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Technische Universität München
• Numeric solution by linearization and iteration
bi
bi
ˆ (ξ
ˆ (ξ , ξ ,...) + H ∆ξ
∇Φ
,
ξ
,...
)
=
∇Φ
i , k +1 b
i ,k
b
ξ i ,k
i
ξ i ,k +1 = ξ i ,k + ∆ξ i
Updated in each iteration
cycle / epoch
H ξ i ,k
ˆ
d bi ∇Φ
=
=
dξ i ξ
i ,k
bi
(
) (
ˆ = (ρ − ρ ) + c(δ − δ ) + Tˆ − Tˆ + Iˆ − Iˆ
∇Φ
i
b
i
b
i
b
i
b
)
ˆ1
 ∂ bi ∇Φ

 ∂xi
ˆ2
 ∂ bi ∇Φ

 ∂xi
M

bi
ˆm
 ∂ ∇Φ
 ∂x
i

ˆ1
∂ bi ∇Φ
∂yi
ˆ2
∂ bi ∇Φ
∂yi
M
bi
ˆm
∂ ∇Φ
∂yi
ˆ

∂ bi ∇Φ
≈ ∂ρˆ i / ∂x i = −e i 
∂x i

Only variation of 
bi
ˆ
T,I neglected 
∂ ∇Φ
≈ 1 (compensated by 
iteration)
∂(cδ i )

June 15, 2009
ˆ1
∂ bi ∇Φ
∂zi
ˆ2
∂ bi ∇Φ
∂zi
M
bi
ˆm
∂ ∇Φ
∂zi
H ξ i ,k
Johannes Traugott
ˆ1 
∂ bi ∇Φ

∂(cδ i ) 
ˆ 2
∂ bi ∇Φ

∂(cδ i ) 
M

bi
m
ˆ
∂ ∇Φ 
∂(cδ i )  x
− e1 T
 i ,k
= M
− e m T
 i ,k
i ,k , yi ,k , z i ,k , cδ i ,k
1

M
1

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− Solution by least squares
(
)
−1
∆ξi = H H H
T
(
T bi
)
~ bi ˆ
∇Φ − ∇Φ(ξi,k ,...)
ξ i ,k +1 = ξ i ,k + ∆ξ i
− Minimization of residuals between measured and computed (modeled)
observations
m
∑ res
j2
= min
j =1
~
ˆj
res j = bi ∇Φ j − bi ∇Φ
− If measurement errors are uncorrelated, unbiased and have constant
variance ∆ξ has minimum variance of all estimates that are linear
combinations of the observations.
June 15, 2009
Johannes Traugott
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− Solution process similar to standard single point positioning
− ξ calculated in each iteration cycle in order to update Jacobian / unit
vectors
− Note: β is “real” solution of the problem (relative positioning)
xi,k
bbik
xb
June 15, 2009
bbik−1
Johannes Traugott
∆x = ∆b
xi,k −1
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Error considerations
MEASUREMENT ERRORS
bi
~ bi ˆ bi
∇Φ = ∇Φ + ∇χ
bi
∇χ = − bi ∇δ S + bi ∇E + bi ∇T − bi ∇I + bi ∇noise + bi ∇mp
satellite clocks
non-modeled measurement
troposphere
noise
ephemeris
non-modeled
multipath
ionosphere
Primary error remaining for time-differenced observables is error drift
S

d
δ
dE dT dI 
 dT dI  bi bi
bi
∇χ =  −
+
+
−  (ti − tb ) + 
−  b + ∇noise + bi ∇mp
dt dt dt  t
dx dx  tb

 dt
b
Dependent of temporal
autocorrelation of error
contributions
June 15, 2009
Present in any kind σ ∇noise = 2σ noise
of differential Single differences only
processing
Johannes Traugott
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Error drift for “clean” static data with best corrections
0.2
0.2
0.15
Eph.:
Sat.-Clk.:
Iono.:
Tropo.:
igs15295.sp3
igs15295.clk_30s
igsg1210.09i
UNB3
3D position error [m]
0.1
C/A SPP with
all corrections
north [m]
0.15
0.1
0.05
0.05
High quality receiver from
IGS reference network
(BRUS)
1Hz, #SV 8, PDOP ~ 1.9
0
-0.05
June 15, 2009
0
east [m]
0.05
0
0
100
100
200
Johannes Traugott
300
400
500
tau [s]
600
700
800
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Error drift for “clean”, static
data with best various
corrections omitted
1.4
1.2
• Initial solution by C/A SPP
• Solution variations typical
compare [Traugott 2008b]
1
3D position error [m]
• Same corrections for initial
solution and time-relative
solution
best
no hr clk
no hr clk / brdc eph
no ion
no trop
0.8
0.6
0.4
0.2
0
0
June 15, 2009
100
100
200
Johannes Traugott
300
400
500
tau [s]
600
700
800
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GEOMETRIC ERROR
Offset in initial position causes “distortion” of base vector
b
x b ,estimated
δb bi
bi*
x*i
~
ˆ (ξ , ξ ) +
∇Φ = bi ∇Φ
i ,0 b,0
+ H ξ i , 0 (∆ξ i − δξ b ) +
δx b
x b , true
bi
& (t − t )δξ + bi ∇χ
+H
i
b
i
1ξ4
4
24
4
3b
b
bi
xi
bi
Effect can be interpreted as additional,
“geometric” range error
No relative error for ti-tb0
No error in initial position no error due to
geometry variation
∇χ geo
− e&1 T
 i ,0
& = M
H
ξi
− e& m T
 i ,0
0

M
0

vsat ≈ 3.9km/s
e& max ≈ 1.9e − 4
Derivation [Traugott 2008], [Ulmer 1995]
June 15, 2009
Johannes Traugott
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Error drift for clean, static data caused by initial offset
7
• uBlox TIM-LP
• 25x25 mm, active patch
antenna
6
best
off 12.5 m
off 25 m
off 50 m
6.45
• Location: Open fields
• Typical results
5
#PRN 7
Compare [Traugott 2008]
1.2
3D position error [m]
1
3D position error [m]
1.4
Initial position as given by
IGS for station BRUS
(including Antenna phase
center corrections)
0.8
4
PDOP ~ 2.1
3.1
3
0.6
2
0.4
1.43
0.2
0
0
1
100
200
300
400
500
tau [s]
600
700
800
• Offset in initial position may
compensate for other errors
June 15, 2009
0.252
0
0
100
Johannes Traugott
200
300
tau [s]
400
500
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Two Approaches to a Precise Trajectory
OVER-ALL SOLUTION
δξb
SPP
ξb tb
βbi
ξi ti
Each epoch independent from previous epochs
reduced computational load for static applications
(if no outlier detection required)
Easier & better estimation of error drift (r.t. later slides)
Change of used constellation / drop-out of satellites steps in trajectory
possible
Same PRN subset at base and rover epoch less observations available
June 15, 2009
Johannes Traugott
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Technische Universität München
ACCUMULATED SOLUTION
ξ n−1 tn−1
δξb
SPP
βn−1,n
ξ n tn
ξb tb
ξi ti
Accumulation of incremental solutions
Max. number of PRN for processing available
Incremental solution anyhow required for outlier detection
Difficult estimation of error drift (see later slides)
Accumulation of errors random walk effects?
June 15, 2009
Johannes Traugott
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COMPARISON
ti tb ξ b
over-all solution
ˆ = const
Φ
b
~
~
Φi − Φb
i
∑
n =b +1
nonlinear
ξ i , 0 = ξ i −1
678
~ b ,i ~
n −1, n
Φ = ∇Φ
1
42∇
4
3
f (∑
n −1, n
) ∑f(
~
∇Φ,... =
n −1, n
~
~
Φ n − Φ n −1
f
nonlinear
ˆ
Φ
n −1
Linearity : f (c1a + c2b ) = c1 f (a ) + c2 f (b )
(
ξ n =ˆ β n −1,n
)
i
∑
β n −1,n
β bi
n =b +1
ξ n −1
t n t n −1
Johannes Traugott
)
)
~
∇Φ,...
exact
ξ n, 0 = ξ n −1
June 15, 2009
(
ξ i =ˆ β bi
f
accumulated solution
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Equivalence of incremental and over-all time differences of observations
i
∑
n −1, n ~
∇Φ =
n =1+1
i
∑
~ ~
Φ n −Φ n −1
n =1+1
=
i
∑ (Φ
n =b +1
n
+ δΦ n ) −(Φ n −1 + δΦ n −1 )
= [(Φ 2 + δΦ 2 ) − (Φ1 + δΦ1 )] + [(Φ 3 + δΦ 3 ) − (Φ 2 + δΦ 2 )]
+ [(Φ 4 + δΦ 4 ) − (Φ 3 + δΦ 3 )] + ...
+ [(Φ i − 2 + δΦ i − 2 ) − (Φ i −3 + δΦ i −3 )] + [(Φ i −1 + δΦ i −1 ) − (Φ i − 2 + δΦ i − 2 )] +
+ [(Φ i + δΦ i ) − (Φ i −1 + δΦ i −1 )]
= [(Φ i + δΦ i ) − (Φ1 + δΦ1 )]
~
1,i
= ∇Φ
δΦ
“usual suspects” in GPS-land
systematic & stochastic
component
June 15, 2009
(IF this sum was computed
Numeric error with double precision)
for 10.000 epochs (~15 min @ 10 Hz)
< 8e-6 m)
Johannes Traugott
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Geometric evidence of equivalence
of over-all and accumulated solution
(“pinned” satellite geometry)
k
1,5
∇Φk =
5
∑
n−1,n
∇Φk
n =2
2,3
1, 2
1,5
∇Φ
k
3, 4
∇Φk
4,5
∇Φk
∇Φk
∇Φk
June 15, 2009
x1=b
Johannes Traugott
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tb=1
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t2
t3
t4
t5
k
1,5
∇Φk =
5
∑
n−1,n
∇Φk
n =2
ρ3
ρ4
ρ5
2,3
1,5
∇Φk
ρ3
∇Φ
k
ρ2
ρ2
1, 2
June 15, 2009
∇Φk
3, 4
∇Φk ρ
4
ρ5
4,5
∇Φk
ρ5
Geometric evidence of equivalence of over-all
and accumulated solution (changing satellite
geometry)
Johannes Traugott
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ρk ,1 = imported
(
(
)
)
~k ~k
ρk ,2 = ρk ,1 + Φ2 − Φ1
~k ~k
~k ~k
~k ~k
ρk ,3 = ρk ,2 + Φ3 − Φ1 = ρk ,1 + Φ2 − Φ1 + Φ3 − Φ1
ρk ,1 = imported
[(
[
(
(
)
]
)]
) (
)
~k
~k
k
ρ = ρk ,1 + Φ2 + δΦ2 − Φ1
~k ~k
~k
~k ~k ~k
'
'
k
k
ρk ,3 = ρk ,2 + Φ3 − Φ2 + δΦ2 = ρk ,1 + Φ2 + δΦ2 − Φ1 + Φ3 − Φ2 + δΦk2
~k ~k
= ρk ,1 + Φ3 − Φ1
'
k ,2
(
)
[(
)
][
(
= ρ k ,3
Error on range measurements does not cause difference
btw. over-all and accumulated position
June 15, 2009
Johannes Traugott
30 / 62
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Comparison of accumulated and over-all solution (clean, static data)
3D position error
over-all
accumulated
0.25
0.06
0.05
0.2
0.04
0.15
[m]
[m]
Receiver: BRUS
3D position difference (over-all - accum.)
0.07
0.03
0.1
0.02
0.05
0
0
0.01
200
400
600
tau[s]
800
0
0
200
400
600
tau [s]
800
• Very same constellation used
within both solutions (8 PRN,
PDOP ~1.9)
• For the time spans of interest, both
solutions virtually coincide
(difference < 10%).
• Difference stemming from second
order effects
• No random walk effects
June 15, 2009
Johannes Traugott
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Comparison of accumulated and over-all solution (not so clean, static data)
1
1
0.8
0.8
0.4
0.6
PRN 31 manually excluded for
obtaining same constellation for
over-all
all solution
3D pos. error [m] (over-all)
3D pos. error [m] (accum.)
#SV × 0.1 (over-all)
#SV × 0.1 (accum.)
0.6
3D pos. error (over-all)
3D pos. error (accum.)
#SV × 0.1 (over-all)
#SV × 0.1 (accum.)
0.4
Drop-out of PRN14
0.2
0.2
0
0
200
400
600
tau [s]
800
0
650
700
750 800
tau [s]
850
• LEA-4T / 1/4 lambda wire antenna • Data screened for outliers (see later slides,
• Open fields
June 15, 2009
dResRms < 0.005 m)
Johannes Traugott
32 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
• Change of used PRN after long processing intervals can cause huge steps
in over-all solution ( each solution independent from previous solutions)
− Not relevant for time spans of interest
− Not possible when using accumulated solution
• Accumulated solution working with more PRN
− continuously better geometry…
− …but here: faster error drift than accumulated solution
− PRN do not drop out “without reason”, poor measurement quality not
compensated by better geometry
Choice of processing method case dependent
For very view satellites in view, accumulated solution often is the only
feasible alternative
June 15, 2009
Johannes Traugott
33 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Digression: Comparison of accumulated solution with speed calculation
• Doppler (range-rate)
measurements
• Velocity equations
 − e1T

 M
− e mT

D = ρ& + cδ& R
− Generated by receiver PLL or FLL?
− Afflicted with (unknown)time-delay?
1
 v
M
λ1δf


1

 D1 − e1T V1 


 

 =
M

  m
T
 D − em V m 


[Montenbruck 2006]
• Alternative strategy using raw phase ranges instead of receiver calculated
Doppler measurement (outline):
n −1, n
∇Φ
n −1, n
f
β
Vel.
equs.
vn
∑
β n −1,n
β bi
1 / (t n − t n −1 )
D
June 15, 2009
∑ v (t
n
n
Johannes Traugott
− t n −1 )
β bi
34 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
• Calculating speed instead of position (increments)
− Unit vectors need to be calculated (by SPP) for each epoch in a
preprocessing step
Increased noise on (base of) unit vectors
Unit vectors not coherent with solution
Additional problems?
As no iteration is required when calculating speed, computational load
might remain unchanged
− Receiver clock drift calculated instead of receiver clock bias.
Additional problems when differentiating and integrating?
− Changed situation when reconstructing trajectory in case of phase
outages
x
x
v
v
t
t
t
t
− No over-all solution possible
− Similar approach but not investigated in the scope of this work
June 15, 2009
Johannes Traugott
35 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Quality Monitoring
OUTLIER DETECTION
RAIM based on RMS of residuals of incremental solution (tn-1,tn)
RMSres =
m
∑
j =1
2
j
resn −1,n
if RMSres > user
defined threshold
if m>4
(over-determined
set of equations)
/ (m − 1)
resnj−1,n
Outlier
detected
=
n −1, n
~ j n −1,n ˆ j
∇Φ −
∇Φ
if m>5
Search afflicted
measurement by
iterative exclusion
Error drift virtually cancels for incremental solution very low residual level
Cycle slip - outlier (multipath) discrimination possible by analysis of resulting
residual of excluded observation (difficult)
Discrimination not necessary for accumulated solution
June 15, 2009
Johannes Traugott
36 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Outlier detection: Flight data
• Dynamic (but not acrobatic) maneuvers
• RTK reference solution (DEOS, TU Delft) available
200
0
-200
north [m]
-400
-600
-800
-1000
• LEA-4T with 25x25 mm passive
patch antenna
-1200
• 10 Hz sampled data
-1400
-1600
June 15, 2009
Johannes Traugott
0
500
east [m]
1000
1500
37 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
• Comparison with outlier corrected RTK solution
0.4
0.4
3D pos. difference [m]
# PRN × 0.01 [-]
0.3
0.3
0.2
0.2
0.1
0.1
20
40
60
80
0.02
0.01
0
0
June 15, 2009
0
0
100
RMSres [m]
0
0
RMSres [m]
Tight outlier detection
Loose outlier detection
20
40
60
tau [s]
80
3D pos. difference [m]
# PRN × 0.01 [-]
20
40
60
80
100
20
40
60
tau [s]
80
100
0.02
0.01
100
Johannes Traugott
0
0
38 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
ERROR (DRIFT) ESTIMATION
Estimation of measurement noise (Least squares rule)
1. Unbiased obs.
σ
2. Uncorrelated obs.
2
~
∇Φ
3. Obs. with constant variance
∑
=
m
j =1
res
m
2
m−4
Error propagation to position domain (DOP)
Cβ = σ
2
∇Φ D
(
D= H H
T
)
−1
Assumptions (1,(2),3) true for
incremental solution
(tn-1, tn)
Estimation of
stochastic component
of position error
June 15, 2009
Assumption 1 violated for
over-all solution
(tb, ti)
Application of
…for the lack of
stochastic concept
to systematic
errors…
better knowledge.
“Confirmed” by
experimental
results.
Estimation of
systematic / drift component
of position error
Johannes Traugott
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Flugsystemdynamik
Technische Universität München
Noise estimation / estimation of stochastic error component
0.25
3D position error
3D estimate of stochastic error component
[m]
0.2
0.15
0.1
0.05
0
0
100
• Clean, static data
200
300
tau [s]
500
3D estimate = σ ∇Φ~ × PDOP
• TIM-LL / active patch antenna
June 15, 2009
400
Incremental solution
Johannes Traugott
40 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Error drift estimation: Accumulated solution
• Problem: increasing over-all residuals (tb, ti) not available
• Observation from analysis of test data:
resij,b =
i
∑
resnj−1,n
n =b +1
• Proposed solution: Reconstruction of over-all RMS of residuals:
RMSres =
2


j

resn −1,n  / (m − 1)

j =1  n =b +1

m
i
∑ ∑
− Accumulated position solution shows no steps (e.g. in case of PRN dropout)
− Error estimate must be smooth as well
− Reset of accumulated residuals required in case of changing number of
satellites
• Remaining problem: Resulting error estimate too pessimistic
June 15, 2009
Johannes Traugott
41 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Error drift estimation: Accumulation of incremental residuals
0.03
11
resPRN
i,b
0.025
11
resPRN
n-1,n
11
SUM(resPRN
)
n-1,n
0.02
[m]
0.015
0.01
0.005
0
-0.005
-0.01
0
June 15, 2009
50
100
tau [s]
Johannes Traugott
150
200
42 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Error drift estimation for clean static data
Over-all solution
0.25
Accum. solution
RMSres
RMSres (reconstructed)
σ
σ
∇Φ
3D error estimate
(PDOP × σ∇Φ)
0.25
3D error
[m]
[m]
0.2
0.15
0.15
0.1
0.1
0.05
0.05
June 15, 2009
3D error estimate
(PDOP × σ∇Φ )
3D error
0.2
0
0
∇Φ
100
200 300
tau [s]
400
500
0
0
Johannes Traugott
100
200 300
tau [s]
400
500
43 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Error drift estimation for not so clean, static data
Accum. TD solution
0.4
RMSres (reconstructed)
3D error estimate
3D error
[m]
0.3
0.2
0.1
[-]
0
June 15, 2009
8
6
4
2
0
0
# PRN
PDOP
100
200
300
400
500
tau[s]
Johannes Traugott
600
700
800
900
44 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Error drift estimation for clean dynamic data
north [m]
0
-50
-100
-300 -250 -200 -150 -100
east [m]
-50
0
50
• Car driving on open fields
• RTK solution from DEOS
available
• TIM-LL with 25 x 25 mm active patch mounted on rooftop
• 10 Hz sampling rate
• Good data quality (7 PRN, DOP 2.15)
June 15, 2009
Johannes Traugott
45 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
• Position difference between accumulated time-difference solution and fixed
ambiguity RTK solution
• Error estimate matches real offset well
0.1
3D position differene
(RTK - accum. TD)
3D error estimate
0.09
0.08
0.07
[m]
0.06
0.05
0.04
0.03
0.02
0.01
0
0
June 15, 2009
20
40
60
80
100
tau [s]
Johannes Traugott
120
140
160
/ 62
46
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Error drift estimation for flight test data
Over-all solution
Accum. solution
0.5
0.45
0.4
0.5
3D position difference
(RTK - TD) [m]
3D error estimate [m]
# PRN × 0.01 [-]
PDOP × 0.01 [-]
0.45
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
20
40
60
tau [s]
80
100
0
0
3D position difference
(RTK - TD) [m]
3D error estimate [m]
# PRN × 0.01 [-]
PDOP × 0.01 [-]
20
40
60
tau [s]
80
100
• Accumulated solution drifts faster from RTK as over-all solution
• Error estimate of accumulated solution too pessimistic.
Problem observed with frequently changing geometry
June 15, 2009
Johannes Traugott
47 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Albatross Data
Equipment of 20 birds during
breeding season
June 15, 2009
20 receivers recovered
16 valid flights = 80% success rate
Johannes Traugott
48 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Five 30 min trajectories with 3D acceleration data
Albatross colony
(breeding site)
June 15, 2009
Johannes Traugott
49 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
11 one to six day trajectories
June 15, 2009
Johannes Traugott
50 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
First 6 days out of
30 day trip
(approx. 3.500 km)
Kerguelen
June 15, 2009
Johannes Traugott
51 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Objective: Analysis of individual dynamic soaring cycles
June 15, 2009
Johannes Traugott
52 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
3D trajectory of exemplary cycle
up × 3 [m]
20
10
0
200
100
north [m]
0
0
20
40
60
80
100
east [m]
June 15, 2009
Johannes Traugott
53 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Time history in vertical direction
10
8
up [m]
6
4
2
0
-2
-4
0
5
10
15
tau [m]
June 15, 2009
Johannes Traugott
54 / 62
Lehrstuhl für
Flugsystemdynamik
1.4
1.2
Technische Universität München
Data analysis
0
3D error estimate [m]
# PRN × 0.1
PDOP × 0.1
EXC
BC
5
NO
1
10
CS
PRN
0.8
BM
15
LE
0.6
20
MD
0.4
REP
25
OK
0.2
30
0
0
5
10
15
20
tau [m]
• Frequently changing geometry
• Multiple outliers
(RMSres limit set to 5 mm )
June 15, 2009
NA
40
60
80 100
tau × 10 [s]
120
140
• Estimate of 3D position drift 1.1 m
(probably too pessimistic)
• Cycle with relatively good data quality!
Johannes Traugott
55 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Data interpolation (here: vertical direction)
10
original data
smoothing spline fit
up [m]
5
0
resid. [m]
noise est. [m]
-5
0
5
10
15
5
10
15
5
10
15
x
0.02
0.01
0
0
0.02
0
-0.02
0
tau [m]
• Cubic smoothing spline
(first shot only)
June 15, 2009
• Residuals correlated with 3D noise
estimate
Johannes Traugott
56 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Speed and acceleration
30
[m/s]
25
20
sog (fit dot)
sog (Doppler SPP)
sog (pos. data diff)
15
10
0
5
10
15
accel (fit ddot)
accel (accelerometer)
accel (pos. data ddiff)
20
• Speed from
differentiation less
noisy than speed
calculated using
Doppler
measurements
(&x& )
A EE
E
15
= M ES ((a Accel )S − (g )S )
(&x& )
2
[m/s ]
A EE
10
= 1g
xS
0
0
gravity = 1g
2g
5
5
10
tau [s]
15
zS
• Absolute values of acceleration measured by accelerometers and derived
from GPS position plausible
June 15, 2009
Johannes Traugott
57 / 62
Lehrstuhl für
Flugsystemdynamik
Total energy
Technische Universität München
E=
1 2
mv + mgh
2
30
E
1 2
=
v +h
mg 2 g
10
total energy [m]
10
0
0
5
10
15
-10
40
40
total energy [m]
250
200
north [m]
20
300
height [m]
speed [m/s]
20
150
100
30
50
20
0
10
0
5
10
15
tau [s]
0
50
100
east [m]
• Already at this point investigation of bird’s total energy management possible
June 15, 2009
Johannes Traugott
58 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Outlook
• Evaluation of individual cycles under
different (environmental) conditions
− Wind strength
− Relative heading w.r.t. wind direction
− Sex / size
• Starting with wind information from global
models (QuickSCAT L3) and boundary
models
• Reconstruction of flight state using
additional field measurements (wing
geometry)
[http://manati.orbit.nesdis.noaa.gov/quikscat]
• …
up × 3 [m]
• (Reconstructing local wind from
trajectories)
500
400
40
300
20
200
0
Confirmation of
0
dynamic soaring theory
and gain of new insights
June 15, 2009
600
100
50
100
0
north [m]
east [m]
Johannes Traugott
C:\DATEN_JT\FSD\02_GPS_Tests\06_Kerguelen\090114_
#10f_Rec7\Processing\1514_347135_347171 clean cycles
59 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Conclusion
Relative accuracy is an absolute value!
Many applications
• Motion analysis
− …for (aircraft) performance and parameter identification
e.g. gliding ratio of gilder planes
− …for sport applications (without antenna shadowing)
−…
• Static measurement of relative position
− …for avalanche rescue
− …for azimuth determination
−…
Time-differences of L1 receiver phase ranges are proposed as a means to
achieve relative precision whilest…
• ….keeping costs really low
• ….keeping field procedures really simple
• ….using miniaturized equipment
June 15, 2009
Johannes Traugott
60 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Thank you for your attention!
[Photo: Traugott]
June 15, 2009
Johannes Traugott
61 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
Acknowledgements
Oliver Montenbruck
Anna Nesterova, Francesco Bonadonna
Alexei Vyssotsky, Giocomo Dell’Omo
Franz Kümmeth, Wolfgang Heidrich
Jan Wendel
Dennis Odijk, Christian Tiberius
AKAModell, TU München
IDAFlieg, AKAFlieg, TU München
…
June 15, 2009
Johannes Traugott
62 / 62
Lehrstuhl für
Flugsystemdynamik
Technische Universität München
References
BirdLife International (2004). Tracking ocean wanderers: the global distribution of albatrosses and petrels. Results from
the Global Procellariiform Tracking Workshop, 1–5 September, 2003, Gordon’s Bay, South Africa. Cambridge, UK:
BirdLife International.
Montenbruck, o. (2006). Lectures on Satellite Navigation. Technische Universität München, 2006
Sachs, G (2004). Minimum shear wind strength required for dynamic soaring of albatrosses. Ibis.
Tickell, W.L.N. (2000). Albatrosses. Yale University Press. New Haven and London.
Traugott, J., G. Dell'Omo, A.L. Vyssotski, D. Odijk, and G. Sachs (2008). A time-relative approach for precise
positioning with a miniaturized L1 GPS logger. In Proceedings of ION GNSS 21th International Technical Meeting of
the Satellite Division, GA, 16 – 19 September 2008. The Institute of Navigation.
Traugott, J., D. Odijk, O. Montenbruck, G. Sachs, and C.C.J.M.Tiberius (2008b). Making a dierence with GPS. GPS
World, 19(5):48 - 57, May 2008.
Ulmer, K., P. Hwang, B. Disselkoen and M. Wagner (1995). Accurate azimuth from a single PLGR+GLS DoD GPS
receiver using time relative positioning. Proc. of ION GPS-95, Palm Springs, CA, 12-15 September 1995, pp. 17331741.
June 15, 2009
Johannes Traugott
63 / 62