Post on 18-Jul-2015
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Shu Ting Goh
Advisor(s): Ossama Abdelkhalik,
Seyed A. (Reza) Zekavat
1
Mechanical Engineering – Engineering Mechanics Department
Spacecraft Formation Flying Multiple spacecraft…
Follow each other
Fly in a formation
Fly through specific trajectory
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Gravity Recovery and Interior Laboratory (GRAIL)
Mission Elapse – 93 DaysLISA Pathfinder
Spacecraft Formation Flying
Applications
3
Gravitational Field
Earth Gravity Recovery and Climate Experiment (GRACE)
Moon Gravity Recovery and Interior Laboratory (GRAIL)
Sun Impact of Sun’s solar storm on Earth (Clusters)
Earth Climate
A Train formation
Why Formation Flying?
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• Cost
• Robustness
• Resolution, accuracy,
precision
VS
Formation Flying Requirements What issues are required to be aware?
Avoid collision between spacecraft
Spacecraft travels at high speed.
Maintain Formation
Orientation, distance, orbit maneuver.
Perturbations
Drag, plasma field and etc.
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Navigation sensors for Formation
Flying
Position
Wireless ranging
with antenna array
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Other
Doppler Tracker
Range Only
Radio Interferometer
Laser Interferometer
Attitude/Direction
VISNAV
Autonomous Formation Flying (AFF)
Vision Based Navigation System
Provides three dimensional position information.
Antenna array technology for space mission focus on
communication purpose.
Bandwidth issue.
Motivation
7
High altitude space mission (GEO):
Poor GPS
Deep space applications:
No GPS
Depends on other instruments:
Sun sensor, star tracker…
Alternative sensor:
Relative position absolute position
Integrate with other sensors, GPS/star tracker/sun sensor
improve navigation performance
Wireless Local Positioning
System (WLPS)
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Dynamic Base Station
(DBS)R, TOA
, DOA
Transponder (TRX)
* WLPS lab, Director: Reza Zekavat, rezaz@mtu.edu,
http://www.ece.mtu.edu/ee/faculty/rezaz/wlps/index.html
Spacecraft Navigation
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DBS
TRX
R, TOA
, DOA
R, TOA
, DOA
Initial Guess
Estimator/Filter
Updated position and velocity
To ground station
Estimation Method
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Kalman Filter
Extended Kalman Filter (EKF)
Smoothing Kalman Filter (SKF)
Unscented Kalman Filter (UKF)
Ensemble Kalman Filter (EnKF)
Measurement Fusion KF (MFKF)
Batch FilterParticle Filter
Offline (Non-real time)
Online (Real time)
Differential Geometric Filter
No linearization required
Monte Carlo
Estimator Comparison
Convergence Rate
Stability Cost Accuracy
EKF Moderate Moderate-Low Low High
UKF Fast High High High
DGF Very Fast High Moderate Moderate-Low
MFKF Moderate Moderate Low High
Particle Filter Fast Dependent Very High Dependent
EnKF Dependent Moderate-Low Very High High
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Future
Research Objective and
Contribution
1. Implementation of WLPS in spacecraft formation flying:a. Navigation performance study
2. Improves the estimation stability and convergence rate:a. Avoid linearization.
Differential Geometric Filter.
3. Improves the estimation accuracy performancea. Applies a constraint into orbit estimation.
b. Integrate the constraint with Kalman Filter Constrained Kalman Filter
4. Propose a relative attitude determination method for spacecraft formation
flying.
5. Lower the estimation computational complexitya. Fuse all weighted WLPS measurement into one.
b. Apply weighted on each WLPS measurement.
Weighted Measurement Fusion Kalman Filter.
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WLPS for Spacecraft Formation
Flying
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WLPS
Extended
Kalman Filter
Absolute
Position
Extended Kalman Filter Implementation
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Model
Gain
Kalman
Filter
Update
Propagate
Gwuxtfx ),,(vxhy )(
),,(ˆ uxtfx
))ˆ(~(ˆˆ xhyKxx
1)( RHHPHPK TT
PKHIP )(
TT GQGPFFPP
Scenario One
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R, TOA
R, TOA
ϕ
Case OneCase Two
Two-spacecraft Formation
Measure:
Range and angles
Estimate:
Absolute Position
DBS
TRX
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Spacecraft Formation Orbit Estimation using
WLPS-based Localization”, International Journal of Navigation and Observation, vol. 2011, Article ID 654057,
12 pages, 2011. doi:10.1155/2011/654057
2 DOA’s RMSE than 1 DOA’s RMSE.
Computational cost consideration 1 DOA case.
RMSE Performance comparison
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Scenario Two
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1r2r
4r 3r
Performance comparison:
GPS only vs GPS+WLPS
Cases:
Number of spacecraft
Formation size
GPS satellites
Case OneCase Two
WLPS improves accuracy.
Number of spacecraft in formation estimation accuracy improves.
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Spacecraft Formation Orbit Estimation using
WLPS-based Localization”, International Journal of Navigation and Observation, vol. 2011, Article ID 654057,
12 pages, 2011. doi:10.1155/2011/654057
Performance Comparison: GPS vs WLPS+GPS
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Impact of Formation Size
Formation Size Setup Ave. RMSE (m)
100km/200km GPS/WLPS
GPS
1.068
2.114
700km/1400km GPS/WLPS
GPS
1.214
2.087
1445km/2450km GPS/WLPS
GPS
1.384
2.042
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Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Spacecraft Formation Orbit Estimation using
WLPS-based Localization”, International Journal of Navigation and Observation, vol. 2011, Article ID 654057,
12 pages, 2011. doi:10.1155/2011/654057
Formation size estimation accuracy when WLPS presents.
Summary Implement WLPS into Spacecraft Formation Navigation.
Feasibility study on the Navigation with only WLPS
We can estimate the spacecraft position with one TOA and either One DOA or Two DOA measurements.
The WLPS improves estimation accuracy
More spacecraft in the formation
Smaller formation size
Published Papers
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Spacecraft Constellation Orbit Estimation via a Novel Wireless Positioning System”, 19TH AAS/AIAA Space Flight Mechanics Meeting, Savannah, Georgia, 2009.
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Spacecraft Formation Orbit Estimation using WLPS-based Localization”, International Journal of Navigation and Observation, vol. 2011, Article ID 654057, 12 pages, 2011. doi:10.1155/2011/654057
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Differential Geometry and
Estimation
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In real life, dynamic model and measurement model are non-linear.
)(xhy
x
y
Czy ),( uxsz
To implement DGF methods,
),( uygy
Nonlinear domain
to linear domain
Transformation
)(xsz )(1 zsx
Mapping and reverse mappingIf additional states that not measured are required in the systems:
Pseudo-measurement
Pseudo-errorWLPS, relative position
Absolute Position
Additional required parameters
Example
Contribution: DGF implementation
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DGF equation of motion:
),( uyBfAzz measurement
If absolute position and relative position measured:B
A
C
y
If only relative position measured:
A
B
??
C
We measure relative position
We estimate absolute position
Transformation: is relative position and velocity.z
12r
13r
14r 1r
rij = relative position between ith spacecraft
and jth spacecraft
Inverse transformation?
If all spacecraft have same absolute distance to earth center.
A and B are linear Matrices
Cases study
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SC 1
SC 2
1r2r
4r 3r
SC 4
SC 3
Scenario 1:
Only Relative Position
Four spacecraft formation
Transformation to relative
position
Scenario 2:
Radar measurement + WLPS
Two spacecraft formation
Both Scenarios
Gaussian Noise
No signal transmission delay
Scenario OneScenario Two
SC 1
SC 2
Scenario One - WLPS only
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Formatioin Size DGF Mean
RMSE
EKF Mean
RMSE
Short ( ~0.25 km) 4.447 103 km 2.657 10-4 km
Medium (~ 60 km) 16.59 km 4.153 10-4 km
Long (~ 1200 km) 0.901 km 7.616 10-3 km
Inverse transformation (linear to nonlinear domain) impacts accuracy performance.
Noise to signal ratio inverse transformation error
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Implementation of Differential Geometric Filter
for Spacecraft Formation Orbit Estimation”, International Journal of Aerospace Engineering, (Accepted).
EKF’s estimation accuracy higher but stability is not guaranteed.
DGF guarantees estimation stability.
DGF has faster convergence rate.
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Implementation of Differential Geometric Filter
for Spacecraft Formation Orbit Estimation”, International Journal of Aerospace Engineering, (Accepted).
Scenario Two - WLPS+Radar
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Summary
Implementation of DGF in spacecraft navigation.
Transformation of nonlinear domain to linear domain.
Absolute position to Relative position, and relative position to absolute position
No linearization required in estimation.
Stability study:
DGF has better stability
Convergence study:
DGF converges faster
Published Papers
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Differential Geometric Estimation for spacecraft formations orbits via a cooperative wireless positioning”, IEEE 2010 Aerospace Conference, Big Sky, MT, 2010.
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Implementation of Differential Geometric Filter for Spacecraft Formation Orbit Estimation”, International Journal of Aerospace Engineering, (Accepted).
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Problem Motivation
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Problem – how to know when spacecraft arrives at apogee and perigee?
Three cases:
1. Circular orbit – constraint always apply.
2. Assume we know when spacecraft arrives at apogee and perigee
3. Assume we are required to estimate the time required by spacecraft to
arrives at apogee and perigee.
For any curve:
First order derivative at maxima, minima are equal to zero
Maxima = Apogee position
Minima = Perigee position
Constrained Kaman Filter
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Initialization
Update estimated states
Predict position at
next time step
Apply the constraints
Measurement
from sensors
If spacecraft arrives at
perigee/apogee position
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Issues:
Covariance convergence faster than
estimation error
Truth error out of predicted error
boundary
Constrained Kaman Filter
Solution:
Introduce alpha and beta
parameters
Reduce convergence rate of
covariance at each constraint updates
Error boundary
Truth Error
Derivation
Cases studies
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SC 1
SC 2
1r2r
4r 3r
SC 4
SC 3
Measure:
• Relative Position
Estimate:
•Absolute Position
Circular Orbit
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CKF estimation accuracy within a certain range of alpha and beta.
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Constraint Estimation of
Spacecraft Positions”, Journal of Guidance, Control, and Dynamics, (Accepted).
EKF Error
CKF ErrorPERF =
Divergence occurs
Known perigee/apogee time
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CKF estimation accuracy within a certain range of alpha and beta.
Improvement guaranteed when beta < 0.8.
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Constraint Estimation of
Spacecraft Positions”, Journal of Guidance, Control, and Dynamics, (Accepted).
EKF Error
CKF ErrorPERF =
Unknown apogee/perigee time
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CKF estimation accuracy when beta < 0.7.
Alpha has less impact on the estimation accuracy improvement.
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Constraint Estimation of
Spacecraft Positions”, Journal of Guidance, Control, and Dynamics, (Accepted).
EKF Error
CKF ErrorPERF =
Summary Constrained Kalman Filter based on apogee and perigee condition is implemented.
Introduce alpha and beta parameters in CKF to avoid discontinuity in covariance
Discontinuity results estimation error diverged.
Three cases are studied:
Circular Orbit
Known perigee/apogee time
Unknown apogee/perigee time
The impact of alpha and beta
Estimation accuracy improve if alpha and beta fall within specific range
Published Paper:
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Constraint Estimation of Spacecraft Positions”, Journal of Guidance, Control, and Dynamics, (Accepted).
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Motivation
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What is the orientation of each spacecraft?Does the spacecraft points toward the desired direction?
Orientation – Attitude Matrix
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Spacecraft/Aircraft’s orientation can be specified in three angles (Euler angle):
1. Row 1st rotation angle
2. Pitch 2nd rotation angle
3. Yaw 3rd rotation angle
Three angles Attitude Matrix
Relative Attitude Determination
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2
2
F
DA1
1
F
DA
Spacecraft 1
Spacecraft 2
Spacecraft 3
φ
θ
φ
cos3
2/3
3
1/3 D
DD
D
DD pp
cos2
2/3
2
2
1
2
1
1/3
1
1 D
DD
F
D
F
F
D
DD
F
D pASpA
1132232233 cossin)(cos)( cbcbcbcbcb
Note: when φ is zero => parallel case.
Out of plane
angle
TF
D
F
D
D
D ASAA 1
1
2
2
2
1
Spacecraft 1Spacecraft 2
Spacecraft 3
Two solutions if φ not zero.
Covariance Analysis
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1ˆˆ FxxxxEPT
xxxxJxx
EF
ˆ,)(
Covariance (expected error boundary)To ensure the determination error stay within expected error when
measurement noise exists.
Fisher Information Matrix
Loss function
Requirement:
Non-singular/
Always invertible
Derivation
Case studies
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S/C1
S/C2
S/C3
φ
θ
Case One:
φ is zero
Case Two:
φ is non zero
Relative Attitude Determination Errorφ is zero
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Shu Ting Goh, Chris Passerello and Ossama Abdelkhalik, “Spacecraft Relative Attitude Determination”, IEEE
2010 Aerospace Conference, Big Sky, MT, 2010.
Errors fall within the three sigma boundaries.
Accuracy of the proposed method always within expected error region.
Two solutions:
True solution
Error within expected error boundary
The other solution
Error out of expected error boundary
Relative Attitude Determination ErrorNon-zero φ
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Summary Relative attitude determination method in spacecraft formation:
Non-parallel case Two unique solutions are always obtained
Covariance study: Parallel case
Determination error falls within expected error boundary
Non-Parallel case True solution’s error fall within expected error boundary
Another solution always out of expect error boundary
Published Paper:
Shu Ting Goh, Chris Passerello and Ossama Abdelkhalik, “Spacecraft Relative Attitude Determination”, IEEE 2010 Aerospace Conference, Big Sky, MT, 2010.
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Motivation: GPS Free Localization
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Beacon, TRX 1 Beacon, TRX 2 Beacon, TRX 4
AWACS, TRX 3
UAV
with
DBS
Each measurement received
at different time.
Apply Kalman Filter at each
measurement reception
High computational
cost
Our contributions:
Weighted Measurement Fusion Kalman Filter
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Fused all measurements
Apply Kalman Filter
Reduce computational costEstimation Update
Estimation Update
Based on DBS TRX distance
Last measurement received UAV’s current position weight
First measurement received UAV’s position at t seconds ago weight
Detail
Case Studies
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Target locationDeparture location
GPS satellites
• Scenario Two:
– GPS and WLPS
• Only WLPS measurements
are fused
• Scenario One:
– WLPS only
Weighted Measurement Fusion Kalman Filter Kalman Filter
The accuracy performance different between WMFKF and EKF is not significant.
The WMFKF estimation error falls within the three sigma boundary
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “A Weighted Measurement
Fusion Kalman Filter Implementation for UAV Navigation”, Aerospace Science and Technology.
(under review)
Scenario One -WLPS only
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Weighted Measurement Fusion Kalman Filter
WMFKF has a better estimation accuracy.
WMFKF estimation error falls within the three sigma boundary
Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “A Weighted Measurement Fusion Kalman
Filter Implementation for UAV Navigation”, Aerospace Science and Technology. (under review)52
Kalman Filter
Scenario Two - WLPS and GPS
Computational Comparison
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For N = 3:
• WMFKF requires 1050 no. of multiplication.
• EKF requires 2700 no. of multiplication.
For N = 8:
• WMFKF requires 1165 no. of multiplication.
• EKF requires 190800 no. of multiplication.
N = no. of TRX.
m = no. of measurement, 3.
n = no. of states, 6.
Summary Proposed a Weighted Measurement Fusion Kalman
Filter method.
Compared to the standard Kalman Filter: Better accuracy performance when GPS presents.
Estimation error falls within three sigma boundary.
Requires Less multiplication computation.
Paper (under review): Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “A Weighted
Measurement Fusion Kalman Filter Implementation for UAV Navigation”, Aerospace Science and Technology.
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Contributions
1. Implement WLPS into spacecraft formation flying:a. Spacecraft formation navigation using only WLPS measurements
b. Integrate WLPS and GPS in spacecraft formation
Improves the navigation performance.
c. Study the impact of the following cases on navigation performance:
Number of spacecraft in formation
Formation size.
2. Implement DGF in SFF navigation:a. Nonlinear to linear domain transformation
b. Avoid linearization – guarantee stability.
c. Faster convergence rate.
3. Develop a constraint estimation method into Kalman Filter process:a. Apply constraint estimation at perigee/apogee position.
b. Introduce alpha and beta parameters to reduce covariance convergence rate
c. Accuracy performance improves for specific alpha and beta
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4. Propose a relative attitude determination method:
a. For both parallel and non-parallel cases.
Two solution always obtained for non-parallel case.
b. Perform covariance analysis for both cases.
c. Determination error fall within expected error boundary.
5. Develop a Weighted Measurement Fusion Kalman Filter:
a. Fuse all WLPS measurements.
b. Lower computational cost.
c. Estimation error within expected error boundary.
d. Better accuracy performance.
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Contributions
PublicationsJournals:
1. Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Spacecraft Formation Orbit Estimation using WLPS-
based Localization”, International Journal of Navigation and Observation, vol. 2011, Article ID 654057, 12 pages, 2011.
doi:10.1155/2011/654057
2. Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Constraint Estimation of Spacecraft Positions”, Journal of
Guidance, Control, and Dynamics, (Accepted).
3. Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Implementation of Differential Geometric Filter for
Spacecraft Formation Orbit Estimation”, International Journal of Aerospace Engineering, (Accepted).
4. Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “A Weighted Measurement Fusion Kalman Filter
Implementation for UAV Navigation”, Aerospace Science and Technology, (Under Review).
Conference Papers:
1. Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Spacecraft Constellation Orbit Estimation via a Novel
Wireless Positioning System”, 19TH AAS/AIAA Space Flight Mechanics Meeting, Savannah, Georgia, 2009.
2. Shu Ting Goh, Ossama Abdelkhalik and Seyed A. (Reza) Zekavat, “Differential Geometric Estimation for spacecraft
formations orbits via a cooperative wireless positioning”, IEEE 2010 Aerospace Conference, Big Sky, MT, 2010.
3. Shu Ting Goh, Chris Passerello and Ossama Abdelkhalik, “Spacecraft Relative Attitude Determination”, IEEE 2010 Aerospace
Conference, Big Sky, MT, 2010.
4. Shu Ting Goh, Seyed A. (Reza) Zekavat and Ossama Abdelkhalik, “Space-Based Wireless Solar Power transfer via a network
of LEO satellites: Doppler Effect Analysis”, IEEE 2012 Aerospace Conference, Big Sky, MT, 2012 (In preparation to submit
final draft).
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Thank you
Question?
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Differential Geometric FilterTransformation Example
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Measure: ,,r Polar coordinates
Estimate: zyx rrr ,, Cartesian coordinates
),,( zyxr rrrhr
),,( zyx rrrh
),,( zyx rrrh
NonlinearLinearization
3
3
32
2
2
)(
xx
hx
x
hx
x
hy
xhy
First order Taylor series expansion
x
y
transform tox z ,,r
Czy
100
010
001
C
Constrained Kaman Filter
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Model
Gain
Kalman
Filter
Update
Constraint
Update
Propagate
Gwuxtfx ),,(vxhy )(
),,(ˆ uxtfx
))ˆ(~(ˆˆ xhyKxx
))ˆ((ˆ xdCLxx
1)( RHHPHPK TT
PKHIP )(
PLDIP )(
TT GQGPFFPP
))ˆ((ˆ xdCLxx PLDIP )(
Covariance Analysis - Parallel
62
12
1ˆˆ FxxxxEP
TD
D
xxxxJxx
EF
ˆ,)(
CovarianceTo ensure the determination error within expected error when
measurement noise exist.
Fisher Information Matrix
Loss function
D11
D12
D13
D21
D22
D23
If the relative
orientation, A,
is known…
Covariance Analysis - Parallel
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13
2
123
1
313
2
123
12
2
122
1
212
2
12211
2
121
1
111
2
121
2
1
2
1
2
1
DADRDAD
DADRDADDADRDADJ
T
TT
Measurement error covariance.
TTTD
D DARDADARDADARDAF
13
2
1
1
313
2
112
2
1
1
212
2
111
2
1
1
111
2
1
2
1
12
1
2
1:Note
D
D
D
D FP
Covariance Analysis – Non Parallel
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S/C1
S/C2
S/C3
φ
θ
Loss Function: cos3
2/3
3
1/3 D
DD
D
DD pp
1
1/2
2
1
2
1/2
D
DD
D
D
D
DD pAp
12
1/2
12
1/2
2
1/2
2
1/3
2
2/3
12
2/3
2
1/3
2
1
TD
DD
D
DD
D
DD
TD
DD
TD
DD
D
DD
D
DD
D
D pRpppRppP
Weighted Measurement Fusion
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y1 y2 y3y4
dt3
dt2
dt1
dt4= 0
time
Measurement received
ii dt1
4
1
2
2
i
i
iiw
Fuse all measurement
4
1i
iii rywy
ri = position between ith TRX
and a specific reference point