The document discusses spacecraft formation flying and orbit estimation techniques. Key points include: - Wireless localization positioning systems (WLPS) can be used to estimate the absolute positions of spacecraft in a formation by measuring their relative positions and angles. - Implementing a differential geometric filter (DGF) for orbit estimation improves stability and convergence compared to an extended Kalman filter by avoiding linearization. - A constrained Kalman filter can be used to estimate when spacecraft reach apogee and perigee in their elliptical orbits by applying constraints at points where the first derivative is zero.

1. Shu Ting Goh
Advisor(s): Ossama Abdelkhalik,
Seyed A. (Reza) Zekavat
1
Mechanical Engineering – Engineering Mechanics Department

2. Spacecraft Formation Flying
 Multiple spacecraft…
 Follow each other
 Fly in a formation
 Fly through specific trajectory
2
Gravity Recovery and Interior Laboratory (GRAIL)
Mission Elapse – 93 DaysLISA Pathfinder

3. 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

4. Why Formation Flying?
4
• Cost
• Robustness
• Resolution, accuracy,
precision
VS

5. 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.
5

6. Navigation sensors for Formation
Flying
Position
Wireless ranging
with antenna array
6
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.

7. 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

8. Wireless Local Positioning
System (WLPS)
8
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

9. Spacecraft Navigation
9
DBS
TRX
R, TOA
, DOA
R, TOA
, DOA
Initial Guess
Estimator/
Filter
Updated
position and
velocity
To ground
station

10. Estimation Method
10
Kalman Filter
Extended Kalman Filter (EKF)
Smoothing Kalman Filter (SKF)
Unscented Kalman Filter (UKF)
Ensemble Kalman Filter (EnKF)
Measurement Fusion KF (MFKF)
Batch Filter
Particle Filter
Offline (Non-real time)
Online (Real time)
Differential Geometric Filter
No linearization required
Monte Carlo

11. 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
11
Future

12. 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 performance
a. 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 complexity
a. Fuse all weighted WLPS measurement into one.
b. Apply weighted on each WLPS measurement.
 Weighted Measurement Fusion Kalman Filter.
12

13. 13

14. WLPS for Spacecraft Formation
Flying
14
WLPS
Extended
Kalman Filter
Absolute
Position

15. Extended Kalman Filter Implementation
15
Model
Gain
Kalman
Filter
Update
Propagate
Gwuxtfx  ),,(
vxhy  )(
),,(ˆ uxtfx 

))ˆ(~(ˆˆ 
 xhyKxx
1
)( 
 RHHPHPK TT

 PKHIP )(
TT
GQGPFFPP 

16. Scenario One
16
R, TOA

R, TOA

ϕ
Case OneCase Two
Two-spacecraft Formation
Measure:
Range and angles
Estimate:
Absolute Position
DBS
TRX

17. 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
17

18. Scenario Two
18
1r
2r

4r
 3r

Performance comparison:
 GPS only vs GPS+WLPS
Cases:
 Number of spacecraft
 Formation size
GPS satellites
Case OneCase Two

19. 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
19

20. 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
20
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.

21. 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
21

23. Differential Geometry and
Estimation
23
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

24. Contribution: DGF implementation
24
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

25. Cases study
25
SC 1
SC 2
1r
2r

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

26. Scenario One - WLPS only
26
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).

27.  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
27

28. 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).
28

29. 29

30. Problem Motivation
30
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

31. Constrained Kaman Filter
31
Initialization
Update estimated states
Predict position at
next time step
Apply the constraints
Measurement
from sensors
If spacecraft arrives at
perigee/apogee position

32. 32
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

33. Cases studies
33
SC 1
SC 2
1r
2r

4r
 3r

SC 4
SC 3
Measure:
• Relative Position
Estimate:
• Absolute Position

34. Circular Orbit
34
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 Error
PERF =
Divergence occurs

35. Known perigee/apogee time
35
 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 Error
PERF =

36. Unknown apogee/perigee time
36
 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 Error
PERF =

37. 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).
37

38. 38

39. Motivation
39
What is the orientation of each spacecraft?
Does the spacecraft points toward the desired direction?

40. Orientation – Attitude Matrix
40
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

41. Relative Attitude Determination
41
2
2
F
DA
1
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 1
Spacecraft 2
Spacecraft 3
Two solutions if φ not zero.

42. Covariance Analysis
42
    1
ˆˆ 
 FxxxxEP
T
xxxxJ
xx
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

43. Case studies
43
S/C1
S/C2
S/C3
φ
θ
Case One:
φ is zero
Case Two:
φ is non zero

44. Relative Attitude Determination Error
φ is zero
44
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.

45. Two solutions:
 True solution
 Error within expected error boundary
 The other solution
 Error out of expected error boundary
Relative Attitude Determination Error
Non-zero φ
45

46. 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.
46

47. 47

48. Motivation: GPS Free Localization
48
Beacon, TRX 1 Beacon, TRX 2 Beacon, TRX 4
AWACS, TRX 3
UAV
with
DBS

49.  Each measurement received
at different time.
 Apply Kalman Filter at each
measurement reception
 High computational
cost
Our contributions:
Weighted Measurement Fusion Kalman Filter
49
 Fused all measurements
 Apply Kalman Filter
 Reduce computational cost
Estimation 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

50. Case Studies
50
Target location
Departure location
GPS satellites
• Scenario Two:
– GPS and WLPS
• Only WLPS measurements
are fused
• Scenario One:
– WLPS only

51. 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
51

52. 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

53. Computational Comparison
53
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.

54. 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.
54

55. 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
55

56. 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.
56
Contributions

57. Publications
Journals:
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).
57

58. Thank you
Question?
58

59. 59

60. Differential Geometric Filter
Transformation Example
60
Measure: ,,r Polar coordinates
Estimate: zyx rrr ,, Cartesian coordinates
),,( zyxr rrrhr 
),,( zyx rrrh 
),,( zyx rrrh 
Nonlinear
Linearization
    

3
3
3
2
2
2
)(
x
x
h
x
x
h
x
x
h
y
xhy






First order Taylor series
expansion
x
y
transform tox z ,,r
Czy 











100
010
001
C

61. Constrained Kaman Filter
61
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 )( 

62. Covariance Analysis - Parallel
62
    12
1
ˆˆ 
 FxxxxEP
TD
D
xxxxJ
xx
EF 









 ˆ,)(
Covariance
To 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…

63. Covariance Analysis - Parallel
63
      
   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

64. Covariance Analysis – Non Parallel
64
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


           1
2
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



65. Weighted Measurement Fusion
65
y1 y2 y3
y4
dt3
dt2
dt1
dt4= 0
time
Measurement received
ii dt1
 
 
 4
1
2
2
i
i
i
iw


Fuse all measurement
 

4
1i
iii rywy
ri = position between ith TRX
and a specific reference point