Spacecraft Formation Flying Navigation via a Novel Wireless Final

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

2

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?

4

• 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.

5

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.

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)

8

Dynamic Base Station

(DBS)R, TOA

, DOA

Transponder (TRX)

* WLPS lab, Director: Reza Zekavat, [email protected],

http://www.ece.mtu.edu/ee/faculty/rezaz/wlps/index.html

Spacecraft Navigation

9

DBS

TRX

R, TOA

, DOA

R, TOA

, DOA

Initial Guess

Estimator/Filter

Updated position and velocity

To ground station

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 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

11

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.

12

13

WLPS for Spacecraft Formation

Flying

14

WLPS

Extended

Kalman Filter

Absolute

Position

Extended Kalman Filter Implementation

15

Model

Gain

Kalman

Filter

Update

Propagate

Gwuxtfx ),,(vxhy )(

),,(ˆ uxtfx

))ˆ(~(ˆˆ xhyKxx

1)( RHHPHPK TT

PKHIP )(

TT GQGPFFPP

Scenario One

16

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

17

Scenario Two

18

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.



12 pages, 2011. doi:10.1155/2011/654057

Performance Comparison: GPS vs WLPS+GPS

19

Impact of Formation Size

Formation Size Setup Ave. RMSE (m)

100km/200km GPS/WLPS

GPS

1.068

2.114


GPS

1.214

2.087


GPS

1.384

2.042

20



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

21

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

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

Cases study

25

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

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).

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

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

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

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

Issues:

Covariance convergence faster than

estimation error

Truth error out of predicted error

boundary


Solution:

Introduce alpha and beta

parameters

Reduce convergence rate of

covariance at each constraint updates

Error boundary

Truth Error

Derivation

Cases studies

33

SC 1

SC 2

1r2r

4r 3r

SC 4

SC 3

Measure:

• Relative Position

Estimate:

•Absolute Position

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 ErrorPERF =

Divergence occurs

Known perigee/apogee time

35

CKF estimation accuracy within a certain range of alpha and beta.

Improvement guaranteed when beta < 0.8.



EKF Error

CKF ErrorPERF =

Unknown apogee/perigee time

36

CKF estimation accuracy when beta < 0.7.

Alpha has less impact on the estimation accuracy improvement.



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).

37

38

Motivation

39

What is the orientation of each spacecraft?Does the spacecraft points toward the desired direction?

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

Relative Attitude Determination

41

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

42

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

43

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

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.

Two solutions:

True solution

Error within expected error boundary

The other solution

Error out of expected error boundary

Relative Attitude Determination ErrorNon-zero φ

45

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

Motivation: GPS Free Localization

48

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

49

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

50

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

51

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

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.

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

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

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

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).

57

Thank you

Question?

58

59

Differential Geometric FilterTransformation Example

60

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


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 )(

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

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

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

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

65

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

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Spacecraft Formation Flying Navigation via a Novel Wireless Final

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