OR.A.SI © Orbit and Attitude

Simulator

Part I

Antonios Arkas Flight Dynamics Engineer

1.1 Orbit Determination

1.1.1 Orbit Determination Module Characteristics

Utilization of a batch weighted least-square estimator.

Enhancement of orbit determination accuracy with a priori information of the state

vector covariance matrix (Bayesian Estimation).

Radiation pressure coefficient Cp , ballistic coefficient and antenna biases for any

number of the Earth stations participating in the localization campaign, can be set as

solve-for parameters (provided that the problem formulation has good observability).

Raw measurement preprocessor for the reduction of noise.

Process of any kind of combination of tracking (azimuth-elevation) and/or range

measurements acquired from an arbitrary number of Earth stations.

Flexibility to configure the weighted least-square estimator with suitable choice

of the following parameters :

Combination of solve-for parameters.

Two different measurement rejection factors. One for the first number of

iterations and another for the subsequent iterations.

Maximum global WRMS (Weighted Root Mean Square) of residuals for

measurement rejection.

Minimum and maximum number of iterations.

Maximum number of divergent iterations.

1.1.2 Orbit Determination Output

Determination of orbital state, reflectivity coefficient Cp, ballistic coefficient and

antenna biases.

Determined orbit validity criteria based on chi square analysis and estimation of the

numerical stability of the normal equation matrix based on its condition which is

calculated with SVD (Single Value Decomposition) method.

Consider covariance computation of the state, the reflectivity coefficient and the

determined antenna biases based on the knowledge of reflectivity coefficient uncertainty,

the antenna bias uncertainties and the relevant measurement noises. The consider

covariance is computed for the state vector (ECI reference frame), the Keplerian

elements and the state vector referred to the local orbital frame RSW of the determined

state.

Propagation of all three forms (ECI – Keplerian and RSW) of the consider covariance of

the determined state.

Computation of the confidence ellipsoid characteristics (semi axes and orientation given

with Euler angles), corresponding both to the determined and propagated consider

covariance matrices, referred to the local orbital frame RSW of the determined state.

1.1.3 Raw Measurements Pre-Processor Interface (Noise Reduction)

Reduction of effective measurement noise be approximately 1𝑁 where N is the number of measurements

in each session

1.1.4 Raw Measurement Pre-Processor - Raw Measurement Graphs

Graph: Raw elevation measurements for a GEO spacecraft located at 39o East, acquired from an Earth

station at φ = 22.6859o and λ = 38.822o East.

1.1.5 Raw Measurement Pre-Processor - Condensed Measurement Graphs

Graph: Condensed elevation measurements corresponding to the raw measurements of the previous slide.

1.1.6 Orbit Determination Input Interface – Main Window (1/2)

Orbit determination configuration interface. Choice of antennas, type of measurements, solve-for

parameters, measurement edition and convergence criteria, covariance matrix for a priori estimation.

1.1.6 Orbit Determination Input Interface – Antenna Characteristics (1/2)

Selection of

1) antenna for localization measurements acquisition

2) antenna characteristics (noise and bias uncertainties for consider covariance analysis) and

3) antenna biases as solve-for parameters

1.1.7 Orbit Determination Output - Results

Results:

1) determined orbit

2) solution validity criteria based on chi square analysis (statistical distribution of residuals)

3) estimation of the numerical stability of the normal equation matrix based on its condition (SVD Single Value

Decomposition)

4) Noise only and consider covariance matrix characteristics (shape and orientation)

1.1.8 Orbit Determination Output – Range Residuals

Production of residuals for both processed and unprocessed measurements.

1.1.9 Orbit Determination Output - Confidence Ellipsoid Referred to the RSW Local Frame

1.2 Orbit Determination Validation

Comparison with COSMIC orbit determination. Actual measurements were acquired

from two distant antennas. From THP2 antenna both range and angular

measurements were used while from CYP antenna only range measurements were

used for orbit determination. COSMIC is the Flight Dynamics software for GEO

spacecrafts developed by AIRBUS (Former EADS-ASTRIUM).

1.2.1 Validation of Orbit Determination – Comparison with COSMIC Results (1/2)

Localization Campaign Characteristics

Localization measurements were acquired from two antennas :

Range, azimuth & elevation from Thermopylae station (THP) at Greece.

Range from Kakoratzia station (CYP) at Cyprus.

Number of sessions : 28

Time interval between sessions : 2 hours.

Solve-for parameters :

State vector.

Azimuth and elevation biases for Thermopylae station.

Range bias for Kakoratzia station.

Radiation pressure coefficient Cp.

COSMIC Results

State Vector

a = 42166.0066 Km

ex = -0.000139

ey = -0.000239

ix = 0.000185 rad

iy = -0.000832 rad

true longitude = 38.99017o

Antenna Biases

THP2 Az Bias = 0.01833o

THP2 El Bias = -0.01245o

CYP Rg Bias = -0.082 Km

Cp = 0.967

OR.A.SI Results

State Vector

a = 42166.0076 Km

ex = -0.000139

ey = -0.000240

ix = 0.000187 rad

iy = -0.000832 rad

true longitude = 38.99014o

Antenna Biases

THP2 Az Bias = 0.01824o

THP2 El Bias = -0.01290o

CYP Rg Bias = -0.082 Km

Cp = 0.961

1.2.2 Validation of Orbit Determination – Comparison with COSMIC Results (2/2)

Range Residuals for Both Antennas

1.3 Consider Covariance Analysis

Why needing consider covariance analysis ?

Methods for handling bias errors in an estimation process

1. Estimated : The state vector is expanded to include dynamic and

measurement model parameters that may be in error.

PROBLEM – Too optimistic covariance matrix in the presence of systematic

no estimated errors.

2. Considered : The state vector is estimated but the uncertainty in the non-

estimated parameter is included in the estimation of the state vector and the

error covariance matrix. This assumes that the no estimated parameters are

constant and their a priori estimate and associated covariance matrix is known.

Consider covariance analysis is a technique to

assess the impact of neglecting to estimate some

parameters on the accuracy of the state estimate

1.3.1 Necessity of Consider Covariance Analysis

1.3.2 Formulation of Consider Covariance Method

1,...mi iii xHy

Linearized observation-state relationship for usual least square method

yi – Observation parameter deviation vector

x

- State parameter deviation vector

m – Number of observations

ει - ith observation error

Linearized observation-state relationship for consider covariance method

1,...mi icxi cHxHyii

c

- Consider parameter deviation vector

Let W the weighting matrix, P the covariance matrix of the usual least square method,

C the a priori covariance matrix of the consider parameter and xlsq the least square solution

TT

x

T

cc

T

x

c WPHCHHWPHPP Consider Covariance

Covariance Matrix

Consider Covariance

Solution cHWPHxx c

T

x

lsq

c

oi

i

i ttX

GH ,

iiii tXGY , Nonlinear observation-state expression and oi tt , the transition matrix

1.4 Formal Consider Covariance

Validation

Monte Carlo Orbit Determination:

Simulate range and angular localization measurements with Gaussian noise, from two

distant antennas. Incorporation of Gaussian uncertainty in the antenna biases and

subsequent orbit determination. Comparison of the formal covariance matrix with

the sample covariance matrix computed from the descriptive statistics of the Monte

Carlo orbit determination iterations.

1.4.1 Validation of Formal Consider Covariance with Monte Carlo (1/2)

Validation Scenario - Both Range and Tracking Measurements from Two Antennas

Antenna

Longitude

[deg East]

Latitude

[deg]

Range Noise

(1-σ) [m]

Azimuth Noise

(1-σ) [deg]

Elevation Noise

(1-σ) [deg]

Range Bias Uncertainty

(1-σ) [m]

Azimuth Bias Uncertainty

(1-σ) [deg]

Elevation Bias Uncertainty

(1-σ) [deg]

THP2 22.6859 38.8224 5.0 0.0023 0.00124 6.0 0.0015 0.0015

CYP 33.3843 34.8592 5.0 0.0129 0.0137 5.0 0.002 0.002

• Geosynchronous spacecraft at 390 East.

• Acquisition of range and tracking measurements from both stations.

• 24 sessions with 2h between range measurements and 10 min between range and tracking measurements.

• Solve-for parameters: State vector – Reflectivity Coefficient Cp – Range Bias for THP2 – Azimuth and

Elevation biases for both antennas.

1.4.1 Validation of Formal Consider Covariance with Monte Carlo (2/2)

Solve-for Antenna Bias Standard Deviations

THP2 Antenna CYP Antenna

R- 1σ T - 1σ N - 1σ Range Bias Azimuth Bias Elevation Bias Azimuth Bias Elevation Bias

[m] [deg]

Formal Computation 17.649 486.787 161.495 16.16 0.001108 0.000363 0.002722 0.002611

Monte Carlo 17.869 487.866 162.529 16.17 0.001104 0.000357 0.002743 0.002598

Number of Monte Carlo Iterations: 3000

0 500 1000 1500 2000 2500 3000

14

16

18

20

22

24

26

Radia

l 1-[m

]

Number of Monte Carlo Iterations

Sigma DR

Convergence of Radial 1-σ

0 500 1000 1500 2000 2500 3000

450

500

550

600

650

700

750

800

Alo

ng T

rack 1

-[m

]

Number of Monte Carlo Iterations

Sigma DT

Convergence of Along Track 1-σ

Consider covariance matrix is absolutely necessary not only for observability

analysis but also for the assessment of collision probability.

0 500 1000 1500 2000 2500 3000

150

160

170

180

190

200

Cro

ss T

rack 1

-[m

]

Number of Monte Carlo Iterations

Sigma DN

Convergence of Cross Track 1-σ

1.5 Case Studies of Various Orbit

Determination Setups Based on

Consider Covariance Analysis

Case Studies for Consider Covariance Analysis – GEO Spacecraft at 39o East (1/2)

Antenna Diameter [m] λ [deg] φ [deg] h [m] σρ [m] σΑ,Ε [deg] σΔρ [m] σΔΑ,ΔΕ [deg]

THP 31 22.68 38.82 70 5 0.003 20 0.004

CYP 4,8 33.384 34.85 215 3 0.02 20 0.03

σR [m] σT [m] σN [m] σVr [m/s] σVt [m/s] σVn [m/s] σa [m]

Noise 75,014 292,233 692,370 0,019 0,005 0,053 11,498

Consider 95,839 994,035 767,935 0,071 0,006 0,054 43,400

0,00049

0,00140

Description (Substantial differences between estimated and consider analysis)

Case A1 One day angle and range measurements (1/2h) from THP Station ; bias parameters considered

σλ [deg]

σR [m] σT [m] σN [m] σVr [m/s] σVt [m/s] σVn [m/s] σa [m]

Noise 76,744 403,478 695,919 0,0276 0,0055 0,0533 16,950

Consider 76,745 673,076 695,919 0,0480 0,0055 0,0533 16,950

0,00062

0,00096

Description (Reduction of the impact of the systematic measurement errors )

Case A2 Same as A1 but with tracking biases estimated

σλ [deg]

Determined State Vector Accuracies Referred to the RSW Local Orbital Frame

σR [m] σT [m] σN [m] σVr [m/s] σVt [m/s] σVn [m/s] σa [m]

Noise 449,689 2008,665 4536,527 0,1357 0,0326 0,3473 27,366

Consider 449,769 2464,346 4537,292 0,1709 0,0326 0,3473 27,368

0,00325

0,00378

Description (Unfavorable tracking geometry and reduced tracking performance )

Case A3 Same as A2 but with CYP Station

σλ [deg]

Case Studies for Consider Covariance Analysis – GEO Spacecraft at 39o East (2/2)

σR [m] σT [m] σN [m] σVr [m/s] σVt [m/s] σVn [m/s] σa [m]

Noise 23,827 79,030 238,741 0,0048 0,0017 0,0187 1,579

Consider 28,333 781,862 241,275 0,0574 0,0019 0,0196 11,968

0,00014

0,00106

Description (Increased accuracy due to double ranging from distant stations )

Case A4 One day ranging (1/2h) from THP and CYP station ; Range biases considered

σλ [deg]

σR [m] σT [m] σN [m] σVr [m/s] σVt [m/s] σVn [m/s] σa [m]

Noise 30,289 301,078 242,589 0,0224 0,0019 0,0201 14,499

Consider 30,290 616,193 242,590 0,0451 0,0019 0,0201 14,500

0,00041

0,00084

Description (Same accuracy as Case A2)

Case A5 One day ranging (1/2h) from THP and CYP station ; CYP range bias estimated

σλ [deg]

σR [m] σT [m] σN [m] σVr [m/s] σVt [m/s] σVn [m/s] σa [m]

Noise 151,366 3494,062 584,922 0,2512 0,0068 0,0800 141,562

Consider 151,366 3535,434 584,922 0,2543 0,0068 0,0800 141,562

0,00482

0,00488

Description (Bud observability due to short data arc )

Case A6 12 hours ranging and tracking measurements (1/30m) from THP ; Tracking biases estimated

σλ [deg]

σR [m] σT [m] σN [m] σVr [m/s] σVt [m/s] σVn [m/s] σa [m]

Noise 159,679 961,330 1782,963 0,0614 0,0121 0,0850 17,040

Consider 160,210 1285,681 1782,995 0,0871 0,0121 0,0850 20,890

0,00127

0,00173

Description (Reasonable accuracy with a very short data arc )

Case A7 6 hours ranging (1/30m) from THP and CYP stations ; Range biases considered

σλ [deg]

1.6. Condition Number, Observability

And Relevance With

Orbit Determination Error Variance

What is the most appropriate setup for orbit determination ?

(Setup ↔ Model : Specific combination of measurement types with solve-for parameters)

Criteria for Quality of Orbit Determination

1. Conditioning : Low sensitivity of the model to antenna noise and

bias uncertainties.

2. Observability : Uniqueness of estimation for all parameters of the state vector.

1,...mi iii xHy

Model : Linearized observation-state relationship for usual least square method

- Observation parameter deviation vector

x

- Solve-for parameters deviation vector

m - Number of observations

- ith observation error

oi

i

i ttX

GH ,

iiii tXGY

, Nonlinear observation-state expression and oi tt , the transition matrix

iy

i

1.6.1 Quality of Orbit Determination

1.6.2 Conditioning and Observability (1/2)

Measurement Model Condition Number κp(Η)

Measure of relative sensitivity of the solve-for parameters x to relative errors in measurement

matrix H and measurement noise ε.

Facts Sensitivity to measurement noise is proportional to κ2.

The higher the value of κ2 the more close to singularity is measurement

matrix H (Relevance to Theoretical Observability).

Number of significant digits lost during inversion of H is grossly log10(κ2(H))

(Relevance to Numerical Observability).

For l2-norm : and

si – singular values of H

and p

p

p

p

p

yH

x

x

p

p

p

p

p

H

HH

x

x

n

i

ixx1

2

2

2

2

12 max

x

AxA

x

1.6.3 Conditioning and Observability (2/2)

Theoretical Observability of the Model y = Ηx + ε

Ability to apply the estimator to a particular system (measurement model) and obtain a unique

estimate for all solve-for parameters. This is equivalent to:

The rank of the m x n matrix H must be n (m > n).

H must have n nonzero singular values (Singular Value Decomposition).

The determinant of HTH must be greater than zero.

Measurement Noise Numerical Error in forming

and inverting normal equations

(Numerical Observability)

Sources of error in the estimated state (Solution of the normal equations)

- Solution of normal equations

- Actual state

Numerical observability may be different from the theoretical one.

Any solution with loss of precision log10(κ2) greater than half the total

floating point precision digits (machine ε), should be highly suspect.

1.6.3 Observability Analysis Module Characteristics (1/2)

Module Initialization (Orbit Determination Simulator)

Production of range, and tracking measurements with identically independently

distributed (iid) errors εi for whatever type of orbit and any number of Earth stations in

the relevant coverage.

Gaussian noise distribution εi with desired mean value E(εi) (systematic error) and

variance E(εΤi εi) = σ2

i.

Configuration of different measurement plan for each ground station with suitable

choice of the following parameters :

Type of acquired measurements.

Error variance.

Time offset of the first localization session.

Time offset between sessions.

Time offset between range and tracking measurements.

Number of sessions.

Antenna bias uncertainties for calculation of consider covariance.

Output epoch for propagation of covariance matrices.

Maneuver characteristics (Epoch , DV, relative error) following orbit determination, for

calculation of propagated covariance matrices.

1.6.4 Observability Analysis Module Characteristics (2/2)

κ2 condition number and rank of HTR-1H normal equations matrix.

Warning for ill-conditioned matrices.

Propagated noise only and consider a posteriori covariance matrices of the Cartesian and

Keplerian state vector forms and the model parameters with respect to the ECI (Earth Centered

Inertial) reference frame.

Propagated noise only and consider a posteriori covariance matrices of the Cartesian state

vector form with respect to the local satellite reference frame RTN (R-Radial, T-Along Track, N-

Cross Track).

Confidence ellipsoid characteristics (semi axes lengths and orientation/Euler angles) with respect

to local satellite reference frame.

Module Output

κ2 condition number and rank of the scaled information matrix:

• R= E[εiεiT ] - Matrix of the measurement covariance.

• R-1/2 - Square root of R = R-T/2 R-1/2

• D - State scaling diagonal nxn matrix with elements

the l2 norm of the corresponding column of

R-1/2 H, that is Di = ||(R-1/2 H):i||2

1.6.5 Observability Dependence on Geometry & Orbit Determination Setup

Antenna Longitude [deg East] Latitude [deg] Range Noise (1-σ) [m] Azimuth Noise (1-σ) [deg] Elevation Noise (1-σ) [deg]

THP2 22.6859 38.8224 4 0.002 0.002

CYP 33.3843 34.8592 4 0.002 0.002

• Same type of measurements have the same noise for both antennas.

• Antenna biases for both antennas have no uncertainties.

• 24 Sessions with 2h between range measurements and 10 min between range and tracking measurements.

• Geosynchronous spacecraft at 390 East.

Noise Only Standard Deviations [m] Scaled Measurement Matrix

Case Setup of Measurements Solve-For Parameters R- 1σ T - 1σ N - 1σ Condition Rank

Single Antenna Measurements

1 THP2 (Rg+Az+El) THP2 (Az+El) + Cp 37.3 110.9 348.0 4.81e2 9

2 CYP (Rg+Az+El) CYP (Az+El) + Cp 36.6 704.75 352.4 3.27e3 9

Double Ranging

3 THP2 (Rg) + CYP (Rg) CYP (Rg) + Cp 18.9 80.67 177.1 5.25e2 8

4 THP2 (Rg) + CYP (Rg) THP2 (Rg) + Cp 18.9 265.5 177.1 1.76e3 8

Double Ranging and Tracking from One Antenna

5 THP2 (Rg+Az+El) + CYP (Rg) THP2 (Az+El) +CYP (Rg) + Cp 17.02 79.7 158.0 5.18e2 10

6 THP2 (Rg+Az+El) + CYP (Rg) THP2 (Rg+Az+El) + Cp 17.02 263.8 158.0 1.74e3 10

Range and Tracking from Both Antennas

7 THP2 (Rg+Az+El) + CYP (Rg+Az+El) CYP (Rg) + Cp 15.26 63.1 143.8 4.1e2 8

8 THP2 (Rg+Az+El) + CYP (Rg+Az+El) THP2 (Rg) + Cp 15.9 109.7 143.5 7.14e2 8

9 THP2 (Rg+Az+El) + CYP (Rg+Az+El) CYP (Rg) + THP2 (Az+El) + Cp 15.4 68.0 143.9 4.4e2 10

10 THP2 (Rg+Az+El) + CYP (Rg+Az+El) THP2 (Rg) + CYP (Az+El) + Cp 15.0 149.9 143.6 9.85e2 10

11 THP2 (Rg+Az+El) + CYP (Rg+Az+El) CYP (Rg+Az+El) + THP2 (Az+El) + Cp 15.6 78.4 144.2 5.12e2 12

12 THP2 (Rg+Az+El) + CYP (Rg+Az+El) CYP (Az+El) + THP2 (Rg+Az+El) + Cp 15.6 261.8 144.1 1.73e3 12

Along track standard deviation with respect to the condition number

0 500 1000 1500 2000 2500 3000 3500

0

100

200

300

400

500

600

700

800A

long T

rack

1-d

ue

to M

easu

rem

ent

Nois

e [m

] Along Track Standard Deviation

Polynomial Fit with Second Order Polynomial

Condition of Scaled Measurement Matrix

Along track standard deviation with respect to loss of significant digits

Singular Value Decomposition (SVD) precision based on log10(κ2) underestimates

the actual loss of precision as the matrix approaches singularity.

3 4 5 6

0

100

200

300

400

500

600

700

800

Alo

ng T

rack

1-d

ue

to M

easu

rem

ent

Nois

e [m

] Along Track Standard Deviation

log10

() (Loss of Significant Digits)

Actual loss of significance

SVD Precision

Underestimation

• 24 Sessions with 2h between range measurements and 10 min between range and tracking measurements.

• Geosynchronous spacecraft at 390 East.

1.6.6 Observability, Consider Covariance & Quality of Orbit Determination (1/2)

Antenna

Longitude

[deg East]

Latitude

[deg]

Range Noise

(1-σ) [m]

Azimuth Noise

(1-σ) [deg]

Elevation Noise

(1-σ) [deg]

Range Bias Uncertainty

(1-σ) [m]

Azimuth Bias Uncertainty

(1-σ) [deg]

Elevation Bias Uncertainty

(1-σ) [deg]

THP2 22.6859 38.8224 4.23 0.003 0.0017 11.4 0.0015 0.0015

CYP 33.3843 34.8592 3.4 0.0147 0.0112 5 0.0025 0.0025

Noise Only Standard Deviations [m] Consider Analysis Standard Deviations [m] Scaled Measurement Matrix

Case Setup of Measurements Solve-For Parameters R- 1σ T - 1σ N - 1σ R- 1σ T - 1σ N - 1σ Condition Rank

Single Antenna Measurements

1 THP2 (Rg+Az+El) THP2 (Az+El) + Cp 34.7 110.6 323.6 34.7 327.1 323.6 4.5e2 9

2 CYP (Rg+Az+El) CYP (Az+El) + Cp 198.6 743.8 2021.4 198.5 825.2 2021.5 3.7e3 9

Double Ranging

3 THP2 (Rg) + CYP (Rg) CYP (Rg) + Cp 18.1 81.5 169.4 18.1 318.4 169.4 5.7e2 8

4 THP2 (Rg) + CYP (Rg) THP2 (Rg) + Cp 18.1 268.5 169.4 18.1 449.4 169.4 1.9e3 8

Double Ranging and Tracking from One Antenna

5 THP2 (Rg+Az+El) + CYP (Rg) THP2 (Az+El) +CYP (Rg) + Cp 16.2 80.4 150.1 16.2 318.12 150.1 5.6e2 10

6 THP2 (Rg+Az+El) + CYP (Rg) THP2 (Rg+Az+El) + Cp 16.1 267.5 150.1 16.1 448.9 150.1 1.9e3 10

Range and Tracking from Both Antennas

7 THP2 (Rg+Az+El) + CYP (Rg+Az+El) CYP (Rg) + Cp 16.0 75.3 149.6 17 298.4 150.7 5.3e2 8

8 THP2 (Rg+Az+El) + CYP (Rg+Az+El) THP2 (Rg) + Cp 15.6 182.0 149.2 17.8 453.7 151.44 1.3e3 8

9 THP2 (Rg+Az+El) + CYP (Rg+Az+El) CYP (Rg) + THP2 (Az+El) + Cp 16.1 80.1 149.7 16.1 316.5 149.7 5.6e2 10

10 THP2 (Rg+Az+El) + CYP (Rg+Az+El) THP2 (Rg) + CYP (Az+El) + Cp 15.6 184.35 149.2 17.9 464.7 151.5 1.3e3 10

11 THP2 (Rg+Az+El) + CYP (Rg+Az+El) CYP (Rg+Az+El) + THP2 (Az+El) + Cp 16.1 80.4 149.7 16.1 318.1 149.7 5.7e2 12

12 THP2 (Rg+Az+El) + CYP (Rg+Az+El) CYP (Az+El) + THP2 (Rg+Az+El) + Cp 16.1 267.3 149.7 16.1 448.8 149.7 1.9e3 12

1.6.7 Observability, Consider Covariance & Quality of Orbit Determination (2/2)

Observability primarily depends on the geometry of the Earth stations participating in

the localization campaign and the orbit determination setup.

Since observability is directly connected to the variance of the along track error, the

Flight Dynamics Engineer can detect the best possible orbit determination setup by

comparing the aforementioned variance corresponding to each different setup.

Observability can’t guarantee best orbit determination performance due to the

additional error dispersion introduced by the uncertainty of the consider parameters.

Conclusions

2,6 2,8 3,0 3,2 3,4 3,6 3,8 4,0 4,2 4,4 4,6 4,8 5,0

0

200

400

600

800

Alo

ng

Tra

ck 1

-d

ue

to M

easu

rem

ent

No

ise

[m]

Actual loss of

significance

log10

() (Loss of Significant Digits)

Along Track Standard Deviation

SVD Precision

Underestimation

0 500 1000 1500 2000 2500 3000 3500 4000

0

100

200

300

400

500

600

700

800

Alo

ng

Tra

ck 1

-d

ue

to M

easu

rem

ent

No

ise

[m]

Condition of Scaled Measurement Matrix

Along Track Standard Deviation

Second Order Polynomial Fit

Equation

y = Intercept + B1*x^1 + B2*x^2

Weight No Weighting

Residual Sum of Squares

1641,0144

Adj. R-Square 0,99487

Value Standard Error

Along Track St Intercept 53,69291 11,0751

Along Track St B1 0,03935 0,01504

Along Track St B2 3,97067E-5 3,82244E-6

1.6.8 Difference Between Theoretical and Numerical Observability

Even when the state vector x is theoretically observable from a given set of measurements,

numerical errors may cause observability tests to fail. Conversely, numerical errors may also

allow observability tests to pass when the system is theoretically unobservable.

Solution Flooding From Numerical Errors in an ill-conditioned setup

Scenario : Acquisition of 2 days range and tracking measurements from

single station THP2 and setting all antenna biases as solve-for

parameters along with the state vector.

- Solution of normal equations

- Actual state Numerical Error in forming

and inverting normal equations

(Numerical Observability)

Along Track 1-σ : 25.5 km

Correction of Range Bias : 2 km Radial 1-σ : 34.7 m Cross Track 1-σ : 324 m

Condition Number : 1.3e5

2.1 Propagation Module

2.1.1 Propagation Module Characteristics (1/2)

Three alternative numerical integrators for orbit propagation of perturbed orbits and

two analytic propagators:

Continuous embedded 4th order Runge-Kutta-Fehelberg method RKF4(5), with adaptive

step size control.

Continuous embedded 8th order Runge-Kutta Dormant-Prince method RKF8(7)-13,

with adaptive step size control.

m-th order Adams-Moulton predictor-corrector method with dense output based on m-th

order Lagrange interpolator.

SGP4/SDP4 propagator for TLE elements (Spacetrack Report No.03).

Analytic solution of the restricted two body problem for unperturbed orbits.

Estimation of the numerical propagator global truncation error in accordance to the

characteristics of the propagator and the type of orbit (eccentricity) which is propagated.

Forward and backward propagation of all types of closed orbits while taking account a

series of triaxial impulsive, continuous thrust maneuvers or a mix of these two type of

maneuvers.

The propagator accounts for the following perturbations:

Sun and Moon gravity.

Earth potential according to GEM10B (order and degree of approximation defined by

the user).

Solar radiation pressure.

Air drag (Jaccia71 density model).

Inertial accelerations due to Luni-Solar and planetary precession and nutation.

2.1.1 Propagation Module Characteristics (2/2)

Flexibility to express the dense output production of the propagated orbital states with

respect to the following reference frames:

True of date.

Mean of Date.

Mean of 1950.

Mean of J2000.

Veis.

OEM ephemeris production in accordance to CCSDS.

Propagation of the state covariance matrix for each spacecraft and computation of the

3x3σ separation from their combined covariance matrix.

Production of Antenna Pointing Data:

Topocentric horizon polar (range, azimuth, elevation) and Cartesian coordinates (x,y,z)

with respect to whatever Earth station in the satellite geographical coverage.

Tropospheric range and elevation correction as functions of local temperature, relative

humidity and barometric pressure (Hopfield model for radio frequencies).

Doppler shift calculation.

Multi-satellite environment.

Sequential propagation of a number of spacecrafts chosen from the database and

computation of all intersatellite characteristics (intersatellite distance, angular

separation, minimum separation, intersatellite separation with respect to the local orbital

frame of each spacecraft, meridian separation, relative velocities and close approach

events) for each pair of spacecrafts.

Easy and straightforward addition or deletion of a spacecraft in the relevant database

with automatic creation of all the relevant directory structure.

2.1.2 Propagation Module Interface (1/2)

Scenario: 14 days propagation of two spacecrafts collocated (inclination-eccentricity separation) GEO

spacecrafts at 39o East .The first spacecraft executes an inclination control and a subsequent

drift/eccentricity control maneuvers.

2.1.2 Propagation Module Interface (2/2) – Selection of Spacecraft Characteristics

2.1.2 Propagation Module Single Satellite Graphs (1/7)

Graph: 14 days evolution of the true and mean longitude evolution for spacecraft No.01

2.1.2 Propagation Module Single Satellite Graphs (2/7)

Graph: 14 days evolution of the mean eccentricity vector for spacecraft No.01

2.1.2 Propagation Module Single Satellite Graphs (3/7)

Graph: 14 days evolution of the inclination node vector for spacecraft No.01

2.1.2 Propagation Module Single Satellite Graphs (4/7)

Graph: 14 days evolution of geocentric latitude vs sub-satellite longitude for spacecraft No.01

2.1.2 Propagation Module Single Satellite Graphs (5/7)

Graph: 14 days evolution of the radial uncertainty (RSW local orbital frame) for spacecraft No.01.

Impact of maneuver execution errors on uncertainty evolution.

Impact of orbit determination executed 2 days after each maneuver execution (initialization of

covariance).

2.1.2 Propagation Module Single Satellite Graphs (6/7)

Graph: 14 days evolution of the along track uncertainty (RSW local orbital frame) for spacecraft No.01.

Impact of maneuver execution errors on uncertainty evolution.

Impact of orbit determination executed 2 days after each maneuver execution (initialization of

covariance).

2.1.2 Propagation Module Single Satellite Graphs (7/7)

Graph: 14 days evolution of the cross track uncertainty (RSW local orbital frame) for spacecraft No.01.

Impact of maneuver execution errors on uncertainty evolution.

Impact of orbit determination executed 2 days after each maneuver execution (initialization of

covariance).

2.1.2 Propagation Module Double Satellite Graphs (1/6)

Graph: 14 days evolution of the sub-satellite longitude for both collocated spacecrafts.

2.1.2 Propagation Module Double Satellite Graphs (2/6)

Graph: 14 days evolution of the linear separation between the collocated spacecrafts.

2.1.2 Propagation Module Double Satellite Graphs (3/6)

Graph: 14 days evolution of the along track vs the radial separation for the collocated spacecrafts.

2.1.2 Propagation Module Double Satellite Graphs (4/6)

Graph: 14 days evolution of the cross track vs the radial separation for the collocated spacecrafts.

2.1.2 Propagation Module Double Satellite Graphs (5/6)

Graph: 14 days evolution of the 3D relative orbit of spacecraft No.02 with respect to the local RSW

orbital frame of spacecraft No.02.

2.1.2 Propagation Module Double Satellite Graphs (6/6)

Graph: 14 days evolution of the superposition of the 2D 3x3 sigma relative position error ellipsoid on the

cross track vs radial separation of the two collocated spacecrafts. Verification of collocation safety.

Execution of an impulsive inclination control maneuver (South) followed by a continuous one (North).

Duration of continuous thrust maneuver : 3.47 hours

Maneuver acceleration of continuous thrust maneuver : 80 μm/s2

Spacecraft mass : 2000 Kgr

2.1.3 Mixing Continuous and Impulsive Thrust Maneuvers (1/2)

2.1.3 Mixing Continuous and Impulsive Thrust Maneuvers (2/2)

Graph: Evolution of inclination node vector under the impact of an impulsive inclination control

maneuver followed by a continuous inclination control maneuver.

Orbit decay of a spacecraft with very eccentric orbit and low altitude perigee

Major Semi Axis = 8600 Km

Period = 2.2 h

Eccentricity = 0.24

Perigee Altitude = 157.87 Km

Ballistic Coefficient CdA/m = 0.08 m2 /Kgr

Orbit Lifetime = 25 days

Orbital Decay Due to

Air Drag

Major Semi Axis Decay

Eccentricity Decay

Geocentric Distance Decay

Y versus X Coordinate Referred to the True of Date Reference Frame

Orbital Lifetime Dependence from Solar Flux and Geomagnetic Indices

0 10 20 30 40 50

6000

6500

7000

7500

8000

8500

9000

Maximum Activity

F10.7 = 330 SFU

F10.7 (90 days) = 240 SFU

Kp = 8o

Typical Activity

F10.7 = 200 SFU

F10.7 (90 days) = 155 SFU

Kp = 4o

Majo

r S

em

i A

xis

(km

)

DOY (ddd.ddd)

Minimum Activity

F10.7 = 70 SFU

F10.7 (90 days) = 70 SFU

Kp = 0o

2.2 OR.A.SI Integrator Validation

Type of tests presented:

1. Step size control adaptation for highly eccentric orbits.

2. Forward propagation with 7 station keeping maneuvers for GEO spacecraft.

3. Backward propagation with 8 station keeping maneuvers for GEO spacecraft.

4. Comparison of stability for the three different type of numerical integrators.

1. Comparison of the numerical solution with an analytic one.

2. Comparison with COSMIC.

3. Comparison with COSMIC.

4. Assessment of global integration error by comparison with an analytic solution.

2.2.1 Demonstration of Step Size Control - Comparison with an Analytic Solution

Utilization of a “steep” problem in order to challenge the integrator’s

capability to adapt its step size.

(the problem doesn’t ought to be physically realizable)

Highly eccentric Keplerian (non-perturbed) orbit with the following characteristics :

a = 65127 Km

e = 0.987

i = 0o

perigee radius = 894.45 Km (Earth’s radius = 6378 Km)

apogee radius = 129407.372 Km

maximum orbital velocity = 28.92 Km/sec (Escape velocity : 11 Km/sec)

Step Size Control vs Radius for a High Eccentric Orbit

Relative Accuracy With Respect to the Analytic Solution

2.2.2 Forward Propagation (Comparison with COSMIC)

Utilization of a series of realistic station keeping maneuvers actually executed for Hellas

Sat II between 16-12-05 and 13-02-06 :

All perturbations taken account.

Total of 7 consecutive maneuvers.

4 South maneuvers coupled with 3 East maneuvers.

1) How accurate is the orbit prediction ?

(Validation of perturbation models)

2) How accurate are the antenna pointing data ?

(Validation of tropospheric corrections model )

Forward Propagation - True and Mean Longitude Evolution

Forward Propagation - Inclination Node Vector Evolution

Forward Propagation - Osculating and Mean Major Semi Axis Evolution

Forward Propagation - ey Eccentricity Component Evolution

Elevation Evolution for Earth Station at φ = 22.6859ο and λ = 38.822ο

Azimuth Evolution for Earth Station at φ = 22.6859ο and λ = 38.822ο

Slant Distance Evolution for Earth Station at φ = 22.6859ο and λ = 38.822ο

Doppler Evolution

2.2.3 Backward Propagation (Comparison with COSMIC)

Utilization of a series of realistic station keeping maneuvers actually executed for Hellas Sat II between

28-10-05 and 18-12-05 :

All perturbations taken account.

Total of 8 consecutive maneuvers.

4 South maneuvers coupled with 3 East maneuvers and a West one.

Backward evolution of inertial coordinate differences between COSMIC and OR.A.SI

Backward Propagation with 8 Maneuvers – True Longitude Evolution

Backward Propagation with 8 Maneuvers – Inclination Node Vector Evolution

2.2.4 Numerical Integration Stability Comparison for the Three Different Types of OR.A.SI Integrators

Assessment of stability through the increase of the global integration error computed by direct comparison

with an analytic solution corresponding to a Keplerian (non perturbed) GEO orbit.

Test Scenario: 30 days forward propagation of a GEO orbit defined from the following state

a = 42166 Km e = 4e-4 i = 0.02o Ω = 123ο ω = 34ο λ = 39ο East

57020 57025 57030 57035 57040 57045 57050 57055

0

200

400

600

800

1000

1200

1400

Alo

ng

Tra

ck E

rro

r W

ith

Re

sp

ect

to t

he

An

aly

tic S

olu

tio

n (

m)

MJD (ddddd.ddd)

4th Order Runge-Kutta-Fehelberg RKF4(5)

8th order Runge-Kutta Dormant-Prince RKF8(7)-13

9th Order Adams-Bashforth

8th Order Adams-Bashforth

3.1 Station Keeping

Maneuver Computation

for GEO Spacecrafts

3.1.1 Inclination Control Maneuver Module Interface

Inclination Target Modes: Secular drift compensation – Specific inclination target –

Maximum cycle curation – Colocation initialization.

3.1.2 Inclination Control Maneuver Module Graphs (1/3)

Graph: Evolution of the inclination node vector depicting the impact of the calculated inclination control

maneuver. Maneuver is long term optimized (calculated in accordance to contemporary secular

drift)

3.1.2 Inclination Control Maneuver Module Graphs (2/3)

Graph: Evolution of modulus of the inclination node vector depicting the impact of the calculated

maneuver.

3.1.2 Inclination Control Maneuver Module Graphs (3/3)

Graph: Evolution of the geocentric latitude.

3.1.3 Inclination Control Maneuver Report

3.1.4 Drift/Eccentricity Control Maneuver Module Interface

Functional for every geographical longitude.

Target Modes: Sun Perigee Pointing – Maneuver on Epoch – Maximize drift period – Target mean

longitude at the end of the cycle.

Radial and tangential effects of the upcoming N/S maneuver are taken into account.

3.1.5 Drift/Eccentricity Control Maneuver Module Graphs (1/5)

Graph: Evolution of the mean and true longitude depicting the impact of the calculated drift/eccentricity

control maneuver as well as the impact of the triaxiality of the upcoming inclination control

maneuver.

3.1.6 Drift/Eccentricity Control Maneuver Module Graphs (2/5)

Graph: Evolution of the mean eccentricity vector depicting the impact of the calculated drift/eccentricity

control maneuver as well as the impact of the triaxiality of the upcoming inclination control

maneuver.

3.1.7 Drift/Eccentricity Control Maneuver Module Graphs (3/5)

Graph: Evolution of the osculating eccentricity vector depicting the impact of the calculated

drift/eccentricity control maneuver as well as the impact of the triaxiality of the upcoming

inclination control maneuver.

3.1.8 Drift/Eccentricity Control Maneuver Module Graphs (4/5)

Graph: Evolution of the mean longitude drift depicting the impact of the calculated drift/eccentricity

control maneuver as well as the impact of the triaxiality of the upcoming inclination control

maneuver.

3.1.9 Drift/Eccentricity Control Maneuver Module Graphs (5/5)

Graph: Mean major semi axis vs mean longitude depicting the impact of the calculated drift/eccentricity

control maneuver as well as the impact of the triaxiality of the upcoming inclination control

maneuver.

3.1.10 Drift/Eccentricity Control Maneuver Report

3.2 Maneuver Computation Validation

Validation through the integration of maneuver computation algorithms in OR.A.SI

module of mission analysis for a GEO spacecraft at 39o East.

Mission analysis duration = 10 years

Station keeping window longitude semi dimension = 0.09o

Station keeping window latitude semi dimension = 0.05o

Maximum allowable eccentricity = 4e-4

Eccentricity tolerance = 5e-5

3.2.1 Inclination Control Maneuvers

Inclination node vector evolution Inclination modulus evolution

Latitude vs longitude evolution Latitude evolution

3.2.1 Drift/Eccentricity Control Maneuvers

Mean eccentricity vector evolution Eccentricity modulus evolution

True longitude evolution Ω+ω evolution

4.1 Maneuver Restitution

4.1.1 Maneuver Restitution Interface

Maneuver restitution based on the comparison of the determined orbit prior to maneuver execution

and the one following the maneuver execution.

4.2 Maneuver Restitution Validation

Comparison with actual inclination control maneuver restitution done with focusGEO

4.2.1 Maneuver Restitution Validation – Comparison of Results

OR.A.SI Results

DVr = 0.0334 m/s

DVt = -0.0094 m/s

DVn = -1.279 m/s

focusGEO Results

DVr = 0.0354 m/s

DVt = -0.0095 m/s

DVn = -1.259 m/s

4.2.2 Maneuver Restitution Validation – Actual screenshot of focusGEO

5. Relocation Maneuvers Module

for GEO Spacecrafts

(Only in console GUI. Pending to be implemented in windows GUI)

5.1 Relocation Maneuvers Module Characteristics

Calculation of velocity increments corresponding to drift setting and drift stop

maneuvers each one of which is performed by means of two tangential maneuvers.

Manual setting of first drift setting and last drift stop maneuver epochs i.e.

adjustment of the drift phase duration and control of the relevant propellant

consumption.

Epoch calculation for the second drift setting maneuver and the first drift stop

maneuver.

Automatic or manual setting of mean eccentricity during drift phase.

Plotting of achievable area for mean eccentricity vector during the drift phase.

Flexibility to choose a specific orbit, for a desired epoch following the last drift

stop maneuver, or initialize a station keeping cycle with desired characteristics (if

reachable).

Relocation from 39o East to 42o East and Initialization of a 14 day Station Keeping Cycle

Input

Initial State : a = 42166.0 Km ex = -0.0002 ey = 0 lo = 39o East

State Epoch : 01/03/2010 00:00:00 UTC

Target State : Initialization of a 14 day station keeping cycle at 42o East

Eccentricity of Control : 0.00035 [Centre of eccentricity circle is (0,0)]

Target Epoch : 28/03/2010 00:00:00 UTC

First Drift Setting Maneuver Epoch : 04/03/2010 00:00:00 UTC

Last Drift Stop Maneuver Epoch : 24/03/2010 00:00:00 UTC (20 days of drift)

Output

First Drift Setting Maneuver : DV1 = -0.262844m/s

Second Drift Setting Maneuver : DV2 = -0.109244m/s

Second Drift Setting Maneuver Epoch : 04/03/10 19:51:48 UTC

First Drift Stop Maneuver : DV3 = 0.469886 m/s

First Drift Stop Maneuver Epoch : 23/03/2010 08:29:39 UTC

Last Drift Stop Maneuver : DV4 = 0.87668 m/s

Relocation from 39o East to 42o East – Longitude Evolution

Relocation from 39o East to 42o East

Detail of Stop Drift and Station Keeping Initialization

Relocation from 39o East to 42o East – Eccentricity Vector Evolution

Relocation from 39o East to 42o East

Mean Major Semi Axis versus Mean Longitude