presentation for the 21th eurostar users conference - june 2013

Observability Analysis and Collision Mitigation with

OR.A.SI© – (Orbit and Attitude Simulator)

Antonios Arkas Flight Dynamics Engineer

1. Condition Number, Observability

and

Orbit Determination Error Variance


OR.A.SI© (Orbit and Attitude Simulator)

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 εrrr

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

- Observation parameter deviation vector

xr

- Solve-for parameters deviation vector

m - Number of observations

- ith observation error

( )oi

i

i ttX

GH ,Φ⋅

∂∂

=

( ) iiii tXGY εrrr

+= , Nonlinear observation-state expression and ( )oi tt ,Φ the transition matrix

iyr

iεr

1.1 Quality of Orbit Determination



1.2 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 grosely 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 εκ

δ⋅≤ ( )

p

p

p

p

p

H

HH

x

x δκ

δ⋅≤

∑=

=n

i

ixx1

2

2

2

2

12 max

x

AxA

x=

=

1.2 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 NoiseNumerical 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.



1.2 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) = σ2i.

� 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.



1.2 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



1.3 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-σ 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 κ2



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

Tra

ck 1

-σ d

ue

to M

easu

rem

ent

No

ise

[m] Along Track Standard Deviation

log10

(κ2) (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.

1.4 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



1.4 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

-σ d

ue

to M

easu

rem

ent

No

ise

[m]

Actual loss of

significance

log10

(κ2) (Loss of Significant Digits)


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

-σ d

ue

to M

easu

rem

ent

No

ise

[m]

Condition of Scaled Measurement Matrix κ2


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



1.5 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



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

� Linearized observation-state relationship for consider covariance method

1,...mi =++= icxi cHxHyii

εrr

cr

- Consider parameter deviation vector

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

and C the a priori covariance matrix of the consider parameter.

( )( )( )TT

x

T

cc

T

x

cWPHCHHWPHPP +=Consider Covariance Matrix

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.



1.6 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

Rad

ial 1

-σ [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

Tra

ck 1

-σ [m

]


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

-σ [m

]


Sigma DN

Convergence of Cross Track 1-σ



2. Close Approach Detection



� Utilization as a standard propagator with either impulsive or continuous maneuver

execution.

� Choice between three different integrators:

� 4th Order Runge-Kutta-Fehelberg RKF4(5) adaptive step size.

� 8th Order Runge-Kutta Dormant-Prince 853 adaptive step size.

� mth Order Adams-Moulton fixed step size.

� Ephemeris and orbital plots for each propagated spacecraft.

� Inter-satellite ephemeris and orbital plots for each pair of propagated spacecrafts.

� Inter-satellite calculations for collocated spacecrafts.

� Inter-satellite distance evolution.

� Eccentricity and inclination separation evolution.

� Evolution of angle between eccentricity-inclination vectors.

� Evolution of geocentric angle between spacecrafts.

� Preliminary detection of close approach (accuracy depended on ephemeris step size)

� Detection of multiple consecutive close approach encounters .

� State vector of each spacecraft on close approach.

� Relative position of the spacecrafts on close approach.

2.1 Multiple Satellite Propagation and Close Approach Module Characteristics



2.2 Multiple Satellite Propagation and Close Approach Module Interface



Raduga 9 Incident: Detection of Intrusion Dates with Sub-Satellite Longitude Graphs



Raduga 9 Incident: Detection of Consecutive Close Encounters with Inter-satellite Distance Graphs



Raduga 9 Incident : Close Approach Details for 10 sec detection step





Raduga 9 Incident: Long Term Prediction of Next Close Approach

3. Collision Mitigation for

High Relative Velocity Encounters



All figure were borrowed from AIAA Paper 05-308 COLLISION AVOIDANCE MANEUVER PLANNING TOOL

by SALVATORE ALFANO

3.1 Geometry and Mathematics of High Relative Velocity Close Encounters (1/2)

Assumptions

� Small encounter time to

ensure constancy of the

individual covariance

matrices and the resulting

combined covariance

matrix.

� High relative velocity to

allow the reduction of the

3D integral to a 2D one.



3.1 Geometry and Mathematics of High Relative Velocity Close Encounters (2/2)

� Maximum probability Pmax corresponds

to xm=0 and a specific minor semi axis of

combined covariance ellipse.

� The probability dilution region is that

region where the standard deviation of

the combined covariance minor semi

axis σx exceeds that which yield Pmax.

If operating within the dilution region,

then the further into this region the

uncertainty progresses the more

unreasonable it becomes to associate low

probability with low risk.

Two-dimensional probability equation in the encounter plane: • OBJ - Combined object radius.

• σx - Projected covariance ellipse

minor semi axis.

• σy - Projected covariance ellipse

major semi axis.

• (xm ,ym) - Projection of miss distance

on covariance frame.



3.2 Collision Mitigation Module Characteristics

� Accurate calculation of close approach characteristics (depended on detection step size):

� Miss distance on TCA (Time of Closest Approach).

� Relative position and velocity of secondary object with respect to primary object

on TCA.

� State vector details of both primary and secondary objects on TCA.

� Collision probability assessment.

� Collision probability based on combined covariance.

� Maximum collision probability for unfavorable orientation and size of the

combined covariance ellipsoid.

� Calculation of probability dilution region.

� Characteristics of the combined error ellipsoid and its projection on the conjunction

plane (combined covariance ellipse)

� Design and optimality testing of along track (East-West) avoidance maneuvers for a

desired range of DVs.

� Collision probability following the execution of each avoidance maneuver .

� Miss distance following the execution of each avoidance maneuver

� Longitude window violation details corresponding to desired avoidance maneuver.

� Monte Carlo simulation of spacecraft collision with or without avoidance maneuver.



3.3 Collision Mitigation Module Interface



3.4 Validation of Miss Distance and Relative Position -Velocity Calculations



Combined Covariance Ellipsoid Characteristics



Combined Covariance Ellipse Characteristics



3.5 Why Probability Calculations are Much Safer than Miss Distance for

Collision Mitigation Decision Making (1/2)

Close Approach Scenario for Miss Distance 1.166 Km and P = 5E-08

Miss Distance – Relative Position – Relative Velocity – Collision Probability



3.5 Why Probability Calculations are Much Safer than Miss Distance for

Collision Mitigation Decision Making (2/2)

Dependence of collision probability on relative position of spacecraft line of sight, on TCA, with respect to combined covariance ellipse



3.6 Validation for Collision Probability P = 0.0164 in the Dilution Region (1/2)

Close Approach Scenario




0 20000 40000 60000 80000 100000 120000

0,010

0,012

0,014

0,016

0,018

0,020

Coll

isio

n P

robab

ilit

y


Collision Probability

Monte Carlo Convergence of Collision Probability

• Theoretical : 0.01645

• Monte Carlo : 0.01639 (121846 iterations)

3.6 Validation for Collision Probability P = 0.0164 in the Dilution Region (2/2)



What is the latest time with respect to

TCA when a moderate (~ 0.04 m/s)

avoidance maneuver is able to

substantially decrease the collision

probability ?



3.7 Avoidance Maneuver Calculation (1/2)

Close Approach Scenario for P = 1.925E-02 in Probability Dilution Region




Miss Distance versus Along Track DV and Centroid



Collision Probability versus Along Track DV and Centroid



3.8 Validation of Collision Mitigation with an Avoidance Maneuver

Collision Probability without Avoidance Maneuver : 1.93E-02

• Theoretical : 1.95E-06 (Probability Dilution)

• Monte Carlo : 2.5E-06 (2 million iterations)

Collision Probability with a 0.04 m/s maneuver 2h prior to TCA



How much does the collision probability

depend on state covariance matrix

norm ?

or else

How strong does the collision probability

depend on observability ?

or else

Can a specific choice of orbit

determination setup reduce the

collision probability ?



3.9 Collision Probability and Observability

Collision probability can be substantially decreased by selecting the orbit

determination setup characterized from the best observability with respect to all

the possible allowable setups.

Best case primary

object’s state

covariance

Worst case primary

object’s state

covariance

Best case state covariance (Best Observability)

Worst case state covariance (Worst Observability)



Best case state

covariance

Dependence of Combined Covariance Ellipsoid Size on Individual Covariance Ellipsoids

Miss Distance : 1.166 Km

Worst case state

covariance



4. Run-Time for

Close Approach Calculations





4.1 Characteristic Run-Time for Close Approach Calculations

Machine Used for OR.A.SI© Execution : Laptop with Intel i7 Processor and 2GB RAM

Preliminary Detection of Close Approach with Multiple Satellite Propagation

and Close Approach Module

• Detection period : 20 days

• Ephemeris Time Step : 1 min

• Propagator : 8th order Adams-Moulton

Run-Time: 22 sec

Accurate Detection of Close Approach with Collision Mitigation Module

• Detection Time Step : 0.01 sec

• Propagator : 8th order Adams-MoultonRun-Time: 6 sec

Avoidance Maneuver Calculation with Collision Mitigation Module

• Detection Time Step : 0.01 sec

• Number of Different Maneuver DV’s : 10

• Number of Centroids for Each DV : 20

• Total Avoidance Maneuver Number : 200

• Propagator : 8th order Adams-Moulton

Run-Time: 3 min 33sec

5. Future Plans for Further

Investigations & Development



5.1 Future Plans for Further Investigations & Development (1/2)

� Theoretical investigation on the feasibility of incorporating consider parameter

uncertainties in the condition number of the scaled measurement matrix to reflect

their impact on observability.

� Development of code for calculation of collision probability for nonlinear low velocity

relative motion, appropriate for encounters of collocated GEO spacecrafts.

� Increase of the time span and accuracy of preliminary detection for close approach

events with utilization of adaptive detection step size.

� Incorporation of cross track (North-South) avoidance maneuver calculation in

collision mitigation module.

� More extensive validation of collision probability module computations with Monte

Carlo simulations.

� Elaboration on the difference between the actual collision probability and the

theoretical one in cases of probability dilution.

� Automatic collision avoidance maneuver calculation for collocated spacecrafts.



� Addition of maneuver increments as sole-for parameters in orbit determination

module.

� Computation of inclination control maneuvers with the optimized long term control

strategy.

� Description of a realistic model for the atmosphere up to the height of 1000 Km in

order to take account the air drug perturbation for LEO calculations (Jacchia

model).

� Orbit determination based on Kalman filter sequential estimator.

5.1 Future Plans for Further Investigations & Development (2/2)



THANK YOU FOR ATTENDING MY PRESENTATION

presentation for the 21th eurostar users conference - june 2013

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