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Comparing Object Correlation Metrics forEffective Space Traffic Management
Marvin Pena Theodore Y. Faust Adam Q. Jaffe Julie Y. Zhang
Institute of Pure and Applied Mathematics, University of California, Los Angeles
Sponsored by The Aerospace Corporation
Industry Mentors: Dr. Daniel P. Lubey, Dr. Andrew J. Binder, Mr. James E. Gidney
Academic Mentor: Minh Pham
August 23, 2018
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 1 / 48
The Aerospace Corporation
Established in 1960 as aCalifornia nonprofit corporation
Operates a federally fundedresearch and development center
Performs objective technicalanalyses and assessments
Specialized laboratories test,analyze, and troubleshootvirtually every aspect of rocketand satellite systems
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 2 / 48
The Aerospace Corporation
Established in 1960 as aCalifornia nonprofit corporation
Operates a federally fundedresearch and development center
Performs objective technicalanalyses and assessments
Specialized laboratories test,analyze, and troubleshootvirtually every aspect of rocketand satellite systems
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 2 / 48
The Aerospace Corporation
Established in 1960 as aCalifornia nonprofit corporation
Operates a federally fundedresearch and development center
Performs objective technicalanalyses and assessments
Specialized laboratories test,analyze, and troubleshootvirtually every aspect of rocketand satellite systems
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 2 / 48
The Aerospace Corporation
Established in 1960 as aCalifornia nonprofit corporation
Operates a federally fundedresearch and development center
Performs objective technicalanalyses and assessments
Specialized laboratories test,analyze, and troubleshootvirtually every aspect of rocketand satellite systems
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 2 / 48
The Aerospace Corporation
Established in 1960 as aCalifornia nonprofit corporation
Operates a federally fundedresearch and development center
Performs objective technicalanalyses and assessments
Specialized laboratories test,analyze, and troubleshootvirtually every aspect of rocketand satellite systems
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 2 / 48
Motivation
Figure 1. Catalogued Objects in Space Surveillance Network
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 3 / 48
Motivation
Figure 2. Collision Probabilities
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 4 / 48
Non-Cooperative Space Traffic Management
Estimate state of objects in orbit without a communication channel
Space-based and ground-based sensors contain measurement error
Consider distribution of possible states
Optimally assign measurements to objects
Comprehensive comparison of likelihood-of-coincidence metrics
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 5 / 48
Non-Cooperative Space Traffic Management
Estimate state of objects in orbit without a communication channel
Space-based and ground-based sensors contain measurement error
Consider distribution of possible states
Optimally assign measurements to objects
Comprehensive comparison of likelihood-of-coincidence metrics
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 5 / 48
Non-Cooperative Space Traffic Management
Estimate state of objects in orbit without a communication channel
Space-based and ground-based sensors contain measurement error
Consider distribution of possible states
Optimally assign measurements to objects
Comprehensive comparison of likelihood-of-coincidence metrics
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 5 / 48
Non-Cooperative Space Traffic Management
Estimate state of objects in orbit without a communication channel
Space-based and ground-based sensors contain measurement error
Consider distribution of possible states
Optimally assign measurements to objects
Comprehensive comparison of likelihood-of-coincidence metrics
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 5 / 48
Non-Cooperative Space Traffic Management
Estimate state of objects in orbit without a communication channel
Space-based and ground-based sensors contain measurement error
Consider distribution of possible states
Optimally assign measurements to objects
Comprehensive comparison of likelihood-of-coincidence metrics
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 5 / 48
Overview
1 Simulation Framework
2 Experiment Design
3 Results
4 Conclusion
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 6 / 48
Simulation Framework
aeropackage
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 7 / 48
True State Propagation
To propagate the true satellite states (positions and velocities) overtime, we numerically solve the following ODE (where r is the positionof the satellite):
r = −µrr3
+ fperturb (1)
fperturb consists of various perturbations that deviate the path of thesatellite from that given in two-body gravity.
Without the fperturb term, models two-body gravity:1 Perfectly spherical earth with symmetric mass distribution2 Only force acting on satellite is Earth’s gravity
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 8 / 48
True State Propagation
To propagate the true satellite states (positions and velocities) overtime, we numerically solve the following ODE (where r is the positionof the satellite):
r = −µrr3
+ fperturb (1)
fperturb consists of various perturbations that deviate the path of thesatellite from that given in two-body gravity.
Without the fperturb term, models two-body gravity:1 Perfectly spherical earth with symmetric mass distribution2 Only force acting on satellite is Earth’s gravity
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 8 / 48
True State Propagation
To propagate the true satellite states (positions and velocities) overtime, we numerically solve the following ODE (where r is the positionof the satellite):
r = −µrr3
+ fperturb (1)
fperturb consists of various perturbations that deviate the path of thesatellite from that given in two-body gravity.
Without the fperturb term, models two-body gravity:1 Perfectly spherical earth with symmetric mass distribution2 Only force acting on satellite is Earth’s gravity
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 8 / 48
True State Propagation
To propagate the true satellite states (positions and velocities) overtime, we numerically solve the following ODE (where r is the positionof the satellite):
r = −µrr3
+ fperturb (1)
fperturb consists of various perturbations that deviate the path of thesatellite from that given in two-body gravity.
Without the fperturb term, models two-body gravity:1 Perfectly spherical earth with symmetric mass distribution
2 Only force acting on satellite is Earth’s gravity
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 8 / 48
True State Propagation
To propagate the true satellite states (positions and velocities) overtime, we numerically solve the following ODE (where r is the positionof the satellite):
r = −µrr3
+ fperturb (1)
fperturb consists of various perturbations that deviate the path of thesatellite from that given in two-body gravity.
Without the fperturb term, models two-body gravity:1 Perfectly spherical earth with symmetric mass distribution2 Only force acting on satellite is Earth’s gravity
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 8 / 48
True State Propagation
In particular,
fperturb = fobl + fdrag + fSRP + f3B
where
fobl is the force due to the oblateness of the Earth
fdrag is the force due to atmospheric dragfSRP is the force due to solar radiation pressure.f3B is the force due to third-body gravity
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 9 / 48
True State Propagation
In particular,
fperturb = fobl + fdrag + fSRP + f3B
where
fobl is the force due to the oblateness of the Earthfdrag is the force due to atmospheric drag
fSRP is the force due to solar radiation pressure.f3B is the force due to third-body gravity
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 9 / 48
True State Propagation
In particular,
fperturb = fobl + fdrag + fSRP + f3B
where
fobl is the force due to the oblateness of the Earthfdrag is the force due to atmospheric dragfSRP is the force due to solar radiation pressure.
f3B is the force due to third-body gravity
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 9 / 48
True State Propagation
In particular,
fperturb = fobl + fdrag + fSRP + f3B
where
fobl is the force due to the oblateness of the Earthfdrag is the force due to atmospheric dragfSRP is the force due to solar radiation pressure.f3B is the force due to third-body gravity
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 9 / 48
Summary of Perturbations
Earth Oblateness: Due to its rotation, the Earth has greater massnear the equator than at the poles.
Atmospheric Drag: The force due to the interaction of atmosphericparticles with the satellite.
Solar radiation pressure (SRP): force caused by the impact, reflection,absorption, and re-emission of photons.
Third-body gravity: The impact on the motion of a satellite by thegravity of other bodies such as the Sun, Moon, or planets.
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 10 / 48
Summary of Perturbations
Earth Oblateness: Due to its rotation, the Earth has greater massnear the equator than at the poles.
Atmospheric Drag: The force due to the interaction of atmosphericparticles with the satellite.
Solar radiation pressure (SRP): force caused by the impact, reflection,absorption, and re-emission of photons.
Third-body gravity: The impact on the motion of a satellite by thegravity of other bodies such as the Sun, Moon, or planets.
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 10 / 48
Summary of Perturbations
Earth Oblateness: Due to its rotation, the Earth has greater massnear the equator than at the poles.
Atmospheric Drag: The force due to the interaction of atmosphericparticles with the satellite.
Solar radiation pressure (SRP): force caused by the impact, reflection,absorption, and re-emission of photons.
Third-body gravity: The impact on the motion of a satellite by thegravity of other bodies such as the Sun, Moon, or planets.
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 10 / 48
Summary of Perturbations
Earth Oblateness: Due to its rotation, the Earth has greater massnear the equator than at the poles.
Atmospheric Drag: The force due to the interaction of atmosphericparticles with the satellite.
Solar radiation pressure (SRP): force caused by the impact, reflection,absorption, and re-emission of photons.
Third-body gravity: The impact on the motion of a satellite by thegravity of other bodies such as the Sun, Moon, or planets.
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 10 / 48
Sensors
Two main types of sensors:
Ground-based: Fixed to specific position (latitude and longitude) onsurface of earthSpace-based: Essentially satellites with sensing capability; movementmodeled by the ODE (1)
Figure 3. Tracks of Sensor Movement
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 11 / 48
Sensors
Two main types of sensors:
Ground-based: Fixed to specific position (latitude and longitude) onsurface of earth
Space-based: Essentially satellites with sensing capability; movementmodeled by the ODE (1)
Figure 3. Tracks of Sensor Movement
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 11 / 48
Sensors
Two main types of sensors:
Ground-based: Fixed to specific position (latitude and longitude) onsurface of earthSpace-based: Essentially satellites with sensing capability; movementmodeled by the ODE (1)
Figure 3. Tracks of Sensor Movement
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 11 / 48
Sensors
Two main types of sensors:
Ground-based: Fixed to specific position (latitude and longitude) onsurface of earthSpace-based: Essentially satellites with sensing capability; movementmodeled by the ODE (1)
Figure 3. Tracks of Sensor Movement
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 11 / 48
Sensors
Additionally, sensors can take different types of measurements of anobject:
Range: The Euclidean distance from the sensor to the satelliteAngle: Represented in azimuth and elevation (also called altitude), theangle (as related to the tangent plane to the Earth at the sensorlocation) of the satellite.
Figure 4. Azimuth/Elevation Diagram
Joshua Cesa (https://commons.wikimedia.org/wiki/File:Azimut altitude.svg), “Azimut altitude”, Text,
https://creativecommons.org/licenses/by/3.0/legalcode
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 12 / 48
Sensors
Additionally, sensors can take different types of measurements of anobject:
Range: The Euclidean distance from the sensor to the satellite
Angle: Represented in azimuth and elevation (also called altitude), theangle (as related to the tangent plane to the Earth at the sensorlocation) of the satellite.
Figure 4. Azimuth/Elevation Diagram
Joshua Cesa (https://commons.wikimedia.org/wiki/File:Azimut altitude.svg), “Azimut altitude”, Text,
https://creativecommons.org/licenses/by/3.0/legalcode
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 12 / 48
Sensors
Additionally, sensors can take different types of measurements of anobject:
Range: The Euclidean distance from the sensor to the satelliteAngle: Represented in azimuth and elevation (also called altitude), theangle (as related to the tangent plane to the Earth at the sensorlocation) of the satellite.
Figure 4. Azimuth/Elevation Diagram
Joshua Cesa (https://commons.wikimedia.org/wiki/File:Azimut altitude.svg), “Azimut altitude”, Text,
https://creativecommons.org/licenses/by/3.0/legalcode
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 12 / 48
Propagating Uncertainty
Due to the error in measurements, we cannot determine the exactstate (position and velocity) of a satellite. Only know the distributionof the state of a satellite
We assume that this state distribution is a multivariate Gaussian:(µ(t0),Σ(t0)) at time t0.
Need to use the dynamics of the system (1) to determine what thedistribution will be at a future time t1 > t0
Use the ODE solver to propagate µ(t0) to time t
Use the State Transition Matrix Φ to propagate Σ(t0) as follows:
Σ(t1) = Φ(t1, t0)Σ(t0)Φ(t1, t0)
T .
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 13 / 48
Propagating Uncertainty
Due to the error in measurements, we cannot determine the exactstate (position and velocity) of a satellite. Only know the distributionof the state of a satellite
We assume that this state distribution is a multivariate Gaussian:(µ(t0),Σ(t0)) at time t0.
Need to use the dynamics of the system (1) to determine what thedistribution will be at a future time t1 > t0
Use the ODE solver to propagate µ(t0) to time t
Use the State Transition Matrix Φ to propagate Σ(t0) as follows:
Σ(t1) = Φ(t1, t0)Σ(t0)Φ(t1, t0)
T .
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 13 / 48
Propagating Uncertainty
Due to the error in measurements, we cannot determine the exactstate (position and velocity) of a satellite. Only know the distributionof the state of a satellite
We assume that this state distribution is a multivariate Gaussian:(µ(t0),Σ(t0)) at time t0.
Need to use the dynamics of the system (1) to determine what thedistribution will be at a future time t1 > t0
Use the ODE solver to propagate µ(t0) to time t
Use the State Transition Matrix Φ to propagate Σ(t0) as follows:
Σ(t1) = Φ(t1, t0)Σ(t0)Φ(t1, t0)
T .
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 13 / 48
Propagating Uncertainty
Due to the error in measurements, we cannot determine the exactstate (position and velocity) of a satellite. Only know the distributionof the state of a satellite
We assume that this state distribution is a multivariate Gaussian:(µ(t0),Σ(t0)) at time t0.
Need to use the dynamics of the system (1) to determine what thedistribution will be at a future time t1 > t0
Use the ODE solver to propagate µ(t0) to time t
Use the State Transition Matrix Φ to propagate Σ(t0) as follows:
Σ(t1) = Φ(t1, t0)Σ(t0)Φ(t1, t0)
T .
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 13 / 48
Propagating Uncertainty
Due to the error in measurements, we cannot determine the exactstate (position and velocity) of a satellite. Only know the distributionof the state of a satellite
We assume that this state distribution is a multivariate Gaussian:(µ(t0),Σ(t0)) at time t0.
Need to use the dynamics of the system (1) to determine what thedistribution will be at a future time t1 > t0
Use the ODE solver to propagate µ(t0) to time t
Use the State Transition Matrix Φ to propagate Σ(t0) as follows:
Σ(t1) = Φ(t1, t0)Σ(t0)Φ(t1, t0)
T .
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 13 / 48
Example of Uncertainty Propagation
Figure 5. Propagation of Mean and Covariance
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 14 / 48
Object Correlation
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 15 / 48
Measurement Space Object Correlation
Figure 6. Object Correlation in Measurement Space
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 16 / 48
Distance Metrics in Measurement Space
Let D1d= N(µ1,Σ1) and D2
d= N(µ2,Σ2), with dimension k.
Mahalanobis:
dM (D1, D2) = (µ2 − µ1)T (Σ1 + Σ2)−1(µ2 − µ1)
Bhattacharyya:
dB(D1, D2) =1
4(µ2 − µ1)T (Σ1 + Σ2)
−1(µ2 − µ1)
+1
2log
(det(Σ1 + Σ2)√det Σ1 det Σ2
)− k
2log 2
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 17 / 48
Distance Metrics in Measurement Space
Let D1d= N(µ1,Σ1) and D2
d= N(µ2,Σ2), with dimension k.
Mahalanobis:
dM (D1, D2) = (µ2 − µ1)T (Σ1 + Σ2)−1(µ2 − µ1)
Bhattacharyya:
dB(D1, D2) =1
4(µ2 − µ1)T (Σ1 + Σ2)
−1(µ2 − µ1)
+1
2log
(det(Σ1 + Σ2)√det Σ1 det Σ2
)− k
2log 2
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 17 / 48
Distance Metrics in Measurement Space
Kullback-Liebler divergence is a nonsymmetric function, so we define twoversions of it.
KL1 :
dKL(D1, D2) =1
2(µ2 − µ1)TΣ−1
2 (µ2 − µ1)
+1
2log
(det Σ2
det Σ1
)+
1
2Tr(Σ−1
2 Σ1)−k
2
KL2 :dKL(D2, D1)
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 18 / 48
Measurement Space Object Correlation
Figure 7. Object Correlation in Measurement Space
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 19 / 48
State Space Object Correlation
Figure 8. Object Correlation in State Space
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 20 / 48
Distance Metrics in State Space
Mahalanobis:
dM (D1, D2) = (µ2 − µ1)T (Σ1 + Σ2 − 2Cov(D1, D2))−1(µ2 − µ1)
Essentially the same as measurement space, but includes a term toaccount for the correlation between the two states
Optimal control distance (OCD):
The effect of a control function u on the system dynamics is
r = f(t, r(t)) + u(t)
The metric is
dOCD((t0, s0), (t1, s1)) = infu∈U
∫ t1
t0
1
2‖u(t)‖2dt
where U is the space of all control functions that take state s0 at timet0 to state s1 at time t1
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 21 / 48
Distance Metrics in State Space
Mahalanobis:
dM (D1, D2) = (µ2 − µ1)T (Σ1 + Σ2 − 2Cov(D1, D2))−1(µ2 − µ1)
Essentially the same as measurement space, but includes a term toaccount for the correlation between the two states
Optimal control distance (OCD):
The effect of a control function u on the system dynamics is
r = f(t, r(t)) + u(t)
The metric is
dOCD((t0, s0), (t1, s1)) = infu∈U
∫ t1
t0
1
2‖u(t)‖2dt
where U is the space of all control functions that take state s0 at timet0 to state s1 at time t1
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 21 / 48
Distance Metrics in State Space
Mahalanobis:
dM (D1, D2) = (µ2 − µ1)T (Σ1 + Σ2 − 2Cov(D1, D2))−1(µ2 − µ1)
Essentially the same as measurement space, but includes a term toaccount for the correlation between the two states
Optimal control distance (OCD):
The effect of a control function u on the system dynamics is
r = f(t, r(t)) + u(t)
The metric is
dOCD((t0, s0), (t1, s1)) = infu∈U
∫ t1
t0
1
2‖u(t)‖2dt
where U is the space of all control functions that take state s0 at timet0 to state s1 at time t1
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 21 / 48
Experiment Design
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 22 / 48
Experiment Design
How do the metrics perform on clusters of satellites?
CLUSTER-OUT: Satellites begin in a cluster at time t0 and disperse asthey are propagated to a later time t1
CLUSTER-IN: Satellites begin dispersed at time t0 and becomeclustered as they are propagated to a later time t1
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 23 / 48
Experiment Design
How do the metrics perform on clusters of satellites?
CLUSTER-OUT: Satellites begin in a cluster at time t0 and disperse asthey are propagated to a later time t1
CLUSTER-IN: Satellites begin dispersed at time t0 and becomeclustered as they are propagated to a later time t1
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 23 / 48
Specifications
10 space-based sensors at equal longitudinal intervals above theequator in geosynchronous orbit
20 ground-based sensors at major cities chosen to have a relativelyeven distribution of latitudes and longitudes
Generate cluster of satellites at time t0 by the following process:
Randomly select cluster mean state µC at a given altitude altCSet the state covariance of the cluster to be
ΣC =
σ2r 0 0 0 0 0
0 σ2r 0 0 0 0
0 0 σ2r 0 0 0
0 0 0 σ2v 0 0
0 0 0 0 σ2v 0
0 0 0 0 0 σ2v
For each i, sample a state deviation ξi ∼ N(~0,ΣC) and set the state ofsatellite i to be ~xi = µC + ξi.
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 24 / 48
Specifications
10 space-based sensors at equal longitudinal intervals above theequator in geosynchronous orbit
20 ground-based sensors at major cities chosen to have a relativelyeven distribution of latitudes and longitudes
Generate cluster of satellites at time t0 by the following process:
Randomly select cluster mean state µC at a given altitude altCSet the state covariance of the cluster to be
ΣC =
σ2r 0 0 0 0 0
0 σ2r 0 0 0 0
0 0 σ2r 0 0 0
0 0 0 σ2v 0 0
0 0 0 0 σ2v 0
0 0 0 0 0 σ2v
For each i, sample a state deviation ξi ∼ N(~0,ΣC) and set the state ofsatellite i to be ~xi = µC + ξi.
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 24 / 48
Specifications
10 space-based sensors at equal longitudinal intervals above theequator in geosynchronous orbit
20 ground-based sensors at major cities chosen to have a relativelyeven distribution of latitudes and longitudes
Generate cluster of satellites at time t0 by the following process:
Randomly select cluster mean state µC at a given altitude altCSet the state covariance of the cluster to be
ΣC =
σ2r 0 0 0 0 0
0 σ2r 0 0 0 0
0 0 σ2r 0 0 0
0 0 0 σ2v 0 0
0 0 0 0 σ2v 0
0 0 0 0 0 σ2v
For each i, sample a state deviation ξi ∼ N(~0,ΣC) and set the state ofsatellite i to be ~xi = µC + ξi.
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 24 / 48
Coarse Parameter Search
Need to determine most impactful parameters
Set to these values when varying:
Observation gap t1 − t0 (seconds): 300, 1800, 3600, 7200Altitude altC (km): 350, 762.5, 1175, 1587.5, 2000Dispersal σ2
r (m2): 106, 108, 1010
Satellite a priori uncertainty σ2ap (m2): 100, 102, 104
Sensor measurement error σ2meas (m2): 100, 102, 104
Set to nominal values when not varying:
Observation Gap: 3600 sAltitude: 750 kmDispersal: 108 m2
Satellite a priori uncertainty: 102 m2
Sensor measurement error: 102 m2
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 25 / 48
Coarse Parameter Search
Need to determine most impactful parameters
Set to these values when varying:
Observation gap t1 − t0 (seconds): 300, 1800, 3600, 7200Altitude altC (km): 350, 762.5, 1175, 1587.5, 2000Dispersal σ2
r (m2): 106, 108, 1010
Satellite a priori uncertainty σ2ap (m2): 100, 102, 104
Sensor measurement error σ2meas (m2): 100, 102, 104
Set to nominal values when not varying:
Observation Gap: 3600 sAltitude: 750 kmDispersal: 108 m2
Satellite a priori uncertainty: 102 m2
Sensor measurement error: 102 m2
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 25 / 48
Coarse Parameter Search
Need to determine most impactful parameters
Set to these values when varying:
Observation gap t1 − t0 (seconds): 300, 1800, 3600, 7200Altitude altC (km): 350, 762.5, 1175, 1587.5, 2000Dispersal σ2
r (m2): 106, 108, 1010
Satellite a priori uncertainty σ2ap (m2): 100, 102, 104
Sensor measurement error σ2meas (m2): 100, 102, 104
Set to nominal values when not varying:
Observation Gap: 3600 sAltitude: 750 kmDispersal: 108 m2
Satellite a priori uncertainty: 102 m2
Sensor measurement error: 102 m2
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 25 / 48
Fine Parameter Analysis
If we pick 2 parameters to vary, we can vary them in higher resolution
Use same nominal values when fixed
Vary across these ranges in 20 equal increments:
Observation gap: [300, 7200] sAltitude: [350, 2000] kmDispersal: [104, 1010] m2 (log scale)Satellite a priori uncertainty: [100, 104] m2 (log scale)Sensor measurement error: [100, 104] m2 (log scale)
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 26 / 48
Fine Parameter Analysis
If we pick 2 parameters to vary, we can vary them in higher resolution
Use same nominal values when fixed
Vary across these ranges in 20 equal increments:
Observation gap: [300, 7200] sAltitude: [350, 2000] kmDispersal: [104, 1010] m2 (log scale)Satellite a priori uncertainty: [100, 104] m2 (log scale)Sensor measurement error: [100, 104] m2 (log scale)
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 26 / 48
Fine Parameter Analysis
If we pick 2 parameters to vary, we can vary them in higher resolution
Use same nominal values when fixed
Vary across these ranges in 20 equal increments:
Observation gap: [300, 7200] sAltitude: [350, 2000] kmDispersal: [104, 1010] m2 (log scale)Satellite a priori uncertainty: [100, 104] m2 (log scale)Sensor measurement error: [100, 104] m2 (log scale)
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 26 / 48
Results
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 27 / 48
CLUSTER-OUT: Determining Impactful Variables
Run 30 trials, with 50 satellites leaving a cluster
For each metric and modality, use Least Absolute Shrinkage andSelection Operator (LASSO) to determine 3 most impactfulparameters
Will run finer pairwise investigation on observation gap, dispersal, andsensor variance
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 28 / 48
CLUSTER-OUT: Determining Impactful Variables
Run 30 trials, with 50 satellites leaving a cluster
For each metric and modality, use Least Absolute Shrinkage andSelection Operator (LASSO) to determine 3 most impactfulparameters
Will run finer pairwise investigation on observation gap, dispersal, andsensor variance
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 28 / 48
CLUSTER-OUT: Determining Impactful Variables
Run 30 trials, with 50 satellites leaving a cluster
For each metric and modality, use Least Absolute Shrinkage andSelection Operator (LASSO) to determine 3 most impactfulparameters
Will run finer pairwise investigation on observation gap, dispersal, andsensor variance
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 28 / 48
CLUSTER-OUT: Grid Plots
Run 30 trials of each test
For each parameter pair, findthe metric with the highestaverage success rate
Conduct a one-sided pairedt-test with α = 0.05 fordetermining whether thewinning metric has asignificantly higher mean thanthe losing metrics
If the null hypothesis is rejectedfor all losing metrics, color thecell according to the metric
Else, color the cell gray since thewin is insignificant
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 29 / 48
CLUSTER-OUT: Grid Plots
Run 30 trials of each test
For each parameter pair, findthe metric with the highestaverage success rate
Conduct a one-sided pairedt-test with α = 0.05 fordetermining whether thewinning metric has asignificantly higher mean thanthe losing metrics
If the null hypothesis is rejectedfor all losing metrics, color thecell according to the metric
Else, color the cell gray since thewin is insignificant
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 29 / 48
CLUSTER-OUT: Grid Plots
Run 30 trials of each test
For each parameter pair, findthe metric with the highestaverage success rate
Conduct a one-sided pairedt-test with α = 0.05 fordetermining whether thewinning metric has asignificantly higher mean thanthe losing metrics
If the null hypothesis is rejectedfor all losing metrics, color thecell according to the metric
Else, color the cell gray since thewin is insignificant
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 29 / 48
CLUSTER-OUT: Grid Plots
Run 30 trials of each test
For each parameter pair, findthe metric with the highestaverage success rate
Conduct a one-sided pairedt-test with α = 0.05 fordetermining whether thewinning metric has asignificantly higher mean thanthe losing metrics
If the null hypothesis is rejectedfor all losing metrics, color thecell according to the metric
Else, color the cell gray since thewin is insignificant
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 29 / 48
CLUSTER-OUT: Results
Figure 9. CLUSTER-OUT Test: Observation Gap vs Cluster Dispersal
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 30 / 48
CLUSTER-OUT: Results
Figure 10. CLUSTER-OUT: Observation Gap vs Sensor Variance
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 31 / 48
CLUSTER-OUT: Results
Figure 11. CLUSTER-OUT: Cluster Dispersal vs Sensor Variance
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 32 / 48
CLUSTER-OUT: Results
Range measurements: Mahalanobis wins for sensor variancelog(σ2meas) < 1.5, and in other regions hard to describe
Angle measurements: KL2 wins for dispersal log(σ2r ) < 6.5
Range and angle measurements: KL2 wins for dispersal log(σ2r ) < 5.5
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 33 / 48
CLUSTER-OUT: Results
Range measurements: Mahalanobis wins for sensor variancelog(σ2meas) < 1.5, and in other regions hard to describe
Angle measurements: KL2 wins for dispersal log(σ2r ) < 6.5
Range and angle measurements: KL2 wins for dispersal log(σ2r ) < 5.5
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 33 / 48
CLUSTER-OUT: Results
Range measurements: Mahalanobis wins for sensor variancelog(σ2meas) < 1.5, and in other regions hard to describe
Angle measurements: KL2 wins for dispersal log(σ2r ) < 6.5
Range and angle measurements: KL2 wins for dispersal log(σ2r ) < 5.5
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 33 / 48
CLUSTER-IN: Determining Impactful Variables
Run 30 trials, with 50 satellites going towards a cluster
For each metric and modality, use LASSO to determine 3 mostimpactful parameters
Dispersal and sensor variance appear in all models
Altitude appears 7 out of the 12 times, with observation gapappearing 5 times
Will run finer pairwise investigation on altitude, dispersal, and sensorvariance
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 34 / 48
CLUSTER-IN: Determining Impactful Variables
Run 30 trials, with 50 satellites going towards a cluster
For each metric and modality, use LASSO to determine 3 mostimpactful parameters
Dispersal and sensor variance appear in all models
Altitude appears 7 out of the 12 times, with observation gapappearing 5 times
Will run finer pairwise investigation on altitude, dispersal, and sensorvariance
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 34 / 48
CLUSTER-IN: Determining Impactful Variables
Run 30 trials, with 50 satellites going towards a cluster
For each metric and modality, use LASSO to determine 3 mostimpactful parameters
Dispersal and sensor variance appear in all models
Altitude appears 7 out of the 12 times, with observation gapappearing 5 times
Will run finer pairwise investigation on altitude, dispersal, and sensorvariance
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 34 / 48
CLUSTER-IN: Determining Impactful Variables
Run 30 trials, with 50 satellites going towards a cluster
For each metric and modality, use LASSO to determine 3 mostimpactful parameters
Dispersal and sensor variance appear in all models
Altitude appears 7 out of the 12 times, with observation gapappearing 5 times
Will run finer pairwise investigation on altitude, dispersal, and sensorvariance
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 34 / 48
CLUSTER-IN: Results
Figure 12. CLUSTER-IN: Altitude vs Cluster Dispersal
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 35 / 48
CLUSTER-IN: Results
Figure 13. CLUSTER-IN: Altitude vs Sensor Variance
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 36 / 48
CLUSTER-IN: Results
Figure 14. CLUSTER-IN: Cluster Dispersal vs Sensor Variance
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 37 / 48
CLUSTER-IN: Results
Range measurements: Mahalanobis wins for dispersal log(σ2r ) > 7, allaltitudes, and sensor variance log(σ2meas) < 2
Angle measurements: KL2 wins for dispersal log(σ2r ) < 7, anyaltitude, and sensor variance log(σ2meas) > 2
Range and angle measurements: KL2 wins for dispersal log(σ2r ) < 6and any altitude
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 38 / 48
CLUSTER-IN: Results
Range measurements: Mahalanobis wins for dispersal log(σ2r ) > 7, allaltitudes, and sensor variance log(σ2meas) < 2
Angle measurements: KL2 wins for dispersal log(σ2r ) < 7, anyaltitude, and sensor variance log(σ2meas) > 2
Range and angle measurements: KL2 wins for dispersal log(σ2r ) < 6and any altitude
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 38 / 48
CLUSTER-IN: Results
Range measurements: Mahalanobis wins for dispersal log(σ2r ) > 7, allaltitudes, and sensor variance log(σ2meas) < 2
Angle measurements: KL2 wins for dispersal log(σ2r ) < 7, anyaltitude, and sensor variance log(σ2meas) > 2
Range and angle measurements: KL2 wins for dispersal log(σ2r ) < 6and any altitude
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 38 / 48
OPTIMAL-CONTROL-DISTANCE
Same experiment setup as CLUSTER-IN, but only with < 5 satellites
Perform object correlation in state space
Consider only Mahalanobis and OCD
OCD always achieves 100% accuracy, while Mahalanobis achievesvaried results across trials
Due to time constraints, it is difficult to examine the behavior ofOCD for larger clusters
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 39 / 48
OPTIMAL-CONTROL-DISTANCE
Same experiment setup as CLUSTER-IN, but only with < 5 satellites
Perform object correlation in state space
Consider only Mahalanobis and OCD
OCD always achieves 100% accuracy, while Mahalanobis achievesvaried results across trials
Due to time constraints, it is difficult to examine the behavior ofOCD for larger clusters
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 39 / 48
SIMULATING-LEO: Experiment Specifications
A more realistic space environment: 1000 satellites
Generate 30 clusters of 25 satellites
15 clusters as in CLUSTER-OUT
15 clusters as in CLUSTER-IN
Generate 250 single satellites with randomly selected elliptical orbits
Run 30 trials of this test, averaging the success rates across trials foreach metric and modality
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 40 / 48
SIMULATING-LEO: Results
Figure 15. SIMULATING-LEO
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 41 / 48
SIMULATING-LEO: Results
Success rates for range are lower than success rates for angle andrange-and-angle by ≈ 1%
Mahalanobis wins for all modalities, but ties with Bhattacharyya forangle
Bhattacharyya performs surprisingly well, given its absence in previousfigures
For range KL1 beats KL2 slightly, while KL2 beats KL1 by a widermargin for angle and range-and-angle
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 42 / 48
SIMULATING-LEO: Results
Success rates for range are lower than success rates for angle andrange-and-angle by ≈ 1%
Mahalanobis wins for all modalities, but ties with Bhattacharyya forangle
Bhattacharyya performs surprisingly well, given its absence in previousfigures
For range KL1 beats KL2 slightly, while KL2 beats KL1 by a widermargin for angle and range-and-angle
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 42 / 48
SIMULATING-LEO: Results
Success rates for range are lower than success rates for angle andrange-and-angle by ≈ 1%
Mahalanobis wins for all modalities, but ties with Bhattacharyya forangle
Bhattacharyya performs surprisingly well, given its absence in previousfigures
For range KL1 beats KL2 slightly, while KL2 beats KL1 by a widermargin for angle and range-and-angle
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 42 / 48
SIMULATING-LEO: Results
Success rates for range are lower than success rates for angle andrange-and-angle by ≈ 1%
Mahalanobis wins for all modalities, but ties with Bhattacharyya forangle
Bhattacharyya performs surprisingly well, given its absence in previousfigures
For range KL1 beats KL2 slightly, while KL2 beats KL1 by a widermargin for angle and range-and-angle
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 42 / 48
Conclusion
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 43 / 48
General Results
CLUSTER-IN and CLUSTER-OUT:
Mahalanobis is the best for range measurements
KL2 is the best for angle and range-and-angle measurements
OPTIMAL-CONTROL-DISTANCE: OCD performs very well, but isextremely slow computationally
SIMULATING-LEO: Mahalanobis is consistently the best metric
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 44 / 48
General Results
CLUSTER-IN and CLUSTER-OUT:
Mahalanobis is the best for range measurements
KL2 is the best for angle and range-and-angle measurements
OPTIMAL-CONTROL-DISTANCE: OCD performs very well, but isextremely slow computationally
SIMULATING-LEO: Mahalanobis is consistently the best metric
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 44 / 48
General Results
CLUSTER-IN and CLUSTER-OUT:
Mahalanobis is the best for range measurements
KL2 is the best for angle and range-and-angle measurements
OPTIMAL-CONTROL-DISTANCE: OCD performs very well, but isextremely slow computationally
SIMULATING-LEO: Mahalanobis is consistently the best metric
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 44 / 48
Deliverables
Simulation framework
Dynamical model
Satellite and sensor classes
Distance metrics and object correlation
Experiment scripts and data processing scripts
Raw and processed data
Final results
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 45 / 48
Deliverables
Simulation framework
Dynamical model
Satellite and sensor classes
Distance metrics and object correlation
Experiment scripts and data processing scripts
Raw and processed data
Final results
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 45 / 48
Deliverables
Simulation framework
Dynamical model
Satellite and sensor classes
Distance metrics and object correlation
Experiment scripts and data processing scripts
Raw and processed data
Final results
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 45 / 48
Deliverables
Simulation framework
Dynamical model
Satellite and sensor classes
Distance metrics and object correlation
Experiment scripts and data processing scripts
Raw and processed data
Final results
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 45 / 48
Future Work
Efficiently implementing optimal control distance
More realistic dynamical system
Range and angle rate sensors
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 46 / 48
Acknowledgements
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 47 / 48
Questions?
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 48 / 48
References
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 48 / 48
Figures
Figure 1. Catalogued Objects in Space Surveillance Network. G. Peterson,M. Sorge, and W. Ailor, Space Traffic Management in the Age ofNew Space, tech. rep., The Aerospace Corporation, April 2018.
Figure 2. Collision Probabilities. G. Peterson, M. Sorge, and W. Ailor,Space Traffic Management in the Age of New Space, tech. rep.,The Aerospace Corporation, April 2018.
Figure 5 Joshua Cesa(https://commons.wikimedia.org/wiki/File:Azimut altitude.svg),“Azimut altitude”, Text,https://creativecommons.org/licenses/by/3.0/legalcode
Figures 3, 4, and 6-15 were generated by our own simulation framework,using Plot.ly and R.
Team Aerospace (IPAM) Object Correlation for Effective STM August 23, 2018 48 / 48
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