Parallel Algorithm for Track Initiation for Optical Space Surveillance

Transcription

1 Parallel Algorithm for Track Initiation for Optical Space Surveillance 3 rd US-China Technical Interchange on Space Surveillance Beijing Institute of Technology Beijing, China May 2013 Dr. Paul W. Schumacher, Jr. Air Force Research Laboratory Dr. Matthew P. Wilkins Dr. Christopher W. T. Roscoe Applied Defense Solutions 1

2 Roscoe presented. 2

3 Outline Motivation: Need to Build the Satellite Catalog to 100,000+ Complexity of Track Initiation with Optical Data Concept of Exact Admissible Regions for Range and Range Rate Partition of the Space of Elements Concept of Explicit Bounds on Range and Range Rate Range Bounds for Single Observations (Angles Only) Elimination of Range Pairs via Plane Limits Elimination of Range Pairs via Special Lambert Solutions Range Bounds for Single Observations (Angles and Rates) Algorithm Summary Two Examples Summary and Conclusions 3

4 Essential Angles-Based Observation Geometry SATELLITE, time 1 STATION 1, time 1 OBSERVATION, time 1 SATELLITE, time 2 At any time: STATION 2, time 2 OBSERVATION, time 2 For building the catalog, typical data sets may have 10 4 to 10 6 observations! 4

5 Concept of Exact Admissible Regions (Angles and Rates) Given observation and partitions [a MIN, a MAX ] and [ e MIN, e MAX ] The admissible region in the plane is defined by: Energy: Angular Momentum: Laplace vector: h = r r cos I MAX h h k cos I MIN These inequalities are polynomial forms in range and range rate. Polynomials have high degree and coupling between range and range rate. Expressions require angle rates. 5

6 Complexity of Track Initiation with Optical Data (Angles Only) Solve for Range Laplace, Gauss, Gooding, Mortari and Karimi, many others All combinations of observations 3 (or more) at a time For N obs, complexity scales like N 3 (or higher) Hypothesize on Range Lambert All combinations of observations 2 at a time For N obs, complexity scales like N 2 Each obs pair is a family of Lambert problems For M range hypotheses per observation, complexity scales like M 2 The number M can be kept small enough by partitioning the space of element sets of interest. Partitioning the element space parallelizes the computation. 6

7 Explicit Bounds on Range and Range Rate (Angles and Rates) e e MAX MAX e MIN a MIN a MAX a MIN MIN MAX Bounds will be explicit expressions. Bounds will apply to individual observations. Bounds are valid for Keplerian elliptical orbits. Range and range rate bounds can be obtained with angle rate data. Bounds will contain entire exact admissible region, plus some. 7

8 Explicit Bounds on Range (Angles Only) e e MAX MAX e MIN a MIN a MAX a MIN MIN MAX Bounds will be explicit expressions. Bounds will apply to individual observations. Bounds are valid for Keplerian elliptical orbits. Range bounds can be obtained without angle rate data. Bounds will contain entire exact admissible region, plus some. 8

9 All observations sent to each partition Embarrassingly Parallel Lambert Solutions (Example Partition: Semimajor Axis and Eccentricity) Angles-only observations Generate range hypotheses for (a,e) Partition 1 { r vectors } LAMBERT candidate orbits Generate range hypotheses for (a,e) Partition K { r vectors } LAMBERT candidate orbits Each element set partition is a pair of closed intervals: semimajor axis [a MIN, a MAX ] eccentricity [e MIN, e MAX ] 9

10 Bounds on Range Implied by Individual Obs (Angles Only) For every observation, require that the orbital radius lie between the minimum possible perigee and the maximum possible apogee, for each element partition: Perigee case: Apogee case: provided that Otherwise eliminate the observation from consideration. 10

11 Restrictions Implied by the Set of Orbital Planes Additional restrictions on allowable values of range come from the orbital plane defined by the choice of a pair of observations using the orbit normal, n, and, k, the north polar unit vector in ECI: r i = R i + ρ i u i Hence, we require that: Construct Orbit Plane Normal Vector cos I = n k k n k n = cos Ω, sin Ω, 0 T n = s r 1 r 2 r 1 r 2 cos I MAX n k cos I MIN Ω MIN tan 1 sin Ω cos Ω Ω MAX 11

12 Restrictions Implied by Lambert s Theorem ( 1 of 3 ) For each pair of observations, and, we solve the family of Lambert Problems. Special solutions of Lambert s Problem allow us to eliminate some pairs of range hypotheses before we compute the actual solution. Minimum possible semimajor axis: Reject the pair if Minimum possible eccentricity: Reject the pair if 12

13 Restrictions Implied by Lambert s Theorem ( 2 of 3 ) Euler s Theorem: Time of flight on parabolic orbit. Separates elliptic and hyperbolic Lambert solutions. where and Reject the pair if NOTE: only pairs of range hypotheses are eliminated. The individual range hypotheses may still be accepted in other combinations. 13

14 Restrictions Implied by Lambert s Theorem ( 3 of 3 ) Solving Lambert s problem for elliptic orbits requires us to specify the number of complete orbit revolutions. Some fraction of a rev must remain after N rev whole revs. We know that the minimum possible orbit period is fixed by geometry: T 0 = 2π a 0 3 μ Hence the time of flight must satisfy: t 2 t 1 N REV T 0 = 2πN REV a 0 3 μ NOTE: only pairs of range hypotheses are eliminated. The individual range hypotheses may still be accepted in other combinations. 14

15 Range and Range Rate Bounds for Single Observations (Angles and Rates) With angle rate data, we can obtain additional bounds on range and on range rate. Require orbital speed to lie between the minimum possible apogee speed and the maximum possible perigee speed: where The level curves of speed-squared are ellipses concentric with respect to a point in the [range, range rate] plane, defined by the observation and element partition. 15

16 Acceptance Algorithm for Range and Range Rate Hypotheses Step Acceptance Criteria for all i, j i, m Step Acceptance Criteria for all i, j i, m 1 5, 2,,

17 Example Angle & Angle Rate Data Observations from station at origin Admissible region contains only those ρ, ρ pairs which map to the given a, e element partition The hypothesis set is somewhat larger 17

18 Example Angle & Angle Rate Data Observations from station on Earth s surface Very different admissible region in this case Hypothesis set still contains the entire admissible region 18

19 Summary and Conclusions Angles-only track initiation can be made to scale like N 2 M 2, where the required number M of range hypotheses per element partition can be controlled by making the partitions smaller. Angle-angle rate track initiation scales like N M 1 M 2, for M 1 range hypotheses and M 2 range rate hypotheses per element partition. Both approaches can be parallelized with respect to element partitions. Explicit bounds on range and range rate are somewhat conservative (relaxed): this is the price for obtaining explicit formulae. Open issue: Efficiency of track initiation with these bounds depends partly on the sampling strategy used in the intervals of range and range rate. Open question: How accurate and precise do the angle rates have to be? 19