Initial Orbit Determination Using Stereoscopic Imaging and Gaussian Mixture Models

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1 Initial Orbit Determination Using Stereoscopic Imaging and Gaussian Mixture Models Keith A. LeGrand K. DeMars, H.J. Pernicka Department of Mechanical and Aerospace Engineering Missouri University of Science and Technology

2 Presentation Overview Introduction Goals and Motivation Stereo Vision Basic Concept The Algorithm Measurements Gaussian Mixtures Linkage Recursive Estimation Results Further Applications Keith A. LeGrand SSC13-VIII-6 2 / 20

Goals and Motivation Introduction Goals and Motivation Small sat close proximity operations Rendezvous RSO characterization Small sat hardware Inexpensive, COTS Low-power Passive (uncooperative)

3 Goals and Motivation Introduction Goals and Motivation Small sat close proximity operations Rendezvous RSO characterization Small sat hardware Inexpensive, COTS Low-power Passive (uncooperative) Small satellites in circumnaviagtion formation. Unobservability The relative state of a nearby spacecraft is unobservable when using only line-of-sight (LOS) measurements from a single camera. Keith A. LeGrand SSC13-VIII-6 3 / 20

4 y Introduction Stereo Vision Space-based Stereo Sensing Unobservability can be rectified by the addition of a second camera Second LOS originating at a known baseline provides depth information of nearby RSO Typical deterministic stereo triangulation schemes result in nonzero mean Gaussian noise x Stereo lines-of-sight. Overall Goal Use stereo LOS information from typical small satellite sensors to perform high-accuracy initial orbital determination of a nearby RSO. Keith A. LeGrand SSC13-VIII-6 4 / 20

Introduction Basic Concept Basic Concept Description t2 800 x x 600 z6[m]6(cross-track) 400 200 t1 0-200 -400-600 2000-800 0 60 100 0 200 300 400 500-2000 600-4000

5 Introduction Basic Concept Basic Concept Description t2 800 x x 600 z6[m]6(cross-track) t y6[m]6(along-track) x6[m]6(radial) (b) Multiple paths postulated. 50 y x (c) Initial pdf formed. (a) Stereo images collected. Keith A. LeGrand SSC13-VIII-6 5 / 20

6 The Algorithm Algorithm Block Diagram Keith A. LeGrand SSC13-VIII-6 6 / 20

7 Measurements The Algorithm Measurements Keith A. LeGrand SSC13-VIII-6 7 / 20

8 The Algorithm Measurements Stereoscopic Imaging Depth information can be extracted from two or more cameras Second camera provides bounds on possible RSO position ρ max ɛ l i ρ l i ρ r i z y ρ min σ θ ɛ r i θ 1 θ 2 COP l b f COP r (d) Stereo geometry. x b Camera 1 Camera 2 (e) LOS bounding. Stereoscopic cameras. Keith A. LeGrand SSC13-VIII-6 8 / 20

9 Gaussian Mixtures The Algorithm Gaussian Mixtures Keith A. LeGrand SSC13-VIII-6 9 / 20

10 Gaussian Mixtures The Algorithm Gaussian Mixtures ρ max Mixing several Gaussian subcomponents enables estimation of uniform uncertainty p(ρ 1 ) = 1 ρ max ρ min, ρ min ρ 1 ρ max 0, otherwise ρ min Gaussian mixture approximating uniform uncertainty [ρ min, ρ max ]. Keith A. LeGrand SSC13-VIII-6 10 / 20

11 Linkage The Algorithm Linkage Keith A. LeGrand SSC13-VIII-6 11 / 20

12 The Algorithm Linkage Linkage: Connecting the dots Need to construct full 6 DOF state from position measurements Relative Lambert solver finds s/c velocity given position measurements at t 1 and t 2 Solver connects the dots between each Gaussian (means, weights, and covariance) θ 1,t1 t 1 Keith A. LeGrand SSC13-VIII-6 12 / 20

13 The Algorithm Linkage Linkage: How it works 1. Gaussians generated for t 1 t 1 θ 1,t1 Keith A. LeGrand SSC13-VIII-6 12 / 20

14 The Algorithm Linkage Linkage: How it works 1. Gaussians generated for t 1 2. Gaussians generated for t 2 t 2 t 1 θ 1,t2 θ 1,t1 Keith A. LeGrand SSC13-VIII-6 12 / 20

15 The Algorithm Linkage Linkage: How it works 1. Gaussians generated for t 1 2. Gaussians generated for t 2 3. Every component l linked to every component j t 2 t 1 r(t) = Φ rrr t1 + Φ rvv t1 v(t) = Φ vrr t1 + Φ vvv t1 v t1 = (Φ rv) 1 (r t2 Φ rrr t1 ) θ 1,t2 θ 1,t1 Keith A. LeGrand SSC13-VIII-6 12 / 20

16 The Algorithm Linkage Linkage: How it works 1. Gaussians generated for t 1 2. Gaussians generated for t 2 3. Every component l linked to every component j t 2 t 1 r(t) = Φ rrr t1 + Φ rvv t1 v(t) = Φ vrr t1 + Φ vvv t1 v t1 = (Φ rv) 1 (r t2 Φ rrr t1 ) θ 1,t2 θ 1,t1 Keith A. LeGrand SSC13-VIII-6 12 / 20

17 The Algorithm Linkage Linkage: How it works 1. Gaussians generated for t 1 2. Gaussians generated for t 2 3. Every component l linked to every component j t 2 t 1 r(t) = Φ rrr t1 + Φ rvv t1 v(t) = Φ vrr t1 + Φ vvv t1 v t1 = (Φ rv) 1 (r t2 Φ rrr t1 ) θ 1,t2 θ 1,t1 Keith A. LeGrand SSC13-VIII-6 12 / 20

18 Recursive Estimation The Algorithm Recursive Estimation Keith A. LeGrand SSC13-VIII-6 13 / 20

19 Recursive Estimation Predictor The Algorithm Recursive Estimation p(x k Y k 1 ) = L l=1 w l,k p g(x k ; m l,k, P l,k ) Each Gaussian component is propagated with linear dynamics Weights are held constant over prediction step Corrector p(x k Y k ) = L l=1 w + l,k p g(x k ; m + l,k, P + l,k ) Corrector is formulated as a Bayesian filter Weights are updated based on a priori agreement to incoming data Means/covariances are updated using nonlinear measurements, just like in an unscented Kalman filter Keith A. LeGrand SSC13-VIII-6 14 / 20

20 Initial pdf Results (f) Position pdf approximation at t 1. (g) Velocity pdf approximation at t 1. Initial position and velocity pdfs. Keith A. LeGrand SSC13-VIII-6 15 / 20

21 Evolution of the position pdf over subsequent measurements. Keith A. LeGrand SSC13-VIII-6 16 / 20 pdf Evolution Results (h) Initial position pdf. (i) Position pdf at measurement 50. (j) Position pdf at measurement 100. (k) Position pdf at measurement 500.

22 Results Sensitivity to Baseline 2 [m] Separation 4 [m] Separation 8 [m] Separation Avg8Position8Error8RSS8[m] Avg3Velocity3Error3RSS3[m/s] Measurement8Number Sensitivity of camera baseline to average position RSS Measurement3Number Sensitivity of camera baseline to average velocity RSS. Keith A. LeGrand SSC13-VIII-6 17 / 20

23 Further Applications Potential Applications Rendezvous/docking Refueling/servicing Advanced camera configurations Sat/sat formation Sat/ground-based optics Constellation - RSO cataloging Geolocation Cameras on separate spacecraft. Keith A. LeGrand SSC13-VIII-6 18 / 20

24 Future Work Future Work Integrate higher fidelity dynamic model and relative Lambert solver Include attitude dynamics Investigate alternative camera configurations Profile algorithm on realistic small sat processors Keith A. LeGrand SSC13-VIII-6 19 / 20

25 Acknowledgments Acknowledgments Frank J. Redd Student Scholarship Program Research Advisors Dr. Kyle DeMars Dr. Henry Pernicka Missouri S&T Space Systems Engineering Lab Missouri S&T AREUS Lab AFRL University Nanosat Program Keith A. LeGrand SSC13-VIII-6 20 / 20

Initial Relative Orbit Determination Using Stereoscopic Imaging and Gaussian Mixture Models

SSC3-VIII-6 Initial Relative Orbit Determination Using Stereoscopic Imaging and Gaussian Mixture Models Keith LeGrand Missouri University of Science and Technology 4 W. 3th St., Rolla, MO; (573) 34-78