Asteroid Rendezvous Uncertainty Propagation Marco Balducci Brandon - PowerPoint PPT Presentation

Asteroid Rendezvous Uncertainty Propagation Marco Balducci ∗ Brandon Jones † ∗ University of Colorado Boulder † The University of Texas at Austin ICATT PRESENTATION MARCH, 2016

Background Motivation Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 2 of 37

Background | Motivation Rendezvous With Asteroid Quantify the Uncertainty of a Rendezvous • Determine Mission Success • Seek to Quantify or Reduce Risks and Costs • Uncertainty Quantification Can Lead to Robust Optimization • Sensitivity of States With Respect to Inputs Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 3 of 37

Background | Motivation State and Uncertainty Estimation Non-linear Propagation • Long propagation times • Large initial uncertainty • Tend to yield non-Gaussian posterior PDFs • Reacquire an object • Desire for non-intrusive approach ◦ legacy software Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 4 of 37

Background | Uncertainty Quantification Traditional Astrodynamic UQ Established Techniques • Monte Carlo • Linearization and the state transition matrix (STM) ◦ Astrodynamics Community • Unscented Transform (UT) ◦ Switching Over These Methods Have Drawbacks Image credit: NRC - Continuing Kepler’s Quest • Convergence rate of MC is slow • STM, as well as UT, rely on Gaussian distribution assumption Therefore, more robust methods must be considered Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 5 of 37

Background | Uncertainty Quantification Proposed Astrodynamic UQ Methods in Development • Polynomial Chaos Expansions (PCE) • Gaussian Mixtures • State Transition Tensors (STT) • Differential Algebra (DA) Image credit: Jones, et al. (2013) Image credit: Horwood, et al. (2011) Image credit: Fujimoto, et al. (2012) Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 6 of 37

Background | Uncertainty Quantification Proposed Astrodynamic UQ - Properties Without mitigation techniques, PCE and Gaussian Mixtures • PCE benefit from surrogate suffer from the curse of properties and fast convergence dimensionality rate • Computation time increases • GMM can leverage existing exponentially with respect to filters and Gaussian techniques input dimensions d • STT and DA methods reduce • Resulting in increased the computation burden computation time or dimension truncation STT Must Solve for Multiple Differential Equations, While DA is an Intrusive Method Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 7 of 37

Separated Representations Separated Representations Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 8 of 37

Separated Representations Separated Representation Premise: Decompose a multi-variate function into a linear combination of the products of uni-variate functions r s l u l  ( ξ  ) u l  ( ξ  ) · · · u l � q ( ξ  , . . . , ξ d ) = d ( ξ d ) + O ( ǫ ) l =  • r is the separation rank • u l i ( ξ i ) are the unknown uni-variate functions/factors • Computation cost dominated by relatively few MC propagations Extensive Background • Chemistry, data mining, imaging, etc • Doostan & Iaccarino 07,09. Nouy 07,10,11,12,13. Koromskij & Schwab 10. Cances et al. 11. Kressner & Tobler 11. Doostan et al. 12,13,14. Beylkin et al. 09 Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 9 of 37

Separated Representations Connections With SVD Singular-value decomposition (SVD): Separated approx: generalization of matrix SVD to tensors: Functions: Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 10 of 37

Separated Representations A Non-Intrusive Implementation Problem Set Up: Given n random samples reconstruct a low-rank separated representation: r s l u l  ( ξ  ) u l  ( ξ  ) · · · u l q ( ξ ) = ˆ � d ( ξ d ) s.t. � q − ˆ q � D ≤ ǫ l =  where, D := 1 N q �  q ( ξ j )  � � ˆ ˆ N j =  Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 11 of 37

Separated Representations Traditional Monte Carlo Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 12 of 37

Separated Representations Traditional Monte Carlo Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 13 of 37

Separated Representations Traditional Monte Carlo Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 14 of 37

Separated Representations Traditional Monte Carlo Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 15 of 37

Separated Representations Traditional Monte Carlo Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 16 of 37

Separated Representations Separated Approach Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 17 of 37

Separated Representations Separated Approach Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 18 of 37

Separated Representations Separated Approach Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 19 of 37

Separated Representations Separated Approach Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 20 of 37

Separated Representations Separated Approach Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 21 of 37

Separated Representations Separated Approach Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 22 of 37

Separated Representations Separated Approach Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 23 of 37

Separated Representations A Non-Intrusive Implementation Spectral Decomposition of Factors P u l c l i ( ξ i ) = � i,p ψ p ( ξ i ) p =  where, � ψ j ( ξ i ) ψ k ( ξ i ) ρ ( ξ i ) dξ i = δ jk Γ i Discrete Approximation � � r � � c l � s l u l  ( · ) u l  ( · ) · · · u l � = arg min � q ( · ) − � d ( · ) � � i,p � { ˆ c l i,p } l =  D Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 24 of 37

Separated Representations Computation Cost Linear Scalability: Required Number of Solution Samples r s l u l  ( ξ  ) u l  ( ξ  ) · · · u l q ( ξ ) = ˆ � d ( ξ d ) l =  Number of unknowns = r · d · P N ∼ O ( r · d · P ) Total Computation Time is Quadratic With Respect to d When ALS is Applied C d ∼ O ( K · r  · d · P  ( N + S )) This cost should be small when compared to the number of required MC propagations Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 25 of 37

Analysis Analysis Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 26 of 37

Analysis | Approach Distribution Characteristics • Nominal Trajectory and Maneuver Found Using Lambert Solver • Uncertainty in asteroid 2006 DN orbital elements and interceptor initial state • Error in magnitude and direction of interceptor maneuver at epoch • Propagated for 1088 days, Dormand-Prince (5)4 Integrator • Estimate heliocentric Cartesian coordinates and velocity Results compared to one million MC samples Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 27 of 37

Analysis | Approach Random Inputs Mean STD Mean STD x (AU) − 0 . 93808 1 . 33691 e − 09 a (AU) 1.38013 3 . 0097 e − 04 y (AU) − 0 . 35197 1 . 33691 e − 09 e 0.27859 1 . 5878 e − 04 z (AU) 1 . 9736 e − 05 1 . 33691 e − 09 inc (deg) 0.26764 1 . 3974 e − 04 x (km/s) ˙ 9.99105 8 e − 03 ω (deg) 101.24110 4 . 3343 e − 03 y (km/s) ˙ -28.00263 8 e − 03 z (km/s) ˙ 2 . 1797 e − 04 8 e − 03 Ω (deg) 96.62356 6 . 6975 e − 03 ∆ V x (km/s) 1 . 51305 0 . 01513 M (deg) 8.69171 0.72173 ∆ V y (km/s) − 3 . 48573 0 . 03485 ∆ V z (km/s) − 0 . 05830 5 . 830 e − 04 θ (deg) 0 1 φ (deg) 0 π Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 28 of 37

Analysis | d = 15 SR Results ( d = 15 ) 15 random inputs required 1200 samples, r = 8 , and P = 3 Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 29 of 37

Analysis | d = 15 SR Results ( d = 15 ) 15 random inputs required 1200 samples, r = 8 , and P = 3 Balducci, et al. | University of Colorado Boulder Asteroid Rendezvous Uncertainty Propagation | 29 of 37