Applied Reachability for Space Situational Awareness and Safety in Spacecraft Proximity Operations

Several existing and emerging applications of Space Situational Awareness (SSA) relate directly to spacecraft Rendezvous, Proximity Operations, and Docking (RPOD) and Formation / Cluster Flight (FCF). Observation correlation of nearby objects, control authority estimation, sensor-track re-acquisition, formation re-conguration feasibility, ‘stuck’ thrusters, and worst-case passive safety analysis are some areas where analytical reachability methods have potential utility. Existing reachability theory is applied to RPOD and FCF regimes. An optimal control policy is developed to maximize the reachability set and optimal control law discontinuities (switching) are examined. Necessary conditions for maximum position reachability are developed, allowing for a reduction in reachable set computation dimensionality. The nonlinear relative equations of Keplerian motion are introduced and used for all reachable position set determinations. The linearized ClohessyWiltshire equations of motion are normalized to accentuate relative control authority for spacecraft propulsion systems at both Low Earth Orbit (LEO) and Geostationary Earth Orbit (GEO). Several examples with traditional and low thrust propulsion systems in LEO and GEO are explored to illustrate the eects of relative control authority on the timevarying reachability set surface using the nonlinear equations of motion. Both monopropellant spacecraft at LEO and Hall thruster spacecraft at GEO are shown to be strongly actuated while Hall thruster spacecraft at LEO are found to be weakly actuated. Weaknesses with the current implementation are discussed and future numerical improvements and analytical eorts are discussed.

[1]  Derek F Lawden.  Analytical Methods of Optimization , 2006 .

[2]  James A. Sethian,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid , 2012 .

[3]  Alexandre M. Bayen,et al.  A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games , 2005, IEEE Transactions on Automatic Control.

[4]  John Lygeros,et al.  On reachability and minimum cost optimal control , 2004, Autom..

[5]  D. Scheeres,et al.  Solving two-point boundary value problems using generating functions: Theory and Applications to optimal control and the study of Hamiltonian dynamical systems , 2003, math/0310475.

[6]  Pravin Varaiya,et al.  Reach Set Computation Using Optimal Control , 2000 .

[7]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[8]  John Lygeros.  On the relation of reachability to minimum cost optimal control , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[9]  Daniel J. Scheeres,et al.  Solving Optimal Continuous Thrust Rendezvous Problems with Generating Functions , 2005 .

[10]  Jonathan P. How,et al.  8 Cooperative Spacecraft Formation Flying: Model Predictive Control with Open-and Closed-Loop Robustness , 2006 .

[11]  H. Seifert,et al.  Rocket Propulsion Elements , 1963 .

[12]  Daniel J. Scheeres,et al.  Determination of optimal feedback terminal controllers for general boundary conditions using generating functions , 2006, Autom..

[13]  P. Lions,et al.  Some Properties of Viscosity Solutions of Hamilton-Jacobi Equations. , 1984 .

[14]  Jonathan P. How,et al.  Safe Trajectories for Autonomous Rendezvous of Spacecraft , 2006 .

[15]  Inseok Hwang,et al.  Applications of polytopic approximations of reachable sets to linear dynamic games and a class of nonlinear systems , 2003, Proceedings of the 2003 American Control Conference, 2003..

[16]  P. Varaiya,et al.  Dynamic Optimization for Reachability Problems , 2001 .

[17]  J. Junkins,et al.  Analytical Mechanics of Space Systems , 2003 .

[18]  Alexandre M. Bayen,et al.  A construction procedure using characteristics for viscosity solutions of the Hamilton-Jacobi equation , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[19]  Christopher Cox,et al.  Intelligent Data Fusion for Improved Space Situational Awareness , 2005 .

[20]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[21]  Owen Brown,et al.  Fractionated Space Architectures: A Vision for Responsive Space , 2006 .

[22]  Jonathan P. How,et al.  Co‐ordination and control of distributed spacecraft systems using convex optimization techniques , 2002 .

[23]  Jeremy D. Schwartz,et al.  Passively safe Receding Horizon Control for satellite proximity operations , 2008, 2008 47th IEEE Conference on Decision and Control.

[24]  Ian M. Mitchell,et al.  A Toolbox of Level Set Methods , 2005 .