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Nonlinear High-Fidelity Solutions for Relative Orbital Motion

This presentation discusses the development of explicit, closed-form analytical expressions for nonlinear translational relative orbital motion, addressing perturbation forces and their implications for space applications relevant to the USAF, DoD, and the broader space community. It covers recent advancements, including the application of Volterra series to establish nonlinear motion solutions and efforts for FY14 and FY15 that aim to improve relative navigation techniques and orbit determination. The goal is to provide more accurate models for close-proximity space scenarios and enhance operational capabilities.

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Nonlinear High-Fidelity Solutions for Relative Orbital Motion

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  1. Nonlinear High-Fidelity Solutions for Relative Orbital Motion 20 May 2014 Dr. T. Alan Lovell AFRL/RVSV Space Vehicles Directorate

  2. Outline • Goals, Motivation, Previous/Recent Work • FY14 Effort • Projected FY15 Effort

  3. Goals • To derive explicit (closed-form) analytical expressions for translational relative orbital motion that capture • Nonlinearities • Perturbation forces • To apply these solutions to space applications of interest to USAF, DoD, & the space community in general • Relative navigation/orbit determination • Orbital targeting/maneuver planning

  4. Relative S/C Dynamics Radial (x) Chief r  0 rC r LVLH frame Deputy Along-track (y) r rD rC • Tschauner-Hempel (T-H) Model • Linear Time-Varying ODE’s • r << r • Any eccentricity Chief • e between chief/deputy small • Hill-Clohessy-Wiltshire (HCW) Model • Linear Time-Invariant ODE’s • r << r • Chief near-circular (e << 1) • e between chief/deputy small Inertial frame • Full 2-Body Dynamics • Nonlinear, Time-Invariant • Any chief/deputy eccentricities • Analytically intractable

  5. Motivation • Relative motion has been subject of many decades of research • Most relative motion models can be categorized based on assumptions inherent in themodel: • What natural forces on each object are to be accounted for? • Are the objects in close proximity, i.e., on closely neighboring orbits? (linear vs nonlinear) • Is the chief on a circular or near-circular orbit? (time-varying vs time-invariant)

  6. Motivation • Summary/categorization of several existing relative motion models • No widely accepted closed-form Cartesian (x, y, z) solution that contains nonlinear terms AND accounts for forces other than two-bodygravity

  7. One Approach • One nonlinear modeling technique involves use of multi-dimensional convolution integrals (e.g. Volterra series)

  8. Recent Solution • Newman (2012) applied Volterraapproach to nonlinear ODEs of 2-body relative orbital motion (assuming circular chief orbit): • Set up problem to retain terms up to 2nd order—quadratic (e.g. x2) and bilinear (e.g. xy) • Resulting solution consists of expressions for x(t), y(t), and z(t) and their rates • Each expression can contain up to 27 terms • For linear solution (e.g. HCW), each expression can contain up to only 6 terms

  9. Recent Solution • Radial (x) component of solution is as follows (15 terms): • y, z expressions are similar (16 & 8 terms, respectively)

  10. Recent Solution • Volterra relative motion solution shows much better agreement w/ 2-body truth compared to linear (HCW) solution • Reveals characteristics not evident in HCW: • coupling between the cross-track (z) motion and the radial/in-track (x-y) motion • secular drift terms in radial direction (HCW indicates in-track driftonly)

  11. FY14 Effort • TASK 1: Apply Volterra solution • Relative navigation/orbit determination • TASK 2: Derive nonlinear closed-form solution by different approach  curvilinear coordinates • Compared to Volterra solution

  12. TASK 1: Apply Volterra Solution • An excellent application of the Volterrasolution is the problem of angles-only relative navigation (or relative orbit determination) • Navigation answers the question “WHERE AM I?” • Relative navigation answers the question “WHERE AM I RELATIVE TO THAT OTHER OBJECT?” (or “WHERE IS IT RELATIVE TO ME?”) • Typically applies to close-proximity scenarios, but can be applied over a LONGER baseline as well

  13. TASK 1: Apply Volterra Solution • This is fundamentally an estimation problem • Critical issue in our scenario is observability • Whether (& how accurately) the states (x, y, z, x-dot, y-dot, z-dot) can be determined from the measurements (LOS) • Unobservabilityimplies ambiguity, i.e. we can’t uniquely determine the relative orbit given our set of measurements • For our scenario, it turns out unobservabilityisguaranteedunder the following conditions: • Linear dynamics assumed • Cartesian coordinate frame • Angles only • No maneuvers • What does this mean? • Two (relative) state vectors that differ by a constant multiple possess identical LOS • When propagated forward with LINEAR, CARTESIAN dynamics, they produce identical LOS histories • We can’t determine is the size of thetrajectory  RANGE AMBIGUITY

  14. TASK 1: Apply Volterra Solution • To alleviate range ambiguity, need to relax at least one of the restricting conditions  one of which was assumption of linear dynamics • Thus, incorporating nonlinear effects in the estimation model should induce observability • Two initial states propagated forward with nonlinear dynamics • LOS histories deviate more with increase in downrange (along-track) separation • (If propagated with linear dynamics, LOS histories would be identical) t1 x t2 x t3 x t1 t2 o o t3 o t4 o t4 x Nonlinear dynamics won’t entirely alleviate ambiguity problem  observability may be WEAK

  15. TASK 1: Apply Volterra Solution • Volterrasolution consists of polynomial expressions in x0, y0, etc • Applying it to “initial relative orbit determination” (IROD) yields SIX 2nd-order polynomial eqn’s in SIX states • Solved via Macaulay resultant matrix (1916)

  16. TASK 1: Apply Volterra Solution • Volterra-based ROD solution provides initial guess for precise/refined solution

  17. TASK 2: Curvilinear Solution • Consider characterizing relative motion in curvilinear coordinates (cylindrical dr, dq, dzor spherical dr, dq, df) • Essentially inertial state differences; no concept of relative frame • Even linearized dynamics accurately capture curvature of drifting motion

  18. TASK 2: Curvilinear Solution • Begin with linearized relative ODEs in spherical coordinates: • These eqn’s possess same form as the linearized Cartesian (“HCW”) equations and are easily solved:

  19. TASK 2: Curvilinear Solution • We can then use the NONLINEAR kinematic relationships between Cartesian & spherical coordinates (& vice versa)…

  20. TASK 2: Curvilinear Solution • …to express the curvilinear solution as a Cartesian solution via “double transformation”:

  21. TASK 2: Curvilinear Solution • Result is a closed-form, nonlinear Cartesian solution for relative motion expressed as a function of the initial states and time: • Possesses many similar properties to Volterra solution… • coupling between cross-track (z) and radial/in-track (x-y) motion • radial secular drift • …but are NOT polynomial expressions in x0, y0, etc

  22. TASK 2: Curvilinear Solution • Generate polynomial expressions from our new nonlinear solution via Taylor series expansion: • Rewriting using basic trig terms yields same form as previously shown for Volterra solution • This allows direct term-by-term comparison of the two solutions

  23. TASK 2: Curvilinear Solution • Comparison of Volterrasolution & curvilinear solution in radial (x) direction: • BLACK: left & right side identical • BLUE: left & right side contain sameelements (e.g. cos(nt), sin(2nt), n2t2) • RED: left & right side do not contain same elements

  24. TASK 2: Curvilinear Solution • Comparison of Volterrasolution & curvilinear solution in along-track (y) direction:

  25. TASK 2: Curvilinear Solution • Comparison of Volterrasolution & curvilinear solution in cross-track (z) direction:

  26. TASK 2: Curvilinear Solution • Plotted results: Case 1—short-term drifting motion xy Orbit Track xy Orbit Track

  27. TASK 2: Curvilinear Solution • Plotted results: Case 1—short-term drifting motion x Error y Error

  28. TASK 2: Curvilinear Solution • Plotted results: Case 2—long-term drifting motion xy Orbit Track xy Orbit Track

  29. TASK 2: Curvilinear Solution • Plotted results: Case 2—long-term drifting motion x Error y Error

  30. TASK 2: Curvilinear Solution • Plotted results: Case 2—very long-term drifting motion xy Orbit Track

  31. TASK 2: Curvilinear Solution • So which of these solutions is preferable? • Answer will likely depend on the application • QV seems to have best short-term propagation accuracy • DTS seems to have best long-term propagation accuracy • Polynomial solutions (QV & ADTS) lend themselves to closed-form IROD solution • DTS/ADTS solutions required far fewer man-hours toderive (approx 1/20th the time)

  32. Projected FY15 Effort • Further comparison of Volterra-based & curvilinear-based models • Derive closed-form nonlinear relative motion solution incorporating J2 gravity (+ perhaps other perturbation forces) • Will consider both candidate methods (Volterra & curvilinear) • Implement curvilinear-based model (ADTS) in IROD • Derive geometric parameter set (“relative orbit elements”) based on 1 or more existing solutions

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