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

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

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

Relative S/C Dynamics Radial (x) Along-track (y)  0 Chief r LVLH frame Deputy Along-track (y) r rD rC r rC  0 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 Chief near-circular (e << 1) Inertial frame Full 2-Body Dynamics Nonlinear, Time-Invariant Any chief/deputy eccentricities Analytically intractable

Motivation Relative motion has been subject of many decades of research Most relative motion models can be categorized based on assumptions inherent in the model: 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)

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-body gravity Common Name Forces Accounted For Close Proximity Assumed? Circular Chief Assumed? Linear or nonlinear? Time-varying or time-invariant? Hill’s-Clohessy-Wiltshire 2-body gravity Y LTI Tschauner-Hempel N LTV Yamanaka-Ankersen Sedwick-Schweighart 2-body gravity & J2 Gim-Alfriend Sabol-McLaughlin (COWPOKE) Lutze-Karlgaard NL

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

Recent Solution Newman (2012) applied Volterra approach 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

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

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 drift only)

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

TASK 1: Apply Volterra Solution An excellent application of the Volterra solution 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

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) Unobservability implies ambiguity, i.e. we can’t uniquely determine the relative orbit given our set of measurements For our scenario, it turns out unobservability is guaranteed under 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 the trajectory  RANGE AMBIGUITY

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 t1 t3 t4 t2 x o 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) Nonlinear dynamics won’t entirely alleviate ambiguity problem  observability may be WEAK

TASK 1: Apply Volterra Solution Volterra solution 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)

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

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

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:

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

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

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

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

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

TASK 2: Curvilinear Solution Comparison of Volterra solution & curvilinear solution in along-track (y) direction:

TASK 2: Curvilinear Solution Comparison of Volterra solution & curvilinear solution in cross-track (z) direction:

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

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

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

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

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

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 to derive (approx 1/20th the time)

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