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) 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

5. 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)

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-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

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

8. 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

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 drift only)

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 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

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)
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

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 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

15. 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)

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, dz or 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 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

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

25. TASK 2: Curvilinear Solution
Comparison of Volterra solution & 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 to derive (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