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Unified Relative Orbital Element Framework for Eccentric Formation Flying

Yuji Takubo, Simone D'Amico

arXiv:2610.09379v1math.DScs.SYeess.SY

TL;DR

For eccentric chiefs, an RN-plane E/I-separation condition does not provide a time-invariant lower bound when a semimajor-axis residual is present. The paper introduces eccentric ROE tied to the VNB relative-motion solution and derives a BN-plane bound; numerical results confirm the bound and show the RN condition can allow keep-out-zone entry.

  • Problem

    For eccentric chiefs, the RN-plane E/I-separation condition does not provide a time-invariant lower bound when the semimajor-axis residual is nonzero.

    Source: §II · p. 6

  • Method

    The paper defines eccentric ROE that reduce to qnsROE at e=0 and correspond to physically normalized integration constants of the Cartesian VNB solution.

    Source: §VII · p. 29

  • Results

    For eccentric chiefs, numerical results confirmed the BN bound, while formations sized by the RN condition entered the keep-out zone.

    Source: §VII · p. 29

  • Takeaways & Limitations

    The proposed EROE provide a BN-plane separation bound for arbitrary E/I-vector orientation with a fixed allowance for semimajor-axis residuals.

    Source: §VII · p. 29

Abstract

from arXiv · show

Quasi-nonsingular relative orbital elements (qnsROE) are among the most widely used state representations for formation flying and rendezvous in near-circular inclined orbits. They admit closed-form perturbed state transition matrices (STMs) and a direct geometric mapping, which underpins eccentricity/inclination (E/I)-vector separation for passive safety. Existing eccentric extensions of the qnsROE resolve this geometry in the radial--tangential--normal frame, where a residual semimajor-axis difference makes the E/I separation margin phase dependent. This paper introduces eccentric ROE that reduce to the qnsROE as \(e\to0\) and whose first-order geometry is resolved in the velocity--normal--binormal (VNB) frame, where the secular drift is confined to the velocity direction. Their first-order position and velocity maps yield a closed-form, phase-independent binormal--normal separation bound with an explicit margin for semimajor-axis residuals. Closed-form Keplerian and secular-\(J_2\) STMs and a decoupled impulsive control input matrix are derived for the same elements, and the elements are recovered as the physically normalized integration constants of the Cartesian equations of relative motion in the VNB frame. Together, these results provide a unified ROE framework for formation flying in general elliptic orbits.

I. Introduction

Existing eccentric extensions of qnsROE retain RTN-resolved geometry, where semimajor-axis residuals make separation phase dependent. The paper introduces qnsROE-like eccentric elements resolved in VNB, with a phase-robust BN separation bound and closed-form propagation and control models.

I. Introduction

  • For eccentric chiefs, RTN-resolved eccentric ROE leave the semimajor-axis-residual separation margin phase dependent.The paper attributes the growing radial drift to the RTN projection rather than the choice of elements.Source: §I · p. 2
  • In VNB, secular drift lies along velocity, so BN separation needs only a fixed margin for bounded binormal offset across 0 ≤ e < 1.The proposed EROE yield a phase-robust BN bound that includes an allowance for semimajor-axis residuals.Source: §I · pp. 2–3
  • The proposed EROE are defined from orbital-element differences, reduce to qnsROE as e→0, and match physically normalized VNB integration constants.Their first-order VNB position map has the harmonic form of the qnsROE map.Source: §I · p. 3
  • Nonlinear simulations found formations sized by the BN bound remained outside the keep-out zone, while the same margin under the RN condition could enter it.The comparison includes a semimajor-axis residual.Source: §I · p. 3
  • Closed-form Keplerian and secular-J2 STMs and an impulsive input matrix are derived; the matrix decouples in-plane and out-of-plane control for 0 ≤ e < 1.These results reduce to their qnsROE counterparts at e=0.Source: §I · p. 3
  • The Cartesian VNB equations yield a closed-form fundamental matrix and analytical inverse, whose modes are normalized to produce the proposed EROE.This normalization makes the EROE-to-Cartesian map and its inverse require no separate transformation.Source: §I · p. 3

II. Frame Dependence of Phase-Robust Geometry

For an eccentric chief, a semimajor-axis residual makes the RN-plane separation condition lack a time-invariant lower bound. Rotating the same relative motion into VNB confines accumulated phase to the velocity direction, leaving BN separation bounded and periodic.

II. Frame Dependence of Phase-Robust Geometry

  • For e > 0, the secular radial component grows linearly away from the apsides, so the RN condition lacks a time-invariant lower bound when the residual semimajor-axis difference is nonzero.The paper identifies this phase dependence as originating in the RTN projection, not the element set.Source: §II · pp. 3, 6
  • In VNB, accumulated phase enters neither B nor N, so BN projection remains bounded and periodic under Keplerian dynamics despite a semimajor-axis residual.The bounded semimajor-axis contribution still shifts binormal geometry and must be included in the margin.Source: §II · p. 6
  • The phase independence follows by rotating the YA solution alone, independently of the relative-element set.Peters and Noomen observed the same independence in C-set coordinates.Source: §II · p. 6

III. Proposed Eccentric Relative Orbital Elements

The proposed EROE adapt relative orbital-element differences to eccentric orbits while recovering the qnsROE in the circular limit. Their VNB geometry confines semimajor-axis drift to velocity and keeps the binormal center bounded.

A. Definition

  • As e→0, the EROE weights approach unity, the energy correction vanishes, and the proposed elements reduce to the qnsROE.They retain δa, scale the inclination pair by η^2, reweight the periapsis contribution to longitude by η^3/χ, and use eccentricity components aligned with the eccentricity-vector axes.Source: A. Definition · p. 7
  • The classical-element form is singular at e=0 because ω is undefined; node-corrected eccentricity differences yield a transformation regular for 0≤e<1.Source: A. Definition · p. 7

B. Geometry in the VNB Frame

  • The first-order VNB maps split each component into a center term and a scaled first harmonic, whose coefficients define the shape components.The maps provide an instantaneous six-state position-and-velocity mapping from the EROE.Source: B. Geometry in the VNB Frame · p. 8
  • For e>0, anomaly-varying shape amplitudes make the projected VB and BN loci non-elliptical, unlike the familiar 2:1 ellipse limit.Source: B. Geometry in the VNB Frame · p. 8
  • In the VNB-frame projection, the drift moves the center only along V, while the trajectory’s binormal coordinate remains between 59 m and 140 m.The nonlinear trajectories agree with the corresponding first-order maps.Source: B. Geometry in the VNB Frame · p. 9

C. Relation to Existing Element Sets

The proposed EROE connect to existing element sets while retaining a qnsROE limit and VNB geometry. In the BN plane, relative semilatus rectum governs crossing and centering, with residual semimajor-axis differences constraining separation.

C. Relation to Existing Element Sets

  • The proposed-to-Delurgio–D’Amico map is smooth for 0≤e<1 and becomes the identity at e=0, whereas the C-set map has no e→0 limit.Source: C. Relation to Existing Element Sets · p. 10
  • The proposed EROE are equivalent to normalized integration constants of the Cartesian VNB equations and reduce to the qnsROE at e=0.Their reference axes follow the line of nodes, unlike the C-set's chief apsidal-line axes, which are undefined at e=0.Source: C. Relation to Existing Element Sets · p. 10

D. Passive Safety in the BN Plane

1. Crossing Conditions and Centering by the Relative Semilatus Rectum

  • For δi>0, the BN curve passes through the chief when δp=±δe cos(φi−φe); for δi=0, crossing occurs when |δp|≤δe.Source: §1 · p. 11
  • Orientation-independent separation is restricted to formations on the order of the semimajor-axis difference; larger formations can be centered by matching semilatus recta.Near the circular limit, the 1/e scaling favors nulling residual δa rather than applying the matching condition.Source: §1 · p. 12

2. Phase-Robust Separation Bound

The proposed bound gives a phase-robust lower bound on binormal–normal separation for elliptic Keplerian formations. It also accounts for semimajor-axis residuals, with a conservative margin away from periapsis.

2. Phase-Robust Separation Bound

  • The BN-plane separation bound holds for every initial longitude and all t ≥ t0 under Keplerian motion, for 0 ≤ e < 1 and δp = 0.At e = 0, the bound recovers the near-circular minimum-distance result.Source: §2 · p. 12
  • The stated lower bound is conservative away from periapsis because it uses the periapsis values of k and q.For parallel vectors, the BN distance can be evaluated at the zeros of ρN or ρB.Source: §2 · p. 13
  • For parallel relative eccentricity and inclination vectors, the bound is exact when δe ≥ (1 + e)δi and δa = 0.In this case, the BN closest approach occurs at periapsis or apoapsis.Source: §2 · p. 13
  • The residual-margin estimate also applies to perturbations acting on ρB alone, including an unmodeled semilatus-rectum error.For a physical semimajor-axis residual, the accompanying δe change makes Eq. (22) sharper than the general δp-error margin.Source: §2 · pp. 13–14
  • The bound accommodates semimajor-axis residuals with a clearance allowance independent of accumulated phase.The accompanying change in δe cancels the 1/k dependence of the residual shift, making this bound sharper than the general δp-error margin.Source: §2 · pp. 12, 14

IV. State Transition Matrices and Control Input Matrix

The section derives closed-form Keplerian and secular-J2 STMs for the proposed EROE and uses the secular-J2 form to examine invariant formation designs. It also identifies a long-horizon station-keeping requirement for maintaining the separation margin.

A. State Transition Matrices

  • The closed-form secular-J2 STM is exact within the first-order secular mean-element model and reduces to the Keplerian STM when J2 = 0.Its form combines the rotation of the eccentricity vector with a linear shear term.Source: A. State Transition Matrices · p. 16
  • With δa0 = 0 and a non-equatorial chief, the rate-matching conditions reduce to eΔe = Δi = 0.For e > 0 this requires equal mean eccentricities and inclinations; at e = 0 the first-order eccentricity constraint vanishes, leaving the qnsROE condition δix0 = 0.Source: A. State Transition Matrices · p. 16
  • Maintaining the E/I-vector separation margin over long horizons generally requires station keeping of δe.Source: A. State Transition Matrices · p. 16

B. Control Input Matrix

The control input map decouples in-plane and out-of-plane dynamics across ellipticities, while the direct Cartesian derivation independently recovers the proposed EROE as normalized integration constants.

B. Control Input Matrix

  • The impulsive control map is decoupled between in-plane and out-of-plane dynamics for all 0 ≤ e < 1.At e = 0, ordering the impulse columns as [T, N, R] recovers the qnsROE input matrix.Source: B. Control Input Matrix · p. 17

V. Direct Cartesian Derivation in the VNB Frame

  • The VNB Cartesian derivation solves the relative variational equations directly, without assuming an element-to-Cartesian map.It identifies the modal constants with the proposed EROE and establishes equivalence between the two formulations.Source: §V · pp. 17–18

A. Relative Equations of Motion in the VNB Frame

B. Scaling of the Variables

Scaling the Cartesian variables simplifies the VNB equations, while solving their modes yields physically normalized integration constants and a closed-form fundamental matrix.

B. Scaling of the Variables

  • Scaling the variables removes the first-derivative term introduced by changing the independent variable, and the out-of-plane equation becomes a harmonic oscillator.The oscillator matches the one in the radius-normalized YA STM derivation.Source: B. Scaling of the Variables · pp. 18–19

C. Integration Constants and Their Responses

  • The six independent integration constants are normalized using Cartesian modal properties, including the phase mode’s first-order time offset.The secular shear is derived from the Cartesian equations rather than assumed from an ROE propagation law.Source: C. Integration Constants and Their… · p. 20

1. Explicit Fundamental Matrix

  • The radius-scaled fundamental matrix is assembled from the responses of the six constants in the order [A, Λ0, Kx, Ky, KNx, KNy].Source: §1 · p. 20

2. Explicit Inverse of the Fundamental Matrix

The analytical inverse recovers the modal constants, and the physical VNB state-transition matrix is nonsingular for 0 ≤ e < 1. The Cartesian and EROE derivations agree under the paper’s physical normalization, which centers the BN geometry on δp rather than δa.

2. Explicit Inverse of the Fundamental Matrix

  • At any reference anomaly, the analytical inverse uniquely recovers the six modal constants from the scaled state.The constants are ordered [A, Λ0, Kx, Ky, KNx, KNy].Source: §2 · p. 21

3. Physical State Transition Matrix

  • The physical VNB state-transition matrix is obtained by eliminating the modal constants between initial and final Cartesian states, and remains nonsingular for 0 ≤ e < 1.The physical matrix is formed by de-scaling the fundamental matrix.Source: §3 · pp. 22–23

E. Identification with the Eccentric Relative Orbital Elements

  • Instantaneous EROE can be recovered from the physical state at any epoch by applying Q^-1 and evaluating the inverse with that epoch as reference.The radius normalization remains confined to the auxiliary Cartesian state.Source: E. Identification with the Eccentric… · p. 23
  • Under the specified physical normalization, the Cartesian and EROE derivations yield the same first-order Keplerian matrix, with δp—not δa—centering the BN geometry.The constant transformation between the two element sets is the identity; the particular-solution constant corresponds to the normalized semilatus-rectum difference δp.Source: E. Identification with the Eccentric… · p. 23

VI. Numerical Results

  • The numerical section compares eccentric formations sized by the RN-plane condition and the BN bound under semimajor-axis residuals, then tests finite-horizon STM accuracy against the YA baseline.The comparison and STM verification are assigned to Sections VI.A and VI.B, respectively.Source: §VI · p. 23

A. Passive-Safety Drift Case Study

With a 100 m semimajor-axis residual, the RN-sized formation loses projected clearance, while the proposed BN formation retains its stated separation guarantee. Nonlinear propagation shows that the former can enter the keep-out sphere, whereas the latter does not in the tested cases.

A. Passive-Safety Drift Case Study

  • The formations were propagated independently under nonlinear two-body dynamics for six chief orbits, with the deliberately large residual chosen to expose secular drift over a short horizon.Separations were computed in the chief’s instantaneous RTN or VNB frame, independently of the first-order position mappings.Source: A. Passive-Safety Drift Case Study · p. 24
  • With the 100 m residual, the RN minimum fell to 26 m and the formation entered the keep-out sphere, while proposed BN minima stayed above 700 m and avoided entry.The BN minima matched first-order predictions within 0.5 m; Table 3 reports six-orbit minimum separations.Source: Table 3 · p. 25, A. Passive-Safety Drift Case Study · p. 25
  • At e = 0.4, the 3D minimum reached 0.66 rKOZ for the RN formation but remained above 1.95 rKOZ for the proposed BN formation.The projected histories show repeated RN keep-out-radius crossings and retention of the BN lower bound.Source: A. Passive-Safety Drift Case Study · p. 25
  • The residual drives a time-growing radial component in the RN case, whereas the proposed formation’s BN offset stays bounded because phase drift remains confined to V.The bounded offset and fixed margin are identified as the behavior predicted by Eqs. (6) and (22).Source: A. Passive-Safety Drift Case Study · p. 25
  • Across the longitude sweep, the RN formation enters the keep-out sphere while the proposed formation retains its bounded BN margin; equal longitude coordinates do not denote identical physical states.The sweep’s curve minima are the worst-case entries reported in Table 3.Source: A. Passive-Safety Drift Case Study · p. 26

B. Finite-Horizon STM Accuracy

The secular-J2 STM errors remain closely matched across the tested sweeps, while Keplerian STMs omit differential J2 drift. The conclusion presents VNB-resolved EROE with phase-independent BN safety geometry, closed-form propagation and control, and a circular-limit correspondence to qnsROE.

B. Finite-Horizon STM Accuracy

  • The comparison uses a nonlinear secular-J2 reference and first-order position maps, excludes short-period terms and mean-to-osculating reconstruction, and holds the physical chief and deputy consistent across sweeps.Source: B. Finite-Horizon STM Accuracy · pp. 26–27
  • The two secular-J2 STM error curves are nearly coincident; their ten-orbit maxima rise from about 5 m at low eccentricity to 25 m at e=0.8.Figure 6 reports maximum position error over ten chief orbits.Source: B. Finite-Horizon STM Accuracy · p. 28, Figure 6 · p. 29
  • Both formulations reach a minimum error of about 3 m near aδλ0=4 km, while the longitude sweep is nonmonotonic.Varying only the initial longitude lets finite-offset truncation terms associated with fixed elements partially cancel.Source: B. Finite-Horizon STM Accuracy · pp. 28–29
  • The Keplerian STMs closely follow nonlinear Keplerian propagation, but their errors are one to two orders of magnitude larger than the J2 STMs because they omit differential J2 drift.Source: B. Finite-Horizon STM Accuracy · p. 29

VII. Conclusion

  • The proposed EROE resolve geometry in VNB, remain regular at e=0, reduce to qnsROE in the circular limit, and coincide with physically normalized integration constants.Source: §VII · p. 29
  • Their BN-plane separation bound applies to arbitrary E/I-vector orientation with fixed clearance for semimajor-axis residuals, and numerical results confirm the bound.Source: §VII · p. 29
  • The paper derives closed-form Keplerian and secular-J2 STMs and a decoupled impulsive control input matrix, each reducing to its qnsROE counterpart at e=0.Source: §VII · p. 29
  • The conclusion reports that formations sized by the RN-plane separation condition can enter the keep-out zone.Source: §VII · p. 29

Appendix A: Regularization at Zero Eccentricity

The classical-element form is singular at zero eccentricity. A node-corrected definition yields a regular conversion for 0≤e<1, while retaining the QNS equatorial singularity.

Appendix A: Regularization at Zero Eccentricity

  • At e=0, the classical-element form is singular in δλ and both components of δe; the regular definition uses node-corrected eccentricity differences.Source: Appendix A · p. 30
  • The transformation is regular for 0≤e<1 and retains only the equatorial singularity of the QNS elements.Source: Appendix A · p. 30
  • Conversion between the proposed EROE and QNS element differences is regular in both directions for 0≤e<1, and the EROE reduce to qnsROE as e→0.Source: Appendix A · p. 30
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