Equinoctial Orbital Elements (Equinoctial Orbital Elements)

Editor Source: 胡敏, 肖金伟, 张天天, 陶雪峰 (2026) "面向中高轨小卫星批量部署的轨道转移飞行器任务规划"
Narayanaswamy S, Damaren C J. Equinoctial Lyapunov control law for low-thrust rendezvous[J]. Journal of Guidance, Control, and Dynamics, 2023, 46(4): 781-795.
Website: https://cislunarspace.cn

Definition

Equinoctial Orbital Elements are a type of non-singular orbital element representation that avoids singularity issues of classical elements in near-circular orbits and small inclination cases through trigonometric transformations.

Modified Equinoctial Elements (MEE) are an improved form with better numerical stability and control performance in Q-law control.

Mathematical Definition

Classical Orbital Elements

Classical orbital elements include:

$a$ : Semi-major axis
$e$ : Eccentricity
$i$ : Orbital inclination
$\omega$ : Argument of periapsis
$\Omega$ : Right ascension of ascending node
$\theta$ : True anomaly

Modified Equinoctial Elements (MEE)

MEE as used by 胡敏等 (2026):

\begin{cases} p = a(1 - e^2) \\ e_1 = e\cos(\omega + \Omega) \\ e_2 = e\sin(\omega + \Omega) \\ h_1 = \tan(i/2)\cos\Omega \\ h_2 = \tan(i/2)\sin\Omega \\ L = \omega + \Omega + \theta \end{cases}

Where:

$p$ : Semi-latus rectum
$e_1, e_2$ : Eccentricity vector components
$h_1, h_2$ : Orbital plane normal vector components
$L$ : True longitude

Advantage Analysis

Numerical Stability

Problem Type	Classical Elements	MEE
Near-circular orbit (e≈0)	Inclination $i$ and argument of periapsis $\omega$ poorly defined	No singularity
Small inclination (i≈0)	Right ascension $\Omega$ poorly defined	No singularity
Large eccentricity	Numerically stable	Numerically stable

Control Performance

胡敏等 (2026) showed that using semi-major axis $a$ instead of semi-latus rectum $p$ provides better control performance:

p = a(1 - e^2)

This modification makes direct control of semi-major axis more natural and stable during optimization.

Geometric Interpretation

MEE components have clear geometric meanings:

$e_1, e_2$ : Eccentricity vector, describing orbital ellipse shape and orientation
$h_1, h_2$ : Describing the orientation of the orbital plane in space
$L$ : True longitude, rapidly changing quantity describing spacecraft instantaneous position

Application in Q-law Control

Q Function Definition

In Q-law control, the Q function is defined as a weighted quadratic form of MEE errors:

Q(Z) = (1 + W_P P) \sum_{i=1}^{5} W_i S_i \left(\frac{\delta Z_i}{\max_L(\dot{X}_i)}\right)^2

Where $\delta Z_i$ are errors in each MEE component.

Normalized Metric

Orbit error normalized metric based on MEE:

\sigma_{oe} = \left(\frac{\Delta a}{R_G}\right)^2 + (\Delta e_1)^2 + (\Delta e_2)^2 + (\Delta h_1)^2 + (\Delta h_2)^2

Where $R_G$ is GEO radius, used for nondimensionalization.

Orbit Insertion Accuracy Criterion

Simulation termination criterion based on orbit insertion accuracy threshold:

\sigma_{oe} \leq \varepsilon

Research uses high-precision standards of $\varepsilon = 1 \times 10^{-6}$ magnitude.

Comparison with Other Orbital Representations

Representation	Dimensions	Singularity	Application
Classical elements	6	Yes (e=0, i=0, i=π)	General orbit analysis
MEE	6	No	Orbit control, optimization
Cartesian	6	No	Dynamics integration
Delaunay	6	Yes	Canonical transformations

References

胡敏, 肖金伟, 张天天, 陶雪峰. 面向中高轨小卫星批量部署的轨道转移飞行器任务规划[J]. 航天器工程, 2026, 25(3): 634-646.
Narayanaswamy S, Damaren C J. Equinoctial Lyapunov control law for low-thrust rendezvous[J]. Journal of Guidance, Control, and Dynamics, 2023, 46(4): 781-795.
Koon W S, Lo M W, Marsden J E, et al. Dynamical systems, the three-body problem and space mission design[M]. 2006.