Connecting the Equinoctial Elements and Rodrigues Parameters: A New Set of Elements

Abstract

A geometric interpretation of the equinoctial elements is given with a connection to orthogonal rotations and attitude dynamics in Euclidean 3-space. An identification is made between the equinoctial elements and classic Rodrigues parameters. A new set of equinoctial elements are developed using the modified Rodrigues parameters, thereby removing the coordinate singularity for retrograde equatorial orbits present in previous versions of these elements. A low-thrust trajectory optimization problem is set up using the new elements to numerically verify convergence for the two-point boundary problem, as compared to their predecessors.

Figures (3)

• Figure 1: Orientation of the perifocal basis (${\hat{{\mathbf{o}}}_i}$), LVLH basis (${\hat{{\mathbf{u}}}_i}$), and equinoctial basis (${\hat{{\mathbf{s}}}_i}$), relative to the inertial reference basis (${\hat{\textbf{\textiota}}_i}$). Orbital plane in orange.
• Figure 2: Time history of co-states for both MRP MEEs (denoted by $\lambda^{MM}$) and MEEs (denoted by $\lambda^{M}$).
• Figure :

Theorems & Definitions (11)

• Remark 2.1: CRP Singularities
• Remark 2.2: MRP singularities
• Remark 2.3: "Shadow" MRPs
• Remark 2.4: Euler angle singularities
• Remark 4.1: meaning of $q_1$ and $q_2$
• Remark 4.2: equatorial orbits
• Remark 4.3: meaning of $l$
• Remark 4.4: meaning of $e_1$ and $e_2$
• Remark 4.5
• Remark 5.1