Distributed situational observer in a displaced orbit: relative dynamics and control

Abstract

We design a distributed situational observer using formation flying in a displaced orbit. The main focus of our investigation is the relative dynamics and control of displaced orbits obtained by low-thrust propulsion. The spatial dynamics in Newtonian form are used to derive the numerical relative motions, and their natural frequencies discovered by eigenvalue decomposition separate from each other at a critical height that differentiates the structural stability, bifurcation, and instability. Using the Jordan decomposition, six fundamental motions are achieved, including the stationary multiequilibria, the periodic oscillations that correspond to the natural frequencies, and the maximum leaving or approaching velocity caused by the different geometric and algebraic multiplicities. Off-axis equilibrium is obtained by a proposed open-loop control, and the motions nearby are proven to be equivalent to the numerical relative motions. The reduced dynamics in Hamiltonian form are used to derive the analytical solutions for linearized relative motions. Bounded relative trajectories with arbitrary initial values are achieved by two extraclosed-loop controls. Using the off-axis equilibrium and resonance of natural frequencies, the applications of a fixed relative baseline vector for interferometric SAR or Fresnel zone lens missions and repeating relative ground tracks for a phased array antenna mission are addressed in terms of the trajectory design. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Linearized relative motion and proximity control of E-sail-based displaced orbits

“…increases from one to infinity [29], the ratio ω3 ⁄ω2 of relative motions with E-sails is different and limited, as shown in Fig. 4d.…”

“…The maximum ratio ω3 ⁄ω2, instead, is limited by the stability constraint, 11 / 17 and its value is about 32. In some resonance cases in which ω3 ⁄ω2=m/n (with m and n positive commutative integers), the bounded relative orbit is periodic instead of quasi-periodic [29]. Thus, Fig.…”

“…Introduce now the inertial reference frame ξ and the orbital reference frame ς [29], so that the relative acceleration in the chief's orbital reference frame ς can be written as ( )…”