New Closed-Form Solutions for Optimal Impulsive Control of Spacecraft Relative Motion

Abstract

This paper addresses the spacecraft relative orbit reconfiguration problem of minimizing the delta-v cost of impulsive control actions while achieving a desired state in fixed time. The problem is posed in relative orbit element (ROE) space, which yields insight into relative motion geometry and allows for the straightforward inclusion of perturbations in linear timevariant form. Reachable set theory is used to translate the cost-minimization problem into a geometric path-planning problem and formulate the reachable delta-v minimum, a new metric to assess optimality and quantify reachability of a maneuver scheme. Next, this paper presents a methodology to compute maneuver schemes that meet this new optimality criteria and achieve a prescribed reconfiguration. Though the methodology is applicable to any linear time-variant system, this paper leverages a state representation in ROE to derive new globally optimal maneuver schemes in orbits of arbitrary eccentricity. The methodology is also used to generate quantifiably sub-optimal solutions when the optimal solutions are unreachable. Further, this paper determines the mathematical impact of uncertainties on achieving the desired end state and provides a geometric visualization of those effects on the reachable set. The proposed algorithms are tested in realistic reconfiguration scenarios and validated in a high-fidelity simulation environment.

Minimum-Fuel Control Strategy for Spacecraft Formation Reconfiguration via Finite-Time Maneuvers

“…They proposed a general methodology, based on the inversion of relative dynamics equations, which led to the straightforward computation of analytical or numerical control solutions. A similar impulsive approach based on the ROE parameterization is developed by Chernick and D'Amico in [3]. Here, the authors extended the results reported in [2], deriving the analytical and semi-analytical solutions for in-plane and out-of-plane reconfigurations, respectively, in near-circular J 2 perturbed and eccentric unperturbed orbits.…”

“…Then, the relative state vector δα consists of the relative semimajor axis δα the relative longitude δλ as well as the coordinates of the relative eccentricity vector δe and relative inclination vector δi, respectively. The preceding relative state is nonsingular for circular orbits (e c 0), whereas it is still singular for strictly equatorial orbits (i c 0) [3]. The averaging theory [13] can be used to derive the variation of the mean ROE due to the Earth's oblateness J 2 [7,14].…”

“…III. To this purpose, the results obtained by the MILP solver are compared with those from the well-known impulsive theory [2,3,12]. In further detail, starting from the scenario described in the previous section, three study cases are investigated, assuming a maneuvering interval of eight chief orbital periods (i.e., u T 16π rad corresponding to about 13.45 h).…”

“…For the in-plane reconfiguration problem in the unperturbed (J 2 0) circular orbit (e c 0), it is possible to compute the minimum delta-v required to achieve the desired formation configuration, as discussed in [2,3,33]. This lower delta-v bound is independent from the number or type of executed maneuvers and it is given by [2] V LB max n c a c Δδe 2 ; n c a c jΔδa j 2 (69)…”

How Many Impulses Redux

“…For the most part, preliminary mission design methods rely on lowfidelity dynamical models, which in turn, frequently leads to analytical propagation of the state dynamics through Keplerian orbit models [20] or by utilizing the solution of Lambert's problem [21][22][23][24][25]. Impulsive maneuvers are also used extensively for formation flight optimal control problems [26][27][28][29][30][31][32][33][34][35][36] and orbit reachability analyses problems [37][38][39][40][41][42][43].…”