Shape-Based Design of Low-Thrust Trajectories to Cislunar Lagrangian Point

Abstract

T HIS Note presents a shape-based method for the design of lowthrust trajectories in the framework of the circular restricted three-body problem (CR3BP). Velocity and thrust profiles are analytically determined as a function of spacecraft position for a particular class of shape functions, derived from the classical exponential sinusoid [1]. An additional term is introduced, which improves performance in terms of velocity increment and transfer time necessary for tracking the prescribed trajectory in the presence of the combined gravitational pull from two different primary bodies. Low-thrust propulsion offers several advantages, thanks to its high specific impulse [2,3], but mission design can become complex and computationally demanding, in the absence of closed-form solutions for the equations of motion [4]. Large CPU time is required for trajectory analysis and optimization. Low-thrust trajectory design relies on optimal control theory, where propellant consumption and/ or transfer time are minimized. Solution methods can be divided into two classes [5]: direct methods [6], where trajectory and control variables are discretized and the resulting minimization problem is solved by means of nonlinear programming techniques; and indirect methods [7], where boundary conditions and first-order necessary conditions for optimality are enforced for a continuous system. Both techniques need an initial guess to start the iterative process, which improves the performance index and constraint fulfillment for direct methods or reduces error on boundary conditions and gradient for indirect ones. Finding a reasonable initial guess, possibly represented by a suboptimal solution, is an important (and often time-consuming) part of mission design, which plays a decisive role for convergence, especially in the presence of localminima. The shape-based approach provides analytical solutions for motion variables for a given class of trajectory shape functions. Once the trajectory shape is selected a priori, the thrust profile is derived a posteriori from the inverse solution of the equations of motion. Total ΔV and transfer time thus become rapidly available bymeans of a simple numerical quadrature. The shape function is defined on the basis of a fewparameters that can be modified by means of an optimization algorithm to improve transfer performance in some sense and guarantee its feasibility. The final solution does not need to be the optimal one, but it should provide a reasonable first guess for more sophisticated optimization procedures. Petropoulos and Longuski employed exponential sinusoids for low-thrust gravity-assist trajectory design [1]. A solution for the lowthrust Lambert’s problem, that is, a fixed-time orbit transfer performed by low-thrust propulsion, was also proposed in terms of exponential sinusoids [8]. Other classes of shape functions are available, such as the logarithmic spiral, Forbes’s spiral, and Lawden’s spiral [9].Wall andConway employed a fifth-order inverse polynomial shape function for time-free orbit transfers and interception and rendezvous trajectories, whereas they proposed sixth-order inverse polynomials for fixed-time problems [10]. The exponential sinusoid remains the most widely used shape function, suitable for interplanetary gravity assists and multiple revolution orbit transfers. The radial distance from the primary body for the exponential sinusoid is expressed as a function of the anomaly θ in the form r θ k0 exp k1 sin k2θ φ , where the parameters k0, k1, k2, andφ are tuned tomatch boundary conditions.A feasibility issue needs to be considered [9,11], because infinite values of thrust and angular rate at periapsis can be encountered. Unfeasible regions in the solution space can be avoided by limiting the value of the product k1k 2 2 [1]. Most of the applications of the shape-based approach are proposed in the framework of the two-body problem, where the spacecraft moves under the action of low thrust and a single primary body. This approach is reasonable for most mission scenarios, during preliminary steps of mission design, when approximate solutions are sufficient and a single primary mass is considered during each mission segment. However sometimes it is necessary to account for gravity pull frommore than a single primary mass. This is the case of missions to Lagrangian points in the Earth–moon and sun–Earth systems, which saw an increasing interest in the last decades [12–14]. In spite of their instability, collinear libration points are attractive locations for a spacecraft, which can orbit around themwith minimal propellant cost along halo, quasi-halo, and Lissajous orbits [15–18]. Moreover, they are characterized by an invariant manifold [17,19,20] that can be employed to design low-energy transfers using little or no fuel [21]. Recently, Arora and Strange [22] proposed a mission design approach that exploits characteristics of both low-thrust and low-energy trajectories, joining an initial multirevolution low-thrust transfer with a low-energy phase, which drives the spacecraft to the target orbit around the desired libration point. In this Note, transfers from low Earth orbit (LEO) to Lagrangian pointL1 of the Earth–moon system are considered, assuming the two primarymasses on a circular relative orbit. Provided that a significant portion of mission time is spent in a region where gravity pull from themoon andEarth is equivalent, a shape-basedmethod is developed, where the exponential sinusoid includes an additional term that deforms its shape in the direction of the line joining the two primary masses m1 and m2 ≤ m1. Solutions obtained for the modified spiral result in shorter transfer time and/or reduced ΔV, if compared with solutions obtained on the basis of the classical formulation for the same boundary conditions. In-plane transfers are considered, where the initial spacecraft orbit lies on moon’s orbital plane. Out-of-plane trajectories are not addressed, because orbit plane changes are inexpensive close to L1, and they do not affect significantly transfer time and cost in terms of ΔV, at a preliminary design stage. The modified spiral can be used as is also for preliminary design of transfer trajectories to equilateral Lagrangian points and to L3. Received 19 July 2013; revision received 2 October 2013; accepted for publication 14 October 2013; published online 8 April 2014. Copyright © 2013 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 1533-3884/14 and $10.00 in correspondence with the CCC. *Ph.D. Student, Department of Mechanical and Aerospace Engineering, Corso Duca degli Abruzzi 24. Professor, Department of Engineering, Ecotekne Campus, Building “O”, Strada per Monteroni. Senior Member AIAA.