Benjamin F. Villac

Classification of the distant stability regions at europa

Long-term stable trajectories around Europa, one of the Galilean moons of Jupiter, are analyzed for their potential applications in spacecraft trajectory design, such as end-of-mission disposal options, backup orbits, or intermediary targets for transfer trajectories. The phase space is analyzed via the computation of families of periodic orbits and the estimation of their stability character. Although the core analysis of the paper uses the circular restricted three-body problem, a selected set of long-term stable solutions is checked by integrating the corresponding initial conditions in an ephemeris model over several years. The current model and methods can be readily applied to other planetary satellites including the other Galileans at Jupiter, the many satellites at Saturn, and the Earths moon.

Adaptive Gravitational Force Representation for Fast Trajectory Propagation Near Small Bodies

approximationusespolynomialinterpolationwithanadaptivespatialdatastructureneartheasteroidandspherical harmonics far from it. These data structures allow us to drive the approximation errors of the model to within userdefined thresholds, while significantly reducing the run time of trajectory integrations about small bodies. This alleviates the computational burden of Monte Carlo simulation for small-body proximity operation and mission design. We conclude with performance tests and models for the asteroid 1998 ML 14.

Dynamical Analysis of 1:1 Resonances Near Asteroids - Application to Vesta

Motivated by the challenging transfer across the 1:1 resonance planned for the Dawn mission, this paper describes the dynamical structure of the near-synchronous orbits and their manifolds near asteroid 4 Vesta. The dynamics are studied in a surface-fixed rotating frame, where equilibrium points representing “geostationary” orbits and near-synchronous libration orbit families are characterized. The invariant manifold structure that arises from the unstable libration orbit families describes a pathway for designing ballistic transfers across the 1:1 resonance. Examples of such transfers across the 1:1 resonance at Vesta are provided.

Heuristic Search-Based Framework for Onboard Trajectory Redesign

This paper presents a framework for automated impulsive transfer design with a view toward onboard application. A database of periodic orbits and other structurally important trajectories is combined with large-scale simulations to create a discrete representation of the system dynamics. A heuristic search is used to quickly find paths in this discretization, which generate a set of arcs and impulses approximating a continuous transfer between boundary conditions specified in the original system. Differential correction and partial optimization are used to create a final, feasible transfer. Each step will be described, with a focus on the construction of a weighted, directed graph representation of the transfer problem, as well as the search process and its relation to the final transfer.

Two Classes of Cycler Trajectories in the Earth-Moon System

Anticipating the need for infrastructure in cislunar space to support telecommunications, navigation, and crew and cargo transportation, the available periodic orbits that encounter both the Earth and the moon, termed cyclers, should be characterized and their utility should be evaluated using appropriate metrics. Using the planar circular restricted three-body problem to model spacecraft motion in the Earth-moon system, a classification scheme based on resonant orbits and homoclinic connections is proposed to summarize two broad cycler classes. One class consists of high-energy near-elliptical cyclers; the second class consists of low-energy cyclers that use the dynamical structure associated with the collinear libration point between the Earth and moon. The first class is organized around resonant cyclers that can be easily characterized and are representative of the whole class. The second class of cyclers is organized by the homoclinic connections associated with the unstable Lyapunov family of periodic orbits around the libration point between the Earth and moon. The homoclinic connections bound sets of cyclers, provide close approximations of the cyclers, and can thus be used to identify their characteristics. Computational methods for both cycler classes, based on continuation and differential correction, are discussed and demonstrated.

Structure Preserving Approximations of Conservative Forces for Application to Small-Body Dynamics

Approximation-based methods, such as the cubetree algorithm, have proven to be significantly faster than traditional methods for complex force evaluations near small irregular bodies. Such methods also hold the promise of simplifying the inclusion of experimental data to update the force model. However, the cubetree algorithm does not preserve intrinsic properties of the gravitational force such as continuity, divergence freedom, or exactness. These properties may be needed for trajectory optimization, for the use of geometric (e.g., symplectic) integrators for long-term propagation, and for other trajectory design problems. This paper presents several adaptive schemes preserving global continuity, exactness, or divergence freedom, and discusses the difficulties involved in preserving all of these properties globally.

Static Solutions of the Hamilton-Jacobi-Bellman Equation for Circular Orbit Transfers

A LYAPUNOV control law was proposed by Chang et al. [1] for transfer between Keplerian orbits. For circular orbits, a Lyapunov function was chosen based on the energy E and the angularmomentumL, and a control lawwas derived in order tomake the derivative of the Lyapunov function negative-definite. However, the optimality of the control law was not analyzed. In this Note, the optimality of these transfers is shown by deriving a cost function for which these transfers are optimal. In particular, the link between the Lyapunov approach and the static solution of the Hamilton–Jacobi– Bellman equation (HJBE) for the two-body orbit transfer is shown. Moreover, it is shown that for long times of flight, constantpropulsive-acceleration transfers can be designed in feedback form with almost the same fuel cost in comparison with fuel-optimal transfers. Bonnard et al. [2] considered the geometric structure of timeoptimal control for spacecraft transfer in elliptic Keplerian orbits. A time-optimal control problem was proposed for constant specific impulse engines and the controllability of the system was analyzed. The methodology to compute the optimal control was based on the traditional indirect method, which was solved numerically by means of a shooting method. Freeman and Kokotovic [3] introduced the concept of a robust control Lyapunov function (rclf) and showed that every rclf satisfies the steady-state Hamilton–Jacobi–Isaacs equation associated with a meaningful game. Indeed, having a control Lyapunov function, it can be used for determination a stabilizing feedback control that is optimal in some sense. This technique is known as inverse optimality method. As an example, Bharadwaj et al. [4] used this method to design globally stabilizing attitude control laws. Primbs et al. [5] investigated the relation between optimal control problems and control Lyapunov functions from a receding-horizon perspective. In that work, it was shown that a broad class of Lyapunov controllers admit natural extensions to recedinghorizon formulations, which are locally optimal. However, these ideas were of a general character and applied only to a simple nonlinear system. In this Note, these ideas are applied specifically for mediumto low-thrust spacecraft orbital transfer in a two-body setting. In particular, a closed-form solution of an infinite-horizon HJBE for two-body, circular-orbit transfers is derived and is compared with fuel-optimal finite time transfers. The solution of the HJBE provides the optimal control at every point in phase space in feedback form. However, the nonlinearity of this partial differential equation (PDE)makes it extremely difficult to solve analytically. For linear systems subject to quadratic cost functions, it is possible to obtain an analytic form for the optimal cost [6], but for general nonlinear systems that are of practical interest, this is not the case in general. The numerical approach to solve this PDE is still challenging, as the solutions are in general not smooth and differentiable at every point. However, the great drawback of dynamic programming is, as Bellman himself calls it, “the curse of dimensionality.” Even recording the solution to a moderately sized problem involves an enormous amount of storage. If only an optimal path from a known initial point is desired, it is wasteful and tedious to find a whole field of extremals [7]. After a review of the main result obtained by Chang et al. [1], the static form of the HJBE is described and the optimality of the Lyapunov control law for a regulized fuel-optimal cost function is shown. The application of this control law is illustrated with a threedimensional transfer from a low Earth orbit (LEO) to the geostationary Earth orbit (GEO). The results are shown to be applicable to both variable specific impulse (VSI) and engines with constantpropulsive-acceleration capability. The examples illustrate the relation between the (infinite-horizon) transfers and actual finite time, fuel-optimal solutions. Finally, the application of thismethod to provide good initial guesses of the costates (relevant to the indirect formulation and solution of the optimal control problem) is explored.

Modified Picard Integrator for Spaceflight Mechanics

This paper investigates the applicability of the Parker–Sochacki Picard iteration method for spaceflight mechanics. Notably, solving initial value problems, parallel integration of multiple trajectories, and high-order state transition tensor computations are addressed. This paper presents a systematic approach to transforming vector fields to polynomial form, which is required for the method investigated. The application of this approach is presented on a few vector fields, including spherical harmonics gravity field. The initial cost of transforming the vector field can be justified by the performance of the initial value problem solver. Additionally, this polynomial form of the vector field is used in the calculation of state transition tensors with differentiation of polynomials only. This reduces the computations required to obtain state transition tensors for some problems and is discussed in this paper. An efficient representation of vector fields in polynomial form benefits the implementation of t...

Mapping Autonomous Constellation Design Spaces Using Numerical Continuation

In response to an anticipated need for space-based navigation and communication infrastructure in the larger cislunar region, this research explores the use of numerical-continuation techniques to support the design of autonomous Earth–moon constellations. Based on the Linked Autonomous Interplanetary Satellite Orbit Navigation technique, the design of such architectures is a challenging task, due in part to the computational effort required to analyze a single-spacecraft configuration. Although an optimization formulation coupled with a continuation-based method was previously developed by the authors to tackle such issues, the direct application of this technique still remained computationally expensive and was only demonstrated using a heuristically reduced cost function. This paper both simplifies and extends the previous approach by carefully studying the spacecraftplacement problem and decoupling the continuation methods into tractable subproblems. Using this decoupling strategy, exhaustive computational bottlenecks are relieved, and a deeper insight into the global behavior of solution manifolds is achieved. Results from two case studies are presented for the halo and axial periodic orbit families in the restricted Earth–moon system.

USING SOLAR ARRAYS FOR ORBITAL CONTROL NEAR SMALL BODIES TRADE-OFFS CHARACTERIZATION

Solar radiation pressure has long been recognized as a force to harness for orbital control using a solar sail, but has otherwise been considered as a perturbation to mitigate. In the case of small body orbiters, however, solar radiation pressure can be on the same order of magnitude as the gravitational attraction, even for spacecraft not equipped with solar sails. In this paper, we characterize the expected magnitude of solar radiation pressure on solar panels and its potential use for orbital control of small body orbiters. In particular, we focus on the trade-offs between control authority, electric power generation, and the potential performance of such systems.