A study was conducted to demonstrate low-thrust minimum-fuel optimization in the circular restricted three-body problem. The study solved the problem of geostationary transfer orbit (GTO)-to-halo transfer for the first time. This result was achieved with an indirect approach and constant specific impulse engine. Thrust-to-mass ratios in agreement with currently available technology were considered. Some effective techniques were applied to cope with problem complexity. These methods involved solving the minimum-fuel, minimum energy, and minimum-time problems, implementing energy-to-fuel homotopy, continuing the maximum thrust magnitude, and computing the analytic Jacobians
Current approaches to uncertainty propagation in astrodynamics mainly refer to linearized models or Monte Carlo simulations. Naive linear methods fail in nonlinear dynamics, whereas Monte Carlo simulations tend to be computationally intensive. Differential algebra has already proven to be an efficient compromise by replacing thousands of pointwise integrations of Monte Carlo runs with the fast evaluation of the arbitrary order Taylor expansion of the flow of the dynamics. However, the current implementation of the DA-based high-order uncertainty propagator fails when the non-linearities of the dynamics prohibit good convergence of the Taylor expansion in one or more directions. We solve this issue by introducing automatic domain splitting. During propagation, the polynomial expansion of the current state is split into two polynomials whenever its truncation error reaches a predefined threshold. The resulting set of polynomials accurately tracks uncertainties, even B Alexander Wittig
The low-thrust version of the low energy transfers to the Moon exploiting the structure of the invariant manifolds associated to the Lagrange point orbits is presented in this paper. A method to systematically produce low-energy, low-thrust transfers executing ballistic lunar capture is discussed. The coupled restricted three-body problems approximation is used to deliver appropriate first guess for the subsequent optimization of the transfer trajectory within a complete four-body model using direct transcription and multiple shooting strategy. It is shown that less propellant than standard low energy transfers to the Moon is required. This paper follows previous works by the same authors aimed at integrating together knowledge coming from dynamical system theory and optimal control problems for the design of efficient low-energy, low-thrust transfers
A method to design ballistic capture orbits in the real solar system model is presented, so extending previous works using the planar restricted three-body problem. In this generalization a number of issues arise, which are treated in the present work. These involve reformulating the notion of stability in threedimensions, managing a multi-dimensional space of initial conditions, and implementing a restricted n-body model with accurate planetary ephemerides. Initial conditions are categorized into four subsets according to the orbits they generate in forward and backward time. These are labelled weakly stable, unstable, crash, and acrobatic, and their manipulation allows us to derive orbits with prescribed behavior. A post-capture stability index is formulated to extract the ideal orbits, which are those of practical interest. Study cases analyze ballistic capture about Mercury, Europa, and the Earth. These simulations show the effectiveness of the developed method in finding solutions matching mission requirements.
A method for the nonlinear propagation of uncertainties in Celestial Mechanics based on differential algebra is presented. The arbitrary order Taylor expansion of the flow of ordinary differential equations with respect to the initial condition delivered by differential algebra is exploited to implement an accurate and computationally efficient Monte Carlo algorithm, in which thousands of pointwise integrations are substituted by polynomial evaluations. The algorithm is applied to study the close encounter of asteroid Apophis with our planet in 2029. To this aim, we first compute the high order Taylor expansion of Apophis' close encounter distance from the Earth by means of map inversion and composition; then we run the proposed Monte Carlo algorithm to perform the statistical analysis.
In this paper we incorporate the low-thrust propulsion in the stable manifold technique to design transfer trajectories to the halo orbits around L 1 and L 2 of the Earth-Moon system. The problem is stated in an optimal control scheme and solved using direct transcription and collocation; the dynamics is discretized over an uniform time grid using a sixth-order linear multi-point method. The resulting transfers are made up by a spiral arc that targets a piece of the stable manifold associated to the final orbit. Thanks to the generality of this approach, halo-to-Moon transfers are also computed combining unstable manifolds and low-thrust. Furthermore, complete Earth-to-Moon transfers via halos can also be constructed. Results show the feasibility of this kind of transfers requiring moderate propellant mass fractions and feasible times of flight.
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