Optimization of Low-Energy Transfers
In this chapter the optimization of low-energy transfers is treated. The concepts of trajectory optimization are recalled, and the direct transcription strategy is briefly sketched. The restricted three- and four-body problems are described, and their pro
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Optimization of Low-Energy Transfers Francesco Topputo and Edward Belbruno
Abstract In this chapter the optimization of low-energy transfers is treated. The concepts of trajectory optimization are recalled, and the direct transcription strategy is briefly sketched. The restricted three- and four-body problems are described, and their properties are discussed. Two different types of low-energy transfers are optimized: impulsive Earth–Moon exterior low-energy transfers and low-energy, low-thrust transfers to the L1 periodic orbits of the Earth–Moon system. Keywords Low-energy transfers • Trajectory optimization • Ballistic capture • Low-thrust propulsion
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Introduction
The patched-conic method represents the classical technique adopted to design a lunar or an interplanetary transfer. In the case of Earth–Moon, the Hohmann transfer takes typically a few days and requires a cost that depends on the altitudes of the initial and final orbits about the Earth and the Moon, respectively. Since the Hohmann transfer is obtained by patching together two different conic arcs (i.e., an ellipse and a hyperbola in the Earth–Moon case) a hyperbolic excess velocity upon Moon arrival arises. The magnitude of this velocity determines the size of the maneuver required to put the spacecraft into a stable, final Moon orbit.
F. Topputo (*) Dipartimento di Ingegneria Aerospaziale, Politecnico di Milano, Via La Masa 34, 20156 Milano, Italy e-mail: [email protected] E. Belbruno Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, New York 10012, USA e-mail: [email protected] G. Fasano and J.D. Pinte´r (eds.), Modeling and Optimization in Space Engineering, Springer Optimization and Its Applications 73, DOI 10.1007/978-1-4614-4469-5_16, # Springer Science+Business Media New York 2013
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F. Topputo and E. Belbruno
The propellant spent to accomplish such transfers is a monotonic function of the sum of all velocity changes or Dv. Low-energy transfers have been found in the attempt of reducing the Dv of Earth–Moon transfers. The idea behind a low-energy transfer is to extend the model in which the orbits are designed. When two or more gravitational attractions simultaneously act on the spacecraft, the more complex dynamics can be exploited to improve the performances of the transfer trajectories. Thus, in a low-energy transfer, the classical Keplerian decomposition is avoided, and the natural dynamics is exploited in a more efficient way. A method to obtain Earth-to-Moon transfers with no hyperbolic excess velocity at Moon arrival was found about two decades ago by exploiting the intrinsic nature of the Sun–Earth–Moon dynamics [1, 3, 4]. The idea of the exterior low-energy transfers to the Moon consists in departing from a given point near the Earth and eventually flying by the Moon to gain enough energy to go at a distance of approximately four Earth–Moon distance units (1. 5 106 km). In this region, due to the high sensitivity to initial conditions, a negligible Dv is use
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