Multiobjective genetic optimization of Earth-Moon trajectories in the restricted four-body problem

In this paper, optimal trajectories of a spacecraft traveling from Earth to Moon using impulsive maneuvers (ΔV maneuvers) are investigated. The total flight time and the summation of impulsive maneuvers ΔV are the objective functions to be minimized. The main celestial bodies influencing the motion of the spacecraft in this journey are Sun, Earth and Moon. Therefore, a three-dimensional restricted four-body problem (R4BP) model is utilized to represent the motion of the spacecraft in the gravitational field of these celestial bodies. The total ΔV of the maneuvers is minimized by eliminating the ΔV required for capturing the spacecraft by Moon. In this regard, only a mid-course impulsive maneuver is utilized for Moon ballistic capture. To achieve such trajectories, the optimization problem is parameterized with respect to the orbital elements of the ballistic capture orbits around Moon, the arrival date and a mid-course maneuver time. The equations of motion are solved backward in time with three impulsive maneuvers up to a specified low Earth parking orbit. The results show high potential and capability of this type of parameterization in finding several Pareto-optimal trajectories. Using the non-dominated sorting genetic algorithm with crowding distance sorting (NSGA-II) for the resulting multiobjective optimization problem, several trajectories are discovered. The resulting trajectories of the presented scheme permit alternative trade-off studies by designers incorporating higher level information and mission priorities.

[1]  Victoria Coverstone-Carroll,et al.  Near-Optimal Low-Thrust Trajectories via Micro-Genetic Algorithms , 1997 .

[2]  K. Yagasaki.  Computation of low energy Earth-to-Moon transfers with moderate flight time , 2004 .

[3]  A. Burbanks,et al.  Dynamics of capture in the restricted three-body problem , 2003, astro-ph/0310618.

[4]  Kazuyuki Yagasaki,et al.  Sun-perturbed earth-to-moon transfers with low energy and moderate flight time , 2004 .

[5]  À. Jorba,et al.  THE LAGRANGIAN POINTS OF THE REAL EARTH-MOON SYSTEM , 2005 .

[6]  Shane D. Ross,et al.  Low Energy Transfer to the Moon , 2001 .

[7]  Guo-Jin Tang,et al.  Optimal Multi-Objective Nonlinear Impulsive Rendezvous , 2007 .

[8]  D. Vallado.  Fundamentals of Astrodynamics and Applications , 1997 .

[9]  Craig A. Kluever,et al.  OPTIMAL EARTH-MOON TRAJECTORIES USING COMBINED CHEMICAL-ELECTRIC PROPULSION , 1996 .

[10]  P. B. Richards,et al.  Some periodic solutions of a four-body problem , 1964 .

[11]  Bion L. Pierson,et al.  Three-stage approach to optimal low-thrust Earth-moon trajectories , 1994 .

[12]  J. K. Miller,et al.  Sun-Perturbed Earth-to-Moon Transfers with Ballistic Capture , 1993 .

[13]  V. Szebehely,et al.  Theory of Orbits: The Restricted Problem of Three Bodies , 1967 .

[14]  Anil V. Rao,et al.  Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method , 2006 .

[15]  M. Lo,et al.  Abstract: Low Energy Interplanetary Transfers Using the Invariant Manifolds of L1, L2, and Halo Orbits , 1998 .

[16]  Bion L. Pierson,et al.  Optimal low-thrust three-dimensional Earth-moon trajectories , 1995 .

[17]  D. D. Mueller,et al.  Fundamentals of Astrodynamics , 1971 .

[18]  Shane D. Ross,et al.  Invariant Manifolds, the Spatial Three-Body Problem and Space Mission Design , 2001 .

[19]  Gerard Gómez,et al.  Dynamics and Mission Design Near Libration Points , 2001 .

[20]  K. H. Tsui.  Satellite capture in a four-body system , 2002 .

[21]  Massimiliano Vasile,et al.  On the Solution of Interplanetary Trajectory Design Problems by Global Optimisation Methods , 2005 .

[22]  Daniel J. Scheeres,et al.  The Restricted Hill Four-Body Problem with Applications to the Earth–Moon–Sun System , 1998 .

[23]  Ya-zhong Luo,et al.  Optimization of multiple-impulse minimum-time rendezvous with impulse constraints using a hybrid genetic algorithm , 2006 .

[24]  Peter J. Fleming,et al.  An Overview of Evolutionary Algorithms in Multiobjective Optimization , 1995, Evolutionary Computation.

[25]  Àngel Jorba,et al.  On the vertical families of two-dimensional tori near the triangular points of the Bicircular problem , 2000 .

[27]  B. Barden,et al.  Application of Dynamical Systems Theory to Trajectory Design for a Libration Point Mission , 1997 .

[28]  Shane D. Ross,et al.  Shoot the Moon , 2000 .

[29]  R. Brusch.  Constrained Impulsive Trajectory Optimization for Orbit-to-Orbit Transfer , 1979 .

[30]  Cesar A. Ocampo,et al.  Indirect Optimization of Spiral Trajectories , 2006 .

[31]  Carlos A. Coello Coello,et al.  An updated survey of GA-based multiobjective optimization techniques , 2000, CSUR.

[32]  Ivo F. Sbalzariniy,et al.  Multiobjective optimization using evolutionary algorithms , 2000 .

[33]  Gianmarco Radice,et al.  Ant Colony Algorithms for Two-Impulse Interplanetary Trajectory Optimization , 2006 .

[34]  John W. Hartmann,et al.  Optimal multi-objective low-thrust spacecraft trajectories , 2000 .

[35]  A. Prado.  Numerical and analytical study of the gravitational capture in the bicircular problem , 2005 .

[36]  E. Pérez-Chavela,et al.  Motions close to escapes in the rhomboidal four body problem , 1993 .

[37]  Z. Makó,et al.  Capture in the Circular and Elliptic Restricted Three-Body Problem , 2004 .

[38]  Gerard Gómez,et al.  Dynamics and Mission Design Near Libration Points: Volume III: Advanced Methods for Collinear Points , 2001 .

[39]  Shinji Suzuki,et al.  Modified Genetic Algorithm for Constrained Trajectory Optimization , 2005 .

[40]  Sunil K. Agrawal,et al.  Optimization of Dynamic Systems , 1999 .

[41]  Kalyanmoy Deb,et al.  Muiltiobjective Optimization Using Nondominated Sorting in Genetic Algorithms , 1994, Evolutionary Computation.

[42]  Kalyanmoy Deb,et al.  Multi-objective optimization using evolutionary algorithms , 2001, Wiley-Interscience series in systems and optimization.

[43]  J Llibre,et al.  Dynamics and Mission Design Near Libration Points: Volume II: Fundamentals: The Case of Triangular Libration Points , 2001 .

[44]  Miguel Àngel Andreu Barrieras.  The quasi-bicircular problem , 1998 .

[45]  Randy L. Haupt,et al.  Practical Genetic Algorithms , 1998 .