Iac-05- C1.5.05 Trajectory Optimization for the Hevelius – Lunar Microsatellite Mission

IAC-05- C1.5.05 TRAJECTORY OPTIMIZATION FOR THE HEVELIUS – LUNAR MICROSATELLITE MISSION Camilla Colombo, Matteo Ceriotti, Ettore Scarì, Massimiliano Vasile Politecnico di Milano Via La Masa, 34, 20156 Milano, Italy [email protected] ABSTRACT In this paper trajectory optimisation for the Hevelius mission is presented. The Hevelius – Lunar Microsatellite Mission – is a multilander mission to the dark side of the Moon, supported by a relay microsatellite, orbiting on a Halo orbit around L2. Three landers, with miniaturized payloads, are transported by a carrier from a LEO to the surface of the Moon, where they perform a semi-hard landing with an airbag system. This paper will present the trajectory optimisation process, focusing, in particular, on the approach employed for Δv manoeuvre optimization. An introduction to the existing methods for trajectory optimization will be presented, subsequently it will be described how these methods have been exploited and originally combined in the Hevelius mission analysis and design. of three landers, with miniaturized payloads, INTRODUCTION that have to be transported by a carrier from a The silvery Moon has gained a renewed LEO to the surface of the Moon. In addition, a interest in recent times, because of both data relay microsatellite has to support the net- technological and scientific implications. lander on the dark side, orbiting on a Halo Therefore, after the Apollo era and the more orbit around L2. recent Clementine (1994) and Lunar To minimize Δv, a number of options have Prospector (1998) missions, a new exploration been devised, exploiting multi-body dynamics. phase of our natural satellite is envisaged in the The concept of using stable manifolds of the near future. On the other hand, the tight restricted three-body problem (RTBP) to constraints on space mission cost and available design low-cost missions has been studied by transportation systems bring about the need for Howell et al. [17] to determine appropriate a reduction in the total mass and consequently solutions for geocentric transfers. By a minimization of the total required Δv. perturbing the insertion conditions in the In order to answer to the need for cheap direction of the stable eigenvector, the scientific missions to the Moon, a multilander spacecraft is placed on the stable manifold mission to her dark side has been recently associated to the periodic orbit, thus permitting studied. The mission, called Hevelius, consists globalization of the trajectory by integrating 1 the equations of motion backward in time to a As operative orbit for the data-relay satellite, position near Earth. several Halo orbits around the point L2 of the A sensible contribute in the study of transfer Earth-Moon system have been investigated by orbits to libration points through manifold the linearization of the equation of motion exploitation has been given by Koon, Lo et al. around L2, followed by a shooting procedure, [18][19], Gómez et al. [12][13], Starchville or by using a third-order-approximated and Melton [23][24]; this strategy has revealed dynamic model, refined with an SQP its effectiveness in the design of low-energy procedure. Trade off of the different orbits was transfers to the Moon. based on maintenance cost, amplitude and slew On the other hand, Belbruno et al. angles. [1][2][3][4] proposed new trajectories, Earth-Moon transfer exploits stable exploiting weak stability boundaries (WSB) of manifolds leaving Halo orbit, calculated by a the Earth-Sun-Moon system. For this kind of backward integration in the RTBP. The total trajectories a long travel time is required but Δv required to insert the spacecraft onto the with a significant reduction in propellant mass. stable manifold has been initially set as the These innovative concepts required the fitness function of a Genetic Algorithm based development of specific tools for trajectory process. This process yielded a number of first design. In particular, the chaotic dynamics guess solutions accurately optimised with a governing those trajectories implies the need SQP solver. In order to phase the departure for methods that assure global convergence at orbit, the drift effect of J2 and the perturbation least to a local optimal solution. A possible effect of the Moon and the Sun have been way to tackle this problem is to generate first exploited. guess solutions by using hybrid methods, that The operative orbit selected for the carrier is combine a global research by Evolutionary a Frozen orbit. The aim of the carrier transfer Programs and a local optimization by orbit is to connect a LEO with the Frozen orbit, Sequential-Quadratic Programming (SQP). with the minimum fuel consumption. This Genetic algorithms have been used to solve transfer exploits the WSB region, in order to difficult problems with objective functions that obtain a free change of inclination and the do not possess a convenient shape (Davis [7], required increase of the perigee with a small Goldberg [9], Holland [16], Michalewictz impulsive manoeuvre. For this problem the [20]). These algorithms maintain and software DITAN [26] has been used. manipulate a population of solutions and implement a “survival of the fittest” strategy in 1. THE HEVELIUS MISSION their search for better solutions, so they can Hevelius project is a pre-phase A analysis provide for a good initial guess for the on a multilander mission to the dark side of the optimization. Besides the method used in the Moon, supervised by a relay microsatellite, SQP optimization is an active set strategy [5] orbiting on a Halo orbit around L2. Three (also known as a projection method), similar to landers, with miniaturized payloads, are that of Gill et al., described in [10] and [11]. transported by a carrier from a LEO to a low- The solution procedure involves two phases. altitude point above the surface of the Moon, The first phase involves the calculation of a from which they perform a semi-hard landing, feasible point (if one exists). The second phase with an airbag system. The transfer exploits the involves the generation of an iterative weak stability boundaries of the Earth-Sun- sequence of feasible points that converge to the Moon system. The landers have been design to solution. withstand the landing impact and the harsh In the design of the Hevelius mission, these thermal environment in order to survive during different methods for trajectory optimization the lunar night. The data relay satellite, and multi-body dynamics have been launched as a secondary payload on the ASAP investigated in order to design low cost platform of Ariane 5, reaches the Halo orbit by trajectories, to reduce the propellant mass and exploiting the L2 manifolds. It has to support to fulfil the launcher requirements. the net-lander on the dark side of the Moon. 2 Figure 1: The three halo orbits 2. L2-HALO ORBIT DESIGN 2.2. Shooting A first guess for the initial state vector has 2.1. Operative orbit selection been computed analytically with a linearization The choice of the final operative orbit for of the equation of motion around L2: the data relay satellite is driven by the ⎧xn$$=+2 yUx $ xx necessity to support a net-lander on the dark ⎪ side of the Moon; particularly in order to create ⎨$$yn=−2 xUy $ + yy ⎪ a constant link between ground stations and the ⎩$$zUz= zz landers, a solution for which both the Earth where UUU,, are constants. The and the far side of the Moon would always be xxyyzz in the spacecraft field of view has to be derived Lissajous orbit is characterized by an investigated. In addition the presence of a in-plane (x-y synodic plane) frequency LODE (Lagrangian Orbit Determination different from the out-of-plane frequency and Experiment) imposes the choice of a periodic corresponds to a non closed solution. orbit around the second collinear libration Additionally it is valid only in a linear point of the Earth-Moon system. Hence a approximation and for restricted amplitudes of restricted three-body problem dynamics has motion. been studied, leading to the design of a Quasi- Since a closed solution is needed, a shooting Halo orbit suitable for the mission. procedure using complete RTBP dynamics is The non linearity of the problem and needed. Halo orbits determined in this way are consequently the strong dependence on the always contained in the x-y plane, i.e. they initial conditions brought the difficult task to have no out of plane motion. find appropriate first guess solutions to be used Since appropriate amplitudes in the y-z with a SQP-based shooting procedure. plane are required, a modified shooting has Three different approaches have been been studied, which assumes a given z initial followed changing constraint conditions and velocity (out-of-plane), and determines initial x objective function. The dynamics is and y velocities and the period of the Halo. adimensionalised in a rotating x, y, z cartesian Greater amplitude Halos have been found reference frame centred in the Earth-Moon increasing progressively the amplitudes of centre of mass, with the x-y plane coinciding motion until the desired values are achieved, with the plane of motion of the primaries and with the objective to minimize the manoeuvre the x-axis pointing along the line connecting required to have a periodic motion. The in the two bodies away from the larger primary. plane amplitudes are not significant while appropriate z-y amplitudes (plane perpendicular to the synodic one) are required 3 4 x 10 3 3000 3000 2000 2 2000 1000 1000 1 0 0 0 Z [Km] Z [Km] Z -1000 Y [Km] -1000 -1 -2000 -2000 -3000 -3000 -2 -4000 -4000 -3 -3 -2 -1 0 1 2 3 4.32 4.34 4.36 4.38 4.4 4.42 4.44 4.46 4.48 4.5 4.52 4.32 4.34 4.36 4.38 4.4 4.42 4.44 4.46 4.48 4.5 4.52 Y [Km] 4 X [Km] 5 X [Km] 5 x 10 x 10 x 10 Figure 2: Projection on three planes of the chosen halo orbit; the hatched (red) line is the first-guess Halo, the continuous (blue) line is the final Halo.

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