Trans. Japan Soc. Aero. Space Sci. Vol. 52, No. 175, pp. 47–54, 2009 Technical Note Trajectory Optimization of Multi-Asteroids Exploration with Low Thrust By Kaijian ZHU, Fanghua JIANG, Junfeng LI and Hexi BAOYIN School of Aerospace, Tsinghua University, Beijing, China (Received May 13th, 2008) Multi-asteroid tour missions require consideration of the visiting sequences and trajectory optimization for each leg, which is a typical global optimization problem. In this paper, the problem is divided into a multi-level optimization prob- lem. Determination of the visiting sequence plays a key part for a tour mission. In this paper, the energy differences between different orbits and phase differences are used to estimate the energy required for a tour mission. First, this paper discusses the relation between fuel cost for transfer and classical orbit elements difference based on the energy relation of Keplerian orbits and the characteristics of low-thrust spacecraft. Second, the phase difference is combined with the energy difference to achieve the rendezvous energy. In addition, the lower and upper bounds of rendezvous time can be estimated by analyzing the phase difference. Very-low-thrust trajectory optimization problems have always been considered difficult problems due to the large time scales. In this paper, a hybrid algorithm of PSO (particle swarm optimization) and DE (differential evolution) is used to achieve a solution to the energy-optimal tour mission. Based on GTOC-3 (Global Trajectory Optimization Competition), this paper determines the exploration sequence and provides the optimal solution. Key Words: Low-Thrust, Trajectory Optimization, Hybrid Algorithm 1. Introduction tion theory. Cui2) used the concept of the Lyapunov feed- back control law to derive semi-analytical expressions for With the development of space technology and particu- suboptimal thrust angles, and obtained a near-optimal solu- larly deep-space exploration technology, asteroid explora- tion that approaches the optimal solution by using sequential tions have come into focus. NASA has achieved the quadratic programming (SQP) to adjust the five weights in multi-purpose exploration of Saturn and Titan with the the Lyapunov function. Ulybyshev3) used the inner-point Cassini spacecraft launched in 1997. The Stardust space- algorithm to optimize the Earth elliptic orbit maneuver craft launched in 1999 successfully made a flyby with the by processing the discrete legs. Betts4) investigated orbit Wild2 comet and brought a comet sample back. In 2004, transfer optimization by utilizing the SQP method, reflect- ESA launched the Rosetta spacecraft to discover the secrets ing the advantages and features of low thrust. Coverstone- of a cometary nucleus and which will take 10 years to ren- Carroll5) used a hybrid optimization algorithm that inte- dezvous with the Comet 67 P/Churyumov-Gerasimenko. grated a multi-objective genetic algorithm with a calculus- The HAYABUSA mission led by the Japan Aerospace Explo- of-variations-based low-thrust trajectory optimizer to iden- ration Agency is designed to explore a small near-Earth tify the optimal trajectory of both Earth-Mars and Earth- asteroid named 25143 Itokawa and to return a sample of Mercury missions. Olds6) indicated that DE was a promising material to Earth for further analysis. The DAWN mission global method suited to trajectory optimization. However was successfully launched in 2007 to explore two asteroids the performance of the method is sensitive to the selection with only one spacecraft, which would be a milestone in of the routine’s tuning parameters. Kluever7) applied the asteroid explorations. ‘‘thrust-coast-thrust’’ flight sequence to design the optimal Based on existing engine technology, the thrust required trajectory for Earth-Moon transfer orbit, simulating the by the rocket and the fuel cost of the spacecraft would case from near-Earth circular orbit to the lunar polar orbit. exceed human tolerances if a spacecraft were designed to Considering the multi-revolution low-thrust trajectory opti- explore a celestial body directly. The trajectory must in- mization with the ‘‘thrust-coast-thrust’’ flight sequence. clude many orbit maneuvers and wide use of a multi- Hull8) converted optimal control problems into parameter impulse transfer, low-thrust spacecraft has advantages of optimization problems and accomplished the method by great efficiency and high precision. A great deal of recent replacing the control and state histories by control and state work involves the trajectory optimization using low-thrust parameters and forming the histories by interpolation. techniques. Many methods and theorems have been used If a spacecraft is designed to explore as many asteroids to solve the problem. Based on Bellman’s principle of opti- as possible, it is important to determine the exploration mality, Ross1) developed an anti-aliasing trajectory optimi- sequence and optimize the flight trajectory so the spacecraft zation method relating low-thrust trajectory optimization can rendezvous with the asteroids in proper time, which problems to the well-know problem of aliasing in informa- has become the focus of research. Generally, the enumera- tion method applied by the STOUR9) software can search Ó 2009 The Japan Society for Aeronautical and Space Sciences 48 Trans. Japan Soc. Aero. Space Sci. Vol. 52, No. 175 all the global optimal solutions. However, the computing parameter optimization problems including the launch date, time and space costs increase as dimensions and problem flight time, stay-time, magnitude and direction of thrust. complexity increase, and cannot be handled by a personal computer. 3. Dynamic Model Based on the Global Trajectory Optimization Compe- tition (GTOC-3)10) organized by the Dipartimento di The motion of a body is described by a system of second- Energetica of the Politecnico di Torino from October to order ordinary differential equations: December 2007, this paper provides an algorithm for select- r r€ þ ¼ a ð2Þ ing candidate asteroids based on the energy and phase rela- r3 p tion and for determining the flight sequence considering constraints on launch window and flight time. Modified where the radius r ¼k r k represents the magnitude of the equinoctial elements are used in numerical integration using inertial position vector. is the gravitational constant of non-dimensional forms. Finally, a heuristic algorithm is the central body. ap is defined as perturbation acceleration. used for global optimization, and a pattern search algorithm For the two-body problem dynamic model, we process the is used for local optimization. low thrust as a perturbation on the small side. In the inertial coordinate, the parameters change rapidly and the classical 2. Problem Description orbit elements exhibit singularities for e ¼ 0, and i ¼ 0, 90. The dynamic model in Cartesian coordinates reduces GTOC-3 chose 140 near-Earth asteroids as candidates the effect of optimization. An appropriate dynamical model from the JPL space objects database. A spacecraft launched not only affects computing time, but also determines the from Earth must rendezvous with three selected asteroids integral accuracy. Kechichian11) analyzed the near-Earth during its tour and return to Earth. The initial mass of the orbit transfer of low-thrust spacecraft where a dynamic spacecraft is 2000 kg which can be used as propulsion fuel. model was defined using a modified set of equinoctial coor- The hyperbolic velocity excess of the spacecraft relative to dinates, which could avoid singularities in the classical ele- the Earth is 0.5 km/s and the direction can be set at will. The ments. However, the integration steps must be increased to maximum magnitude of the engine is 0.15 N with specified maintain the integration speed and lower integration preci- direction and a specific impulse of 3000 s. The launch win- sion. We applied the non-dimensional modified equinoctial dow is from 2016 to 2025 and the whole flight time is less elements to avoid singularities during the numerical integra- than 10 years. The stay-time at each of the three asteroids tion. The equations of motion are described as follows:12) must be at least 60 days. The effect of Earth and the aste- h_ ¼ h Á n Á ft roids are neglected in the whole trajectory, and Earth flybys _ at altitudes exceeding 6871 km can be considered. The f ¼ h Á sin L Á fr þ½h Á cos L þ n ðcos L þ f Þ Á ft objective of the optimization is to maximize the nondimen- À n Á X Á g Á fn sional quantity: g_ ¼h Á cos L Á fr þ½h Á sin L þ n ðsin L þ gÞ Á ft m ð Þ þ n Á X Á f Á fn f minj¼1;3 j ð3Þ J ¼ þ K ð1Þ 1 mi max h_ ¼ Á n Á s2 Á cos L Á f 2 n where mi and mf are the initial and final mass of the space- 1 k_ ¼ Á n Á s2 Á sin L Á f craft, respectively. j (j ¼ 1; 2; 3) represents the stay-time at 2 n the j-th asteroid in the rendezvous sequence and minj¼1;3ðjÞ 1 _ ¼ ¼ L ¼ n Á X Á fn þ is the shortest asteroid stay-time. K 0:2, max 10 years n2 Á h Á is the journey time. h First, we analyze the objective function. During the long where, n ¼ , X ¼ h sin L À k cos L, journey, we must try to save the fuel and prolong the stay- 1 þ f cos L þ g sin L pffiffiffiffiffiffiffi time at the rendezvous bodies. Only one long stay-time s2 ¼ 1 þ h2 þ k2, h ¼ p. p is defined by p ¼ að1 À 2 has no effect on the object function because the shortest e Þ. fr, ft and fn are defined as the thrust accelerations in a stay-time will be utilized. If the stay-time of the spacecraft local radial-tangential-normal (RTN) rotating frame.