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The direct collocation method transforms a trajectory optimization problem into a nonlinear programming (NLP) problem by discretizing both control and state variables. During the NLP solution process, repeated calculations of the first and second derivatives of the NLP and the values of the dynamic system at each discrete point are required, leading to great computational complexities. Therefore, this paper proposes the following method: First, the hyper-dual number method is introduced to accurately identify the sparsity of the second-derivative matrix of the NLP and to determine the locations of the non-zero elements. Then, a multi-core parallel approach is used to rapidly compute the non-zero elements of the first and second derivatives of the NLP as well as the values of the dynamic system at each discrete point. Finally, OpenMP is employed for programming calculation in the C++ environment to further enhance computational efficiency from the perspective of programming language. Simulation results demonstrate that the proposed method effectively improves the efficiency of trajectory optimization and its computational efficiency without compromising accuracy. 直接配点法通过对控制变量和状态变量都进行离散将轨迹优化问题转化为非线性规划(nonlinear programming, NLP)进行求解。在求解NLP时, 需要反复计算NLP的一阶/二阶偏导数和动力学系统在各离散点处的值, 计算量比较大。针对该问题, 提出如下解决策略: 引入超对偶数方法准确识别NLP二阶偏导数矩阵的稀疏型, 确定其中非零元素的位置; 采用多核并行方式快速计算NLP的一阶/二阶偏导数的非零元素以及动力学系统在各离散点处的值; 在C++环境下采用OpenMP方式进行编程计算, 从编程语言角度进一步提高计算效率。仿真结果表明, 文中方法给出的策略在不影响精度的情况下, 均能显著提高轨迹优化效率。

Most of the existing studies use constant force to reduce springback while researching stretch force. However, variable stretch force can reduce springback more efficiently. The current research on springback prediction in stretch bending forming mainly focuses on artificial neural networks combined with the finite element simulation. There is a lack of springback prediction by support vector regression（SVR）. In this paper, SVR is applied to predict springback in the three-dimensional stretch bending forming process, and variable stretch force trajectory is optimized. Six parameters of variable stretch force trajectory are chosen as the input parameters of the SVR model. Sixty experiments generated by design of experiments（DOE） are carried out to train and test the SVR model. The experimental results confirm that the accuracy of the SVR model is higher than that of artificial neural networks. Based on this model, an optimization algorithm of variable stretch force trajectory using particle swarm optimization（PSO） is proposed. The springback amount is used as the objective function. Changes of local thickness are applied as the criterion of forming constraints. The objection and constraints are formulated by response surface models. The precision of response surface models is examined. Six different stretch force trajectories are employed to certify springback reduction in the optimum stretch force trajectory, which can efficiently reduce springback. This research proposes a new method of springback prediction using SVR and optimizes variable stretch force trajectory to reduce springback.

Significant propellant mass saving can be obtained with the use of complex multiple intermediate flyby maneuvers for conventional propulsion systems, and trip time also decreases for a portion of the proper solar sail missions. This paper discusses the performance of gravity assist （GA） in the time-optimal control problem of solar sailing with respect to sail lightness number and the energy difference between the initial and final orbit in the rendezvous problem in a two-body model, in which the GA is modeled as a substantial change in the velocity of the sailcraft at the GA time. In addition, this paper presents a method to solve the time-optimal problem of solar sailing with GA in a full ephemeris model, which introduces the third body＇s gravity in a dynamic equation. This study builds a set of inner constraints that can describe the GA process accurately. Finally, this study presents an example for evaluating the accuracy and rationality of the two-body model＇s simplification of GA by comparison with the full ephemeris model.

The lunar probe often has some remaining fuel on completing the predefined Moon exploration mission and may carry out some additional tasks from the Moon orbit using the fuel. The possibility for the lunar probe to escape from the Moon and the Earth is analyzed. Design and optimization of the trajectory from the Moon orbit to the Near Earth Asteroids （NEAs） using the spacecraft＇s residual fuel is studied. At first, the semi-major axis, inclinations and the phase relations with the Earth of all the numbered NEAs are investigated to preliminarily select the possible targets. Based on the Sun-centered two-body problem, the launch window and the asteroid candidates are determined by calculating the minimum delta-v for two-impulse rendezvous mission and one-impulse flyby mission, respectively. For a precise designed trajectory, a full ephemeris dynamical model, which includes gravities of the Sun, the planets and the Moon, is adopted by reading the JPL ephemeris. The departure time, arrival time, burning time duration and thrust angles are set as variables to be designed and optimized. The optimization problem is solved via the Particle Swarm Optimization （PSO） algorithm. Moreover, two feasible NEA flyby missions are presented.