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 引用本文: 张哲, 代洪华, 冯浩阳, 汪雪川, 岳晓奎. 初值约束与两点边值约束轨道动力学方程的快速数值计算方法. 力学学报, 2022, 54(2): 503-516.
Zhang Zhe, Dai Honghua, Feng Haoyang, Wang Xuechuan, Yue Xiaokui. Efficient numerical method for orbit dynamic functions with initial value and two-point boundary-value constraints. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(2): 503-516.
 Citation: Zhang Zhe, Dai Honghua, Feng Haoyang, Wang Xuechuan, Yue Xiaokui. Efficient numerical method for orbit dynamic functions with initial value and two-point boundary-value constraints. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(2): 503-516.

## EFFICIENT NUMERICAL METHOD FOR ORBIT DYNAMIC FUNCTIONS WITH INITIAL VALUE AND TWO-POINT BOUNDARY-VALUE CONSTRAINTS

• 摘要: 轨道动力学快速计算是航天工程中的基础问题, 广泛存在于轨道设计、空间抓捕以及深空探测等任务中. 基于有限差分原理的经典数值积分算法, 由于精度严重依赖小积分步长, 难以满足航天器在轨快速计算需求. 针对该问题, 提出一种局部配点反馈迭代算法, 该算法能高效解算受到初值约束和两点边值约束的轨道动力学方程. 基于Picard迭代公式建立数值算法以避免计算雅可比矩阵, 引入误差反馈以加快迭代收敛速度. 通过时域配点法消除迭代公式推导中的复杂符号运算, 使迭代公式更为简洁. 结合拟线性化法及叠加法, 算法能对两点边值约束下的Lambert问题实现高效解算. 基于ph网格细化法建立计算参数自适应调节算法, 能进一步增强局部配点反馈迭代法的大步长计算优势. 通过求解二体动力学模型下的递推轨道, 摄动Lambert问题以及限制性三体动力学模型下的转移轨道验证了算法有效性. 仿真结果表明, 在相同计算精度下, 局部配点反馈迭代算法计算速度比拟线性化-局部变分迭代法提高1.5倍以上, 引入变参数方案能够进一步将算法计算速度提高6倍以上.

Abstract: It is a fundamental problem to efficiently solve the orbit dynamic systems occurred in the process of orbit design, space capture, deep space exploration and many other aerospace engineering missions. The traditional numerical integration methods, which are based on finite difference method, can hardly meet the requirement of low-latency computation in aerospace missions for their strict needs of small integration step size. In this paper, the high performance local collocation feedback iteration (LCFI) method is presented for orbit dynamic functions with initial value and two-point boundary-value constraints. LCFI does not need to estimate Jacobian matrix during the calculation process for that it is based on Picard iteration method, and it is able to save convergence time via combining error-feedback strategy. Besides, time domain collocation method is used to transfer the symbolic operations into algebraic operations, thus make the iterative formula of LCFI concise. In addition, LCFI is able to solve Lambert’s problem efficiently via combining quasi linearization method and superposition method, and its parameters can be adaptively adjusted via an adopted ph mesh refinement method to better play its ability of calculating with large step size. The validity of LCFI is verified via solving the orbit propagation problem, the perturbed Lambert’s problem, and the transfer trajectory in the circular restricted three-body model. Simulation results show that the computational efficiency of LCFI is higher than 1.5 times that of quasi linearization local variational iteration method, and the parameter adjustment method based on ph mesh refinement method is able to further increase the calculation speed of LCFI by more than 6 times.

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