On the solution to every Lambert problem
On the solution to every Lambert problem
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DOI：
10.1007/s10569-019-9927-z
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发表时间：
2019-10
影响因子：
1.6
通讯作者：
R. Russell
中科院分区：
文献类型：
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作者：
R. Russell
兰伯特问题是开普勒动力学的两点边值问题。针对零旋转和多次旋转问题的参数和解空间进行了调查，包括详细研究通常困扰兰伯特求解器的应力情况。与公式无关的问题域对于每个旋转情况都显示为矩形，这使得难以捉摸的初始猜测和解决方案本身适合插值。实施双五次样条以实现连续导数和快速评估。生成的函数可以直接用作低保真度解决方案，也可以与单次更新迭代一起使用，而无需采取任何保护措施。提出了兰伯特问题的简洁、改进的 Vercosine 公式，包括新的无奇点和精度节省方程。插补方案最多适用 100 转。考虑的域包括所有实际上可以想象的飞行时间，以及除了问题的唯一物理奇点附近的小区域之外的所有可能的几何形状：相等的终端向量情况。解决方案已存档并针对准确性、内存占用和速度进行基准测试。对于典型场景，用户可以使用插值解而无需迭代来期望更多位数的速度矢量精度。使用单个、不受保护的迭代可以得到在整个域（包括最极端的情况）上具有接近机器精度的解决方案。根据所需的分辨率，每个转数情况下系数文件的大小从 65 MB 不等。评估运行时间比行业基准 Gooding 算法快 5 倍。系数文件和驱动程序在线提供。虽然该方法目前在 Vercosine 公式中得到了演示，但二维插值方案将使所有兰伯特问题公式受益。
Lambert’s problem is the two-point boundary value problem for Keplerian dynamics. The parameter and solution space is surveyed for both the zero- and multiple-revolution problems, including a detailed look at the stress cases that typically plague Lambert solvers. The problem domain, independent of formulation, is shown to be rectangular for each revolution case, making the elusive initial guess and the solution itself amenable for interpolation. Biquintic splines are implemented to achieve continuous derivatives and quick evaluation. Resulting functions may be used directly as low-fidelity solutions or used with a single update iteration without safeguards. A concise, improved vercosine formulation of the Lambert problem is presented, including new singularity-free and precision-saving equations. The interpolation scheme is applied for up to 100 revolutions. The domain considered includes all practically conceivable flight times, and every possible geometry except a small region near the only physical singularity of the problem: the equal terminal vector case. The solutions are archived and benchmarked for accuracy, memory footprint, and speed. For typical scenarios, users can expector more digits of velocity vector accuracy using an interpolated solution without iteration. Using a single, unguarded iteration leads to solutions with near machine precision accuracy over the full domain, including the most extreme scenarios. Depending on desired resolution, coefficient files vary in size fromto 65 MB for each revolution case. Evaluation runtimes vary fromto 5 times faster than the industry benchmark Gooding algorithm. The coefficient files and driver routines are provided online. While the method is currently demonstrated on the vercosine formulation, the 2D interpolation scheme stands to benefit all Lambert problem formulations.