Classical techniques for initial orbit determination (IOD) require the analyst to find a body’s orbit given only observations such as bearings, range, and/or position. In these cases, one of the goals is often to solve for the unknown velocity vector at one or more of the observation times to fully define the orbit. Recently, however, a new class of IOD problems has been proposed that switches the knowns and unknowns in these classic IOD problems. Specifically, the objective is to find the unknown position vectors given only velocity measurements. This paper presents a detailed assessment of the geometric properties of this new family of velocity-only IOD problems. The primary tool for this geometric analysis is Hamilton’s orbital hodograph, which is known to be a perfect circle for all orbits obeying Keplerian dynamics. This framework is used to produce intuitive and efficient algorithms for IOD from three velocity vectors (similar structure to Gibbs problem) and for IOD from two velocity vectors and time-of-flight (similar structure to Lambert’s problem). Performance of these algorithms is demonstrated through numerical results.

中文翻译：

---

基于速度的轨道确定问题的几何解

初始轨道确定（IOD）的经典技术要求分析人员仅在观察到诸如方位，范围和/或位置等观察值的情况下才能找到其轨道。在这些情况下，目标之一通常是在一个或多个观察时间处求解未知速度矢量，以完全定义轨道。然而，近来，已经提出了新种类的IOD问题，其在这些经典的IOD问题中切换了已知和未知。具体而言，目的是在仅进行速度测量的情况下找到未知的位置矢量。本文提出了这个新的仅速度IOD问题系列的几何特性的详细评估。进行这种几何分析的主要工具是汉密尔顿的轨道全息图，它是所有遵循开普勒动力学的轨道的理想圆。该框架用于为三个速度矢量（与吉布斯问题相似的结构）的IOD和两个速度矢量和飞行时间（与Lambert问题的结构相似）的IOD生成直观有效的算法。通过数值结果证明了这些算法的性能。