A central problem in orbit transfer optimization is to determine the number, time, direction, and magnitude of velocity impulses that minimize the total impulse. This problem was posed in 1967 by T. N. Edelbaum, and while notable advances have been made, a rigorous means to answer Edelbaum’s question for multiple-revolution maneuvers has remained elusive for over five decades. We revisit Edelbaum’s question by taking a bottom-up approach to generate a minimum-fuel switching surface. Sweeping through time profiles of the minimum-fuel switching function for increasing admissible thrust magnitude, and in the high-thrust limit, we find that the continuous thrust switching surface reveals the N-impulse solution. It is also shown that a fundamental minimum-thrust solution plays a pivotal role in our process to determine the optimal minimum-fuel maneuver for all thrust levels. Remarkably, we find that the answer to Edelbaum’s question is not generally unique, but is frequently a set of equal-Δv extremals. We further find, when Edelbaum’s question is refined to seek the number of finite-duration thrust arcs for a specific rocket engine, that a unique extremal is usually found. Numerical results demonstrate the ideas and their utility for several interplanetary and Earth-bound optimal transfers that consist of up to eleven impulses or, for finite thrust, short thrust arcs. Another significant contribution of the paper can be viewed as a unification in astrodynamics where the connection between impulsive and continuous-thrust trajectories are demonstrated through the notion of optimal switching surfaces.

中文翻译：

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有多少冲动

轨道转移优化中的核心问题是确定使总脉冲最小化的速度脉冲的数量，时间，方向和大小。这个问题是由TN Edelbaum于1967年提出的，尽管取得了显着进步，但在过去的五十多年中，一直没有一种严格的方法来回答Edelbaum关于多次革命演习的问题。我们通过采用自下而上的方法生成最小燃料转换面来重新审视Edelbaum的问题。扫描最小燃料转换函数的时间曲线以增加可允许的推力幅度，并在高推力极限中，我们发现连续的推力转换表面揭示了N脉冲解。它也表明，一个根本最小推力解决方案在我们确定所有推力水平的最佳最小燃料机动性的过程中起着关键作用。值得注意的是，我们发现，答案Edelbaum的问题不是一般唯一的，但常常是一组等于-Δ的v极端 我们进一步发现，当对Edelbaum的问题进行改进以寻找特定火箭发动机的有限持续时间推力弧的数量时，通常会发现一个独特的极值。数值结果证明了这种思想及其在几种行星际和与地球结合的最佳传递中的实用性，这些传递包括多达11个脉冲，或者对于有限推力而言，是短推力弧。本文的另一个重要贡献可以看作是天体动力学的统一，其中通过最佳切换面的概念证明了脉冲和连续推力轨迹之间的联系。