How Many Impulses Redux

Ehsan Taheri,John L Junkins

doi:10.1007/s40295-019-00203-1

Ehsan Taheri, John L Junkins

Open Access

https://doi.org/10.1007/s40295-019-00203-1

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Abstract

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.

Highlights

Most space trajectory design algorithms make use of low-fidelity dynamical models and idealized control input assumptions to make the search space tractable [1,2,3,4,5,6,7,8,9]
We have shown that there exists a fundamental minimum-thrust solution that plays a pivotal role in determining fundamental switching surface, and an associated Nr∗ev, which in turn reveals the “optimal” number of thrust arcs for any finite specification of maximum thrust
The fundamental switching surface is generated by a homotopic sweep of the maximum thrust away from the minimum-thrust extremal among all minimum-thrust solutions, considering all feasible multi-revolution solutions

Summary

Introduction

Most space trajectory design algorithms make use of low-fidelity dynamical models and idealized control input assumptions to make the search space tractable [1,2,3,4,5,6,7,8,9]. Traditional impulsive-based trajectory analysis tools, typically with inverse-square gravity models, hold a special place for preliminary mission design. The output of preliminary mission design studies is the starting design for higher-fidelity optimization. Making these approximations in preliminary mission design is driven by practicality, which is natural, given the level of complexity of the overall mission design challenge. Preliminary mission design methods rely on low-fidelity dynamical models, which in turn, frequently lead to analytical propagation of the state dynamics through Keplerian orbit models [20] or by utilizing the solution of Lambert’s problem [21,22,23,24,25]. Impulsive maneuvers are used extensively for solving formation flight optimal control problems [26,27,28,29,30,31,32,33,34,35,36] and orbit reachability analyses problems [37,38,39,40,41,42,43]

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