How Many Impulses Redux

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 the answer to Edelbaum's question is not generally unique, but is frequently a set of equal-∆v 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. NOMENCLATURE f = vector function of unforced dynamics B = control influence matrix c = exhaust velocity, m/s g 0 = Earth gravitational acceleration, m/s 2 h = specific angular momentum vector, km 2 /s H = Hamiltonian I sp = specific impulse, s J = cost functional

Advances in Trajectory Optimization for Space Vehicle Control

“…A promising modern indirect method family relies on homotopy in order to solve the TPBVP (Pan et al, 2016(Pan et al, , 2019Pan and Pan, 2020;Taheri and Junkins, 2019;Trélat, 2012;Rasotto et al, 2015;Lizia et al, 2014). Homotopy aims to address the aforementioned challenges of slow convergence, initial guess quality, and active constraint specification.…”

A novel approach for optimal trajectory design with multiple operation modes of propulsion system, part 2

“…This indicates that the particular configuration of the propulsion system is more capable and the trajectory starts and terminates on the socalled late-departure and early-arrival boundaries. These boundaries will trace a curve and are revealed as part of the optimal switching surfaces introduced by Taheri and Junkins [56]. Another important point is that the particular number of engines associated with each operation mode is just an indication of the number and operation power setting of potential engines.…”

“…The final value, l T , is the updated target value for the final true longitude taking into account any additional number of revolutions. Previous studies [56,57] have shown that for each number of revolutions in the feasible set of N rev integers, there is one local extremal for fuel-optimal trajectories.…”

Optimal Space Trajectories with Multiple Coast Arcs Using Modified Equinoctial Elements

“…In this context, orbit transfer optimization, aimed at minimizing propellant consumption, reduces to a nonlinear programming problem. However, in the presence of moderate or low thrust levels, the impulsive approximation loses accuracy, and the general properties of optimal finite-thrust trajectories can no longer be inferred from an impulsive solution [34,49]. In these cases, the optimal path of interest must be found as the solution of a continuous-time optimal control problem.…”

“…This consolidated property was proven in the 60 s [30] and since then a vast amount of literature was dedicated to investigating minimum-fuel space trajectories. A very interesting recent contribution is due to Taheri and Junkins [49], who analyzed the relations between impulsive transfers and finite-thrust paths using optimal control theory. They point out the existence of optimal trajectories with a variety of structures (i.e.…”