Polynomial-based method for determining coast-terminating zero of fuel-optimal time-fixed trajectory

Abstract

This paper presents a fail-safe polynomial-based method for determining the coast-terminating zero (CTZ) of fuel-optimal, time-fixed, low-thrust trajectories. The CTZ of the fuel-optimal bang-bang control is decided by the zeros of the switching function (SF). The state and costate differential equations along the coast arcs of the optimal trajectory have closed-form solutions; thus efficiently determining the CTZ is important. Existing methods suffer potential failures or are limited to time-free problems. The polynomial-based method consists of three steps. First, the SF is simplified by a coordinate transformation that eliminates the three orientation-related orbital elements. Then, the SF is rewritten as a combination of power and trigonometric functions of the argument eccentric anomaly, and the coefficients in the SF are simply evaluated using the states and costates at the periapsis. Finally, the SF is described as a polynomial of the eccentric anomaly by replacing the trigonometric functions with polynomials. The roots of the polynomial can provide an accurate estimation of the CTZ. Based on a Monte Carlo simulation, SF curves whose greatest possible numbers of zeros are $1,2,\ldots,7$ are listed. For each curve, the CTZ is solved by both the presented polynomial-based method and the existing sampling-searching method. The performance comparison demonstrates that the polynomial-based method provides superior reliability and efficiency.