Analytical Sensitivity Matrix for Near-Optimal Solution to Elliptical Orbit Transfer

Orbit transfer problem plays an essential role in space missions. The sensitivity matrix, describing the first-order relationship between the piecewise constant thrust (where thrust acceleration is a piecewise constant function of time) and the orbital element changes, has been widely used for solving the orbit transfer problem due to its ability for simplifying the nonlinear orbit transfer problem into a linear one regarding piecewise constant thrust (PCT). However, it is hard to be analytically computed because it is defined with the form of differential equations in time, which brings expensive computation burden. To address this limitation, this paper presents an analytical approach to calculate the sensitivity matrix for solving the elliptical orbit transfer problem. Specifically, the sensitivity matrix's differential equations, expressed in terms of eccentric anomaly, are derived and can be analytically integrated using mathematical techniques involving the elliptic integral and Taylor expansions regarding eccentricity. Furthermore, based on the analytical sensitivity matrix, a fuel near-optimal PCT solution for the elliptical orbit transfer problem is obtained through a self-adaptive sequential optimization method. In this process, the PCT solution is sequentially evolved, leading to a reduction in fuel consumption. This optimization is self-adaptively terminated once further decreases in fuel consumption become negligible. Compared with the optimal solution from the conventional method, the proposed near-optimal PCT solution significantly reduces the computation time with only a marginal increase in fuel consumption, making it a practical option for engineering applications.