Assuming that there exist the bodies A and B in Keplerian orbits around a single gravitational center and a spacecraft transfers from A to B,a new model called v∞-transfer-orbit(VTO)-problem is proposed for determining the spacecraft's transfer orbit.In the VTO-problem,the escaping time t0 and the escaping velocity v∞departing from A are selected as the spacecraft's orbital determination parameters.According to the spatial relative positions between A and B,the VTO-problem is divided into three cases:A/B is nonplanar,A/B is coplanar,and A/B is co-orbital,and there exist three types of solutions:General-VTO,Backflip-VTO and Resonant-VTO.In this paper,a uniform geometric analysis method for solving the VTO-problem is introduced,in which the position constraint of the spacecraft's arrival at B is decomposed into orbital constraint and time constraint,the spacecraft's orbital parameters are resolved by a single variable based on the orbital constraint,and an equation referring to this single variable is constructed based on the time constraint.According to the geometric analysis method,the VTO-problem is transformed into a one-variable equation-rooting problem.Firstly,the one-variable equation for General-VTO is derived in response to the cases of A/B nonplanar,A/B coplanar,and A/B co-orbital,and the intervals of the variable and an efficient equation-rooting algorithms based on the cubic spline interpolation are elaborated.Secondly,the different one-variable equation-rooting problem for Backflip-VTO is derived,and another set of equation-rooting algorithms are described on the basis of analyzing the equation function properties,such as monotonicity,extreme points and inflection points.Thirdly,the analytic solution is given directly for Resonant-VTO.Finally,examples are given to expound the solution multiplicity of the VTO-problem.
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