Tangent Orbit Technique in Three Dimensions

Abstract

T HE two-body two-point boundary value problem is well known as Lambert’s problem when the transfer time is assigned. The transfer time can be written as a function of different transfer orbital parameters as independent variables. Classic methods for Lambert’s problem are to solve the semimajor axis in Lagrange’s equation of transfer time [1]. Other orbital parameters, including the flight-path angle [2] and the transverse eccentricity vector component [3], are proposed to solve Lambert’s problem. The multiple-revolution Lambert’s problem (MRLP) can also be solved with these orbital parameters [4–6]. There are some other results for MRLP by using the semimajor axis as an independent variable [7–9]. If the velocity directions of the transfer orbit at the boundary points, instead of the assigned transfer time, are required to be specified, the two-point boundary value problem can be solved as the tangent transfer orbit problem. For two coplanar tangent orbits, the velocity vectors at their single common point are collinear. If two spacecraft move in the same direction, the task of nulling the relative velocity would be dramatically simplified for an orbital rendezvous problem. By investigating the geometric characteristics of twoimpulse cotangential transfers between coplanar elliptic orbits, Adamyan et al. [10] derived the transfer orbital parameters and the magnitudes of the velocity impulses in explicit form. Moreover, based on the orbital hodograph theory, Thompson et al. [11] solved the tangent transfer orbit problem, but a numerical iterative technique was implemented for the cotangent transfer. Furthermore, Zhang et al. [12] provided closed-form solutions for the tangent transfer orbit and its solution-existence conditions. In addition, Zhang et al. [13] studied a tangent orbit rendezvous problem which requires the same flight time and the same direction of terminal velocities. All the preceding methods for the tangent transfer orbit are only valid for the coplanar case, where two orbits are tangent to each other at the common point if and only if the flight-direction angles are equal. Thus, the tangent to initial/final orbit problem can be transformed into the specified departure/arrival flight-direction angle problem whose flight-direction angle is equal to that of the initial/ final orbit. The velocity directions with the same flight-direction angle for two orbits at their common point, which is the tangency condition for noncoplanar orbits defined in [11], produce a cone such that they cannot be viewed as tangency in geometry. Actually, two different curves are tangent at the common point if and only if they share the same tangent unit vector. With this definition, if the initial and final orbits are noncoplanar, there is no transfer orbit tangent to the initial/final orbit. Thus, a new definition of “tangent” orbits in three dimensions (3D) is required. In this Note, based on a definition of tangent orbits in 3D, the flight-direction angle of the transfer orbit tangent to the initial/final orbit will be obtained. Then the tangent to initial/final orbit problem in 3D can be solved. A novel and simple relationship between the transfer angle and the initial true anomaly is provided. Then the cotangent transfer problem in 3D will be solved by the secant method.