Lambert's problem

Abstract: In this paper, the definition of the primer vector is extended to include nonoptimal as well as optimal trajectories. With this definition, simple tests are developed which determine how a given trajectory can be improved (in terms of velocity requirements). This problem arose in the study of the use of impulsive trajectories to generate approximate adjoint initial conditions for finite thrust vehicles. To do this, the optimum fixed-time impulsive trajectory must be found. However, since many mission analyses are done on an impulsive basis, a wider application is foreseen. Necessary conditions are developed for when an additional impulse can improve the trajectory; how interior impulses of a multi-impulse trajectory can be moved so as to decrease the cost; and when initial and/or final coasts improve the trajectory. In the case of transfers between circular, coplanar orbits a geometric interpretation is given. For the case of an inverse-square gravitational field, the components of the primer vector can be calculated analytically. Using Floquet theory, a convenient form of this solution is presented.

Abstract: A procedure is described that provides a universal solution for Lambert's problem. Based on the approach of Lancaster and his colleagues, the procedure uses Halley's cubic iteration process to evaluate the unknown parameter, x, at the heart of the approach, initial estimates for x being selected so that three iterations of the process always suffice to yield an accurate value. The overall procedure has been implemented via three Fortran-77 subroutines, listings of which are appended to the paper, and the way in which the subroutines have been tested is outlined.

Abstract: Theminimum- ¢V,e xed-time,two-impulserendezvousbetweentwo spacecraftorbitingalong two coplanarunidirectional circular orbits (moving-target rendezvous )is studied. To reach thisgoal, the minimum- ¢V, e xed-time, two-impulse transfer problem between two e xed points on two circular orbits is e rst solved. This e xed-endpoint transferisrelated to the moving-target rendezvousproblem by a simple transformation. The e xed-endpoint transfer problem is solved using the solution to the multiple-revolution Lambert problem. A solution procedure is proposed based on the study of an auxiliary transfer problem. When this procedure is used, the minimum ¢V of the moving-target rendezvous problem without initial and terminal coasting periodsis obtained for a range of separation angles and timesofe ight. Thus, a contour plot ofthecostvs separation angleand transfertime isobtained. This contour plot, along with a sliding rule, facilitates the task of e nding the optimal initial and terminal coasting periods and, hence, obtaining the globally optimal solution for the moving-target rendezvous problem. Numerical examples demonstrate the application of the methodology to multiple rendezvous of satellite constellations on circular orbits.

Abstract: The orbital boundary value problem, also known as Lambert problem, is revisited. Building upon Lancaster and Blanchard approach, new relations are revealed and a new variable representing all problem classes, under L-similarity, is used to express the time of flight equation. In the new variable, the time of flight curves have two oblique asymptotes and they mostly appear to be conveniently approximated by piecewise continuous lines. We use and invert such a simple approximation to provide an efficient initial guess to an Householder iterative method that is then able to converge, for the single revolution case, in only two iterations. The resulting algorithm is compared, for single and multiple revolutions, to Gooding’s procedure revealing to be numerically as accurate, while having a significantly smaller computational complexity.