A new approach to impulsive rendezvous near circular orbit

Abstract

A new approach is presented for the problem of optimal impulsive rendezvous of a spacecraft in an inertial frame near a circular orbit in a Newtonian gravitational field. The total characteristic velocity to be minimized is replaced by a related characteristicvalue function and this related optimization problem can be solved in closed form. The solution of this problem is shown to approach the solution of the original problem in the limit as the boundary conditions approach those of a circular orbit. Using a form of primer-vector theory the problem is formulated in a way that leads to relatively easy calculation of the optimal velocity increments. A certain vector that can easily be calculated from the boundary conditions determines the number of impulses required for solution of the optimization problem and also is useful in the computation of these velocity increments.Necessary and sufficient conditions for boundary conditions to require exactly three nonsingular non-degenerate impulses for solution of the related optimal rendezvous problem, and a means of calculating these velocity increments are presented. If necessary these velocity increments could be calculated from a hand calculator containing trigonometric functions. A simple example of a three-impulse rendezvous problem is solved and the resulting trajectory is depicted.Optimal non-degenerate nonsingular two-impulse rendezvous for the related problem is found to consist of four categories of solutions depending on the four ways the primer vector locus intersects the unit circle. Necessary and sufficient conditions for each category of solutions are presented. The region of the boundary values that admit each category of solutions of the related problem are found, and in each case a closed-form solution of the optimal velocity increments is presented. Some examples are simulated. Similar results are presented for the simpler optimal rendezvous that require only one-impulse. For brevity degenerate and singular solutions are not discussed in detail, but should be presented in a following study Although this approach is thought to provide simpler computations than existing methods, its main contribution may be in establishing a new approach to the more general problem.

Optimization control for the far-distance rapid cooperative rendezvous of spacecraft with different masses

“…The relations between classic orbital elements and non-singular point orbital elements are expressed as follows: (6) In this study, the thrust is assumed to be a continuous large thrust. Then, the change in mass of the spacecraft obeys the dynamics equation of a variable mass body as follows:…”

A New Approach to Optimal Impulsive Transfer Near Circular Orbit

“…In recent work [6] we defined y ′ 1 = y 2 . We find that (7) used here is more convenient in geometric work [10].…”

“…The work follows a similar approach as that of optimal rendezvous near a nominal circular orbit [6]. The state variables are transformed so that the differential equations of motion are linear [6,7].…”

“…The work follows a similar approach as that of optimal rendezvous near a nominal circular orbit [6]. The state variables are transformed so that the differential equations of motion are linear [6,7]. The problem therefore can be solved using a theory that was originally developed for linearized relative-motion problems [8].…”

“…We modify previous theorems [8] to fit the present problem. Because of the assumption that the original orbits are near a circular orbit, we may replace increments of certain variables by differentials as in recent work [6] resulting in a final theorem yielding very simple formulas for singular and nonsingular solutions. The accuracy of the results can be justified mathematically, and is tested also through two examples with known solutions [9].…”