Overview

The animations on this page illustrate our continuation framework for optimal low-thrust trajectory design, which couples pseudo-arclength continuation (PALC) with a stationary-condition–based termination (SCBT) criterion. In each stage, we track a zero-curve of a parameterized shooting function and halt exactly when the transversality (stationary) condition for the active continuation parameter is satisfied—yielding an optimal solution to the final desired problem.

Results are showcased for one of the two examples from the paper; a minimum-fuel transfer between $\text{L}_2$ and $\text{L}_1$ halo orbits, where the videos step through freeing the terminal-halo parameter $(s_1)_f$, the initial-halo parameter $(s_1)_0$, and finally the final time $t_f$. The resulting solution reorganizes thrust/coast structure and exploits the unstable invariant manifold of the initial L2 halo orbit.

Collectively, the clips make visible how the trajectory and optimal control cost function changes while each parameter is freed from a solution to an initial “fixed-parameter” problem to a solution of the desired one. In each animation, the panels showing the SCBT function (i.e., $c_i$ $i\in\{0,1,2\}$) and the corresponding optimal-control cost (i.e., $\Delta m$) include a moving marker that traces the current solution point along the zero curve, synchronized with all other panels.

Transfer Between Halo Orbits with Minimum-Fuel

This example corresponds to the minimum-fuel $\text{L}_2$ to $\text{L}_1$ halo orbit transfer discussed in Section IV-B of the paper. The sequence of animations illustrates how the stationary-condition–based continuation method transitions the initial fixed-parameter problem to the desired problem formulation. Each stage frees one parameter of the optimal control problem while pseudo-arclength continuation tracks the corresponding zero-curve and terminates automatically at points satisfying the transversality condition corresponding to the current continuation parameter.

Continuation of $(s_1)_f$ parameter

This first stage frees the terminal-halo parameter $(s_1)_f$, allowing the trajectory to converge toward the optimal arrival location on the $\text{L}_1$ halo. As shown in the animation, the insertion point on the target halo orbit gradually shifts as the continuation parameter is varied, leading to noticeable changes in the overall trajectory. The placement and timing of the thrust arcs also evolve throughout the process, as evident in the top-center panel.

Continuation of $s_0$ parameter

The second stage frees the initial-halo parameter $(s_1)_0$, enabling optimization of both the departure from the $\text{L}_2$ orbit and the arrival at the $\text{L}_1$ orbit. As the continuation parameter $(s_1)_0$ is varied, the exit point from the initial halo orbit moves steadily, producing corresponding adjustments in the overall trajectory. Interestingly, the final insertion point into the target halo also shifts to remain optimal—ultimately returning toward the initially prescribed location.

Continuation of $t_f$ parameter

In the final stage, the continuation frees the final-time parameter $t_f$, completing the transition to the fully optimal transfer solution. As $t_f$ increases along the zero curve, the departure point from the $\text{L}_2$ halo orbit changes significantly, and the structure of the thrust arcs adapt accordingly. One thrust arc disappears entirely at the point where the slope of the $c_2$ is discontinuous, while the initial thrust arc diminishes until it closely resembles an impulsive maneuver.