⚡ electrical engineering

A Direct–Indirect Homotopic Approach for Fuel-Optimal Low-Thrust DRO-to-NRHO Transfer in Cislunar Space

This paper presents a direct–indirect homotopic method that solves a fuel-optimal low-thrust transfer from a distant retrograde orbit to a near-rectilinear halo orbit in the Earth–Moon system by using a smooth energy-optimal solution to initialize and guide an indirect shooting formulation through a homotopy parameter.

Original authors: Hongxiang Luo, Zhongtao Zhang, Yasheng Zhang, Xuefeng Tao, Wenhua Cheng, Yuan Zeng

Published 2026-08-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hongxiang Luo, Zhongtao Zhang, Yasheng Zhang, Xuefeng Tao, Wenhua Cheng, Yuan Zeng

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to guide a tiny, super-efficient spaceship from one cosmic dance floor to another. In the vast emptiness between Earth and the Moon, gravity isn't just a simple pull; it's a chaotic tangle of forces from both planets pulling at once. This is the "Circular Restricted Three-Body Problem" (CR3BP), a fancy name for a tricky gravitational tug-of-war. To navigate this, spacecraft use "low-thrust" engines. Think of these not as the massive, explosive rockets that launch satellites, but as gentle, continuous nudges—like a sailboat catching a steady breeze rather than a motorboat chugging through water. These engines are incredibly fuel-efficient, but because they push so gently, the path to the destination is incredibly sensitive. If you nudge the steering wheel a tiny bit wrong at the start, you might miss the target entirely.

The big challenge for scientists is finding the "fuel-optimal" path. This means using the absolute minimum amount of fuel to get from Point A to Point B in a set amount of time. The problem is that the best way to use fuel often looks like a "bang-bang" switch: the engine is either fully on or completely off, with very little in between. Finding this perfect on/off pattern is like trying to solve a puzzle where the pieces keep changing shape, and the math is so sensitive that computers often get lost before they even start. This paper tackles exactly that headache: how to get a spaceship from a distant, looping orbit (a Distant Retrograde Orbit, or DRO) to a very specific, wobbly orbit near the Moon (a Near-Rectilinear Halo Orbit, or NRHO) using the least amount of fuel possible, without the computer crashing.

The Paper's Solution: A Two-Step Dance

The authors, a team from the Space Engineering University in Beijing, propose a clever "Direct–Indirect Homotopic Approach." To understand their method, imagine you are trying to teach a robot to walk across a room without falling.

First, they use a Direct Method (specifically a "Radau pseudospectral method" solved by software called GPOPS-II). Think of this as the "smooth practice run." Instead of forcing the robot to walk with stiff, jerky steps (on/off), they ask it to walk with a gentle, continuous flow. It's like asking the robot to glide on ice. This is mathematically easy for the computer to solve because there are no sudden switches. However, this "gliding" solution isn't the most fuel-efficient way to walk; it's just a safe, smooth path that gets the robot to the right spot.

Here is the magic trick: Once the computer solves this smooth, easy problem, it extracts a "map" of the robot's internal state (called "costates"). This map is like a reference guide that tells the computer where the robot should be looking to make the best decisions.

Next, they switch to an Indirect Method. This is the "real deal" math that demands the robot walk with the strict, jerky "on/off" steps required for maximum fuel efficiency. But this method is notoriously difficult; if you give the robot the wrong starting instructions, it falls over immediately. This is where the reference guide from the first step comes in. The authors use the smooth solution's "map" to give the difficult method a perfect head start.

Then, they use a technique called Homotopy. Imagine you have a dimmer switch for the robot's walking style. You start with the switch all the way up (smooth, easy gliding). Then, you slowly, slowly turn the dimmer down. As you turn it down, the robot's walking style gradually changes from a smooth glide to a jerky, on/off stride. Because you are changing the rules very slowly, the computer can adjust its path step-by-step, never getting lost. If you tried to jump straight from "smooth" to "jerky," the computer would fail. But by taking this slow, winding path, they successfully guide the math to the perfect, fuel-saving solution.

What They Found

In their simulations, the team tested this method on a specific mission: moving a 500 kg spacecraft from a DRO to an NRHO in exactly 5 days. They used a tiny engine that pushes with 0.3 Newtons of force (about the weight of a small apple) and has a very high efficiency (1000 seconds of specific impulse).

The results were successful. The "dimmer switch" approach worked perfectly.

The Transition: They showed that the computer could smoothly transition the engine's behavior from a gentle, continuous push to the sharp "full-on, full-off" pattern needed for fuel savings.
The Fuel Savings: By finding the perfect starting and ending times (phases) on the orbits, they identified a specific path that used only 4.343 kg of fuel for the entire 5-day trip.
The Map: They created a "fuel-consumption map," which is like a weather map for space travel. It shows that if you leave at the wrong time, you might use much more fuel, but if you leave at the exact right moment (the "sweet spot" they found), you save a significant amount.

Why It Matters

This paper doesn't claim to have solved every space travel problem, nor does it say this method works for every single type of orbit. Instead, it demonstrates a reliable way to solve a very specific, difficult math problem that has been a bottleneck for designing future lunar missions. By combining the robustness of a "smooth" computer solution with the precision of a "strict" fuel-saving solution, they provide a new tool for engineers. This is crucial for missions like the Gateway lunar station, where every gram of fuel counts, and where the gravitational dance between Earth and Moon makes finding the right path incredibly tricky. The authors suggest that this approach could be expanded in the future to handle even more complex scenarios, but for now, they have proven that this "two-step dance" works beautifully for getting from a distant loop to a lunar halo orbit.

Technical Summary: A Direct–Indirect Homotopic Approach for Fuel-Optimal Low-Thrust DRO-to-NRHO Transfer

Problem Statement
The paper addresses the challenge of designing fuel-optimal, low-thrust transfers between periodic orbits in cislunar space, specifically from a Distant Retrograde Orbit (DRO) to a Near-Rectilinear Halo Orbit (NRHO) within the Earth–Moon Circular Restricted Three-Body Problem (CR3BP). The core difficulty lies in the nature of the optimal control solution: for fixed-time transfers, the minimum-fuel throttle profile approaches a "bang-bang" structure. This results in a highly sensitive indirect two-point boundary-value problem (TPBVP) where the convergence of the shooting method is critically dependent on the accuracy of the initial costate (adjoint) estimates. Standard indirect methods often struggle to find these initial guesses without a smooth starting point.

Methodology
The authors propose a direct–indirect homotopic approach to bridge the gap between robust direct transcription methods and precise indirect optimal control. The workflow consists of three main stages:

Direct Energy-Optimal Initialization:
- A smooth auxiliary problem is formulated to minimize energy (the integral of the square of the throttle) rather than fuel.
- This problem is solved using a Radau pseudospectral transcription implemented in GPOPS-II.
- Crucially, the Lagrange multipliers associated with the collocation constraints in the direct solution are mapped to continuous costates using the pseudospectral covector-mapping theorem. This provides a high-quality estimate of the initial costates for the subsequent indirect stage.
Indirect Homotopy Continuation:
- The estimated costates from the direct stage initialize an indirect shooting formulation based on Pontryagin's Minimum Principle (PMP).
- A homotopy parameter ( $\varepsilon$ ) is introduced to the performance index, creating a family of problems that transitions smoothly from the energy-optimal case ( $\varepsilon = 1$ ) to the fuel-optimal case ( $\varepsilon = 0$ ).
- The solution is continued by gradually decreasing $\varepsilon$ . At each step, the converged costates from the previous step serve as the initial guess for the next, allowing the solver to track the solution path from the smooth energy-optimal regime to the near bang-bang fuel-optimal regime.
Numerical Implementation:
- The indirect shooting problem is solved using fmincon for optimization and ode113 for integrating the coupled state and costate dynamics.
- The method handles fixed-time, free-final-mass rendezvous problems where the endpoints are specific states on the DRO and NRHO.

Key Results
The methodology was validated on a representative 5-day transfer scenario with the following parameters:

Spacecraft: Initial mass of 500 kg, maximum thrust of 0.3 N, and specific impulse of 1000 s.
Costate Reconstruction: The study demonstrated that the costates recovered from the direct energy-optimal solution closely matched the initial conditions required for the indirect shooting problem, successfully initializing the TPBVP.
Homotopic Evolution: The continuation process successfully tracked the evolution of the initial costates and the throttle profile. As $\varepsilon$ decreased, the throttle profile transitioned from a smooth curve to a bang-bang structure with distinct full-thrust and coast arcs, consistent with theoretical expectations for minimum-fuel problems.
Fuel Consumption: The final fuel-optimal trajectory for the specific 5-day transfer was computed. Furthermore, an all-phase fuel-consumption map was generated by sampling various departure and arrival phases. This map identified a specific phase pair that minimized fuel consumption to 4.343 kg.

Significance and Claims
The paper claims that its primary contribution is a robust initialization strategy for indirect optimal control in cislunar transfers. By leveraging the robustness of direct pseudospectral methods to generate costate seeds and the physical interpretability of indirect PMP formulations to recover the true bang-bang control structure, the approach overcomes the sensitivity issues typically associated with fuel-optimal low-thrust trajectory design.

The authors position this method as particularly effective for fixed-time, phase-specified transfers between cislunar periodic orbits where direct solutions are available but indirect shooting lacks reliable initial costates. The study concludes that this hybrid workflow successfully verifies the costate initialization, continuation behavior, and final trajectory reconstruction, providing a practical tool for mission design in the Earth–Moon system. Future work is noted to focus on higher-fidelity dynamics, variable transfer times, and operational constraints, but the current paper limits its scope to the CR3BP model and fixed-time transfers.

Technical Summary: A Direct–Indirect Homotopic Approach for Fuel-Optimal Low-Thrust DRO-to-NRHO Transfer

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