Reachability for Low-Thrust Trajectories via Maximum Initial Mass

This paper introduces a dual formulation for low-thrust trajectory reachability that determines the maximum allowable initial mass for a successful transfer, enabling the construction of efficient, data-driven neural network surrogates to rapidly evaluate mission feasibility without the computational cost of classical forward simulations.

The Gist

Imagine you are planning a road trip for a very special, fuel-hungry car. This car doesn't use gas; it uses a tiny, gentle push (low-thrust) that keeps it moving for months or years. The big question for your trip planner is: "Can this car actually make it to the destination with the fuel we have?"

Traditionally, to answer this, engineers would run thousands of computer simulations. They would try to drive the car to every possible spot on a map, one by one, to see if it makes it. If the car runs out of fuel before arriving, that spot is marked "No." If it arrives, it's marked "Yes."

The problem? This is incredibly slow and computationally expensive. It's like trying to map a whole country by walking every single street individually. Also, the line between "Yes" and "No" is often jagged and messy, making it hard for computers to learn the pattern.

The New Idea: The "Maximum Weight" Test

The authors of this paper propose a clever twist, a "dual" way of looking at the problem. Instead of asking, "Can this specific car make it?", they ask:

"What is the heaviest car we could possibly send to this destination and still make it?"

Think of it like a bridge. Instead of testing if a specific 2-ton truck can cross, you calculate the maximum weight limit of the bridge.

• If your truck weighs 1.5 tons, and the bridge holds 2 tons, you know instantly: Yes, it can cross.
• If your truck weighs 2.5 tons, the answer is No.

In space terms, they calculate the Maximum Initial Mass.

• If your spacecraft is lighter than this calculated limit, the trip is possible.
• If it's heavier, it's impossible.

This turns a messy, jagged "Yes/No" map into a smooth, flowing landscape (like a topographic map showing elevation). This smoothness makes it much easier for computers to understand and predict.

The Solar Sail Twist

They also tested this on "Solar Sail" spacecraft. These don't burn fuel at all; they use the pressure of sunlight to push. Since they don't lose mass, the question changes slightly. Instead of "How heavy can the ship be?", they ask: "How strong does the sail need to be to make the trip?"

If the required sail strength is low, it means even a small, weak sail could do it (so it's reachable). If the required strength is huge, it's likely impossible for current technology.

The "Cheat Sheet" (Machine Learning)

Even with this new, smoother method, calculating the exact "Maximum Weight" or "Sail Strength" for every possible destination still takes a lot of computer power. It's like calculating the bridge limit for every single truck that ever existed.

To speed this up, the authors trained AI models (neural networks) to act as a "Cheat Sheet."

1. They first did the hard math (using advanced physics rules called Pontryagin's Principle) for thousands of trips to create a dataset.
2. They taught an AI to look at the start and end points of a trip and guess the answer instantly.

The Winner: The "Residual Network"

They tried different types of AI architectures to see which one learned the best.

• Plain AI: Like a standard student trying to memorize a textbook. It struggled with the complex patterns.
• SIREN AI: A very fancy student good at high-frequency details but got confused and unstable with this specific problem.
• Residual Network (ResNet): This was the winner.

The Analogy: Imagine a ResNet as a student who learns by making small corrections to a simple guess. Instead of trying to memorize the whole answer from scratch, they start with a basic idea and then add tiny "fixes" layer by layer. This made the AI much more stable, accurate, and faster to train.

The Results

• For Electric Thrust: The AI could predict if a trip was possible with 97.8% accuracy. It was particularly good at knowing exactly where the "edge" of possibility was.
• For Solar Sails: The AI was even better, achieving 99.4% accuracy.

Why This Matters (According to the Paper)

The paper concludes that by combining this "Maximum Mass" math trick with the "Residual Network" AI, mission planners can now instantly check if a destination is reachable. They don't need to run slow, heavy simulations for every single idea. It turns a difficult, hours-long calculation into a split-second check, helping engineers design better space missions faster.

In short: They turned a hard "Can I get there?" question into an easier "How heavy can I be?" question, and then taught a smart AI to answer that instantly.

Technical Summary

Technical Summary: Reachability for Low-Thrust Trajectories via Maximum Initial Mass

Problem Statement
Reachability analysis is fundamental to low-thrust spacecraft trajectory optimization, determining which target states are achievable given constraints on time, thrust, and propellant. Traditional methods construct reachable sets by solving numerous optimal control problems over grids of terminal states. While effective, this approach is computationally expensive and often impractical for high-dimensional systems or complex nonlinear dynamics (e.g., cis-lunar environments). Furthermore, classical minimum-time formulations introduce strong nonlinearities and sensitivity near the reachable-set boundary, leading to convergence difficulties in two-point boundary-value problems (TPBVPs) and complicating the training of machine learning surrogates.

Methodology
The authors propose a dual formulation of the reachability problem that shifts the focus from explicitly constructing the reachable set to evaluating a scalar field: the Maximum Initial Mass (MIM).

1. Dual Formulation (MIM): Instead of fixing the initial mass ($m_0$) and asking if a target is reachable, the method fixes the initial state ($x_0$), final state ($x_f$), and transfer time ($t_f$), and solves for the maximum initial mass that permits a successful transfer.

   • For electric low-thrust, the objective is to maximize $m_0$ subject to the mass-flow law and thrust constraints.
   • For solar sails, where mass is constant, the objective is to minimize the sail-strength parameter ($k_0$), which is inversely proportional to the allowable sail mass for a given area.
   • This transforms the binary feasibility problem into a continuous scalar optimization, yielding a well-behaved scalar field that encodes the same feasibility information as the classical reachable set but with smoother numerical behavior.

2. Indirect Optimization: The authors derive indirect formulations using Pontryagin's Minimum Principle (PMP) for both scenarios:

   • Electric Propulsion: A forward-backward shooting method is employed to solve the TPBVP. The control structure follows a full-thrust arc, and the shooting vector includes initial/final costates and the initial mass.
   • Solar Sailing: A single-shooting formulation is used, augmented with a parameter stationarity condition for the constant sail-strength parameter $k_0$. The optimal sail orientation is derived in closed form.

3. Machine Learning Surrogates: To avoid repeated indirect solves, data-driven surrogate models are trained to approximate the MIM scalar field (or the derived binary reachability indicator).

   • Architectures: The study compares fully connected Multi-Layer Perceptrons (MLPs), Sinusoidal Representation Networks (SIREN), and Residual Networks (ResNet).
   • Activation Functions: Various smooth activations (Softplus, SiLU, SELU, GELU) are tested.
   • Training: The models are trained on datasets generated by solving the PMP problems for various orbital transfers. For electric propulsion, the task is binary classification (reachable/unreachable based on a threshold mass). For solar sails, the task is regression of the minimum sail-strength parameter.

Key Results

• Numerical Behavior: The MIM scalar field exhibits significantly smoother gradients and better conditioning than the minimum-time value function, particularly near the reachable-set boundary. This makes it a superior target for regression and classification.
• Surrogate Performance: Residual Networks (ResNet) with smooth activation functions, specifically Softplus, consistently outperformed plain MLPs and SIREN networks across both problem domains.
  • Electric Propulsion: The best ResNet-Softplus model (64 hidden units) achieved a validation accuracy of 97.77% and an F1-score of 0.9772. Increasing network width yielded diminishing returns.
  • Solar Sailing: The same architecture achieved a validation accuracy of 99.45% and an F1-score of 0.9924 on the GTOC13 simulation environment.
• Error Distribution: Misclassifications occurred predominantly near the feasibility threshold (the operational mass limit), indicating that the surrogate correctly captures the "hard" decision boundary while maintaining high confidence for clearly reachable or unreachable targets.

Significance and Contributions
The paper claims two primary contributions:

1. Formulation: It establishes the Maximum Initial Mass (or minimum sail-strength) as a robust, dual formulation for low-thrust reachability. This approach provides an implicit scalar-field representation of the reachable set that is numerically more stable and regular than traditional minimum-time formulations.
2. Surrogate Modeling: It demonstrates that Residual Networks with smooth activations provide the optimal trade-off between accuracy, training stability, and complexity for learning these dual reachability indicators.

The authors conclude that combining indirect MIM formulations with neural surrogates offers a practical, scalable framework for preliminary mission design. This approach enables rapid feasibility assessment without the computational cost of repeated optimal-control solves, serving as an efficient "reachability oracle" for trade-space exploration in low-thrust and solar sail missions.