Reachability analysis is a fundamental concept in low-thrust spacecraft trajectory optimization, providing insights into which target states a spacecraft can reach given constraints on time, thrust, and available propellant. Understanding the boundaries of the reachable set can simplify greatly mission planning and design, particularly for complex interplanetary or multi-body missions, as well as for small body interception problems. Typically, reachable sets are computed by solving a large number of time-optimal control problems over a grid of possible terminal states thus constructing the domain of attainable positions and velocities. This process is carried out via forward computations where the initial position velocity and mass are fixed. This is not only computationally demanding but can become prohibitive as the dimensionality of the system increases or when highly nonlinear dynamics are considered, such as in cislunar or solar sail environments.

In this work, we propose a dual perspective on reachability, resulting in reduced computational cost. Instead of directly computing the full reachable set, we reformulate the following dual problem: for a fixed transfer time and fixed initial and final position and velocity, we determine the maximum initial mass that allows a spacecraft to successfully reach the target. The target is then reachable if the spacecraft’s initial mass does not exceed this threshold. By reducing the problem to a single scalar optimization for each target state, this approach avoids the need to compute the entire set of reachable states and produces a problem that is generally smoother and better behaved numerically than previous formulations.

We demonstrate the generality of this maximum initial mass reachability approach by applying it across multiple dynamical environments, including the classical two-body problem, the circular restricted three-body problem, and trajectories influenced by solar sailing forces, inspired by the GTOC13 competition scenario. We show that the methodology is independent of the specific governing dynamics and can also be used in cases where constraints on the direction and magnitude of the controller are present, making the method applicable to a wide range of mission scenarios. 

The dual reachability problem thus formulated, while efficient, does not reconstruct the entire classical reachable set, which, in some application, might be still desirable. We thus explore the use of surrogate models based on machine learning techniques to approximate the maximum initial mass. These models provide a fast means of estimating reachable sets, without requiring repeated solutions of complex optimal control problems.

Our results demonstrate that this dual perspective on reachability provides a computationally efficient framework for assessing trajectory feasibility and planning low-thrust missions. Moreover, it simplifies the problem of learning approximators for the reachable set, making such models more accurate and easier to train. By shifting the focus from a direct computation of the whole reachable set to determining the maximum initial mass for each candidate target, this approach offers mission designers a tool to rapidly evaluate interception or rendezvous feasibility.