Technical Summary: Interception-Driven Inverse Reachability for Engagement Zone Construction

Problem Statement

Autonomous vehicles operating in contested environments often face adversarial pursuers whose launch locations are unknown. Traditional engagement analysis typically assumes prior knowledge of pursuer parameters (launch position, range, maneuverability) or relies on prescribed probability distributions. However, in many operational settings, threat locations must be inferred from sparse, indirect, or event-driven observations (e.g., binary interception events). This creates a coupled estimation–planning problem: agents must gather informative data about a threat while simultaneously generating trajectories that remain safe during information acquisition. Existing engagement zone (EZ) formulations generally fail in this context because they fundamentally rely on known launch locations or pre-existing uncertainty models, rather than constructing EZs directly from interception data.

Methodology

The paper proposes a geometric framework that infers a feasible pursuer launch region directly from observed interception events and propagates this region to construct deterministic and probabilistic engagement zones.

1. Geometric Inverse Reachability

The core inference mechanism relies on the geometric constraints imposed by interception events. If a pursuer intercepts an agent at location $\mathbf{x}_{int}$, the pursuer must have launched from a location capable of reaching $\mathbf{x}_{int}$ within its travel budget (maximum range $R$ and capture radius $r$).

• Constraint Generation: Each interception induces a disc constraint on the unknown launch location. If only the interception location is known, the feasible launch region is a disc of radius $R+r$ centered at the interception point. If launch time is also measured, the radius is tightened to $d_{trav} + r$, where $d_{trav}$ is the distance traveled based on the time difference.
• Feasible Region Construction: The feasible launch region ($P_{pot}$) is the intersection of all such discs generated by $N_{int}$ interception events. This region is convex and bounded by circular arcs. An exact algorithm computes the boundary by identifying valid intersection points between discs and retaining arc segments that lie within all other discs.

2. Engagement Zone (EZ) Construction

Once the feasible launch region $P_{pot}$ is inferred, the paper constructs Engagement Zones to define safe operating spaces for evaders.

• Deterministic EZ: The reachable region (RR) is computed as the Minkowski sum of the feasible launch region and a disc of radius $R+r$. The EZ is then derived by translating this RR opposite to the evader's heading by a distance proportional to the speed ratio ($\nu R$). Any trajectory remaining outside this EZ is guaranteed safe against all pursuers originating within the feasible set.
• Probabilistic EZ (PEZ): To allow for risk-aware planning, the paper introduces a probabilistic extension. Assuming a uniform distribution over the feasible launch region, the probability of interception at a given evader position is calculated as the normalized area of overlap between the launch region and the shifted capture disc. This yields graded risk fields rather than binary safe/unsafe boundaries.

3. Information-Driven Sacrificial Planning

To accelerate the localization of the pursuer, the framework employs "sacrificial" (attritable) agents designed to induce interceptions.

• Objective Function: The trajectory of a sacrificial agent is optimized to maximize expected information gain.
  • Pre-Observation (No interceptions yet): The objective is to maximize the "launch-coverage fraction," i.e., the portion of the initial feasible launch region that could successfully engage the trajectory, thereby maximizing the probability of inducing an initial interception.
  • Post-Observation: The objective shifts to maximizing the expected contraction of the feasible launch region area. Trajectories are selected to induce interceptions at locations that, when intersected with the current feasible set, yield the largest reduction in area.
• Optimization: Trajectories are parameterized as B-splines and optimized using interior-point solvers (IPOPT) subject to kinematic constraints (curvature, turn rate) and the specific information-driven objectives.

Key Contributions

The paper outlines four primary contributions:

1. Geometric Inverse-Reachability Method: A method to infer a feasible pursuer launch region for a single pursuer directly from interception events without prior knowledge of the launch location.
2. Deterministic EZ Construction: A reachable-region construction that maps bounded launch uncertainty into deterministic EZs, providing explicit worst-case safety guarantees for trajectory planning.
3. Probabilistic Extension: A probabilistic engagement zone formulation that quantifies engagement likelihood via geometric overlap, enabling risk-aware planning where some probability of interception is tolerable.
4. Information-Driven Planner: A trajectory objective for sacrificial agents that selects paths to maximize the expected contraction of the feasible launch region, actively accelerating threat localization.

Results

The framework was evaluated using 1,000 Monte Carlo simulations across three pursuer commitment doctrines (passive, nominal, aggressive) and four sensing/planning configurations (naive straight-line vs. optimized spline; with vs. without launch-time measurement).

• Region Contraction: Optimized spline trajectories reduced the feasible launch region area significantly faster than naive straight-line trajectories. For an aggressive pursuer, the feasible area contracted by 99.25% after only three sacrificial deployments.
• Path Efficiency: As the feasible launch region contracted, the length of safe trajectories for high-value agents decreased, approaching the optimal path achievable with perfect knowledge of the pursuer's location. For the aggressive pursuer, the normalized safe-path time ratio dropped to within 0.29% of the perfect-information optimum after three deployments.
• Impact of Sensing: Measuring launch time provided substantial benefits for passive and nominal pursuers by tightening constraints, but offered minimal additional benefit for aggressive pursuers who typically intercept near their maximum range (where fixed-radius and time-dependent constraints are similar in size).
• Convergence: The results indicate that only a small number of well-placed sacrificial deployments are required to recover most of the achievable operational benefit in terms of path optimality.

Significance and Claims

The paper claims to demonstrate that a relatively small number of sacrificial agents can substantially increase operational awareness and improve safety in the presence of an unknown pursuer. The significance lies in the ability to:

• Eliminate the need for prior knowledge of the adversary's launch location.
• Construct engagement zones directly from sparse, event-driven observations.
• Provide explicit worst-case safety guarantees (deterministic) or graded risk assessments (probabilistic) without assuming a specific probability distribution for the threat.
• Actively accelerate threat localization through information-driven trajectory planning, allowing high-value assets to transition from conservative, long trajectories to near-optimal paths rapidly.

The authors note that the framework assumes maximum-capability pursuers and bounded range, and future work could extend this to multiple pursuers, uncertain capabilities, and more complex 3D geometries.