Inferring Turn-Rate-Limited Engagement Zones with Sacrificial Agents for Safe Trajectory Planning

1. Problem Statement

In modern adversarial environments, autonomous agents must navigate airspace while avoiding threats (pursuers) whose kinematic capabilities (speed, turn rate, range, position, and heading) are often unknown or partially known. Traditional methods for defining Engagement Zones (EZs)—regions where a target can be intercepted—rely on perfect knowledge of pursuer parameters or assume simplified geometric shapes (e.g., cardioids) that do not reflect physical dynamics.

This paper addresses the challenge of inferring unknown pursuer parameters using only sparse, binary observations: whether a "sacrificial" (decoy) agent was intercepted or survived. The goal is to:

1. Reconstruct the pursuer's Reachable Region (RR) and Engagement Zone (EZ) based on these binary outcomes.
2. Use the inferred uncertainty to plan safe, time-optimal trajectories for high-value agents that avoid all feasible engagement regions.

2. Methodology

The proposed framework consists of three main components: a geometric model, a learning algorithm, and a trajectory selection strategy.

A. Geometric Modeling (CSBEZ)

The authors utilize the Curve-Straight Basic Engagement Zone (CSBEZ) model, which accounts for turn-rate limitations (Dubins kinematics).

• Reachable Region (RR): The set of points a pursuer can reach given its initial state, speed ($v_P$), minimum turn radius ($a$), and maximum range ($R$).
• Engagement Zone (EZ): A shifted version of the RR, accounting for the evader's motion during the time it takes the pursuer to traverse its range.
• Assumptions: The pursuer is stationary at launch; evaders follow known straight-line trajectories; interception is deterministic based on kinematic constraints.

B. Parameter Inference via Optimization

The core learning algorithm estimates the pursuer parameter vector $\theta_P = [x_P, y_P, \psi_P, a, R, v_P]^\top$ using a gradient-based multi-start optimization approach.

• Loss Functions: Two distinct formulations are introduced based on interception assumptions:
  1. Boundary Interception ([A1-strict]): Assumes interception occurs exactly on the RR boundary. The loss function penalizes deviations of the observed interception point from the boundary and ensures non-intercepted trajectories remain outside the RR.
  2. Interior Interception ([A1-relaxed]): Assumes interception can occur anywhere inside the RR. The loss function ensures intercepted points are strictly inside the RR and surviving points are strictly outside.
• Speed Estimation: Since speed is unobservable from binary outcomes alone, Case 3 introduces a launch-time measurement (e.g., from acoustic/visual sensors) to make speed identifiable.
• Multi-Start Strategy: To handle the non-convexity of the loss landscape (where many parameter sets can explain the same data), the algorithm uses Latin Hypercube Sampling (LHS) to generate multiple initial guesses. These are optimized in parallel using IPOPT (an interior-point solver) with gradients computed via JAX (automatic differentiation).
• Iterative Refinement: The algorithm maintains a set of feasible parameter vectors. As new sacrificial agents are deployed, the set is updated, and the standard deviation of the estimates is monitored until convergence.

C. Sacrificial Trajectory Selection

To maximize information gain with minimal agents, two trajectory selection strategies are proposed:

1. Geometric Heuristic (Boundary Case): Selects trajectories that maximize the spatial spread of predicted interception points along the inferred RR boundary, ensuring uniform exploration.
2. Bayesian Experimental Design (Interior Case): Uses a D-optimality criterion based on the Gauss-Newton information matrix. It selects trajectories that maximize the expected curvature of the loss function (information gain), specifically targeting regions where the loss is most sensitive to parameter changes (near the boundary).

D. Safe Trajectory Planning

Once a set of feasible pursuer parameters is learned, a safe path planner generates trajectories for high-value agents.

• Formulation: A nonlinear optimization problem minimizing path time (equivalent to length at constant speed).
• Constraints:
  • Kinematic feasibility (unicycle model, bounded turn rate).
  • Safety: The trajectory must satisfy $\varphi_{EZ} \geq 0$ for all parameter vectors in the learned feasible set, ensuring the agent avoids every possible engagement zone consistent with observations.
• Representation: Trajectories are modeled using B-splines for smoothness and differentiability.

3. Key Contributions

1. Inference Framework: An optimization-based, multi-start framework to estimate pursuer parameters (position, heading, turn radius, range, speed) directly from binary interception outcomes.
2. Differentiable Loss Models: Two novel loss functions tailored for boundary and interior interception assumptions, enabling gradient-based learning of complex kinematic engagement zones.
3. Active Learning Strategies: Two complementary methods for selecting sacrificial trajectories (geometric spread vs. Bayesian D-optimality) to accelerate parameter convergence.
4. Integrated Planning: A method to directly incorporate the learned set of uncertain pursuer parameters into a time-optimal trajectory planner, guaranteeing safety against all feasible threat configurations.

4. Results

The authors conducted 500 Monte Carlo simulations across three learning cases (varying unknown parameters and noise levels) and two interception assumptions.

• Parameter Convergence:
  • Boundary Interception: Accurate parameter recovery was achieved with 5–7 sacrificial agents in noise-free scenarios and slightly more in noisy scenarios.
  • Interior Interception: Required more agents (9–20) due to the ambiguity of interior capture points, but launch-time measurements (Case 3) significantly accelerated convergence.
• Geometric Consistency: The union of all learned Reachable Regions consistently covered >95% of the true RR area in over 94% of deployment steps, even with noisy measurements.
• Path Planning Efficiency:
  • Even with a single sacrificial agent, the safe path length decreased significantly compared to planning with zero knowledge (which requires avoiding the entire theoretical maximum range).
  • With 3–5 agents, the planned safe paths approached the optimality of paths planned with perfect knowledge of the pursuer.
• Robustness: The method remained effective under additive Gaussian noise in trajectory measurements and launch times.

5. Significance

This work bridges the gap between active sensing and safe motion planning in adversarial environments.

• Data Efficiency: It demonstrates that sparse binary data (intercept/survive) is sufficient to constrain complex kinematic models, eliminating the need for continuous sensing or prior probability distributions.
• Safety Guarantees: By maintaining an ensemble of feasible parameters rather than a single point estimate, the framework provides robust safety guarantees that account for inference uncertainty.
• Practical Application: The framework is applicable to scenarios involving radar mapping, path clearance, and autonomous navigation in contested airspace where threat capabilities are initially unknown.
• Future Directions: The paper outlines extensions to mobile pursuers, probabilistic interception models, and coordinated multi-agent probing, establishing a foundation for more complex adaptive adversarial interactions.