Probabilistic Weapon Engagement Zones for a Turn Constrained Pursuer

TL;DR

The paper tackles safe trajectory planning under adversarial uncertainty by extending the curve-straight basic engagement zone (CSBEZ) to a probabilistic CSPEZ framework. It develops four uncertainty-propagation approaches—MCCSPEZ, LCSPEZ, QCSPEZ, and NNCSPEZ—and integrates them into a B-spline trajectory-optimization pipeline with IPOPT, using automatic differentiation for gradient information. Empirical results show NNCSPEZ offers the closest alignment to a Monte Carlo baseline and yields the safest, near-optimal paths with manageable computation. The work advances robust planning in contested environments and lays a groundwork for future 3D extension and multi-threat scenarios.

Abstract

Curve-straight probabilistic engagement zones (CSPEZ) quantify the spatial regions an evader should avoid to reduce capture risk from a turn-rate-limited pursuer following a curve-straight path with uncertain parameters including position, heading, velocity, range, and maximum turn rate. This paper presents methods for generating evader trajectories that minimize capture risk under such uncertainty. We first derive an analytic solution for the deterministic curve-straight basic engagement zone (CSBEZ), then extend this formulation to a probabilistic framework using four uncertainty-propagation approaches: Monte Carlo sampling, linearization, quadratic approximation, and neural-network regression. We evaluate the accuracy and computational cost of each approximation method and demonstrate how CSPEZ constraints can be integrated into a trajectory-optimization algorithm to produce safe paths that explicitly account for pursuer uncertainty.

Figures (6)

• Figure 1: Reachable set (brown) and CSBEZ (green). The evader’s start position $E$ and projected position $F$ are shown, along with the pursuer’s start $P$, point $G_\ell$, and turn center $C_\ell$. The four vectors $\boldsymbol{v}_{1,\ell}$–$\boldsymbol{v}_{4,\ell}$ used in the path-length derivation are also illustrated.
• Figure 2: Box plot of absolute errors for each CSPEZ approximation relative to the MCCSPEZ baseline; NNCSPEZ shows the lowest error.
• Figure 3: Probability level sets for an example CSPEZ scenario. The pursuer’s mean heading is $\pi/2$ (red marker). NNCSPEZ most closely matches the MCCSPEZ baseline.
• Figure 4: Absolute error between each approximation and the MCCSPEZ baseline for the example scenario. The pursuer’s mean heading is $\pi/2$; RMSE, mean, and max errors are shown in each plot.
• Figure 5: Absolute error versus the trace of the pursuer covariance. As uncertainty increases, LCSPEZ and QCSPEZ degrade while NNCSPEZ remains accurate.