PoKER: A Probability of Kill Estimation Rate Model for Air-to-Air Missiles Using Machine Learning on Stochastic Targets

Joao P. A. Dantas; Andre N. Costa; Diego Geraldo; Marcos R. O. A. Maximo; Takashi Yoneyama

cs.LGOct 23, 2026

PoKER: A Probability of Kill Estimation Rate Model for Air-to-Air Missiles Using Machine Learning on Stochastic Targets

Authors: Joao P. A. Dantas, Andre N. Costa, Diego Geraldo, Marcos R. O. A. Maximo, Takashi Yoneyama

Organizations: Decision Support Systems Subdivision, Institute for Advanced Studies Brazil · Computer Science Division, Aeronautics Institute of Technology, Brazil · Electronic Engineering Division, Aeronautics Institute of Technology, Brazil

Abstract

This work introduces PoKER, a novel probabilistic model engineered to optimize missile launch effectiveness in air-to-air scenarios, specifically within Beyond Visual Range (BVR) air combat. Unlike conventional Weapon Engagement Zone (WEZ) models that delineate zones based on static distances such as maximum, minimum, and no-escape ranges, PoKER applies machine learning to predict kill probabilities more accurately by integrating the stochastic behaviors of targets and missile miss distances. This model dynamically evaluates target behavior, greatly expanding the predictive capabilities of engagement analysis. By factoring in elements such as warhead lethality, target and shooter orientations, and the specific conditions of engagement, PoKER provides important insights into engagement dynamics and quantifies success probabilities. Consequently, it can potentially be an important tool for BVR air combat pilots, improving operational decision-making within this specialized combat domain.

Figures & tables

Figure 1: Diagram illustrating aerial combat WEZ, detailing ranges for missile engagement from maximum kinematic reach to minimum arming distance, including the airspace where the target has limited evasion options.

Figure 2: Missile engaging a target performing a +5 G maneuver, illustrating the missile’s adaptive trajectory in response to the target’s high-G evasive actions.

Figure 3: Metrics from the simulation include target velocity (North, East), acceleration (North, East), altitude, and heading over time. Note that in this case, the target maneuver was considered to be leveled, and the target kept a constant altitude while evading the missile, resulting in a flat altitude profile. There is also a delay time in the pilot’s reaction, causing all of the charts to present a flat profile for the first few seconds.

Figure 4: Simulation metrics for the missile, showing thrust, mass, lift, drag, total alpha, pitch, heading, Mach number, seeker angle, and acceleration (X, Y, Z) over time.

Figure 5: Diagram illustrating the heading, off-boresight, and pitch angles between two aircraft in aerial maneuvering with respect to their centers of mass (CM).

Parameter	Min	Max	Unit
Shooter altitude ( alt_sht )	1,000	45,000	ft
Shooter velocity ( vel_sht )	300	700	kt
Shooter pitch ( pit_sht )	-30	30	deg
Target altitude ( alt_tgt )	1,000	45,000	ft
Target velocity ( vel_tgt )	300	700	kt
Target heading ( hdg_tgt )	-180	180	deg

Table 1: Input simulation data with the respective intervals.

Figure 6: Impact of Detonation Distance on Warhead Lethality. The figure shows the relationship between detonation distance and $P_{\text{kill}}$ for a proximity-fuzed warhead. The lethal radius represents the optimal distance for maximum lethality, contrasted with the decreasing kill probability as the distance increases.

Variable	Value	Unit
Explosive Data
Explosive Type	PBX-9404	-
Explosive Density ( $\rho_{\text{e}}$ )	1710	$\text{kg/m}^{3}$
Explosive Mass ( $M_{\text{e}}$ )	8.94	kg
Detonation Velocity	9500	m/s
Gurney Constant ( $G$ )	2895	m/s

Table 2: Missile’s warhead data including structural, fragment, and explosive properties.

Figure 7: Illustration of ten diverse scenarios of defensive turn maneuvers. Each scenario shows a target attempting to maximize distance from the launcher. The dark red points (smaller) indicate the initial position of the target, while the orange (thinner) and red arrows (thicker) represent the initial heading and the suggested turn direction, respectively. The dark circle (larger) represents the shooter. The blue arc indicates the turn degree ( turn_dg ) performed by the target, which varies between 0° and 180°.

Figure 8: Graph illustrating the target’s delay time as a function of its orientation with respect to the shooter.

Model	PR	ANN	XGBoost
Hyperparameters	degree=4	hidden_layers=3, units=128	max_depth=14, learning_rate=0.01, subsample=0.9, colsample_bytree=0.9, colsample_bylevel=0.6, min_child_weight=5, gamma=0.1
MAE	11.52	2.33	5.04
MSE	296.26	51.90	117.14
RMSE	17.21	7.20	10.82
R²	67.94	94.38	87.31
Training Time (s)	532.23	4722.17	6098.33

Table 3: Comparative evaluation of machine learning models based on the proposed metrics. The best results for each metric are highlighted in bold red text.

Metric	Friedman Statistic	p-value
MAE	10.0	0.0067
MSE	10.0	0.0067
RMSE	10.0	0.0067
$R^{2}$	10.0	0.0067
Training Time	8.4	0.0150
Inference Time	4.8	0.0907

Table 4: Friedman Test Results. Significant p-values are highlighted in bold red text.

Metric	Comparison	p-value
MAE	PR vs ANN	0.0045
	XGBoost vs PR	0.2541
	XGBoost vs ANN	0.2541
MSE	PR vs ANN	0.0045
	XGBoost vs PR	0.2541
	XGBoost vs ANN	0.2541

Table 5: Nemenyi post-hoc test p-values for pairwise comparisons among models. Significant p-values are highlighted in bold red text.

Panel of Figure 9 — Figure 9: Comparative analysis of neural network performance metrics. (a) illustrates how model configurations, varying in units and hidden layers, impact accuracy through the mean R² value. (b) delineates the effect of model complexity on inference time, measured in seconds.

Figure 10: Enhanced MFD representation incorporating the integration of $P_{\text{kill}}$ alongside the maximum range and no-escape zones, providing a multifaceted view of the engagement zone to support informed decision-making in missile launch scenarios.

BVR Sim: An Open and High-Throughput Environment for Heterogeneous Air-Combat Reinforcement Learning
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Beijing University of Posts and Telecommunications · Air Force Engineering University · Aviation Engineering School, Air Force Engineering University Xi’an 710038, China
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May 24, 2026Chengwei Li, Junlin Liu, Yang GaoMulti-Agent Coordination Reinforcement Learning
Institute of Automation, Chinese Academy of Sciences · School of Artificial Intelligence, University of Chinese Academy of Sciences

Aug 26, 2026J/K move · Enter open · S save

BVR Sim: An Open and High-Throughput Environment for Heterogeneous Air-Combat Reinforcement Learning

Haocheng Sun, Mulai Tan

Beijing University of Posts and Telecommunications · Air Force Engineering University · Aviation Engineering School, Air Force Engineering University Xi’an 710038, China

RL Benchmarks Reinforcement Learning Heterogeneous MARL

Open paper AskPDF arXiv

Beyond-visual-range (BVR) air combat is a challenging reinforcement-learning domain characterized by partial observability, long-horizon decision making, energy management, and limited weapons. We present BVR Sim, an open-source Gymnasium-style environment designed for heterogeneous air-combat reinforcement learning. BVR Sim supports multiple JSBSim aircraft models, including the F-15, F-16, F/A-18, and F-22, with configurable weapons, sensors, controllers, and opponents. A unified tactical action interface specifies desired heading, altitude, speed, and weapon release above aircraft-specific inner-loop controllers, enabling policies to operate across heterogeneous platforms. The environment provides interchangeable Python and accelerated C++ backends, entity-oriented observations, compositional rewards, scripted opponents, replay and visualization, and adapters for multi-agent learning frameworks. At a 0.4-s decision interval, the C++ backend achieves 104 simulated seconds per wall-clock second in 1-vs-1 and remains practical through 10-vs-10 scenarios. A policy trained only on the F-16 transfers without retraining to four unseen aircraft, reaching a 45.5% mean win rate with aircraft-specific controller adaptation. MAPPO and HAPPO experiments further verify end-to-end compatibility with standard multi-agent reinforcement-learning pipelines.

Abstract

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