Fox 2 Interceptor Simulation

Abstract

This project is a python-based high fidelity model for simulating infrared guided missile engagments. In 3-DOF, the project moves beyond basic logic to implement professional-grade guidance and numerical integration standards.

Technical Architecture

The simulation utilizes a 4th-Order Runge-Kutta (RK4) integrator to propagate the entity state vector $S = [x, y, z, v_x, v_y, v_z, m]$.

Unlike Euler integration, which has a local truncation error of $O(\Delta t^2)$, RK4 samples the derivatives at four points to achieve $O(\Delta t^5)$ local error, ensuring stability during high-G terminal maneuvers.

$\rightarrow y_{n+1} = y_n + \frac{h}{6}(k_1 + 2k_2 + 2k_3 + k_4)$

Guidance & Navigation

The interceptor utilizes Gravity-Compensated Proportional Navigation (PN) analogous to an operational sidewinder missile to prevent "sagging" due to underestimating gravity. The commanded acceleration $a_c$ is derived from the Line-of-Sight (LOS) rate $\dot{\lambda}$ and closing velocity $V_c$.

$\rightarrow a_c = N \cdot V_c \cdot \dot{\lambda}$

Propulsion/Dynamic Mass

The model couples kinematics with a variable-mass propulsion system. Mass $m$ is treated as a dynamic variable, depleting according to the rocket equation based on thrust $T$ and specific impulse $I_{sp}$. $\rightarrow \dot{m} = -\frac{T}{g_0 I_{sp}}$ This ensures the $F=ma$ calculations account for the significantly increased maneuverability of the interceptor in the post-burn "dart" phase.

Aerodynamics & Atmospheric Modeling

• ISA Model: Air density $\rho$ is calculated using the International Standard Atmosphere model for accurate drag profiles across altitudes.
• Transonic Drag Rise: Implements Mach-dependent drag coefficients to simulate wave drag increases between Mach 0.8 and 1.2.
• $\rightarrow F_d = \frac{1}{2} \rho v^2 C_d A$

Seeker + Countermeasure (flare) logic

The IR seeker model includes: - Gimbal Limits: Rigid FOV and seeker head constraints - Signal-to-Noise Modeling: Detection probability based on target IR signature, range, and aspect angle. - IRCCM: Flare susceptibility logic based on IR intensity ratios and spatial separation/

Monte Carlo Probability Analysis & $P_k$

The effectiveness of each engagement is evaluated via Monte Carlo iteration. by injecting Gaussian noise into initial launch conditions and target state estimates, the system generates a statistical probability of kill ($P_k$). $\rightarrow P_k = \frac{\sum \text{Intercepts}}{\sum \text{Total Runs}}$

Technical Note: This is a 3-DOF point-mass simulation. Future iterations may focus on 6-DOF by implementing rotation mechanics.