Proportional Navigation Guidance — Interactive Simulator

Proportional Navigation (PN) is the most widely used guidance law in homing missiles. The core idea is elegantly simple: steer so that the line-of-sight (LOS) rotation rate goes to zero. If the LOS angle stops changing, a collision course is guaranteed.

The Three Laws

Law	Command	Notes
TPN — True PN	nc = N′ · Vc · λ̇	Uses closing velocity; optimal for constant-Vc
PPN — Pure PN	nc = N′ · VM · λ̇	Uses missile speed; simple, robust
APN — Augmented PN	nc = N′ · Vc · λ̇ + ½N′ · nT	Feedforward target acceleration term

N′ is the effective navigation ratio (typically 3–5), λ̇ is the LOS rate, and nT is target normal acceleration.

Simulator

The simulator below integrates the 2D engagement equations with 4th-order Runge-Kutta (RK4). Set initial conditions, pick a guidance law, and hit RUN SIMULATION. Use Play or the scrubber to animate the intercept.

State Equations

The missile and target states [xM, yM, θM, xT, yT, θT] evolve as:

ẋM = VM cos θM,   ẏM = VM sin θM,   θ̇M = nc / VM
ẋT = VT cos θT,   ẏT = VT sin θT,   θ̇T = nT / VT

LOS geometry:

R  = √[(xT−xM)² + (yT−yM)²]
λ  = atan2(yT−yM, xT−xM)
λ̇  = [(xT−xM)(ẏT−ẏM) − (yT−yM)(ẋT−ẋM)] / R²
Vc = −[(xT−xM)(ẋT−ẋM) + (yT−yM)(ẏT−ẏM)] / R

Key Observations

• Miss distance — the minimum R achieved. A good intercept gives <1 m.
• Acceleration demand — TPN typically produces lower peak nc than PPN in high-speed tail-chase scenarios.
• APN advantage — when the target maneuvers with a delayed onset, APN’s feedforward term significantly reduces miss distance compared to TPN/PPN.
• Navigation ratio N′ — values below 3 may miss; above 5 rarely help and demand more control effort.