Homing Guidance

Many missiles are guided to their targets by "homing guidance" laws.  These laws are mathematical algorithms that are designed to guide the missile to the target in some efficient and effective way.  Perhaps the most ubiquitous of the homing guidance laws is one that is usually called "Proportional Navigation", even though it is not, strictly-speaking, a navigational algorithm, but is a guidance algorithm.

Proportional Navigation (abbr. PN) uses the measured angular rate of the vector between missile and target to calculate how much the missile should accelerate in a direction normal to this "line-of-sight" (abbr. LOS) vector
.  In its simplest form in a single plane, the PN guidance law calculates this maneuver acceleration as:
                aM = N SC dL/dt                                                                 (1)

where N is the PN gain (a value between 3 and 5 in most cases), SC  
is the closing speed of the missile with respect to the target, and dL/dt  is the LOS angular rate.  Under certain ideal conditions, the application of this guidance law minimizes the missile maneuver energy required to intercept a non-maneuvering target.  An augmented form of the PN law should be applied if the target accelerates normal to the LOS:

                aM = N SC dL/dt  + (N/2) aT                                                              (2)

where aT is the target's acceleration normal to the LOS.  The form in equation (2) is termed "augmented proportional navigation" (abbr. APN).

The PN and APN laws were originally derived without regard for the effects of gravity, and the accelerations in the formulas are "sensible" accelerations that can be measured, at least theoretically, by Accelerometers.  Happily, even in a gravity field, the formulas are still optimal to the degree that the gravitational acceleration of the missile and target are the same.  In any likely scenario, the small difference in gravitational accelerations of the missile and target during homing are negligible.

The PN law should only be used outside the atmosphere against a non-maneuvering ballistic target, which has zero sensible acceleration.  The APN law should be used for all other applications.

PN, instead of APN, is sometimes wrongly used in anti-aircraft missiles.  Even the non-maneuvering aircraft target has an upward one-g sensible acceleration, and the APN formula is needed.  Instead, a so-called "gravity compensation" term is sometimes added to the right side of the PN formula to compensate for the downward gravitational acceleration of the missile.  This term typically has a value of one g.  In the APN formula, the non-maneuvering aircraft's one-g sensible acceleration is multiplied by N/2, producing a more efficient intercept.

In most applications, the missile cannot accelerate normal to the LOS.  For example, an anti-aircraft missile most often uses aerodynamic control surfaces to maneuver, and its maneuver acceleration is approximately normal to the missile's longitudinal axis.  And, in most engagement scenarios as the missile closes on the target, the missile's longitudinal axis is canted with respect to the LOS.  The sine of this cant angle is proportional to the ratio of target speed to missile speed and to the sine of the angle between the target's velocity and the LOS.  Therefore, this cant angle is greatest for high-speed crossing targets.

The missile's guidance computer must take into account the cant angle between the missile's longitudinal axis and LOS in determining the required maneuver acceleration normal to the missile's longitudinal axis.  This maneuver acceleration must combine with any acceleration along the longitudinal axis (due to thrust and/or aerodynamic axial force) to produce an acceleration normal to the LOS that is equal to the aM value calculated by the PN or APN guidance law.