TRAJECTORY SOLUTION Treatises
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
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
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
aM = N SC dL/dt + (N/2) aT
is the target's acceleration normal to the LOS. The form in
equation (2) is termed "augmented proportional navigation" (abbr. APN).
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
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.
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.
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
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.