Proportional navigation

For example, if the line of sight rotates slowly from north to east, the missile should turn to the right by a certain factor faster than the LOS-rate. This factor is N.

A missile (red) intercepts a target (blue) by maintaining constant bearing to it. Note that the lines of sight (grey) are parallel throughout the flight.

Proportional navigation (also known as PN or Pro-Nav) is a guidance law (analogous to proportional control) used in some form or another by most homing air target missiles. ^[1] It is based on the fact that two vehicles are on a collision course when their direct line-of-sight does not change direction as the range closes. PN dictates that the missile velocity vector should rotate at a rate proportional to the rotation rate of the line of sight (Line-Of-Sight rate or LOS-rate), and in the same direction. For example, if the target is seen moving from the left to the front, thus moving to the right as seen from the missile, then the missile should turn to the right until this angular motion stops again.

A rather simple hardware implementation of this guidance law can be found in early AIM-9 Sidewinder missiles. These missiles use a rapidly rotating parabolic mirror as a seeker. Simple electronics detect the directional error the seeker has with its target (an IR source), and apply a moment to this gimballed mirror to keep it pointed at the target. Since the mirror is in fact a gyroscope it will keep pointing at the same direction if no external force or moment is applied, regardless of the movements of the missile. The voltage applied to the mirror while keeping it locked on the target is then also used (although amplified) to deflect the control surfaces that steer the missile, thereby making missile velocity vector rotation proportional to line of sight rotation. Although this does not result in a rotation rate that is always exactly proportional to the LOS-rate (which would require a constant airspeed), this implementation is effective and simple to implement.

The basis of proportional navigation was first discovered at sea, and was used by navigators on ships to avoid collisions. Referred to as constant bearing, decreasing range, the concept remains useful: If an object viewed from a ship is getting closer but maintaining bearing, a collision will occur unless action is taken.

Description

${\displaystyle a_{n}=N{\dot {\lambda }}V}$ (pure proportional navigation)

Where ${\displaystyle a_{n}}$ is the acceleration perpendicular to the missile's instantaneous velocity vector, ${\displaystyle N}$ is the proportionality constant generally having an integer value 3-5 (dimensionless), ${\displaystyle {\dot {\lambda }}}$ is the line of sight rate, and V is the closing velocity.

Since the line of sight is not in general co-linear with the missile velocity vector, the applied acceleration does not necessarily preserve the missile kinetic energy. In practice, in the absence of engine throttling capability, this type of control may not be possible.

Proportional navigation can also be achieved using an acceleration normal to the instantaneous velocity difference:

${\displaystyle {\vec {a}}=N{\vec {V}}_{r}\times {\vec {\Omega }}}$

where ${\displaystyle \Omega }$ is the rotation vector of the line of sight:

${\displaystyle {\vec {\Omega }}={\frac {{\vec {R}}\times {\vec {V}}_{r}}{{\vec {R}}\cdot {\vec {R}}}}}$

and ${\displaystyle {\vec {V}}_{r}={\vec {V}}_{t}-{\vec {V}}_{m}}$ is the target velocity relative to the missile and ${\displaystyle {\vec {R}}={\vec {R}}_{t}-{\vec {R}}_{m}}$ is the range from missile to target. This acceleration depends explicitly on the velocity difference vector, which may be difficult to obtain in practice. By contrast, in the expressions that follow, dependence is only on the change of the line of sight and the magnitude of the closing velocity. If acceleration normal to the instantaneous line of sight is desired (as in the initial description), then the following expression is valid:

${\displaystyle {\vec {a}}=-N|{\vec {V}}_{r}|{\frac {\vec {R}}{|{\vec {R}}|}}\times {\vec {\Omega }}}$ (true proportional navigation) ^[2]

If energy conserving control is required (as is the case when only using control surfaces), the following acceleration, which is orthogonal to the missile velocity, may be used:

${\displaystyle {\vec {a}}=-N|{\vec {V}}_{r}|{\frac {{\vec {V}}_{m}}{|{\vec {V}}_{m}|}}\times {\vec {\Omega }}}$

Variants

The types described above are "pure" and "true" PN. There exist also:

• Generalized proportional navigation, where the acceleration ${\displaystyle {\vec {a}}}$ is not necessarily normal to either the LOS vector, but maintains a constant angle to it. ^[2]
• Augmented proportional navigation, which adds a term to compensate for target acceleration (if it can be measured). ^[3]

PN is not suitable for every solution; more complicated guidance laws may be required. For example, Lin has derived a guidance law for midcourse guidance that maximizes missile velocity while minimizing miss distance using optimal control theory. ^[4]

In biology

Holcocephala fusca and Coenosia attenuata are two species of predatory flies that use proportional navigation to reach their prey. The former uses N ≈ 3 with a time delay of ≈ 28 ms, which is suitable for its long-range intercepts and minimizes the control effort required. The latter uses N ≈ 1.5 with a time delay of ≈ 18 ms, which is adapted to its short-range hunts and helps reduce overcompensation. ^[5]

A guidance law resulting in motion camouflage is used by a number of predator species. By setting up the chase so that the predator either appears stationary relative to the background while growing larger (real-point motion camouflage), or always appears at a fixed bearing (infinite-point motion camouflage), the predator reduces its chance of being detected. Such a guidance law is also mathematically related to proportional navigation and similarly provides an efficiency benefit over pure pursuit guidance. The infinite-point case (or "parallel navigation") can be viewed as pure proportional navigation with a distance-dependent N. ^[6]

See also

• Motion camouflage

Bibliography

• Yanushevsky, Rafael. Modern Missile Guidance. CRC Press, 2007. ISBN   978-1420062267.

References

1. Yanushevsky, page 3.
2. Shukla, U.S.; Mahapatra, P.R. (March 1990). "The proportional navigation dilemma-pure or true?". IEEE Transactions on Aerospace and Electronic Systems. 26 (2): 382–392. doi:10.1109/7.53445.
3. Berglund, Erik (2001). "Guidance and Control Technology".
4. Palumbo, NF; Blauwkamp, RA; Lloyd, JM (2010). "Basic Principles of Homing Guidance" (PDF). Johns Hopkins APL Technical Digest. 29 (1): 39.
5. Fabian, Samuel T.; Sumner, Mary E.; Wardill, Trevor J.; Rossoni, Sergio; Gonzalez-Bellido, Paloma T. (October 2018). "Interception by two predatory fly species is explained by a proportional navigation feedback controller". Journal of the Royal Society Interface. 15 (147): 20180466. doi:10.1098/rsif.2018.0466. PMC   6228472 .
6. Justh, E.W; Krishnaprasad, P.S (8 December 2006). "Steering laws for motion camouflage". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 462 (2076): 3629–3643. arXiv: math/0508023 . doi: 10.1098/rspa.2006.1742 .