Missile guidance | Combine

### Missile guidance

Missile guidance is a field of control theory that has been studied thoroughly the last 70 years, see for example (Zarchan, 2013). It is an interesting field of control theory that has many practical applications. Interestingly there are also many parallels in nature. It turns out that evolution has optimized how to pursue targets, i.e. prey. These optimized strategies in nature are in many cases very similar to the strategies used in missile guidance.

The engineering needed for controlling a missile is comprised of many separate fields: control theory, aerodynamics, propulsion, material science, etc… Here only control theory is discussed, and only a subset of that.

A typical architecture for missile guidance and control can be described as follows:

Figure 1. Missile guidance and control.

The target state is measured by sensor(s). This measurement together with the missile state is fed into the missile control system. This system can be split into two parts, guidance and autopilot. The guidance portion determines what the optimal maneuver is for the missile. The autopilot performs that maneuver by controlling the missile, typically with control surfaces such as rudders. In this discussion the guidance is considered, i.e. how should the missile maneuver in an optimal way. The sensors, autopilot and missile dynamics are assumed to be ideal; without latency, noise or other issues.

# Guidance principles and strategies

Event though there exist many different strategies for missile guidance they can be dived into two major ones to consider when designing modern missiles, Proportional Navigation and Command-to-Line-Of-Sight.

Proportional Navigation (PN) is a guidance law that exploits the fact that two vehicles that have constant Line-of-Sight (LOS) to each other are on a collision course. In other words, if the LOS to the target does not rotate seen from the missile, it is on an intercept course. This has been known in shipping for hundreds of years and is used to avoid collisions between ships.

PN tries to achieve a constant LOS angle by accelerating the missile towards the rotation of the LOS, and thereby eliminating the rotation. Basically, the assumption is that the best guess on future target trajectory is that it will continue its current course.

The guidance law in its simplest form can be described as:

$$a_m =N\dot{\lambda} |V_c|$$

$$a_m$$: Commanded acceleration perpendicular to LOS
$$\dot{\lambda}$$: rotation of LOS
$$|V_c|$$: closing speed of missile relative to target

In a missile with an onboard target tracking system, a camera or an IR-sensor,  is easily available.   can in some cases be measured but is often approximated. Since the missile should have much higher speed than the target a good approximation can be missiles speed.

N is a design parameter, typically in the range 3-6 and always >2. A high value guides the missile on to an intercept course faster but requires higher accelerations and makes the missile more sensitive to noise. In practice it is common to have a varying value of N, a low value at launch and higher when closer to the target.

Note that the range to the target is not needed. This is an important property that has contributed to the widespread use of this principle.

The above guidance law cannot guarantee interception against a maneuvering target, that requires an extension of the guidance law where the target acceleration is considered, called augmented proportional navigation (APN):

$$a_m=N\dot{\lambda}|V_c|+\frac{Na_{\bot}}{2}$$
$$a_{\bot}$$:target acceleration normal to the LOS

The acceleration is seldom possible to measure by a sensor and must be estimated, which can be difficult and create high noise. Note also that the required acceleration by the missile is proportional to N. This means that choosing a more responsive missile, i.e. high N, requires more acceleration from the missile.

## Command-to-Line-Of-Sight

Command-to-Line-Of-Sight (CLOS) works by keeping the missile on the line seen from the sight towards the target. If the missile is closing on the target it will eventually intercept its path regardless of the target range.

Figure 3. Command-to-Line-of-sight. The missile is kept on the line between the sight and the target.

The resulting flight path is one where the missile leads the target more and more the closer it is to intercept. This can be seen in Figure 3 as an increasing angle between the LOS and missile velocity vector.

At launch the missile flies straight at the target and near intercept the missile matches the targets angular velocity. This is intuitively a good strategy: at launch it is hard to predict where the target is heading and how it will maneuver in the future, so a good guess is its current direction. But closer to intercept, the target has little time left to maneuver and the intercept point can be predicted, i.e. use full lead angle.

Note that this guidance principle also doesn’t require the range to the target. All that is needed is some way to measure the missiles position relative to the LOS as seen from the sight. Also, the missile is guaranteed to intercept the target if kept on the LOS, regardless of the target maneuver. The missile does not, in theory, need to maneuver more than the target, which is the case for PN and APN.

## Other guidance principles

There exist other principles than PN and CLOS. Pure Pursuit is a principle that has been used. Pursuit works by just pointing the missile velocity vector straight at the target. If the target is not stationary, or very close to stationary, then this will result in a missile trajectory that requires very high, in theory infinite, accelerations close to the target. The principle is chosen where simplicity is more important than performance, i.e. against targets with negligible movements.

## Simulation

The properties of the principles can be examined by some simulation examples. In the simulations the missile has constant speed which is unrealistic, but it helps to clearly show the properties of the guidance principles. The missile is initialized with a velocity pointing straight at the target.

In the first scenario the target is travelling with a constant velocity, from right to left. CLOS and PN results in the following missile trajectories.

In Figure 4 it is clear that at launch PN accelerates towards a straight intercept course. In comparison, CLOS does not accelerate as much at launch but the trajectory requires more acceleration closer to intercept with the target. For a target with a constant velocity PN generally travel a shorter distance.

In the next simulation the target starts with a constant velocity (right to left), but will after some time do a 90° maneuver, and then continue with a constant velocity.

At launch and until the target maneuver (#1 – #2) the missiles behave as the previous example. PN maneuvers towards an intercept course assuming the target will continue with a constant velocity. This results in a greater course change for PN during and after the target maneuver (#2 – #3) since the new predicted intercept course has moved. CLOS on the other hand did not fully commit to the intercept course before the target maneuver and requires smaller course corrections after the target maneuver.

Note also that PN is not guaranteed to intercept the target during its maneuver, but CLOS is (in theory).

Both CLOS and extended PN are useful as guidance principles. Which one that’s optimal is, as always, a matter of how “optimal” is defined. CLOS is in practice only used for shorter ranges since the target must be seen by the sight at all times.
Missiles using PN typically has the angular measuring sensor in the missile which gives increasing accuracy and precision when closing on the target. CLOS has the sensor in the sight which requires a better sensor to achieve an acceptable performance, because of the longer range to target. However, a sensor in the missile needs to be small, cheap, and disposable, whereas a sensor in the sight can be designed with less compromises.

In practice the type of guidance principle is chosen with a number of considerations. Considerations such as: what is the kinematic performance of the missile? What are the intended targets? What kind of sensors are available?

# Path optimization in nature

In nature there exist several techniques used by predators to pursue their prey. The optimal strategies can vary depending on the goal. As for missile guidance, the chance to catch a prey can be optimized but there can also be other optimization variables. Animals have evolved to detect motion, predators can therefore try to minimize their movement against the perceived background to limit the available reaction time the prey has until it is caught (Zamani & Amador Kane, 2014; Mizutani, Chahl, & Srinivasan , 2003).

These techniques that have been shown to be used in nature are very similar to modern missile guidance laws[1].

When the background, e.g. trees and bushes, is sufficiently close, minimizing movement against that background is accomplished by staying on the line between a landmark and the prey. This is of course very similar to CLOS, where the sight is exactly behind the missile as seen from the target.

When the background is far away, as the sky is for birds attacking from above, the strategy instead becomes “Parallel Navigation” where the line between the predator and prey has a constant bearing.

The same strategy has also been seen with bats, where they keep a constant bearing towards their prey. But since bats hunt at night this is not to avoid detection but rather because it is an efficient strategy, and quite close to optimal for catching erratically moving insects  (Ghose, Horiuchi, Krishnaprasad, & Moss, 2006). Bats and their interaction with prey is interesting in many aspects. The echolocating sonar they use has made some of their prey evolve countermeasures against it, where they emit sound to “jam” the sonar. This has then caused the bats to evolve a more complex sonar to counteract the jamming. Compare this to the military techniques of ECM and ECCM.

# Conclusion

Choosing the strategy for guiding an object to intercept another object is an interesting engineering problem. A theoretical analysis is useful to show the practical application. Choosing the “optimal” principle is only possible if there are stated goals and requirements, as well as knowing prerequisites and limitations.

Choosing PN as a guidance principle can seem to be optimal if looking at the problem with some specific conditions, such as having a target traveling with constant velocity. But when the target maneuvers CLOS seems to be a better choice. But PN can be modified to Augmented PN, where it may give better performance.

The analysis can show the basic properties of the principles, but the final choice needs to consider all aspects, including such things as development cost and time. But this is all part of engineering!

# References

Armstrong, R. E., Drapeau, M. D., Loeb, C. A., & Valdes, J. J. (2010). Bio-inspired Innovation and National Security.

Ghose, K., Horiuchi, T. K., Krishnaprasad, P. S., & Moss, C. F. (2006). Echolocating Bats Use a Nearly Time-Optimal Strategy to Intercept Prey. PLOS Biology.

Mizutani, A., Chahl, J. S., & Srinivasan , M. V. (2003). Motion camouflage in dragonflies. Nature.

Zamani, M., & Amador Kane, S. (2014). Falcons pursue prey using visual motion cues: new perspectives from animal-borne cameras. Journal of Experimental Biology.

Zarchan, P. (2013). Tactical and Strategic Missile Guidance.

[1] https://www.newscientist.com/article/dn3870-dragonfly-trick-makes-missiles-harder-to-dodge/