How to create an “intercept missile” for a game?

Question

I have a game I am working on that has homing missiles in it. At the moment they just turn towards their target, which produces a rather dumb looking result, with all the missiles following the target around.

I want to create a more deadly flavour of missile that will aim at the where the target "will be" by the time it gets there and I am getting a bit stuck and confused about how to do it.

I am guessing I will need to work out where my target will be at some point in the future (a guess anyway), but I can't get my head around how far ahead to look. It needs to be based on how far the missile is away from the target, but the target it also moving.

My missiles have a constant thrust, combined with a weak ability to turn. The hope is they will be fast and exciting, but steer like a cow (ie, badly, for the non HitchHiker fans out there).

Anyway, seemed like a kind of fun problem for Stack Overflow to help me solve, so any ideas, or suggestions on better or "more fun" missiles would all be gratefully received.

Next up will be AI for dodging them ...

Answer 1

What you are suggesting is called "Command Guidance" but there is an easier, and better way.

The way that real missiles generally do it (Not all are alike) is using a system called Proportional Navigation. This means the missile "turns" in the same direction as the line-of-sight (LOS) between the missile and the target is turning, at a turn rate "proportional" to the LOS rate... This will do what you are asking for as when the LOS rate is zero, you are on collision course.

You can calculate the LOS rate by just comparing the slopes of the line between misile and target from one second to the next. If that slope is not changing, you are on collision course. if it is changing, calculate the change and turn the missile by a proportionate angular rate... you can use any metrics that represent missile and target position.

For example, if you use a proportionality constant of 2, and the LOS is moving to the right at 2 deg/sec, turn the missile to the right at 4 deg/sec. LOS to the left at 6 deg/sec, missile to the left at 12 deg/sec...

In 3-d problem is identical except the "Change in LOS Rate", (and resultant missile turn rate) is itself a vector, i.e., it has not only a magnitude, but a direction (Do I turn the missile left, right or up or down or 30 deg above horizontal to the right, etc??... Imagine, as a missile pilot, where you would "set the wings" to apply the lift...

Radar guided missiles, which "know" the rate of closure. adjust the proportionality constant based on closure (the higher the closure the faster the missile attempts to turn), so that the missile will turn more aggressively in high closure scenarios, (when the time of flight is lower), and less aggressively in low closure (tail chases) when it needs to conserve energy. Other missiles (like Sidewinders), which do not know the closure, use a constant pre-determined proportionality value). FWIW, Vietnam era AIM-9 sidewinders used a proportionality constant of 4.

Answer 2

I've used this CodeProject article before - it has some really nice animations to explain the math.

"The Mathematics of Targeting and Simulating a Missile: From Calculus to the Quartic Formula": http://www.codeproject.com/KB/recipes/Missile_Guidance_System.aspx

(also, hidden in the comments at the bottom of that article is a reference to some C++ code that accomplishes the same task from the Unreal wiki)

Answer 3

Take a look at OpenSteer. It has code to solve problems like this. Look at 'steerForSeek' or 'steerForPursuit'.

Answer 4

Have you considered negative feedback on the recent change of bearing over change of time?

Details left as an exercise.

The suggestions is completely serious: if the target does not maneuver this should obtain a near optimal intercept. And it should converge even if the target is actively dodging.

Need more detail?

Solving in a two dimensional space for ease of notation. Take \vec{m} to be the location of the missile and vector \vec{t} To be the location of the target. The current heading in the direction of motion over last time unit: \vec{h} = \bar{\vec{m}_i - \vec{m}_i-1}}. Let r be the normlized vector between the missile and the target: \vec{r} = \bar{\vec{t} - \vec{m}}. The bearing is b = \vec{r} \dot \vec{h} Compute the bearing at each time tick, and the change thereof, and change heading to minimize that quantity.

The math is harrier in 3d because of the need to find the plane of action at each step, but the process is the same.

Answer 5

You'll want to interpolate the trajectory of both the target and the missile as a function of time. Then look for the times in which the coordinates of the objects are within some acceptable error.