Colision angle between guided missile and target

[zatoichi]

Hi!
I'm designing a game. Given a target moving on a straight line, with constant speed, followed by a guided missile with a constant speed. The missile is faster than the target. It is always headed towards the target. No deviation due to inertia or whatsoever. Supposing the target is a sphere of radius r1. What is the direction of movement of missile when it reaches hits target( distance between is r1)?

 

[russ_watters]

Does the missile start out directly behind the object? If not, that's sometimes going to be an impossible tracking system you've described, since the missile will have to turn in an ever-tightening circle to make the collision happen. It will eventually pull in behind the target and hit it from behind (assuming r1 is relatively small).

You may want to have your missile lead the target. The missile could measure the rate of change of the angle to the target (target angle) and try to keep that angle constant by turning faster than the change in target angle.

 

[zatoichi]

Unfortunately, the missile has to follow the target, not anticipate it. The missile starts from the origin, so for a speed greater then that of the target it will eventually hit it. This of course, for the case I have: the target doesn't change it's course one bit. Keeps going in straight line. In the game, missile scope is not to explode, but to deviate direction of target. For this I have to know the angle of colision.

 

[berkeman]

Unfortunately, the missile has to follow the target, not anticipate it. The missile starts from the origin, so for a speed greater then that of the target it will eventually hit it. This of course, for the case I have: the target doesn't change it's course one bit. Keeps going in straight line. In the game, missile scope is not to explode, but to deviate direction of target. For this I have to know the angle of colision.

That is not how missle systems work. Why are you putting that dumbed-down constraint on the missle?

 

[zatoichi]

Do you have tthe solution for that case? I think it only makes the problem harder.
This is what I did so far: I considered polar coordinates. So there will be two accelerations, straight line and rotation, given by:
dr/dt=-Vm-Vtcos(Betha-Alpha)
rdBetha/dt=Vtsin(Betha-Alpha)
where:r=distance between missile and target
Vm=missile speed
Vt=target speed
Betha=angle between orisontal axis and Vm
Alpha=angle between orizontal axis and Vt
(Note that if the missile anticipates the target, the two equations will be a bit more complex)
I divided them one by another, got rid of time. both 1/sin and cos/sin integrals turned into natural logarithm after integration, and so did 1/r. However, I cannot get the formula for the Betha as function of final r(the dimensions of the target), and the other constants given. The differential equation was solved, but the result is so nonlinear I cannot get Betha, and I really tried.
Another try I gave, was to compute dr from first equation, replace in second to get dBetha, compute the new dr, new dBetha and so on. However, the final Betha keeps growing with every decrease of dt. Wat did I do wrong?

 

[berkeman]

Do you have tthe solution for that case? I think it only makes the problem harder.
This is what I did so far: I considered polar coordinates. So there will be two accelerations, straight line and rotation, given by:
dr/dt=-Vm-Vtcos(Betha-Alpha)
rdBetha/dt=Vtsin(Betha-Alpha)
where:r=distance between missile and target
Vm=missile speed
Vt=target speed
Betha=angle between orisontal axis and Vm
Alpha=angle between orizontal axis and Vt
(Note that if the missile anticipates the target, the two equations will be a bit more complex)
I divided them one by another, got rid of time. both 1/sin and cos/sin integrals turned into natural logarithm after integration, and so did 1/r. However, I cannot get the formula for the Betha as function of final r(the dimensions of the target), and the other constants given. The differential equation was solved, but the result is so nonlinear I cannot get Betha, and I really tried.
Another try I gave, was to compute dr from first equation, replace in second to get dBetha, compute the new dr, new dBetha and so on. However, the final Betha keeps growing with every decrease of dt. Wat did I do wrong?

Yes, it does complicate the math, but it gives a much more optimum intercept. You use the target speed and direction at any moment to give you a vector to the projected intercept point. You know the target's speed and direction, and you know the missle speed, so that gives you an angle to the point where you would intercept the target if everything stays the same. If the target changes its direction, you need to re-calculate the intercept point.

 

[zatoichi]

It will be no longer a guided missile, but is so simple that I'll do it. I am designing general optimization techniques for games, and in the end I'm going to publish a paper. Though simple your idea helped me, and I would like to mention it . May I know your real name?

 

[berkeman]

It will be no longer a guided missile, but is so simple that I'll do it. I am designing general optimization techniques for games, and in the end I'm going to publish a paper. Though simple your idea helped me, and I would like to mention it . May I know your real name?

It's definitely a guided missle, just a more intelligent one. There is nothing new about this type of guidance system -- no need to give me any credit for it. If anything, you can just point to this thread here if you want to give attribution to where you learned about the idea.

 

[Bob S]

It's definitely a guided missle, just a more intelligent one. There is nothing new about this type of guidance system -- no need to give me any credit for it. If anything, you can just point to this thread here if you want to give attribution to where you learned about the idea.

One of the best and oldest guided missiles, but perhaps not the most intelligent one (intelligence is not always an asset), is the Sidewinder missle (a sidewinder is a type of rattlesnake), developed by some rogue scientists at China Lake, in the middle of the Mojave Desert, in the mid 1950's. The Lab was not given any funds for this project, so the scientists used "spare funds" from other projects. The Sidewinder's principle was very simple. It had a reddish lens (for transmitting IR light), maybe 2 inches in diameter, right on the tip of the missle tied to the guidance system. It's lens followed (and caught) anything emitting IR light, like jet engines. I was at China Lake in 1957, and saw a box in a gimble with a lens on it, sitting on a table. The speaker had a cigarette. The "eye" followed him as he walked around the table.

[Added in edit] The IR lens focussed the image of the target on a small 4-quadrant IR photodetector (I don't know how they did this in 1957). The quadrants were labelled up, down, left, right. In the Sidewinder, the signals were amplified by the guidance system and sent to the Sidewinders 4 canards (control surfaces at front of missile).