# Inverse kinematics for object trajectory in 2D (control theory?)

## Main Question or Discussion Point

Drone - grey, Asteroid - brown.

The goal is to steer the drone into the asteroid. The force acting upon the drone ("thrust") for movement can be in any direction, varies in magnitude, and can be applied at any discrete point in time.
The drone's future location is deterministic. I'm confused as to whether there is an analytic solution for this problem or whether I must brute force.

I've already used an iterative approach (using PID control) to do something similar (see http://www.kongregate.com/games/mahirigames/asteroids-rox-teehee" [Broken]) but I'd like to take this a step further and look for a complete analytic solution. The brute force approach would evaluate a huge amount of permutations of thrust vectors and possibly look for one matching the criteria given some heuristic. I can't implement it btw.

The future location of the drone is dependent on 3 things – the initial (current) velocity, any future forces, and the time before reaching the destination.

Should I try using langrange multipliers or PDE's. How would I go about finding the shortest path (aka minimize time to collision). I actually expected this to be done already but google didn't turn anything up.

Any enlightening ideas or guidance would be greatly appreciated.

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