Imagine a "drone" and a target point on a 2d plane. Assuming the target is stationary, there are eight parameters:
P = my position Q = target's position V = my velocity I = my moment of inertia w = my angular velocity s = my angular position T = max thrust U = max torque
The drone's job is to get to the target as fast as possible, obeying max torque and max thrust. There are only two ways to apply the torque, since this is only in a 2d plane. Thrust is restricted to only go in one direction relative to the orientation of the craft, and cannot be aimed without rotating the drone. Neglect any resistance, you can just pretend it is floating around in 2d outer space. Let's say the drone checks an equation at time interval
t (maybe something like every .01 seconds), plugs in the parameters, and adjusts its torque and thrust accordingly.
- What should the equations for thrust and torque be?
What have we tried?
We know that the time it takes for the drone to reach the target in the x-direction has to be the same for the same time in the y-direction. There is going to have to be some integral over time in each dimension to account for the changing thrust based on total thrust, and total thrust in each direction given the changing angular position. I have no idea how to tie the torque and thrust together in a practical way where a function can just be called to give what thrust and torque should be applied over the interval
t unless there is some other technique.