Looked around a bit and couldn't seem to find a similar question. (either that or it was worded with vocabulary above the multivariable calculus I've taken. :))


Roughly worded: I would like to develop an algorithm (either in the form of "action to take each discrete time step" or "do these actions at exactly these times") for navigating a rigid body vehicle time-optimally in a frictionless environment from point A to point B.


This specific instance happens to be a Spaceship, whom I want to get from point A to point B utilizing Forward thrusters, reverse thrusters (for braking), torque thrusters, and an omnidirectional thruster for minor position/velocity corrections. The environment is 2 dimensional, though if someone knows preexisting work in 3 dimensions I can extrapolate a simpler solution from that. I've looked around the web a bit, and was unable to find anything other than some work on steering behaviors, which always assume point particles and thus do not factor torque into the equations.


I've worked for a couple of days on this problem using standard Newtonian equations, (p(t) = at^2/2 + vt + p(0)), but the entire problem is polluted by the torque calculations, such that torque is applied to turn towards and then slow down and stop at an angle that changes based on the objects velocity and the time it would take to turn to that angle. :-S 9 pages of scribbling and several frustrated nights later, a friend Reccomend I ask here.


A generic solution (which incorporates initial velocity and angle) with a high degree of accuracy would be preferred, though in the end if I have to just "fake it" to look nice (space based RTS), that would be fine too.



Navigation solution for frictionless vehicles.
The optimal solution is given by the proportional navigation guidance law, see for example the Wikipedia page: [on proportional navigation](http://en.wikipedia.org/wiki/Proportional_navigation). This solution applies for the general case where the target point B is moving. In our case it is fixed which doesn't change the character of the solution. The principle of the solution is as follows: If the velocity vector happens to be in the direction of the line of sight that is in the direction connecting the current position with the point B, then the vehicle should proceed in the same direction, otherwise, the line of sight has a rate of rotation which can be computed using using the instantaneous geometry; in this case the navigation law requires accelarting the vehicle perpendicular to the line of sight by a force proportional to rate of rotation of the line of sight. This solution leads to a minimum time to reach the point B.


