I have two points and corresponding tangents in the plane. I'm looking for the shortest path that connects the points and is aligned with the tangents at these points. At any point on the path, the absolute curvature needs to be capped by a given maximum absolute curvature. Furthermore, for any two points on the path, the difference of the curvatures is at most the arc length between the points multiplied by a given factor. Thus, the optimal solution would consist of a spline containing clothoids (Euler-spiral segments), circle segments, and/or line segments.
The path generation needs to happen in real-time, so I'm looking for a quick and dirty solution, knowing that computing anything close to the optimal solution would probably take too much time. I did a quick search in literature on robot path generation and the underlying math (clothoid spline fitting, calculus of variations), but haven't found any papers touching this specific problem. I may have been looking in the wrong places, so I'm asking here if someone could point me to some good sources on the subject. Any help is greatly appreciated.
Edit: I'm guessing that a solution could work as follows: Start with a path laid out by the chord and subdivide it into a number of segments. Of course, this path violates the tangent constraints, or rather, the curvature constraints at the end points. Now increase the arc length by increasing the length of each segment by some uniform value. Use the excess length to reduce the curvature error. For this, you could use a steepest descent approach. Repeat this step gradually increasing arc length until the curvature constraints are met. Ideally you would want to combine the two nested loops into a single loop. Depending on the growth factor and the number of segments, you should be able to get arbitrary close to a (locally) optimal solution. If someone here knows whether this or a similar approach has been applied to solving shortest path problems, I'd be happy to know about it.