Hi All,
Please see the image linked above, because it will make it much clearer what I'm talking about.
I have a non-holonomic robot (i.e. it cannot turn on the spot - e.g. a car), and I need to plan a path from it's current position $(x_r, y_r)$ to a goal position $(x_g, y_g)$. I am assuming that the robot always turns at it's maximum turning angle, creating a turning circle with a known radius $r_t$ (sorry I forgot to label that on the diagram). The heading of the robot, $\theta$, is also known (defined from the x-axis).
The two variables that I need equations for are the turning angle, $\alpha$, and the length of the straight path segment, $d$.
I've tried solving the problem myself, but I keep getting very complex simultaneous equations that I cannot solve. I feel that there is probably a simpler and more intuitive way of solving it that I am not seeing.
Also, please note that this problem cannot be solved using the same method as a Dubins path, as here the goal heading is not specified.
Let me know if you have any ideas.
Thanks, Greg
$d$ can be found using the right angled triangle between the centre of the turning circle, the goal position, and the point on the turning circle where the robot starts to go straight (as the other two lengths of this triangle are known). The internal angles of the same triangle can then be used to find $\alpha$. Credit goes to my mate Tom.