Direct Localization

uses the distances to the goals and the angle between the goal vectors to determine the robot's position and orientation. First, the vectors $\vec{a}$ and $\vec{b}$ from the perception origin to the goals have to be mapped to local world coordinates. If $\vec{a_w}$ and $\vec{b_w}$ denote these mapped vectors, we define $\vec{c_w}:=\vec{a_w}-\vec{b_w}$. Now $\delta'$ is the angle between $\vec{c_w}$ and $\vec{b_w}$. Hence, the position $p$ of the robot lies on a line at an angle of $\delta'$ to the line connecting the goals. Knowing the distance to one goal determines $p$. The orientation is also known, because the angle $\alpha$ at which the other goal appears in the image in respect to the robot's viewing direction is preserved by the optical mapping.