How many landmark measurements are required to localize a robot (i.e. find its x,y position and orientation) moving in a 2D world using 1. only range measurements 2. range plus bearing measurements?

  • $\begingroup$ is this a school assignment? $\endgroup$ – jsotola Feb 2 at 19:52
  • $\begingroup$ try using a pencil and paper to solve the question ... you can also use coffee cups and salt&pepper shakers on a kitchen table $\endgroup$ – jsotola Feb 2 at 19:55

1) Only range measurements

Minimal landmarks is 3 for position $[x,y]$. This one is pretty easy to visualize. Just draw the distance from the landmarks as a circle. The intersection of the 3 circles is your position. You can see the problem with 2 landmarks below as it has 2 valid solutions.

Localization with 2 landmarks

Orientation($\theta$) is impossible with just landmarks. You must have odometry information. So you must be able to calculate $p_{R_1,R_2}$ and $\theta_{R_1,R_2}$ which is the movement from robot pose $R_1$ to $R_2$ as estimated from your odometry.

2) Range + Bearing

Minimum is 2 points. With 1 landmark you only have two knowns ($r$,$\phi$) and 3 unknowns ($x,y,\theta$). Geometrically you can again imagine the distance circle. Now set the robot heading always tangent to the circle. By moving the robot around this circle you change the state $\theta$ , but the measurement $\phi$ stays the same.

With odometry information you can get away with 1 landmark.

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