# How many landmark measurements to localize a planar robot?

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?

• is this a school assignment? – jsotola Feb 2 '20 at 19:52
• try using a pencil and paper to solve the question ... you can also use coffee cups and salt&pepper shakers on a kitchen table – jsotola Feb 2 '20 at 19:55

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.
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.
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.