A simple two dimensional robot localisation problem using bearing information can be formulated as an optimisation

problem.

Suppose a mobile robot is navigating in an environment and it observes $4$ landmarks. The xy positions of the $4$ landmarks

are known. The robot can only observe the bearings to the $4$ landmarks. The bearing means the direction of the landmark

with respect to the heading of the robot. The problem is to estimate the robot position and orientation using the $4$ bearings

and the landmark positions.

$(1)$ Generate the bearing data using the robot pose and $4$ landmark positions as shown in the following figure.

$(2)$ Derive the model of the localisation problems with $4$ landmarks and solve the problem; apply at least two

different optimisation techniques$/$software and compare the results.

$(3)$ Add some small noises on the bearing data and solve the problem again.

$(4)$ Change the number of landmarks and their positions, formulate and solve another two problems.

$(5)$ Discuss the conditions when the problem can or cannot be solved.

Bearing only localisation problem with four landmarks $(1$ to $4)$

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