Mathematics, Python, Logics, and Coding

this project is regarding the "tracking", "mapping", and "moving logics", and the programming of a robot. (1) the "tracking" means that I need to know the position of the robot while the robot is traveling inside a working area. the working area is a small open area, for example about 10 x 10 sq meter2, without any obstacles, but the shape of the working area may be irregular, which means the shape is not a simple square or circle. However, to start the project, you may assume a simple shape to start with, then move on to more complex irregular shapes. (2) the "mapping" means that the robot knows where is the border of the working area, so the robot won't travel to outside. Robot knows where is the border, not because robot already travel to the border and hit or touch something. Even though the robot is in the moddle of the working, it knows where is the border, and i will explan in the attached file. (3) the "moving logics" means that i want the robot to move around, and the moving route shall be able to cover entire working area without too much overlapping. Ideally i want to have full control of the moving route. To know more about the project, please read attached file. I will define what are knowns, what are unknowns, and what I want. Thanks for your reading.