Assignment III $-$ Simultaneous Localization and Mapping $($SLAM$)$ A $2$D indoor mobile robot mounted with a $2$D Lidar sensor is navigating in a room containing three landmarks with reflectors which are fixed on the walls in unknown places to the robot. The robot attempts to make a straight path between two points A and $B($in blue$)$ and the positions of the landmarks are shown in red as: $B$ A There are encoders that measure the speed for cach wheel. The robot. is placed in an unknown initial pose and starts exploring the room in order to map it$,$ i$.$c$.,$ to learn the position of the landmarks. At the same time as it maps the environment, it also localizes itself relative to the initial position. This is the problem of Simultancous Localization and Mapping $($SLAM$).1.$ Find the dynamic model equations of the evolution model that allows to recursively esti$-$mate the current pose $(xr,Yr,Br)$ of the robot and the positions of the center for each of the three landmarks $($X$,$Y$),($X$,$Y$.).($Xig$,$ Yig$)$ over time. Your inputs are the lincar wheels speeds v and Up$,$ the distance between the wheels is $L,$ and the sampling time between measurements of the sensors is AT$.2.$ The LIDAR sensor is equipped with a low$-$level processing algorithm which allows it to detect the landmarks because they have a high reflective power. Each landmark detection comes with two values: a distance $r$ and a bearing angle a with respect to the orientation of the robot. Assuming that the LIDAR is placed between the center of axis linking the two wheels of the robot, find the observation$/$measurement equations linking the distance rand o for each landmark to the state of the robot, and give the observation model for $r$ and $cr.3.$ Write the stochastic representation of the linearized state and observation models $($i$.$e$.,$ Jacobians$)$ and indicate what the model noises represent. $4.$ Please assume that the pairing of each observation $(r,a)$ and its corresponding land$-$mark association is given by an oracle. How would you initialize the positions and the uncertaintics of the landmarks? a $5.$ We consider now the scenario where the association of the landmarks and the observations are not given. Propose a method $($spatial distance criteria$)$ to associate one measurement $(r,$ a$)$ with one of the three landmarks. Hint: Please remember of the distance metric discussed in the class for considering the data uncertaintics. $6.$ Write a python program $($Jupyter notebook$)$ that simulates the landmarks and robot motion $($following a straight line$).$ You can select three $2D$ positions along a rectangular room of your choice in the simulation. For creating your measurements you can add Gaussian noise in the observations and inputs fr$$m the well encoders. $7.$ Implement an Extended Kalman Filter $($EKF$\text{'})$ for localizing the robot and to estimate the positions of your static landmarks over time. Please show the plots of their $2$D position and uncertainty $($covariance$)$ given by the filter. Please adopt the initialization of the state and covariance following your answer in the item $4).$ Please choose the sampling time and number of iterations $($time steps$)$ accordingly for letting your initial state estimation to converge $8.$ Now make some changes in your simulation and filtering to consider a more challenging scenario. Your sensor measuring the landmarks distance and angle stops working after $5$ measurements $($time steps$).$ Then it stays broken during $100$ time steps and then it starts working again during $100$ time steps. Please show the plots of how the uncertainties of the robot position evolves in this scenario.