Please solve all the parts.
1. (7 points) Apply one iteration of the discrete Bayes filter algorithm for updating the belief of the robot about its state. Assume that the environment is two-dimensional and tessellated into 16 equally spaced cells as shown in the figure below. The robot's initial belief bel(x0) is represented in the figure below. Here, x0 represents the initial state of the robot.
The robot can move in any direction by one cell with equal probability. What will be the predicted robot belief distribution bel(c) after this movement?
Let us now assume that the robot uses its onboard range finder and measures the distance from the origin (left bottom corner). Assume that the measurement model p(zi) is given as follows: The distance of the robot from the origin can be 3 units in the x-direction and 1 to 4 units in the y-direction with equal probability. Find the belief distribution bel( after the measurement update step.
2.5 points: Consider the discrete-Bayes Linear filter algorithm for belief prediction. Let x denote the state, zt denote the measurement vector. Let pk,t denote the actual belief at t for the k-th cell and let Pk,t denote the predicted belief at t for the k-th cell. Assume that p(x|ut,x) denotes the state transition probability from state x to x. Show that Pk,t = E[p(xt|ut,xi) * pi,t-1].
