\fAt each time step I, the robot am either move from one node to another by traveling over a graph edge, or choose to remain in the node it is located at. However, paths connecting nodes are in bad condition, so the motion control commands result into random outcomes. Suppose there are two control commands that the robot can implement, "move" (I1; = l}, and 9'do not moveII (ll, = U) at each time step L The transition probabilities of the robot Lu'uler the "move" command are the following: (i) When the robot is initially at A: p(x;+1= All}; = 1' x; = A) = 0.25, p{x+1= BIu, = 1, x; = A) = 0.?5. (ii) When the robot is initially at B: p(xt+l = Ail-'1. = 11 x3 = B) = 025: P(xs+l = Bill: = 1, I: = B) = 0-25: p(x;+l = CI\"; = 1, so = B) = (1.5. (iii) When the robot is initially at C: 9(xa+1= Ci\": =1, 3': = C) = 0-5: P(xs+l = Dill: =1, 3': = C) = \"-5' (iv) When the robot isat D, p(x,+1 = B'u; = 11 x; = D) = l. The robot may choose to remain in the section it is by choosing a \"do not move\" command. In that case, the transitions are naturally p(x'+1 = "I\": = 0, x1 = n) = II for an. E {A1 3' C, D}. Answer the following: (i) (4 points) Derive the state transition probability matrices that describe the motion of the robot for u; = 1 and u; = 0. (ii) (4 points) If the initial beliefon the location of the robot is given by: hel(x0 = A) = 0.5, hel(xn = C) = 0.5 What is the belief on the location of the robot after the sequence ofcommands has been implemented: Ill = 1, [12 = , andlla = 1? (iii) (2 points) If the robot starts with the initial beliefhoxn = A) = 1, and implements I1' = (l for all times, what is the predicted beliefat all future times t