Consider a robot navigating through a grid world as the one depicted in the following figure,...

   

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Consider a robot navigating through a grid world as the one depicted in the following figure, The experiment starts at time t=0 with the robot placed on the blue tile, and the goal is to reach the exit represented as a green tile as fast as possible. As the robot is exploring to learn the environment, it moves according to the following prob- abilistic navigation policy P(W)=0, P(S) = 0.2, P(E)=0.5, P(N) = 0.3. (1) That is, starting from the blue tile at time t= 0 the robot moves in one of four possible directions, west, south, east, or north, with the probabilities given in (1). Then the robot advances in that direction and arrives at the contiguous tile at time t = 1. The navigation process continues recursively at time t starting from this new tile. The robot selects a new direction with the same probabilities and moves again. The directions in subsequent steps are chosen independently. If at some point the selected movement directs the robot to the surrounding wall, represented in gray in the figure, then the robot crashes and continues from the previous tile at time t+1. For instance, if the robot starts from the blue tile at t = 0 and the first move is south, then the robot will remain on the blue tile at time t = 1. The experiment ends if the robot crashes twice, or in the event that it reaches the green exit tile, which is represented by the set G. (a) If C₁ denotes the event of crashing to the wall on the first move, and C₂ of crashing on the second move. Compute the probabilities P(C₁), P(C₂) and P(C, UC₂). Are the events C₁ and C, independent? Explain intuitively and formally. (b) Compute the probability of reaching the exit depicted in green, in T = 4 moves following the path shown in the figure and without crashing. (c) Compute the probability P((T=4)nG) of reaching the exit depicted in green in T = 4 moves starting from the blue tile and not crashing. Hint: List all possible paths to reach the green tile in T = 4 moves. (d) Compute the probability P((T=5)nG) of reaching the exit depicted in green in T = 5 moves starting from the blue tile and crashing once. (e) Compute the probability of reaching the exit P(G) and the probability of crashing twice P(GC). (f) Compute the probability P(T= n|G) for n = 5, and n = 6. (g) Compute and sketch pric(n) and Fra(n), those are the probability mass function and the cumulative distribution of T conditioned to G. (h) Compute the expected value E[TG]. (i) Compute P(G) and E[TG] again if the navigation policy is now P(W) = 0, P(S) = 0.2. P(E)=0.3, P(N) = 0.5. (2) Determine which is the best navigation policy in terms of the probability of reaching the goal and reaching it faster. Justify your answer. Consider a robot navigating through a grid world as the one depicted in the following figure, The experiment starts at time t=0 with the robot placed on the blue tile, and the goal is to reach the exit represented as a green tile as fast as possible. As the robot is exploring to learn the environment, it moves according to the following prob- abilistic navigation policy P(W)=0, P(S) = 0.2, P(E)=0.5, P(N) = 0.3. (1) That is, starting from the blue tile at time t= 0 the robot moves in one of four possible directions, west, south, east, or north, with the probabilities given in (1). Then the robot advances in that direction and arrives at the contiguous tile at time t = 1. The navigation process continues recursively at time t starting from this new tile. The robot selects a new direction with the same probabilities and moves again. The directions in subsequent steps are chosen independently. If at some point the selected movement directs the robot to the surrounding wall, represented in gray in the figure, then the robot crashes and continues from the previous tile at time t+1. For instance, if the robot starts from the blue tile at t = 0 and the first move is south, then the robot will remain on the blue tile at time t = 1. The experiment ends if the robot crashes twice, or in the event that it reaches the green exit tile, which is represented by the set G. (a) If C₁ denotes the event of crashing to the wall on the first move, and C₂ of crashing on the second move. Compute the probabilities P(C₁), P(C₂) and P(C, UC₂). Are the events C₁ and C, independent? Explain intuitively and formally. (b) Compute the probability of reaching the exit depicted in green, in T = 4 moves following the path shown in the figure and without crashing. (c) Compute the probability P((T=4)nG) of reaching the exit depicted in green in T = 4 moves starting from the blue tile and not crashing. Hint: List all possible paths to reach the green tile in T = 4 moves. (d) Compute the probability P((T=5)nG) of reaching the exit depicted in green in T = 5 moves starting from the blue tile and crashing once. (e) Compute the probability of reaching the exit P(G) and the probability of crashing twice P(GC). (f) Compute the probability P(T= n|G) for n = 5, and n = 6. (g) Compute and sketch pric(n) and Fra(n), those are the probability mass function and the cumulative distribution of T conditioned to G. (h) Compute the expected value E[TG]. (i) Compute P(G) and E[TG] again if the navigation policy is now P(W) = 0, P(S) = 0.2. P(E)=0.3, P(N) = 0.5. (2) Determine which is the best navigation policy in terms of the probability of reaching the goal and reaching it faster. Justify your answer.

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The images youve provided describe a problem involving a robot navigating through a grid world The robot moves according to a certain policy defined b... View the full answer