1 Integrating multiple sensor readings

A robot is moving in an environment where there are two doors. One door is blue, the other is red. The robot is equipped with a sensor that can return R, B, or N to indicate, respectively, that the robot is facing the red door, the blue door, or no door. The sensor model is given below, and let Z_t be the sensor reading returned at time t (so, Z_t {R,B,N}). The state of the robot is X {X_B,X_R,X_N}. X = X_B means the robot is facing the blue door, X = X_R means the robot is facing the red door, and X = X_N means the robot is not facing any door. The robot queries the sensor three times and no motion happens between the readings. Assume the prior is

Pr[.

1. If the sensor returns (in sequence) R,R,B what is the posterior after the three sensor readings have been integrated?

Table 1: Sensor model. Values in the table give the conditional probabilities for the sensor readings. For example Pr[Z = R|X = X_R] = 0.8, Pr[Z = N|X = X_B] = 0.2, and so on.

2 Unidimensional Kalman Filter

Consider a scenario similar to example 6.8 in the lecture notes with a robot moving along a rail with the following transition and sensor models:

xt = xt1 + 2ut

1

z_t = 2x_t

Assume x₀ N(0,1), R N(0,1) and Q N(0,0.2). Let u_t = 2 and z_t = 5. Compute one full iteration of the Kalman Filter, i.e., prediction and update, and draw the diagram as in Figure 6.12 in the lecture notes.

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