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 Zt be the sensor reading returned at time t (so, Zt ∈ {R,B,N}). The state of the robot is X ∈ {XB,XR,XN}. X = XB means the robot is facing the blue door, X = XR means the robot is facing the red door, and X = XN 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?
X = XR
X = XB
X = XN
Z = R
0.8
0.2
0.2
Z = B
0.05
0.6
0.1
Z = N
0.15
0.2
0.7
Table 1: Sensor model. Values in the table give the conditional probabilities for the sensor readings. For example Pr[Z = R|X = XR] = 0.8, Pr[Z = N|X = XB] = 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 = xt−1 + 2ut
1
zt = 2xt
Assume x0 ∼ N(0,1), R ∼ N(0,1) and Q ∼ N(0,0.2). Let ut = 2 and zt = 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.
