CSE 180 – Introduction to Robotics Homework N. 4 solved

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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[X = XN ] = Pr[X = XR] = Pr[X = XB] = 1
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
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.