CSE { Introduction to Robotics Homework N. 4 Solution

Description

• 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 2 fR; B; Ng). The state of the robot is X 2 fX_B; X_R; X_N g. 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[X = X_N ] = Pr[X = X_R] = Pr[X = X_B] = ¹₃ .

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 = RjX = X_R] = 0:8, Pr[Z = NjX = X_B] = 0:2, and so on.

• 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:

x_t = x_{t 1} + 2u_t

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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