1. Part 3 of 3: Beliefs in Probabilistic Robotics

2. References and Sources of Figures Part 1: Stuart Russell and Peter Norvig, Artificial Intelligence, 2 nd ed., Prentice Hall, Chapter 13 Part 2: Stuart Russell and Peter Norvig, Artificial Intelligence, 2 nd ed., Prentice Hall, Chapter 14 Part 3: Sebastian Thrun, Wolfram Burgard, and Dieter Fox, Probabilistic Robotics, Chapter 2

3. Revisit the Mobile Robot Example

4. Scenario a mobile robot uses its camera to detect the state of the door (open or closed) camera is noisy: –if the door is in fact open: the probability of detecting it open is 0.6 –if the door is in fact closed: the probability of detecting it closed is 0.8

5. Scenario the robot can use its manipulator to push open the door if the door is in fact closed: the probability of robot opening it is 0.8

6. Scenario At time t 0, the probability of the door being open is 0.5 Suppose at t 1 the robot takes no control action but it senses an open door, what is the probability of the door is open?

7. Scenario Using Bayes Filter, we will see that: –at time t 1 the probability of the door is open is: 0.75 after taking a measurement –at time t 2 the probability of the door is open is ～ 0.984 after the robot pushes open the door and takes another measurement

8. Belief Distribution probability distribution over the state x t at time t, conditioned on all past measurements z 1:t and all past controls u 1:t

9. Belief Distribution probability distribution over the state x t at time t, conditioned on all past measurements z 1:t-1 (i.e. before incorporating z 1:t ) and all past controls u 1:t

10. often referred to as prediction in the context of probabilistic filtering because it predicts the state ( x ) at time t based on the previous state posterior, before incorporating the measurement ( z ) at time t

11. Algorithm of Bayes Filter Bayes_filter( bel(x t-1 ), u t, z t ): for all x t do endfor return bel(x t ) Predict x after exerting u: Update belief of x after making a measurement:

13. Example: A Mobile Robot Estimating the State of a Door At t 0 :

14. Noisy Sensors

15. Uncertainty from Manipulator

27. As you see, the degree of belief changes (is updated) over time as actuations are performed and measurements are made.

28. Summary Reviewed of probability theory, Bayes' rule, product rule How random variables and their causal relations are represented in DAGs— Bayesian Networks (BN) Dynamic Bayesian Networks (DBN): Adding the temporal aspect to Bayesian Networks How DBN can be used to characterize the evolution of states ( x t ), controls ( u t ), and measurements ( z t ) in robotics Through DBN, discussed two overarching steps in localization filters: predict and update beliefs Demonstrated how these two steps work in the algorithm of Bayes filter Explained where the Bayes' rule is used in the Bayer filter