1. Computer Vision Group Prof. Daniel Cremers Autonomous Navigation for Flying Robots Lecture 5.3: Reasoning with Bayes Law Jürgen Sturm Technische Universität München

2. The State Estimation Problem We want to estimate the world state from 1.Sensor measurements and 2.Controls (or odometry readings) We need to model the relationship between these random variables, i.e., Jürgen Sturm Autonomous Navigation for Flying Robots 2

3. Causal vs. Diagnostic Reasoning  is diagnostic  is causal  Diagnosic reasoning is typically what we need  Often causal knowledge is easier to obtain  Bayes rule allows us to use causal knowledge in diagnostic reasoning Jürgen Sturm Autonomous Navigation for Flying Robots 3

4. Bayes Rule  Definition of conditional probability  Bayes rule Jürgen Sturm Autonomous Navigation for Flying Robots 4 observation likelihoodprior on world state prior on sensor observations

5. Normalization  Direct computation of can be difficult  Idea: Compute improper distribution, normalize afterwards  Step 1:  Step 2:  Step 3: Jürgen Sturm Autonomous Navigation for Flying Robots 5

6. Background Knowledge  Same derivation also works in the presence of background knowledge  Bayes rule with background knowledge Jürgen Sturm Autonomous Navigation for Flying Robots 6

7. Example: Sensor Measurement  Quadrotor seeks the landing zone  Landing zone is marked with many bright lamps  Quadrotor has a brightness sensor Jürgen Sturm Autonomous Navigation for Flying Robots 7

8. Example: Sensor Measurement  Binary sensor  Binary world state  Sensor model  Prior on world state  Assume: Robot observes light, i.e.,  What is the probability that the robot is above the landing zone? Jürgen Sturm Autonomous Navigation for Flying Robots 8

9. Example: Sensor Measurement  Sensor model  Prior on world state  Probability after observation (using Bayes) Jürgen Sturm Autonomous Navigation for Flying Robots 9

10. Combining Evidence  Suppose our robot obtains another observation (either from the same or a different sensor)  How can we integrate this new information?  More generally, how can we estimate ? Jürgen Sturm Autonomous Navigation for Flying Robots 10

11. Combining Evidence  Suppose our robot obtains another observation (either from the same or a different sensor)  How can we integrate this new information?  More generally, how can we estimate ?  Bayes formula (with background knowledge) gives us Jürgen Sturm Autonomous Navigation for Flying Robots 11

12. Recursive Bayesian Updates Markov Assumption: is independent of if we know Jürgen Sturm Autonomous Navigation for Flying Robots 12

13. Example: Sensor Measurement  Quadrotor seeks the landing zone  Landing zone is marked with many bright lamps and a visual marker Jürgen Sturm Autonomous Navigation for Flying Robots 13

14. Example: Second Measurement  Sensor model  Previous estimate  Assume robot does not observe marker  What is the probability of being home? Jürgen Sturm Autonomous Navigation for Flying Robots 14 The second observation lowers the probability that the robot is above the landing zone!

15. Actions/Controls (Motions)  World state changes over time because of  actions carried out by the robot…  actions carried out by other agents…  or just time passing by… …change the world  How can we incorporate actions? Jürgen Sturm Autonomous Navigation for Flying Robots 15

16. Typical Actions  Quadrotor accelerates by changing the speed of its motors  Position also changes when quadrotor does “nothing” (and drifts..)  Actions are never carried out with absolute certainty  In contrast to measurements, actions generally increase the uncertainty of the state estimate Jürgen Sturm Autonomous Navigation for Flying Robots 16

17. Action Models  To incorporate the outcome of an action into the current state estimate (“belief”), we use the conditional pdf  This term specifies the probability that executing the action u in state x will lead to state x’ Jürgen Sturm Autonomous Navigation for Flying Robots 17

18. Example: Take-Off  Action:  World state: Jürgen Sturm Autonomous Navigation for Flying Robots 18 ground air 0.1 0.99 0.01 0.9

19. Integrating the Outcome of Actions  Discrete case  Continuous case Jürgen Sturm Autonomous Navigation for Flying Robots 19

20. Example: Take-Off  Prior belief on robot state: (robot is located on the ground)  Robot executes “take-off” action  What is the robot’s belief after one time step? Jürgen Sturm Autonomous Navigation for Flying Robots 20

21. Lessons Learned  Bayes rule  Data fusion of sensor observations  Data fusion of actions/motion commands  Next: Bayes filter Jürgen Sturm Autonomous Navigation for Flying Robots 21