1. Computer Vision Group Prof. Daniel Cremers Autonomous Navigation for Flying Robots Lecture 6.1: Bayes Filter Jürgen Sturm Technische Universität München

2. Markov Assumption  Observations depend only on current state  Current state depends only on previous state and current action Jürgen Sturm Autonomous Navigation for Flying Robots 2

3. Markov Chain  A Markov chain is a stochastic process where, given the present state, the past and the future states are independent Jürgen Sturm Autonomous Navigation for Flying Robots 3

4. Underlying Assumptions  Static world  Independent noise  Perfect model, no approximation errors Jürgen Sturm Autonomous Navigation for Flying Robots 4

5. Bayes Filter  Given:  Sequence of observations and actions  Sensor model  Action model  Prior probability of the system state  Wanted:  Estimate of the state of the dynamic system  Posterior of the state is also called belief Jürgen Sturm Autonomous Navigation for Flying Robots 5

6. Bayes Filter Algorithm For each time step, do 1.Apply motion model 2.Apply sensor model Jürgen Sturm Autonomous Navigation for Flying Robots 6

7. Notes  Bayes filters also work on continuous state spaces (replace sum by integral)  Bayes filter also works when actions and observations are asynchronous Jürgen Sturm Autonomous Navigation for Flying Robots 7

8. Example: Localization  Discrete state  Belief distribution can be represented as a grid  This is also called a histogram filter Jürgen Sturm Autonomous Navigation for Flying Robots 8 P(x) = 1.0 P(x) = 0.0

9. Example: Localization  Action  Robot can move one cell in each time step  Actions are not perfectly executed Jürgen Sturm Autonomous Navigation for Flying Robots 9

10. Example: Localization  Action  Robot can move one cell in each time step  Actions are not perfectly executed  Example: move east 60% success rate, 10% to stay/move too far/ move one up/move one down Jürgen Sturm Autonomous Navigation for Flying Robots 10

11. Example: Localization  Binary observation  One (special) location has a marker  Marker is sometimes also detected in neighboring cells Jürgen Sturm Autonomous Navigation for Flying Robots 11

12. Example: Localization  Let’s start a simulation run… (shades are hand-drawn, not exact!) Jürgen Sturm Autonomous Navigation for Flying Robots 12

13. Example: Localization  t=0  Prior distribution (initial belief)  Assume we know the initial location (if not, we could initialize with a uniform prior) Jürgen Sturm Autonomous Navigation for Flying Robots 13

14. Example: Localization  t=1, u=east, z=no-marker  Bayes filter step 1: Apply motion model Jürgen Sturm Autonomous Navigation for Flying Robots 14

15. Example: Localization  t=1, u=east, z=no-marker  Bayes filter step 2: Apply observation model Jürgen Sturm Autonomous Navigation for Flying Robots 15

16. Example: Localization  t=2, u=east, z=marker  Bayes filter step 1: Apply motion model Jürgen Sturm Autonomous Navigation for Flying Robots 16

17. Example: Localization  t=2, u=east, z=marker  Bayes filter step 2: Apply observation model  Question: Where is the robot? Jürgen Sturm Autonomous Navigation for Flying Robots 17

18. Lessons Learned  Markov assumption allows efficient recursive Bayesian updates of the belief distribution  Useful tool for estimating the state of a dynamic system  Bayes filter is the basis of many other filters  Grid/histogram filter  Kalman filter  Particle filter  … Jürgen Sturm Autonomous Navigation for Flying Robots 18