1. 3/11/2016CS225B Kurt Konolige Probabilistic Models of Sensing and Movement Move to probability models of sensing and movement Project 2 is about complex behavior using sensing Sensor interpretation is difficult – simple interpretation in this section Artifacts [goal-directed motion] and reactive behaviors Lectures Probabilistic sensor models Probabilistic representation of uncertain movement Particle filter implementation Project PF for motion model Markov localization with PF Stretch – feature-based localization Slides thanks to Steffen Gutmann

2. 3/11/2016CS225B Kurt Konolige Robot Motion Robot motion is inherently uncertain. How can we model this uncertainty?

3. 3/11/2016CS225B Kurt Konolige Dynamic Bayesian Network for Controls, States, and Sensations

4. 3/11/2016CS225B Kurt Konolige Probabilistic Motion Models To implement the Bayes Filter, we need the transition model p(x | x’, u). The term p(x | x’, u) specifies a posterior probability, that action u carries the robot from x’ to x. In this section we will specify, how p(x | x’, u) can be modeled based on the motion equations. We concentrate on wheel-based robots; for legged ones, similar equations hold.

5. 3/11/2016CS225B Kurt Konolige Coordinate Systems In general the configuration of a robot can be described by six parameters. Three-dimensional Cartesian coordinates plus three Euler angles pitch, roll, and tilt. Throughout this section, we consider robots operating on a planar surface. The state space of such systems is three- dimensional (x,y,).

6. 3/11/2016CS225B Kurt Konolige Typical Motion Models In practice, one often finds two types of motion models: Odometry-based Velocity-based (dead reckoning) Odometry-based models are used when systems are equipped with encoders that can measure the actual path traveled. Velocity-based models have to be applied when no encoders are given. They calculate the new pose based on the velocities and the time elapsed.

7. 3/11/2016CS225B Kurt Konolige Example Wheel Encoders These modules require +5V and GND to power them, and provide a 0 to 5V output. They provide +5V output when they "see" white, and a 0V output when they "see" black. These disks are manufactured out of high quality laminated color plastic to offer a very crisp black to white transition. This enables a wheel encoder sensor to easily see the transitions. Source: http://www.active-robots.com/

8. 3/11/2016CS225B Kurt Konolige Dead Reckoning Derived from “ deduced reckoning. ” Mathematical procedure for determining the present location of a vehicle. Achieved by calculating the current pose of the vehicle based on its velocities and the time elapsed, over small time intervals

9. 3/11/2016CS225B Kurt Konolige Reasons for Motion Errors bump ideal case different wheel diameters carpet and many more …

10. 3/11/2016CS225B Kurt Konolige Odometry Model Robot moves from to. Odometry information.

11. 3/11/2016CS225B Kurt Konolige The atan2 Function Extends the inverse tangent and correctly copes with the signs of x and y.

12. 3/11/2016CS225B Kurt Konolige Noise Model for Odometry The measured motion is given by the true motion corrupted with noise.

13. 3/11/2016CS225B Kurt Konolige Variances and Deviations For independent errors, variances add. If errors are specified using std, the length over which the error occurs must be given: 6 cm in 1 m => 36 cm 2 in 1 m 3 deg in 360 deg => 9 deg 2 in 360 deg Consider to specify a variance s2s2 s2s2 2s 2

14. 3/11/2016CS225B Kurt Konolige Typical Distributions for Probabilistic Motion Models Normal distributionTriangular distribution

15. 3/11/2016CS225B Kurt Konolige Calculating the Posterior given x, x ’, and u 1.Algorithm motion_model_odometry(x,x’,u) 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.return p 1 · p 2 · p 3 odometry values (u) values of interest (x,x’)

16. 3/11/2016CS225B Kurt Konolige Application Typical banana-shaped distributions obtained for 2d-projection of 3d posterior. x’ u p(x|u,x’) u x’