Probabilistic Robotics: Motion and Sensing

1. Probabilistic Robotics: Motion and Sensing Sebastian Thrun & Alex Teichman Stanford Artificial Intelligence Lab Slide credits: Wolfram Burgard, Dieter Fox, Cyrill Stachniss, Giorgio Grisetti, Maren Bennewitz, Christian Plagemann, Dirk Haehnel, Mike Montemerlo, Nick Roy, Kai Arras, Patrick Pfaff and others

2. Bayes Filters

3. Bayes Filters Sensing Action

4. Probabilistic Motion Models

5. Bayes Filters Action: p(x’ | u, x)

6. Robot Motion Robot motion is inherently uncertain. How can we model this uncertainty?

7. 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.

8. Locomotion of Wheeled Robots Locomotion (Oxford Dict.): Power of motion from place to place • Differential drive (AmigoBot, Pioneer 2-DX) • Car drive (Ackerman steering) • Synchronous drive (B21) • Mecanum wheels, XR4000

9. ICC Instantaneous Center of Curvature For rolling motion to occur, each wheel has to move along its y-axis

10. v l v r Differential Drive y w ICC q R x (x,y) l/2

11. Differential Drive: Forward Kinematics ICC R P(t+dt) P(t)

12. Differential Drive: Forward Kinematics ICC R P(t+dt) P(t)

13. Mecanum Wheels

14. Example

15. ICC XR4000 Drive y vi(t) • q x wi(t)

16. XR4000 [courtesy by Oliver Brock & Oussama Khatib]

17. Tracked Vehicle: Urban Robot

18. Tracked Vehicle: OmniTread [courtesy by Johann Borenstein]

19. Odometry

20. different wheeldiameters ideal case carpet bump Reasons for Motion Errors and many more …

21. 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,).

22. 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 wheel encoders. • Velocity-based models have to be applied when no wheel encoders are given. • They calculate the new pose based on the velocities and the time elapsed.

23. 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/

24. 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.

25. Odometry Model Robot moves from to . Odometry information .

26. The atan2 Function Extends the inverse tangent and correctly copes with the signs of x and y.

27. Noise Model for Odometry • The measured motion is given by the true motion corrupted with noise.

28. Typical Distributions for Probabilistic Motion Models Normal distribution Triangular distribution

29. Calculating the Probability (zero-centered) • For a normal distribution • For a triangular distribution • Algorithm prob_normal_distribution(a,b): • return • Algorithm prob_triangular_distribution(a,b): • return

30. values of interest (x,x’) odometry values (u) Calculating the Posterior Given x, x’, and u • Algorithm motion_model_odometry(x,x’,u) • return p1 · p2 · p3

31. Examples

32. Application • Repeated application of the sensor model for short movements. • Typical banana-shaped distributions obtained for 2d-projection of 3d posterior. p(x|u,x’) x’ x’ u u

33. Sample-based Density Representation

34. Sample-based Density Representation

35. How to Sample from Normal or Triangular Distributions? • Sampling from a normal distribution • Sampling from a triangular distribution • Algorithm sample_normal_distribution(b): • return • Algorithm sample_triangular_distribution(b): • return

36. Normally Distributed Samples 106 samples

37. 103 samples 104 samples 106 samples 105 samples For Triangular Distribution

38. sample_normal_distribution Sample Odometry Motion Model • Algorithm sample_motion_model(u, x): • Return

39. Start Sampling from Our Motion Model

40. Examples

41.  Map-Consistent Motion Model Approximation:

42. Summary • We discussed motion models for odometry-based motion (see book for velocity-based systems) • We discussed ways to calculate the posterior probability p(x| x’, u). • We also described how to sample from p(x| x’, u). • Typically the calculations are done in fixed time intervals t. • In practice, the parameters of the models have to be learned. • We also discussed an extended motion model that takes the map into account.

43. Probabilistic Sensor Models

44. Bayes Filters Sensing: p(z | x)

45. Sensors for Mobile Robots • Contact sensors:Bumpers, Whiskers • Internal sensors • Accelerometers (spring-mounted masses) • Gyroscopes (spinning mass, laser light) • Compasses, inclinometers (earth magnetic field, gravity) • Proximity sensors • Sonar (time of flight) • Radar (phase and frequency) • Laser range-finders (triangulation, tof, phase) • Infrared (intensity) • Stereo • Visual sensors:Cameras, Stereo • Satellite-based sensors: GPS

46. Polaroyd 6500 Ultrasound Sensors • Emit an ultrasound signal • Wait until they receive the echo • Time of flight sensor

47. Time of Flight sensors v: speed of the signal t: time elapsed between broadcast of signal and reception of the echo. emitter object

48. Properties of Ultrasounds • Signal profile [Polaroid]

49. Sources of Error • Opening angle • Crosstalk • Specular reflection

50. Typical Ultrasound Scan