Probabilistic Robotics Motion and Sensing Sebastian Thrun Alex

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 4 -

Bayes Filters 4 -2

Bayes Filters Sensing Action 4 -3

Probabilistic Motion Models 4 -4

Bayes Filters Action: p(x’ | u, x) 4 -5

Robot Motion Robot motion is inherently uncertain. How can we model this uncertainty? 4 -6

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. 4 -7

Locomotion of Wheeled Robots Locomotion (Oxford Dict. ): Power of motion from place to place • Differential drive (Amigo. Bot, Pioneer 2 -DX) • Car drive (Ackerman steering) • Synchronous drive (B 21) • Mecanum wheels, XR 4000 4 -8

Instantaneous Center of Curvature ICC For rolling motion to occur, each wheel has to move along its y-axis 4 -9

Differential Drive ICC y w vl R x (x, y) vr l/2 4 -10

Differential Drive: Forward Kinematics ICC R P(t+dt) P(t) 4 -11

Differential Drive: Forward Kinematics ICC R P(t+dt) P(t) 4 -12

Mecanum Wheels 4 -13

Example 4 -14

XR 4000 Drive y vi(t) wi(t) x ICC 4 -15

XR 4000 [courtesy by Oliver Brock & Oussama Khatib] 4 -16

Tracked Vehicle: Urban Robot 4 -17

Tracked Vehicle: Omni. Tread [courtesy by Johann Borenstein] 4 -18

Odometry 4 -19

Reasons for Motion Errors ideal case bump different wheel diameters carpet and many more … 4 -20

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 threedimensional (x, y, ). 4 -21

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. 4 -22

Example Wheel Encoders These modules require +5 V and GND to power them, and provide a 0 to 5 V output. They provide +5 V output when they "see" white, and a 0 V 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/ 4 -23

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. 4 -24

Odometry Model Robot moves from Odometry information to . .

The atan 2 Function Extends the inverse tangent and correctly copes with the signs of x and y. 4 -26

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

Typical Distributions for Probabilistic Motion Models Normal distribution Triangular distribution

Calculating the Probability (zero-centered) • For a normal distribution 1. Algorithm prob_normal_distribution(a, b): 2. return • For a triangular distribution 1. Algorithm prob_triangular_distribution(a, b): 2. return 4 -29

Calculating the Posterior Given x, x’, and u 1. Algorithm motion_model_odometry(x, x’, u) 2. odometry values (u) 3. 4. 5. 6. values of interest (x, x’) 7. 8. 9. 10. 11. return p 1 · p 2 · p 3 4 -30

Examples 4 -31

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

Sample-based Density Representation

Sample-based Density Representation 4 -34

How to Sample from Normal or Triangular Distributions? • Sampling from a normal distribution 1. Algorithm sample_normal_distribution(b): 2. return • Sampling from a triangular distribution 1. Algorithm sample_triangular_distribution(b): 2. return 4 -35

Normally Distributed Samples 106 samples 4 -36

For Triangular Distribution 103 samples 104 samples 105 samples 106 samples 4 -37

Sample Odometry Motion Model 1. Algorithm sample_motion_model(u, x): 1. 2. 3. 4. 5. sample_normal_distribution 6. 7. Return

Sampling from Our Motion Model Start

Examples

Map-Consistent Motion Model Approximation:

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. 4 -42

Probabilistic Sensor Models 4 -43

Bayes Filters Sensing: p(z | x) 4 -44

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 4 -45

Ultrasound Sensors • Emit an ultrasound signal • Wait until they receive the echo • Time of flight sensor Polaroyd 6500 4 -46

Time of Flight sensors emitter object v: speed of the signal t: time elapsed between broadcast of signal and reception of the echo. 4 -47

Properties of Ultrasounds • Signal profile [Polaroid] 4 -48

Sources of Error • Opening angle • Crosstalk • Specular reflection 4 -49

Typical Ultrasound Scan 4 -50

Laser Range Scanner 4 -51

Properties • High precision • Wide field of view • Approved security for collision detection 4 -52

Robots Equipped with Laser Scanners Zora: Groundhog: Herbert: 4 -53

Typical Scans 4 -54

Proximity Sensors • The central task is to determine P(z|x), i. e. , the • • probability of a measurement z given that the robot is at position x. Question: Where do the probabilities come from? Approach: Let’s try to explain a measurement. 4 -55

Beam-based Sensor Model • Scan z consists of K measurements. • Individual measurements are independent given the robot position. 4 -56

Beam-based Sensor Model 4 -57

Typical Measurement Errors of an Range Measurements 1. Beams reflected by obstacles 2. Beams reflected by persons / caused by crosstalk 3. Random measurements 4. Maximum range measurements 4 -58

Proximity Measurement • Measurement can be caused by … • • a known obstacle. cross-talk. an unexpected obstacle (people, furniture, …). missing all obstacles (total reflection, glass, …). • Noise is due to uncertainty … • • in measuring distance to known obstacle. in position of known obstacles. in position of additional obstacles. whether obstacle is missed. 4 -59

Beam-based Proximity Model Measurement noise 0 zexp zmax Unexpected obstacles 0 zexp zmax 4 -60

Beam-based Proximity Model Random measurement 0 zexp zmax Max range 0 zexp zmax 4 -61

Resulting Mixture Density How can we determine the model parameters? 4 -62

Raw Sensor Data Measured distances for expected distance of 300 cm. Sonar Laser 4 -63

Approximation • Maximize log likelihood of the data • Search space of n-1 parameters. • • Hill climbing Gradient descent Genetic algorithms … • Deterministically compute the n-th parameter to satisfy normalization constraint. 4 -64

Approximation Results Laser Sonar 300 cm 4 -65

Example z P(z|x, m) 4 -66

Discrete Model of Proximity Sensors • Instead of densities, consider discrete steps along the • sensor beam. Consider dependencies between different cases. Laser sensor Sonar sensor 4 -67

Approximation Results Laser Sonar 4 -68

Influence of Angle to Obstacle 4 -69

Influence of Angle to Obstacle 4 -70

Influence of Angle to Obstacle 4 -71

Influence of Angle to Obstacle 4 -72

Summary Beam-based Model • Assumes independence between beams. • Justification? • Overconfident! • Models physical causes for measurements. • Mixture of densities for these causes. • Assumes independence between causes. Problem? • Implementation • Learn parameters based on real data. • Different models should be learned for different angles at which the sensor beam hits the obstacle. • Determine expected distances by ray-tracing. • Expected distances can be pre-processed. 4 -73

Scan-based Model • Beam-based model is … • not smooth for small obstacles and at edges. • not very efficient. • Idea: Instead of following along the beam, just check the end point. 4 -74

Scan-based Model • Probability is a mixture of … • a Gaussian distribution with mean at distance to closest obstacle, • a uniform distribution for random measurements, and • a small uniform distribution for max range measurements. • Again, independence between different components is assumed. 4 -75

Example Likelihood field Map m P(z|x, m) 4 -76

San Jose Tech Museum Occupancy grid map Likelihood field 4 -77

Scan Matching • Extract likelihood field from scan and use it to match different scan. 4 -78

Scan Matching • Extract likelihood field from first scan and use it to match second scan. ~0. 01 sec 4 -79

Properties of Scan-based Model • Highly efficient, uses 2 D tables only. • Smooth w. r. t. to small changes in robot position. • Allows gradient descent, scan matching. • Ignores physical properties of beams. • Will it work for ultrasound sensors? 4 -80

Landmarks • Active beacons (e. g. , radio, GPS) • Passive (e. g. , visual, retro-reflective) • Standard approach is triangulation • Sensor provides • distance, or • bearing, or • distance and bearing. 4 -81

Distance and Bearing 4 -82

Probabilistic Model 1. Algorithm landmark_detection_model(z, x, m): 2. 3. 4. 5. Return 4 -83

Distributions 4 -84

Distances Only No Uncertainty X’ a P 1 P 2 d 1 x P 1=(0, 0) P 2=(a, 0) 4 -85

Bearings Only No Uncertainty P 3 D 2 P 2 D 1 P 1 D 3 P 2 a z 2 a D 1 z 2 z 1 z 3 b a P 1 z 1 Law of cosine 4 -86

Bearings Only With Uncertainty P 3 P 2 P 1 Most approaches attempt to find estimation mean. 4 -87

Summary of Sensor Models • Explicitly modeling uncertainty in sensing is key to robustness. • In many cases, good models can be found by the following approach: 1. Determine parametric model of noise free measurement. 2. Analyze sources of noise. 3. Add adequate noise to parameters (eventually mix in densities for noise). 4. Learn (and verify) parameters by fitting model to data. 5. Likelihood of measurement is given by “probabilistically comparing” the actual with the expected measurement. • This holds for motion models as well. • It is extremely important to be aware of the underlying assumptions! 4 -88