1. City College of New York 1 Dr. John (Jizhong) Xiao Department of Electrical Engineering City College of New York jxiao@ccny.cuny.edu Course Summary Introduction to ROBOTICS

2. City College of New York 2 Mobile Robot

3. City College of New York 3 Mobot System Overview Abstraction level Motor Modeling : what voltage should I set now ? Control (PID) : what voltage should I set over time ? Kinematics : if I move this motor somehow, what happens in other coordinate systems ? Motion Planning : Given a known world and a cooperative mechanism, how do I get there from here ? Bug Algorithms : Given an unknowable world but a known goal and local sensing, how can I get there from here? Mapping : Given sensors, how do I create a useful map? low-level high-level Localization : Given sensors and a map, where am I ? Vision : If my sensors are eyes, what do I do?

4. City College of New York 4 Mobile Robot Locomotion Swedish Wheel Locomotion: the process of causing a robot to move  Tricycle  Synchronous Drive  Omni-directional  Differential Drive R  Ackerman Steering

5. City College of New York 5 Differential Drive Nonholonomic Constraint Property: At each time instant, the left and right wheels must follow a trajectory that moves around the ICC at the same angular rate , i.e., Kinematic equation

6. City College of New York 6 Differential Drive Basic Motion Control Straight motion R = Infinity V R = V L Rotational motion R = 0 V R = -V L R : Radius of rotation

7. City College of New York 7 Tricycle d: distance from the front wheel to the rear axle Steering and power are provided through the front wheel control variables: –angular velocity of steering wheel w s (t) –steering direction α(t)

8. City College of New York 8 Tricycle Kinematics model in the world frame ---Posture kinematics model

9. City College of New York 9 Synchronous Drive All the wheels turn in unison –All wheels point in the same direction and turn at the same rate –Two independent motors, one rolls all wheels forward, one rotate them for turning Control variables (independent) –v(t), ω(t)

10. City College of New York 10 Ackerman Steering (Car Drive) The Ackerman Steering equation: –: R

11. City College of New York 11 Car-like Robot non-holonomic constraint: : forward velocity of the rear wheels : angular velocity of the steering wheels l : length between the front and rear wheels X Y   l ICC R  Rear wheel drive car model: Driving type: Rear wheel drive, front wheel steering

12. City College of New York 12 Robot Sensing Collect information about the world Sensor - an electrical/mechanical/chemical device that maps an environmental attribute to a quantitative measurement Each sensor is based on a transduction principle - conversion of energy from one form to another Extend ranges and modalities of Human Sensing

13. City College of New York 13 Solar Cell Digital Infrared Ranging Compass Touch Switch Pressure Switch Limit Switch Magnetic Reed Switch Magnetic Sensor Miniature Polaroid Sensor Polaroid Sensor Board Piezo Ultrasonic Transducers Pyroelectric Detector Thyristor Gas Sensor Gieger-Muller Radiation Sensor Piezo Bend Sensor Resistive Bend Sensors Mechanical Tilt Sensors Pendulum Resistive Tilt Sensors CDS Cell Resistive Light Sensor Hall Effect Magnetic Field Sensors Compass IRDA Transceiver IR Amplifier Sensor IR Modulator Receiver Lite-On IR Remote Receiver Radio Shack Remote Receiver IR Sensor w/lens Gyro Accelerometer IR Reflection Sensor IR Pin Diode UV Detector Metal Detector

14. City College of New York 14 Sensors Used in Robot Resistive sensors: –bend sensors, potentiometer, resistive photocells,... Tactile sensors: contact switch, bumpers… Infrared sensors –Reflective, proximity, distance sensors… Ultrasonic Distance Sensor Motor Encoder Inertial Sensors (measure the second derivatives of position) –Accelerometer, Gyroscopes, Orientation Sensors: Compass, Inclinometer Laser range sensors Vision, GPS, …

15. City College of New York 15 Incremental Optical Encoders A B A leads B Incremental Encoder: Encoder pulse and motor direction It generates pulses proportional to the rotation speed of the shaft. Direction can also be indicated with a two phase encoder:

16. City College of New York 16 Motion Planning –Configuration Space –Motion Planning Methods Roadmap Approaches Cell Decomposition Potential Fields Bug Algorithms Path Planning: Find a path connecting an initial configuration to goal configuration without collision with obstacles

17. City College of New York 17 Motion Planning Motion Planning Methodololgies – Roadmap – Cell Decomposition – Potential Field Roadmap – From C free a graph is defined (Roadmap) – Ways to obtain the Roadmap Visibility graph Voronoi diagram Cell Decomposition – The robot free space (C free ) is decomposed into simple regions (cells) – The path in between two poses of a cell can be easily generated Potential Field – The robot is treated as a particle acting under the influence of a potential field U, where: the attraction to the goal is modeled by an additive field obstacles are avoided by acting with a repulsive force that yields a negative field Global methods Local methods

18. City College of New York 18 Full-knowledge motion planning approximate free space represented via a quadtree Cell decompositionsRoadmaps exact free space represented via convex polygons visibility graph voronoi diagram

19. City College of New York 19 Potential field Method Usually assumes some knowledge at the global level The goal is known; the obstacles sensed Each contributes forces, and the robot follows the resulting gradient.

20. City College of New York 20 Dr. John (Jizhong) Xiao Department of Electrical Engineering City College of New York jxiao@ccny.cuny.edu Mobile Robot Mapping Introduction to ROBOTICS

21. City College of New York 21 Sonar sensing Why is sonar sensing limited to between ~12 in. and ~25 feet ? “The sponge” Polaroid sonar emitter/receivers sonar timeline 0 a “chirp” is emitted into the environment 75  s typically when reverberations from the initial chirp have stopped.5s the transducer goes into “receiving” mode and awaits a signal... after a short time, the signal will be too weak to be detected Sonar (sound navigation and ranging): range sensing using acoustic (i.e., sound) signal Blanking timeAttenuation

22. City College of New York 22 Using sonar to create maps What should we conclude if this sonar reads 10 feet? 10 feet there is something somewhere around here there isn’t something here Local Map unoccupied occupied or... no information

23. City College of New York 23 Using sonar to create maps What should we conclude if this sonar reads 10 feet... 10 feet and how do we add the information that the next sonar reading (as the robot moves) reads 10 feet, too? 10 feet

24. City College of New York 24 Several answers to this question have been tried: It’s a map of occupied cells. It’s a map of probabilities: p( o | S 1..i ) p( o | S 1..i ) It’s a map of odds. The certainty that a cell is occupied, given the sensor readings S 1, S 2, …, S i The certainty that a cell is unoccupied, given the sensor readings S 1, S 2, …, S i The odds of an event are expressed relative to the complement of that event. odds( o | S 1..i ) = p( o | S 1..i ) The odds that a cell is occupied, given the sensor readings S 1, S 2, …, S i o xy cell (x,y) is occupied cell (x,y) is unoccupied ‘83 - ‘88 pre ‘83 What is it a map of ? probabilities evidence = log 2 (odds)

25. City College of New York 25 Combining evidence So, how do we combine evidence to create a map? What we want -- odds( o | S 2  S 1 ) the new value of a cell in the map after the sonar reading S 2 What we know -- odds( o | S 1 ) the old value of a cell in the map (before sonar reading S 2 ) p( S i | o ) & p( S i | o ) the probabilities that a certain obstacle causes the sonar reading S i

26. City College of New York 26 p( S 2 | o ) p( S 1 | o ) p(o) Combining evidence odds( o | S 2  S 1 ) = p( o | S 2  S 1 ). = p( S 2  S 1 | o ) p(o) p( S 2 | o ) p( S 1 | o ) p(o) Update step = multiplying the previous odds by a precomputed weight. def’n of odds Bayes’ rule (+) conditional independence of S 1 and S 2 p( S 2 | o ) p( o | S 1 ) Bayes’ rule (+) previous odds precomputed values the sensor model

27. City College of New York 27 Dr. John (Jizhong) Xiao Department of Electrical Engineering City College of New York jxiao@ccny.cuny.edu A Taste of Localization Problem Introduction to ROBOTICS

28. City College of New York 28 Bayes Formula Posterior probability distribution Prior probability distribution If y is a new sensor reading Generative model, characteristics of the sensor     Does not depend on x

29. City College of New York 29 Bayes Rule with Background Knowledge

30. City College of New York 30 Markov assumption (or static world assumption)

31. City College of New York 31 Markov Localization Measurement Action

32. City College of New York 32 Measurement: Update Phase ab c

33. City College of New York 33 Measurement: Update Phase

34. City College of New York 34 Recursive Bayesian Updating Markov assumption: z n is independent of z 1,...,z n-1 if we know x.

35. City College of New York 35 Action: Prediction Phase The robot turns its wheels to move The robot uses its manipulator to grasp an object Plants grow over time… Actions are never carried out with absolute certainty. In contrast to measurements, actions generally increase the uncertainty. How can we incorporate such actions?

36. City College of New York 36 Modeling Actions To incorporate the outcome of an action u into the current “belief”, we use the conditional pdf P(x|u,x’) This term specifies the pdf that executing u changes the state from x’ to x.

37. City College of New York 37 Integrating the Outcome of Actions Continuous case: Discrete case:

38. City College of New York 38 Summary Measurement Action

39. City College of New York 39 Bayes Filters: Framework Given: –Stream of observations z and action data u: –Sensor model P(z|x). –Action model P(x|u,x’). –Prior probability of the system state P(x). Wanted: –Estimate of the state X of a dynamical system. –The posterior of the state is also called Belief:

40. City College of New York 40 Markov Assumption Underlying Assumptions Static world, Independent noise Perfect model, no approximation errors State transition probability Measurement probability   Markov Assumption: –past and future data are independent if one knows the current state

41. City College of New York 41 Bayes Filters Bayes z = observation u = action x = state Markov Total prob. Markov

42. City College of New York 42 Bayes Filters are Family Bayes rule allows us to compute probabilities that are hard to assess otherwise. Under the Markov assumption, recursive Bayesian updating can be used to efficiently combine evidence. Bayes filters are a probabilistic tool for estimating the state of dynamic systems.

43. City College of New York 43 Thank you! Next Class: Final Exam Time: 6:30pm-8:30pm, Dec. 9, 2013, Coverage: Mobile Robot Close-book with 1 page cheat sheet, but Do Not Cheat