1. Fuzzy Logic

2. Priyaranga Koswatta Mundhenk and Itti, 2007

3. Advantages of Fuzzy Controllers Minimal mathematical formulation Can easily design with human intuition Smoother controlling Faster response

4. Agenda General Definition Applications Formal Definitions Operations Rules Fuzzy Air Conditioner Controller Structure

5. General Definition Fuzzy Logic - 1965 Lotfi Zadeh, U.C. Berkeley superset of conventional (Boolean) logic that has been extended to handle the concept of partial truth central notion of fuzzy systems is that truth values (in fuzzy logic) or membership values (in fuzzy sets) are indicated by a value on the range [0.0, 1.0], with 0.0 representing absolute Falseness and 1.0 representing absolute Truth. deals with real world vagueness

7. Applications ABS Brakes Expert Systems Control Units Bullet train between Tokyo and Osaka Video Cameras Automatic Transmissions

8. Formal Definitions Definition 1: Let X be some set of objects, with elements noted as x. X = {x}. Definition 2: A fuzzy set A in X is characterized by a membership function mA(x) which maps each point in X onto the real interval [0.0, 1.0]. As mA(x) approaches 1.0, the "grade of membership" of x in A increases. Definition 3: A is EMPTY iff for all x, mA(x) = 0.0. Definition 4: A = B iff for all x: mA(x) = mB(x) [or, mA = mB]. Definition 5: mA' = 1 - mA. Definition 6: A is CONTAINED in B iff mA  mB. Definition 7: C = A UNION B, where: mC(x) = MAX(mA(x), mB(x)). Definition 8: C = A INTERSECTION B where: mC(x) = MIN(mA(x), mB(x)).

9. http://www.seattlerobotics.org/Encoder/mar 98/fuz/flindex.html http://www.cs.cmu.edu/Groups/AI/html/faq s/ai/fuzzy/part1/faq.html http://plato.stanford.edu/entries/logic-fuzzy/

12. Operations A B A  B A  B  A

13. Example: Using Fuzzy Logic for a Line Following Robot

14. Mechanical Design of a very inexpensive Line-Following Robot

15. Basic Motions of a Differential Drive robot

16. Input Membership Functions

17. Sample Fuzzy Rule Base

18. Output Membership Function

23. Example: Using Fuzzy Logic for an Obstacle Avoiding Robot

26. Very Basic Control Theory Your speed towards a goal or away from an object should be proportional to the distance from it. If you want to get to a goal in an optimal amount of time you want to move quickly. However, you need to decelerate as you grow near the target so you can have more control. Speed ∝ distance-to-target

27. Very Basic Control Theory In systems with momentum (i.e. the real world) we have to worry about more complex acceleration and deceleration. We can overshoot our target or stop short! You increase your rate of deceleration based on how close you are to a goal or obstacle. You can also integrate over the distance to a goal to create a steady state. This is the basic idea behind a PID controller. Proportional Integral Derivative The physical derivation of PID can be tricky, we will avoid it for now. However this part of an extremely interesting topic!

28. IDEA! Lets just hack a fuzzy controller together and avoid some math. The gods will curse us…. But if it works, that may be all that matters! Derive rule of thumb ideas for speed and direction If I’m 6 meters from the obstacle, am I far from it?

30. Try some fuzzy rules… Lets look at adjusting trajectory first then we will look at speed… If an obstacle is near and center, turn sharp right or left. If an obstacle is far and center, turn soft left or right. If an obstacle is near, turn slightly right or left, just in case. Etc…

37. The robot works in continuous time The fuzzy rules should change slightly at each time step. We don’t want the robot to jerk to a new trajectory too quickly. Smooth movements tend to be better. How often we need to update the controller is dependant on how fast we are moving. For instance: If we update the controller 30 times a second and we are moving < 1 kph then movement will be smooth. Ideally, the commands issued from the fuzzy controller should create an equilibrium with the observations.

38. Our robot has momentum We have somewhat implicitly integrated the notion of momentum. This is why we would slow down as we get closer to an obstacle What about more refined control of speed and direction? Use the derivative of speed and trajectory to increase or decrease the rate of change. Thus, if it seems like we are not turning fast enough, then turn faster and visa versa.

41. Controller Structure Fuzzification –Scales and maps input variables to fuzzy sets Inference Mechanism –Approximate reasoning –Deduces the control action Defuzzification –Convert fuzzy output values to control signals

42. Rule Base Air Temperature Set cold {50, 0, 0} Set cool {65, 55, 45} Set just right {70, 65, 60} Set warm {85, 75, 65} Set hot { , 90, 80} Fan Speed Set stop {0, 0, 0} Set slow {50, 30, 10} Set medium {60, 50, 40} Set fast {90, 70, 50} Set blast { , 100, 80}

43. Rules Air Conditioning Controller Example: IF Cold then Stop If Cool then Slow If OK then Medium If Warm then Fast IF Hot then Blast default: The truth of any statement is a matter of degree default: The truth of any statement is a matter of degree Membership function is a curve of the degree of truth of a given input value

44. Fuzzy Air Conditioner

45. Mapping Inputs to Outputs

46. EXAMPLE: Using Fuzzy Logic for a SWERVING ROBOT

47. Motivating Example: Swerving Robot