2. A great man who created a complete new mathematics with many practical applications.

3. Fuzzy Logic practical easy examples

4. How to combine various, even conflicting pieces of advise?
Linguistic
How to combine various, even conflicting pieces of advise?

5. Examples of Operations on the same variable
A  B A  B A

6. Example 1: Using Fuzzy Logic for a Line Following Robot

7. Mechanical Design of a very inexpensive Line-Following Robot
In our example below:
Line white, background black

8. Basic Motions of a Differential Drive robot
Reminder = Braitenberg Vehicles

9. Calculations for input rules

10. Defuzzifier of the “sum”
Structure of this type of the system: Two Levels, various membership functions in each, shared
MIN
MIN
Defuzzifier of the “sum”
Fuzzified inputs
MIN
MIN
MIN
Fuzzy values of combined first level groups
One input Defuzzifier
MIN
MAX
MIN
Level of input membership functions
Level of output membership functions
Reminder = general simple scheme for Fuzzy Logic

11. Input Membership Functions
Characteristics of two identical sensors
We have three membership functions: Black, Gray and White

12. Sample Fuzzy Rule Base – for input rules
Here is an example how you can create your own rules for a robot of your choice.
This table decides what to do for every combination of data from both input sensors SR
X = don’t know = undetermined
The system will “learn” values for these don’knows.
Descriptions such as this table:
Generalize Truth Table
Generalize Karnaugh Map

13. Output Membership Function
Output of the controller can be negative or positive

14. This slide explains how to apply rules for left sensor
We calculate for given measurement for the left sensor
For black membership function
white
black
gray
For white membership function
black
Reading the data from left sensor

15. How to apply input rules?

16. How to calculate output?
Here are some examples of rule calculations
This calculated 0.7 for SL
This calculated 0.2 for F
Right sensor
Left sensor

17. Calculations for output rules

18. How to calculate Output Membership Function
Calculated in previous slide
0.7 for SL
Calculated similarly for SR
0.2 for F
Note: this is some kind of weighted sum
This is one example of defuzzification that calculates the speed of the motor

19. Structure of this type of the system: Two Levels, various membership functions in each, shared
Sensor L
This is fuzzy “strength” of the fuzzy output value SL
white SL
0.7
Defuzzifier
0.7
MIN
Note: this is some kind of weighted sum
gray
This is fuzzy “strength” of the fuzzy output value F
0.8 SL F
black
-20
MIN F
white
other
MIN HL
…….
gray
Fuzzy values of combined first level groups
black
Sensor R
Level of input membership functions
Level of output membership functions
speed of the motor

20. To remember
The previous slide was only a special case of defuzzification in a sum gate
Many other methods and shapes are possible.
Remember the general structure of the circuit. There are many variants of this circuit. With many defuzzification methods.

21. Example 2: Using Fuzzy Logic for an Obstacle Avoiding Robot

22. In this example the principles are the same as before, but the way of calculating is somewhat different.
We have no table, just we directly write rules.
This is just for didactic reasons.

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

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

27. IDEA! Heuristic experimental rules
Lets just hack a fuzzy controller together and avoid some math.
The formal mathematicians will hate us,
but if it works, that may be all that matters!
Derive rule of thumb ideas for speed and direction
If I am 6 meters from the obstacle, am I far from it?

28. If near, turn more

29. Experimental heuristics in robot design
Try some fuzzy rules…
Let us 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…
You have to experiment with these rules

30. Degrees of angle
Degrees of angle
meters

32. Implication of the rules

33. Defuzzification

35. Get closer, turn more
Many rules of thumb like this

36. Advise on creating rules
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.
These are just examples.
You decide for your robot and your sensors

37. Advise on creating rules
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.

40. Remember the general structure
MIN
MIN
Defuzzifier of the “sum”
Fuzzified inputs
MIN
MIN
MIN
Fuzzy values of combined first level groups
One input Defuzzifier
MIN
MAX
MIN
Level of input membership functions
Level of output membership functions
Reminder = general simple scheme for Fuzzy Logic

41. EXAMPLE 3: Fan control

42. Controller Structure Fuzzification Inference Mechanism Defuzzification
Scales and maps input variables to fuzzy sets
Linear or not
Single input single output
Inference Mechanism
Approximate reasoning
Deduces the control action
Various shapes of membership functions
Various operators, not only MIN, MAX and NOT.
Defuzzification
Convert fuzzy output values to control signals
Defuzzification can be a single output linear or nonlinear function with fuzzy input and crisp output
Defuzzification can be built-into the “output sum” which can be other function than MAX or SUM or TRUNCATED SUM.

43. Rule Base for Fan Control
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}

44. Defuzzifier of the “sum”
Structure of this type of the system: Two Levels, various membership functions in each, shared
MIN
MIN
Defuzzifier of the “sum”
Fuzzified inputs
MIN
MIN
MIN
Fuzzy values of combined first level groups
One input Defuzzifier
MIN
MAX
MIN
Level of input membership functions
Level of output membership functions

45. Rules Air Conditioning Controller Example: IF Cold then Stop
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
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

46. Fuzzy Air Conditioner outputs
This is another useful visualization in two-dimensional space.
inputs

47. Mapping Inputs to Outputs

48. EXAMPLE 4: Using Fuzzy Logic for a SWERVING ROBOT
This is a more detailed analysis of a simple version of Example 1

49. Motivating Example: Swerving Robot

53. proximity
motion
We take into account only PROXIMITY

59. DEFUZZIFICATION has many faces
Blended Centroid
Largest Centroid
Weighted Means

60. Defuzzification

61. Defuzzification: BLENDED CENTROID

63. Defuzzification: Largest Centroid

64. Defuzzification: Weighted Means

65. Back to Swerve Now we also take into account where is open for motion

66. Open can be rightside or leftside

67. Examples for evaluation with PROXIMITY and OPEN

68. evaluation
open

69. Variant which uses NOT in a rule

70. Variant which uses hedges in a rule
Hard_Right
Very Hard_Right

71. Summary on Fuzzy Controllers for simple robots

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

73. Few More real-life Applications
ABS Brakes
Expert Systems
Control Units
Bullet train between Tokyo and Osaka
Video Cameras
Automatic Transmissions

75. Advise based on our previous projects
Fuzzy logic is easy to program
Fuzzy logic can be easily combined with any other method from this and next class
In particular , there is a very good match to evolutionary ideas.
Much software exist, you do not have to write from scratch.
You should concentrate on combining ideas and analyzing results.
I am not happy if you only give code but do not discuss, analyze and illustrate how it actually works on a robot, its part or your simulated model.
With more time, it is good to modify the model step by step based on more experiments on a real robot
Look to many examples from my 478, 479, and 510 AER webpages
There are also many free books and reports on Internet.
All this can be completely formalized, new math created and you can write a MS or PHD on these topics.
Every branch of mathematics or calculus can be rewritten
This happens always when we deal with something as fundamental as sets and probabilities. The same is in quantum. Quantum Fuzzy logic has been also created.

76. Defuzzifier of the “sum”
Structure of this type of the system: Two Levels, various membership functions in each, shared
MIN
MIN
Defuzzifier of the “sum”
Fuzzified inputs
MIN
MIN
MIN
Fuzzy values of combined first level groups
One input Defuzzifier
MIN
MAX
MIN
Level of input membership functions
Level of output membership functions

77. Questions and Problems
Draw the complete logic diagram for Example 1.
Write software for Example 1.
Draw the complete logic diagram for Example 2.
Write software for Example 2.
Draw the complete logic diagram for Example 3.
Write software for Example 3.
Draw the complete logic diagram for Example 4.

79. Priyaranga Koswatta
Mundhenk and Itti, 2007