Fuzzy Fuzzy LogicLogic

Priyaranga Koswatta

• Mundhenk and Itti, 2007

• Advantages of Fuzzy Controllers

• Minimal mathematical formulation

• Can easily design with human intuition

• Smoother controlling

• Faster response

Agenda

• General Definition• Applications• Formal Definitions• Operations• Rules• Fuzzy Air Conditioner• Controller Structure

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

Applications

• ABS Brakes

• Expert Systems

• Control Units

• Bullet train between Tokyo and Osaka

• Video Cameras

• Automatic Transmissions

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)).

• http://www.seattlerobotics.org/Encoder/mar98/fuz/flindex.html

• http://www.cs.cmu.edu/Groups/AI/html/faqs/ai/fuzzy/part1/faq.html

• http://plato.stanford.edu/entries/logic-fuzzy/

Operations

A B

A B A B A

Example:Example:Using Fuzzy Using Fuzzy

Logic for a Line Logic for a Line Following RobotFollowing Robot

Mechanical Design of a very Mechanical Design of a very inexpensive Line-Following Robotinexpensive Line-Following Robot

Basic Motions of a Basic Motions of a Differential Drive robotDifferential Drive robot

Input Membership FunctionsInput Membership Functions

Sample Fuzzy Rule BaseSample Fuzzy Rule Base

Output Membership FunctionOutput Membership Function

Example:Example:Using Fuzzy Logic Using Fuzzy Logic

for an Obstacle for an Obstacle Avoiding RobotAvoiding Robot

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∝

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!

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?

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…

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.

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.

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

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}

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

Membership function is a curve of the degree of truth of a given input value

Fuzzy Air Conditioner

Stop

Slow

Medium

Fast

Blast

0

10

20

30

40

50

60

70

80

90

100

0

1

45 50 55 60 65 70 75 80

0C

old

C

ool

85 90

Just

Rig

ht

W

arm

Hot

if Coldthen Stop

IF CoolthenSlow

If Just Rightthen

Medium

If WarmthenFast

If HotthenBlast

Mapping Inputs to Outputs1

0

10

20

30

40

50

60

70

80

90

100

0

1

45 50 55 60 65 70 75 80

0

85 90

t

EXAMPLE:EXAMPLE:

Using Fuzzy Logic Using Fuzzy Logic for a SWERVING for a SWERVING

ROBOTROBOT

Motivating Example: Motivating Example: Swerving RobotSwerving Robot