Integrating Fuzzy Qualitative Trigonometry with Fuzzy Qualitative Envisionment

George M. Coghill, Allan Bruce, Carol Wisely & Honghai Liu

Overview

Introduction Fuzzy Qualitative Envisionment

Morven Toolset

Fuzzy Qualitative Trigonometry Integration issues Results and Discussion Conclusions and Future Work

The Context of Morven

PredictiveAlgorithm

Vector Envisionment

FuSim

Qualitative

Reasoning

P.A. V.E.

QSIM

TQA & TCP

Morven

The Morven Framework

ConstructiveNon-constructive

Simulation

Envisionment

Synchronous

Asynchronous

Quantity Spaces

+

0

-

μA

(x)

10 x-1 0.2 0.4 0.6 0.8-0.8 -0.6 -0.4 -0.2

n-top n-large n-medium n-small zero p-small p-medium p-large p-top

Basic Fuzzy Qualitative Representation

4-tuple fuzzy numbers (a, b, ) precise and approximate useful for computation μ

a x

x a

x a x a a

x a b

b x x b b

x b

( )

( ) [ ]

[ ]

( ) [ ]

=

< −

− + ∈ −

∈

+ − ∈ +

> +

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

−

−

0

1

0

1

1

x

μA(x)

1

0 a x

(a)

μA(x)

1

0 a b x

(b)μA(x)

1

0 a- a xa+

(c)

μA(x)

1

0 a- b+a b

(d)

FQ OperationsThe arithmetic of 4-tuple fuzzy numbers

• Approximation principle

Single Tank System

h

qi

qo

h

t

+ - +

+ o o

+ + -

Plane 0qo = f(h)h’= qo - qi

Plane 1q’o = f’(h).h’h’’= q’o - q’i

Fuzzy Vector Envisionment

h

t

72

6

510

16p-small

p-medium

p-large

p-max

Fuzzy Vector Envisionment

Standard Trigonometry

Sine = opp/hyp = yp

Cos = adj/hyp = xp

Tan = opp/adj = sin/cos

Pythagorean lemma

sin2cos2

P = (xp, yp)

0 x

y

r = 1

xp

yp

FQT Coordinate systems

Quantity spaces

Let p=16, q[x]= q[y]=21

FQT Functions

Sine example

Consider the 3rd FQ angle:[0.1263, 0.1789, 0.0105, 0.0105]

Crossing points with adjacent values:0.1209 and 0.1842

Convert to deg or rad: 0.1209 -> 0.7596 & 0.1842 -> 1.1574

Sine of crossing points:sin(0.7596) = 0.6886 & sin(1.1574) = 0.9158

Sine example (2)

Map back (approximation principle):

sin(Qsa(3)) = 0.7119 0.7996 0.0169 0.01690.8136 0.8983 0.0169 0.01690.9153 1.000 0.0169 0

Cosine calculated similarly Gives 5 possible values.

Pythagorean example

Global constraint:sin2(QSa(pi)) + cos2(QSa(pi)) = [1 1 0 0]

Third angle value Sin has 3 values & cos has 5 values

=> 15 possible values Only 9 values consistent with global constraint

FQT RulesFQT supplementary valueFQT complementary valueFQT opposite valueFQT anti supplementary valueFQT sine ruleFQT cosine rule

FQT Triangle TheoremsAAA theoremAAS theoremASA theoremASS theoremSAS theoremSSS theorem

Integrating Morven and FQT

Fairly straightforward Morven - dynamic systems - differential planes FQT - kinematic (equilibrium) systems - scalar

Introduces structure:Eg: y = sin(x) becomes y’ = x’.cos(x) at first diff. plane;Need auxiliary variables:

d = cos(x)y’ = d.x’

Example: A One Link Manipulator

Plane 0:

x’1 = x2

x’2 = p.sin(x1) - q.x1 + rPlane 1:

x’’1 = x’2

x’2 = p.x’1.cos(x1) - q.x’1 + r’

p= q/l; q = k/m.l2; r = 1/m.l2

mg

k

T

x

l

Example cont’d

FQ model requires nine auxiliary variables 9 quantities used Constants (l, m, g, & are real 1266 (out of a possible 6561) states generated 14851 transitions in envisionment graph. Settles to two possible values:

Pos3: [0.521 0.739 0.043 0.043] Pos4: [0.783 1.0 0.043 0]

Results Viewer

Directed Graph for State Transitions Behaviour paths easily observed

Conclusions and Future Work

Fuzzy qualitative values can be utilised for qualitative simulation of dynamic systems

Integration is successful but just beginning; initial results are encouraging.

Extend to include complex numbers More complex calculations required Started with MSc summer project.

Acknowledgements

Dave Barnes

Andy Shaw

Eddie Edwards