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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