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Integrating Fuzzy Qualitative Trigonometry with Fuzzy Qualitative Envisionment George M. Coghill, Allan Bruce, Carol Wisely & Honghai Liu

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Overview Introduction Fuzzy Qualitative Envisionment l Morven Toolset Fuzzy Qualitative Trigonometry Integration issues Results and Discussion Conclusions and Future Work

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The Context of Morven Predictive Algorithm Vector Envisionment FuSim Qualitative Reasoning P.A. V.E. QSIM TQA & TCP Morven

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The Morven Framework Constructive Non-constructive Simulation Envisionment Synchronous Asynchronous

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Quantity Spaces + 0 -

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Basic Fuzzy Qualitative Representation 4-tuple fuzzy numbers (a, b, ) precise and approximate useful for computation x A (x) 1 0 a x (a) A (x) 1 0 a b x (b) A (x) 1 0 a- a x a+ (c) A (x) 1 0 a- b+ ab (d)

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FQ Operations The arithmetic of 4-tuple fuzzy numbers Approximation principle

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Single Tank System h qiqi qoqo h t + - + + o o + + - Plane 0 qo = f(h) h= qo - qi Plane 1 qo = f(h).h h= qo - qi

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Fuzzy Vector Envisionment

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Standard Trigonometry Sine = opp/hyp = y p Cos = adj/hyp = x p Tan = opp/adj = sin/cos Pythagorean lemma sin 2 cos 2 P = (x p, y p ) 0 x y r = 1 xpxp ypyp

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FQT Coordinate systems

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

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FQT Functions

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Sine example Consider the 3 rd 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

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Sine example (2) Map back (approximation principle): sin(Qs a (3)) = 0.7119 0.7996 0.0169 0.0169 0.8136 0.8983 0.0169 0.0169 0.9153 1.000 0.0169 0 Cosine calculated similarly l Gives 5 possible values.

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Pythagorean example Global constraint: sin 2 (QS a (p i )) + cos 2 (QS a (p i )) = [1 1 0 0] Third angle value l Sin has 3 values & cos has 5 values => 15 possible values l Only 9 values consistent with global constraint

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FQT Rules FQT supplementary value FQT complementary value FQT opposite value FQT anti supplementary value FQT sine rule FQT cosine rule

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FQT Triangle Theorems AAA theorem AAS theorem ASA theorem ASS theorem SAS theorem SSS theorem

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Integrating Morven and FQT Fairly straightforward l Morven - dynamic systems - differential planes l 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

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Example: A One Link Manipulator Plane 0: x 1 = x 2 x 2 = p.sin(x 1 ) - q.x 1 + r Plane 1: x 1 = x 2 x 2 = p.x 1.cos(x 1 ) - q.x 1 + r p= q/l; q = k/m.l 2 ; r = 1/m.l 2 mg k T x l

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Example contd 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: l Pos3: [0.521 0.739 0.043 0.043] l Pos4: [0.783 1.0 0.043 0]

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Results Viewer Directed Graph for State Transitions l Behaviour paths easily observed

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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 l More complex calculations required l Started with MSc summer project.

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Acknowledgements Dave Barnes Andy Shaw Eddie Edwards

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