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

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

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

4. The Morven Framework Constructive Non-constructive Simulation Envisionment Synchronous Asynchronous

5. Quantity Spaces + 0 -

6. 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)

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

8. 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

9. Fuzzy Vector Envisionment

11. 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

12. FQT Coordinate systems

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

14. FQT Functions

15. 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

16. 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.

17. 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

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

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

20. 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

21. 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

22. 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]

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

24. 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.

25. Acknowledgements Dave Barnes Andy Shaw Eddie Edwards