Integrating Fuzzy Qualitative Trigonometry with Fuzzy Qualitative Envisionment

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

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

3. The Context of Morven Q u a l i t a t i v e T Q A & T C P V . E . P . A . Morven P r e d i c t i v e V e c t o r E n v i s i o n m e n t A l g o r i t h m F u S i m Q S I M R e a s o n i n g

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

5. 0 - + Quantity Spaces

6.  (x)  (x) A A  (x)  (x) A A  b+ a- a+  a-   Basic Fuzzy Qualitative Representation • 4-tuple fuzzy numbers (a, b, ) • precise and approximate • useful for computation 1 1 x 0 0 x b a a ( a ) ( b ) 1 1 0 0 x a a b x ( c ) ( d )

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

8. qi h h + - + qo + o o + + - t Single Tank System Plane 0 qo = f(h) h’= qo - qi Plane 1 q’o = f’(h).h’ h’’= q’o - q’i

9. Fuzzy Vector Envisionment

10. Fuzzy Vector Envisionment

11. Standard Trigonometry • Sine = opp/hyp = yp • Cos = adj/hyp = xp • Tan = opp/adj = sin/cos • Pythagorean lemma sin2q + cos2q = 1 y P = (xp, yp) yp r = 1 q 0 x xp

12. FQT Coordinate systems

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

14. FQT Functions

15. 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

16. Sine example (2) • Map back (approximation principle):sin(Qsa(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 • Gives 5 possible values.

17. 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

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 • 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’

21. k x mg l T Example: A One Link Manipulator Plane 0: x’1 = x2 x’2 = p.sin(x1) - q.x1 + rt Plane 1: x’’1 = x’2 x’2 = p.x’1.cos(x1) - q.x’1 + rt’ p= q/l; q = k/m.l2; r = 1/m.l2

22. Example cont’d • FQ model requires nine auxiliary variables • 9 quantities used • Constants (l, m, g, & t)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]

23. Results Viewer • Directed Graph for State Transitions • 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 • More complex calculations required • Started with MSc summer project.

25. Acknowledgements Dave Barnes Andy Shaw Eddie Edwards