1 / 25

Integrating Fuzzy Qualitative Trigonometry with Fuzzy Qualitative Envisionment

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

brant
Download Presentation

Integrating Fuzzy Qualitative Trigonometry with Fuzzy Qualitative Envisionment

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related