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Mathematics’ Mortua Manus:Discovering Dexterity

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From Linear Operators to Computational Biology

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Abstract

Dexterous manipulation, a major subfield of robotics and manufacturing, experienced a mathematical rebirth in the mid 80’s, when this nascent field established many beautiful connections to convexity theory and computational geometry. Jack Schwartz played a seminal role in its inception and development. Here, I speculate on where Jack might have liked this field to go in the future.

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Notes

  1. 1.

    The cross product τ=p×f is defined as

  2. 2.

    If one allows unbounded objects then in 3-D, we have to include unbounded prisms and helical objects and in 2-D an unbounded strip of constant width. These objects in 3-D describe the so-called Reuleux pairs, studied almost a century ago.

References

  1. Bicchi, A.: Hands for dextrous manipulation and robust grasping: a difficult road towards simplicity. IEEE Trans. Robot. Autom. 16(6), 652–662 (2000)

    Article  Google Scholar 

  2. Carathéodory, C.: Über den Variabilitätsbereich der Koeffizienten von Potenzreihen die gegebene Werte nicht annehmen. Math. Ann. 64, 95–115 (1907)

    Article  MathSciNet  MATH  Google Scholar 

  3. Danzer, L., Grünbaum, B., Klee, V.: Helly’s theorem and its relatives. Convexity 7, 101–180 (1963)

    Google Scholar 

  4. Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Berlin (1987)

    Book  MATH  Google Scholar 

  5. Kirkpatrick, D., Mishra, B., Yap, C.: Quantitative Steinitz’s theorem with applications to multifingered grasping. Discrete Comput. Geom. 7, 295–318 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Klee, V.: Facet Centroids and Volume Minimization (1986)

    Google Scholar 

  7. Markenscoff, X., Ni, L., Papadimitriou, C.: The geometry of grasping. Int. J. Robot. Res. 9, 61–74 (1990)

    Article  Google Scholar 

  8. Mason, M., Srinivasa, S., Vazquez, A.: Generality and simple hands. In: International Symposium on Robotics Research (2009)

    Google Scholar 

  9. Megiddo, N.: Linear programming in linear time when the dimension is fixed. J. ACM 31(1), 114–127 (1984)

    MathSciNet  MATH  Google Scholar 

  10. Mishra, B.: Grasp Metrics: Optimality and Complexity, pp. 137–166. AK Peters, Wellesley (1995)

    Google Scholar 

  11. Mishra, B., Schwartz, J., Sharir, M.: On the existence and synthesis of multifinger positive grips. Algorithmica 2, 541–558 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Mishra, B., Silver, N.: Some discussion of static gripping and its stability. IEEE Trans. Syst. Man Cybern. 19, 783–796 (1989)

    Article  Google Scholar 

  13. Mishra, B., Teichmann, M.: Reactive algorithms for 2 and 3 finger grasping. In: Proceedings of the 1994 International Workshop on Intelligent Robots and Systems, IRS 94 (1994)

    Google Scholar 

  14. Mishra, B., Teichmann, M.: Reactive algorithms for grasping using a modified parallel jaw gripper. In: Proceedings of the 1994 IEEE International Conference on Robotics and Automation, ICRA 94 (1994)

    Google Scholar 

  15. Mishra, B., Teichmann, M.: The Power of Friction: Quantifying the ‘Goodness’ of Frictional Grasps, pp. 311–320. AK Peters, Wellesley (1997)

    Google Scholar 

  16. O’Rourke, J.: Computational Geometry in C. Cambridge University Press, Cambridge (1994)

    MATH  Google Scholar 

  17. O’Rourke, J., Aggarwal, A., Maddila, S., Baldwin, M.: An optimal algorithm for finding minimal enclosing triangles. J. Algorithms 7(2), 258–269 (1986). doi:10.1016/0196-6774(86)90007-6

    Article  MathSciNet  MATH  Google Scholar 

  18. Schwartz, J.: The Pernicious Influence of Mathematics on Science, pp. 230–235. Springer, Berlin (2006)

    Google Scholar 

  19. Technolgies, B.: The Barrett Hand. http://www.barrett.com/robot/products-hand.html

  20. Teichmann, M.: Grasping and fixturing: a geometric study and an implementation. Ph.D. thesis, New York University, New York (1995)

    Google Scholar 

  21. Wigner, E.: The unreasonable effectiveness of mathematics in the natural sciences. Commun. Pure Appl. Math. 13, 1–14 (1960)

    Article  MATH  Google Scholar 

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Acknowledgements

The paper has improved considerably following many insightful suggestions from several colleagues: most notably, S. Kleinberg and E. Schonberg of NYU, M. Mason of CMU and M. Wigler of CSHL.

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  1. Courant Institute, 251 Mercer St., New York, NY, USA

    B. Mishra

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  1. B. Mishra
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Correspondence to B. Mishra .

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  1. Courant Institute of Mathem. Science, New York University, Mercer Street 251, New York, 10012, New York, USA

    Martin Davis

  2. Courant Institute of Mathem. Science, New York University, Mercer Street 251, New York, 10012, New York, USA

    Edmond Schonberg

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Mishra, B. (2013). Mathematics’ Mortua Manus:Discovering Dexterity. In: Davis, M., Schonberg, E. (eds) From Linear Operators to Computational Biology. Springer, London. https://doi.org/10.1007/978-1-4471-4282-9_7

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  • DOI: https://doi.org/10.1007/978-1-4471-4282-9_7

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-4281-2

  • Online ISBN: 978-1-4471-4282-9

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