American Institute of Mathematical Sciences

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March  2012, 7(1): 71-111. doi: 10.3934/nhm.2012.7.71

Robot's finger and expansions in non-integer bases

Anna Chiara Lai 1, and  Paola Loreti 2,

1. 

Sapienza Università di Roma, Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sezione Matematica, Via A. Scarpa n.16 00161 Roma, Italy
2. 

Sapienza Università di Roma, Sapienza Università di Roma, Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Sezione Matematica, Via A. Scarpa n.16 00161 Roma, Italy

Received  July 2011 Revised  December 2011 Published  February 2012

We study a robot finger model in the framework of the theory of expansions in non-integer bases. We investigate the reachable set and its closure. A control policy to get approximate reachability is also proposed.
Keywords: discrete control, expansions in non-integer bases, Robot finger, expansions in complex bases..
Mathematics Subject Classification: 70E60, 11A6.
Citation: Anna Chiara Lai, Paola Loreti. Robot's finger and expansions in non-integer bases. Networks & Heterogeneous Media, 2012, 7 (1) : 71-111. doi: 10.3934/nhm.2012.7.71
References:
[1]

A. Bicchi, Robotic grasping and contact: A review,, Proc. IEEE Int. Conf. on Robotics and Automation, (2000), 348. Google Scholar

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Y. Chitour and B. Piccoli, Controllability for discrete systems with a finite control set,, Mathematics of Control Signals and Systems, 14 (2001), 173. doi: 10.1007/PL00009881. Google Scholar

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P. Erd\Hos and V. Komornik, Developments in non-integer bases,, Acta Math. Hungar., 79 (1998), 57. doi: 10.1023/A:1006557705401. Google Scholar

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K. J. Falconer, "Fractal Geometry,", Mathematical Foundations and Applications, (1990). Google Scholar

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W. J. Gilbert, The fractal dimension of sets derived from complex bases,, Canad. Math. Bull., 29 (1986), 495. doi: 10.4153/CMB-1986-078-1. Google Scholar

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W. J. Gilbert, Complex bases and fractal similarity,, Ann. Sci. Math. Québec, 11 (1987), 65. Google Scholar

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P. S. Heckbert, ed., "Graphics Gems IV,", Academic Press, (1994). Google Scholar

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J. Easudes C. J. H. Moravec and F. Dellaert, Fractal branching ultra-dexterous robots (bush robots),, Technical report, (1996). Google Scholar

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J. Hutchinson, Fractals and self-similarity,, Indiana Univ. Math. J., 30 (1981), 713. doi: 10.1512/iumj.1981.30.30055. Google Scholar

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K.-H. Indlekofer, I. Kátai and P. Racskó, Number systems and fractal geometry,, in, 80 (1992), 319. Google Scholar

[12]

A. C. Lai, "On Expansions in Non-Integer Base,", Ph.D thesis, (2010). Google Scholar

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W. Parry, On the $\beta $-expansions of real numbers,, Acta Math. Acad. Sci. Hungar., 11 (1960), 401. doi: 10.1007/BF02020954. Google Scholar

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J. Pineda, A parallel algorithm for polygon rasterization,, Proceedings of the 15th annual conference on Computer graphics and interactive techniques, 22 (1988), 17. Google Scholar

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A. Rényi, Representations for real numbers and their ergodic properties,, Acta Math. Acad. Sci. Hungar, 8 (1957), 477. doi: 10.1007/BF02020331. Google Scholar

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B. Siciliano and O. Khatib, "Springer Handbook of Robotics,", 2008., (). Google Scholar

show all references

References:
[1]

A. Bicchi, Robotic grasping and contact: A review,, Proc. IEEE Int. Conf. on Robotics and Automation, (2000), 348. Google Scholar

[2]

Y. Chitour and B. Piccoli, Controllability for discrete systems with a finite control set,, Mathematics of Control Signals and Systems, 14 (2001), 173. doi: 10.1007/PL00009881. Google Scholar

[3]

P. Erd\Hos and V. Komornik, Developments in non-integer bases,, Acta Math. Hungar., 79 (1998), 57. doi: 10.1023/A:1006557705401. Google Scholar

[4]

K. J. Falconer, "Fractal Geometry,", Mathematical Foundations and Applications, (1990). Google Scholar

[5]

W. J. Gilbert, Geometry of radix representations,, in, (1981), 129. doi: 10.1007/978-1-4612-5648-9_7. Google Scholar

[6]

W. J. Gilbert, The fractal dimension of sets derived from complex bases,, Canad. Math. Bull., 29 (1986), 495. doi: 10.4153/CMB-1986-078-1. Google Scholar

[7]

W. J. Gilbert, Complex bases and fractal similarity,, Ann. Sci. Math. Québec, 11 (1987), 65. Google Scholar

[8]

P. S. Heckbert, ed., "Graphics Gems IV,", Academic Press, (1994). Google Scholar

[9]

J. Easudes C. J. H. Moravec and F. Dellaert, Fractal branching ultra-dexterous robots (bush robots),, Technical report, (1996). Google Scholar

[10]

J. Hutchinson, Fractals and self-similarity,, Indiana Univ. Math. J., 30 (1981), 713. doi: 10.1512/iumj.1981.30.30055. Google Scholar

[11]

K.-H. Indlekofer, I. Kátai and P. Racskó, Number systems and fractal geometry,, in, 80 (1992), 319. Google Scholar

[12]

A. C. Lai, "On Expansions in Non-Integer Base,", Ph.D thesis, (2010). Google Scholar

[13]

W. Parry, On the $\beta $-expansions of real numbers,, Acta Math. Acad. Sci. Hungar., 11 (1960), 401. doi: 10.1007/BF02020954. Google Scholar

[14]

J. Pineda, A parallel algorithm for polygon rasterization,, Proceedings of the 15th annual conference on Computer graphics and interactive techniques, 22 (1988), 17. Google Scholar

[15]

A. Rényi, Representations for real numbers and their ergodic properties,, Acta Math. Acad. Sci. Hungar, 8 (1957), 477. doi: 10.1007/BF02020331. Google Scholar

[16]

B. Siciliano and O. Khatib, "Springer Handbook of Robotics,", 2008., (). Google Scholar

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