Robot's finger and expansions in non-integer bases

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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

Abstract
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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 and 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), 348-353.
[2]	Y. Chitour and B. Piccoli, Controllability for discrete systems with a finite control set, Mathematics of Control Signals and Systems, 14 (2001), 173-193. doi: 10.1007/PL00009881.
[3]	P. Erd\Hos and V. Komornik, Developments in non-integer bases, Acta Math. Hungar., 79 (1998), 57-83. doi: 10.1023/A:1006557705401.
[4]	K. J. Falconer, "Fractal Geometry," Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990.
[5]	W. J. Gilbert, Geometry of radix representations, in "The Geometric Vein," Springer, New York-Berlin, (1981), 129-139. doi: 10.1007/978-1-4612-5648-9_7.
[6]	W. J. Gilbert, The fractal dimension of sets derived from complex bases, Canad. Math. Bull., 29 (1986), 495-500. doi: 10.4153/CMB-1986-078-1.
[7]	W. J. Gilbert, Complex bases and fractal similarity, Ann. Sci. Math. Québec, 11 (1987), 65-77.
[8]	P. S. Heckbert, ed., "Graphics Gems IV," Academic Press, 1994.
[9]	J. Easudes C. J. H. Moravec and F. Dellaert, Fractal branching ultra-dexterous robots (bush robots), Technical report, NASA Advanced Concepts Research Project, 1996.
[10]	J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.
[11]	K.-H. Indlekofer, I. Kátai and P. Racskó, Number systems and fractal geometry, in "Probability Theory and Applications," Math. Appl., 80, Kluwer Acad. Publ., Dordrecht, (1992), 319-334.
[12]	A. C. Lai, "On Expansions in Non-Integer Base," Ph.D thesis, Sapienza Università di Roma and Université Paris Diderot, 2010.
[13]	W. Parry, On the $\beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416. doi: 10.1007/BF02020954.
[14]	J. Pineda, A parallel algorithm for polygon rasterization, Proceedings of the 15th annual conference on Computer graphics and interactive techniques, 22 (1988), 17-20.
[15]	A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, 8 (1957), 477-493. doi: 10.1007/BF02020331.
[16]	B. Siciliano and O. Khatib, "Springer Handbook of Robotics," 2008.

show all references

References:

[1]	A. Bicchi, Robotic grasping and contact: A review, Proc. IEEE Int. Conf. on Robotics and Automation, (2000), 348-353.
[2]	Y. Chitour and B. Piccoli, Controllability for discrete systems with a finite control set, Mathematics of Control Signals and Systems, 14 (2001), 173-193. doi: 10.1007/PL00009881.
[3]	P. Erd\Hos and V. Komornik, Developments in non-integer bases, Acta Math. Hungar., 79 (1998), 57-83. doi: 10.1023/A:1006557705401.
[4]	K. J. Falconer, "Fractal Geometry," Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester, 1990.
[5]	W. J. Gilbert, Geometry of radix representations, in "The Geometric Vein," Springer, New York-Berlin, (1981), 129-139. doi: 10.1007/978-1-4612-5648-9_7.
[6]	W. J. Gilbert, The fractal dimension of sets derived from complex bases, Canad. Math. Bull., 29 (1986), 495-500. doi: 10.4153/CMB-1986-078-1.
[7]	W. J. Gilbert, Complex bases and fractal similarity, Ann. Sci. Math. Québec, 11 (1987), 65-77.
[8]	P. S. Heckbert, ed., "Graphics Gems IV," Academic Press, 1994.
[9]	J. Easudes C. J. H. Moravec and F. Dellaert, Fractal branching ultra-dexterous robots (bush robots), Technical report, NASA Advanced Concepts Research Project, 1996.
[10]	J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.
[11]	K.-H. Indlekofer, I. Kátai and P. Racskó, Number systems and fractal geometry, in "Probability Theory and Applications," Math. Appl., 80, Kluwer Acad. Publ., Dordrecht, (1992), 319-334.
[12]	A. C. Lai, "On Expansions in Non-Integer Base," Ph.D thesis, Sapienza Università di Roma and Université Paris Diderot, 2010.
[13]	W. Parry, On the $\beta $-expansions of real numbers, Acta Math. Acad. Sci. Hungar., 11 (1960), 401-416. doi: 10.1007/BF02020954.
[14]	J. Pineda, A parallel algorithm for polygon rasterization, Proceedings of the 15th annual conference on Computer graphics and interactive techniques, 22 (1988), 17-20.
[15]	A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar, 8 (1957), 477-493. doi: 10.1007/BF02020331.
[16]	B. Siciliano and O. Khatib, "Springer Handbook of Robotics," 2008.

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