Article Contents
Article Contents

# Robot's finger and expansions in non-integer bases

• 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.
Mathematics Subject Classification: 70E60, 11A63.

 Citation:

•  [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.