Page 52 - Textos de Matemática Vol. 43

40 JANUSZ JAKUBIAK
4. Conclusions
In this paper we studied an application of splines defined on the torus to the motion planning task for the double pendulum arm. It was shown that although we are not able to find a closed-form solution for geodesics, it is possible to use numerical solutions of the equations and plug them into spline generation algorithm.
Comparison of the N-splines with the X-splines typically used in robotic path planning for a continuous velocity case has shown similar shape of the constructed curves and the average velocity of the N-spline lying between results for the X- splines. Certainly, a choice of di↵erent velocities in the intermediate points changes the resulting N-spline, but the ability of setting velocities in those points is an advantage of the proposed algorithm.
References
[1] F. Egebrand, C. F. Martin, and M. Egerstedt. Smoothing splines on the torus. In Mathemat- ical Theory of Networks and Systems, Budapest, Hungary, July 2010.
[2] G. Farin. Curves and Surfaces for Computer Aided Geometric Design. Academic Press, San Diego, 1993.
[3] S. M. Gomes, A. K. Kushpel, J. Levesley, and D. L. Ragozin. Interpolation on the torus using sk-splines with number theoretic knots. Journal of Approximation Theory, 98(1):56 – 71, 1999.
[4] T. N. T. Goodman, S. L. Lee, and A. Sharma. Approximation and interpolation by com- plex splines on the torus. Proceedings of the Edinburgh Mathematical Society (Series 2), 32(02):197–212, 1989.
[5] A. Gray. Modern di↵erential geometry of curves and surfaces with Mathematica. CRC Press, 1998.
[6] J. Jakubiak, F. Silva Leite, and R.C. Rodrigues. A two-step algorithm of smooth spline generation on Riemannian manifolds. J. Comput. Appl. Math., 194(2):177–191, 2006.
[7] S. M. LaValle. Planning Algorithms. Cambridge University Press, Cambridge, U.K., 2006. Available at http://planning.cs.uiuc.edu/.
[8] S. Lee and W. Tang. Complex homogeneous splines on the torus. Approximation Theory and its Applications, 5:31–42, 1989. 10.1007/BF02836067.
[9] C.S. Lin and P.R. Chang. Joint trajectories of mechanical manipulators for Cartesian path approximation. IEEE Trans. Syst. Man Cyber., 13(6):1094–1102, 1983.
[10] J. Oprea. Di↵erential geometry and its applications. MAA, 2007.
[11] F. Park and B. Ravani. B´ezier curves on Riemannian manifolds and Lie groups with kinematic
applications. ASME J. Mechan. Design, 117:36–40, 1995.
[12] R.C. Rodrigues, F. Silva Leite, and J. Jakubiak. A new geometric algorithm to generate
smooth interpolating curves on Riemannian manifolds. LMS J. Comput. Math., 8:251–266,
2005.
[13] J. Saranen and J. Anttila. The boundary element spline collocation for nonuniform meshes
on the torus. Numerische Mathematik, 106:289–302, 2007. 10.1007/s00211-006-0055-9.
[14] J. T. Schwartz, M. Sharir, and J. Hopcroft. Planning, Geometry, and Complexity of Robot
Motion. Ablex, Norwood, NJ, 1987.
[15] L. Sciavicco and B. Siciliano. Modelling and Control of Robot Manipulators. Springer-Verlag,
Berlin, 1996.