Abstract
The problem of generating uniform deterministic samples over the rotation group, SO(3), is fundamental to many fields, such as computational structural biology, robotics, computer graphics, astrophysics. We present the best-known method to date for constructing incremental, deterministic grids on SO(3); it provides the: 1) lowest metric distortion for grid neighbor edges, 2) optimal dispersion-reduction with each additional sample, 3) explicit neighborhood structure, and 4) equivolumetric partition of SO(3) by the grid cells. We also demonstrate the use of the sequence on motion planning problems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Chirikjian, G.S., Kyatkin, A.B.: Engineering Applications of Noncommutative Harmonic Analysis. CRC Press, Boca Raton (2001)
Diaconis, P., Shahshahani, M.: The subgroup algorithm for generating uniform random variables. Prob. in Eng. and Info. Sci. 1, 15–32 (1987)
Fishman, G.F.: Monte Carlo: Concepts, Algorithms, and Applications. Springer, Berlin (1996)
Górski, K.M., Hivon, E., Banday, A.J., Wandelt, B.D., Hansen, F.K., Reinecke, M., Bartelmann, M.: HEALPix: a framework for high-resolution discretization and fast analysis of data distributed on the sphere. arXiv:astro-ph/0409513 622, 759–771 (April 2005)
Hardin, D.P., Saff, E.B.: Discretizing manifolds via minimum energy points. Notices of the American Mathematical Society 51(10), 1186–1194 (2004)
Kavraki, L.E., Svestka, P., Latombe, J., Overmars, M.H.: Probabilistic roadmaps for path planning in high-dimensional configuration spaces. IEEE Trans. Robot. & Autom. 12(4), 566–580 (1996)
Kuffner, J.: Effective sampling and distance metrics for 3D rigid body path planning. In: IEEE Int. Conf. Robot. & Autom. IEEE, Los Alamitos (2004)
Lindemann, S.R., LaValle, S.M.: Incremental Low-Discrepancy lattice methods for motion planning. In: IEEE Int’l Conf. on Robotics and Automation, pp. 2920–2927 (2003)
Lindemann, S.R., Yershova, A., LaValle, S.M.: Incremental grid sampling strategies in robotics. In: Workshop on the Algorithmic Foundations of Robotics (2004)
Mitchell, J.C.: Sampling rotation groups by successive orthogonal images. SIAM J. Sci. Comput. 30(1), 525–547 (2007)
Niederreiter, H.: Random Number Generation and Quasi-Monte-Carlo Methods. Society for Industrial and Applied Mathematics, Philadelphia (1992)
Ramamoorthy, S., Rajagopal, R., Ruan, Q., Wenzel, L.: Low-discrepancy curves and efficient coverage of space. In: Proc. of the Workshop on Algorithmic Foundations of Robotics (2006)
Rote, G., Tichy, R.F.: Spherical dispersion with applications to polygonal approximation of the curves. Anz. Österreich. Akad. Wiss. Math.-Natur, Kl. Abt. II 132, 3–10 (1995)
Rovira, J., Wonka, P., Castro, F., Sbert, M.: Point sampling with uniformly distributed lines. In: Point-Based Graphics, 2005. Eurographics/IEEE VGTC Symposium Proc. (June 2005)
Shoemake, K.: Animating rotation with quaternion curves. In: Proc. of the 12th Annual Conference on Computer Graphics and Interactive Techniques. ACM Press, New York (1985)
Shoemake, K.: Uniform random rotations. In: Kirk, D. (ed.) Graphics Gems III, pp. 124–132. Academic Press, London (1992)
Sternberg, M.J.E., Moont, G.: Modelling protein-protein and protein-DNA docking. In: Lengauer, T. (ed.) Bioinformatics – From Genomes to Drugs, pp. 361–404. Wiley-VCH, Weinheim (2002)
Sun, X., Chen, Z.: Spherical basis functions and uniform distribution of points on spheres. J. Approx. Theory 151(2), 186–207 (2008)
Wagner, G.: On a new method for constructing good point sets on spheres. Journal of Discrete and Computational Geometry 9(1), 119–129 (1993)
Wales, D., Doye, J.: Global optimization by Basin-Hopping and the lowest energy structures of Lennard-Jones clusters containing up to 110 atoms. J. Phys. Chem. AÂ 101 (1997)
Yan, A.K., Langmead, C.J., Donald, B.R.: A Probability-Based similarity measure for saupe alignment tensors with applications to residual dipolar couplings in NMR structural biology. Int. J. Robot. Res. 24(2-3), 162–182 (2005)
Yershova, A., LaValle, S.M.: Deterministic sampling methods for spheres and SO(3). In: Proc. IEEE International Conference on Robotics and Automation (2004)
Zyczkowski, K., Kus, M.: Random unitary matrices. J. Phys. A 27(12), 4235–4245 (1994)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Yershova, A., LaValle, S.M., Mitchell, J.C. (2009). Generating Uniform Incremental Grids on SO(3) Using the Hopf Fibration. In: Chirikjian, G.S., Choset, H., Morales, M., Murphey, T. (eds) Algorithmic Foundation of Robotics VIII. Springer Tracts in Advanced Robotics, vol 57. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00312-7_24
Download citation
DOI: https://doi.org/10.1007/978-3-642-00312-7_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00311-0
Online ISBN: 978-3-642-00312-7
eBook Packages: EngineeringEngineering (R0)