Double-coset decomposition for Euclidean motion group

Matlab implementation for quantizing Euclideam motions using double-coset decomposition

Quantizing Euclidean motions via double-coset decomposition

Matlab implementation for quantizing Euclideam motions using double-coset decomposition

Authors:

• Christian Wülker: Philips Research Hamburg
• Ruan Sipu (repository maintainer): National University of Singapore, [e-mail]
• Greogory S. Chirikjian: National University of Singapore, [e-mail]

Introduction

Concepts from mathematical crystallography and group theory are used here to quantize the group of rigid-body motions, resulting in a "motion alphabet" with which to express robot motion primitives. From these primitives it is possible to develop a dictionary of physical actions. Equipped with an alphabet of the sort developed here, intelligent actions of robots in the world can be approximated with finite sequences of characters, thereby forming the foundation of a language in which to articulate robot motion. In particular, we use the discrete handedness-preserving symmetries of macromolecular crystals (known in mathematical crystallography as Sohncke space groups) to form a coarse discretization of the space SE(3) of rigid-body motions. This discretization is made finer by subdividing using the concept of double-coset decomposition. More specifically, a very efficient, equi-volumetric quantization of spatial motion can be defined using the group-theoretic concept of a double-coset decomposition of the form Γ\SE(3)/Δ, where Γ is a Sohncke space group and Δ is a finite group of rotational symmetries such as those of the icosahedron. The resulting discrete alphabet is based on a very uniform sampling of SE(3) and is a tool for describing the continuous trajectories of robots and humans. An efficient coarse-to-fine search algorithm is presented to round off any motion sampled from the continuous group of motions to the nearest element of our alphabet. It is shown that our alphabet and this efficient rounding algorithm can be used as a geometric data structure to accelerate the performance of other sampling schemes designed for desirable dispersion or discrepancy properties. Moreover, the general "signals to symbols" problem in artificial intelligence is cast in this framework for robots moving continuously in the world.

Test Scripts

• alphaber_conceptual_plot.m Conceptual plot for alphabet and round-off for continuous Euclidean motions, reproduction of Figure 4
• decomp_SE2_p1.m, decomp_SE2_p4.m Single coset decomposition of SE(2) by p1, p4 symmetry groups, reproduction of Figure 7
• motion_example.m Example of discretization of SE(2) motion sequence, Equation (46)
• voronoi_2D_p1.m, voronoi_2D_p2.m, voronoi_2D_p3.m, voronoi_2D_p4.m, voronoi_2D_p6.m Center Voronoi-like cell of Fundamental domain of SE(2) divided by p1, p2, p3, p4, p6 symmetry groups, reproduction of Figure 5
• voronoi_2D_p4_c5_double_coset.m Center Voronoi-like cell of Fundamental domain of SE(2) divided by p4 and C5 groups, reproduction of Figure 6
• voronoi_icosahedron_group.m voronoi_octahedron_group.m voronoi_tetrahedron_group.m Center Voronoi cell of Fundamental domain of Γ\SO(3) where Γ is icosahedral, octahedral, tetrahedral rotational symmetry groups respectively, reproduction of Figure 1
• voronoi_ico_ico_double_coset.m voronoi_ico_octa_double_coset.m voronoi_ico_tetra_double_coset.m Center Voronoi cell of Fundamental domain of Γ\SO(3)/Δ where Γ is icosahedral group and Δ is icosahedral, octahedral, tetrahedral groups respectively, reproduction of Figure 2
• non_voronoi_ico_ico_double_coset.m Fundamental domain of H\SO(3)/H where H is icosahedral group, reproduction of Figure 3
• sampling_SO3/comp_connectivity.m sampling_SO3/comp_consistency.m sampling_SO3/comp_density.m sampling_SO3/comp_dispersion sampling_SO3/comp_uniformity Performance comparisons for SO(3) sampling using our double-coset decomposition, Euler angles (ZYZ) and Hopf fibration methods: number of connected neighbors, consistency, density of neighborhood, dispersion, uniformity, results in Section 8.1.1
• sampling_SO3/comp_query_accuracy.m sampling_SO3/comp_query_time.m Benchmarks for SO(3) sampling using our double-coset decomposition, Euler angles (ZYZ) and Hopf fibration methods: Nearest neighbor search accuracy, running time, results in Section 8.1.2 (including reproduction of Figure 8)
• sampling_SO3/demo_fast_serach.m sampling_SO3/demo_fast_search_hk.m Demonstration of fast hybrid nearest neighbor search algorithm via our double-coset decomposition method: using double cosets H\SO(3)/H, H\SO(3)/K where H is icosahedral group and K is defined in Eq. (47), results in Section 8.2

Paper

Please cite the associated paper as follows:

• Wülker, C., Ruan, S. and Chirikjian, G.S., 2019. Quantizing Euclidean motions via double-coset decomposition. Research, 2019.

@article{wulker2019quantizing,
  title={Quantizing Euclidean motions via double-coset decomposition},
  author={W{\"u}lker, Christian and Ruan, Sipu and Chirikjian, Gregory S},
  journal={Research},
  volume={2019},
  year={2019},
  publisher={Science Partner Journal}
}