Preprint - cyphynets

Fig. 9. ψ 1 using different thresholds t on magnitudes of eigenvectorcomponents. [1 st row left] t = 1 [1 st row right] t = 0.8. [2 nd rowleft] t = 0.7 [2 nd row right] t = 0.6 [3 rd row left] t = 0.5 [3 rd rowright] t = 0.3.changes sign right at the necks. This is shown in Figure 4for ψ 3 for three obstructions (bottom left) and ψ 4 for fourobstructions (bottom right).V. CONCLUSIONS AND FUTURE WORKIn this paper, we have presented a way to characterize theproperties of the configuration space using spectral analysison the PRM graph. It is evident that the spectral methodis a useful, easily computable and visualizable alternativeto direct geometrical methods of characterizing obstructionssuch as expansiveness. Eigenvalues and eigenvectors of thegraph Laplacian reveal important information about the sizeand number of narrow passages and have a direct relationshipwith expansiveness. Particularly, the smallest nonzeroeigenvalue detects the severity of configuration spacedistortion and the spectral gap helps identify the number ofobstructions. Thus far we have only demonstrated an empiricalrelationship albeit very strong one between spectra andmotion planning difficulties. Work in underway in makingconcrete analysis of these connections. Finally this workdemonstrates the utility of spectral graph theory and spectralgeometry methods in robot motion planning.REFERENCES[1] S. M. LaValle, Planning Algorithms, Cambridge University Press,2006.[2] Jean Claude Latombe, Robot Motion Planning, Kluwer AcademicPublishers, Boston, MA, 1991.[3] H. Choset, K.M. Lynch, S. Hutchinson, G. Kantor, W. Burgard,L.E. Kavraki, and S. Thrun, ”Principles of Robot Motion : Theory,Algorithms, and Implementations,” The MIT Press, 2005.[4] L.E. Kavraki, P. Svestka, J.C. Latombe, and M.H. 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