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Principal Kinematic Inequalities

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Advances in Robot Kinematics 2018 (ARK 2018)

Part of the book series: Springer Proceedings in Advanced Robotics ((SPAR,volume 8))

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Abstract

One hundred years ago, Blaschke and Poincaré derived exact closed-form formulas for integrals over SE(n) of Euler characteristics of intersections of n-dimensional bodies for \(n=2,3\). Three decades thereafter Chern extended these formulas to the n-dimensional case. These results form a core tool in the fields of convex geometry, integral geometry, geometric probability, and stochastic geometry. These results, often referred to as “Principal Kinematic Formulae,” are extended here to the case of integrals of set-indicator functions, resulting in inequalities for the case of non-convex bodies. These results are relevant to assessing the frequency of occurrence of collisions that occur in sample-based robot motion planning.

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Author information

Authors and Affiliations

  1. Johns Hopkins University, Baltimore, MD, 21218, USA

    Gregory S. Chirikjian

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  1. Gregory S. Chirikjian
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Correspondence to Gregory S. Chirikjian .

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Editors and Affiliations

  1. J. Stefan Institute, Ljubljana, Slovenia

    Jadran Lenarcic

  2. Department of Industrial Engineering, University of Bologna, Bologna, Italy

    Vincenzo Parenti-Castelli

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Chirikjian, G.S. (2019). Principal Kinematic Inequalities. In: Lenarcic, J., Parenti-Castelli, V. (eds) Advances in Robot Kinematics 2018. ARK 2018. Springer Proceedings in Advanced Robotics, vol 8. Springer, Cham. https://doi.org/10.1007/978-3-319-93188-3_3

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