Parts Entropy and the Principal Kinematic Formula

  • Gregory S. Chirikjian
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Automated (robotic) assembly systems that are able to function in the presence of uncertainties in the positions and orientations of feed parts are, by definition, more robust than those that are not able to do so. This can be quantified with the concept of “parts entropy,” which is a statistical measure of the ensemble of all possible positions and orientations of a single part confined to move in a finite domain. In this chapter the concept of parts entropy is extended to the case of multiple interacting parts. Various issues associated with computing the entropy of ensembles of configurations of parts with excluded-volume constraints are explored. The rapid computation of excluded-volume effects using the “principal kinematic formula” from the field of Integral Geometry is illustrated as a way to potentially avoid the massive computations associated with brute-force calculation of parts entropy when many interacting parts are present.


Convex Body Constant Curvature Euler Characteristic Integral Geometry Assembly Planning 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler, R., Taylor, J., Random Fields and Geometry, Springer, New York, 2007.Google Scholar
  2. 2.
    Ambartzumian, R.V., Combinatorial Integral Geometry with Applications to Mathematical Stereology, John Wiley and Sons, Somerset, NJ, 1982.Google Scholar
  3. 3.
    Ambartzumian, R.V., “Stochastic geometry from the standpoint of integral geometry,” Adv. Appl. Probab., 9(4), pp. 792–823, 1977.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Baccelli, F., Klein, M., Lebourges, M., Zuyev, S., “Stochastic geometry and architecture of communication networks,” Telecommun. Syst., 7, pp. 209–227, 1997.CrossRefGoogle Scholar
  5. 5.
    Baddeley, A., “Stochastic geometry: An introduction and reading-list,” Int. Statist. Rev. / Rev. Int. Statist., 50(2), pp. 179–193, 1982.MathSciNetMATHGoogle Scholar
  6. 6.
    Baddeley, A.J., Jensen, E.B.V., Stereology for Statisticians, Monographs on Statistics and Applied Probability Vol. 103. Chapman & Hall/CRC, Boca Raton, FL, 2005.Google Scholar
  7. 7.
    Baryshnikov, Y., Ghrist, R., “Target enumeration in sensor networks via integration with respect to Euler characteristic,” SIAM J. Appl. Math. 70, pp. 825–844, 2009.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Beneˇs, V., Rataj, J., Stochastic Geometry: Selected Topics, Kluwer Academic, Boston, 2004.Google Scholar
  9. 9.
    Bernig, A., “A Hadwiger-type theorem for the special unitary group,” Geom. Funct. Anal., 19, pp. 356–372, 2009 (also arXiv:0801.1606v4, 2008).Google Scholar
  10. 10.
    Bernig, A., Fu, J.H.G., “Hermitian integral geometry,” Ann. of Math., 173, pp. 907–945, 2011 (also arXiv:0801.0711v9, 2010).Google Scholar
  11. 11.
    Blaschke, W., “Einige Bemerkungen ¨uber Kurven und Fl¨achen konstanter Breite,” Ber. Kgl. S¨achs. Akad. Wiss. Leipzig, 67, pp. 290–297, 1915.Google Scholar
  12. 12.
    Blaschke, W., Vorlesungen ¨uber Integralgeometrie, Deutscher Verlag der Wissenschaften, Berlin, 1955.Google Scholar
  13. 13.
    Bonnesen, T., Fenchel, W., Theorie der Konvexen K¨orper, Springer Verlag, Heidelberg, 1934.Google Scholar
  14. 14.
    Boothroyd G., Assembly Automation and Product Design, 2nd ed., CRC Press, Boca Raton, FL, 2005.Google Scholar
  15. 15.
    Boothroyd, G., Redford, A.H., Mechanized Assembly: Fundamentals of Parts Feeding, Orientation, and Mechanized Assembly, McGraw-Hill, London, 1968.Google Scholar
  16. 16.
    Br¨ocker, L., “Euler integration and Euler multiplication,” Adv. Geom., 5(1), pp. 145–169, 2005.Google Scholar
  17. 17.
    Brothers, J.E., “Integral geometry in homogeneous spaces,” Trans. Am. Math. Soc., 124, pp. 408–517, 1966.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Buffon, G.L.L., “Comte de: Essai d’Arithm´etique Morale,” In: Histoire naturelle, g´en´erale et particuli`ere, Suppl´ement 4, pp. 46–123. Imprimerie Royale, Paris, 1777.Google Scholar
  19. 19.
    Chen, C.-S., “ On the kinematic formula of square of mean curvature,” Indiana Univ. Math. J., 22, pp. 1163–1169, 1972–3.Google Scholar
  20. 20.
    Chern, S.-S., “On the kinematic formula in the Euclidean space of N dimensions,” Am. J. Math., 74(1), pp. 227–236, 1952.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Chern, S.-S., “On the kinematic formula in integral geometry,” J. Math. Mech., 16(1), pp. 101–118, 1966.MathSciNetMATHGoogle Scholar
  22. 22.
    Chirikjian, G.S., “Parts entropy, symmetry, and the difficulty of self-replication,” Proceedings of the ASME Dynamic Systems and Control Conference, Ann Arbor, Michigan, October 20–22, 2008.Google Scholar
  23. 23.
    Chirikjian, G.S., “Parts Entropy and the Principal Kinematic Formula,” Proceedings of the IEEE Conference on Automation Science and Engineering, pp. 864–869, Washington, DC, August 23–26, 2008.Google Scholar
  24. 24.
    Chirikjian, G.S., “Modeling loop entropy,” Methods Enzymol, C, 487, pp. 101–130, 2011.Google Scholar
  25. 25.
    Crofton, M.W., “Sur quelques th´eor`emes de calcul int´egral,” C.R. Acad. Sci. Paris, 68, pp. 1469–1470, 1868.Google Scholar
  26. 26.
    Crofton, M.W., “Probability.” In: Encyclopedia Britannica, 9th ed., 19, pp. 768–788. Cambridge University Press, Cambridge, 1885.Google Scholar
  27. 27.
    Czuber, E., Geometrische Wahrscheinlichkeiten und Mittelwerte, B. G. Teubner, Leipzig, 1884 (reprinted in 2010 by BiblioBazaar/Nabu Press, Charleston, South Carolina, 2010).Google Scholar
  28. 28.
    de Mello, L.S.H., Lee, S., eds., Computer-Aided Mechanical Assembly Planning, Kluwer, Boston, 1991.Google Scholar
  29. 29.
    Erdmann, M.A., Mason, M.T., “An exploration of sensorless manipulation”, IEEE J. Robot. Autom., 4(4), pp. 369–379, 1988.CrossRefGoogle Scholar
  30. 30.
    Federer, H., “Some integralgeometric theorems,” Trans. Am. Math. Soc., 72(2), pp. 238– 261, 1954.MathSciNetCrossRefGoogle Scholar
  31. 31.
    Fournier, J.J.F., “Sharpness in Young’s inequality for convolution,” Pacific J. Math., 72(2), pp. 383–397, 1977.MathSciNetMATHGoogle Scholar
  32. 32.
    Fu, J.H.G., “Kinematic formulas in integral geometry,” Indiana Univ. Math. J., 39(4), pp. 1115–1154, 1990.MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Fu, J.H.G., “The two faces of Blaschkean integral geometry,” Internet notes, August 22, 2008, Scholar
  34. 34.
    F¨uhr, H., “Hausdorff–Young inequalities for group extensions,” Can. Math. Bull., 49(4), pp. 549–559, 2006.Google Scholar
  35. 35.
    Glasauer, S., “A generalization of intersection formulae of integral geometry,” Geom. Dedicata, 68, pp. 101–121, 1997.MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Glasauer, S., “Translative and kinematic integral formulae concerning the convex hull operation,” Math. Z., 229, pp. 493–518, 1998.MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Goodey, P.,Weil, W., “Translative integral formulae for convex bodies,” Aequationes Mathematicae, 34, pp. 64–77, 1987.MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Goodey, P., Weil, W., “Intersection bodies and ellipsoids,” Mathematika, 42, pp. 295–304, 1995.MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Groemer, H., “On translative integral geometry,” Arch. Math., 29, pp. 324–330, 1977.MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Hadwiger, H., Vorlesungen ¨uber Inhalt, Oberfl¨ache und Isoperimetrie., Springer-Verlag, Berlin, 1957.Google Scholar
  41. 41.
    Hadwiger, H., Altes und Neues ¨uber Konvexe K¨orper, Birkh¨auser-Verlag, Basel, 1955.Google Scholar
  42. 42.
    Harding, E.F., Kendall, D.G., Stochastic Geometry: A Tribute to the Memory of Rollo Davidson, John Wiley and Sons, London, 1974.Google Scholar
  43. 43.
    Howard, R., “The kinematic formula in Riemannian homogeneous spaces,” Mem. Am. Math. Soc., 106(509), pp. 1–69, 1993.Google Scholar
  44. 44.
    Karnik, M., Gupta, S.K., Magrab, E.B., “Geometric algorithms for containment analysis of rotational parts,” Computer-Aided Design, 37(2), pp. 213–230, 2005.CrossRefGoogle Scholar
  45. 45.
    Kendall, M.G., Moran, P.A.P., Geometrical Probability, Griffin’s Statistical Monographs, London, 1963.Google Scholar
  46. 46.
    Klain, D.A., Rota, G.-C., Introduction to Geometric Probability, Cambridge University Press, Cambridge, 1997.Google Scholar
  47. 47.
    Langevin, R., Integral Geometry from Buffon to Geometers of Today, Internet notes, 2009, 03 introdintegral.pdf.Google Scholar
  48. 48.
    Langevin, R., Shifrin, T., “Polar varieties and integral geometry,” Am. J. Math., 104(3), pp. 553–605, 1982.MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Liu, Y., Popplestone, R.J., “Symmetry Groups in Analysis of Assembly Kinematics,” ICRA 1991, pp. 572–577, Sacramento, CA, April 1991.Google Scholar
  50. 50.
    Mani-Levitska, P., “A simple proof of the kinematic formula,” Monatsch. Math., 105, pp. 279–285, 1988.MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Miles, R. E. “The fundamental formula of Blaschke in integral geometry and geometrical probability, and its iteration, for domains with fixed orientations,” Austral. J. Statist., 16, pp. 111–118, 1974.MathSciNetCrossRefMATHGoogle Scholar
  52. 52.
    Nijenhuis, A., “On Chern’s kinematic formula in integral geometry,” J. Diff. Geom., 9, pp. 475–482, 1974.MathSciNetMATHGoogle Scholar
  53. 53.
    Ohmoto, T., “An elementary remark on the integral with respect to Euler characteristics of projective hyperplane sections,” Rep. Fac. Sci. Kagoshima Univ., 36, pp. 37–41, 2003.MathSciNetMATHGoogle Scholar
  54. 54.
    Poincar´e, H., Calcul de Probabilit´es, 2nd ed., Gauthier-Villars, Imprimeur-Libraire, Paris, 1912. (reprinted by BiblioLife in 2009).Google Scholar
  55. 55.
    P´olya, G., “ ¨Uber geometrische Wahrscheinlichkeiten,” S.-B. Akad. Wiss. Wien, 126, pp. 319–328, 1917.Google Scholar
  56. 56.
    Rataj, J., “A translative integral formula for absolute curvature measures,” Geom. Dedicata, 84, pp. 245–252, 2001.MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Rataj, J., Z¨ahle, M., “Mixed curvature measures for sets of positive reach and a translative integral formula,” Geom. Dedicata, 57, pp. 259–283, 1995.Google Scholar
  58. 58.
    Ren, D.-L., Topics in Integral Geometry, World Scientific Publishing, Singapore, 1994.Google Scholar
  59. 59.
    Rother, W., Z¨ahle, M., “A short proof of the principal kinematic formula and extensions,” Trans. Am. Math. Soc., 321, pp. 547–558, 1990.Google Scholar
  60. 60.
    Sanderson, A.C., “Parts entropy methods for robotic assembly system design,” Proceedings of the 1984 IEEE International Conference on Robotics and Automation (ICRA ’84), Vol. 1, pp. 600–608, March 1984.Google Scholar
  61. 61.
    Santal´o, L., Integral Geometry and Geometric Probability, Cambridge University Press, Cambridge, 2004 (originally published in 1976 by Addison-Wesley).Google Scholar
  62. 62.
    Schneider, R., “Kinematic measures for sets of colliding convex bodies,” Mathematika 25, pp. 1–12, 1978.MathSciNetCrossRefGoogle Scholar
  63. 63.
    Schneider, R., Weil, W., “Translative and kinematic integral formulas for curvature measures,” Math. Nachr. 129, pp. 67–80, 1986.MathSciNetCrossRefMATHGoogle Scholar
  64. 64.
    Schneider, R., Weil, W., Stochastic and Integral Geometry, Springer-Verlag, Berlin, 2008.Google Scholar
  65. 65.
    Schneider, R., “Integral geometric tools for stochastic geometry,” in Stochastic Geometry, A. Baddeley, I. B´ar´any, R. Schneider, W. Weil, eds., pp. 119–184, Springer, Berlin, 2007.Google Scholar
  66. 66.
    Shifrin, T., “The kinematic formula in complex integral geometry,” Trans. Am. Math. Soc., 264, pp. 255–293, 1981.MathSciNetCrossRefMATHGoogle Scholar
  67. 67.
    Schuster, F.E., “Convolutions and multiplier transformations of convex bodies,” Trans. Am. Math. Soc., 359(11), pp. 5567–5591, 2007.MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    Slavski˘i, V.V., “On an integral geometry relation in surface theory,” Siberian Math. J., 13(3), pp. 645–658, 1972.Google Scholar
  69. 69.
    Solanes, G., “Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces,” Trans. Am. Math. Soc., 358(3), pp. 1105–1115, 2006.MathSciNetCrossRefMATHGoogle Scholar
  70. 70.
    Solomon, H.,Geometric Probability, SIAM, Philadelphia, 1978.Google Scholar
  71. 71.
    Stoyan, D., Kendall, W.S., Mecke, J., Stochastic Geometry and its Applications, 2nd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley and Sons, Chichester, UK, 1995.Google Scholar
  72. 72.
    Stoyan, D., “Applied stochastic geometry: A survey,” Biomed. J., 21, pp. 693–715, 1979.MathSciNetMATHGoogle Scholar
  73. 73.
    Taylor, J.E., “A Gaussian kinematic formula,” Ann. Probab., 34(1), pp. 122–158, 2006.MathSciNetCrossRefMATHGoogle Scholar
  74. 74.
    Teufel, V.E., “Integral geometry and projection formulas in spaces of constant curvature,” Abh. Math. Sem. Univ. Hamburg, 56, pp. 221–232, 1986.MathSciNetCrossRefMATHGoogle Scholar
  75. 75.
    Viro, O., “Some integral calculus based on Euler characteristic,” Lecture Notes in Mathematics, Vol. 1346, pp. 127–138, Springer-Verlag, Berlin, 1988.Google Scholar
  76. 76.
    Wang, Y., Chirikjian, G.S., “Error propagation on the Euclidean group with applications to manipulator kinematics,” IEEE Trans. Robot., 22(4), pp. 591–602, 2006.CrossRefGoogle Scholar
  77. 77.
    Weil, W., “Translative integral geometry,” in Geobild ’89, A. H¨ubler et al., eds., pp. 75–86, Akademie-Verlag, Berlin, 1989.Google Scholar
  78. 78.
    Weil, W., “Translative and kinematic integral formulae for support functions,” Geom. Dedicata, 57, pp. 91–103, 1995.MathSciNetCrossRefMATHGoogle Scholar
  79. 79.
    Whitney, D.E., Mechanical Assemblies, Oxford University Press, New York, 2004.Google Scholar
  80. 80.
    Wolf, J.A., Spaces of Constant Curvature, Publish or Perish Press, Berkeley, CA, 1977.Google Scholar
  81. 81.
    Young,W.H. “On the multiplication of successions of Fourier constants,” Proc. Soc. London A, 87, pp. 331–339, 1912.CrossRefMATHGoogle Scholar
  82. 82.
    Zhang, G., “A sufficient condition for one convex body containing another,” Chin. Ann. Math., 9B(4), pp. 447–451, 1988.Google Scholar
  83. 83.
    Zhou, J., “A kinematic formula and analogues of Hadwiger’s theorem in space,” Contemporary Mathematics, Vol. 140, pp. 159–167, American Mathematical Society, 1992.Google Scholar
  84. 84.
    Zhou, J., “When can one domain enclose another in R3?,” J. Austral. Math. Soc. A, 59, pp. 266–272, 1995.CrossRefMATHGoogle Scholar
  85. 85.
    Zhou, J., “Sufficient conditions for one domain to contain another in a space of constant curvature,” Proc. AMS, 126(9), pp. 2797–2803, 1998.CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringThe Johns Hopkins UniversityBaltimoreUSA

Personalised recommendations