Parts entropy and the principal kinematic formula

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. Sanderson quantified this with the concept of ldquoparts entropy,rdquo 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 paper 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 ldquoPrincipal Kinematic Formulardquo 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.

[1]  S. Glasauer.  A Generalization of Intersection Formulae of Integral Geometry , 1997 .

[2]  Euler Integration and Euler Multiplication , 2005 .

[3]  E. Teufel.  Integral geometry and projection formulas in spaces of constant curvature , 1986 .

[4]  Satyandra K. Gupta,et al.  Geometric algorithms for containment analysis of rotational parts , 2005, Comput. Aided Des..

[5]  Translative and kinematic integral formulae for support functions , 1995 .

[6]  W. Weil.  Integral Geometric Tools for Stochastic Geometry , 2007 .

[7]  T. Ohmoto.  AN ELEMENTARY REMARK ON THE INTEGRAL WITH RESPECT TO EULER CHARACTERISTICS OF PROJECTIVE HYPERPLANE SECTIONS : Dedicated to Professor Shoji Tsuboi on the occasion of his 60th birthday , 2003 .

[8]  Joseph A. Wolf.  Spaces of Constant Curvature , 1984 .

[9]  P. Mani-Levitska.  A simple proof of the kinematic formula , 1988 .

[10]  A. Bernig.  A Hadwiger-Type Theorem for the Special Unitary Group , 2008, 0801.1606.

[11]  R. E. Miles.  THE FUNDAMENTAL FORMULA OF BLASCHKE IN INTEGRAL GEOMETRY AND GEOMETRICAL PROBABILITY, AND ITS ITERATION, FOR DOMAINS WITH FIXED ORIENTATIONS , 1974 .

[12]  G. Boothroyd,et al.  Assembly Automation and Product Design , 1991 .

[13]  E. Czuber.  Geometrische Wahrscheinlichkeiten und Mittelwerte , 1884 .

[14]  V. Benes,et al.  Stochastic Geometry: Selected Topics , 2004 .

[15]  Yuliy Baryshnikov,et al.  Target Enumeration via Euler Characteristic Integrals , 2009, SIAM J. Appl. Math..

[16]  W. Weil,et al.  Stochastic and Integral Geometry , 2008 .

[17]  H. Hadwiger.  Vorlesungen über Inhalt, Oberfläche und Isoperimetrie , 1957 .

[18]  G. Solanes.  Integral geometry and the Gauss-Bonnet theorem in constant curvature spaces , 2005 .

[19]  Aoyong Wuhan.  A SUFFICIENT CONDITION FOR ONE CONVEX BODY CONTAINING ANOTHER , 1988 .

[20]  R. Adler,et al.  Random Fields and Geometry , 2007 .

[21]  Ambartzumian,et al.  Combinatorial Integral Geometry: With Applications to Mathematical Stereology , 1982 .

[22]  A. H. Redford,et al.  Mechanized assembly : fundamentals of parts feeding, orientation, and mechanized assembly , 1968 .

[23]  A short proof of principal kinematic formula and extensions , 1990 .

[24]  E. Harding,et al.  Stochastic Geometry: A Tribute to the Memory of Rollo Davidson , 1974 .

[25]  Arthur C. Sanderson,et al.  Parts entropy methods for robotic assembly system design , 1984, ICRA.

[26]  H. Führ.  Hausdorff–Young Inequalities for Group Extensions , 2006, Canadian Mathematical Bulletin.

[27]  R. Langevin,et al.  Polar Varieties and Integral Geometry , 1982 .

[28]  R. Ambartzumian.  Stochastic geometry from the standpoint of integral geometry , 1977, Advances in Applied Probability.

[29]  S. Glasauer.  Translative and kinematic integral formulae concerning the convex hull operation , 1998 .

[30]  Mixed curvature measures for sets of positive reach and a translative integral formula , 1995 .

[31]  Susan A. Murphy,et al.  Monographs on statistics and applied probability , 1990 .

[32]  R. Schneider.  Kinematic measures for sets of colliding convex Bodies , 1978 .

[33]  Adrian Baddeley,et al.  Stochastic Geometry: An Introduction and Reading-List , 1982 .

[34]  Jati K. Sengupta,et al.  Introduction to Information , 1993 .

[35]  P. Goodey,et al.  Intersection bodies and ellipsoids , 1995 .

[36]  A. Nijenhuis.  On Chern's kinematic formula in integral geometry , 1974 .

[37]  W. Weil,et al.  Translative and Kinematic Integral Formulae for Curvature Measures , 1986 .

[38]  Adrian Baddeley,et al.  Stereology for Statisticians , 2004 .

[39]  L. Santaló.  Integral geometry and geometric probability , 1976 .

[40]  T. Bonnesen,et al.  Theorie der Konvexen Körper , 1934 .

[41]  W. H. Young.  On the Multiplication of Successions of Fourier Constants , 1912 .

[42]  Yanxi Liu,et al.  Symmetry groups in analysis of assembly kinematics , 1991, Proceedings. 1991 IEEE International Conference on Robotics and Automation.

[43]  Sufficient conditions for one domain to contain another in a space of constant curvature , 1998 .

[44]  De-lin Ren,et al.  Topics in Integral Geometry , 1994 .

[45]  Matthew T. Mason,et al.  An exploration of sensorless manipulation , 1986, Proceedings. 1986 IEEE International Conference on Robotics and Automation.

[46]  Gregory S. Chirikjian,et al.  Entropy, symmetry, and the difficulty of self-replication , 2022, Artificial Life and Robotics.

[47]  J. Fournier.  Sharpness in Young’s inequality for convolution , 1977 .

[48]  D. Stoyan,et al.  Applied Stochastic Geometry: A Survey (Invited Paper) , 1979 .

[49]  H. Groemer.  On translative integral geometry , 1977 .

[50]  Sukhan Lee,et al.  Computer-Aided Mechanical Assembly Planning , 1991 .

[51]  Masud Mansuripur,et al.  Introduction to information theory , 1986 .

[52]  H. Hadwiger.  Altes und Neues über konvexe Körper , 1955 .

[53]  Daniel A. Klain,et al.  Introduction to Geometric Probability , 1997 .

[54]  Jiazu Zhou.  When can one domain enclose another in 3? , 1995, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[55]  A Translative Integral Formula for Absolute Curvature Measures , 2001 .

[56]  S. Chern.  On the Kinematic Formula in the Euclidean Space of N Dimensions , 1952 .

[57]  T. Shifrin.  The kinematic formula in complex integral geometry , 1981 .

[58]  François Baccelli,et al.  Stochastic geometry and architecture of communication networks , 1997, Telecommun. Syst..

[59]  Geoffrey Boothroyd.  Assembly Automation and Product Design, Second Edition (Manufacturing Engineering and Materials Processing) , 2005 .

[60]  Oleg Viro,et al.  Some integral calculus based on Euler characteristic , 1988 .

[61]  D. Stoyan,et al.  Stochastic Geometry and Its Applications , 1989 .

[62]  Shiing-Shen Chern,et al.  On the Kinematic Formula in Integral Geometry , 1966 .

[63]  Ralph Howard,et al.  The kinematic formula in Riemannian homogeneous spaces , 1993 .

[64]  H. Fédérer.  Some integralgeometric theorems , 1954 .

[65]  Franz E Schuster.  Convolutions and multiplier transformations of convex bodies , 2007, 1207.7252.

[66]  Gregory S. Chirikjian,et al.  Error propagation on the Euclidean group with applications to manipulator kinematics , 2006, IEEE Transactions on Robotics.

[67]  P. Goodey,et al.  Translative integral formulae for convex bodies , 1987 .

[68]  W. Blaschke.  Vorlesungen über Integralgeometrie , 1937 .

[69]  Jonathan E. Taylor.  A gaussian kinematic formula , 2006, math/0602545.

[70]  G. Chirikjian,et al.  Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups , 2000 .