The partially wrapped normal distribution for SE(2) estimation

We introduce a novel probability distribution on the group of rigid motions SE(2) and we refer to this distribution as the partially wrapped normal distribution. Describing probabilities on SE(2) is of interest in a wide range of applications, for example, robotics, autonomous vehicles, or maritime navigation. We derive some important properties of this novel distribution and propose an estimation scheme for its parameters based on moment matching. Furthermore, we provide a qualitative comparison to a recently published approach based on the Bingham distribution, and show that there are complementary advantages and disadvantages of the two approaches.

[1]  Eric R. Ziegel,et al.  Statistical Theory and Data Analysis II , 1990 .

[2]  G. S. Watson,et al.  Equatorial distributions on a sphere , 1965 .

[3]  Nicholas I. Fisher,et al.  Statistical Analysis of Spherical Data. , 1987 .

[4]  J. Kent.  The Fisher‐Bingham Distribution on the Sphere , 1982 .

[5]  J. M. Selig.  Geometric Fundamentals of Robotics , 2004, Monographs in Computer Science.

[6]  Nicholas I. Fisher,et al.  Statistical Analysis of Spherical Data. , 1987 .

[7]  K. V. Mardia,et al.  Linear-Circular Correlation Coefficients and Rhythmometry , 1976 .

[8]  Christopher Bingham.  An Antipodally Symmetric Distribution on the Sphere , 1974 .

[9]  Richard A. Johnson,et al.  Some Angular-Linear Distributions and Related Regression Models , 1978 .

[10]  Gerhard Kurz,et al.  A new probability distribution for simultaneous representation of uncertain position and orientation , 2014, 17th International Conference on Information Fusion (FUSION).

[11]  R. Johnson,et al.  Measures and models for angular correlation and angular-linear correlation. [correlation of random variables] , 1976 .

[12]  Neil J. Gordon,et al.  A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking , 2002, IEEE Trans. Signal Process..

[13]  Kanti V. Mardia,et al.  A Model for Cylindrical Variables with Applications , 1978 .

[14]  Kanti V. Mardia,et al.  A multivariate von mises distribution with applications to bioinformatics , 2008 .

[15]  Harshinder Singh,et al.  Probabilistic model for two dependent circular variables , 2002 .

[16]  R. Fisher.  Dispersion on a sphere , 1953, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[17]  S. R. Jammalamadaka,et al.  Topics in Circular Statistics , 2001 .

[18]  Gerhard Kurz,et al.  Bivariate angular estimation under consideration of dependencies using directional statistics , 2014, 53rd IEEE Conference on Decision and Control.

[19]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[20]  Richard Kempter,et al.  Quantifying circular–linear associations: Hippocampal phase precession , 2012, Journal of Neuroscience Methods.

[21]  Gerhard Kurz,et al.  Nonlinear measurement update for estimation of angular systems based on circular distributions , 2014, 2014 American Control Conference.

[22]  Petar M. Djuric,et al.  Gaussian particle filtering , 2003, IEEE Trans. Signal Process..

[23]  Gerhard Kurz,et al.  Recursive nonlinear filtering for angular data based on circular distributions , 2013, 2013 American Control Conference.

[24]  Nicholas I. Fisher,et al.  Statistical Analysis of Circular Data , 1993 .

[25]  S. R. Jammalamadaka,et al.  Directional Statistics, I , 2011 .

[26]  Kaare Brandt Petersen,et al.  The Matrix Cookbook , 2006 .