Nonholonomic Motion Planning Using Diffusion of Workspace Density Functions

2003 ◽  
Author(s):  
Yu Zhou ◽  
Gregory S. Chirikjian
Keyword(s):  
Planck Equation ◽  
Probability Density Functions ◽  
Time Dependent ◽  
Density Functions ◽  
Value Of Time ◽  
Nonholonomic Mobile Robots ◽  
Planning Algorithm ◽  
The Fourier Transform ◽  
Nonholonomic Motion Planning ◽  
Closed Form Approximation

This paper introduces a trajectory planning algorithm for nonholonomic mobile robots which operate in an environment with obstacles. An important feature of our approach is that the planning domain is the workspace of the mobile robot rather than its configuration space. The basic idea is to imagine the robot being subjected to Brownian motion forcing, and to generate evolving probability density functions (PDF) that describe all attainable positions and orientations of the robot at a given value of time. By planning a path that optimizes the value of this PDF at each instant in time, we generate a feasible trajectory. The PDF of robot pose can be constructed by solving the corresponding Fokker-Planck equation using the Fourier transform for SE(N). A closed-form approximation of the resulting time-dependent PDF is then used to plan a trajectory based on the observation that the evolution of this “workspace density” is a diffusion process. Examples are provided to illustrate the algorithm.

scholarly journals Time-Dependent Probability Density Functions and Attractor Structure in Self-Organised Shear Flows

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 613 ◽  
Author(s):  
Quentin Jacquet ◽  
Eun-jin Kim ◽  
Rainer Hollerbach
Keyword(s):  
Probability Density ◽  
Shear Flows ◽  
Numerical Solutions ◽  
Planck Equation ◽  
Probability Density Functions ◽  
Time Dependent ◽  
Parameter Study ◽  
Density Functions ◽  
Stochastic Forcing ◽  
Mean Values

We report the time-evolution of Probability Density Functions (PDFs) in a toy model of self-organised shear flows, where the formation of shear flows is induced by a finite memory time of a stochastic forcing, manifested by the emergence of a bimodal PDF with the two peaks representing non-zero mean values of a shear flow. Using theoretical analyses of limiting cases, as well as numerical solutions of the full Fokker–Planck equation, we present a thorough parameter study of PDFs for different values of the correlation time and amplitude of stochastic forcing. From time-dependent PDFs, we calculate the information length ( L ), which is the total number of statistically different states that a system passes through in time and utilise it to understand the information geometry associated with the formation of bimodal or unimodal PDFs. We identify the difference between the relaxation and build-up of the shear gradient in view of information change and discuss the total information length ( L ∞ = L ( t → ∞ ) ) which maps out the underlying attractor structures, highlighting a unique property of L ∞ which depends on the trajectory/history of a PDF’s evolution.

scholarly journals A Stochastic Two Species Competition Model: Nonequilibrium Fluctuation and Stability

2011 ◽  
Vol 2011 ◽  
pp. 1-7 ◽  
Author(s):  
G. P. Samanta
Keyword(s):  
Planck Equation ◽  
Competition Model ◽  
Probability Density Functions ◽  
Density Functions ◽  
Species Competition ◽  
Symmetric Competition ◽  
Competitive Species ◽  
The Stability ◽  
The Relationship ◽  
Nonequilibrium Fluctuation

The object of this paper is to study the stability behaviours of the deterministic and stochastic versions of a two-species symmetric competition model. The logistic parameters of the competitive species are perturbed by colored noises or Ornstein-Uhlenbeck processes due to random environment. The Fokker-Planck equation has been used to obtain probability density functions. Here, we have also discussed the relationship between stability behaviours of this model in a deterministic environment and the corresponding model in a stochastic environment.

scholarly journals Consistency issues in PDF methods

2015 ◽  
Vol 23 (3) ◽  
pp. 187-208
Author(s):  
N. Suciu ◽  
L. Schüler ◽  
S. Attinger ◽  
C. Vamoș ◽  
P. Knabner
Keyword(s):  
Contaminant Transport ◽  
Evolution Equations ◽  
Chemical Species ◽  
Planck Equation ◽  
Probability Density Functions ◽  
Random Environments ◽  
Density Functions ◽  
Ito Process ◽  
Concentration Statistics ◽  
The One

Abstract Concentrations of chemical species transported in random environments need to be statistically characterized by probability density functions (PDF). Solutions to evolution equations for the one-point one-time PDF are usually based on systems of computational particles described by Itô equations. We establish consistency conditions relating the concentration statistics to that of the Itô process and the solution of its associated Fokker-Planck equation to that of the PDF equation. In this frame, we use a recently proposed numerical method which approximates PDFs by particle densities obtained with a global random walk (GRW) algorithm. The GRW-PDF approach is illustrated for a problem of contaminant transport in groundwater.

scholarly journals Investigating Information Geometry in Classical and Quantum Systems through Information Length

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 574 ◽  
Author(s):  
Eun-jin Kim
Keyword(s):  
Stochastic Processes ◽  
Information Geometry ◽  
Quantum Systems ◽  
Probability Density Functions ◽  
Time Dependent ◽  
Density Functions ◽  
Mathematical Tool ◽  
Initial Condition ◽  
Disciplinary Boundaries ◽  
Information Change

Stochastic processes are ubiquitous in nature and laboratories, and play a major role across traditional disciplinary boundaries. These stochastic processes are described by different variables and are thus very system-specific. In order to elucidate underlying principles governing different phenomena, it is extremely valuable to utilise a mathematical tool that is not specific to a particular system. We provide such a tool based on information geometry by quantifying the similarity and disparity between Probability Density Functions (PDFs) by a metric such that the distance between two PDFs increases with the disparity between them. Specifically, we invoke the information length L(t) to quantify information change associated with a time-dependent PDF that depends on time. L(t) is uniquely defined as a function of time for a given initial condition. We demonstrate the utility of L(t) in understanding information change and attractor structure in classical and quantum systems.

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