Probability-Based Optimal Path Planning for Two-Wheeled Mobile Robots

2015 ◽  
Author(s):  
Jaeyeon Lee ◽  
Wooram Park
Keyword(s):  
Probability Density Function ◽  
Path Planning ◽  
Mobile Robots ◽  
Probability Density ◽  
Density Function ◽  
Deterministic Model ◽  
Optimal Path ◽  
Planck Equation ◽  
Practical Implementation ◽  
Wheeled Mobile Robots

Most dynamic systems show uncertainty in their behavior. Therefore, a deterministic model is not sufficient to predict the stochastic behavior of such systems. Alternatively, a stochastic model can be used for better analysis and simulation. By numerically integrating the stochastic differential equation or solving the Fokker-Planck equation, we can obtain a probability density function of the motion of the system. Based on this probability density function, the path-of-probability (POP) method for path planning has been developed and verified in simulation. However, there are rooms for more improvements and its practical implementation has not been performed yet. This paper concerns formulation, simulation and practical implementation of the path-of-probability for two-wheeled mobile robots. In this framework, we define a new cost function which measures the averaged targeting error using root-mean-square (RMS), and iteratively minimize it to find an optimal path with the lowest targeting error. The proposed algorithm is implemented and tested with a two-wheeled mobile robot for performance verification.

Deterministic epidemic waves

1976 ◽  
Vol 80 (2) ◽  
pp. 315-330 ◽  
Author(s):  
C. Atkinson ◽  
G. E. H. Reuter
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Deterministic Model ◽  
Removal Rate ◽  
Uniform Density ◽  
Space And Time ◽  
Time Variables ◽  
Image Position

In the well-known deterministic model for the spread of an epidemic, one considers a population of uniform density along a line and divides the population into three classes: susceptible but uninfected, infected and infectious, infected but removed. If we denote space and time variables by s, t and let x(s, t), y(s, t), z(s, t) be the proportions of the population at (s, t) in these three classes, then x + y + z = 1 and we suppose thatHere Ῡ(s, t) denotes a space average ∫ y(s + σ) p(σ) dσ, where p is a probability density function; b is the removal rate; the scale of t has been adjusted to remove a constant that would otherwise occur in (1).

The fBm-driven Ornstein-Uhlenbeck process: Probability density function and anomalous diffusion

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Caibin Zeng ◽  
YangQuan Chen ◽  
Qigui Yang
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Anomalous Diffusion ◽  
Planck Equation ◽  
Viscosity Coefficient ◽  
Infinite Variance ◽  
Nonlinear Stochastic Differential Equations ◽  
The Fourier Transform ◽  
Fractional Ito Formula

AbstractThis paper deals with the Ornstein-Uhlenbeck (O-U) process driven by the fractional Brownian motion (fBm). Based on the fractional Itô formula, we present the corresponding fBm-driven Fokker-Planck equation for the nonlinear stochastic differential equations driven by an fBm. We then apply it to establish the evolution of the probability density function (PDF) of the fBm-driven O-U process. We further obtain the closed form of such PDF by combining the Fourier transform and the method of characteristics. Interestingly, the obtained PDF has an infinite variance which is significantly different from the classical O-U process. We reveal that the fBm-driven O-U process can describe the heavy-tailedness or anomalous diffusion. Moreover, the speed of the sub-diffusion is inversely proportional to the viscosity coefficient, while is proportional to the Hurst parameter. Finally, we carry out numerical simulations to verify the above findings.

Nonstationary Response Envelope Probability Densities of Nonlinear Oscillators

2006 ◽  
Vol 74 (2) ◽  
pp. 315-324 ◽  
Author(s):  
P. D. Spanos ◽  
A. Sofi ◽  
M. Di Paola
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Planck Equation ◽  
Nonlinear Oscillators ◽  
Basis Functions ◽  
Time Dependent ◽  
Fokker Planck Equation ◽  
Response Envelope ◽  
Fokker Planck

The nonstationary random response of a class of lightly damped nonlinear oscillators subjected to Gaussian white noise is considered. An approximate analytical method for determining the response envelope statistics is presented. Within the framework of stochastic averaging, the procedure relies on the Markovian modeling of the response envelope process through the definition of an equivalent linear system with response-dependent parameters. An approximate solution of the associated Fokker-Planck equation is derived by resorting to a Galerkin scheme. Specifically, the nonstationary probability density function of the response envelope is expressed as the sum of a time-dependent Rayleigh distribution and of a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. These functions are the eigenfunctions of the boundary-value problem associated with the Fokker-Planck equation governing the evolution of the probability density function of the response envelope of a linear oscillator. The selected basis functions possess some notable properties that yield substantial computational advantages. Applications to the Van der Pol and Duffing oscillators are presented. Appropriate comparisons to the data obtained by digital simulation show that the method, being nonperturbative in nature, yields reliable results even for large values of the nonlinearity parameter.

scholarly journals Investigation of Phase-Locked Loop Statistics via Numerical Implementation of the Fokker–Planck Equation

2020 ◽  
Vol 10 (7) ◽  
pp. 2625
Author(s):  
Dah-Jing Jwo
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Additive White Gaussian Noise ◽  
Phase Error ◽  
Planck Equation ◽  
Phase Locked Loop ◽  
Comprehensive Treatment ◽  
Error Statistics ◽  
Fokker Planck

The goal of this paper is to explore the effect of various parameters on the information geometric structure of the phase-locked loop (PLL) statistics, both transient and stationary. Comprehensive treatment on the behavior of PLL statistics will be given. The behavior of the phase-error statistics of the first-order PLL, in the presence of additive white Gaussian noise (WGN) is investigated through solving the differential equations known as the Fokker–Planck (FP) equation using the implicit Crank–Nicolson finite-difference method. The PLL is one of the most commonly used circuits in electrical engineering. A full knowledge of probability density functions (PDFs) of the phase-error statistics becomes essential in understanding the PLLs. Several illustrative examples are presented to provide profound insights on understanding the PLL statistics both qualitatively and quantitatively. Results covered include the transient and stationary statistics for the nonmodulo-2π probability density function, modulo-2π probability density function, and cycle slipping density function, of the phase error. Various numerical settings of PLL parameters are involved, including the detuning factor and signal-to-noise ratio (SNR). The results presented in this paper elucidate the link between various parameters and the information geometry of the phase-error statistics and form a basis for future investigation on PLL designs.

scholarly journals The positive occupation time of Brownian motion with two-valued drift and asymptotic dynamics of sliding motion with noise

2014 ◽  
Vol 14 (04) ◽  
pp. 1450010 ◽  
Author(s):  
D. J. W. Simpson ◽  
R. Kuske
Keyword(s):  
Brownian Motion ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Planck Equation ◽  
Occupation Time ◽  
Dimensional System ◽  
Sliding Motion ◽  
Long Time ◽  
Long Time Asymptotics

We derive the probability density function of the positive occupation time of one-dimensional Brownian motion with two-valued drift. Long time asymptotics of the density are also computed. We use the result to describe the transitional probability density function of a general N-dimensional system of stochastic differential equations representing stochastically perturbed sliding motion of a discontinuous, piecewise-smooth vector field on short time frames. A description of the density at larger times is obtained via an asymptotic expansion of the Fokker-Planck equation.

Random Vibrations of Plates With Large Amplitudes

1965 ◽  
Vol 32 (3) ◽  
pp. 547-552 ◽  
Author(s):  
R. E. Herbert
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Joint Probability ◽  
Planck Equation ◽  
Middle Surface ◽  
Arbitrary Boundary ◽  
Surface Displacements ◽  
Non Gaussian ◽  
Joint Probability Density

The theory of the Markoff process and the associated Fokker-Planck equation is used to investigate the large vibrations of beams and plates with arbitrary boundary conditions subjected to while-noise excitation. An expression for the joint probability-density function of the first N-coefficients of series expansions of the middle surface displacements is obtained. Detailed calculations presented for simply supported beams and plates show that the probability-density function of the modal amplitudes is non-Gaussian and statistically dependent. Numerical computations for the plate indicate a significant reduction of the mean-squared displacement for values of the parameters well inside the range of practical considerations. Furthermore, for the square plate, the percent reduction is greatest.

Sign in / Sign up

Export Citation Format

Share Document