Discretely Actuated Manipulator Workspace Generation by Closed Form Convolution

1998 ◽  
Vol 120 (2) ◽  
pp. 245-251 ◽  
Author(s):  
Imme Ebert-Uphoff ◽  
Gregory S. Chirikjian
Keyword(s):  
Closed Form ◽  
Density Function ◽  
Unit Volume ◽  
Density Functions ◽  
Brute Force ◽  
Euclidean Group ◽  
Planar Case ◽  
Continuous Density ◽  
Group 3 ◽  
Special Euclidean Group

We determine workspaces of discretely actuated manipulators using convolution of real-valued functions on the Special Euclidean Group. Each workspace is described in terms of a density function that provides the number of reachable frames inside any unit volume of the workspace. A manipulator consisting of n discrete actuators each with K states can reach Kn frames in space. Given this exponential growth, brute force representation of discrete manipulator workspaces is not feasible in the highly actuated case. However, if the manipulator is of macroscopically-serial architecture, the workspace can be generated by the following procedure: (1) partition the manipulator into B kinematically independent segments; (2) approximate the workspace of each segment as a continuous density function on a compact subset of the Special Euclidean Group; (3) approximate the whole workspace as a B-fold convolution of these densities. We represent density functions as finite Hermite-Fourier Series and show for the planar case how the B-fold convolution can be performed in closed form. If all segments are identical only O(log B) convolutions are necessary.

Discretely Actuated Manipulator Workspace Generation by Closed-Form Convolution

1996 ◽  
Author(s):  
Imme Ebert-Uphoff ◽  
Gregory S. Chirikjian
Keyword(s):  
Closed Form ◽  
Density Function ◽  
Computation Time ◽  
Density Functions ◽  
Brute Force ◽  
Euclidean Group ◽  
Planar Case ◽  
Group 3 ◽  
Special Euclidean Group

Abstract We discuss the determination of workspaces of discretely actuated manipulators using convolution of real-valued functions on the Special Euclidean Group. Each workspace is described in terms of a density function that provides for any unit taskspace volume of the workspace the number of reachable frames therein. A manipulator consisting of n discrete actuators each with K states can reach Kn frames in space. Given this exponential growth, brute force representation of discrete manipulator workspaces is not feasible in the highly actuated case. However, if the manipulator is of macroscopically-serial architecture, the workspace can be generated by the following procedure: (1) partition the manipulator into segments; (2) approximate the workspace of each segment as a continuous density function on a compact subset of the Special Euclidean Group; (3) approximate the whole workspace as an n-fold convolution of these densities. We represent density functions as finite Hermite-Fourier Series and show for the planar case how the n-fold convolution can be performed in closed form requiring O(n) computation time. If all segments are identical the computation time reduces to O(logn).

A CLOSED FORM FOR THE DENSITY FUNCTIONS OF RANDOM WALKS IN ODD DIMENSIONS

2015 ◽  
Vol 93 (2) ◽  
pp. 330-339 ◽  
Author(s):  
JONATHAN M. BORWEIN ◽  
CORWIN W. SINNAMON
Keyword(s):  
Random Walk ◽  
Probability Density Function ◽  
Random Walks ◽  
Closed Form ◽  
Probability Density ◽  
Density Function ◽  
Dimensional Space ◽  
Density Functions ◽  
Piecewise Polynomial ◽  
Odd Dimensions

We derive an explicit piecewise-polynomial closed form for the probability density function of the distance travelled by a uniform random walk in an odd-dimensional space.

scholarly journals Nonlinear stochastic position and attitude filter on the Special Euclidean Group 3

2019 ◽  
Vol 356 (7) ◽  
pp. 4144-4173 ◽  
Author(s):  
Hashim A. Hashim ◽  
Lyndon J. Brown ◽  
Kenneth McIsaac
Keyword(s):  
Euclidean Group ◽  
Group 3 ◽  
Special Euclidean Group

scholarly journals Polynomial Invariants of the Euclidean Group Action on Multiple Screws

2021 ◽  
Author(s):  
◽  
Deborah Crook
Keyword(s):  
Group Action ◽  
Three Dimensions ◽  
Euclidean Group ◽  
Polynomial Invariants ◽  
Generating Sets ◽  
Special Euclidean Group ◽  
The One

<p>In this work, we examine the polynomial invariants of the special Euclidean group in three dimensions, SE(3), in its action on multiple screw systems. We look at the problem of finding generating sets for these invariant subalgebras, and also briefly describe the invariants for the standard actions on R^n of both SE(3) and SO(3). The problem of the screw system action is then approached using SAGBI basis techniques, which are used to find invariants for the translational subaction of SE(3), including a full basis in the one and two-screw cases. These are then compared to the known invariants of the rotational subaction. In the one and two-screw cases, we successfully derive a full basis for the SE(3) invariants, while in the three-screw case, we suggest some possible lines of approach.</p>

scholarly journals On the Density Functions of Integrals of Gaussian Random Fields

2013 ◽  
Vol 45 (02) ◽  
pp. 398-424 ◽  
Author(s):  
Jingchen Liu ◽  
Gongjun Xu
Keyword(s):  
Closed Form ◽  
Random Fields ◽  
Random Variables ◽  
Gaussian Random Fields ◽  
Density Functions ◽  
Exponential Functions ◽  
Asymptotic Bounds

In the paper we consider the density functions of random variables that can be written as integrals of exponential functions of Gaussian random fields. In particular, we provide closed-form asymptotic bounds for the density functions and, under smoothness conditions, we derive exact tail approximations of the density functions.

scholarly journals Normalized multi-bump solutions for saturable Schrödinger equations

2019 ◽  
Vol 9 (1) ◽  
pp. 1259-1277
Author(s):  
Xiaoming Wang ◽  
Zhi-Qiang Wang
Keyword(s):  
Variational Method ◽  
Density Function ◽  
Global Maximum ◽  
Schrodinger Equations ◽  
Density Functions ◽  
Concentration Compactness ◽  
Schrödinger Equations ◽  
Compactness Method ◽  
Normalized Solutions

Abstract In this paper, we are concerned with the existence of multi-bump solutions for a class of semiclassical saturable Schrödinger equations with an density function: $$\begin{array}{} \displaystyle -{\it\Delta} v +{\it\Gamma} \frac{I(\varepsilon x) + v^2}{1+I(\varepsilon x) +v^2} v =\lambda v,\, x\in{{\mathbb{R}}^{2}}. \end{array}$$ We prove that, with the density function being radially symmetric, for given integer k ≥ 2 there exist a family of non-radial, k-bump type normalized solutions (i.e., with the L2 constraint) which concentrate at the global maximum points of density functions when ε → 0+. The proof is based on a variational method in particular on a convexity technique and the concentration-compactness method.

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