# Fredholm integral equations on the Euclidean motion group

@article{Chirikjian1996FredholmIE, title={Fredholm integral equations on the Euclidean motion group}, author={G. Chirikjian}, journal={Inverse Problems}, year={1996}, volume={12}, pages={579-599} }

In this work, methods for the solution of Fredholm equations of the first kind with convolution kernel are presented, where all the functions in the integral equation are functions on the Euclidean motion group, and the convolution product is defined relative to the group operation. An application in which such equations arise is examined in detail. The properties of the Fourier transform of scalar-valued functions on the Euclidean motion group are reviewed and applied to an exactly solvable… Expand

#### 22 Citations

Regularized Solutions of a Nonlinear Convolution Equation on the Euclidean Group

- Mathematics
- 1998

In this paper we apply Fourier analysis on the two and three dimensional Euclidean motion groups to the solution of a nonlinear convolution equation. First, we review the theory of the irreducible… Expand

An operational calculus for the euclidean motion group with applications in robotics and polymer science

- Mathematics
- 2000

In this article we develop analytical and computational tools arising from harmonic analysis on the motion group of three-dimensional Euclidean space. We demonstrate these tools in the context of… Expand

Deconvolution on the Euclidean motion group SE(3)

- Mathematics
- 2011

Deconvolution is an important class of inverse problems. The goal of this paper is to develop the optimality properties of deconvolution density estimators on the noncommutative and noncompact… Expand

NUMERICAL SYNTHESIS OF BINARY MANIPULATOR WORKSPACES USING THE FOURIER TRANSFORM ON THE EUCLIDEAN MOTION GROUP

- 1998

In this paper we apply the Fourier transform on the Euclidean motion group to solve problems in kinematic design of binary manipulators. We begin by reviewing how the workspace of a binary… Expand

Numerical convolution on the Euclidean group with applications to workspace generation

- Mathematics, Computer Science
- IEEE Trans. Robotics Autom.
- 1998

By partitioning a discretely actuated manipulator into P modules, and approximating the workspace of each module as a density function on a compact subset of the special Euclidean group, the whole workspace can be approximated as an P-fold convolution of these densities. Expand

Applications of Fourier Methods on the Motion Group in Robot Kinematics

- Computer Science
- 1998

The fast Fourier transform is applied to the problem of finding the workspace density of discretely actuated manipulators to compute convolution-like integrals that arise in robot kinematics and motion planning. Expand

Regularized deconvolution on the 2D-Euclidean motion group

- Mathematics
- 2008

This paper considers a class of inverse problems involving deconvolution density estimation over the Euclidean motion group. Group representations of the Euclidean motion group are used to break… Expand

Computation of Robot Configuration and Workspaces via the Fourier Transform on the Discrete-Motion Group

- Mathematics, Computer Science
- Int. J. Robotics Res.
- 1999

Fourier transforms for the discrete-motion group of the plane are developed and implemented to apply fast Fourier transform methods to the computation of convolution-like integrals that arise in robot kinematics and motion planning. Expand

Metrics on Motion and Deformation of Solid Models

- Mathematics
- 1998

Recently, the importance of metrics on the group of rigid body motions has been addressed in a number of works in the kinematics and robotics literature. This paper defines new metrics on motion… Expand

Deconvolution on the Euclidean motion group and planar robotic manipulator design

- Mathematics, Computer Science
- Robotica
- 2009

This work considers a serial-chain robot arm as a concatenation of segments and examines how statistics of known segments can be used to select the remainder of the structure so as to attain the desired statistical properties of the whole structure, a deconvolution density estimation problem for the Euclidean motion group. Expand

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