Projection Metrics for Rigid-Body Displacements

2005 ◽  
Author(s):  
Pierre M. Larochelle ◽  
Andrew P. Murray
Keyword(s):  
Rigid Body ◽  
Robot Control ◽  
Research Question ◽  
Singular Value ◽  
Invariant Metric ◽  
Euclidean Group ◽  
Robot Calibration ◽  
Coordinate Frames ◽  
Open Research ◽  
Spatial Displacements

An open research question is how to define a useful metric on SE(n) with respect to (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances. We present two techniques for approximating elements of the special Euclidean group SE(n) with elements of the special orthogonal group SO(n+1). These techniques are based on the singular value and polar decompositions (denoted as SVD and PD respectively) of the homogeneous transform representation of the elements of SE(n). The projection of the elements of SE(n) onto SO(n+1) yields hyperdimensional rotations that approximate the rigid-body displacements (hence the term projection metric. A bi-invariant metric on SO(n+1) may then be used to measure the distance between any two spatial displacements. The results are PD and SVD based projection metrics on SE(n). These metrics have applications in motion synthesis, robot calibration. motion interpolation, and hybrid robot control.

A Distance Metric for Finite Sets of Rigid-Body Displacements via the Polar Decomposition

2006 ◽  
Vol 129 (8) ◽  
pp. 883-886 ◽  
Author(s):  
Pierre M. Larochelle ◽  
Andrew P. Murray ◽  
Jorge Angeles
Keyword(s):  
Singular Value Decomposition ◽  
Rigid Body ◽  
Polar Decomposition ◽  
Research Question ◽  
Singular Value ◽  
Invariant Metric ◽  
Euclidean Group ◽  
Coordinate Frames ◽  
Open Research ◽  
Value Decomposition

An open research question is how to define a useful metric on the special Euclidean group SE(n) with respect to: (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances that is useful for the synthesis and analysis of mechanical systems. We discuss a technique for approximating elements of SE(n) with elements of the special orthogonal group SO(n+1). This technique is based on using the singular value decomposition (SVD) and the polar decompositions (PD) of the homogeneous transform representation of the elements of SE(n). The embedding of the elements of SE(n) into SO(n+1) yields hyperdimensional rotations that approximate the rigid-body displacements. The bi-invariant metric on SO(n+1) is then used to measure the distance between any two displacements. The result is a left invariant PD based metric on SE(n).

A Displacement Metric for Finite Sets of Rigid Body Displacements

2008 ◽  
Author(s):  
Venkatesh Venkataramanujam ◽  
Pierre Larochelle
Keyword(s):  
Euclidean Space ◽  
Rigid Body ◽  
Orthogonal Group ◽  
Polar Decomposition ◽  
Coordinate Frame ◽  
Invariant Metric ◽  
Motion Generation ◽  
Euclidean Group ◽  
Finite Set ◽  
Potential Applications

There are various useful metrics for finding the distance between two points in Euclidean space. Metrics for finding the distance between two rigid body locations in Euclidean space depend on both the coordinate frame and units used. A metric independent of these choices is desirable. This paper presents a metric for a finite set of rigid body displacements. The methodology uses the principal frame (PF) associated with the finite set of displacements and the polar decomposition to map the homogenous transform representation of elements of the special Euclidean group SE(N-1) onto the special orthogonal group SO(N). Once the elements are mapped to SO(N) a bi-invariant metric can then be used. The metric obtained is thus independent of the choice of fixed coordinate frame i.e. it is left invariant. This metric has potential applications in motion synthesis, motion generation and interpolation. Three examples are presented to illustrate the usefulness of this methodology.

scholarly journals Photovoltaics Enabling Sustainable Energy Communities: Technological Drivers and Emerging Markets

Energies ◽  
2021 ◽  
Vol 14 (7) ◽  
pp. 1862
Author(s):  
Alexandros-Georgios Chronis ◽  
Foivos Palaiogiannis ◽  
Iasonas Kouveliotis-Lysikatos ◽  
Panos Kotsampopoulos ◽  
Nikos Hatziargyriou
Keyword(s):  
State Of The Art ◽  
Research Question ◽  
Economic Benefits ◽  
Energy Market ◽  
Small Scale ◽  
Local Energy ◽  
Community Members ◽  
Open Research ◽  
Energy Community

In this paper, we investigate the economic benefits of an energy community investing in small-scale photovoltaics (PVs) when local energy trading is operated amongst the community members. The motivation stems from the open research question on whether a community-operated local energy market can enhance the investment feasibility of behind-the-meter small-scale PVs installed by energy community members. Firstly, a review of the models, mechanisms and concepts required for framing the relevant concepts is conducted, while a clarification of nuances at important terms is attempted. Next, a tool for the investigation of the economic benefits of operating a local energy market in the context of an energy community is developed. We design the local energy market using state-of-the-art formulations, modified according to the requirements of the case study. The model is applied to an energy community that is currently under formation in a Greek municipality. From the various simulations that were conducted, a series of generalizable conclusions are extracted.

Bézier Curves on Riemannian Manifolds and Lie Groups With Kinematics Applications

1994 ◽  
Author(s):  
Frank C. Park ◽  
Bahram Ravani
Keyword(s):  
Riemannian Manifold ◽  
Rigid Body ◽  
Lie Groups ◽  
Riemannian Manifolds ◽  
Group Structure ◽  
Recursive Algorithm ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Spatial Displacements ◽  
Natural Way

Abstract In this article we generalize the concept of Bézier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljau’s algorithm for constructing Bézier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their algebraic group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bézier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bezier sense) using this recursive algorithm. We apply this algorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body.

Computation of Spatial Displacements From Geometric Features

1993 ◽  
Vol 115 (1) ◽  
pp. 95-102 ◽  
Author(s):  
B. Ravani ◽  
Q. J. Ge
Keyword(s):  
Computational Geometry ◽  
Clifford Algebra ◽  
Theoretical Foundation ◽  
Geometric Features ◽  
Theoretical Interest ◽  
Robot Calibration ◽  
Minimum Number ◽  
The Common ◽  
Spatial Displacements ◽  
Three Space

This paper develops the theoretical foundation for computations of spatial displacements from the simple geometric features of points, lines, planes, and their combinations. Using an oriented projective three space with a Clifford Algebra, all these three features are handled in a similar fashion. Furthermore, issues related to uniqueness of computations and minimum number of required features are discussed. It is shown that contrary to the common intuition, specification of a minimum of four points (planes) or three lines are necessary for computation of a unique displacement. Only when the sense of the orientations of these features are specified then the minimum number of required features reduces to three for points and planes and two for lines. The results, in addition to their theoretical interest in computational geometry of motion, have application in robot calibration.

scholarly journals Robot Control Using Alternative Trajectories Based on Inverse Errors in the Workspace

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Quoc Khanh Duong ◽  
Thanh Trung Trang ◽  
Thanh Long Pham
Keyword(s):  
Control System ◽  
High Resolution ◽  
Robot Control ◽  
Low Cost ◽  
Transmission System ◽  
Measuring System ◽  
Robot Calibration ◽  
Measuring Device ◽  
Calibration Solution ◽  
System Errors

It is easy to realize that most robots do not move to the desired endpoint (Tool Center Point (TCP)) using high-resolution noncontact instrumentation because of manufacturing and assembly errors, transmission system errors, and mechanical wear. This paper presents a robot calibration solution by changing the endpoint trajectories while maintaining the robot’s control system and device usages. Two independent systems to measure the endpoint positions, the robot encoder and a noncontact measuring system with a high-resolution camera, are used to determine the endpoint errors. A new trajectory based on the measured errors will be built to replace the original trajectory. The results show that the proposed method can significantly reduce errors; moreover, this is a low-cost solution and easy to apply in practice and calibration can be done cyclically. The only requirement for this method is a noncontact measuring device with high-resolution and located independently with the robot in calibration.

Designing Planar Mechanisms Using a Bi-Invariant Metric in the Image Space of SO (3)

1994 ◽  
Author(s):  
Pierre Larochelle ◽  
J. Michael McCarthy
Keyword(s):  
Rigid Body ◽  
Numerical Results ◽  
Image Space ◽  
Dimensional Synthesis ◽  
Invariant Metric ◽  
Planar Mechanisms ◽  
Position And Orientation ◽  
Position Synthesis

Abstract In this paper we present a technique for using a bi-invariant metric in the image space of spherical displacements for designing planar mechanisms for n (> 5) position rigid body guidance. The goal is to perform the dimensional synthesis of the mechanism such that the distance between the position and orientation of the guided body to each of the n goal positions is minimized. Rather than measure these distances in the plane, we introduce an approximating sphere and identify rotations which are equivalent to the planar displacements to a specified tolerance. We then measure distances between the rigid body and the goal positions using a bi-invariant metric on the image space of SO(3). The optimal linkage is obtained by minimizing this distance over all of the n goal positions. The paper proceeds as follows. First, we approximate planar rigid body displacements with spherical displacements and show that the error induced by such an approximation is of order 1/R2, where R is the radius of the approximating sphere. Second, we use a bi-invariant metric in the image space of spherical displacements to synthesize an optimal spherical 4R mechanism. Finally, we identify the planar 4R mechanism associated with the optimal spherical solution. The result is a planar 4R mechanism that has been optimized for n position rigid body guidance using an approximate bi-invariant metric with an error dependent only upon the radius of the approximating sphere. Numerical results for ten position synthesis of a planar 4R mechanism are presented.

Computation of Spatial Displacements From Geometric Features

1991 ◽  
Author(s):  
B. Ravani ◽  
Q. J. Ge
Keyword(s):  
Computational Geometry ◽  
Clifford Algebra ◽  
Theoretical Foundation ◽  
Geometric Features ◽  
Theoretical Interest ◽  
Robot Calibration ◽  
The Common ◽  
Spatial Displacements ◽  
Three Space

Abstract This paper develops the theoretical foundation for computations of spatial displacements from the simple geometric features of points, lines, planes and their combinations. Using an oriented projective three space with a Clifford Algebra, all these three features are handled in a similar fashion. Furthermore, issues related to uniqueness of computations and minimal number of required features are discussed. It is shown that contrary to the common intuition, specification of a minimum of four points (planes) or three lines (each pair being non-planar) are necessary for computation of a unique displacement. Only when the sense of the orientations of these features are specified then the minimal number of required features reduces to three for points and planes and two for lines. The results, in addition to their theoretical interest in computational geometry of motion, have application in robot calibration.

Reconciling Distance Metric Methods for Rigid Body Displacements

2009 ◽  
Author(s):  
Anurag Purwar ◽  
Qiaode Jeffrey Ge
Keyword(s):  
Rigid Body ◽  
Polar Decomposition ◽  
Fast Method ◽  
Robot Calibration ◽  
Mechanism Synthesis ◽  
Orthogonal Matrices ◽  
Higher Dimensional ◽  
Motion Approximation ◽  
Value Decomposition ◽  
And Control

In the last twenty years, researchers have proposed a few different methods to establish a norm (or, metric) for both planar and spatial rigid body displacements. Desire to meaningfully quantify a displacement composed of rotation and translation stems from a requirement to ascertain “distance” between two given displacements in applications, such as motion approximation and interpolation, mechanism synthesis, collision avoidance, positioning, and robot calibration and control. In this paper, we show that the various seemingly different shape independent norm calculation methods based on approximating displacements with higher dimensional rotations via orthogonal matrices, or polar decomposition (PD) and singular value decomposition (SVD) can be reconciled and unified in the mathematically compact and elegant framework of biquaternions. In the process, we also propose an elegant and fast method for such norm calculations.

Local Governance in a Developing Country

2014 ◽  
Vol 12 (4) ◽  
pp. 750-774 ◽  
Author(s):  
Mustafa Kemal Öktem
Keyword(s):  
Inner City ◽  
Developing Country ◽  
Random Sampling ◽  
Sampling Method ◽  
Developing World ◽  
Local Governance ◽  
Research Question ◽  
Citizen Involvement ◽  
The Public ◽  
Open Research

How the implementation of local governance can be improved in the developing world is an open research question. This study discusses the difficulty of transitioning toward local governance in Turkey. To analyze the basic difficulties of local governance, a survey was conducted using a random sampling method within the inner city municipalities of Ankara. The findings indicate that improving local governance by enhancing transparency and building mechanisms of e-governance is the first step to motivating the public to participate in and to move toward a system of local governance. In general, each of these strategies would likely increase overall citizen involvement and, in particular, would increase the involvement of those citizens between 26 and 35 years old.

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