The Geometry of Virtual Work Dynamics in Screw Space

Steven Peterson

doi:10.1115/1.2899535

The Geometry of Virtual Work Dynamics in Screw Space

Journal of Applied Mechanics ◽

10.1115/1.2899535 ◽

1992 ◽

Vol 59 (2) ◽

pp. 411-417 ◽

Cited By ~ 2

Author(s):

Steven Peterson

Keyword(s):

Tangent Space ◽

Screw Theory ◽

Virtual Work ◽

Rigid Bodies ◽

Dynamic Equations ◽

Coordinate Space ◽

Dynamical Equations ◽

Least Action ◽

System Of Rigid Bodies ◽

Configuration Manifold

In this paper, screw theory is employed to develop a method for generating the dynamic equations of a system of rigid bodies. Exterior algebra is used to derive the structure of screw space from projective three space (homogeneous coordinate space). The dynamic equation formulation method is derived from the parametric form of the principle of least action, and it is shown that a set of screws exist which serves as a basis for the tangent space of the configuration manifold. Equations generated using this technique are analogs of Hamilton’s dynamical equations. The freedom screws defining the manifold’s tangent space are determined from the contact geometry of the joint using the virtual coefficient, which is developed from the principle of virtual work. This results in a method that eliminates all differentiation operations required by other virtual work techniques, producing a formulation method based solely on the geometry of the system of rigid bodies. The procedure is applied to the derivation of the dynamic equations for the first three links of the Stanford manipulator.

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On the Integrability of Screw Spaces

Journal of Mechanical Design ◽

10.1115/1.2916939 ◽

1992 ◽

Vol 114 (2) ◽

pp. 251-256 ◽

Cited By ~ 2

Author(s):

S. W. Peterson

Keyword(s):

Tangent Space ◽

Screw Theory ◽

Joint Space ◽

Rigid Bodies ◽

Joint Contact ◽

A Chain ◽

Configuration Manifold ◽

Relationship Of ◽

The Relationship ◽

Line Space

In the development of a method for generating the dynamic equations of a chain of rigid bodies in elliptic line space, it has been discovered that the joint freedom spaces from the tangent space of the system’s joint space, called the configuration manifold by dynamicists. The freedom space of a joint can be calculated from the joint contact geometry using reciprocal screw theory. The present paper describes the relationship of the joint freedom spaces to the tangent space of the configuration manifold, and determines the integrability conditions which must be satisfied if the freedom space represents a valid configuration manifold. The paper also shows that the integrability of the tangent space of a chain of rigid bodies is established once the integrability of each joint freedom space has been demonstrated. These conditions are used to define valid generalized coordinates for describing the chain’s configuration. Cylindrical and spherical joints are treated as examples.

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Virtual Work & d’Alembert’s Principle

10.1093/oso/9780198822370.003.0013 ◽

2018 ◽

Author(s):

Peter Mann

Keyword(s):

Euler Equations ◽

Virtual Work ◽

Lagrange Equation ◽

Classical Mechanics ◽

Rigid Bodies ◽

Gibbs Function ◽

Constraint Force ◽

Appell Equations ◽

Jourdain's Principle ◽

D'alembert's Principle

This chapter discusses virtual work, returning to the Newtonian framework to derive the central Lagrange equation, using d’Alembert’s principle. It starts off with a discussion of generalised force, applied force and constraint force. Holonomic constraints and non-holonomic constraint equations are then investigated. The corresponding principles of Gauss (Gauss’s least constraint) and Jourdain are also documented and compared to d’Alembert’s approach before being generalised into the Mangeron–Deleanu principle. Kane’s equations are derived from Jourdain’s principle. The chapter closes with a detailed covering of the Gibbs–Appell equations as the most general equations in classical mechanics. Their reduction to Hamilton’s principle is examined and they are used to derive the Euler equations for rigid bodies. The chapter also discusses Hertz’s least curvature, the Gibbs function and Euler equations.

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Improving Techniques in Statically Equivalent Serial Chain Modeling for Center of Mass Estimation

Volume 5B: 38th Mechanisms and Robotics Conference ◽

10.1115/detc2014-34286 ◽

2014 ◽

Author(s):

Bingjue Li ◽

Andrew P. Murray ◽

David H. Myszka

Keyword(s):

Center Of Mass ◽

Original System ◽

Rigid Bodies ◽

Joint Angles ◽

Practical Applications ◽

Serial Chain ◽

System Of Rigid Bodies ◽

Final Development ◽

Data Error ◽

Insight Into

Any articulated system of rigid bodies defines a Statically Equivalent Serial Chain (SESC). The SESC is a virtual chain that terminates at the center of mass (CoM) of the original system of bodies. A SESC may be generated experimentally without knowing the mass, CoM, or length of each link in the system given that its joint angles and overall CoM may be measured. This paper presents three developments toward recognizing the SESC as a practical modeling technique. Two of the three developments improve utilizing the technique in practical applications where the arrangement of the joints impacts the derivation of the SESC. The final development provides insight into the number of poses needed to create a usable SESC in the presence of data collection errors. First, modifications to a matrix necessary in computing the SESC are proposed. Second, the problem of generating a SESC experimentally when the system of bodies includes a mass fixed in the ground frame are presented and a remedy is proposed for humanoid-like systems. Third, an investigation of the error of the experimental SESC versus the number of data readings collected in the presence of errors in joint readings and CoM data is conducted. By conducting the method on three different systems with various levels of data error, a general form of the function for estimating the error of the experimental SESC is proposed.

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Reduction of Hamiltonian Mechanical Systems With Affine Constraints: A Geometric Unification

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4034729 ◽

2016 ◽

Vol 12 (2) ◽

Cited By ~ 2

Author(s):

Robin Chhabra ◽

M. Reza Emami ◽

Yael Karshon

Keyword(s):

Symmetry Group ◽

Mechanical Systems ◽

Hamiltonian Formalism ◽

Geometrical Approach ◽

Constraint Forces ◽

Dynamical Equations ◽

Relative Configuration ◽

D’Alembert Equation ◽

Dynamical Reduction ◽

Configuration Manifold

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

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Kinematics and dynamics of a 4- P RUR Schönflies parallel manipulator by means of screw theory and the principle of virtual work

Mechanism and Machine Theory ◽

10.1016/j.mechmachtheory.2017.12.022 ◽

2018 ◽

Vol 122 ◽

pp. 347-360 ◽

Cited By ~ 11

Author(s):

Jaime Gallardo-Alvarado ◽

Ramón Rodríguez-Castro ◽

Pedro J. Delossantos-Lara

Keyword(s):

Parallel Manipulator ◽

Screw Theory ◽

Virtual Work ◽

Kinematics And Dynamics ◽

Principle Of Virtual Work

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Cosmology from a new nonconservative gravity

International Journal of Modern Physics D ◽

10.1142/s0218271818410067 ◽

2018 ◽

Vol 27 (06) ◽

pp. 1841006 ◽

Cited By ~ 7

Author(s):

Júlio C. Fabris ◽

Hermano Velten ◽

Thiago R. P. Caramês ◽

Matheus J. Lazo ◽

Gastão S. F. Frederico

Keyword(s):

Fluid Model ◽

Gravitational Theory ◽

Dissipative Process ◽

The Novel ◽

Least Action Principle ◽

Dynamical Equations ◽

First Order ◽

Least Action ◽

Dissipative Effects ◽

Cosmic Background

In this paper, we present a cosmological model arising from a nonconservative gravitational theory proposed in [M. J. Lazo, J. Paiva, J. T. S. Amaral and G. S. F. Frederico, Phys. Rev. D 95 (2017) 101501.] The novel feature where comparing with previous implementations of dissipative effects in gravity is the possible arising of such phenomena from a least action principle, so they are of a purely geometric nature. We derive the dynamical equations describing the behavior of the cosmic background, considering a single fluid model composed by pressureles matter, whereas the dark energy is conceived as an outcome of the “geometric” dissipative process emerging in the model. Besides, adopting the synchronous gauge, we obtain the first-order perturbative equations which shall describe the evolution of the matter perturbations within the linear regime.

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