scholarly journals On the Hamel Coefficients and the Boltzmann–Hamel Equations for the Rigid Body

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Andreas Müller
Keyword(s):  
Rigid Body ◽  
Nonholonomic Systems ◽  
Rigid Bodies ◽  
Body Motion ◽  
Floating Body ◽  
Lagrange Equations ◽  
Local Coordinates ◽  
Dynamics And Control ◽  
Boltzmann Hamel Equations ◽  
And Control

AbstractThe Boltzmann–Hamel (BH) equations are central in the dynamics and control of nonholonomic systems described in terms of quasi-velocities. The rigid body is a classical example of such systems, and it is well-known that the BH-equations are the Newton–Euler (NE) equations when described in terms of rigid body twists as quasi-velocities. It is further known that the NE-equations are the Euler–Poincaré, respectively, the reduced Euler–Lagrange equations on SE(3) when using body-fixed or spatial representation of rigid body twists. The connection between these equations are the Hamel coefficients, which are immediately identified as the structure constants of SE(3). However, an explicit coordinate-free derivation has not been presented in the literature. In this paper the Hamel coefficients for the rigid body are derived in a coordinate-free way without resorting to local coordinates describing the rigid body motion. The three most relevant choices of quasi-velocities (body-fixed, spatial, and hybrid representation of rigid body twists) are considered. The corresponding BH-equations are derived explicitly for the rotating and free floating body. Further, the Hamel equations for nonholonomically constrained rigid bodies are discussed, and demonstrated for the inhomogenous ball rolling on a plane.

Dynamics and Control of a Translating Flexible Beam With a Prismatic Joint

1992 ◽  
Vol 114 (3) ◽  
pp. 422-427 ◽  
Author(s):  
Sivakumar S. K. Tadikonda ◽  
Haim Baruh
Keyword(s):  
Rigid Body ◽  
The Other ◽  
Body Motion ◽  
Rigid Base ◽  
Flexible Beam ◽  
Positioning Control ◽  
Dynamics And Control ◽  
And Control ◽  
Tip Mass ◽  
Free Beam

The complete dynamic model of a translating flexible beam, with a tip mass at one end and emerging from or retracting into a rigid base at the other, is presented. The model considers the effect of elastic and translational motions of the beam on each other. The properties of the eigenfunctions of a fixed-free beam are exploited to obtain closed-form expressions for several domain integrals that arise in the model. It is shown that neglecting the effect of elastic motion on the rigid body motion leads to inaccuracies in positioning control. Issues associated with the feedback control of such a beam are discussed.

On Higher Order Point-Line and the Associated Rigid Body Motions

2006 ◽  
Vol 129 (2) ◽  
pp. 166-172 ◽  
Author(s):  
Yi Zhang ◽  
Kwun-Lon Ting
Keyword(s):  
Rigid Body ◽  
Higher Order ◽  
Rigid Bodies ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Dual Quaternion ◽  
Displacement Operator ◽  
Screw Motion ◽  
Rigid Body Motions ◽  
Point Line

This paper presents a study on the higher-order motion of point-lines embedded on rigid bodies. The mathematic treatment of the paper is based on dual quaternion algebra and differential geometry of line trajectories, which facilitate a concise and unified description of the material in this paper. Due to the unified treatment, the results are directly applicable to line motion as well. The transformation of a point-line between positions is expressed as a unit dual quaternion referred to as the point-line displacement operator depicting a pure translation along the point-line followed by a screw displacement about their common normal. The derivatives of the point-line displacement operator characterize the point-line motion to various orders with a set of characteristic numbers. A set of associated rigid body motions is obtained by applying an instantaneous rotation about the point-line. It shows that the ISA trihedrons of the associated rigid motions can be simply depicted with a set of ∞2 cylindroids. It also presents for a rigid body motion, the locus of lines and point-lines with common rotation or translation characteristics about the line axes. Lines embedded in a rigid body with uniform screw motion are presented. For a general rigid body motion, one may find lines generating up to the third order uniform screw motion about these lines.

Geometric Spectral Algorithms for the Simulation of Rigid Bodies

2019 ◽  
Vol 14 (12) ◽  
Author(s):  
Yiqun Li ◽  
Razikhova Meiramgul ◽  
Jiankui Chen ◽  
Zhouping Yin
Keyword(s):  
Lie Group ◽  
Spectral Methods ◽  
Rigid Bodies ◽  
Geometric Control ◽  
Homogeneous Manifolds ◽  
Practical Algorithms ◽  
Group Methods ◽  
Dynamics And Control ◽  
And Control ◽  
Geometric Dynamics

Abstract Lie group methods are an excellent choice for simulating differential equations evolving on Lie groups or homogeneous manifolds, as they can preserve the underlying geometric structures of the corresponding manifolds. Spectral methods are a popular choice for constructing numerical approximations for smooth problems, as they can converge geometrically. In this paper, we focus on developing numerical methods for the simulation of geometric dynamics and control of rigid body systems. Practical algorithms, which combine the advantages of Lie group methods and spectral methods, are given and they are tested both in a geometric dynamic system and a geometric control system.

Dynamics and control of capture of a floating rigid body by a spacecraft robotic arm

2014 ◽  
Vol 33 (3) ◽  
pp. 315-332 ◽  
Author(s):  
Xiao-Feng Liu ◽  
Hai-Quan Li ◽  
Yi-Jun Chen ◽  
Guo-Ping Cai ◽  
Xi Wang
Keyword(s):  
Rigid Body ◽  
Robotic Arm ◽  
Dynamics And Control ◽  
And Control

Nonaxisymmetric Acoustic Radiation and Scattering From Rigid Bodies of Revolution Using the Surface Variational Principle

1998 ◽  
Vol 120 (1) ◽  
pp. 95-103 ◽  
Author(s):  
J. H. Ginsberg ◽  
Kuangcheng Wu
Keyword(s):  
Variational Principle ◽  
Rigid Body ◽  
Acoustic Radiation ◽  
Axisymmetric Body ◽  
Rigid Bodies ◽  
Body Motion ◽  
Convergence Properties ◽  
Bodies Of Revolution ◽  
Computer Codes ◽  
Acoustic Fluid

The surface variational principle (SVP), which represents the surface response as a series of basis functions spanning the entire surface, provides a global description of acoustic fluid-structure interaction that has many of the benefits associated with analytical methods. This paper describes the extension of SVP to model the interaction between the velocity and pressure on the surface of an axisymmetric body subjected to nonaxisymmetric excitation. Problems addressed are radiation due to arbitrary rigid body motion, and scattering associated with arbitrary incidence of a plane wave on a stationary rigid body. Numerical results are presented for flat-ended and hemi-capped cylinders. These results are compared to those obtained from the CHIEF-88 and SHIP-92 computer codes. The convergence properties of SVP are examined in detail, particularly for its requirements when ka is in the upper part of the mid-frequency range.

scholarly journals A unified approach to rigid body rotational dynamics and control

2011 ◽  
Vol 468 (2138) ◽  
pp. 395-414 ◽  
Author(s):  
Firdaus E. Udwadia ◽  
Aaron D. Schutte
Keyword(s):  
Rigid Body ◽  
Closed Form ◽  
Equations Of Motion ◽  
Fundamental Equation ◽  
Rotational Dynamics ◽  
Constrained Motion ◽  
Dynamics And Control ◽  
Common Framework ◽  
The Stability ◽  
And Control

This paper develops a unified methodology for obtaining both the general equations of motion describing the rotational dynamics of a rigid body using quaternions as well as its control. This is achieved in a simple systematic manner using the so-called fundamental equation of constrained motion that permits both the dynamics and the control to be placed within a common framework. It is shown that a first application of this equation yields, in closed form, the equations of rotational dynamics, whereas a second application of the self-same equation yields two new methods for explicitly determining, in closed form, the nonlinear control torque needed to change the orientation of a rigid body. The stability of the controllers developed is analysed, and numerical examples showing the ease and efficacy of the unified methodology are provided.

scholarly journals Towards a compact and computer-adapted formulation of the dynamics and stability of multi rigid body systems

2002 ◽  
Vol 12 (1) ◽  
pp. 64-70 ◽  
Author(s):  
Hooshang Hemami
Keyword(s):  
Nonlinear Systems ◽  
Rigid Body ◽  
Computer Simulations ◽  
Euler Method ◽  
Rigid Bodies ◽  
Lyapunov Methods ◽  
Full Detail ◽  
Stability And Control ◽  
Euler Methods ◽  
And Control

The dynamics of rigid bodies coupled by homonymic and non-homonymic constraints are formulated by the Newton - Euler method - employing a compact notation. The compact notation involves the use of two three by three matrices A and ? and the totality of constraint vector C. The Lagrangian and Newton - Euler methods are related for a one - link rigid body in order to introduce the methodology of the paper in full detail. Stability and control of the resulting nonlinear systems are investigated by the use of Lyapunov methods. Digital computer simulations of typical movements are carried out in order to demonstrate feasibility of the formulation and the approach.

A Computationally Efficient Planar Rigid Body Velocity Measure

2009 ◽  
Author(s):  
Luis E. Criales ◽  
Joseph M. Schimmels
Keyword(s):  
Rigid Body ◽  
Rigid Bodies ◽  
Body Motion ◽  
Computationally Efficient ◽  
Straight Line ◽  
Body Velocity ◽  
Velocity Measure ◽  
Rigid Body Motions ◽  
Line Path ◽  
Planar Motions

A planar rigid body velocity measure based on the instantaneous velocity of all particles that constitute a rigid body is developed. This measure compares the motion of each particle to an “ideal”, but usually unobtainable, motion. This ideal motion is one that would carry each particle from its current position to its desired position on a straight-line path. Although the ideal motion is not a valid rigid body motion, this does not preclude its use as a reference standard in evaluating valid rigid body motions. The optimal instantaneous planar motions for general rigid bodies in translation and rotation are characterized. Results for an example planar positioning problem are presented.

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